Time delay

Physics Reports 364 (2002) 83 – 174 www.elsevier.com/locate/physrep

Time delay C.A.A. de Carvalhoa , H.M. Nussenzveigb; ∗; 1 a

Instituto de F sica, UFRJ, C. P. 68528, 21945-970 Rio de Janeiro, RJ, Brazil NASA Goddard Space Flight Center, Code 913, Greenbelt, MD 20771, USA

b

Received June 2001; editor: J: Eichler Contents 1. Introduction 2. Early treatments 2.1. Group velocity derivations 2.2. Asymptotic behavior of r 2.3. Stationary dwell time 2.4. Discussion 3. Average dwell time 3.1. Spherical wave packets 3.2. Plane wave trains 3.3. One-dimensional scattering 3.4. Electromagnetic scattering 4. Properties and extensions 4.1. Classical time delay and the virial 4.2. Quantum virial relation 4.3. Time delay and density of states 4.4. The time delay operator 4.5. Spectral property and Levinson’s theorem 5. Applications to statistical mechanics 5.1. Connection with the second virial coe>cient 5.2. Time delay and correlations

85 86 86 89 91 94 94 94 102 107 112 120 120 121 123 126 130 133

6. Application to tunneling 6.1. Tunneling time 6.2. Group velocity 6.3. ‘Clocks’ and related approaches 6.4. Path summations 6.5. Conditional dwell time 6.6. Average dwell time in tunneling 7. Applications to Mie scattering 7.1. Quantized Mie modes 7.2. Resonance and background contributions 7.3. Further applications 8. Time delay and chaotic scattering 8.1. Brief introduction to chaos 8.2. Classical chaotic scattering 8.3. Quantum chaotic scattering 9. Conclusion Acknowledgements References

133 136

∗

Corresponding author. E-mail address: [email protected] (H.M. Nussenzveig). 1 Permanent address: Instituto de FCDsica, UFRJ, C. P. 68528, 21945-970 Rio de Janeiro, RJ, Brazil. c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter  PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 9 2 - 8

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Abstract The concepts of time delay and dwell time in quantum mechanics, and their applications to regular and chaotic scattering, to statistical mechanics, and to the tunneling time problem, among others, are reviewed. c 2002 Elsevier Science B.V. All The emphasis is on physical concepts and on a pedagogical presentation.
rights reserved. PACS: 03.65.−w; 03.80.+r; 11.55.−m; 05.45.+b

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85

1. Introduction This report aims at presenting a pedagogical survey of time delay and related concepts, including their manifold applications to physics. Time delay and its cousin, dwell time, characterize the duration of collision processes, the lifetime of unstable systems, the response of a scattering system to perturbations, the dynamics of tunneling and the subtleties of chaotic scattering. They are directly related to the density of states, and to the virial expansion in statistical mechanics. In spite of the fundamental character of this subject, earlier comprehensive surveys do not seem to be available. The main connections with scattering theory, from a mathematical physics viewpoint, have been thoroughly reviewed (Martin, 1981). Readers looking for rigorous derivations are referred to Martin’s excellent review. Our emphasis is on a critical presentation of the basic ideas, hopefully in readily accessible terms, and on their applications to a variety of Kelds, including more recent developments in scattering theory. Time plays a peculiar role in quantum mechanics. Usually—and this is also our attitude—it is treated as a parameter in the dynamical evolution (Landau and Lifshitz, 1965), rather than as a quantum observable, associated with a Hermitian operator. Physical time is deKned by clocks, and the role of clocks will be brieMy touched upon in Section 6.3. However, we do not deal with the still controversial questions associated with the deKnition of a time observable (Pauli, 1958; Salecker and Wigner, 1958, Oppenheim et al., 1997; Casher and Reznik, 2000) or with the observability of arrival times (Aharonov et al., 1998a,b; Muga and Leavens, 2000). Most of the problems we treat are non-relativistic, although some relativistic extensions are included. The concept of time delay was introduced by Eisenbud (1948) and Wigner (1955), in the context of s-wave quantum scattering. We begin (Section 2.1) with their derivation, based on the group velocity of spherical wave packets, i.e., on following the ‘peak’ of a wave packet. A direct extension of their argument to the total scattering amplitude leads to the concept of angular time delay. An attempt to improve these derivations, by employing average position, rather than peak following, is described in Section 2.2. Unsatisfactory features of these early contributions are pointed out in Section 2.4. An important new method for dealing with time delay was proposed by Smith (1960). He introduced the concept of the dwell time of a particle in a given spatial region (Section 2.3). Restricting his analysis to stationary scattering, he rederived the Eisenbud–Wigner time delay, as a diOerence between interacting and free dwell times. A general time-dependent approach, based on average wave packet dwell time, gets rid of the di>culties found in previous treatments and is uniformly applicable to all cases (Section 3). Beginning with Kxed angular momentum (Section 3.1), the results are Krst extended to the global three-dimensional situation (Section 3.2) and then specialized to one-dimensional scattering (Section 3.3), as a prelude to the tunneling applications of Section 6. A relativistic extension, to electromagnetic scattering, is given in Section 3.4. Section 4 deals with general properties of the time delay and with extensions to more general situations. In Section 4.1, a connection with classical mechanics and with the Clausius virial function is established, and extended to quantum mechanics (Section 4.2). The next section introduces one of the most important properties of time delay: its connection with the density of states (spectral property). Formal results from the mathematical physics treatments are quoted without proofs in Section 4.4. They include the general deKnition of the time delay operator and extensions to more

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general situations. A connection between the time delay operator and the response of a scattering system to a perturbation is also established. Section 4.5 gives a general formulation of the spectral relation. Levinson’s theorem, a well-known property of the phase shifts for a given partial wave, and its extension to the total scattering amplitude, yield sum rules for the time delay. Applications to statistical mechanics are discussed in Section 5. There is a direct relationship with the theory of virial coe>cients (Section 5.1). General connections with Finite Temperature Field Theory lead to expressions in terms of correlation functions, that may be employed in perturbative or semiclassical approximations (Section 5.2). An application that has led to an extensive and highly controversial literature, the problem of tunneling time, is the object of Section 6. In Sections 6.2– 6.5, we give a critical review of the main approaches that have been proposed to deal with this problem: group velocity, clocks, path summations, and conditional dwell time. The average one-dimensional wave packet dwell time approach discussed in Section 3.3 allows us to reinterpret previous results in a fully consistent way, characterizing the tunneling time problem as an ill-posed one. In Section 7, we apply the results to a speciKc example, Mie scattering. In this example, the connection with the density of states provides explicitly computable illustrations of several properties and applications of time delay. Connections with the Goos–HPanchen eOect and with the speed of light in resonant media are also discussed. Finally, this model illustrates sensitive dependence on initial conditions in scattering, introducing the transition to chaotic scattering. Applications to chaotic scattering are the subject of Section 8. Section 8.1 is a brief introduction to chaos: no prior knowledge is assumed. Time delay as a signature of classical chaotic scattering is treated in Section 8.2. Quantum chaotic scattering and some illustrative examples are discussed in Section 8.3. Section 9 contains concluding remarks. 2. Early treatments 2.1. Group velocity derivations Early results on time delay were based on Kelvin’s Principle of Stationary Phase (Lamb, 1953), often employed to derive the group velocity of a wave packet in a dispersive medium. Given a one-dimensional wave packet,
(x; t) = A(k) exp{i[kx − !(k)t]} d k ; (2.1.1) where A(k) = |A(k)| exp[i(k)], with |A(k)| peaked around a wave number k0 , one may ask, in a short-wavelength regime, around what value of x will the wave packet be peaked, at a given time t. Regarding (2.1.1) as a superposition of monochromatic waves, one argues that rapid phase variation will generally lead to destructive interference, so that the most favorable situation will occur when k0 coincides with a point where the phase is stationary, x − x0 − (d!=d k)k0 t = 0 ;

(2.1.2)

where x0 =−(d=d k)k0 . The wave packet peak thus propagates with the group velocity vg =(d!=d k)k0 , which diOers from the phase velocity v = (!=k)k0 if the medium is dispersive.

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The simplest version of quantum time delay has to do with s-wave scattering by a spherically symmetric scattering center. At large distances from the center,   
∞ E p dE (r → ∞) (2.1.3) |A(E)| exp i − r − t + (E) r in ≈ ˝ ˝ 0 is an incoming wave packet centered at the stationary phase point r t = − + t0 ; v evaluated at the peak of |A(E)|. Here, t0 = ˝(d=dE), and  2 d p p dE = = v≡ dp dp 2m m is the particle velocity, for a non-relativistic particle of mass m. The corresponding outgoing wave packet,   
∞ E p r out ≈ r − t + 2(E) + (E) dE |A(E)| exp i ˝ ˝ 0

(2.1.4)

(r → ∞) ;

(2.1.5)

where S(E) = exp[2i(E)] is the S-function, is centered at r t = + t0 + Ut ; v

(2.1.6)

(2.1.7)

where Ut = 2˝

dS d = −i˝S −1 (E) dE dE

(2.1.8)

is the Eisenbud–Wigner time delay 2 for s-waves (Eisenbud, 1948; Wigner, 1955). For scattering by a hard core of radius a, with S(E) = exp(−2ika), where k ≡ p=˝ ≡ (mv)=˝, (2.1.8) gives a time advance, Ut = −2a=v ;

(2.1.9)

corresponding to direct reMection from the surface. In classical theory, for a quasi-monoenergetic wave packet and a scattering center of radius a, this would represent the earliest arrival time expected from causality. In quantum theory, Wigner (1955) employed a causality condition to derive the inequality Ut ¿ − (2a +
)=v ; 2

(2.1.10)

Wigner’s derivation employed a superposition of two monoenergetic beams of slightly diOerent energies as a substitute for a wave packet. The stationary phase derivation was given by Bohm (1951) and (following an argument by van Kampen) by Lane and Thomas (1958).

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where the extra term
≡ 1=k, of the order of the de Broglie wavelength, arises from the uncertainty principle, as will be discussed below. In the neighborhood of a sharp isolated resonance at energy Er , with halfwidth , where the S-function can be represented by a one-pole approximation, S(E) ≈

E − Er − 12 i ; E − Er + 12 i

(2.1.11)

one gets Ut =

˝ ; (E − Er )2 + 14 2

(2.1.12)

which is a rapidly varying Lorentzian function of the peak energy E of the incoming wave packet. At exact resonance, E = Er , we Knd Ut = 4˝= ;

(2.1.13)

which is 4 times the resonance lifetime. This is taken to imply that the particle is delayed by the scatterer for a time lapse of this order. The extension of these results to higher-order partial waves is immediate, since they depend only on the asymptotic behavior of incoming and outgoing wave packets at large distances, respectively given by (2.1.3) and (2.1.5), and these diOer only by trivial factors for higher angular momentum. Thus, for the lth partial wave, the Eisenbud–Wigner time delay is given by Ut = 2˝

dl : dE

(2.1.14)

A further extension, to the total scattering amplitude, was given by Froissart, Goldberger and Watson (1963). Let us consider an incident plane wave train 3   
∞ p E |A(E)| exp i z − t + (E) dE ; (2.1.15) 0 (z; t) = ˝ ˝ 0 √ where p = 2mE (positive square root) and |A(E)| is peaked around some energy. Its arrival time at a given plane z is, by stationary phase, t=

z + t0 ; v

v≡

p ; m

t0 = ˝

d ; dE

all evaluated at the peak energy. The corresponding scattered wave packet in the direction  is, asymptotically,  
∞ iEt exp(ikr) exp − + i(E) dE (r → ∞) ; |A(E)|f(k; ) s (r; ; t) = r ˝ 0 3

(2.1.16)

(2.1.17)

These authors considered the more complicated case of a three-dimensional wave packet, deriving also a lateral (spatial) displacement (see Sections 2.2 and 7.3).

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where f(k; ) is the total scattering amplitude, and spherical symmetry of the scattering center has been assumed. With f(k; ) = |f(k; )| exp[i arg f(k; )] ;

(2.1.18)

the corresponding arrival time at (r; ) is r t = + t0 + Ut() ; v where

(2.1.19)

9 arg f(k; ) (2.1.20) 9E is called the angular time delay. However, it must be noted that it is not really a delay in the same sense as before: it is not compared with the arrival time (2.1.16) in the absence of the scatterer, except in the exactly forward direction, where, however, it is not expected to be applicable, as the scattered wave interferes with the incident one for  = 0. By considering an incident wave packet that is also laterally bounded, and applying a similar stationary phase argument, Froissart et al. (1963) concluded that the scattered wave packet also undergoes a spatial lateral displacement, given by the derivative of arg f with respect to the momentum transfer that takes place in the scattering process. Ut() = ˝

2.2. Asymptotic behavior of r Instead of characterizing the position of the scattered wave packet by its peak, it was proposed by Brenig and Haag (1959) to evaluate the expectation value r at large times, outside the range of the interaction. Under these conditions, the scattered wave behaves like a free-particle wave packet. To evaluate r , let us consider a generic three-dimensional free-particle wave packet,   
2 p i −3=2 ˜ (p) exp p·r− t d3 p ; (2.2.1) t (r) = (2˝) ˝ 2m normalized to unity, so that | ˜ (p)|2 is the momentum space probability density. Its asymptotic behavior at large t and r follows from the three-dimensional version of the stationary-phase method (Braun, 1956). The stationary phase point is pW = mr=t ;

(2.2.2)

and a Taylor expansion of the phase as well as of ˜ (p) around this point, up to second order, leads to the stationary-phase result,    2  3 m 3=2 ˜ mr i˝  m 5=2 ˜ W − − (Up )p=pW + · · · : (2.2.3) (p) t (r) ≈ exp i 2˝t 4 t 2 t

If |(Ur)2 | denotes the spatial width of the initial wave packet, (2.2.3) is an expansion into powers of t=ts , where ts is the spreading time of the wave packet (the time it takes to double its initial

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width, because of the momentum uncertainty), given by ts ∼ m(Ur)2 =˝ :

(2.2.4)

By (2.2.2), the relevant momentum component at (r; t) is that associated with the classical free time of Might from the origin to this point. The asymptotic t −3 decay law compensates for the volume scaling factor r=t. We now employ the asymptotic form (2.2.3) to evaluate the expectation value of r = |r|. The result is r = v t + b + O(t −1 ); where p 1 v = = m m and b=−

i˝ 2







t→∞;

| ˜ (p)|2 |p| d 3 p ;

∗ ∗ p[ ˜ (p)U ˜ (p) − ˜ (p)U ˜ (p)] d 3 p :

By partial integration, we get



˜∗ ˜ (p) i˝ 9 i˝ 9 ∗9 ˜ ∗ ˜ b= − ˜ d3 p = [ ˜ (p)]2 d3 p : 2 9p 9p 2 9p ˜ ∗ (p) With ˜ (p) ≡ | ˜ (p)| exp [i arg ˜ (p)], this becomes  
9 9 2 3 b = −˝ | ˜ (p)| arg ˜ (p) d p = −˝ arg ˜ (p) 9p 9p   9 ˜ = −˝ v arg (p) ; 9E

(2.2.5)

(2.2.6)

(2.2.7)

(2.2.8)

(2.2.9)

where v = p=m; E = p2 =(2m) (free particles). If the energy spectrum of the wave packet is su>ciently sharply peaked, we can replace v by v and (2.2.5) becomes, asymptotically, r = v (t − ) ; where

 =˝

(2.2.10) 

9 arg ˜ (p) 9E

;

(2.2.11)

which generalizes the s-wave result following (2.1.3). Let ˜ i (E; ) denote the momentum space wave function of the incident wave, where  stands for its angular dependence, sharply concentrated around the direction 0 of the average incident

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momentum p0 . The corresponding wave function for the scattered wave is (Brenig and Haag, 1959)

˜ (E;  ) d ; ˜ (E; ) = |T | ; E ˜ (E;  ) d ≈ |T |0 ; E (2.2.12) s i i where T denotes the T -matrix and |T |0 ; E is proportional to the scattering amplitude in the direction . Substituting into (2.2.11) and assuming that arg ˜ i (E;  ) does not vary too rapidly with  , we get s =  i +  d ;

(2.2.13)

where, for a spherically symmetric scatterer, the Brening–Haag time delay d is given by   9 d = ˝ arg f(E; ) : (2.2.14) 9E Here, the angular brackets denote an angular average over all directions , in accordance with (2.2.6). This is the angular average of the Froissart–Goldberger–Watson angular time delay (2.1.19). Brenig and Haag discuss the conditions for the validity of their result, pointing out that it requires not only that the eOective interaction must already be negligible at times of the order of d , but also that, since (2.2.3) is an expansion into powers of t=ts , one must have tts ;

(2.2.15)

where ts is the spreading time of the wave packet. 2.3. Stationary dwell time A new deKnition of time delay was proposed by Smith (1960), in terms of the concept of dwell time, the time spent by the particle within a distance R of the scattering center. The time delay is deKned as the diOerence between the dwell times in the presence and in the absence of the scatterer, usually evaluated asymptotically, for large R. Smith considered only stationary scattering states. For such states, the dwell time tR within the sphere of radius R is deKned as the ratio of the probability of Knding the particle inside to the inward (same as outward) Mux through the surface:  2 3 |r|6R | | d r tR = ; (2.3.1) #R where



˝ rˆ · jin d r = − #R = − 2im |r|=R 2

 

9 in − in 9r ∗

9 in∗ in 9r

 r=R

R2 d

is the Mux of the inward probability current jin through the sphere surface. From the stationary SchrPodinger equation H (E; r) = E (E; r) ;

(2.3.2)

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and its derivative with respect to E, 9 (E; r) 9 (E; r) =E + (E; r) ; 9E 9E

H we get

9 9 − (H )∗ 9E 9E  9 9 ˝2 ∇ = − div ∗ (∇ ) − 2m 9E 9E

∗

∗

=

H

so that the numerator of (2.3.1) becomes
  9 9 ∗ ˝2 2 2 3 | | d r= R − 2m 9E 9r |r|6R

∗

 ;

∗

92 9E9r

(2.3.3)  r=R

d :

(2.3.4)

By (2.3.2) and (2.3.4), both the numerator and the denominator of (2.3.1) can be evaluated at large enough R so that the asymptotic form of the wave function can be employed. Thus, for the lth partial wave, we substitute in

→ Al (; )

→ 2i

exp(−ikr) r

l+1 Al (; )

r

(r → ∞)

  l exp[il (k)] sin kr + + l (k) 2

(r → ∞) ;

(2.3.5)

and we Knd, for large enough R, #R = and

˝k |Bl |2 = v|Bl |2 ; m




2

|r|6R

2

|Bl | =




  dl (−)l 1 − sin[2kR + 2l (k)] + O ; R+ dk 2k R



| | d r = 2|Bl |

where 2

3

(2.3.6)

d|Al (; )|2 ;

(2.3.7)

(2.3.8)

and we have employed 9=9E = (˝v)−1 9=9k :

(2.3.9)

Substituting (2.3.6) and (2.3.7) into (2.3.1), we Knd, for the dwell time for the lth partial wave.   2R 2 dl (−)l 1 + − sin [2(kR + l )] + O : (2.3.10) tR; l = v v dk kv R

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To get rid of the third term, which is an oscillatory function of R, we take an average over R, deKned, e.g., as
1 2R · · · dR : (2.3.11) · · · R ≡ R R For the free dwell time in the absence of the scatterer, this yields, for R → ∞, t 0R; l R = 2R=v ;

(2.3.12)

the time taken by the incoming wavefront at R to collapse to the origin and get back out to R. Averaging (2.3.10) over R and subtracting the free dwell time (2.3.12), we Knally get, taking into account (2.3.9), Utl ≡ lim [tR; l R − t 0R; l R ] = 2˝ R→∞

dl ; dE

(2.3.13)

the same as the Eisenbud–Wigner partial-wave time delay. For s-waves (l = 0), the free dwell time follows from (2.3.10) without the last term, t 0R; 0 =

2R sin[2(kR)] − ; v kv

(2.3.14)

an exact expression, valid for all R. It follows from this expression that, in the low-energy limit kR1, in which s-waves are dominant, t 0R; 0 =

2R 2R O[(kR)2 ] v v

(kR1) :

(2.3.15)

This clariKes the role of the oscillatory terms in (2.3.10) and (2.3.14): they are related to the uncertainty principle (Nussenzveig, 1969). Indeed, in the above limit, the de Broglie wavelength is much larger than R, and a particle with a well-deKned wavelength cannot be localized within a domain much smaller than this wavelength. Denoting by ul (E; r)=r the radial SchrPodinger wave function for energy E = (˝k)2 =(2m), we know that   l(l + 1) 2m  + 2 V (r) ul = k 2 ul ; (2.3.16) − ul + r2 ˝ where primes denote derivatives with respect to r and ul (E; 0) = 0. We choose to normalize ul so that ul (E; r) → wl (E; r) ≡ sin[kr − 12 l + l (k)]

(r → ∞) :

(2.3.17)

Let ul; 1 ≡ ul (E1 ; r) and ul; 2 ≡ ul (E2 ; r), for two diOerent energies E1 ; E2 , and similarly for wl . Then, it follows from (2.3.16) that d (ul; 1 ul; 2 − ul; 1 ul; 2 ) = (k12 − k22 )ul; 1 ul; 2 : dr

(2.3.18)

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Subtracting the corresponding equation for wl and integrating from 0 to ∞, we get
∞ wl; 1 (0)wl; 2 (0) − wl; 1 (0)wl; 2 (0) (ul; 1 ul; 2 − wl; 1 wl; 2 ) dr = ; k12 − k22 0

(2.3.19)

where we have employed the boundary conditions at the origin and at inKnity. The right-hand side is explicitly known by (2.3.17). Letting E1 → E2 and employing l’Hospital’s rule, we Knally obtain
∞ dl l sin[2l (k)] = (−) +2 [ul2 (E; r) − wl2 (E; r)] dr : (2.3.20) dk 2k 0 We apply this to s-wave scattering by a cutoO potential, V (r) = 0

for r ¿ a :

(2.3.21)

In this case, u0 = w0 for r ¿ a, and (2.3.20) yields
a sin[2(ka + 0 )] d0 = −a + u02 (E; r) dr : +2 dk 2k 0

(2.3.22)

Combining this with (2.3.13), we get Wigner’s causal inequality (2.1.10). We also see that large positive time delays must arise from a large probability to Knd the particle inside the scatterer, i.e., from resonances. 2.4. Discussion Each of the above treatments has unsatisfactory features (Nussenzveig, 1972a). The group velocity method assumes that one can identify the particle position with the peak of its representative wave packet, evaluated by stationary phase. It does not take into account that reshaping by dispersion can distort the wave packet so much that a peak cannot be taken as a good position indicator. For further critique of the group velocity, see Section 6.2. Brenig and Haag’s treatment, based on the expectation value of the distance from the scattering center, applies only for times much larger than the spreading time of the wave packet, an unrealistic assumption in usual scattering experiments (Goldberger and Watson, 1964, p. 65). Smith’s dwell time was derived in connection with a stationary scattering situation, a very restrictive setting. In the next section, which contains the core results, we discuss its extension to time-dependent problems. This will allow us to deal with general situations.

3. Average dwell time 3.1. Spherical wave packets We start again with an incoming s-wave wave packet   
∞ E p ’in (r; t) ≡ r in ≈ A(E) exp i − r − t dE ˝ ˝ 0

(r → ∞) :

(3.1.1)

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

We normalize it to represent one incident particle, by taking
∞  lim #R (t) dt = 1 ; −∞ R→∞

95

(3.1.2)

where #R (t), as in (2.3.2), is the inward probability Mux through a sphere of radius R. We Knd  
∞
∞ p + p  A∗ (E  )A(E) #R (t) ≈ 2 dE dE m 0 0   i   ×exp − [(p − p )r + (E − E )t] (R → ∞) ; (3.1.3) ˝ so that (3.1.2) yields, with v ≡ p=m,
∞ 82 ˝ |A(E)|2 v dE = 1 :

(3.1.4)

0

The total wave function is (r; t) = (’in + ’out )=r ;

(3.1.5)

where
’out (r; t) ≈ −

∞

0

   p E S(E)A(E) exp i r − t dE ˝ ˝

(r → ∞) :

(3.1.6)

By probability conservation, the probability, at time t, to Knd the particle inside a sphere of radius r centered at the origin is given by
∞ | (r  ; t)|2 r 2 dr  : (3.1.7) Pi (r; t) = 1 − 4 r

The average dwell time of the wave packet in this region is
∞ Pi (r; t) dt : T(r) = −∞

(3.1.8)

The diOerence between this and the corresponding quantity T0 (r) in the absence of the scatterer, evaluated, as in (3.1.1) and (3.1.6), at large enough r for the interaction to be negligible, is the average (dwell) time delay in the scattering process:
∞
∞ Ut(r) = 4 dt dr  [|’0 (r  ; t)|2 − |’(r  ; t)|2 ] : (3.1.9) −∞

r

We see that only the asymptotic behavior in the interaction-free region is involved.

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Substituting ’ = ’in + ’out , where ’in and ’out are respectively given by (3.1.1) and (3.1.6), and ’0 by the same expressions with S(E) = 1, we get  
∞
∞
∞
∞ i    ∗  UT(r) = 4 dt dr dE dE A (E )A(E) exp (E − E)t ˝ r 0 0 −∞    i × [1 − S ∗ (E  )S(E)] exp (p − p )r  ˝     i i   ∗    : − [1 − S(E)] exp (p + p )r − [1 − S (E )] exp − (p + p )r ˝ ˝ (3.1.10) The integration with respect to r  is performed with the help of the following distribution identities (cf., e.g., Nussenzveig, 1972b, Appendix A):  
∞
0 i 1 + ; exp(ikx) d x = exp(−ikx) d x ≡ 2 (k) = = (k) + iP k + i0 k 0 −∞  
∞
0 1 −i − = (k) − iP ; (3.1.11) exp(−ikx) d x = exp(ikx) d x ≡ 2 (k) = k − i0 k 0 −∞ where P denotes the Cauchy principal value. The result is  
∞
∞
∞ i   UT(r) = 4i˝ dt dE dE exp (E − E)t A∗ (E  )A(E) ˝ 0 0 −∞    i [1 − S ∗ (E  )S(E)]  exp (p − p )r × p − p + i0 ˝     i [1 − S ∗ (E  )] i [1 − S(E)]   exp − (p + p )r : exp (p + p )r − − p + p + i0 ˝ p + p − i0 ˝ According to the identities (3.1.11), we may replace   1 1 →P : p ± p ± i0 p ± p

(3.1.12)

(3.1.13)

Indeed, the -function terms do not contribute, either because of the unitarity of the S function or because p and p are non-negative. Since the integration over t in (3.1.12) will produce 2˝(E − E  ), it su>ces to consider the behavior of the integrand as E → E  ,   dS   S(E) = S(E ) + (E − E ) + O[(E − E  )2 ] ; dE E


C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

and

 lim

E
→E

E − E p − p



97

dE =v : dp

=

Thus, (3.1.12) becomes
UT(r) = 82 ˝

∞

−i˝S ∗ (E)

0

   sin(2kr) − sin[2(kr + )] dS + |A(E)|2 v dE : dE kv

(3.1.14)

The Krst term within the curly brackets, by (2.1.8), is the Eisenbud–Wigner partial-wave time delay Ut(E) at energy E. To discuss the other terms, let us Krst evaluate the average s-wave free-particle dwell time within the sphere of radius R,
∞
R 0 T (R) = 4 dt dr|’0 (r; t)|2 −∞


= 16

∞

−∞

0


dt

0

∞

dE




0

∞



i dE A (E ) A(E) exp (E  − E)t ˝ ∗



Performing the integration with respect to t, we get 
∞ 2R sin(2kR) 0 2 − |A(E)|2 v dE = t 0R; 0 (E) in ; T (R) = 8 ˝ v kv 0


0

R

sin(k  r) sin(kr) dr : (3.1.15)

(3.1.16)

where  in denotes the spectral average with respect to the energy spectrum of the incident wave packet, with the normalization (3.1.4), and t 0R; 0 (E) is the stationary free dwell time (2.3.14), the physical interpretation of which has already been given. The oscillatory terms within square brackets in (3.1.14) are also related with the quantal localization uncertainty. They are of the order of magnitude of the free time of Might across a de Broglie wavelength
= 1=k. We may get rid of them, as was done in (2.3.11), by spatial averaging over R, leading to   d TUT(R)UR = 2˝ : (3.1.17) dE in Thus, the (spatially smoothed) average dwell time delay is the spectral average over the incident wave packet of the Eisenbud–Wigner stationary time delay (Nussenzveig, 1972). Since the derivation depends only on the asymptotic behavior for R → ∞, it extends immediately to higher-order partial waves. Combining (3.1.14) and (3.1.16), we Knd the average interaction dwell time within the sphere of radius R,  
∞ 2R sin[2(kR + )] d T(R) = 82 ˝ + − |A(E)|2 v dE : 2˝ (3.1.18) dE v kv 0 The distance R must be such that the interaction can be neglected. For s waves, if the interaction is cutoO at r = a, we can already take R = a. Since the dwell time is positive semideKnite, (3.1.18) leads again to Wigner’s causal inequality.

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3.1.1. Example: the delta-shell potential An instructive soluble example (Nussenzveig, 1961, 1972b) is the repulsive delta-shell potential, A (r − a); 2a

V (r) =

A¿0 ;

(3.1.19)

where, for this example, we take units such that ˝ = m = 1. The dimensionless parameter A measures the opacity of the shell, which becomes impenetrable in the limit A → ∞. We denote by indices 1 and 2 the interior and exterior of the shell, respectively, with corresponding s-wave radial wave functions ’j (r; t) = r j (r; t);

(j = 1; 2) :

(3.1.20)

The wave function satisKes the free-particle SchrPodinger equation inside and outside and is continuous at the shell, but (3.1.19) leads to a discontinuity in the radial derivative, @’2 @’1 A (a; t) − (a; t) = ’(a; t) : @r @r a

(3.1.21)

To describe scattering, we deKne the solution by the initial condition ’1 (r; 0) = 0 ; and we Knd that
’2 (r; t) =

∞

a

(3.1.22)

G22 (r; r  ; t)’2 (r  ; 0) dr  ;

(3.1.23)

where ’2 (r; 0) is the initial wave packet, and G22 is the external propagator. Similarly, the decay of a wave packet initially conKned entirely within the shell would be described, in the internal region, by
a ’1 (r; t) = G11 (r; r  ; t)’1 (r  ; 0) dr  : (3.1.24) 0

In the impenetrable limit A → ∞, the internal propagator G11 can be expanded into the stationary (bound) eigenstates of the cavity, ∞

2 G11 (r; r ; t) = sin(kn r) sin(kn r  ) exp(− 12 ikn2 t) ; a n=1 

(3.1.25)

where kn = n=a. On the other hand, by applying Jacobi’s theta function transformation formula, it can also be expressed as (Nussenzveig, 1969) G11 (r; r  ; t) = G1 (|r − r  |; t) − G1 (r + r  ; t) ;

(3.1.26)

where G1 (0; t) =

∞  n=−∞

U (0 − 2na; t) ;

(3.1.27)

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

99

and  2 i0 e−i=4 exp U (0; t) = √ 2t 2t

(3.1.28)

is the free-particle SchrPodinger propagator, in the chosen units. The physical interpretation of (3.1.26) is immediate: a delta-like pulse produces an inKnite series of images in the two ‘mirrors’, one at the origin (where ‘incoming’ turns into ‘outgoing’) and the other one at r = a. This representation converges well for short times, when only a few mirror reMections have occurred, whereas the stationary-state expansion converges well for large times, after a large number of reMections. ‘Short’ and ‘large’ time scales are deKned by reference to  = 2a=k1 , the oscillation period of the ground state. For Knite A1, the shell becomes partially transparent, and the bound states are converted into resonances, whose lifetimes can be rendered arbitrarily large by adjusting A. They are associated with poles of the S function, S(k) = −

A + 2i2 − A exp(−2i2) ; A − 2i2 − A exp(2i2)

where 2 ≡ ka. For nA, the poles are approximated by      n 2 1 n 3 2n = n 1 − −i ; +O A+1 A A

(3.1.29)

(3.1.30)

which corresponds to the lifetime n ≈ 12 (n)−3 A2 a2 :

(3.1.31)

This lifetime has an immediate interpretation in terms of the multiple reMection picture. It is the ratio of the period 2a=vn (vn = n=a = internal velocity) to the transmissivity of the shell, 3n = 4(n=A)2 (escape probability). On the average, it takes the particle 1=3n periods to escape. The stationary-state expansion of the propagators can be extended, by deriving an expansion in terms of the S-matrix poles. This is done by applying the Laplace transform to solve the initial-value problem, and by employing a Cauchy-type Mittag-LeYer expansion (Titchmarsh, 1958) of the inverse of the S-matrix denominator. The result is expressed in terms of transient mode propagators. The transient mode propagator M (x; kn ; t) associated with a complex pole kn (Im kn ¡ 0) of the S-matrix is the one-dimensional free-particle SchrPodinger wave function that reduces, for t = 0, to the cutoO exponential wave packet M (x; kn ; 0) = (−x) exp(ikn x) ;

(3.1.32)

( is the Heaviside step function). For a discussion of its properties and for the derivation of the transient mode expansion of the propagators, see Nussenzveig (1972b). For the application to dwell time, we consider the outgoing wave packet deKned by (3.1.21), (3.1.22), with the initial wave

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C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

function ’2 (r; 0) = exp[ − ik0 (r − a)];

k0 ≡ k1 − i60

(60 ¿ 0) ;

(3.1.33)

where 21 = k1 a = (k1 − i61 )a, given by (3.1.30), is the lowest-order pole of the S function. Since Re k0 = Re k1 , we call this resonant excitation. We also take 60 aA, so that no other resonance is appreciably excited. Under these conditions, it can be shown (Nussenzveig, 1961, 1972b) that only the immediate neighborhood of the resonant level contributes, and the S function is well represented by a single-pole approximation, k − k1∗ S(k) exp(2ika) ≈ : (3.1.34) k − k1 For times t greater than the resonance lifetime 1 , but much smaller than the ‘spreading time’ ts ≡ min(60−2 ; 61−2 ), one Knds that the outgoing wave packet ’2; out has a diOuse wavefront around r − a ≈ k1 t, with a width of order (2t)1=2 [in ordinary units, (2˝t=m)1=2 ]. Within this region, quantum interference eOects known as ‘diOraction in time’ (Moshinsky, 1952) take place. We consider only the behavior well behind this region, so that 7 ≡ k1 t − (r − a)(2t)1=2 : We then Knd

 ’2; out (r; t) ≈ e−60 7 +

 261 (e−60 7 − e−61 7 ) exp[i(k1 r − 12 k12 t)] : (60 − 61 )

(3.1.35)

(3.1.36)

The Krst term within square brackets arises from hard sphere scattering. The other one is the resonance term. The one-level approximation (3.1.34) yields d d 61 ; (3.1.37) =k = −a +  dk dE (k − k1 )2 + (61 )2 and the initial wave function (3.1.22), (3.1.33), normalized by (3.1.4), corresponds to the Lorentzian energy spectrum 60 : (3.1.38) v|A(E)|2 ≈ 3  8 k1 [(k − k1 )2 + (60 )2 ] Thus, translating a time delay Ut into a spatial retardation Ur = k1 Ut, the Eisenbud–Wigner time delay (2.1.8) amounts to a spatial retardation 2 (Ur)E−W = −2a + ; (3.1.39) 61 whereas the average dwell time delay (3.1.17) yields an average spatial retardation 2 Ur ≈ −2a + : (3.1.40) 60 + 6 1 In both cases, by our assumption that 61 a1; 60 a1, the Krst (direct reMection) term is negligible with respect to the second one. In Fig. 3.1, we plot the probability densities associated with the Krst (hard sphere) term in (3.1.36) (dotted line), as well as the total outgoing wave packet (full line), for excitation by a broad spectrum

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

101

Fig. 3.1. Probability density proKles for the hard sphere (direct reMection) component (: : : : : :) and for the total outgoing wave packet (—) plotted against x ≡ 60 7 (7 = distance behind outgoing wavefront), for a broad line (60 = 561 ), complex resonance (60 = 61 ) and a narrow line 60 = 15 61 . The vertical lines (−· − ·−) indicate the excitation and resonance decay lengths. The thick horizontal bar ( ) next to the origin indicates the average dwell time retardation.

line (60 61 ) and by a narrow line (60 61 ), as well as for ‘complex resonance’ (60 =61 ). The dotted curve, neglecting the direct reMection time advance, may be thought of as the outgoing probability density in the absence of the scatterer, i.e., the reference proKle against which to measure spatial retardation.

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C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

In all cases, a sharp peak due to direct reMection at the surface (hard sphere term) is followed by the growth, then exponential decay, of the resonant tail. The probability density decay lengths 1 −1 1 −1 6 ; 2 61 are indicated by vertical dash-dotted lines in Fig. 3.1. Destructive interference between 2 0 hard sphere and resonance terms produces a dip in the surface-reMected wave, corresponding to the penetration of the incoming wave packet within the shell, to build up the resonance contribution. The associated probability transfer contributes to the average dwell time retardation (3.1.40), represented by the thick horizontal bar extending from the origin. The most naPDve view, that would associate the delay with the peak of the outgoing wave packet, is obviously untenable: apart from diOraction in time oscillations, which have been omitted, the peak undergoes the direct reMection time advance (because we have taken an initial wave packet with a sharp front). For excitation by a narrow line, the average dwell time retardation is smaller than the Eisenbud– Wigner result 2=61 [4 times the decay length, as in (2.1.13)], because of the destructive interference dip. As the line becomes narrower (60 → 0), the approximation improves, but the retardation becomes much smaller than the position uncertainty associated with the wave packet. This is a general consequence of the energy-time uncertainty relation: in order to sharply deKne the energy associated with the delay, approaching the stationary limit, we must broaden the time span covered by the wave packet, so that the delay itself becomes less and less signiKcant in comparison with the uncertainty in arrival time for the packet. At complex resonance, the interference dip reduces the average retardation to one half of the Eisenbud–Wigner result. For excitation by a broad line, the comparatively slow decay of the resonant tail still yields appreciable retardation. We see that wave packet reshaping strongly restricts the applicability of the Eisenbud–Wigner time delay. In contrast, the average dwell time gives a reasonable account of what happens, in all cases. 3.2. Plane wave trains We now take an incident plane wave train, deKned by
∞ A(E) exp[i(kz − !t)] dE ; 0 (z; t) = 0

(3.2.1)

where k = p=˝;

! = E=˝;

E = p2 =(2m) :

This corresponds to a total incident particle Mux per unit (x; y) area given by 
∞
∞ ˝ 9 0∗ d#in ∗9 0 = − 0 dt = 2˝ |A(E)|2 v dE ; 0 d9 2mi −∞ 9z 9z 0

(3.2.2)

(3.2.3)

where v = p=m [cf : (3:1:4)]. The total incident Mux on a sphere of radius R is #R; in = R2

d#in : d9

(3.2.4)

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

103

Let (r; t) be the total wave function, and j(r; t) the corresponding probability current density, associated with the incident wave train (3.2.1). By probability conservation, the probability, at time t, to Knd the particle within a distance R from the scattering center, is given by t   −∞ #R (t ) dt P(R; t) = ; (3.2.5) #R; in where #R (t) is the inward probability Mux through |r| = R,     9 ∗ 2 i˝ ∗9 − r d : #R (t) = − rˆ · j(r; t)r 2 d = 2m |r|=R 9r 9r |r|=R The average dwell time within the sphere of radius R is
∞ T(R) = P(R; t) dt ; −∞

(3.2.6)

(3.2.7)

which should be compared with Smith’s stationary dwell time deKnition (2.3.1). The average free-particle dwell time T0 (R) (no scatterer) is obtained by replacing by (3.2.1) in (3.2.5) – (3.2.6). This yields

∞ ˝ 2 ∞  #0; R (t) = −2i R dE dE A∗ (E  )A(E)(k + k  )j1 [(k − k  )R] exp[ − i(! − ! )t] ; m 0 0 (3.2.8) where j1 is the spherical Bessel function of order 1. The time integration in (3.2.5) can be performed employing (3.1.11),  −i(!−!
)t 
t e −i(!−!
)t
 + (! − ! ) ; e dt = iP  ! − ! −∞

(3.2.9)

where the  function does not contribute and the principal value may be omitted, since j1 (z) → 13 z for z → 0. We Knd
t
∞
∞ j1 [(k − k  )R]   2  #0R (t ) dt = 4R dE dE A∗ (E  )A(E) exp[ − i(! − ! )t] : (k − k  ) −∞ 0 0 The time integration in (3.2.7) yields another  function, leading to
∞ 8 |A(E)|2 dE : #R; in T0 (R) = 2 R3 ˝ 3 0 Taking into account (3.2.3) and (3.2.4), this Knally yields 4 R ; T0 (R) = 3 v where

∞ v|A(E)|2 dE v = 0 ∞ |A(E)|2 dE 0

is the average velocity of the wave train (3.2.1).

(3.2.10)

(3.2.11)

(3.2.12)

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C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

The result (3.2.11) has a simple interpretation. Consider a homogeneous beam of classical particles with average velocity v crossing, along the z direction, a sphere of radius R. The average travel time across, for particles at distances between 0 and 0 + d0 from the center, is 2(R2 − 02 )1=2 =v ; and the fraction of such particles is 20 d0=(R2 ), so that the classical average free time of Might is
R 4 4R classical T0 = (R2 − 02 )1=2 0 d0 = ; (3.2.13) 2 v R 0 3v to be compared with (3.2.11). To go over from free to interacting particles, we must add to packet, given by (2.1.16). The change in (3.2.5) is

0

the corresponding scattered wave

#R; in [P(R; t) − P0 (R; t)]
t

∞
∞ ˝ d dt  dE  dE A∗ (E  )A(E): =− 2m −∞ 0 0 ×{kRf(k; ) exp[i(k − k  cos )R] + kR cos f∗ (k  ; ) exp[i(k cos  − k  )R] + kf∗ (k  ; )f(k; ) exp[i(k − k  )R]} ×exp[ − i(! − ! )t  ] + c:c: ;

(3.2.14)

where c.c. denotes the complex conjugate, and R is large enough that the interaction at distances ¿ R may be neglected. By (3.2.9), this becomes, integrating over t  ;

∞ ˝2 d dE k|A(E)|2 {|f(k; )|2 #R; in [P(R; t) − P0 (R; t)] = − m 0 1 + R(1 + cos )[f(k; )eikR(1−cos ) + f∗ (k; )e−ikR(1−cos ) ]} 2
∞

∞ A∗ (E  )A(E) dE  dE exp[ − i(! − ! )t] 2 : −iP d k − k 2 0 0 ×{R(k + k  cos )f(k; )eiR(k −k




cos )

+ R(k  + k cos )f∗ (k  ; ) eiR(k cos −k )


+(k + k  )f∗ (k  ; )f(k; ) ei(k −k )R }:


(3.2.15)

We now employ the asymptotic formula (Jones, 1952, Messiah, 1965) exp(ik · R) = exp(ik R cos )     1 2 [(k − R ) − (k + R )] + O ; = ikR (kR)2

R→∞;

ˆ and k and R denote the directions of k and R, respectively. where k = k z,

(3.2.16)

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

105

Substituting into (3.2.15), and evaluating the -function contributions (from  =0 and from  =), we get 

˝2 ∞ 2i #R; in [P(R; t) − P0 (R; t)] = − [f(k; 0) − f∗ (k; 0)] d|f(k; )|2 + dE k|A(E)|2 m 0 k
∞
∞ −iP dE  dE A∗ (E  )A(E) : 0

0


exp[i(k − k  )R] df∗ (k  ; )f(k; ) × k − k    2i f(k; 0) f∗ (k  ; 0) f(k; ) i(k+k
)R − + 2i − e k k k + k k  f∗ (k  ; ) −i(k+k
)R exp[ − i(! − ! )t] : + e k 

It follows from the optical theorem (Nussenzveig, 1972b),
4 Im f(k; 0) ; 9t (E) = |f(k; )|2 d = k

(3.2.17)

where 9t is the total cross section, that the Krst integral is identically zero and that the principal value in the other integral may be removed. Thus, integrating over t from −∞ to ∞, we get, according to (3.2.7),
∞ 2 #R; in [T(R) − T0 (R) ] ≡ #R; in UT(R) = 2i˝ Re dE v|A(E)|2 : 




9 × 9E 
2 −2 ˝

0

∞

0



f(k; 0) f∗ (k  ; 0) df (k ; )f(k; ) + 2i − k k ∗



dE |A(E)|2 [f(k; )e2ikR + f∗ (k; )e−2ikR ] ; k2

 E
=E

(3.2.18)

where UT(R) is the average dwell time delay within the sphere of radius R, and l’Hospital’s rule has been employed to evaluate the -function contribution in the Krst integral. In terms of the partial-wave expansion, ∞ i f(k; ) = (l + 12 ){1 − exp[2il (k)]}Pl (cos ) ; k

(3.2.19)

l=0

the last term of (3.2.18) has an oscillatory R dependence, behaving like the terms in sin(2kR) and sin(2kR + l ) found in the treatment of partial-wave time delay. We dispose of them by taking an average [indicated as in (3.1.17)] over a distance of the order of the mean de Broglie wavelength

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in R, leading to

  9f∗ (k; ) 9f(k; ) ∗ d f(k; ) #R; in TUT(R)UR = i˝ dE v|A(E)| − f (k; ) 9E 9E 0
∞ 2 9 |A(E)| [k Re f(k; 0)] : (3.2.20) + 42 ˝2 dE v k 2 9E 0 2




∞

2




We note that   i 9f∗ 9 ∗ 9f f −f = |f|2 arg f : 2 9E 9E 9E

(3.2.21)

Taking into account (3.2.3) and (3.2.4), (3.2.20) Knally yields

where

TUT(R)UR = R (E) in ;

(3.2.22)

∞ v|A(E)|2 F(E) dE F(E) in ≡ 0  ∞ v|A(E)|2 dE 0

(3.2.23)

denotes a spectral average over the incident wave train [cf. (3.1.16)], and  
2˝ 9 1 9 2 arg f(k; ) d + 2 [k Re f(k; 0)] : |f(k; )| ˝ R (E) = 2 R 9E k 9E Substituting the partial-wave expansion (3.2.19) into (3.2.24), we get   ∞   (2l + 1)
2 dl : 2˝ R (E) = R2 dE

(3.2.24)

(3.2.25)

l=0

As is well known, (2l + 1)
2 may be regarded as the area of a circular zone associated with the lth partial wave in the incident beam. The factor within square brackets is the corresponding fraction of the target area presented by the sphere of radius R. Thus, the dwell time delay for the plane wave train is the sum of its partial-wave components. Similarly, in (3.2.24), dPs =

d9=d |f(k; )|2 d = d R2 R2

(3.2.26)

is the probability of scattering into d. In the triple average sense of (3.2.22) (averages over direction, incident energy spectrum and distance), therefore, the ‘angular time delay’ (2.1.19), ˝(9=9E) arg f(k; ), for  = 0, represents the ‘time delay in the direction ’. Of course, this does not imply validity of the pointwise result. In particular, the forward direction must be excluded. Its contribution to the dwell time delay is given by the last term of (3.2.24). This is related with the subtle interference eOects between the forward scattered wave and the incident wave, that give rise to the optical theorem (3.2.17). While this involves the imaginary part of the forward amplitude, the time delay associated with the real part is connected with the refractive index of a medium, as will be discussed in Section 3.4.

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

107

For incident beam particles with energy E that hit the sphere of radius R, the total scattering probability is
dPs 9t (E) Ps (E; R) = ; (3.2.27) d = d R2 and the quantity (corresponding to a conditional probability) TUT(R)UR

Ps (E; R) in

=

R2 R (E) in 9t (E) in

(3.2.28)

is R-independent, providing a diOerent measure of the average dwell time delay. These results, derived in (Nussenzveig, 1972a), agree with the expectation value, for the incident plane wave train (3.2.1), of the global dwell time delay operator, that will be discussed in Section 4.4. 3.3. One-dimensional scattering In condensed matter applications, particularly in connection with thin semiconductor layers, onedimensional scattering models are often considered. In order to extend the treatment of average dwell time to this situation, we employ a suitable formulation of one-dimensional potential scattering theory (Nussenzveig, 2000). Assuming that V (x) and xV (x) are absolutely integrable, the asymptotic behavior of standard scattering solutions for unilateral incidence from left or right is, respectively for (x → −∞; x → +∞), → (k; x)

≈ (eikx + R− (k)e−ikx ; T (k)eikx ) ;

(3.3.1)

← (k; x)

≈ (T (k)e−ikx ; e−ikx + R+ (k)eikx ) ;

(3.3.2)

where equality of T (k) follows from time-reversal invariance. The S-matrix is given by (Faddeev, 1964; Chadan and Sabatier, 1989)    T (k) R+ (k)   S(k) =   R− (k) T (k)  ;

(3.3.3)

where, for real k, we have the symmetry relations Sij (−k) = Sij∗ (k) ; and, by unitarity, we can employ the parametrization R∓ (k) ≡ ±0(k) exp{i[’(k) ± ;(k)]};

T (k) ≡ (k) exp[i’(k)] ;

(3.3.4)

where 02 (k) + 2 (k) = 1 :

(3.3.5)

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C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

In particular, if V is symmetric, V (−x) = V (x), parity implies R− = R+ , so that ;(k) = −=2;

[V (−x) = V (x)] :

(3.3.6)

A symmetric V is the one-dimensional analog of a central potential in three dimensions. The general deKnition of the time-delay matrix (Smith, 1960) is a direct extension of (2.1.8): T ≡ −i˝S−1

dS = T† : dE

(3.3.7)

In terms of the parametrization (3.3.4), one Knds that        2  −i;    0; − i e ’ +0 ;   0    T = ˝    i;  2    e 0; + i  ’ − 0 ;   0 where f ≡ df=dE. Thus, the eigenvalues of T are   (Ut)± = ˝’ ± ˝ (0; )2 + (=0)2 [(ln ) ]2 :

(3.3.8)

(3.3.9)

For the special case of a symmetric potential, by (3.3.6),   (Ut)sym ± = ˝’ ± ˝(=0)(ln ) ;

(3.3.10)

where the superscript (sym) is employed to denote results that hold only when V (−x) = V (x). By analogy with the group velocity derivation of the Eisenbud–Wigner time delay in Section 2.1, one would be led to identify ˝’ as a transmission group delay [since ’ is the phase of the complex transmission amplitude T (k)], and ˝(’ ± ; ) as the (left or right) reMection group delay. However, to Knd the physical interpretation of these results, we again consider general wave packets and their average dwell times. The general stationary solution is a superposition of left and right incidence basis functions, (3.3.1) – (3.3.2). However, this basis is related with the transfer matrix (Chadan and Sabatier, 1989), rather than with the scattering matrix. To deal with general wave packets in terms of the S-matrix elements, we go over to new basis functions, ( s ; a ), respectively associated with symmetric and antisymmetric bilateral incidence, s (k; x)

≡

a (k; x)

→ (k; x)

≡

+

→ (k; x)

−

← (k; x)

;

← (k; x)

:

(3.3.11)

For a symmetric potential, these can be thought of as one-dimensional analogs of s and p waves. The most general one-dimensional wave packet is a superposition,
∞ (x; t) = [As (E) s (k; x) + Aa (E) a (k; x)]e−iEt=˝ dE ; (3.3.12) 0

where E = (˝k)2 =(2m) = 12 mv2 (v is the asymptotic velocity). To gain more insight into this representation, let us relate it with the expansion into momentum eigenfunctions for the special case of

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

a free-particle wave packet,  
∞ E dk : a(k) exp i kx − t 0 (x; t) = ˝ −∞ Rewriting it in terms of (3.3.12), we Knd that   m a(k) ± a(−k) : As; a (E) = 2 2˝ |k|

109

(3.3.13)

(3.3.14)

Unilateral incidence, according to (3.3.11) and (3.3.12), corresponds to As (E) = ±Aa (E) ≡ 12 A(E)

(+ for →; − for ←) :

(3.3.15)

On the other hand, one needs bilateral incidence to represent even wave packets that appear to travel unidirectionally, such as the well-known example of a free Gaussian wave packet (its Gaussian momentum spectral decomposition extends from −∞ to ∞ in k). Similarly, by analyticity, any wave packet that is, at some time, unilaterally conKned (e.g., by a shutter) corresponds to bilateral incidence. We now extend the treatment of Sections 3.1 and 3.2, by evaluating the average dwell time Td (X ) of the general wave packet (3.3.12) within the domain − 12 X 6 x 6 12 X , where X is in the asymptotic scattering region. For a symmetric potential, the origin is taken at its center of symmetry, so that this domain is the analog of the spherical region taken for central potentials in three dimensions. As before, we normalize the wave packet to represent one incident particle, by requiring [cf. (3.1.2)]
∞ lim [jin (− 12 X; t) − jin ( 12 X; t)] dt = 1 ; (3.3.16) X →∞

−∞

where jin (x; t)=(˝=m) Im( in∗ 9 in =9x), with in given by the incoming component of the wave packet, respectively on the left and on the right, is the incident probability current. This yields
∞ 4˝ [|As (E)|2 + |Aa (E)|2 ]v dE = 1 ; (3.3.17) 0

which should be compared with (3.1.4). With this normalization, the probability to Knd the particle inside the domain (−X=2; X=2) at time t is given by
| (x; t)|2 d x ; (3.3.18) P(X; t) = 1 − |x|¿X=2

and the average dwell time spent by the particle within this domain is
∞ P(X; t) dt ; Td (X ) = −∞

which, by (3.3.18), is deKned entirely in terms of asymptotic quantities.

(3.3.19)

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Denoting by 0 (x; t) the free-particle wave packet (for V =0) associated with (3.3.12), the average dwell time delay in the scattering process is
∞
Utd = lim dt [| 0 (x; t)|2 − | (x; t)|2 ] d x : (3.3.20) X →∞

−∞

|x|¿X=2

The average wave packet dwell time within the interaction region is Td (X ) = Td0 (X ) + Utd (X ) ;

(3.3.21)

where X is large enough to be in the asymptotic scattering region. Here, Td0 (X ) is the average free-particle dwell time in the domain (−X=2; X=2), given by
∞
X=2 0 Td (X ) = dt | 0 (x; t)|2 d x ; (3.3.22) −X=2

−∞

where, for free particles, s and a in (3.3.12) are respectively proportional to cos(kx) and sin(kx). Substituting into (3.3.22), we Knd 
∞ 0 2 2 X 2 2 sin(kX ) v dE : (3.3.23) [|As (E)| + |Aa (E)| ] + [|As (E)| − |Aa (E)| ] Td (X ) = 4˝ v kv 0 This result should be compared with (3.1.16). In view of (3.3.17), the Krst contribution is the spectral average of the free time of Might X=v, taken over the energy spectrum of the incident wave packet. The oscillatory contributions (of the order of the free time of Might across a de Broglie wavelength
= 1=k) are again related to the uncertainty principle. The evaluation of (3.3.20) makes use of the distribution identities (3.1.11). Combining the contributions from the integration ranges (−∞; − 12 X ) and ( 12 X; ∞), delta-function terms get canceled on account of the unitarity relations (3.3.4), (3.3.5). As was discussed following (3.1.13), the energy conservation delta function arising from time integration also leads to the elimination of principal value denominators and gives rise to energy derivatives of the S-matrix elements. The result, for a symmetric potential, is (Nussenzveig, 2000)
∞ sym sym 2 [|As (E)|2 (Ut)sym Utd = 4˝ + + |Aa (E)| (Ut)− ] 0

cos(U’) kv   sin(U’) sin(kX ) 2 2 − v dE ; + [|As (E)| + |Aa (E)| ]  kv kv − [|As (E)|2 + |Aa (E)|2 ]0

(3.3.24)

where U’ ≡ kX + ’ ; sym

(3.3.25)

and (Ut)± are the eigenvalues (3.3.9) of the time delay matrix. sym This provides the sought-for physical interpretation: (Ut)sym + [(Ut)− ] is associated with symmetric (antisymmetric) bilateral incidence, and the result for an arbitrary wave packet is a spectral average, taken with the normalization (3.3.17). The oscillatory terms, again, arise from the uncertainty relation.

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

111

In the most general case of an asymmetric potential, the result is somewhat more complicated, involving also interference terms between As and Aa . On the other hand, according to (3.3.15), unilateral incidence on a symmetric potential is a particular case, with As (E) = ±Aa (E) ≡ 12 A(E) (+ for →, − for ←), yielding  
∞ cos(U’) sym 2  v dE = Utd sym |A(E)| ˝’ − 0 (3.3.26) Utd → = 2˝ ← ; kv 0 which (apart from the oscillatory term) is the spectral average [note that (3.3.17) is also changed] of the transmission group delay. This result is a generalization of an identity derived (Hauge et al., 1997) under more restrictive assumptions on the wave packet. 3.3.1. Example: rectangular potential To illustrate these results, we consider a rectangular potential, V (x) = V0 (−a=2 ¡ x ¡ a=2); =0 otherwise, where V0 may be positive (barrier) or negative (well). We take

E ¿ V0 , reserving the discussion of a tunneling situation for a later section. We deKne N ≡ 1 − (V0 =E) (refractive index),  ≡ 2Nka (double traversal phase), > ≡ (N − 1)ka (single traversal phase shift). Then, T2 exp(i>) ; 1 − R2 exp(i)

2iR exp(i>) sin(Nka) ; (3.3.27) 1 − R2 exp(i) √ where R = (N − 1)=(N + 1), T = 2 N =(N + 1) are the reMection and transmission amplitudes, respectively, for a potential step of height (depth) |V0 |. Since T=

R=

1 = 1 + R2 exp(i) + [R2 exp(i)]2 + · · · ; 1 − R exp(i) 2

(3.3.28)

there is an immediate and well-known interpretation in terms of an inKnite series of multiple internal reMections and transmissions (Fabry–Perot interferometer). A similar interpretation holds for the time-dependent propagator (Nussenzveig, 1972b). Since V has compact support, we can take X = a. Around a transmission resonance, Nka = m (m = 1; 2; 3; : : :), (3.3.21) and (3.3.26) lead to   R2 2a a Td (a) → ≈ ; (3.3.29) + v1 T2 v1 where v1 = Nv is the internal velocity. For a tall barrier (N 1) and around a near-barrier-top resonance, the dwell time is a=v1 (classical time of Might), because of the large average number (≈ R2 =T2 ) of oscillations before emergence. One-dimensional scattering has some peculiarities. One of them is that the free SchrPodinger propagator (‘heat pole’ Green function) decays like O(t −1=2 ) as |t| → ∞, instead of the three-dimensional O(t −3=2 ) decay. This entails some restrictions on admissible wave packets, in order to ensure that the free-particle dwell time over a Knite interval be bounded (Muga et al., 1995). According to (3.3.23), a necessary condition is that As; a (E) must be square integrable in a neighbourhood of the origin. By (3.3.14), this requires that the free-particle wave packet should have no zero-velocity component, i.e., the space average of the initial wave packet must vanish.

112

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In the presence of interaction, faster decay has been found in several examples (Muga et al., 1995). Indeed, for the rectangular potential, the average dwell time at zero energy is well-behaved. 3.4. Electromagnetic scattering We now go over from non-relativistic to electromagnetic scattering (Nussenzveig, 1997). To simplify the presentation, we consider a classical electromagnetic wave Keld, but the results also apply to one-photon wave packets, as pointed out below (Section 7). We consider the average dwell time of the electromagnetic energy within a given region for an incident plane wave train, extending the treatment of Section 3.2. The incident train, traveling in the z direction, is taken to be linearly polarized with its electric Keld E0 in the x direction, so that [cf. (3.2.1)]
∞ E! exp[i(kz − !t)] d! xˆ ; (3.4.1) E0 (r; t) = −∞

where k = !=c, carets denote unit vectors and E0 must be real, so that E−! = E!∗ :

(3.4.2)

The corresponding magnetic Keld is B0 (r; t) = zˆ × E0 (r; t)=c :

(3.4.3)

The incident energy density w0 and Poynting vector S0 are respectively given by w0 = @0 E02 ;

S0 = cw0 zˆ :

(3.4.4)

The total incident energy Mux per unit area #0 is
∞
∞ |E! |2 d! ; zˆ · S0 dt = 2c@0 #0 = −∞

(3.4.5)

−∞

and the total energy incident on a sphere of radius R is R2 #0 . By the continuity equation, the total incident energy present within such a sphere at time t is

t 

t 3  9w0 3  rˆ · S0 (r; t  )R2 d : w0 (r; t) d r = dt  d r = − dt (3.4.6) 9t |r|6R |r|6R −∞ −∞ |r|=R We now evaluate the average free dwell time of the energy (in the absence of a scatterer) within this sphere. For this purpose, as in (3.2.5), we consider the fraction of the total energy incident on the sphere that is present inside it, at a time t, 
w0 (r; t) d 3 r (R2 #0 ) ; (3.4.7) P0 (R; t) = |r|6R

which we interpret [cf. (3.2.5)] as the probability to Knd the energy within this sphere at time t. Then, the average energy dwell time within this sphere is  
∞
∞
t   2 P0 (R; t) dt = − dt dt (3.4.8) rˆ · S0 (r; t )R d (R2 #0 ) : T0 (R) = −∞

−∞

−∞

|r|=R

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

From (3.4.1) to (3.4.4), we get 
rˆ · S0 (r; t  ) d = 4ic@0 |r|=R

∞

−∞

d!




∞

−∞

113

d! E! E!∗
· exp[ − i(! − ! )t  ]j1 [(! − ! )R=c] ; (3.4.9)

where j1 (x) is the spherical Bessel function of order one. As in Section 3.2, the integration over t  in (3.4.8) can be performed, with the help of (3.2.9), in which the  function does not contribute, and the principal value may be omitted, since j1 (x) → 1 x (x → 0). Substituting the result in (3.4.8), and performing the integration over t, we Knd 3 
∞
∞
t − dt dt  |E! |2 d! ; (3.4.10) rˆ · S0 (r; t  ) d = 43 (R=c)2c@0 −∞

−∞

|r|=R

−∞

so that (3.4.5) and (3.4.8) lead to T0 (R) =

4R ; 3c

(3.4.11)

to be compared with (3.2.11). This agrees with the average free time of Might of a parallel beam of classical particles, with velocity c, through a spherical region of radius R. 3.4.1. Average dwell time delay We assume that the scatterer is spherically symmetric, non-absorbing and reAection-invariant. For a linearly polarized unit amplitude incident wave, the asymptotic behavior of the electric Keld, at large distances R from the scattering center, is then given by (van de Hulst, 1957; Bohren and HuOman, 1983) E(R; ; ’; t) ≈ exp(−i!z )xˆ + F! (; ’) exp(−i!R )=(ikR) ;

(3.4.12)

z ≡ t − (z=c);

(3.4.13)

where R ≡ t − (R=c) ;

F! (; ’) = sin ’ S1 (!; )’ˆ − cos ’ S2 (!; )ˆ :

(3.4.14)

In (3.4.14), S1 is the perpendicularly polarized (magnetic multipole) scattering amplitude and S2 is the parallel polarized (electric multipole) scattering amplitude. Note that these amplitude functions are dimensionless, diOering, by the factor 1=(ik) in the last term of (3.4.12), from the quantum-mechanical deKnition (2.1.16) of the scattering amplitude. For the incident wave train (3.4.1), the total asymptotic electric Keld is
∞
∞ c d! ; (3.4.15) E(R; ; ’; t) ≈ E! exp(−i!z ) d! xˆ + E! F! (; ’) exp(−i!R ) iR ! −∞ −∞

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C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

where, corresponding to (3.4.2), F−! = F!∗ : The corresponding asymptotic magnetic Keld is

1 ∞ ∗ 1 ∞ ∗ d! B(R; ; ’; t) ≈ E! exp(i!z ) d! yˆ + E! rˆ × F!∗ (; ’) exp(i!R ) ; c −∞ iR −∞ !

(3.4.16)

(3.4.17)

where we have employed (3.4.2) and (3.4.16). The asymptotic Poynting vector follows from (3.4.15) and (3.4.17). Its radial component, taking into account (3.4.14), is found to be

∞ ic2 @0 ∞ rˆ · S = rˆ · S0 − d! d! E!∗
E! R −∞ −∞  1 × exp[i(! R − !z )][S1∗ (! ; ) sin2 ’ + S2∗ (! ; ) cos  cos2 ’] !  1  2 2 − exp[i(! z − !R )][S1 (!; ) cos  sin ’ + S2 (!; ) cos ’] !

c3 @0 ∞ d! ∞ d! ∗ + 2 E E exp[ − i(! − ! )R ]  !
! R −∞ ! −∞ ! ×[S1 (!; )S1∗ (! ; ) sin2 ’ + S2 (!; )S2∗ (! ; ) cos2 ’] ; where S0 is the incident Poynting vector. Integrating this over the surface |r| = R, and taking into account (3.4.5) – (3.4.7), we get R2 #0 [P(R; t) − P0 (R; t)]

∞ d! 2 2 2 |E! | d sin  = i c @0 R 0 −∞ ! ×{exp[ − ikR(1 − cos )][S1∗ (!; ) + S2∗ (!; ) cos ] − exp[ikR(1 − cos )][S1 (!; ) cos  + S2 (!; )]}
∞
 d! 2 3 2 |E! | d sin [|S1 (!; )|2 + |S2 (!; )|2 ] − c @0 2 0 −∞ !

∞
∞ exp[ − i(! − ! )t] ∗ E d! d! E d sin  −c2 @0 RP
! ! (! − ! ) 0 −∞ −∞  1 exp[iR(k cos  − k  )][S1∗ (! ; ) + S2∗ (! ; ) cos ] × !

(3.4.18)

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

 1 exp[ − iR(k  cos  − k)][S1 (!; ) cos  + S2 (!; )] !

∞ d! ∞ d! exp[ − i(! − ! )R ] ∗ 3 E!
E! − ic @0 P  (! − ! ) −∞ ! −∞ !
 × d sin [S1 (!; )S1∗ (! ; ) + S2 (!; )S2∗ (! ; )] ;

115

−

0

(3.4.19)

where we have employed (3.2.9). We now let R increase along the Kxed direction , and employ (3.2.16). Because of the assumed rotation and reMection symmetry of the scatterer, we have (van de Hulst, 1957) S1 (!; 0) = S2 (!; 0) ≡ S(!; 0);

S1 (!; ) = −S2 (!; ) ≡ S(!; ) :

(3.4.20)

Thus, performing the angular integrations over the delta-function terms arising from (3.2.16), we Knd R2 #0 [P(R; t) − P0 (R; t)]   
∞ 4 2 2ikR 2 −2ikR ∗ 9t − 2 Re S(!; 0) − 2 [e S(!; ) − e d!|E! | S (!; )] =− c@0 k k −∞

∞ d! ∞ d! exp[ − i(! − ! )R ] ∗ 3 E!
E ! −ic @0 P (! − ! ) −∞ ! −∞ ! 
  ∗  ∗  ∗  × d sin [S1 (!; )S1 (! ; ) + S2 (!; )S2 (! ; )] − 2[S(!; 0) + S (! ; 0)] 0

+ 2ic3 @0 P




∞

−∞

d! !




∞

−∞

d! exp[ − i(! − ! )t] ∗ E!
E! ! (! − ! )

×{S(!; ) exp[i(k + k  )R] − S ∗ (! ; ) exp[ − i(k + k  )R]} ;

(3.4.21)

where 9t is the total cross section. The optical theorem (van de Hulst, 1957) 9t = (4=k 2 ) Re S(!; 0)

(3.4.22)

implies that the Krst bracket on the r.h.s. of (3.4.22) vanishes. The integral of the second bracket also vanishes, because, by the symmetry relations (3.4.2) and (3.4.16), its integrand is an odd function. The remaining integrals, as a consequence, are regular at ! = ! ; so that the principal value sign can be removed. With the change of variables ! − ! = u;

1 (! 2

+ ! ) = v

(3.4.23)

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(3:4:21) becomes R2 #0 [P(R; t) − P0 (R; t)]
∞
∞ du dv E∗ 1 E 1 = − ic3 @0 exp(−iuR ) 2 − 1 u2 v− 2 u v+ 2 u u v −∞ −∞ 4 
 × d sin [S1 (v + 12 u; )S1∗ (v − 12 u; ) + S2 (v + 12 u; )S2∗ (v − 12 u; )] 0

− 2[S(v +

1 u; 0) 2

∗

+ S (v −

 1 u; 0)] 2

;

(3.4.24)

where the last integral in (3.4.21), which would give rise to the term
∞
∞ du dv 3 1 E 1 exp(−iut) 2ic @0 E 1 2 −v+ 2 u v+ 2 u 2 −∞ u −∞ v − 4 u ×{S(v + 12 u; ) exp(2ivR=c) − S(−v + 12 u; ) exp(−2ivR=c)} ; does not contribute, because its integrand, again by the symmetry relations, is odd. Note that the terms within the curly brackets correspond to the rapidly oscillating terms containing exp(±2ikR) in the last integral of (3.2.18). They are identically canceled out here, whereas, in the non-relativistic expression (3.2.18), it was necessary to dispose of them by averaging over a distance of the order of the mean de Broglie wavelength. Integrating both sides of (3.4.24) with respect to t from −∞ to ∞, and recalling (3.4.8), we get
∞ dv 2 2 2 3 R #0 [T(R) − T0 (R) ] ≡ R #0 Utd (R) = −2i c @0 |Ev |2 2 v −∞   
 1 · ×lim d sin [S1 (v + 12 u; )S1∗ (v − 12 u; ) + S2 (v + 12 u; )S2∗ (v − 12 u; )] u→0 u 0  ∗ 1 1 − 2[S(v + 2 u; 0) + S (v − 2 u; 0)] ; (3.4.25) or, employing 1’Hospital’s rule and reverting to the previous notation (3.4.23),
∞ d! 2 2 3 R #0 Utd (R) = −i c @0 |E! |2 2 ! −∞ 
  9 9 − × d sin [S1∗ (! ; )S1 (!; ) + S2∗ (! ; )S2 (!; )] 9! 9! 0  ∗  − 2[S(!; 0) + S (! ; 0)] : (3.4.26) !=!


Substituting #0 by its expression (3.4.5) and taking into account the relation (3.2.21), we Knd Utd (R) = UR (!) in ;

(3.4.27)

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

where

∞ F(!) in ≡

2

−∞ |E! | F(!) d! ∞ 2 −∞ |E! | d!

denotes a spectral average over the incident wave train, and 
  9 1 arg S1 (!; ) |S1 (!; )|2 UR (!) = (kR)2 9! 0   d 2 9 arg S2 (!; ) sin  d − 2 Im S(!; 0) + |S2 (!; )| 9! d!

117

(3.4.28)

(3.4.29)

is the spectral time delay. As was done in connection with (3.2.24), the result (3.4.29) can also be interpreted in terms of the probabilities for scattering. The diOerential cross section for scattering into solid angle d = sin  d d’ is given by (van de Hulst, 1957) d9(; ’) = k −2 (i1 sin2 ’ + i2 cos2 ’) d ;

(3.4.30)

ij ≡ |Sj (!; )|2

(3.4.31)

where (j = 1; 2) :

Integrating over ’, we Knd that  d9j = 2 ij sin  d (3.4.32) k is the diOerential cross section for scattering into d with polarization j. Thus, the probability for scattering from the sphere of radius R into d, with polarization j, is given by d9j 1 |Sj (!; )|2 sin  d ; (3.4.33) dPj = 2 = R (kR)2 and (3.4.29) may therefore be rewritten as  2
   dPj 9 d [arg Sj (!; )] d − 2 Im S(!; 0) ; UR (!) = d 9! d! j=1 0 where the Krst term is an angular average over . The average dwell time delay (3.4.27) can therefore be expressed as   2  d Ut(R) = Im S(!; 0) Utj (!; ) ; R in − 2 ; d! in j=1

(3.4.34)

(3.4.35)

where, in the double angular bracket, the inner one denotes the angular average over the sphere of radius R, weighted by the probability of scattering from this sphere, and the outer one is the spectral average (3.4.23) over the incident wave train. In (3.4.35), 9 Utj (!; ) = [arg Sj (!; )] (3.4.36) 9!

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therefore represents, in this double average sense, the angular time delay, for scattering in the direction  = 0; at frequency !; with polarization j. However, just as was found in connection with (3.2.24), this is not true in the forward direction. The forward time delay, connected with the last term of (3.4.35), is aOected by subtle interference eOects with the incident wave. For light propagation through a rareKed medium containing N scatterers per unit volume, the forward delay gives rise to the real refractive index, as follows from the relation (van de Hulst, 1957) 2N n(!) − 1 = 3 Im S(!; 0) : (3.4.37) k 3.4.2. Partial-wave expansion The partial-wave expansion of the polarized scattering amplitudes Sj (!; ) is given by (Nussenzveig, 1992) ∞ 1 Sj (!; ) = {[1 − Sl( j) (!)]t1 (cos ) + [1 − Sl(i) (!)]pl (cos )} (i; j) = 1; 2; i = j ; 2 l=1

(3.4.38)

where tl (cos ) =

(2l + 1) d 1 P (cos ); l(l + 1) d l

tl (1) = pl (1) = l + with


1 2



0

1 2

0



(2l + 1) Pl1 (cos ) ; l(l + 1) sin 

;

(3.4.39) (3.4.40)

tl (cos )pl
(cos ) sin  d = 0;




pl (cos ) =

∀(l; l ) ;

[tl (cos )tl
(cos ) + pl (cos )pl
(cos )] sin  d = (2l + 1)l; l
:

(3.4.41) (3.4.42)

Substituting (3.4.38) into (3.4.29), and taking into account these orthonormality relations, we Knd 
 2 9 2 9 arg S1 (!; ) + |S2 (!; )| arg S2 (!; ) sin  d |S1 (!; )| 9! 9! 0 2  ∞  d (2l + 1) {Im[i exp(il( j) ) sin l( j) ]} ; (3.4.43) = d! j=1 l=1

where Sl( j) (!) = exp[2il( j) (!)] ;

(3.4.44)

so that l( j) is the phase shift associated with the lth partial wave in polarization j. Similarly, taking into account (3.4.40),   2  ∞ ( j)  d ( j) dl −2 Im S(!; 0) = : (3.4.45) (2l + 1) [1 − cos(2l )] d! d! j=1 l=1

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

It follows from (3.4.29), (3.4.43) and (3.4.45) that  ∞ (1)  d(2) (2l + 1) dl l : UR (!) = + (kR)2 d! d!

119

(3.4.46)

l=1

The factor (2l + 1)=(kR)2 in this expression is the fraction of the total beam area R2 incident on the sphere that is associated with the lth partial wave. Indeed, by the localization principle (van de Hulst, 1957), this corresponds to a circular ring with area (2l + 1)=k 2 . Comparing (3.4.46) with (3.2.25), we see that the electromagnetic result diOers from the nonrelativistic one by the substitution (1)

(2)

d d dl → l + l ; (3.4.47) dE d! d! so that the partial-wave time delay for each polarization is one-half of the Eisenbud–Wigner time delay for non-relativistic scattering. If there is a sharp isolated resonance for the lth partial wave at frequency !0 with halfwidth ;, the corresponding phase shift is dominated, near the resonance frequency, by a resonant term  1 ; ( j) −1 2 l; res = −tan : (3.4.48) ! − !0 2˝

The corresponding time delay at resonance,   d ( j) 2 l; res = d! ; !=!0

(3.4.49)

is twice the lifetime associated with the resonance. We can attribute one-half of this result to the excitation rise time and the other half to the decay. 3.4.3. Example: totally reAecting sphere As an explicit application, we consider electromagnetic scattering by a perfectly conducting sphere, of radius a much larger than the wavelength, so that 2 ≡ ka1 ;

(3.4.50)

where 2 is the dimensionless size parameter. Under these conditions, the dominant term in the polarized scattering amplitudes, outside of the penumbra region, i.e., for 2−1=3 , is given by the WKB approximation (van de Hulst, 1957; Nussenzveig, 1992)    1 S1 (!; ) ≈ S2 (!; ) ≈ − i2 exp −2i2 sin (2−1=3 ) ; (3.4.51) 2 2 so that, neglecting the subdominant contribution from the penumbra,

 2  9 d sin  |Sj |2 sin  sin(=2) d arg Sj (!; ) = −(a=c)22 9! 0 0 j=1 = − 43 (a=c)22 :

(3.4.52)

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Since the dominant term of the forward scattering amplitude is S(!; 0) ≈ 12 22 ;

(3.4.53)

the forward contribution in (3.4.29) may be neglected relative to that from (3.4.52), which yields 4 a  a 2 UR = − : (3.4.54) 3c R Since a is much larger than the wavelength and there is no interaction beyond r = a, we may already apply (3.4.54) at R = a, yielding Ua = − 43 (a=c) :

(3.4.55)

This result has a very simple interpretation. As we see from (3.4.52), it represents a time advance, which is the angular average of the geometrical optic time advance −(a=c) sin(=2), associated with an incident ray that is directly reMected from the surface into the direction . The same result may be obtained from the partial-wave expansion (3.4.46) (Nussenzveig, 1997). 4. Properties and extensions 4.1. Classical time delay and the virial Consider, in classical mechanics, a particle of mass m, scattered by a central potential V (r). Its energy E and angular momentum L are respectively given by E = P 2 =(2m);

L = Pb ;

(4.1.1)

where P is the asymptotic momentum and b is the impact parameter. Taking the origin of time t = 0 at the distance of closest approach, the particle radial momentum at time t at position r is 1=2  b2 2mV (r) ; (4.1.2) p = mr˙ = @(t)P 1 − 2 − r P2 where @(t) is the sign function. For free particles (V = 0), this yields r → ±(P=m)t (t → ±∞) ;

(4.1.3)

where ± signs correspond to each other. Assuming that V (r) = O(r −1− ); (r → ∞;  ¿ 0) ;

(4.1.4)

which excludes Coulomb potentials, it follows from (4.1.2) that r → ±(P=m)t ± s

(t → ±∞) ;

so that the classical time delay is given by m Ut = −2 s : P

(4.1.5) (4.1.6)

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121

We now relate this to the virial function introduced by Clausius in classical kinetic gas theory (ter Haar, 1955). For this purpose (Robinson and Hirschfelder, 1963), we start from the Poisson bracket relation dw = (H; w) ; (4.1.7) dt where w is any function of the generalized coordinates and momenta, and H is the Hamiltonian, given here by L2 P2 p2 + : + V (r) = 2m 2mr 2 2m Choosing for w the function H=

(4.1.8)

w(r; p) = r(p − P) ;

(4.1.9)

we Knd

  dw P dV = (P − p) − 2V + r ; dt m dr

(4.1.10)

where (4.1.8) has been employed. Integrating (4.1.10) along the trajectory, from t = 0 to t = T , we get 
T dV P2 [r(p − P)]T0 = T − [rP]T0 − dt : 2V + r m dr 0 Letting T → ∞ in this result, we note that, by (4.1.2) and (4.1.4), the contribution from the upper limit on the left-hand side vanishes. Taking into account (4.1.5), we obtain the virial theorem for classical scattering (Demkov, 1961) 
∞ dV dt = −Ps : (4.1.11) 2V + r dr 0 Together with (4.1.6), this leads to the classical relationship between the time delay and the virial, 
∞ P2 dV Ut = dt : (4.1.12) 2V + r m dr −∞ We now extend this relationship to quantum scattering. 4.2. Quantum virial relation Let

(k; r) be a solution of the stationary SchrPodinger equation   ˝2 h2 2 H (k; r) = − B + V (r) (k; r) = k (k; r) : 2m 2m

Applying Green’s second identity to a volume V bounded by a surface 9V, we Knd  
 ˝2 2 ˝2 ∗ 3 k ;W H− d r=− [ ∗ ∇(W ) − W ∇ ∗ ] · dS ; 2m 2m V 9V

(4.2.1)

(4.2.2)

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where W is an operator on

. Choosing

9 9 −k ; 9r 9k we Knd for the commutator     ˝2 2 dV k ;W H− = − 2V + r 2m dr W =r

(4.2.3)

;

so that the left-hand side of (4.2.2) is real, yielding  
 dV ˝2 ∗ d3 r = Re 2V + r [ ∗ ∇(W ) − W ∇ dr 2m V 9V

(4.2.4)

∗

] · dS :

We now take for V a sphere of radius R, centered at the origin, and for wave function, with asymptotic behavior

(4.2.5) the stationary scattering

eikR (R → ∞) : (4.2.6) R Assuming that V (r) is of su>ciently rapid decrease at inKnity, so that the integral converges, this leads to  

 dV ˝2 9 ∗ 3 d r=− lim Re 2V + r d sin  [kf(k; )] dr m R→∞ 9k 0 (k; r) ≈ exp(ikR cos ) + f(k; )

×{[ikR(1 + cos ) − 1] exp[ikR(1 − cos )] + 2ikf∗ (k; ) + o(1)} : Employing integration by parts and the identity (3.2.21), this yields  

4 9 1 dV 9 ∗ 3 d r = 2 d|f(k; )|2 arg f(k; ) + 2 [k Re f(k; 0)] ; 2V + r E dr 9k k 9k

(4.2.7)

where E = (˝k)2 =(2m). Thus, 9=9k = ˝v 9=9E

(v = ˝k=m) ;

and (4.2.7) may be rewritten as   1 dV 1 V+ r = R2 vR (E) ; E 2 dr

(4.2.8)

(4.2.9)

where R2 R (E), given by (3.2.24), plays the role of a global dwell time delay at energy E and is independent of R, as shown by (3.2.28). The expectation value on the left is taken with respect to the total wave function (4.2.6), which is dimensionless, so that both sides of (4.2.9) have dimensions of volume. Corresponding partial-wave results are obtained by substituting (3.2.25) on the right-hand side of (4.2.9), and employing the partial-wave expansion of the total wave function, (k; r) =

∞  l=0

(2l + 1)il ’l (k; r)Pl (cos ) ;

(4.2.10)

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123

where ’l (k; r) is the regular solution, with asymptotic behavior ’l (k; r) ≈ exp(il ) sin(kr − 12 l + l )=(kr) (r → ∞) : We Knd 4m ˝2 v


0

∞

dV 2V + r dr



|’l (k; r)|2 r 2 dr = 2˝

dl ; dE

(4.2.11)

(4.2.12)

which expresses the Eisenbud–Wigner partial-wave time delay in terms of the virial. This result, due to Demkov (1953), provides some qualitative insight into the connection between time delay and the potential. Thus, a monotonically decreasing potential, for which dV=dr ¡ 0, is not likely to produce a large positive delay, as expected (one needs a ‘pocket’ to hold the particle and produce a resonance: see Section 7.2 for an example). 4.3. Time delay and density of states This connection between time delay and density of states, one of its most important properties, was brought out, independently, by Friedel (1952) and Lifshitz (1952); the latter contribution is reviewed in Section 4.5. Friedel was dealing with the eOect of substitutional impurities in a metal on the charge distribution of electrons. For simplicity, he considered electrons in a spherically symmetric potential V (r), assumed to represent the eOect of the impurities in a Hartree–Fock description. He related the change in the density of states, relative to free electrons, with the phase shifts in continuum electronic levels. That led to the connection with time delay. As a preamble, consider a free-particle wave packet,  
1 E ˝2 k 2 (0) 3 d (r; t) = kA(k) exp i k · r − t ; E = : (4.3.1) (2)3=2 ˝ 2m The density of states 0(0) (k) per unit volume of momentum space, for this wave packet, is deKned by

3 (0) (0) (0) d k0 (k) ≡  | = d 3 r| (0) (r; t)|2 ; (4.3.2) which yields 0(0) (k) = |A(k)|2 :

(4.3.3)

This gives the probability density for Knding a state of momentum ˝k within a cell d 3 k of momentum space. Note that, in the inKnite volume limit, the momentum eigenstates in the packet, (0) k (r)

= r|

(0) k

=

exp(ik · r) ; (2)3=2

(4.3.4)

(0)  satisfy a continuum normalization  k(0)
| k = (k − k), and that their density of states diverges as 3 V=(2) , where V is the spatial volume. We now compute the change in the density of states per unit energy interval induced by the potential. Consider initially that the electrons are enclosed within a large sphere of radius R. If free,

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they are described, for given angular momentum l, by spherical multipole waves,  2 (0) (0) (4.3.5) r| klm = ’l (r)Ylm (; ) = jl (kr)Ylm (; ) ;  where jl is the spherical Bessel function. Asymptotically, for large r,  1  sin kr − l (r → ∞) ; (4.3.6) ≈ ’(0) l r 2 whereas, in the presence of a potential V (r),  1  ’l ≈ sin kr − l + l (r → ∞) : (4.3.7) r 2 By imposing a Dirichlet boundary condition, ’l (R) = 0, we discretize the energy levels, through  (4.3.8) kR − l + l (k) = nl  ; 2 where nl is an integer. The level density per unit energy is therefore   dl dnl (2l + 1) d k = R+ ; (4.3.9) 0l (E) ≡ (2l + 1) dE  dE dE taking into account the degeneracy (2l + 1). This quantity diverges as R → ∞, as before. However, the total change in the density of states due to the potential, ∞ 1 dl U0 = 0(E) − 0(0) (E) = (2l + 1) ; (4.3.10)  dE l=0

should be Knite. By referring to (2.1.13) and (3.2.25), the connection with time delay already becomes apparent. However, before discussing it, let us rederive the result, directly from SchrPodinger’s equation. We begin by Knding the partial-wave expansion of the free density of states. For this purpose, we substitute into (4.3.4) the expansion exp(ik · r) = 4

∞  l 

∗ il jl (kr)Ylm (k ; k )Ylm (r ; r ) :

(4.3.11)

l=0 m=−l

Comparing the result with (4.3.5), we get |

(0) k

=

∞  l  il ∗ Y (k ; k )| k lm l=0 m=−l

(0) klm

:

(4.3.12)

Substituting into (4.4.2) and integrating | k(0) (r)|2 over (k ; k ) and over the sphere of radius R, we Knd
R ∞ 2 (0) 0R (k) = (2l + 1) dr(kr)2 jl2 (kr) : (4.3.13)  0 l=0

Comparing this with (4.3.5), we obtain
R ∞  (0) 2 2 0R (k) = (2l + 1) |’(0) l (r)| r dr ; l=0

0

(4.3.14)

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

so that we may write 0R (k) −

0(0) R (k)

=

∞ 


(2l + 1)

l=0

|r|6R

d 3 r(| l |2 − |

(0) 2 l | )

;

where l (r) ≡ ’l (r)Ylm (; ). We now make use of (2.3.7) to transform (4.3.15) into   ∞  (2l + 1) dl (−)l (0) + cos l sin(2kR + 2l ) : 0R (k) − 0R (k) =  dk k

125

(4.3.15)

(4.3.16)

l=0

Averaging over R to get rid of the oscillatory term, as usual, and letting R → ∞ (again, the divergent term R in (2.3.7) cancels out in the diOerence), we recover (4.3.10). It follows from (4.3.10) that the change in the density of states for the lth partial wave, U0l = (1=) dl =dE, is related to the corresponding time delay Utl = 2˝ dl =dE, by Utl = hU0l ;

(4.3.17)

which is the dwell time delay in a phase space cell, since U0l is the probability of occupying such a cell. By (3.2.25), (4.3.10) is therefore the relationship between the total change in the density of states and the global dwell time delay (sum over all partial waves). We now return to Friedel’s argument concerning the change in the charge distribution arising from the impurity. Denoting by k and k (0) , respectively, the wave numbers with and without the potential, we Knd from (4.3.8) k − k (0) ≈ −(k)=R :

(4.3.18)

Thus, there is a one-to-one correspondence between such wave numbers: the potential just shifts the corresponding energy levels. For free particles, as R → ∞, the levels lie in a continuum. For V = 0, if the potential is attractive enough, it may produce a bound (localized) state. Then, since l (∞) = 0, it follows from Levinson’s theorem, discussed in Section 4.5, that the s-wave zero-energy phase shift will be 0 (0)=. Therefore, from (4.3.18), while k = 0 will be shifted from k (0) = =R, the state k (0) = 0 will not correspond to any real value of k: it will correspond to the bound state. If we now assume that there are N electrons in the sphere of radius R, and that K (0) is the corresponding maximum wave number for free electrons, we have, in each partial wave l, K − K (0) ≈ −l (K)=R :

(4.3.19) (0)

The number of states between K and K is obtained by multiplying (4.3.19) by (2l + 1)R=, their density in k-space, and by summing over l and over both spin orientations, ∞ 2 UN = − (2l + 1)l (K (0) ) ; (4.3.20)  l=0

where we have replaced K by K (0) , as l does not vary rapidly near K (0) . This is the number of electron states that are localized by the impurity, in order to exactly compensate for the charge it introduces. This follows from a general result on asymptotic independence of normal modes from boundary conditions (von Laue, 1914).

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For electrically neutral impurities, like atoms, the charged core will bind electrons. Thus, additional electrons will have to be introduced into the gas to preserve neutrality, with the end result that the electron gas will conserve the same maximum (Fermi) momentum ˝K (0) . The result (4.3.20) is known as the Friedel sum rule, and it plays an important role in condensed matter physics (Ziman, 1964). Lloyd (1967) generalized it to allow for the inclusion of a Knite cluster of non-overlapping mu>n-tin potentials. The basic formula (4.3.10) is known as the spectral property of the time delay for a spherically symmetric potential. In Section 4.5, it will be extended to a general operator formulation. 4.4. The time delay operator In the present section, we brieMy review the formal operator theory of time delay, outlining its logical development and quoting the main results, without proofs. We skip most of the mathematical subtleties. For a rigorous presentation, the reader is referred to Martin’s review (Martin, 1981). The principal advantage of the operator theory is its greater generality. We specify the domain of applicability of the main results. We assume familiarity with the basic concepts of formal scattering theory (Taylor, 1972; Newton, 1982). Throughout the present section, we set ˝ = 1. For two-body interactions, we retain the notation m for the reduced mass. 4.4.1. General deEnition The evolution operators respectively associated with the free Hamiltonian H0 and with the interaction Hamiltonian H = H0 + V are U0t ≡ exp(−iH0 t);

Ut ≡ exp(−iHt) :

(4.4.1)

The MHller operators ± are deKned by ± ≡ lim (Ut† U0t ) t →∓∞

(4.4.2)

where the limit is taken in the strong sense. To any given initial normalized state (wave packet) G in the Hilbert space H, we can associate the freely evolving state ’t = U0t G :

(4.4.3)

The corresponding fully interacting state Gt , which converges toward ’t as t → −∞, is given by Gt = Ut + G = + U0t G = + ’t ;

(4.4.4)

where we have used the intertwining relation H+ = + H0 : For background on these results, see the above references on scattering theory.

(4.4.5)

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

Let PR be the projection operator which acts on functions G(r) as  G(r) for |r| ¡ R ; PR G = 0 for |r| ¿ R :

127

(4.4.6)

Then, the dwell time within the sphere HR of radius R, in the presence of interaction, is
∞ TR [Gt ] = dt(Gt ; PR Gt ) ;

(4.4.7)

and similarly for the corresponding free-particle dwell time. The dwell time delay within HR due to scattering, associated with G, is therefore
∞ UTR G = TR [Gt ] − TR [’t ] = dt[(Gt ; PR Gt ) − (’t ; PR ’t )] :

(4.4.8)

Taking (4.4.4) into account, this may be rewritten as
∞ † UTR G = dt(G; U0t† [+ PR + − PR ]U0t G) :

(4.4.9)

−∞

−∞

−∞

This leads to a deKnition (Goldberger and Watson, 1964; Jauch et al., 1972) of the dwell time delay operator Q, through its expectation value in a state G, by
∞ † (G; QG) = lim UTR G = lim dt(G; U0t† [+ PR + − PR ]U0t G) : (4.4.10) R→∞

R→∞

−∞

One can represent Q by a momentum space kernel p|Q|p , through
∗ ˜  ) d 3 p d 3 p : (G; QG) = G˜ (p)p|Q|p G(p

(4.4.11)

Q is energy-conserving; its on-shell restriction q(E) veriKes p|Q|p =

(E − E  ) ˆ p|q(E)| pˆ ; mp

(4.4.12)

where E = p2 =(2m), E  = p2 =(2m), the carets denote unit vectors and the denominator arises from a Jacobian. The on-shell S-matrix s(E) is similarly deKned p|S|p =

(E − E  ) ˆ p|s(E)| pˆ : mp

(4.4.13)

The √ operators q and s act on wave functions labeled by the direction pˆ in momentum space, with p = 2mE. It is found (Jauch et al., 1972; Amrein et al., 1977) that the Hermitian operator q is given by q(E) = −is† (E)

d s(E) ; dE

which should be compared with (3.3.7).

(4.4.14)

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In the particular case of a spherically symmetric potential, the matrix elements of the reduced S-matrix and of the time delay operator become (Osborn and BollCe, 1977) 

ˆ p|s(E)| pˆ = (4)

−1

∞ 

(2l + 1)Sl (E)Pl (pˆ · pˆ ) ;

(4.4.15)

(2l + 1)2(dl =dE)Pl (pˆ · pˆ ) ;

(4.4.16)

l=0

ˆ p|q(E)| pˆ = (4)−1

∞  l=0

where Sl = exp(2il ) is the S-function. Taking into account the identity ∞

1 (2l + 1)Pl (cos ) = (cos  − 1) ; 2 l=0

we Knd that (4.4.15) may be rewritten as p ˆ f(E; pˆ · pˆ ) ; p|s(E)| pˆ = (pˆ − pˆ ) − 2i

(4.4.17)

where f denotes the total scattering amplitude (3.2.19) and cos  = pˆ · pˆ . The expectation value pˆ0 |q(E)|pˆ0 deKnes the global time delay associated with an incident plane wave with incident direction pˆ0 and energy E,      
 d  ˆ (4.4.18) pˆ0 ∗ pˆ  s(E) pˆ0 d pˆ ; pˆ0 |q(E)|pˆ0 = Re −i p|s(E)| dE   ˆ · ·) = dp (· · ·) is the integral over solid angles in momentum space. With the help where d p(· of (3.2.21), one can verify (BollCe and Osborn, 1976) that (4.4.18) is equivalent to (3.2.28), so that the average dwell time delay found in Section 3.2 corresponds to the global time delay deKned by (4.4.18). This also follows from (4.4.16) and (3.2.25). 4.4.2. Time delay and response to a perturbation Consider the eOect of adding to the potential V (r), in the Hamiltonian H , a localized perturbation KW (r), where W is bounded and has compact support, and the parameter K is also bounded. Thus, H (K) = H0 + V (r) + KW (r) :

(4.4.19)

The evolution operator for the interacting system becomes Ut (K) = exp[ − iH (K)t] ;

(4.4.20)

and the MHller operators (4.4.2) become ± (K) = lim ±; t (K); t →∞

±; t (K) ≡ U∓† t (K)U0∓t :

(4.4.21)

The scattering operator is deKned by (Newton, 1982) † (K)+ (K) = lim St (K) ; S(K) = − t →∞

(4.4.22)

C.A.A. de Carvalho, H.M. Nussenzveig / Physics Reports 364 (2002) 83 – 174

129

where † † St (K) ≡ − ; t (K)+; t (K) = U0t U2t (K)U0−t :

(4.4.23)

We now apply (Martin and Sassoli de Bianchi, 1992; Jaworski and Wardlaw, 1992) the usual time-dependent perturbation theory (Dyson expansion) to Krst order:  
t †  † St (K) = −; t (0) 1 − iK dt Ut
(0)WUt (0) +; t (0) + O(K2 ) ; (4.4.24) −t

where the argument 0 in the operators refers to K = 0 in (4.4.18), i.e., to the scattering system without the localized perturbation KW . It follows from (4.4.24) that  
t   9 †  †  St (K) = −i−; t (0) dt Ut
(0)WUt
(0) +; t (0) : (4.4.25) 9K −t

K=0

Now let t → ∞ and assume that one can interchange this limit with the derivative (the limit must be understood in the weak sense: cf. Martin and Sassoli de Bianchi, 1992). We get  
∞   9 † †  dtUt (0)WUt (0) + (0) : (4.4.26) S(K) = −i− (0) 9K −∞

K=0

† † = + , we get Multiplying on the left by iS † (0) and noting that, by (4.4.22), S † − 
∞ dS  † ≡ T [W ] = dtU0t† + W+ U0t ; iS †  dK K=0

−∞

(4.4.27)

where we have employed the intertwining relations. In particular, if we apply this to the projection operator PR deKned by (4.4.6), taking into account (4.4.9), we get (G; T [PR ]G) = TR [Gt ]

(4.4.28)

which is the dwell time within the sphere |r| ¡ R of the particle scattered by the potential V (r), with incoming state G. By localizing the perturbation KW through the projection operator PR , we can employ the perturbation as a ‘clock’ for determining dwell time, with the help of (4.4.27). Examples will be provided in Section 6. 4.4.3. Extensions The operator formulation of time delay has been extended to more general situations. For two-body potential scattering by a central potential V (r), su>cient conditions for the validity of the above results are that V (r) falls oO faster than r −3 at inKnity and is less singular than r −3=2 at the origin; it can have a Knite number of Knite discontinuities. The Coulomb potential does not satisfy these conditions, and the corresponding time delay is logarithmically divergent (Martin, 1981). However, one can consider perturbations of the Coulomb behavior and deKne relative time delay with respect to the long-range part of the interaction (BollCe et al., 1983).

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The time delay operator (4.4.14) can also be applied to non-spherically symmetric potentials and even to nonlocal interactions. However, in the deKnition (4.4.10), an expanding spherical region should always be employed in the limiting process (Sassoli de Bianchi and Martin, 1992). Already in the original formulation of the stationary dwell time concept (Smith, 1960), the generalization to multichannel scattering was considered, with (4.4.14) extended to the multichannel S-matrix. A phenomenological description in terms of time delay in an optical model was given by Martin (1981). Formal extensions to three-body (Osborn and BollCe, 1975) and multiparticle scattering (BollCe and Osborn, 1976, 1979) have been given. 4.5. Spectral property and Levinson’s theorem 4.5.1. Spectral property We now go over to the general operator formulation of the spectral property discussed in Section 4.3. For a spherically symmetric potential, this property is expressed by (4.3.10). On the other hand, (4.4.16) yields, in this case,
∞  dl ˆ ˆ dp = 2 tr q(E) = p|q(E)| p (2l + 1) ; (4.5.1) dE l=0

so that we must evaluate the trace of the on-shell time delay operator. Concerning the density of states, it is convenient to introduce (Krein, 1953; Birman and Krein, 1962) the spectral family e(@) associated with H , such that e(@)H represents all states in Hilbert space (both bound and scattering states) with energy ¡ @,
H = @ de(@) ; (4.5.2) where the integral is a Stieltjes one. A similar decomposition with e0 (@) holds for the free Hamiltonian H0 , with @ denoting kinetic energy. In terms of the spectral families, the resolvent operators R(z) = (H − z)−1 ; R0 (z) = (H0 − z)−1 have the representations
R(z) = (@ − z)−1 de(@); Im z = 0 ;
R0 (z) = (@ − z)−1 de0 (@); Im z = 0 : (4.5.3) For E ¿ 0 (scattering states), the spectral shift B(E) is deKned by B(E) ≡ tr[e(E) − e0 (E)] :

(4.5.4)

If we were to discretize the spectrum by box normalization, tr[e(E)] would be just the number of eigenstates with energy less than E, and B(E) would represent the change in this number due to the interaction. This change remains Knite for the continuous spectrum, although each term separately would be inKnite. It follows that dB(E)=dE is the change in the density of states at E due to the interaction. The spectral shift function was introduced by Krein (1953, 1983), motivated by the discussion given by Lifshitz (1952, 1964), at the same time as Friedel’s, of the eOect of impurities on the density of states in a crystal.

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ˆ we have From (4.5.3), for any state ’(E; p) 
(’; Im[R(E + i) − R0 (E + i)]’) = Im

1 @ − E − i



d (’; [e(@) − e0 (@)]’) d@ : d@

131

(4.5.5)

Letting  → +0 and employing (3.1.11), we get d (4.5.6) (’; Im[R(E + i0) − R0 (E + i0)]’) =  (’; [e(E) − e0 (E)]’) : dE Applying (4.5.6) to each of a complete orthonormal set of states ’, and summing over the set, we Knd dB(E) : (4.5.7) Im tr[R(E + i0) − R0 (E + i0)] =  dE It can be shown (Buslaev, 1972; Tsang and Osborn, 1975; Osborn and Tsang, 1976; Osborn et al., 1980) that Im tr[R(E + i0) − R0 (E + i0)] = 12 tr[q(E)] ;

(4.5.8)

where the trace is deKned as in (4.5.1). Combining (4.5.7) and (4.5.8), we Knally get the operator form of the spectral property, dB(E) 1 = tr[q(E)] : (4.5.9) dE 2 For a central potential, this reduces to (4.3.10). It follows from (4.4.14) and the unitarity of the S-matrix that q(E) is (−i) times the logarithmic derivative of S. The matrix relation tr ln M = ln det M

(4.5.10)

indicates the connection with the Birman–Krein (1962) representation for the spectral shift, i ln det S(E) ; (4.5.11) B(E) = 2 showing that the spectral property (4.5.9) relates the total time delay to the energy derivative of the phase of det S(E). Applications to condensed matter physics are discussed by Faulkner (1977). The role (4.5.4) of B in eigenstate counting has led to a generalization, for scattering problems, of Weyl’s celebrated formula for the asymptotic distribution of eigenvalues inside a cavity with impenetrable walls (Baltes and Hilf, 1976; HPormander, 1983). A generalized relationship has been derived by Melrose (1988). 4.5.2. Levinson’s theorem For a central potential and a single partial wave l, Levinson’s theorem (Levinson, 1949) states that (Newton, 1982)
∞ d [l (E)] dE = l (∞) − l (0) = (nl + 12 s) ; (4.5.12) dE 0 where nl is the number of bound states with angular momentum l, and s=1 if there is a ‘half-bound’ state at zero energy, s = 0 otherwise. A ‘half-bound’ state can occur only for l = 0.

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For the contribution to the resolvent operator from the sum over a Knite number of partial waves, one can show that tr[R(z) − R0 (z)], which is analytic in the z plane, cut along the positive real axis, goes to zero faster than 1=|z|, uniformly, as |z| → ∞. This function has simple poles at the bound state energies. Therefore, by applying Cauchy’s theorem, it would follow that (Tsang and Osborn, 1975)
∞ 2i dE Im tr[R(E + i0) − R0 (E + i0)] = −2i(N + 12 s) ; (4.5.13) 0

where N is the total number of bound states, counted with their multiplicities, for all partial waves (a Knite number). However, the fast decay at inKnity is no longer valid for the sum over all partial waves. As is well-known (Newton, 1982), the asymptotic high-energy behavior of the scattering amplitude is given by Born’s approximation. The trace involves the forward amplitude, which, at high energies, is given by the forward Born approximation, proportional to
V˜ = V (r) d 3 r : (4.5.14) Indeed, one Knds (Osborn and BollCe, 1977; Martin, 1981)  1=2 1 m3 v˜ V˜ √ ≡ −√ tr[q(E)] → − (E → ∞) :  2 E E

(4.5.15)

Therefore, one needs to subtract out this term in order to apply Cauchy’s theorem, with the result 
 1 ∞ v˜ tr[q(E)] + √ dE = −(N + 12 s) ; (4.5.16) 2 0 E where N is the total number of bound states, each counted with its multiplicity. This is the extended version of Levinson’s theorem, which, in view of (4.4.18), may also be regarded as a sum rule for the average (global) time delay. It follows that, if there are sharp resonances giving large positive contributions to the global time delay, there must also exist compensating time advances at other energies. Su>cient assumptions on the potential, for the validity of the extended Levinson theorem, are that it is both absolutely integrable and square integrable, i.e., that it belongs to L1 ∩ L2 . According to a well-known estimate by Bargmann (1952), this implies that the total number of bound states N is Knite. An intuitive graphical interpretation of Levinson’s theorem (4.5.12), for an attractive potential (Nussenzveig, 1972b), relates the phase shift increment of  for each additional bound state with the extra oscillation in the wave function that must be accommodated within the potential well. This shows how scattering data can provide spectral information on bound ‘internal’ states. An interesting generalization of this idea is the inside-outside duality for cavities with impenetrable walls (Doron and Smilansky, 1992; Dietz and Smilansky, 1993; Smilansky, 1992, 1994). The idea can be illustrated by two-dimensional scattering from a circular obstacle of radius a with Dirichlet boundary conditions, for which the partial-wave S matrix is Sl (k) = −Hl(2) (ka)=Hl(1) (ka) :

(4.5.17)

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The obstacle becomes ‘transparent’ for wave numbers kn such that Sl (kn a) = 1. These, however, are precisely the spectral eigenvalues inside the circular billiard, Jl (kn a) = 0. A precise formulation of this duality is contained in the following theorem, valid for a simply connected domain with a smooth boundary and Dirichlet boundary condition (Eckmann and Pillet, 1995): If, at kn ¿ 0, the S matrix has an eigenvalue 1 of multiplicity m, then −B has an eigenvalue kn2 of multiplicity at least m. The corresponding eigenfunctions can be extended to bounded solutions of the Helmholtz equation in the plane. Illustrations and additional discussion are given in Dietz et al. (1995). 5. Applications to statistical mechanics In a physical theory, quantities are ultimately expressed in terms of correlations. Time delay is no exception. It can be expressed in terms of correlation functions, through its connection with other physical quantities, such as the density of states (Section 4.4) or the second virial coe>cient (Beth and Uhlenbeck, 1936, 1937). Indeed, the former is related to the quantum-mechanical Green function, and the latter to Keld-theoretical temperature Green functions or, ultimately, to S-matrix elements (Dashen et al., 1969). Thanks to these connections, one may take advantage of various approximate methods, developed for computing Green functions, in order to calculate the time delay. In particular, semiclassical approaches (Balian and Bloch, 1971, 1974) may be employed. In this section, we establish the relation between time delay and the already mentioned physical quantities and, Knally, we express it in terms of correlation functions. 5.1. Connection with the second virial coeGcient The connection between time delay and the second virial coe>cient is based on the work of Beth and Uhlenbeck (1936, 1937) and on the relation of that coe>cient to two-body scattering (Gropper, 1937; Kahn, 1938). We derive this relation for the special case of a monatomic gas of spinless particles, following the procedure of Landau and Lifshitz (1967). The thermodynamic potential for the gas is given by the Gibbs distribution formula  

  (N )  = −PV = −kB T ln e2NN ; (5.1.1) e−2En N

n

where 2 ≡ 1=(kB T ); kB is Boltzmann’s constant, N is the number of atoms and En(N ) denotes the nth energy level of the N -atom system. The second virial coe>cient introduces corrections to ideal gas behavior arising from the inclusion of two-body interactions. Therefore, in (5.1.1), it su>ces to consider the sum up to N = 2,  ≈ −kB T ln(1 + O1 + O2 ) ; (5.1.2)  (N ) with ON ≡ exp(N2N) n exp(−2En ). The energy of the two-atom system may be written as the sum of center-of-mass (CM) and relative motion contributions, P2 E (2) = +@ ; (5.1.3) 2M

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where P is the CM momentum, M = 2m, with m being the mass of an atom, and @ is the energy of relative motion, that of a particle of reduced mass m∗ = m=2, moving in a central potential U (r); r being the interatomic separation. CM motion will be treated classically, while relative motion is quantized. This allows us to sum CM coordinates and momenta over phase space cells of size h3 , to obtain O2 = e22N ZCM Zrel ;



(5.1.4)

where ZCM =V (MkB T=2˝2 )3=2 ; V being the volume, and Zrel = exp(−2@) is the sum over quantum states of relative motion. (0) (0) Writing Zrel = Zrel + Zint in (5.1.4), where Zrel is the ideal gas result, inserting this into (5.1.2), and keeping terms up to exp(22N), we get  ≈ 0 − kB T e22N ZCM Zint ;

(5.1.5)

where 0 denotes the ideal gas result up to the same order. We are using exp(2N)1, which, as indicated below, can be translated into 1=2  N 2˝2 1 ; (5.1.6)  ; Q≡ V Q3 mkB T where Q is the thermal wavelength and Q−3 is the quantum limiting density for the gas. Indeed, the virial expansion is an expansion into powers of NQ3 =V . Expanding the result for the ideal Bose gas, we obtain (Landau and Lifshitz, 1967)  0 ≈ B −

1 V kB T e22N 3 ; 25=2 Q

(5.1.7)

where B = −kB T e2N V=Q3 is the Boltzmann result. Inserting (5.1.7) into (5.1.5), and using ZCM = 23=2 V=Q3 , leads to V 1 kB T e22N 3 (1 + 16Zint ) : (5.1.8) 5=2 2 Q In (5.1.8), the second term arises from the eOective exchange interaction induced by Bose statistics in the present case, whereas the third one arises from the direct interaction of two atoms, vanishing when U = 0. For spin-1=2 atoms, Fermi statistics would change the signs of both these terms. To obtain the equation of state, we Krst use (9=9N)T; V =−N to relate N with T; V , and N [in the Boltzmann case, this yields e2N = NQ3 =V ]; then, we compute F =  + NN and use (9F=9V )T; N = −P, to Knd the virial expansion to this order,  ≈ B −

PV 1 NQ3 (1 + 16Zint ) ; = 1 − 5=2 NkB T 2 V

(5.1.9)

which identiKes the second virial coeGcient B(T ) as Q3 16Q3 Zint − ; (5.1.10) 25=2 25=2 where the negative sign of the free-particle (ideal gas) exchange term entails pressure reduction, arising from the eOective attraction, due to Bose statistics. B(T ) = Bexch; 0 + Bint = −

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We now compute Zint . In general, this involves a sum over discrete levels @n ¡ 0 and continuum states of relative motion @ = p2 =(2m∗ ) = p2 =m. The total contribution to the sum over states from the continuum is obtained by multiplying exp(−@=kB T ) by the density of states and integrating over @. This diverges in the inKnite-volume limit, as seen in (4.3.9). However, for the interaction contribution, only the change in the density of states due to the interaction is relevant, and this is given by (4.3.10). Thus, we Knd
∞  1  dl −p2 =mkB T |@n |=kB T Zint = e e + (2l + 1) dE ; (5.1.11)  dE 0 n l even

where the restriction to even l arises from the exchange symmetry. In terms of the dwell time delay (4.3.17) for the lth partial wave, the continuum contribution to Bint in (5.1.10) may therefore be written as
 16Q3 1 ∞ Utl −@=kB T cont e Bint = − 5=2 d@ (2l + 1) : (5.1.12) 2  0 2˝ l even

Taking into account (3.2.25), and with √ |A(E)|2T = Ee−E=kB T

(5.1.13)

in (3.2.23), we see that  √ 2 kB T cont R T : Bint = −4 R m

(5.1.14)

The average speed of relative motion for a Boltzmann distribution is given by vWT = 8kB T=(m∗ ). Thus, we may write cont Bint = −R2 vWT R T :

(5.1.15)

These results show that the continuum contribution to the second virial coe>cient of a quantum system is proportional to the average dwell time delay in a two-particle collision, in which the incident wave packets have a spectral distribution given by the canonical ensemble. This is to be expected, since contributions to the partition function, from each phase space domain, are weighted by the time the system spends in that domain. For a gas of hard spheres of radius a, there are no bound states, and (5.1.14) allows us to compute the dominant contribution to the high-temperature behavior of B(T ). For this purpose, we rewrite the partial-wave summation in (5.1.11) as a sum over all l plus the diOerence between even l and odd l terms, which represents an (interacting) exchange eOect. This leads to a decomposition which diOers from (5.1.10): B(T ) = Bdirect + Bexch ; where Bdirect ≡

(5.1.16)

∞ 

(2l + 1)Bint; l ;    (2l + 1)Bint; l ; ≡ Bexch; 0 + −

(5.1.17)

l=0

Bexch

l even

l odd

and Bint; l is the contribution from a single partial wave l in (5.1.12).

(5.1.18)

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The direct term corresponds to the result for a classical hard sphere gas. Its high-temperature behavior arises from the short-wavelength domain ka1; k = p=˝, and it can be asymptotically evaluated (Nussenzveig, 1972a) by applying results from diOraction theory. One Knds a global direct-reMection time advance    R (@) m∗ a 1 =− 1+O ; (5.1.19) Ps (@; R) 3˝k ka where Ps (@; R) = 9T =R2 , with 9T = 2a2 the total high-energy cross section, is the total scattering probability. For the exchange contribution, it has been shown by path integral methods (Lieb, 1967) that, both for bosons and for fermions, it is exponentially small relative to the ideal gas case,  3   Bexch  a 2 a 2=3 : (5.1.20) = exp − +O Bexch; 0 2 Q Q Thus, averaging (5.1.19) over the thermal distribution, we get    Q 2a3 2 1+O : R R T = − 3vW a Taking into account (5.1.10) and (5.1.15), this yields    Q 2 3 Bhs (T ) = a 1 + O : 3 a

(5.1.21)

(5.1.22)

Up to corrections of order Q=a, which are small at high temperature, this is just the excluded volume per particle, which, to leading order, is one half the volume of a sphere. The general results for B(T ), for two-body systems of spin s, may be written as B(s) =

(s + 1)BBE + sBFD ; 2s + 1

even s ;

B(s) =

sBBE + (s + 1)BFD ; 2s + 1

odd s :

(5.1.23)

where BBE denotes the Bose–Einstein result (5.1.10), and BFD is the Fermi–Dirac one, in which the sign of the exchange term is changed to plus. In the Fermi–Dirac case, the sum over l in (5.1.11) runs over odd values. The connection with time delay has also been extended to higher virial coe>cients (Osborn and Tsang, 1976). 5.2. Time delay and correlations The relationships among time delay, density of states and virial coe>cients can be viewed as direct consequences of an S-matrix formulation of statistical mechanics (Dashen et al., 1969). This formulation related them to the correlation functions involved in scattering processes, allowing for extensions to relativistic systems. To attain this purpose, the virial expansion was combined with the Keld-theoretic method of Knite temperature Green functions (Abrikosov et al., 1975; Fetter and

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Walecka, 1971), and diagrammatics was used to relate those functions to S-matrix elements. Nowadays, the path integrals of Knite temperature Keld theory (Kapusta, 1989; Le Bellac, 1996) provide a natural derivation of this result, thus accomplishing the main task of statistical mechanics: to obtain thermodynamic quantities from microscopic information. Again, we start from the thermodynamic potential  = −kB T ln Tr exp [ − 2(H − NN )] : Taking the sum over the spectrum of N yields ∞  2NN −2H ; e Tr N e  = −kB T ln

(5.2.1)

(5.2.2)

N =0

where Tr N is taken over N -particle states. If we subtract the ideal gas contribution,

 ∞  e2NN (Tr N e−2H − Tr N e−2H0 ) ;  − 0 = −kB T ln 1 + e20

(5.2.3)

N =0

with H0 the free Hamiltonian operator, and expand the right-hand side into powers of exp(2N), we obtain the virial expansion,  − 0 = −kB T

∞ 

BN e2NN ;

(5.2.4)

N =1

where we deKne B1 to be the ideal gas term, B1 = −20 e−2N . In order to relate (5.2.4) to the form PV=(NkB T ) = [1 + B(T )(N=V ) + · · · ], one must express e2N in terms of N; V , and T , through the equation (9=9N)T; V = −N , as was done before. Classically, B1 = V=Q3 , whereas in quantum mechanics it involves all the exchange corrections. From (5.2.3) and (5.2.4), B2 = Tr 2 e−2H − Tr 2 e−2H0 : In the notation of Section 5.1, B2 = ZCM Zint , or, equivalently,
d@ e−2@ [0(@) − 00 (@)] ; B2 = ZCM

(5.2.5)

(5.2.6)

which recovers the connections with the density of states and time delay. The argument of the logarithm in (5.2.3) corresponds to O Tr exp [ − 2(H − NN )] : = O0 Tr exp [ − 2(H0 − NN )]

(5.2.7)

This ratio may be expanded in terms of the interaction potential through a Feynman–Dyson expansion or, equivalently, it may be represented as a path integral at Knite temperature, for which we may substitute the corresponding perturbative series. Because of the normalization relative to the free Hamiltonian, all of the resulting Feynman diagrams depend only on the interaction potential. Taking the logarithm gets rid of all graphs that

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can be written as products, leaving only connected diagrams (Dashen et al., 1969; Das, 1997; Itzykson and Zuber, 1985; DrouOe and Itzykson, 1989; Zinn-Justin, 1993). Thus, we may write  − 0 = −kB T [Tr e−2(H −NN ) ]c ;

(5.2.8)

where the subscript denotes that only connected diagrams with the interaction potential acting at least once are to be kept. The connected diagrams contain both statistical and dynamical information. In order to disentangle them and single out the latter, through its connection with the S-matrix, we introduce the resolvent operators (our notation here is slightly diOerent from that in Section 4.5) G(E) = (E − H )−1 ; G0 (E) = (E − H0 )−1 ; so that Tr e−2(H −NN ) =



dE −2(E −NN ) e Tr G(E) ; 2i

(5.2.9)

(5.2.10)

where the counterclockwise path, in the complex E plane, encloses the spectrum of H . Assuming that no bound states are present, for simplicity, one can reduce (5.2.10) to an integral over the real axis, as was done in Section 4.5. Furthermore, as G also admits a perturbative expansion, one can show that the equality still holds if only connected diagrams are kept on both sides. Thus,
dE −2(E −NN ) [Tr e−2(H −NN ) ]c = − Im[Tr G(E)]c ; (5.2.11) e  where the integral extends over the spectrum of H , and it is understood that the argument of G has a small imaginary part that approaches 0+ after we let the volume tend to inKnity. It remains to express [Tr G]c in terms of the S-matrix. Denoting the interaction Hamiltonian by V ≡ H − H0 , and introducing the operators T (E) = V + VG(E)V; S(E) = G0 (E ∗ )G −1 (E ∗ )G(E)G0−1 (E) ;

(5.2.12)

which are related by [cf. (4:4:17)] S = I − 2i(E − H0 )T ;

(5.2.13)

where I is the identity operator, one obtains, after some manipulations (Dashen et al., 1969),  ↔ 9 (5.2.14) Tr S −1 S = −4i Im Tr(G − G0 ) ; 9E which should be compared with (4.5.8). Concentrating again on connected graphs that depend on the interaction, one can show from the diagrammatics that this relation holds for the ‘on-shell’ S operator, which corresponds to the physical

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S-matrix elements, and that bound states as well as exchange terms are also accounted for in the end result (Dashen et al., 1969),

 ↔ 
dE 9 : (5.2.15) − 2( − 0 ) = [Tr e−2(H −NN ) ]c = − e−2(E −NN ) Tr S −1 S 4i 9E c

This relationship expresses the thermodynamic potential in terms of the S-matrix elements. These elements can be obtained through Green functions computed in various approximation schemes, such as perturbative or semiclassical expansions (Dashen et al., 1969; Balian and Bloch, 1971, 1974; DeWitt-Morette, 1972; de Carvalho et al., 1999); (5:2:10) gives a deKnite prescription for computing higher-order virial coe>cients. A useful consequence of these results is the possibility of writing a representation for the second virial coe>cient in terms of the poles of the S-matrix (Nussenzveig, 1973), which is particularly suitable to study its low-temperature behavior. Indeed, for a cutoO two-particle interaction, one can show, under very general assumptions (Nussenzveig, 1972b), that, for each partial wave, Sl (k) = exp[2il (k)] is a meromorphic function of k with the canonical product expansion   1 − (k=k ∗ )  −2ikd nl ; (5.2.16) Sl (k) = e 1 − (k=k nl ) n where d is the cutoO distance (smallest interparticle separation beyond which the interaction vanishes), and the product runs over all the poles in the complex k-plane, in order of increasing absolute value. The logarithmic derivative of (5.2.16) yields  Im knl dl = −d − : dk |k − knl |2 n

(5.2.17)

The poles knl can be classiKed into three categories (Nussenzveig, 1972b): (i) bound-state poles knl = iKb; nl , with Kb; nl ¿ 0 and Eb; nl = −˝2 Kb;2 nl =m; (ii) antibound (virtual)-state poles knl = −iKa; nl ,  − iK , with k  = 0 and K ¿ 0. The latter occur in with Ka; nl ¿ 0; (iii) complex poles kc; nl = knl nl nl nl pairs (kc; nl ; −kc;∗ nl ), whose contributions must be grouped together; if su>ciently far apart and close to the real axis, they may be interpreted as resonances. We may now separate the contributions of each type of pole to (5.2.17) and employ (5.1.16) to derive

      ∗    ikc; nl 1  d iKb; nl iKa; nl −Eb; nl =kB T Bl = − − : − e −√ w w w KT 2 KT KT 2Q b; nl b; nl a; nl c; nl (5.2.18)

√ where Q is the thermal wavelength, KT = 2=Q, and the function w can be written as (Abramowitz and Stegun, 1965)
2iz ∞ exp(−t 2 ) w(z) = − dt; Im z ¿ 0 : (5.2.19)  0 t 2 − z2

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Combining the Krst and third terms of (5.2.13), and employing a property of the w function (Abramowitz and Stegun, 1965), we get

       Kb;2 nl iKb; nl iKb; nl iKb; nl 1 1 1 −Eb; nl =kB T w − 2 exp − w = = : (5.2.20) − w e 2 KT 2 KT 2 KT KT2 With this, we arrive at the S-matrix pole representation of Bl ,  ∗ 1 d knl ; − w Bl = − √ KT 2Q 2 n

(5.2.21)

where the sum extends over all the poles knl of Sl (k). It includes bound, antibound and resonant states, all treated on an equal footing. 6. Application to tunneling 6.1. Tunneling time For a particle transmitted through a potential barrier, can one tell how long it took to cross it? An a>rmative answer leads to the concept of ‘tunneling time’. The question was posed soon after the formulation of quantum mechanics (MacColl, 1932), but it has been revisited in connection with the tunneling of carriers through thin layers, giving rise to a huge and highly controversial literature. Reviews (Hauge and StHvneng, 1994; Landauer and Martin, 1994; Chiao and Steinberg, 1997) and entire conference proceedings (Mugnai et al., 1997) have been devoted to this problem. Controversial aspects include the concept itself and seemingly paradoxical results deriving from it, particularly the possibility of superluminal propagation. We give here a brief critical review of the main proposals, and discuss the results obtained (Nussenzveig, 2000) by applying the above treatment of average dwell time to this problem. Three main approaches have been proposed in the analysis of tunneling time: (I) group velocity; (II) ‘clocks’; (III) path summations. In (I), one tries to follow the peak of a wave packet across the barrier. In (II), a small perturbation, supposed to play the role of a clock, is superimposed upon the barrier. In (III), an average is taken over some set of trajectories. We analyze each of these proposals successively. In essentially all discussions based on these approaches, one-dimensional models are considered, with a unilaterally incident particle. 6.2. Group velocity For incidence from the left, the asymptotic behavior of stationary scattering solutions is given by (3.3.1). The complex transmission amplitude is represented by (3.3.4), T (k) = |T (k)|exp[i’(k)], and the group velocity argument given in Section 2.1 would imply that the peak of a transmitted wave packet undergoes the transmission group delay d’ Utg = ˝ : (6.2.1) dE This is interpreted as the tunneling time in the group velocity approach.

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Consider as an example a rectangular potential barrier of height V0 and width a, centered at the origin. The stationary solution within the barrier is of the form c exp(−6x) + d exp(6x), where 1=6 ¿ 0 is the penetration depth. For a very opaque barrier (6a1), (6.2.1) leads to Utg ≈

2 2m = ; ˝k6 v6

(6.2.2)

which is the free time of Might of the incident particle through two penetration depths. Since the result is independent of the barrier width, a su>ciently thick barrier would lead to superluminal propagation. This is known as the Hartman eHect (Hartman, 1962). However, there are several reasons why the transmission group delay is not a meaningful indicator of propagation across the barrier. To begin with, the incident wave packet must satisfy a number of constraints: it should be initially conKned well to the left of the barrier, recognizably peaked, spectrally centered around the energy at which (6.2.1) is evaluated and such that transmission predominantly arises from tunneling, rather than from energies above the barrier. The main problems with group delay arise from wave packet reshaping eOects. Both free space as well as the region inside the barrier behave like dispersive media. Spectrally selective attenuation, arising from strong reMection by a nearly opaque barrier, distorts the transmitted shape. In view of these eOects, there need be no causal relationship between an incident peak and a transmitted one. Indeed, faster spectral components of the incident packet are transmitted more eOectively than slower ones, so that the front end is favored in transmission. It is therefore possible to build up a wave packet for which the transmitted peak precedes the arrival of the incident one at the barrier (Landauer and Martin, 1992). Thus, group velocity may become superluminal and even negative, in apparent violation of causality. Pulse reshaping leading to apparent superluminality is found in absorbing or amplifying media. In inverted (non-linear) media, one can even have superluminal solitons: the weak leading edge of a pulse unleashes the energy already stored in the medium, building up the advanced peak, like a fuse setting up an explosion. It has been shown (da Costa, 1970) that these self-transparent solitons are rigorously causal: the domain of dependence and the domain of inMuence of the solution, at a given spacetime point, are respectively contained within the backward and the forward light cones. The apparent superluminality is an optical illusion, just like the well-known rotating searchlight eOect. It has recently been shown that quantum noise, in a gain medium, also contributes toward reducing an operationally deKned signal velocity to values less than c (Kuzmich et al., 2001). General arguments based on unitarity and causality show that, for superluminal group velocity, the peak of the transmitted pulse is reconstructed from a small leading edge of the incident one (Segev et al., 2000). Several experiments on superluminal group velocities have been performed (Chiao and Steinberg, 1997; Dogariu et al., 2001). 6.3. ‘Clocks’ and related approaches 6.3.1. The Larmor times ˆ we assume that a For incident electrons with spin polarized along their direction of propagation x, ˆ is introduced within the small perturbing uniform magnetic Keld B, of magnitude B0 and direction z, barrier. Inside this region, the spin component in the (xy) plane, precessing at the Larmor frequency

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!L , plays the role of a clock. The total precession angle y deKnes the Baz–Rybachenko (1967) measure of ‘tunneling time’, the in-plane Larmor time Ty , given by y = !L Ty :

(6.3.1)

It was pointed out, however (BPuttiker, 1983), that an incident x-polarized spin is a superposition of up and down polarizations in z, which, by the Zeeman splitting, see diOerent eOective barriers, favoring transmission parallel to the Keld. This tends to align the spin with B, leading to an equivalent out-of-plane z rotation (much larger than the y one, for an opaque barrier) z = !L Tz ;

(6.3.2)

which deKnes the out-of-plane Larmor time Tz . The full 3-dimensional rotation is associated with the total traversal time  Tx = (Ty )2 + (Tz )2 ≡ TT :

(6.3.3)

This is interpreted as the barrier interaction (tunneling) time for particles that get transmitted. A similar analysis for reMected particles (BPuttiker, 1983) leads to corresponding reAection times. For a rectangular barrier of height V0 and width a, the following results are found: Ty = −˝

9’ ; 9V0

Tz = −˝

9 ln  ; 9V0

(6.3.4)

where T =  exp(i’) is the complex transmission amplitude deKned in (3.3.4). For an opaque barrier (1), Tz is dominant and is approximately given by Tz ≈ a=|v|;

|v| ≡ ˝6=m :

(6.3.5)

Here |v| is the magnitude of the (imaginary) intrabarrier ‘velocity’ and a=|v| is called the ‘bounce time’. The results (6.3.4) may also be regarded as illustrating the connection between time delay and linear response to a perturbation, discussed in Section 4.4. The perturbation here would correspond to altering the height of the barrier, and the derivatives with respect to V0 correspond to the diOerentiation with respect to K in (4.4.27) (Jaworski and Wardlaw, 1992). 6.3.2. Modulated barrier or incident wave Another possible perturbation of the barrier potential consists in introducing a small oscillatory modulation of its height (BPuttiker and Landauer, 1982). For slow modulation frequencies, the transmission follows the modulation adiabatically. The order of magnitude of the modulation period for which a departure from adiabaticity begins to appear is taken to provide an estimate of the ‘barrier interaction (tunneling) time’. An alternative proposal (BPuttiker and Landauer, 1985) is a modulation of the incident wave, by superposing energies E and E + UE. Here, one looks for a departure from adiabatic following at a modulation period ˝=UE. The resulting barrier interaction time is given by |TE |;

where TE ≡ −i˝

9 [ln T (E)] ; 9E

(6.3.6)

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where T (E) is the complex transmission amplitude. Note that ReTE = Utg is the transmission group delay (6.2.1), and that, for a rectangular barrier, |UTE | diOers from the total Larmor traversal time (6.3.3) only be the substitution of 9=9V0 ↔ −9=9E :

(6.3.7)

In particular, for an opaque rectangular barrier, (6.3.6) is also dominated by the bounce time. Landauer and Martin (1994) also proposed a variant of the modulated incident wave approach in which the two superimposed incident waves have opposite spins, leading to an oscillatory spin modulation as the ‘clock’ signal, with the same result (6.3.6). 6.4. Path summations In proposals inspired by Feynman’s path integral formulation of quantum mechanics (Sokolovski and Baskin, 1987; Fertig, 1990), one deKnes, for a classical path x(t), the time spent under the barrier by
t tcl [x(t)] ≡ dt  B [x(t  )] ; (6.4.1) 0

where B = 1 under the barrier, B = 0 otherwise. Then, the ‘Feynman’ tunneling time TF is deEned as the functional average of tcl [x(t)] over paths that start far to the left of the barrier and end far to the right, with the weight function exp{iS[x(t)]=˝}, where S is the action associated with the path x(t). The result is a complex ‘tunneling time’ TF = Ty − iTz ;

(6.4.2)

where Ty and Tz are the in-plane and out-of-plane Larmor times. As has been pointed out by several authors (cf. Landauer and Martin, 1994), this approach falls outside Feynman’s formulation. Indeed, the weighting by exp(iS=˝) is as little justiKed as trying to deKne an ‘eOective position’ by 

xeO ≡ x (x) d x (x) d x : (6.4.3) One is replacing probability density by probability amplitude. It is therefore not surprising that the result is not real. Further discussion (Yamada, 1999) leads to the conclusion that a probability distribution of tunneling times cannot be deKned (is ‘unspeakable’). Another non-conventional approach is based on Bohm trajectories (Leavens and Aers, 1993). Since such trajectories do not cross, one is led to a very peculiar picture of the transmission process, arising entirely from the leading edge, regardless of packet width. Problems also arise in an approach based on Wigner distribution paths (cf. Landauer and Martin, 1994). 6.5. Conditional dwell time The restriction only to particles that get transmitted amounts to choosing a subensemble, preselected (by preparing particle incidence only from the left)—and postselected (by including only particles emerging on the right). The ‘tunneling time’ would be a property of the system during the time interval between the initial and Knal situations.

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Aharonov et al. (1988) and Aharonov and Vaidman (1990) treated such conditional (weak) measurements. The following (Steinberg, 1995) is a simpliKed (non-rigorous!) version of their argument: assume that, given an initial state |i and a Knal state |f , the conditional probability P(Ai |f) for the particle to be found in an eigenstate Ai , given that it is subsequently found in |f , is P(Ai |f) = Proj(f)Proj(Ai ) Proj(f) ;

where Proj(f) ≡ |f f| :

(6.5.1)

Note that the product of projectors need not be Hermitian! The weak value of A is then deKned by A fi ≡

f|A|i : f|i

(6.5.2)

This is generally complex, because of the non-hermiticity. Weak values of observables have many paradoxical properties: the spin component of a spin 1=2 particle in a given direction may be arbitrarily large, superluminal velocities are found, etc. Smith’s stationary barrier dwell time Td , the expectation value of the projector onto the region within the barrier, can readily be evaluated for a rectangular barrier (BPuttiker, 1983). It is found to be equal to the in-plane Larmor time Ty . It becomes very small for an opaque barrier, when most particles are reMected. However, Td tends to be dismissed as a measure of tunneling time, since it does not distinguish between transmitted and reMected particles. Steinberg [18] used weak measurements to evaluate the conditional dwell time Td; c , deKned as the weak value of the projector onto the region within the barrier, for particles that Knally get transmitted. The result is complex, and identical to (6.4.2): Td; c = Ty − iTz = TF :

(6.5.3)

By considering the weak value in terms of a von Neumann-style measurement, Steinberg was led to the conclusion that Ty is indeed a property of the tunneling process, regardless of how it is observed, while Tz has the character of a susceptibility, describing the sensitivity of the tunneling probability to small perturbations. Because of ensemble postselection, weak measurements tend to amplify the eOects of rare Muctuations, yielding very imprecise results for each measurement and leading to the above-mentioned paradoxical properties (note that transmission is a rare event for an opaque barrier). In particular, they predict negative values for the average time spent by reMected particles on the right side of the barrier (Iannaconne, 1997). 6.6. Average dwell time in tunneling In Section 3.3, the average dwell time within a rectangular potential barrier was evaluated for energies above the barrier top. In

order to extend the results to tunneling (E ¡ V0 ), it su>ces to replace the refractive index N = 1 − (V0 =E) by i|N |. In the notations of Section 3.3, the following

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145

results are found (Nussenzveig, 2000): Td (a) → = Ty ; 

 Td (a) s; a = Ty ± Tz 0

(6.6.1)  ;

(6.6.2)

where Ty and Tz are BPuttiker’s in-plane and out-of-plane Larmor times of Section 6.3, respectively. For an opaque barrier (1); Td (a) → is very small, consistent with the fact that most particles get reMected. BPuttiker’s out-of-plane Larmor reAection time is (BPuttiker, 1983) TzR = −(=0)2 Tz ;

(6.6.3)

which diOers from the last term of (6.6.2) by an additional factor =0. For above-barrier transmission resonances, (6.6.3) is divergent, while (6.6.2) remains Knite. The result (6.6.2) leads to a reinterpretation of the Larmor times: 1 Ty = (Td s + Td a ) ; 2    1 Tz = (Td s − Td a ) : 0 2

(6.6.4)

In agreement with the results reported in Section 6.5, we Knd that Ty is indeed an average barrier dwell time, while the diOerential character of Tz shows that it behaves instead like a susceptibility, associated with the diOerence between symmetric and antisymmetric excitation, which aOects the interference between transmission and reMection. The results (6.6.4) apply to a rectangular potential. In order to extend them to a general symmetric potential, getting back to the treatment in Section 3.3, let us introduce the notations X T˜ E ≡ T (k)eikX = (E) exp[iU’(E)] ;

(6.6.5)

X R˜ E ≡ ieikX R(E) = 0(E) exp[iU’(E)] ;

(6.6.6)

where R (E) = R+ (E) ≡ R(E) for a symmetric potential, and the free change of phase U’ [cf. (3.3.25)] over the interaction region (− 12 X; 12 X ) has been absorbed into the exponent. The corresponding BPuttiker–Landauer barrier interaction time, in the modulated incident wave approach (6.3.6), is |TEX |, where 9 [ln T˜ X (E)] : (6.6.7) 9E The general results for average dwell time in a symmetric potential, (3.3.23) and (3.3.24), can be rewritten as   1
˜X sym X sym (Td s + Td a ) = Re TE − RE = Td sym → = Td ← ; 2 v    
˜X 1 sym X (Td s − Td a ) = −Im TE − RE ; (6.6.8) 2 0 v TEX = −i˝

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which generalizes the relations (6.6.4). The reMection amplitude terms in (6.6.8) are typical oscillatory contributions of the order of the free time of Might over a de Broglie wavelength
. We can now sum up the above discussion of diOerent approaches to the ‘tunneling time’ problem. Wave packet reshaping by dispersion, and strong selective transmission, remove dynamical signiKcance from the group velocity. Attempts to enforce the restriction to transmitted particles alone give rise to typical ‘weak measurement’ di>culties. Also, as already follows from the Fabry–Perot picture (inKnite number of internal reMections), one cannot separate ‘to-be-transmitted’ and ‘to-be-reMected’ portions of paths. Interference (in time!) between paths that spend diOerent amounts of time within the barrier does not allow one to deKne a ‘probability distribution of tunneling times’ (Yamada, 1999). In conclusion, we concur with the judgment of previous authors (Hauge et al., 1997) that the ‘tunneling time’ problem is ill-posed. In contrast, the average wave packet dwell time is always well deKned, remaining within the conventional treatment of quantum theory. In terms of (6.6.4), it leads to an entirely ‘standard’ physical interpretation of the Larmor times for the rectangular potential, showing that in-plane and out-of-plane times relate to quite diOerent features of the interaction with the barrier. Similar remarks apply to (6.6.8), with respect to the BPuttiker–Landauer times for a general symmetric potential. The average dwell time applies to arbitrary wave packets, thus accounting for shape eOects. It leads to reasonable results both for tunneling and for above-barrier energies. It does not distinguish between reMected and transmitted particles. Though usually taken as a defect, this is actually a virtue, since transmission and reMection are inextricably intertwined. The average dwell time is, in principle, experimentally accessible, although the detection of subtle interference eOects would require coherent bilateral excitation. Indeed, as suggested by Pippard (BPuttiker, 1990), the average dwell time can be measured by making the barrier weakly absorbing: absorption also does not distinguish (Iannaconne, 1997) between to-be-reMected and to-be-transmitted portions of the wave packet within the barrier.

7. Applications to Mie scattering 7.1. Quantized Mie modes We consider scattering by a transparent sphere of radius a and refractive index N . The quantized (transverse) electric Keld is given by ! 
∞ ˝!k ( j) ( j) E⊥ (r; t) = dk [alm (k)vl( j) (k; r)Xlm (; ’) + h:c:] ; (7.1.1) 2@0 0 lmj

( j) are vector spherical where the polarization index j (=1; 2) is deKned as in (3.4.14), !k = ck; Xlm ( j) harmonics, and h.c. denotes the hermitian conjugate. The operators alm obey the commutation rules
†

( j) [alm (k); a(l
jm)
(k  )] = jj
ll
mm
(k − k  ) :

(7.1.2)

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147

The mode functions vl( j) (k; r) are identical, for j = 1, to the functions 1

( j) jl (Nkr) ul( j) (k; r) = (2)− 2 kLkl

(0 6 r 6 a) ;

1

( j) (1) = (2)− 2 k[h(2) l (kr) + Sl (k)hl (kr)]

(r ¿ a) ;

(7.1.3)

and, for j = 2, they involve both these functions and their derivatives with respect to kr. In (7.1.3), the S-matrix elements are given by

 (2)  (2)  7 (2) 7 (2) − Ne ln (S) ln j l l ; (7.1.4) Sl( j) (k) = − l(1) 7l (2) ln 7l(1) (2) − Nej ln l (S) where l (z) ≡ zjl (z); 7l( j) (z) ≡ zhl( j) (z) are the Ricatti–Bessel functions, ln denotes the logarithmic derivative, and 2 ≡ ka;

S ≡ Nka;

e1 = 1;

e2 = 1=N 2 :

(7.1.5)

The amplitudes of the radial functions inside the sphere are √ 2i ej ( j) Lkl = (1) : 7l (2) l (S)[ln 7l(1) (2) − Nej ln l (S)]

(7.1.6)

7.1.1. Local density of states The local density of states at r; 0(!; r), is deKned in terms of the Wiener–Khintchine theorem (Mandel and Wolf, 1995), by (+) (− ) (r; t − t  ) ≡ @(r)0|E⊥ (r; t) · E⊥ (r; t  )|0
= 0(!; r) 12 ˝! exp[ − i!(t − t  )] d! ;

(7.1.7)

where the superscripts in the Kelds denote their creation and annihilation components, and @(r) = N 2 (r ¡ a); @(r) = 1 (r ¿ a), is the dielectric constant. In free space (no sphere), this leads to the familiar vacuum local spectral density of states per unit volume, 00 (!; r) =

!2 : 2 c3

(7.1.8)

Substituting (7.1.1) into (7.1.7), one Knds, for the Mie local spectral density of states per unit volume, 0(!; r) =

2  ∞  j=1 l=1

0l( j) (!; r) ;

where 0l( j) (!; r)

(2l + 1) @(r) = 8c x2



2|xul( j) |2

(7.1.9)

 1 d2 ( j) 2 + [|xul | ] ; 2 d x2

x ≡ kr :

(7.1.10)

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The corresponding global spectral densities, integrated over all space, diverge like the volume, but the diOerence 0 − 00 is Knite, and it is related to the time delay by the spectral property (4.3.10), with the substitution (3.4.47), appropriate for electromagnetic scattering. The explicit Mie solution allows us to analyze in detail the contributions from diOerent spatial regions, as well as their physical interpretation. For this purpose, we integrate the spectral density over the volume V of the scatterer, |r| 6 a, and normalize the result, by taking its ratio G c (2; N ) to the corresponding integrated vacuum spectral density. We call this ratio the cavity gain factor (relative to vacuum): 

G c (2; N ) ≡ 0(!; r) d 3 r 00 (!; r) d 3 r : (7.1.11) |r|6a

|r|6a

The result is (cf. also Ching et al., 1987)

  ∞ 2 l(l + 1) 3N 2   ( j) 2 1− (2l + 1)|Lkl | G (2; N ) = 1622 j=1 S2 c

2 l (S)



+ [ l (S)]

2

 ;

(7.1.12)

l=1

where the notations (7.1.5) – (7.1.6) are employed. 7.2. Resonance and background contributions For (N − 1)21, Mie scatterers have a large number of very sharp resonances. Their physical interpretation becomes readily apparent by employing the analogy between wave optics and quantum mechanics, which associates the scatterer with a rectangular central potential well, of depth proportional to (N 2 − 1)22 (Nussenzveig, 1992). The corresponding eOective radial potential for the lth partial wave, the sum of this well with the centrifugal barrier, is represented in Fig. 7.1(a). It is a potential pocket surrounded by a barrier,

Fig. 7.1. (a) EOective radial potential for partial wave l, the sum of the associated square well, of depth (N 2 − 1)k 2 , with the centrifugal barrier (K=r)2 , where K ≡ l + 12 . For an ‘energy’ k 2 , and impact parameter b ¿ a, the radial turning points are at r = b=N , a, and b. Resonances are located between the top T and the bottom B of the potential pocket, corresponding to impact parameters a and Na, respectively. Radial wave functions for two resonances, associated with mode orders (number of radial nodes) n = 0 and n = 1, are sketched. (b) Corresponding ray optics picture, for a resonant impact parameter b ¿ a, showing whispering gallery modes.

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a typical situation known to lead to long-lived resonances. For a given wave number k, the outermost radial turning point is at the impact parameter   1 k ; (7.2.1) bl = l + 2 and there is an internal turning point at bl =N , with the third one, r = a, in between. Tunneling (frustrated total internal reMection) takes place across the barrier, between a and bl . Resonances are associated with complex poles of Sl( j) (k). Since the denominators of (7.1.4) and (7.1.6) are the same, the internal intensity builds up to large values at their real parts. The wave functions associated with resonant levels are sketched in Fig. 7.1(a): they correspond to (E) or (M) modes, with a mode order n analogous to a principal quantum number. In the limit of vanishing barrier transmissivity, they would be bound states of light. The corresponding ray picture is illustrated in Fig. 7.1(b). Rays with resonant impact parameter bl , with a ¡ bl ¡ Na in the tunneling range, between the top T and the bottom B of the barrier, tunnel from outside, and are captured internally, in orbits that undergo multiple reMections, between r = a and a caustic at the internal turning point bl =N . They hit the internal sphere surface beyond the critical angle, so that they would be totally reMected in geometrical optics, where the resonances would be associated with whispering gallery modes. The leakage through the barrier represents spoiling of total reMection by surface curvature. Since the lifetime of a resonance is determined by tunneling, it is exponentially dependent on 2 and bl . Very small impact parameter changes, in the range a ¡ bl ¡ Na, drastically change the time delay, both because of the smallness of resonance widths (and their variable range) and on account of the contrast with the oO-resonant situation. This instability is related to one of the signatures of chaos, ‘sensitive dependence to initial conditions’. However, one does not yet have chaotic scattering (cf. Sections 7.3 and 8). Because of the exponential growth of resonance lifetimes with size, high-transparency materials in the visible, such as fused silica and water, yield optical cavities with extremely high Q factors. Values of order 1010 have been observed in silica microspheres (Gorodetsky et al., 1996). They correspond to large time delays, representing huge numbers of internal reMections before emergence. In ordinary Mie scattering, the resonances are responsible for the rapid ‘ripple’ Muctuations that occur in all cross sections (Nussenzveig, 1992). In the density of states, sharp resonances give rise to Lorentzian peaks, that would approach delta functions in the limit of vanishing width. The area under each peak approaches the degeneracy factor (2l + 1), as it would for the contribution of a true bound state. A plot of the internal local density of states, given by (7.1.6) with r ¡ a, at a sharp resonance, shows (Ching et al., 1987) a large peak near the rim, in the layer between the internal caustic bl =N and a. The strong internal Keld within this layer can give rise to nonlinear optical eOects, such as lasing in liquid droplets (Barber and Chang, 1988; Chang and Campillo, 1996). The contribution from each l value in (7.1.12) may be ascribed to a corresponding incident ray, with impact parameter given by (7.2.1) (localization principle). The contributions from rays passing outside the sphere, with bl ¿ Na [below the bottom B of the pocket in Fig. 7.1(a)], are damped faster than exponentially, and may be neglected. Resonant impact parameters in the domain a ¡ bl ¡ Na c (between B and T in the Kgure) are responsible for the resonance contribution Gres to the cavity gain, which we identify with the sum over the domain 2 ¡ l ¡ S in (7.1.12).

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c The contribution Gback from the terms with l ¡ 2 in (7.1.12), corresponding to all those rays that hit the sphere (bl ¡ a), will be called the background contribution. In terms of dwell time delay, it represents a prompt response, of the order of the time of Might across the diameter, in contrast with resonance lifetimes associated with whispering gallery paths, that can be many orders of magnitude larger. In Fig. 7.2, the resonance contributions to the gain are plotted, in the interval 24 ¡ 2 ¡ 25. Resonances are labeled by their polarization (E or M), subscript l and superscript n (mode order). We see that the resonant gain reaches values of order 100 even in this relatively low range of size. In contrast, the background contribution is small (≈ 2) and slowly-varying. By applying complex angular momentum techniques (Nussenzveig, 1992), one can compute size parameter averages of these quantities, c c G c = Gback + Gres ;

(7.2.2)

taken over a size parameter range U2 = , so as to smooth over rapid oscillations. One can also employ these techniques to derive (Nussenzveig, unpublished) the asymptotic high-frequency behavior of these averages. For an empty cavity with impenetrable walls, the asymptotic behavior of the mean cavity density of (discrete) eigenvalues is given by the famous ‘Weyl rule’ (Baltes and Hilf, 1976). It is dominated by a term proportional to the volume, with a universal (shape-independent) coe>cient, identical to the vacuum density (7.1.8). The next-order term is proportional to the surface area of the cavity. If the cavity is Klled with a homogeneous medium with refractive index N , the velocity of light in (7.1.8) is replaced by c=N , so that the coe>cient of the volume term is multiplied by N 3 . In the Mie scattering problem, we have an open ‘cavity’, where the energy is transmitted to the outside region. Nevertheless, we Knd that the ‘Weyl rule’ remains valid: G c ≈ N 3 + O(2−1 )

(21) ;

(7.2.3)

where the fact that the next-order term remains proportional to the surface area results from a nontrivial cancellation of contributions from the ‘edge domain’ |l − 2| = O(21=3 ). The asymptotic mean background gain, originating from rays that hit the scatterer (l ¡ 2), is found to be c ≈ N 3 − (N 2 − 1)3=2 Gback

(21)

(7.2.4)

c The behavior of Gback for N = 1:45 and 2 ¡ 2 ¡ 50 is plotted in Fig. 7.3, where the approach to the asymptotic limit (7.2.4) can be veriKed. It follows from (7.2.2) to (7.2.4) that the asymptotic mean resonance contribution to the cavity gain is given by c Gres ≈ (N 2 − 1)3=2 :

(7.2.5)

For N = 1:33, the asymptotic resonance contribution to the mean cavity density of states is therefore (1 − N −2 )3=2 ≈ 29%. Note that this large contribution arises entirely from incident rays external to the sphere, that would not be included in geometrical optics! In the limit of very large N , where the resonances approach discrete eigenvalues, (7.2.5) tends to the Weyl result (7.2.3), so that resonances dominate the density of states. The smoothed asymptotic

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151

Fig. 7.2. Background (- - - - -) and resonance (——) contributions to the cavity gain factor for N = 1:45. Fig. 7.3. Mean background gain (——) and its asymptotic limit (- - - - - -) for N = 1:45.

density of resonant states (degeneracy included) is given by   4 dNres ≈ (N 2 − 1)3=2 22 d2 3

(7.2.6)

7.3. Further applications 7.3.1. Deformed droplets and classical chaos Dye-doped lasing liquid droplets in free fall acquire spheroidal (prolate and oblate) deformations, and observations (Qian et al., 1986) show that lasing is then conKned to speciKc angular domains, around the droplet periphery. The eOect of such deformations was investigated (Mekis et al., 1995) at the level of ray optics, the analog of classical mechanics. At this level, internal reMection beyond the critical angle is total, and whispering gallery modes become trapped orbits (bound states). It was found that orbits with high enough axial angular momentum (near the equatorial plane) still hit the internal surface beyond the critical angle, thus remaining conKned. However, for large enough deformation, and lower values of axial angular momentum, the angle of internal incidence falls below the critical angle, allowing the rays to escape outside the droplet. This transition turns out to correspond to the Kolmogorov–Arnold–Moser=Lazutkin transition to chaos in Hamiltonian dynamics (Arnold and Avez, 1968; Moser, 1973; Lazutkin, 1973). The chaotic behavior is conKrmed by plotting PoincarCe surfaces of section, for typical trajectories. The onset of chaos spoils the Q factor and contributes to an explanation of the preferential lasing regions observed around and droplet rim. 7.3.2. Spherical Goos–HJanchen eHect As was mentioned at the end of Section 2.1, Froissart et al. (1963), analyzing the scattering of laterally bounded wave packets, derived a spatial analog of time delay: a transverse displacement, given by the derivative of the phase of the scattering amplitude with respect to momentum transfer.

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In the total reMection of a light beam at a plane interface, there is a lateral displacement of the reMected beam, already pointed out by Isaac Newton in his “Opticks” (Newton, 1982) and known as the Goos–HPanchen shift (Goos and HPanchen, 1947). It arises from light tunneling into the optically less dense medium, and traveling along the interface. The stationary-phase expression for the displacement (Artmann, 1948) is equivalent to the Froissart, Goldberger and Watson result. In Mie scattering by a spherical cavity (e.g., an air bubble in water), a similar tunneling eOect takes place, near the critical angle. A stationary phase argument would predict an angular displacement U, for critically incident rays, given by (Fiedler-Ferrari and Nussenzveig, 1987) U = 2(d=dK)K0 ;

(7.3.1)

where (k; K) is the phase shift as a function of (continuous) angular momentum K, and K0 corresponds to critical incidence. A more detailed investigation (Fiedler-Ferrari et al., 1991), however, shows that the stationary-phase approximation leads to incorrect results also in the present case. The tunneling angular displacement is not given by (7.3.1); one must take into account new diOraction eOects, that take place near the critical angle. Experiments (Tran et al., 1995) are in good agreement with the improved treatment. 7.3.3. EHects on light propagation We have already seen, through (3.4.37), that cumulative forward time delay, in light propagation through a medium containing many scatterers, is related to the reduction in phase velocity, characterized by the change in real refractive index, n(!) − 1. Since there are no bound states of light (strongly localized photons) in ordinary media, one also expects a sum rule, analogous to Levinson’s theorem, for the global time delay,
∞ [n(!) − 1] d! = 0 : (7.3.2) 0

Indeed, this follows from causality and from the high-frequency behavior of n(!)−1 (Altarelli et al., 1972). It is a macroscopic expression of the compensation between time delays and time advances, when averaged over the whole spectrum. Dwell time also plays an important role in determining the speed of light in random media, taking into account the eOects of resonant multiple scattering (for a review, see Lagendijk and van Tiggelen, 1996). In a strongly scattering random medium, the speed vE of energy transport diOers from both the phase velocity vp and from the group velocity vg . In a medium containing a large number of identical, randomly distributed scatterers, it is given by (van Albada et al., 1991; van Tiggelen et al., 1992). c ; (7.3.3) vE ≈ 1 + (Td =mf ) where Td is the dwell time within each scatterer, and mf = l=vp is the mean free time between successive scatterings (l is the mean free path). It follows from (7.3.3) that sharp resonances, in a close-packed random medium, can drastically reduce the speed of light propagation in the medium. This reduction reMects the large dwell time delays within each scatterer, associated with long-lived resonances. Another possible eOect of long scattering time delays, in an active random medium, is to produce feedback, contributing to enhance the gain and thus replacing the action of the mirrors in usual

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lasers. This can lead to mirrorless ‘random lasers’ (Letokhov, 1968; Lawandy et al., 1994; Jiang and Soukoulis, 2000; van Soest et al., 2001; Cao et al., 2001). 8. Time delay and chaotic scattering We now establish the connection between time delay and chaotic scattering. No prior knowledge of the physics of chaos is assumed. We deKne the basic concepts required, borrowing from various books (Ott, 1993; Gutzwiller, 1990; Giannoni et al., 1992). The aim is to convey just the physical basis for the connection, simplifying the discussion as much as possible, and referring the reader to the literature for more thorough mathematical treatments. 8.1. Brief introduction to chaos In this and in following subsections, we use material from Ott (1993), where one Knds a very clear account of the physics of chaos. A system that evolves in time according to a deterministic mathematical prescription is called dynamical. The state of a dynamical system is characterized by time-dependent (dynamical) variables, such as the positions and momenta of classical mechanics. As these evolve in time, the state traces out an orbit (trajectory, path) in the space of all possible values of the dynamical variables, the phase space of the system. Dynamical systems may evolve either continuously in time, or by discrete time steps. In the former case, they are called Aows; in the latter, maps. Typically, a Mow corresponds to a set of N autonomous Krst-order diOerential equations, dX = F[X(t)] ; (8.1.1) dt for the N dynamical variables encoded in X(t). Thus, X and F are N -dimensional vectors. A map, on the other hand, is simply given by Xn+1 = M(Xn ) ;

(8.1.2)

which relates steps n and n + 1 in time evolution. Again Xn and M are N -dimensional vectors. Dynamical systems may exhibit chaotic behavior, which is characterized by the combination of exponentially sensitive dependence on initial conditions with boundedness of phase space. One needs both of these conditions. In a chaotic Mow, time evolution will cause two inKnitesimally close initial states to trace out bounded orbits, which stretch out exponentially away from each other. Letting X(t) and Y(t) denote two such orbits, we deKne 0(t) ≡ X(t) − Y(t) :

(8.1.3)

The orbits are bounded within a large phase space radius R, |X(t)| ¡ R;

|Y(t)| ¡ R ;

(8.1.4)

yet we have

   0(t)   ∼ exp(ht);  lim B(0)→0  0(0) 

h ¿ 0; t ;

(8.1.5)

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for some typical time . In a chaotic map, it is this combination of stretching (exponential sensitivity to small perturbations) with folding, arising from the boundedness of the orbits, that tends to produce mixing (Arnold and Avez, 1968) and leads to chaos. A simple illustration of how this comes about is provided by the “tent” map, deKned by x n+1 = M (x n ) ≡ 1 − 2|x n − 12 | :

(8.1.6)

It is easily seen that successive iterations of this map take any point outside the interval [0; 1] to −∞, while the interval [0; 1] is mapped onto itself. The stretching takes x n to 2x n , if x n ¡ 12 ; however, if x n ¿ 12 , we obtain 2(1 − x n ), which folds us back to [0; 1]. After m iterations, it should be clear (by graphing the various “tents”) that, even if we know that an initial condition x0 lies within the interval [x0 − 2−m ; x0 + 2−m ], of size 2−(m−1) , its mth iterate may be anywhere in [0; 1]. For large m, an inKnitesimal change in the initial conditions may cause a change of order one after m time intervals. This example shows that chaotic behavior may occur for one-dimensional maps. In fact, noninvertible one-dimensional maps may exhibit chaos, which is the case here, since the “tent” map is not one-to-one. It can be shown that invertible maps can only be chaotic for (phase space) dimension N ¿ 2. This results from a general theorem (Hirsch and Smale, 1974), which states that invertible Aows require N ¿ 3 to exhibit chaotic behavior. One may construct a map from any Mow by making use of the Poincare surface of section. One chooses a surface in phase space, and one records the points where the orbit crosses that surface, in increasing time order. Alternatively, one may sample the Mow at discrete, equally spaced times, tn = t0 + nT , to produce the so-called time-T map. For an N -dimensional Mow, the dimension of the maps we obtain is N − 1. Thus, chaos for invertible Mows with N ¿ 3 leads to chaos for invertible maps with N ¿ 2. Dynamical systems are either conservative or dissipative. In the former case, the volume V of N -dimensional phase space enclosed by a (N − 1)-dimensional closed surface S(0), at t = 0, is preserved, as the surface evolves in time to S(t), i.e., V (t) = V (0). This is what happens for Hamiltonian systems (by Liouville’s theorem). For dissipative systems, damping intervenes, leading to V (t) = V (0). A dissipative system may exhibit attractors, i.e., subsets of phase space to which it will evolve asymptotically. Typical examples are points (dim 0), or limit cycles (closed lines of dim 1). However, there may exist attractors of fractal (noninteger) dimension, called strange attractors. The concept of fractal dimension is most easily captured if we introduce the box-counting dimension, a simpliKed version of the HausdorO dimension (Ott, 1993). Given a set in an N -dimensional Cartesian space, we count the number N(@) of N -dimensional hypercubes, of side @, needed to cover the set. The box-counting dimension D0 is then   ln N(@) : (8.1.7) D0 ≡ lim @→0 ln(1=@) It is easy to see that D0 = 0 for a point, D0 = 1 for a line, and D0 = 2 for an area on a plane. However, there are sets for which D0 is non-integer. An example is the middle third Cantor set: starting with [0; 1], divide this interval into three equal parts and discard the middle one; then repeat the operation time and time again; keep the set that remains in the limit of an inKnite number of iterations. If one chooses a sequence {@n } of values for the sides, with @n = (1=3)n , then N(@n ) = 2n ,

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and D0 =ln 2=ln 3 ≈ 0:63. The middle third Cantor set is self-similar, i.e., any portion of it, magniKed, reproduces the entire set. An example of a dynamical system that yields just the middle third Cantor set is given by the map x n+1 = 3x n

if x n ¡ 1=2;

x n+1 = 3x n − 2

if x n ¿ 1=2 :

(8.1.8)

Points outside [0,1] are mapped to +∞, if x n ¿ 1, or to −∞, if x n ¡ 0. The iterate of the map in [0,1] will take ( 13 ; 12 ] to x n+1 ¿ 1, and [ 12 ; 23 ) to x n+1 ¡ 0. On subsequent iterations they will then be driven to ±∞. Clearly, the overall eOect is to exclude middle thirds at every iteration. As n → ∞, we are left with the middle third Cantor set. Note that, if we start with a uniform distribution of points, 00 (x) = 1, x ∈ [0; 1]; 00 (x) = 0 outside, after n iterations we will have 00 (x) = ( 23 )n . The fraction of points remaining in [0; 1] is
1 d x0n (x) = ( 23 )n = exp(−n ln 32 ) : (8.1.9) 0

It decays exponentially with the discrete time n, with a decay rate ; = ln 32 , so that  = ;−1 is a typical decay time. 8.2. Classical chaotic scattering The Cantor set of the previous subsection is invariant under the action of the map: initial conditions belonging to it will remain forever in the set. However, it is a set of (Lebesgue) measure zero; almost all initial conditions (apart from that set of measure zero) are not attracted to it, as they eventually leave (go oO to ±∞). The amount of (discrete) dwell time that a given orbit spends in [0,1] is extremely sensitive to initial conditions (take 13 −  and 13 + ,  → 0, as initial points), one of the signatures of chaos. Orbits may spend a long time near the non-attracting invariant set, also called chaotic repeller (or strange repeller), before leaving (KadanoO and Tang, 1984), with transient times characterized by the average dwell time  , also known as the classical escape time. This is a simple example of a non-attracting chaotic invariant set. In chaotic scattering, as in this example, there is also a non-attracting chaotic invariant set. It is the existence of such a set that brings in the chaotic ingredient, and that also allows for the connection with time delay. One might wonder whether chaos can occur in a scattering process, since it depends both on the stretching that signals sensitivity to initial conditions, and on the folding within a bounded phase space. Such boundedness, an essential ingredient, is apparently missing in scattering processes. However, it is quickly recovered, by employing appropriate scattering variables, such as the scattering angle, which is naturally bounded, as it is deKned modulo 2. NaPDve intuition would also lead one to expect no chaos in scattering, since asymptotic incoming and outgoing motion are both free, and therefore simple. However, this presupposes a Knite dwell time in the interaction region. If paths can become trapped within this region (inKnite time delay), scattering is not deKned for them, and singularities in scattering functions may be expected (Eckhardt, 1988). A well-known example is the phenomenon of orbiting (Berry and Mount, 1972; Nussenzveig, 1992), in which the particle energy sits at the top of an eOective potential barrier, corresponding

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to an unstable circular orbit, around which it can keep circulating at precisely this energy. Rather than at such a single isolated point, unstable trapping, in classical chaotic scattering, typically occurs on a Cantor set, the strange repeller. Paths can spend very long times in the neighborhood of this set, which randomizes them and leads to large Muctuations in scattering (Smilansky, 1992). We see, therefore, that the dwell time plays a central role in the very deKnition of chaotic scattering. In order to explore the emergence of chaos in scattering, we consider the classical problem of a point particle, moving in a potential V that is conKned to a Knite spatial region. Coming from outside, where it moves freely along a straight line, the particle enters the scattering region, where it interacts with the potential V = 0. After spending some time in the interaction region, it Knally escapes, moving oO to spatial inKnity along a straight line. The relationship between incoming and outgoing variables deKnes a reaction function, which will be our object of study. For motion in the (x; y) plane, we may take as incoming variable the impact parameter b with respect to the x-axis, and as outgoing variable the scattering angle  (angle between the Knal velocity and the x-axis). The reaction function is then (b). Bleher et al. (1990) considered a speciKc potential (Ott, 1993; Ott and TCel, 1993), V (x; y) = x2 y2 exp[ − (x2 + y2 )] :

(8.2.1)

This potential has peaks at four points: (1; 1), (1; −1), (−1; 1) and (−1; −1), with maximum value Em = e−2 . For incident energies E ¿ Em , the function (b) becomes smoother and smoother, as we increase the number of b values in the numerical plot [Fig. 8.1(a)]. For E=Em ¡ 1, however, numerical plots of the reaction function reveal regions of poor resolution. For E=Em = 0:260 [Fig. 8.1(b)], they occur at −0:6 ¡ b ¡ − 0:2, and at 0:2 ¡ b ¡ 0:6. If we magnify one such region, by increasing the number of plotted values by a large factor, the new Kgure will alternate subregions of poor and good resolution, within the original one. Magnifying still more produces a pattern of self-similarity, illustrated in Fig. 8.2. The corresponding behavior of the time delay as a function of the impact parameter (Fig. 8.3) clearly illustrates the prevalence of very long and sensitively varying dwell times in chaotic scattering. The strange repeller in this problem emerges as the set of orbits whose initial points {bs } separate regions of poor and good resolution, in the limit of inEnite magniEcation. These {bs } are singular points of the reaction function, since inside [bs − ; bs + ] there are always points whose scattering angles diOer by amounts of order one, even as  → 0. These points belong to a fractal set, whose dimension was estimated (Bleher et al., 1990) as D0 = 0:67. The strange repeller illustrates well the features of chaotic scattering, and it provides the anticipated connection with time delay: the bs values are just those initial conditions that cause the particle to spend an inKnite amount of time in the interaction region, unable to escape to spatial inKnity, undergoing multiple reMections oO the potential, conKning it to the region V = 0. InKnite time delay therefore characterizes the orbits in the chaotic repeller. For initial conditions close to the ones leading to these orbits, the dwell time will be very long, but the particle eventually escapes. Furthermore, two initial points very close to a given bs value may correspond to n and n+1 bounces oO the potential, respectively, with n very large. The extra bounce or few bounces may be enough to lead to completely diOerent scattering angles. Fig. 8.4 corroborates the above statements. A picture is worth a thousand words: a very striking visual illustration of chaotic scattering, showing both sensitivity to initial conditions and fractal structure, appears in the scattering of light

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Fig. 8.1. Scattering angle  versus impact parameter b: (a) for E=Em = 1:626, and (b) for E=Em = 0:260 (Bleher et al., 1990).

from a pyramid of four touching mirrored balls, so disposed that their centers form a tetrahedron (Sweet et al., 1999). The light goes into an “inner” region through one of the four openings, and it may leave through any one of them. The collection of light rays leaving through a particular opening deKnes its “basin”. Using colored light to identify the basin where a light ray originated, and aiming through one of the openings, one gets a color picture, in which one can literally see the fractal nature of the basin boundaries. We may relate the scattering process to the general discussion of Section 8.1 by regarding it as a Mow, where X = (x; y; vx ; vy ) and F = (vx ; vy ; −9x V; −9y V ). Conservation of energy, E = 12 m(vx2 + vy2 ) + V (x; y) ;

(8.2.2)

shows that only three of the four phase space variables are independent. We may take them to be (x; y; ), with  ≡ vx =|v|.

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Fig. 8.2. Successive magniKcations of the plot in Fig. 8.1(b), within increasingly smaller subintervals (Bleher et al., 1990).

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Fig. 8.3. Time delay versus impact parameter for E=Em = 0:260: (a) within the range in Fig. 8.1(a); (b) magniKcation within the same subinterval as in Fig. 8.2(a) (Bleher et al., 1990). Fig. 8.4. Orbits for particles with impact parameters diOering only by 10−8 ; the one in (b) undergoes about 4 more bounces and emerges almost in the opposite direction (Bleher et al., 1990).

If we take an orbit Xs (t) in the invariant set and perturb it slightly, a linearized perturbation problem follows from writing X(t) = Xs (t) + X(t) ;

(8.2.3)

and expanding (8.1.1) in X. Since Xs (t) satisKes (8.1.1), we derive d Xi (t) = [9j Fi ]s Xj (t) + O(2) ; (8.2.4) dt where [9j Fi ]s ≡ 9Fi =9Xj (Xs ) denotes the Jacobian matrix 9Fs on points of the invariant orbit, and O(2) denotes higher-order terms.

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Keeping only Krst-order terms yields a linearized problem, dY = L(t)Y ; dt

(8.2.5)

deKned by the real N × N matrix L. For periodic motion of period T (which may be ∞), L(t) = L(t + T ), we may use a Floquet Ansatz Y(t) = est E(t);

with E(t) = E(t + T ) :

This deKnes a linear problem, with eigenvalues {sj } and eigenvectors {Ej }. It is even simpler to  consider a PoincarCe surface of section (say, y = 0), and the resulting map Xn+1 = M(Xn ), where X lives in dimension (N − 1). The map will be periodic, with period p. If p = 1, this will yield a Kxed point Xs of the map, around which we can linearize, to directly  obtain Yn+1 = L Yn . Writing Yn = Kn E , we get the eigenvalue equation L E = KE ;

(8.2.6)

whose eigenvalues are the roots Kj of det(L −KI)=0, with corresponding eigenvectors Ej . Directions with |Kj | ¿ 1 are unstable; those with |Kj | ¡ 1 are stable; those with |Kj | = 1 are marginal. The invariant subspace generated by the eigenvectors that correspond to unstable directions is the unstable manifold; analogously, we deKne the stable and center manifolds, respectively. If p ¿ 1, it su>ces to consider (L )p . Then, any one of the p points within a period will be a Kxed point of (L )p . We can linearize around it, and repeat the analysis. If Xl is one such point, the chain rule implies 9[(L )p ](Xl ) = 9L (Xl −1 ) · · · 9L (X0 )9L (Xp −1 ) · · · 9L (Xl ) : Finally, we may write Kj = exp(sj T ), to relate eigenvalues and Floquet indices. We remark that there is always a zero value for the Floquet index, corresponding to displacements along the orbit, that is not included in the previous discussion, since the surface of section is not altered by such a displacement. The eigenvalues of the linearized map deKne invariant subspaces, which make up the tangent space at a Kxed point of the map; they are tangent vectors. Tangent spaces without center manifolds, i.e., those that are direct sums of a stable and an unstable manifold, are called hyperbolic. This notion can be deKned not only for Kxed points, but, more generally, for invariant sets of a map. In that case, the invariant set is hyperbolic if the tangent space of any of its points is a direct sum of invariant stable and unstable manifolds, such that, for K ¿ 0 and 0 ¡ 0 ¡ 1, and M denoting the original map (8.1.2), |9Mn · Y| ¡ K0n |Y| ;

(8.2.7)

for Y in the stable subspace, whereas |9M−n · Y| ¡ K0n |Y|

(8.2.8)

for Y in the unstable subspace. This means that, as the discrete time n grows, points in the stable manifold approach the invariant set in the far future, whereas points in the unstable manifold approach it in the remote past.

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Hyperbolic invariant sets are quite common in chaotic scattering. They are structurally stable against small perturbations, and mathematically well described. The example (8.2.1) exhibits hyperbolic dynamics: the invariant set is the intersection of its stable and unstable manifolds, which are fractal sets with dimensions between 2 and 3. To establish the connection between time delay and hyperbolic chaotic scattering, we consider a compact set B which contains the chaotic repeller of our example. We randomly choose N (0) of its points as initial conditions. After a time t, only those initial conditions that lie on the invariant set or on its stable manifold will remain in B, so that the number of points N (t) still in B decays with an average decay time  ,   1 1 N (0) = lim lim ln : (8.2.9)  t →∞ t N (0)→∞ N (t) This decay time can be related to quantitative signatures of chaos, such as Lyapunov exponents and entropies. For a map M, initial condition X0 , and an inKnitesimal displacement along a tangent vector Y0 , with unit vector uˆ0 = Y0 =|Y0 |, the displacement evolves as Yn+1 = 9M(Xn ) · Yn ; and the Lyapunov exponent h(X0 ; uˆ0 ) is given by 1 Yn 1 h(X0 ; uˆ0 ) ≡ lim ln = lim ln 9Mn (X0 ) · uˆ0 : n→∞ n Y0 n→∞ n

(8.2.10)

(8.2.11)

This parameter quantiKes the stretching of an initial displacement. Chaos itself can be quantiKed by positive entropies: the metric entropy (Kolmogorov, 1958; Sinai, 1959), for instance, measures the rate of creation of information (Ott, 1993) as a chaotic orbit evolves. Chaotic orbits create information because, as they undergo exponential separation, they allow one to distinguish better and better among very close initial conditions. The stretching in phase space, and the limitation to a bounded region, allow chaotic orbits to probe more and more of that space as time evolves. One can derive formulae relating the above decay time to the Lyapunov exponents and information dimensions of both stable and unstable manifolds, in the preceding example of hyperbolic scattering. The information dimensions are obtained from the natural measures associated with the stable and unstable spaces, which allow one to calculate entropies. For the case at hand, the Lyapunov exponents have two possible values, h1 ¿ 0 ¿ h2 . If the initial unit displacement vector is tangent to the stable manifold, then h=h2 , while, if it is randomly chosen, h=h1 . The following relations can be obtained (Ott, 1993): 1 ds = 1 − ;  h1   1 1 ; d u = h1 −  |h2 |    1 1 1 d = d s + du = h1 − ; (8.2.12) −  h1 h2 where ds ; du and d are the information dimensions for the stable, unstable and invariant manifolds, respectively.

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It should be noted, however, that, for nonhyperbolic chaotic scattering, instead of N (t) ˙ exp(−t=), we may have algebraic decay, N (t) ˙ t −S . One important diOerence between these two regimes is that, for hyperbolic dynamics, all periodic orbits are unstable, with no Kolmogorov–Arnold–Moser (KAM) tori (as are found in Hamiltonian systems); for nonhyperbolic chaotic scattering, there are KAM tori in the scattering region (Lau et al., 1991). The ‘stickiness’ near the boundary of regular regions gives rise to the algebraic decay (Meiss and Ott, 1985). 8.3. Quantum chaotic scattering ‘Quantum chaos’ (Gutzwiller, 1992) is the name given to the quantum-mechanical description of classically chaotic systems. The wave aspects of quantum mechanics tend to smooth away many features of classical chaos (Berry, 1987), so that early approaches to the subject tended to the conclusion that ‘quantum chaos’ is a contradiction in terms. The main concern then became to Knd and characterize manifestations of the underlying classical chaos of physical systems in their quantum behavior. The path integral formulation of quantum mechanics is particularly suitable to accomplish this goal, as it relies on sums over phase-sensitive paths to compute quantum amplitudes, and these sums are, in the semiclassical limit, dominated by classical trajectories. The quantum description of classically chaotic scattering will be the main topic of interest in this section. In a quantum description of scattering, cross sections and other physical quantities are obtained by Krst summing over phase-sensitive amplitudes, and then taking the modulus square of a total amplitude. In path integral language, the semiclassical limit provides a natural connection with classically chaotic scattering, as the sums over paths are dominated by classical trajectories. On the other hand, the multiplicity of scattering trajectories in the interaction region, that is typical of classical chaotic scattering, gives rise to random features, due to interferences among wildly Muctuating phases. These features can be obtained from a random matrix treatment, of the type proposed by Wigner (1967) for the distribution of energy eigenvalues, in his study of the statistical properties of nuclear spectra. For a discussion of the subtle relationship between chaotic properties of the S matrix and integrability properties of an underlying Hamiltonian, we refer to a review by Jung and Seligman (1997). Corresponding to these approaches, two ingredients are commonly used to extract signatures of underlying classical chaos from a quantum picture: semiclassical formulae (Balian and Bloch, 1971; Gutzwiller, 1990) and random matrix theory (Mehta, 1967). They are not unrelated: connections have been established (Bogomolny, 1998) between statistical properties of energy eigenvalues and the distribution of classical periodic orbits of deterministic systems. For chaotic scattering, these connections are made through the distributions of resonances and time delays. Although some aspects still do not seem to be completely clariKed, the search for underlying classical chaos involves the concepts of time delays and their statistical correlations. We begin our discussion with the work of BlPumel and Smilansky (1988). They considered the average two-point correlation of the S-matrix elements, linking an incoming state I with an outgoing state I  (I and I  are action variables associated to speciKc channels), between two diOerent energies, CII
(@) = SII∗
(E)SII
(E + @) E ;

(8.3.1)

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where the brackets denote an average over an energy range that is classically small, but quantummechanically large enough for SII
to undergo many oscillations. Their results relate the semiclassical approximation to CII
with the classical time delay distribution PII
(E; t). The latter quantity is obtained from N (t), introduced in (8.2.9), suitably normalized at a given energy E. PII
(E; t) dt is the probability that a randomly chosen orbit, linking I to I  , has a classical time delay between t and t + dt. Then,
CII
(@) ≈ dtPII
(E; t) E exp (i@t=˝) ; (8.3.2) where · · · E again denotes an average over a classically small energy interval UE. This establishes a connection between a quantum S-matrix correlation CII
and a classical time delay distribution. For exponentially decaying distributions, such as those in hyperbolic scattering, this implies a Lorentzian shape for CII
, with a width inversely proportional to the mean classical time delay [cf. (8.2.9)]. In other cases, power laws appear. Therefore, classically chaotic systems leave a diOerent quantum signature than regular ones in the average correlation of S-matrix elements at diOerent energies. From the S-matrix autocorrelation, one can get the autocorrelation for cross-section Muctuations. The result, for hyperbolic systems, is again a Lorentzian associated with the mean time delay, reproducing the expression found by Ericson (1960, 1963) for the correlations in nuclear cross sections. Thus, Ericson Muctuations can be interpreted in terms of chaotic scattering. Instead of looking at the autocorrelation function of the S-matrix, one can look at the autocorrelation function of the time delay, with special attention to resonant contributions. Resonances play a crucial role in quantum chaotic scattering. Indeed, as discussed in Section 8.2, unstable trapped periodic orbits, in classical chaotic scattering, can randomize neighboring paths by delaying them for long and critically sensitive times. The number of such orbits proliferates exponentially with increasing period (Smilansky, 1992). Their quantum counterparts are resonances, that yield long time delays and produce strong scattering Muctuations. The ‘ripple’ Muctuations in Mie scattering (Section 7.2), though not chaotic, are similarly related to the trapped ray optics paths illustrated in Fig. 7.1. By combining (4.3.10) with (5.2.17), one could obtain, for cutoO interactions, a representation for the time delay in terms of the poles of the S-matrix, in which the time delay would be represented as a superposition of Lorentzians, with positions and widths respectively determined by the real and imaginary parts of the poles; for the Mie density of states, such a representation was mentioned in Section 7.2. More generally, for real energies, the resonant part of the time delay (projecting out the regular part; cf. Lehmann et al., 1995) can be viewed as a superposition of Lorentzians, one for each resonance. Indeed, for the resonant part SR of the S-matrix, one has   E − E∗  n ; (8.3.3) det SR (E) = E − E n n where En = En − 12 in . This leads to the representation  n R (E) = ; (E − En )2 + 14 n2 n

(8.3.4)

which agrees with the discussion in Section 7.2. As we have seen in Section 5.2, the time delay is also related to Green’s function G(E) = (E − H)−1 , deKned for an eOective non-Hermitian Hamiltonian

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H, such that its poles coincide with the En , (E) = −2 Im TrG(E) :

(8.3.5)

Signatures of chaotic behavior can be read oO from the study of the autocorrelation function of the time delay, which measures the collective eOect of all resonances. It is convenient to work with the corresponding √ spatial delay W (with dimensions of length), represented as a function of the wave number k = 2mE=˝. One considers the resonant (Muctuating) part of , W subtracting out the smooth background, and projected out with a window function w(k), that selects a given segment, Ww (k) = WR (k)w(k). The autocorrelation is deKned by
∞  6  6 Ww k + dk (8.3.6) C(6) ≡ K Ww k − 2 2 −∞ with the constant K Kxed by the normalization C(0) = 1, so that C is dimensionless. It is this correlation that exhibits traces of the underlying classical chaos. We focus the discussion on this quantity, whose properties have been investigated by a semiclassical approach that sums over periodic orbits (Eckhardt, 1993) or by random matrix methods (Lehmann et al., 1997), applied to multichannel scattering problems. We shall not attempt to relate the two approaches, a step that could clarify certain inconsistencies, but which is still lacking. 8.3.1. Semiclassical approach The semiclassical approximation to the time delay was obtained by Balian and Bloch (1971; 1974), employing the techniques that led to the Gutzwiller trace formula for the density of states (Gutzwiller, 1990). The result is (k) W = W0 (k) + WF (k) = W0 (k) + 2 Re

∞  p

r=1

Lp

exp[ir(kLp − (=2)Np )] |det(I − Mpr )|1=2

(8.3.7)

where W0 (k) is a smooth part, while WF (k) is the Muctuating contribution, determined by the periodic orbits. In (8.3.7), Lp denotes the period p orbits primitive (single traversal) length, Np is a Maslov index, that keeps track of phase changes across caustics (Maslov, 1962), I is the identity matrix, Mp is the Jacobian of the mapping perpendicular to the orbit, and the sum over r takes into account multiple traversals of the orbits. There is good agreement between the semiclassical formula and exact quantum results in various examples, so that it can be used to locate resonance positions (except for the lowest few, for which the stationary phase approximation has large corrections). If kn = sn − i;n denote the complex wave numbers associated with the resonances, ;n 1 ; (8.3.8) WF (k) = WS (k) + WR (k) = WS (k) +  n (k − sn )2 + ;2n where WS (k) is a slowly-varying part, and WR (k) is the contribution of the resonances. Comparison with (8.3.8) allows one to determine the kn . The small 6 behavior of the autocorrelation C(6), a probe of the widths of the Lorentzians, can be obtained by inserting the second term of (8.3.8) into (8.3.6), using a window that selects a Knite

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number N of resonances, to be summed over. We obtain K  ;n + ; m C(6) = : N n; m [6 − (sn − sm )]2 + (;n + ;m )2

165

(8.3.9)

For separations much larger than the widths, |sn − sm |;n + ;m , the dominant contribution for small 6 comes from the diagonal term n = m, 1 2;q K  2;n ≈ ; (8.3.10) C(6) ≈ 2 2 N n 6 + 4;n  62 + 4;2q where the Knal approximation holds for narrow distributions of decay rates ;n , and deKnes an average resonance width ;q . For intermediate and large values of 6, it is convenient to look at the Fourier transform of C(6), the form factor L(K),
L(K) ≡ d6C(6) cos (6K) : (8.3.11) The behavior of C(6) for large and intermediate 6 follows from that of the form factor for small and intermediate K. For small K; L(K) has a series of sharp isolated peaks (Eckhardt, 1993), which grow denser and weaker as K grows, and whose widths depend on the length of the 6 interval used in (8.3.12). As K increases, the peaks overlap more and more. Starting from (8.3.8), one may derive a semiclassical expression for L(K) which sums over periodic orbits. For very small K, the form factor is a sum of contributions from individual periodic orbits, as nondiagonal terms involving orbits of diOerent periods do not overlap. This regime is restricted, since periodic orbits become exponentially denser as the period increases. As K increases, orbits do overlap. In the intermediate K regime, oO-diagonal terms can still be neglected, but the sum becomes restricted to periodic orbits with typical lengths Lp close to K. In this regime, one Knds L(K) ≈ K exp (−;c K) ;

(8.3.12)

(v )−1

(v is the particle velocity) is the classical escape rate (inverse length). in which ;c = Finally, for larger values of K, interferences among periodic orbits can no longer be neglected. One eventually reaches the Lorentzian behavior (8.3.10) for small 6. This corresponds to a quantum regime which is hard to obtain from the periodic orbit picture (here it followed from general considerations about resonances). In this Lorentzian limit, L(K) ≈ exp (−;q K), to be compared with the classical expression (8.3.12). The transition between the classical and quantum regimes, just mentioned, occurs at the Heisenberg length KH , related to the mean resonance separation Us by KH ≈ 2=Us. The classical expression (8.3.12) for the form factor leads to C(6) = ;2c

(;2c − 62 ) : (;2c + 62 )2

(8.3.13)

This expression is obtained in the limit ˝ → 0, when the density of states diverges. It involves the classical escape rate, that emerged in the classical description of chaotic orbits. Many results obtained from the periodic orbit picture can be veriKed in the example of a point particle scattered

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oO three disks in a plane (Eckhardt et al., 1995), for various values of the distance to radius ratio d=R. The reader is referred to Eckhardt (1993) for detailed comparisons. The three-disc problem was treated in classical, semiclassical and full quantum terms in a series of three papers by Gaspard and Rice (1989), which provide explicit illustrations of many aspects of classical and quantum chaotic scattering. Their work was extended to scattering by a system of n non-overlapping and disconnected hard disks (“pinball machine”) in a thorough review article by Wirzba (1999), who derived the exact multiscattering S-matrix and employed it for a critical evaluation of existing semiclassical proposals. 8.3.2. Random matrix approach An alternative approach consists in treating models of chaotic resonance scattering based upon random matrix methods. The underlying hypothesis is that the statistical properties of the S-matrix for ˝ → 0 are determined by Dyson’s theory for the orthogonal ensemble of random unitary matrices (Dyson, 1962; Mehta, 1967; Guhr et al., 1998). Following Lehmann et al. (1997), we assume that there exist M open scattering channels, which couple to N resonant states. The time delay is then deKned as an average over channels (Section 4.4), i˝ 9 (E) = − ln det S ; (8.3.14) M 9E and the S-matrix has additional indices associated with such channels. The eOective non-Hermitian Hamiltonian H = H − igW couples, with strength g, the M channels to the N resonant states, through W = VV T , and Vmn is the transition amplitude between channel m and resonance n. Consistently with the above hypothesis, we assume that Hnn
belongs to the Gaussian orthogonal ensemble (GOE), and that the Vmn are also statistically independent Gaussian variables. As before, we introduce the time delay autocorrelation "  @  @ # @  @ # "  C(E; @) = F E + F E − =  E+  E− 2 2 2 2 "  @ # @ # "   E− ; (8.3.15) −  E+ 2 2 where the brackets denote ensemble averages. Restricting our attention to the center E=0 of the GOE ˜ spectrum, and deKning a dimensionless quantity C(@) ≡ C(0; @)=(0) 2 , the ensemble averaging can be performed via supersymmetry techniques (Lehmann et al., 1995). The average time delay (0) −1 can then be computed: it is proportional to W ≡ 200 =(MT ), where 00 ≡ 0(E = 0) is the density of resonant levels projected onto the real axis (Lehmann et al., 1997) and T is the transmission coe>cient. We now consider the behavior of C˜ for Kxed V ≡ M=N; N → ∞. ˜ By construction, for a closed system, C(@) is proportional to the density–density correlation of the GOE, which has a singular delta term plus a smooth (Dyson) contribution. With v1, as 00 W → 0, we recover the GOE result, because, in the limit @W , the coupling to the continuum channels can be neglected: an open system cannot be distinguished from a closed one for t˝=W . Therefore, in the limit of isolated resonances, 00 W 1, we have a Lorentzian term coming from the GOE delta function, plus terms that tend to the smooth Dyson contribution as 00 W → 0; these are the terms one would obtain for g = 0, the closed system case.

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167

On the other hand, the limit of overlapping resonances yields 42 (2 − @2 ) ˜ C(@) = 2 W2 2W M T (W + @2 )2

(8.3.16)

which is of the same form as (8.3.13), with the classical  replaced by W . For V Knite, the same expression emerges, but with a G instead of W . This G is the gap between the distribution of resonance poles in the complex E plane and the real axis. It appears to be a typical feature of classical hyperbolic scattering. The overlapping regime strongly suppresses time delay Muctuations, as can be seen by comparing the result for isolated resonances, ˜ C(0) =

1 1 ; 00 W

(8.3.17)

with the result for overlapping ones, ˜ C(0) =

1 1 : (00 W )2

(8.3.18)

Although we have described results derived using a supersymmetric technique, other random matrix methods have been used to obtain directly the distribution of eigenvalues of the time delay matrix in chaotic scattering (Brouwer et al., 1997). Although the eigenvalue distribution for the S-matrix had been known to follow the so-called circular ensemble introduced by Dyson (1962), Brouwer et al. show that the time delay eigenvalues are distributed according to the so-called Laguerre ensemble of random matrix theory. A comparison with the periodic orbit picture shows that the limit of overlapping resonances should correspond to the classical one, because the typical time ˝=W is much greater than the Heisenberg time tH ≈ ˝00 . However, 00 is a free parameter in the random matrix picture, which renders it impossible to have an automatic divergence of the density of states as ˝ → 0, as desired. Thus, the classical rate ;c (or the corresponding energy scale c ≡ ˝v;c ) never appears, in contrast with what happens in the periodic orbit picture. It is replaced by the gap G . It has been argued (Gaspard and Rice, 1989) that this replacement might in fact occur in certain chaotic systems. Despite these diOerences between the two approaches, the qualitative aspects of resonant scattering are present in both, and their relation to classical chaotic scattering is made evident either through the classical escape rate in the time delay autocorrelation, or through the gap that comes from a random average. An alternative derivation of the semiclassical relation between quantum time delay and properties of the set of trapped periodic orbits in the chaotic repeller can be found in Vallejos et al. (1998), where a comparison with the predictions of random matrix theories is made. A comprehensive review of chaotic scattering and resonances is found in Gaspard (1993); see also Haake (1991). 8.3.3. Recent theoretical and experimental studies In this subsection, we list some of the recent investigations on chaotic quantum scattering, which involve either theoretical work or experiments. As examples of theoretical investigations, we can list: (i) The discussion, employing random matrix theory, of the decay law for quantum systems with a small number of open channels (Dittes, 2000),

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e.g., two-dimensional billiards (systems in which particles move freely within a region of the plane, conKned by perfectly reMecting walls) connected to inKnite leads (waveguides). (ii) The statistics of time delays and resonance widths for a Bloch particle in the presence of dc and ac Kelds (GlPuck et al., 1999), a system which exhibits classical chaos. As long as the Bloch period and that of the ac Keld are commensurate, the results are shown to agree with the predictions of a random matrix approach. (iii) Those statistics, for one-dimensional quasiperiodic SchrPodinger equations, that exhibit fractal spectra for the closed (no coupling of resonances to open channels) system (Steinbach et al., 2000). The authors Knd power law decays, whose exponents are related to the fractal dimensions of the spectra. (iv) Numerical studies of quantum transport through ballistic cavities, represented, as above, by billiards with leads (Huckenstein et al., 2000). The studies relate the behavior of time delay to the degree of chaos in phase space. (v) The theoretical analysis of chaotic scattering on graphs (Kottos and Smilansky, 2000), which allows for a semiclassical treatment, and also shows good agreement with the predictions of random matrix theory, evidence that the two descriptions are compatible in this particular case. (vi) A detailed theoretical analysis of time delay distributions for one-dimensional random potentials (Texier and Comtet, 1999), using the resonance tunneling picture, which shows good agreement with numerical calculations. (vii) Finally, employing random matrix theory and the supersymmetry approach, a discussion of the distribution of proper time delays (eigenvalues of the time delay matrix) in chaotic scattering (Sommers et al., 2001). In terms of experimental realizations, a number of studies are worthy of mention: (i) Microwave experiments (Doron et al., 1990; Richter, 1998), investigating the transmission of waves through waveguide junctions, which act as an interaction region. Because the SchrPodinger equation reduces to the Helmholtz one [(∇2 + k 2 )G = 0] for free particles, the study of ballistic quantum systems (billiards) can be mapped onto the study of acoustic, elastic or electromagnetic waves. The wavenumber k, rather than the energy, is the natural variable here, which explains the use of ‘time’ delays measured in units of length, in the previous subsections. Chaotic scattering takes place here, as well as in many other systems where the dimensions of the scatterer are much greater than the scattered wave−1 lengths. Semiclassical estimates of the average ‘time’ delay yield (k) W k ≈ (1 + 1=N );c , where N is the number of propagating modes. Besides recovering the classical result for large N , good agreement with the experiment is obtained for 1 6 N 6 6, indicating a quantum enhancement of the time delay by roughly a factor of two. (ii) Experiments with microwaves (Lu et al., 1999), incident on a two-dimensional n-disk geometry, that yield frequencies and widths of low-lying resonances, comparable to semiclassical estimates, as well as a spectral autocorrelation function [see (8.3.1)], with a decay rate well matched by the classical escape rate, for small energies. For intermediate energies, non-universal oscillations are well described by the periodic orbit picture. (iii) Instead of employing microwaves, experiments with atom-optics billiards have recently been done, by conKning ultracold atoms, with optical potentials induced by laser beams (Milner et al., 2001; Friedman et al., 2001). Measurements of the decay of the number of trapped atoms through a hole in the boundary Knd that it behaves exponentially only for classically chaotic situations, showing a clear distinction between chaotic and regular systems. (iv) Even more recently (Dembowski et al., 2001), microwave experiments have tested mixed systems, i.e., systems exhibiting both chaotic and regular features, as well as their limiting cases, fully chaotic or fully regular behavior. They checked the validity of a trace formula (Tomsovic et al., 1995; Ullmo et al., 1996) that interpolates between these two extremes. Apart from these recent examples of quantum chaotic scattering, various others can be found in Ott and TCel (1993), Gaspard (1991), and Smilansky (1992).

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8.3.4. Coherence from decoherence In the last few years, a coherent picture of the relation between classical chaos and its corresponding quantum description, the main objective of our discussion of chaotic scattering, has provided important new insights, based on the recent results on the role of decoherence in the quantum-to-classical transition (Zurek, 1991). In non-linear systems, a classical trajectory, initially centered on a quantum wave packet, can quickly diverge from the motion of its centroid, because of coherent interference among fragments of the wave packet. For generic initial conditions, this is expected to occur, for integrable systems, on a time scale inversely proportional to a power of ˝, whereas, for chaotic systems, it occurs on a much smaller logarithmic time scale, th ≈ ln(C=˝). Comparison between Wigner distribution functions evolved quantum-mechanically, and classical phase space distributions (Habib et al., 1998), shows that the discrepancy between quantum and classical evolution is greatly reduced even by a weak coupling of the system to the environment, which leads to decoherence in the quantum case. Decoherence eOects destroy quantum interference in Wigner distributions; and the coupling to the environment washes out Kne structure in classical distributions. The net result is to bring quantum and classical descriptions closer together. In the speciKc case of a quartic double-well, coupled to a bath of harmonic oscillators, a system which is classically chaotic, it has been shown (Monteoliva and Paz, 2000) that the rate of entropy production is proportional to the diOusion coe>cient (Kxed by the coupling) for short times, but it is Kxed by a Lyapunov exponent for longer times, before equilibrium is reached. Conditions for the quantum state vector, projected out by a measurement, to remain localized, and to follow a trajectory characterized by classical Lyapunov exponents, have been studied (Battacharya et al., 2000). Furthermore, connections between the decoherence rate and Lyapunov exponents have also been established (Jalabert and Pastawski, 2001). In short, the analysis of decoherence eOects in quantum scattering may be the road to follow in order to relate the quantum description to classical chaos. This is a subject of current theoretical and experimental (Friedman et al., 2001) interest, which will probably add signiKcantly to our present understanding. 9. Conclusion The fundamental role played by the variation of the phase of wave functions with respect to physical parameters is well illustrated by the preeminence of gauge Kelds in present-day theoretical physics. Time delay has to do with the phase response to energy variation. It is not surprising, therefore, to Knd that it has an ubiquitous role in quantum mechanics. This is also rendered apparent through its close relationship with the density of states. Experimental aspects have been mostly ignored in the present report. Direct experimental observations employing wave packets tend to be very di>cult, because of the invasive eOects of quantum measurements. However, indirect detection of time delay eOects takes place, e.g., through the measurement of decay lifetimes, of the density of states, and of chaotic scattering. In macroscopic wave scattering, measuring of time delay, combined with tomography and Doppler techniques, are the basis of radar and lidar ranging, as well as of a variety of imaging techniques. An intriguing possibility, not pursued here, would be combining the gravitational and

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electromagnetic interactions, for a scattering theory derivation of the Shapiro gravitational time delay (Will, 1993). Acknowledgements We acknowledge the partial support of the Brazilian agencies Conselho Nacional de Desenvolvimento CientCDKco e TecnolCogico (CNPq), Fundabca˜ o de Amparo ac Pesquisa do Estado do Rio de Janeiro (FAPERJ) and Fundabca˜ o UniversitCaria JosCe BonifCacio (FUJB). The Knal stages in the preparation of this work were performed while one of the authors (HMN) held a National Research Council Research Associateship Award at NASA Goddard Space Flight Center. We thank an anonymous referee for suggesting the inclusion of some additional useful references. References Abramowitz, M., Stegun, I.A., 1965. Handbook of Mathematical Functions. Dover, New York, p. 297. Abrikosov, A.A., Gorkov, L.P., Dzyaloshinski, I.E., 1975. Methods of Quantum Field Theory in Statistical Physics. Dover, New York. Aharonov, Y.D., Albert, Z., Vaidman, L., 1988. Phys. Rev. Lett. 60, 1351. Aharonov, Y.D., Vaidman, L., 1990. Phys. Rev. A 14, 11. Aharonov, Y.D., Oppenheim, J., Popescu, S., Reznik, B., Unruh, W.G., 1998aa. Phys. Rev. A 57, 4130. Aharonov, Y.D., Reznik, B., Stern, A., 1998bb. Phys. Rev. Lett. 81, 2190. Altarelli, M., Dexter, D.L., Nussenzveig, H.M., Smith, D.Y., 1972. Phys. Rev. B 6, 4502. Amrein, W.O., Jauch, J.M., Sinha, K.B., 1977. Scattering Theory in Quantum Mechanics. W.A. Benjamin, Reading, MA. Arnold, V.I., Avez, A., 1968. Ergodic Problems of Classical Mechanics. W.A. Benjamin, New York. Artmann, K., 1948. Ann. Phys. (Leipzig) 2, 87. Balian, R., Bloch, C., 1971. Ann. Phys. (N.Y.) 63, 592; ibid. 85 (1974), 514. Baltes, H.P., Hilf, E.R., 1976. Spectra of Finite Systems. B-I Wissenschaftsverlag, Mannheim. Barber, P.W., Chang, R. (Eds.), 1988. Optical EOects Associated with Small Particles. World ScientiKc, Singapore. Bargmann, V., 1952. Proc. Natl. Acad. Sci. USA 38, 961. Baz’, A.I., 1967, Yad. Fiz. 4, 252 [Sov. J. Nucl. Phys. 4, 229]; Rybachenko, V.F., ibid. 5, 895 [ibid. 5, 635]. Berry, M.V., 1987. Proc. Roy. Soc. London A 413, 183. Berry, M.V., Mount, K.E., 1972. Rep. Prog. Phys. 35, 315. Beth, E., Uhlenbeck, G.E., 1936. Physica 3, 729; 1937, ibid. 4, 915. Birman, M.S., Krein, M.G., 1962. Dokl. Acad. Nauk 144, 268 [Sov. Math. Dokl. 3, 707]. Bleher, S., Grebogi, C., Ott, E., 1990. Physica D 46, 87. BlPumel, R., Smilansky, U., 1988. Phys. Rev. Lett. 60, 477. Bogomolny, E., 1998. Doc. Math. J., Extra Volume ICM, 99. Bohm, D., 1951. Quantum Theory. Prentice-Hall, Englewood CliOs, NJ, p. 257. Bohren, C.F., HuOman, D.R., 1983. Absorption and Scattering of Light by Small Particles. Wiley, New York. BollCe, D., Osborn, T.A., 1976. Phys. Rev. D 13, 299. BollCe, D., Osborn, T.A., 1979. J. Math. Phys. 20, 1121. BollCe, D., Gesztesy, F., Grosse, H., 1983. J. Math. Phys. 24, 1529. Braun, G., 1956. Acta Phys. Austr. 10, 8. Brenig, W., Haag, R., 1959. Fortschr. Phys. 7, 183. Brouwer, P.W., Frahm, K.M., Beenakker, C.W.J., 1997. Phys. Rev. Lett. 78, 4737. Buslaev, V.S., 1972. In: Birman, M.S. (Ed.), Spectral Theory and Wave Processes. Consultants Bureau, London. BPuttiker, M., 1983. Phys. Rev. B 27, 6178. BPuttiker, M., Landauer, R., 1982. Phys. Rev. Lett. 49, 1739.

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