- Email: [email protected]
Resonance fluorescence in a tailored vacuum
1 July 1995
OPTICS COMMUNICATIONS ELSEVIER
Optics
Communications 118 (1995) 143-153
Full length article
Resonance fluorescence in a tailored vacuum C.H. Keitela, P.L. Knighta, L.M. Narduccibyl, M.O. Scullyb~* a Optics Section, Blacken Laboratory, Imperial College, London SW7 282 UK b Max-Planck Institut fiir Quantenoptik, Hans-Kopfermann Strasse 1, 85748 Garching bei Miinchen, Germany Received 17 January 1995
Abstract
We consider the resonance fluorescence of a two-level atom with an emphasis on the dependence of this phenomenon on the modal density of the environment. We use an approach of the Weisskopf-Wigner type to investigate the interaction of the dressed states of the atom-driving field system to the surrounding environment, and calculate the linewidth of the resonance fluorescence spectrum in terms of the decay rate of the dressed coherences. Unlike the conventional theory, we deal first with the exact interaction of the atom with the driving laser field, and then consider the coupling of this combined system to the modes of the environment within the Weisskopf-Wigner approximation. This modification of the traditional sequence in which the interactions are handled leads to a sensitive dependence of the fluorescence spectrum on the density of modes and suggests that a driven atom placed in a suitable environment, such as for example a high-Q resonator, even if Markovian, might behave quite differently from an identical system in free space. The significance of this effect is shown explicitly for the case of a strongly driven two level atom in an environment with a modal density which is substantially frequency dependent on the scale of the Rabi frequency of the driving laser field.
Spontaneous emission has been a well understood phenomenon since the pioneering work by Weisskopf and Wigner [ 1 ] who clarified its physical origin as the result of the interaction of an excited atom with the infinite number of electromagnetic modes of the vacuum. During the late 1960’s Mollow discovered that the fluorescence spectrum of a two-level atom may develop sidebands if the atomic transition is driven by a coherent single-mode field [ 21. The origin of this phenomenon can be traced to the energy level structure of the interacting atom-field system, whose eigenstates, the so-called dressed states as they are commonly known, are organized along an infinite ladder of doublets with a singlet ground state [ 31. Cohen-Tannoudji and Reynaud provided an especially appealing description of the physics of resonance fluorescence with an approach based on the dressed states formalism [4], and after appropriate generalization for the case of multi-level atoms discovered rich additional features also in the corresponding fluorescence spectra of driven three-level systems [ 51. The main virtue of their approach is that, in the limit of large driving Rabi frequencies, it provides the positions and widths of the various spectral components in analytic form, following consideration of the time dependence of the dressed coherences. More recently Narducci and collaborators found that three-level atoms 1Permanent address: Department of Physics and AtmosphericSciences, Drexel University, Philadelphia, PA 19104, USA. * Also: Department of Physics, Texas A & M University, College Station, TX 77843, USA and Texas Laser Laboratory, HARC, The Woodlands,
TX 77381, USA.
0030.4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00161-1
144
C.H. Keitel et al. /Optics
Communications 118 (1595) 143-153
in a V-configuration, when driven coherently on both allowed atomic transitions, can produce fluorescence spectra which are characterized by sub-natural linewidths [ 61, a prediction that was confirmed experimentally by Gauthier et al. [ 71. The underlying physical mechanism of the line narrowing can also be understood fairly readily with the help of the dressed states of the coupled atom-field system [6,8]. When one places an atom into a high Q-resonator or a photon band-gap material, for example, spontaneous emission can be suppressed or enhanced relative to the rate of decay in free space [ 9- 111. From a theoretical point of view this situation can be modeled by a two-level atom coupled to the nearest resonator mode (the other modes of the cavity can be assumed to be quite removed in frequency), which in turn interacts with the modes of the surrounding environment, as discussed, for example in Ref. [ 121. Dynamical suppression of spontaneous emission has been predicted also for a driven two-level atom in a cavity field as a result of the finite response time of the environment [ 131. This effect, which is connected with non-Markovian features, was tested experimentally quite recently [ 141 and interpreted theoretically as a consequence of the renormalization of the cavity-induced spontaneous emission by the strong driving field [ 151. Other decay studies have been published, which indicate that the traditional Born-Markov assumption of essentially unperturbed source evolution needs careful attention. Some approaches to this problem have involved a modification of the Scully-Lamb laser theory with the inclusion of atom-field correlation effects during the atomic relaxation process [ 16 3 ; others have raised the possibility of inconsistencies in the approach of the Jaynes-Cummings model to thermal equilibrium if the effects of atom-field interactions are ignored in the description of the field damping process [ 171. More recently a generalized set of Maxwell-Bloch equations has been proposed to describe semiclassically the limit of strong atom-field coupling [ 181; in common with the present quantum mechanical treatment, the analysis by Kocharovskaya and collaborators [ 181 applies the Born-Markov approximation and predicts a sensitive dependence of the relaxation terms in the master equation of driven atomic systems on the modal density of the surrounding environment. For the most part, until now, the common approach to the description of resonance fluorescence has involved consideration of the interaction between the atom and the coherent driving field separately from the process of atomic decay, the latter part of the problem being modeled by the conventional interaction of the bare two-level atom with a suitable reservoir. The few studies that have taken into account the atom-field coupling together with the atomic damping process, however, have investigated the parameter domain beyond the Born-Markov approximation and have stressed the importance of the response of the environment [ 13-161. In this paper we show that interesting effects arise already in the domain in which the Born-Markov approximation is justified, i.e. we carry out this approximation, and, as shown schematically in Fig. 1, we consider the coupling to infinitely many modes of the reservoir only after taking into account the dressing operation of the atom. The order in which these two interactions are handled in the calculation is not an obvious matter, and surely the traditional approach and the one developed in this paper are not equivalent because of the approximate nature of the irreversible part of the calculation, based as it is on the Weisskopf-Wigner approach. From a conceptual point of view it appears more correct to apply the Weisskopf-Wigner approximation to the degrees of freedom of the strongly interacting subsystems, in this case the atom and the driving field regarded as a single entity. In fact, as it turns out, some details of the resonance fluorescence spectrum are sensitive to the order in which the calculation is carried out. The difference between the final results of the traditional approach and the one adopted in this work is especially evident in the argument of the density of modes which enters as a multiplicative factor in all the spontaneous emission rates: in the traditional treatment the density of modes is evaluated at the atomic resonance frequency, while in our approach it must be calculated at the various dressed states resonances. In free space, this difference is not very substantial, given the slowly varying character of the vacuum density of states, but in a high-Q resonator it can be quite large, In order to emphasize this point, we introduce the notion of “tailored vacuum” to indicate that the magnitude of the vacuum fluctuations at a particular frequency can be modified with a suitable design of the surrounding environment. In this paper we construct the Heisenberg equations of motion for the operators that describe the interaction of the combined atom-driving field system with an infinite collection of reservoir modes, and derive decoupled
C.H. Keitel et al. /Optics Conventional
Communications
118 (1995) 143-153
145
treatment:
laser mode 6.1, -
Correct treatment:
/I-
reservoir
Fig. 1. Usually the damping rates which are added phenomenologically to the equation of motion of the density matrix elements are due to the interaction of the bare atom with the reservoir in spite of the presence of the additional driving field. Here we consider the coupling of the reservoir to the atom-driving field system and perform the usual Weisskopf-Wigner approximation after the interaction of the atom with the driving field has been taken into account. The box indicates which subsystems in each approach are considered first in a pre-diagonalization: atom-reservoir or atom-laser field.
_
Ib.n+l>
-
la,n-l>
Fig. 2. The left hand side displays the bare two-level atoms with the upper level a and the lower level b and n denotes the photon number of the driving field. The right hand side shows the corresponding dressed states.
equations for the dressed coherence operators. In the specific case of a driven two-level system, we perform the Weisskopf-Wagner approximation for the corresponding dressed states operators and obtain the linewidths and weights of the various spectral components as functions of the density of modes at the dressed states resonances. The deviation from the traditional Mallow spectrum is here put forward in a simple analytic form in terms of the modal density of the environment and graphically for several realistic examples. Finally parameters are pointed out which seem suitable to demonstrate this effect experimentally. The Hamiltonian HO describing the subsystem consisting of a two-level atom with upper level a and lower level b and the coherent driving field mode with frequency OL and the photon number n can be written in terms of the dressed states as
with
Il,n) =cosela,n)
+sinf?lb,n+
l),
12,n) = -sinO)a,n)
+cosB(b,n
+ l),
(2)
and
G.n=noL*iti12,
{+fori=l,-fori=2},
012 = (LP + S2)j,
tan(20) = O/s.
(3)
146
C.H. Keitel et al. /Optics
Communications 118 (1995) 143-153
Here 0 denotes the Rabi frequency of the driving field and 6 = tiOb - WL the detuning between the atomic transition and the driving field frequency. Furthermore, we have assumed the laser field to be intense and the corresponding photon number to obey Poissonian statistics. As a consequence the width of the photon number distribution is small compared to the mean number of photons, yet still large as compared to unity, so that we can neglect the n-dependence of the Rabi frequency over a wide range around the average [ 4,5]. Thus, the dressed states energy level diagram can be viewed as consisting of periodically repeating doublets in this range; in Fig. 2 we have displayed the bare states on the left and the corresponding dressed states on the right for one particular value of the driving field photon number. The atomic dipole operator ,&,which is the source of the radiated field and whose time dependence determines the shape of the spectrum, can be represented conveniently, if somewhat clumsily, as a superposition of dressed states projectors as follows
iiL=~~li,nj~i,nlPl~,m)(~,nl=~[~,1I1, n- l)(l,nl +p22p,nj,m +p21)2,n
l)(Znl
+p12(1,n-
1)(294
n
i,n -
l)(Lnll
(4)
+h.C.,
with PII =
-CL22
= COS 6 Sin 8 p&,
CL12 =
-sin28 &&b,
P21 = C0S28 ,U&,
(5)
and where i,j E { 1,2}, m,n E { 1,2, . ..} and &b = (bl,k]u) is the atomic dipole matrix element that couples the upper level a to the lower level b. The advantage of this representation, as shown more explicitly below, is that in the secular approximation, particular lines of the fluorescence spectrum can be associated with excellent approximation to particular dressed states projectors; thus, for each sideband only one projector is relevant (if we ignore the sum over the field photon number), while for the central component of the spectrum two projectors oscillate at this frequency and must be considered simultaneously. The spectrum is governed by the two-time correlation function of the raising and lowering contributions of the dipole operator. Following the regression theorem, this reduces to a sum of one-time dipole operator expectation values with complex two-time weighting factors which are identical to those that govern the evolution of the dipole operator itself. In the dressed states basis these summands give rise to well separated spectral peaks if the driving field is sufficiently large. As a consequence, the one-time weighting factors that distinguish the time evolution of the dipole correlation function from that of the dipole operator itself play a central role in assigning the intensities of the various peaks but virtually no role in setting the linewidths of the spectral components. Thus, for example, the summand (2,n - l)(l, II I in the dipole operator of Eq. (4) leads to a spectral peak at OL + ~12 and its damping rate determines the linewidth. Our interest in the following is the calculation of the relaxation rates of these dressed coherences. The complete Hamiltonian H which includes the interaction of the dipole operator with the vacuum modes, whose annihilation operators are denoted by {ak}, takes the form H=~o+Gl,,
(6)
with
n - l)(l,nl +g22kp,IZ - l)Q,nl
H WC n +g12k]l,n
k -
I)(294
+g21k[%n
-
l)(l,nll
+h.C.
+ xbk&k,
(7)
k
where the coupling COllStZiMS gijk ZiR defined as gijk = pij&k/h and 8k = ( fUl)k/kO)f denotes the so-called electric field per photon, V is the quantization volume and Wk is the frequency of the kth vacuum mode.
-7-
C.H. Keifel et al. /Optics Commu~catiu~
I18 ff995j
143-153
147
The time dependence of the dressed population and coherence operators l’Ii,n;j,m= Ii, n>(j, ml is governed by the Heisenberg equations of motion $fli.n:j,nt = i [ Hv fli,n;j,ml~
(8)
with analogous equations describing the evolution of the field operators u&(t). For the coherence operator K7~.n;2,,r-~(i),for example, we obtain $L,:*,.-1 + i
= i(wL -t
~2)n1,~;2,~-1
Ca:lgldh-b2,n-l
+ g2ikni,n-k2,n-1
-
g22kJ11,n;2,n
-
g2uXi,n~~,nl
k
In the next step we integrate the Heisenberg equations for the vacuum field operators ak( t), insert the resulting expressions into Eq. (9), and assume a continuous density of modes by replacing the sum over the vacuum modes by an integral over their density D( wk) (in free space equal to Vw$‘(c3$)), i.e. +CC r_:
gijkgnrlk
k
---f
J --cc
+oO
dmk D(~k)gijkgntlk
=
J --oo
dwk D(@k)&jPdt
(10)
where r)( Ok) = D( Ok) ( &k/h)2 which is eq.tal to oi/( ko&c3) in free space. We assume the temperature of the reservoir to be absolute zero and thus remove terms containing the operators a&O) and CZ~ (0) by taking their vacuum exp~tation value. This step reduces the number of con~~butions that govern the equation of motion of 171,,;2,,_1 [Eq. (9) ] to 32, in addition to the free evolution term i( OJ_f ~12)ZIl,~;2,~__1;a typical contribution has the form I
al T=-
dwkB(Wk) s
0
.Ic 0
d7
exp[iMt-
7~ll~~~~j~~~,~~~,~-~~7~ffi,~--1~2,~-~~~~.
(11)
m
In general, this equation is still too complicated to be handled analytically. As usual in dressed states computations, a major simplification can be achieved in the secular limit, in which the Rabi frequency of the driving field is much larger than all the decay constants of the system. In this case I7i,n;j,moscillates mainly because of its free evolution frequency Oij + (n - M)OL with ~021= -4~12 and 011 = 022 = 0, and we can neglect terms in its equation of motion whose oscillation frequency differs by at least the Rabi frequency of the driving field. If we adopt tilda superscripts to denote the slowiy varying operators, the term T in Eq. ( 11) can then be written in the form co T = -_Im112exp[i(wL
I exp[i(wk
+ wdtl
-
wL)(t
-
711
m
(12) The integrals over Ok and T contribute significantly only for Ok -+ WI. and 7 -+ t, and the sum reduces to its element m = n because I?r~,m;~,m_~(t)~~,n_-1;2,n_~(t)= Il,m)(l,m - l\(t) Il,n - 1)(2,?I - 1\(t) = (1, rn) (2, n - 11(t) S,,,, [ 191. For this assumption, usually refered to as the Weisskopf Wigner approximation,
C.H. Keitel et al. /Optics
148
Communications 118 (199-V 143~153
to be valid, the modal density needs to vary slowly on the scale of the respective dressed states transition linewid~, however not on the scale of the Rabi frequency of the driving field. The sample term in Rq. (12) is one of six out of the complete set of thirty-two terms that oscillate with the frequency WL f ~1.5 the other terms have frequencies that differ by at least 012, and can be neglected Furthermore, we can assume (nt,,~,,+r) = (flr,,+~:& in view of the strong driving field assumption, so that, finally, we obtain the following equation for the damping of the dressed coherence
( $IL2,n-I) =
ti(wL
&
=~[~(~~)(l~~~~2
+ ~12)
-
~51(J71,~;2,~-1),
+ llu2212--~WPI&~))
=?rll.tab1*[4c0s28Sin281S(~~)
(13)
~-~~(wL+w,~)ILL~,/~~~(wL
-l-cos48b(w~-l-~12)
Jrsin”&B(wr,-ut2)].
-wz)JP~~~~I
(14)
As already pointed out we only need to know the time dependence of the dressed coherence IJI,~;~,,+~ in order to calculate the linewid~ of the outer sideband in the secular approximation. Eq. (13) can be integrated at once for this purpose. The Fourier transform of the exponential function gives a Lorentzian line centered at the frequency WL -t- 0~12with L, as the corresponding half width at half maximum. In free space with M D(w,~) we obtain the well-known linewidth of a driven two-level %D(WL-t012)MD(WL--012) b(@L) system for the outer sideband, i.e. LS = ~~(~~~)1~~*~2( 1 -I- 2cos28sin28) [47. Thus, the results in Ref. 141, which assume a negligible frequency dependence of the modal density, can be generalized by multiplying the dressed states matrix elements by the density of modes at the corresponding dressed states resonance frequency rather than the atomic resonance frequency. In an environment with a strong frequency dependence such as a high-Q resonator the situation changes signific~tly in such a way that we cannot assume a negligible frequency dependence of the density of modes. Furthermore we should note, that the coupling constant of the atom to such a cavity must be rather small because we have assumed an infinite response time for the enviro~ent and this rules out dynamical effects due to the interaction with the resonator. Thus, effects such as vacuum Rabi splitting 1201 are not expected in this treatment. However, from studies of spontaneous emission in cavities [ 9- II], the possibility of suppressed and enhanced emission is well known and its influence on resonance fluorescence should not come as a surprise. The features of the resonance fluorescence in a cavity are much richer, however. For example we can envisage ch~ging signific~tly the fluorescence spectrum by slightly altering the frequency or the intensity of the coherent driving field rather than the cavity resonance. The calculation for the other sideband is completely analogous and leads to the identical linewidth. For the central com~nent of the spectrum around the laser driving frequency WL, we have an extra implication due to the fact that two dressed coherences IJl,n;t,n_t and D2,n;2,n-t contribute, because both oscillate at the same frequency OL. In the semiclassical limit of a strong driving field, such that (fl~,n;i,n-t) M (Qn+t;& M exp( iWt.t) p(n) (fli;,) with normalization for the photon number distribution C,p(n) = 1 and (ntii> being the semiclassical population of the two semiclassical dressed states 1 and 2, we have a significant number of cancellations of terms and find, in an analogous way to the calculation for the sideband:
with
C.H. Keitel et al. /Oprics
149
Communicarions 118 (1995) 143-153
where we have included the definition of ~11 and ~22 for later purposes. The differential equations ( 15) represent a coupled system of first order differential equations in the unknown variables (nl,nn;,,_l) and (172,n;2,n_l) which can be solved to give the central component of the dipole operator (fiC) that oscillates at the frequency WL: ,=11I l,n-
(G)=(C[
l)(l,nl+~22(2,+
1)(2,4
= Pll
= PII ~((17,..;,,n-,)
+h.c.l >
n
Y12 - Y2’ exp(iW) ( Yl2 + Y2l
+
Kexp
C-(792
+
+
exp(iwLt)
7v2l)t]
-
(nz,n;2,n-1))
+c.c.
” c.c.,
(17)
>
where the integration constant K can be fixed to match the initial condition of the atom. We note that the Fourier transform of the two summands of the above expression leads to two fundamentally different structures. The first term whose time dependence is assigned by exp( i@Lt) only yields a delta-function component at the frequency CIQ_which is usually known as coherent scattering, or Rayleigh peak. The linewidth of this peak, which, ideally, is vanishingly small in the case of negligible fluctuations of the driving laser field, is unaffected by the environment. The second contribution in Eq. (171, i.e. the one whose time dependence has a damped sinusoidal behaviour generates a spectral component centered at WL with half width at half maximum Lc given by L =
Y12 +
y21 = 2d
l/w212mwL
= 2rr(E*.ub\2[sin4BD(w L -
w12)
012) +
+
cos4e
Ip2,
12mwL
D(OL
+
+ Wl2)
w2)
1
I,
(18)
which again agrees with the result in Ref. [ 41 for D( OL) M D( OL + ~12) z D ( wL - ~12) M D ( uab). This spectral component together with the two sidebands has a non vanishing linewidth, and is usually refered to as the incoherent contribution to the spectrum. As a very interesting feature in the strong field regime we find that the linewidth of the central component depends only on the density of modes at the dressed states resonances corresponding to the two sidebands. After the discussion of the linewidths we now turn our attention to the influence of the modal density on the intensities of the three incoherent lines and the coherent scattering peak. For this purpose we consider the equations of motion of the dressed populations. These can be derived following the same procedure already described in connection with the dressed coherences, and their form is given explicitly by (&I:l,n)
= _(Yll
+ Y21) (~1,“;l.“)
+ Yll(~l,n+l;l,n+l)
+ Y12(~2,*+1;2,“+1)~
(fi2,n:2.n)
= -(Y22
+ Y12) (fl2.n;2.n)
+ Y22(172,n+1;2,n+l)
+ Y2l (~l,“+l;l,“+l)~
(19)
with yii given in Eq. ( 16). These equations can be interpreted as the description of the various spontaneous emission processes between dressed states: from 11, n + 1) to 11, n) at the rate 711, from 12, n + 1) to 12, n) at the rate ~22, from ) 1, n + 1) to 12,n) at the rate ~21 and from 12,n + 1) to 11, n) at the rate ~12. In the semiclassical limit the decay from (i, n + 1) to Ii, n) occurs at the same rate as the decay from Ii, n) to (i, n - 1) and thus it has no effect on (ZZi,n;i,n), as a consequence the dressed populations are only affected by the density of modes at the transition frequencies of the Rabi sidebands 3 . Thus, in the semiclassical limit (ni,n;;,n) M (ni,,+,;i,n+l) x p(n)(ni;i) with (Z7l;l) + (D23) = 1 we can easily calculate the steady state values of the dressed populations with the results (nl:l(00))
=
Y12 Y2l + Yl2’
(n2;2(0))
=
Y21 Y21 + Y12’
3 A similar argument applies for the coherences (Lli,n;i.n_l) which can be related to the dressed populations via the relationship exp(ioLt) (Ili.n;i,n) in the semiclassical limit. Thus, we also understand that the linewidths of the central component x of the spectrum are only dependent on the density of modes at the resonance transitions of the Rabi sidebands. tnr,n;i,n-I)
150
C.H. Keitel et al. /Optics
Commrrnications I18 (1995) 143-153
Following Cohen-T~noudji and Reynaud [4] (see also Ref. 1211) the number of photons emitted on the transitions WL and WL f ~12 are equal to the total number of transitions between the corresponding dressed states. Thus, the stationary weights of the peak corresponding to the transition from Ii, n + I) to Ij, n) are given by the product of the upper level population (L’i,n;i,n(co)), the transition rate yji and a proportionality factor r that depends on the measurement time (assumed large compared to the inverse of all decay rates of the system), the detector efficiency and the number of atoms. For the sideband ffuorescence weights W, therefore, we obtain
(21) We note that the intensities of both sidebands are always equal and depend on the product of the density of modes on the resonance transitions of both Rabi sidebands. As a consequence the fluorescence intensity of both Rabi sidebands may be quenched or greatly reduced just because the modal density is very small at one sideband frequency, even if it happens to be large at the other. It may be surprising, at first glance, to have a symmetrical spectrum when the modal density is ~y~e~c~. The dressed states decay rate does indeed increase on the transition with larger modal density (Eq. ( 16)) and therefore enhance the generated fluorescence intensity on this transition frequency, this, however, is just compensated for by a reduced upper dressed population on this transition (see Eqs. (20), (21)). For the intensity of the central component, which still contains both its coherent and incoherent contribution, we find WWL)
=~(~~;(~~;l(~))
+~22(J72;2(00)))
=wl
(22)
~~~~~~~~~~~~~~~~~~~~~~~~
coherent component of scattered intensity WC&,is obtained from the absolute value squared of the steady state dipole operator Eq. ( 17), which only contributes at the central component and appears as the summand with the undamped time dependence exp( iwtt) only. The Fourier transform of the correlation function corresponding to this term gives us as expected the S-function in frequency and the integral over frequency the corresponding intensity:
The
=
sin48b(q--12) 2rr7cos28 sin2Bb( ot_) 1~~~~1~ sin48B(w_-w2)
2
-cos~~~~(w~+w~~) +~0~~6Ls(0~+0,~)
>
(23)
*
In the limit in which the driving Rabi frequency is much larger than the detuning of the field from resonance, i.e. when 8 = 71.14,and if the modal density is weakly frequency dependent, we find from Eq. (23) that the coherent scattered intensity becomes virtually zero. From the above expressions, however, we also see that this is not the case in a tailored vacuum, where B( wt. - ~12) differs from b ( wt_+ ~12). This is a clear consequence of the influence of the environment and should be observable experimentally. In order to obtain the incoherent contribution of the scattered spectrum at the central frequency Winc(#~) we must substract the coherently scattered intensity from the total fluorescence at this frequency, by taking advantage of Eqs. (22) and (23). The result is
C.H. Keitel etal. /Optics
Conmunications
151
I18 (1995) 143-153
W
mu-3
fmw==Y
Fig. 3. Resonance fluorescence spectrum in the secular limit in {a) ordinary vacuum b W and (b) a tailored vacuum with B(ot_) = 2 B, and L)(wL i: 042) = 0.5 6, and (c) a tailored vacuum with b(wL) = 0.5& and bfo~ f 012) = 2 d, and (d) a tailored vacuum with Ij(wLf = 0.5 B, and @WI_ - wra) = 2 B MCand ii{ wt, + 012) = 2/ 13 6,. The modal density is indicated in a dashed line and is displayed in units of the modal density of vacuum b W. The frequency axis is relative to the laser frequency OL and scaled in terms of the decay rate of the two level system in ordinary vacuum y = 27rj~L,42bW(w,b), the measurement time was chosen such that r = loo-’ and the Rabi frequency of the driving field Q is assumed huge compared to the detuning S of the driving field to resonance, such that @= ?r/4. Y12Y21
wnc(@L)
= W(@L)
- Wcoh(@L)
=47'11 (Yl2 fY21)2
=
sin%cos%b(q
~~mw_)Ipab12
1sin48B(WL-c0i2)
- ~12) D(wL + wt2)
(24)
+cos~~D(w~+o~~)]~'
which agrees with the result presented in Ref. [4] when L)(wL) M D(WL t 0~12)M D(WL - 012) x D(w@~) as do all the previous weights and linewidths. Now we are in the position to write down the entire spectrum S(W) in the secular limit:
s(w)
=wcoh(@L)~(~-~L)
+ ~nc(6&,>
(w_
U7.r oL12 +L2 C
with all indicated parameters derived in previous equations and 6(0 - WL) reffering to the S-function centered around the driving laser frequency wt. In order to emphasize the influence of the modal density on the fluorescence spectrum we have displayed in Fig. 3a the traditional Mollow spectrum in vacuum with a virtually uniform density of modes, and compared it with situations where the environment causes a change in the modal density by a small factor over a frequency range of the order of the Rabi frequency of the driving laser field. Thus, we see in Fig. 3b that the central line can strongly dominate the sidebands in the case of an enhanced
C.H. Keitel et al. /Optics Communications 118 (1595) 143-153
152
modal density at the laser frequency or, viceversa, the sidebands can become more intense than the central line when the modal density of both sideband frequencies is increased with respect to that at the center of the spectrum (Fig. 3~). In the case of an asymmetric density of modes with respect to the laser frequency WL, as in Fig. 3d, we note that the spectrum remains symmetric and that the coherent scattering reappears even in the secular and resonant limit. We finally would like to point out various expressions which show most clearly the effect of the frequency dependence of the modal density and which seem most favorable to test experimentally. Those seem to be the ratios RL, Rw, and RH of the linewidths, weights and maximal heights of the central incoherent spectral component compared to those of the sideband peaks. In the simple case when the driving Rabi frequency .f2 is much larger than the detuning S of the driving field from resonance, i.e. when 19= 7r/4, we obtain
L
2
~(~L--l2)/2+B(~L+~,2)/2
26(0~)/3
RL = Z
= 3
R
Wnc(WL)
= w
W(WL
&z-=3 Rw RL
f
w12)
+ ~(WL - ~12)/6+ =2
_
D(WL + ~12)/6’
RWL)
D(wL-w~~)/~+~(wL+w,~)/~'
~CWL)@&~L)/~+&~L-
w12)/6+
(B(oL--w12)/2+~(~L+w12)/2)2
~(oL+
~12)/6)
(26c) '
where we have separated the free space from the possibly substantial modal correction factor. In conclusion we have presented a novel treatment of resonance fluorescence in the Born-Markov approximation by direct coupling of the reservoir to the dressed states. We have also stressed the sensitive influence of the density of modes with the possibility of suppression and enhancement of the fluorescence. For the particular example of the driven two-level system we have put forward explicit expressions for the linewidths and weights of the spectral components which show the rather drastic role of the environment and could give a clear comparison for an experimental observation. The support of the Office of Naval Research, The Welch foundation and the Texas Advanced Research Project, the European Community and the U.K. Engineering and Physical Sciences Research Council is gratefully acknowledged. One of us (CHK) would also like to acknowledge Dr. M. Fleischhauer, Dr. M.B. Plenio, and Dr. S.Y. Zhu for helpful discussions and in particular Dr. 0. Kocharovskaya for valuable suggestions in the initial phase of this work.
References [ I] V. Weisskopf and E. Wigner, Z. Phys. 63 ( 1930) 54; 65 ( 1931) 18. [2] B.R. Mollow, Phys. Rev. 188 (1969) 1969; A 2 (1970) 76; for the experimental observation of this effect see: W. Hartig and H. Walther, Appl. Phys. 1 ( 1973) 171; F. Schuda, CR. Stroud Jr. and M. Hercher, J. Phys. B 7 (1974) L198; F.Y. Wu. R.E. Grove and S. Ezekiel, Phys. Rev. Lett. 35 (1975) 1426; R.E. Grove, EY. Wu and S. Ezekiel, Phys. Rev. A 15 (1975) 227; H. Walther, Proc. Second Spectroscopy Conf., Megeve, France, 1974, eds. S. Haroche, J.C. Pebay-Peyroula, T.W. HBnsch and S.E. Harris (Springer, Berlin, 1975) p. 358. [3] S. Haroche, Ann. Phys. (Paris) 6 (1971) 189, 327. [4] C. Cohen-Tannoudji and S. Reynaud, Multiphoton Processes, Proc. Intern. Conf., University of Rochester, Rochester, June 1977, eds. J. Eberly and P. Lambropoulos (Wiley, New York, 1978) p. 103; for a more detailled study see also, C. Cohen-Tannoudji, Frontiers in Laser Spectroscopy, L-es Houches Summer School 1975, Session 27, eds. R. Balian, S. Haroche and S. Liberman (North-Holland, Amsterdam, 1976) pp. 5-104. [ 5 ] C. Cohen-Thtmoudji and S. Reynaud, J. Phys. B 10 (1977) 345, 365, 2311.
C. H. Keitel et al. /Optics 161 17) [ 81 [9 ] [ 101
Communications
118 (1995) 143-153
L.M. Narducci, M.O. Scully, G.-L. Oppo, P Ru and J.R. Tredicce, Phys. Rev. A 42 (1990) 1630. D.J. Gauthier, Y. Zhu and T.W. Mossberg, Phys. Rev. Lett. 66 ( 1991) 2460. C.H. Keitel, L.M. Narducci and M.O. Scully, Appl. Phys. B 60 (1995) 153. E.M. Purcell, Phys. Rev. 69 (1946) 681. D. Kleppner, Phys. Rev. Lett. 47 (1981) 232. ( I I 1 S. Haroche and D. Kleppner, Phys. Today 42 ( 1989) 24. [ 121 PL. Knight, Frontiers in Quantum Optics, eds. S. Sarkar and E.R. Pike (Adam Hilger, Bristol, 1986); B. Piraux, R. Bhatt and PL. Knight, Phys. Rev. A 41 (1990) 6296. [ 131 M. Lewenstein, T.W. Mossberg and R.J. Glauber, Phys. Rev. Lett. 59 (1987) 775; M. Lewenstein and T.W. Mossberg, Phys. Rev. A 37 ( 1988) 2048. I 141 W. Lange and H. Walther, Phys. Rev. A 48 (1993) 4551. [ 151 G.S. Agarwal, W. Lange and H. Walther, Phys. Rev. A 48 (1993) 4555; M. Elk and P. Lambropoulos, Phys. Rev. A 50 (1994) 1490. [ 161 H.J. Carmichael and D.F. Walls, Phys. Rev. A 9 (1974) 2686; J. Phys. A 6 (1973) 1552. [ 171 J.D. Cresser, J. Mod. Optics 39 (1992) 2187. I 18 1 0. Kocharovskaya, S.-Y. Zhu, M.O. Scully, P Mandel and Y.V. Radeonychev, Phys. Rev. A 49 ( 1994) 4928. [ 191 M. Sargent Ill, M.O. Scully and W.E. Lamb Jr., Laser Physics (Addison-Wesley, Reading, MA, 1974). 1201 M.G. Raizen, R.J. Thompson, R.J. Brecha, H.J. Kimble and H.J. Carmichael, Phys. Rev. Lett. 63 (1989) 240. 121 I P. Meystre and M. Sargent, Elements of Quantum Optics (Springer, Berlin, 1990) Sect. 15.3.
153
OPTICS COMMUNICATIONS ELSEVIER
Optics
Communications 118 (1995) 143-153
Full length article
Resonance fluorescence in a tailored vacuum C.H. Keitela, P.L. Knighta, L.M. Narduccibyl, M.O. Scullyb~* a Optics Section, Blacken Laboratory, Imperial College, London SW7 282 UK b Max-Planck Institut fiir Quantenoptik, Hans-Kopfermann Strasse 1, 85748 Garching bei Miinchen, Germany Received 17 January 1995
Abstract
We consider the resonance fluorescence of a two-level atom with an emphasis on the dependence of this phenomenon on the modal density of the environment. We use an approach of the Weisskopf-Wigner type to investigate the interaction of the dressed states of the atom-driving field system to the surrounding environment, and calculate the linewidth of the resonance fluorescence spectrum in terms of the decay rate of the dressed coherences. Unlike the conventional theory, we deal first with the exact interaction of the atom with the driving laser field, and then consider the coupling of this combined system to the modes of the environment within the Weisskopf-Wigner approximation. This modification of the traditional sequence in which the interactions are handled leads to a sensitive dependence of the fluorescence spectrum on the density of modes and suggests that a driven atom placed in a suitable environment, such as for example a high-Q resonator, even if Markovian, might behave quite differently from an identical system in free space. The significance of this effect is shown explicitly for the case of a strongly driven two level atom in an environment with a modal density which is substantially frequency dependent on the scale of the Rabi frequency of the driving laser field.
Spontaneous emission has been a well understood phenomenon since the pioneering work by Weisskopf and Wigner [ 1 ] who clarified its physical origin as the result of the interaction of an excited atom with the infinite number of electromagnetic modes of the vacuum. During the late 1960’s Mollow discovered that the fluorescence spectrum of a two-level atom may develop sidebands if the atomic transition is driven by a coherent single-mode field [ 21. The origin of this phenomenon can be traced to the energy level structure of the interacting atom-field system, whose eigenstates, the so-called dressed states as they are commonly known, are organized along an infinite ladder of doublets with a singlet ground state [ 31. Cohen-Tannoudji and Reynaud provided an especially appealing description of the physics of resonance fluorescence with an approach based on the dressed states formalism [4], and after appropriate generalization for the case of multi-level atoms discovered rich additional features also in the corresponding fluorescence spectra of driven three-level systems [ 51. The main virtue of their approach is that, in the limit of large driving Rabi frequencies, it provides the positions and widths of the various spectral components in analytic form, following consideration of the time dependence of the dressed coherences. More recently Narducci and collaborators found that three-level atoms 1Permanent address: Department of Physics and AtmosphericSciences, Drexel University, Philadelphia, PA 19104, USA. * Also: Department of Physics, Texas A & M University, College Station, TX 77843, USA and Texas Laser Laboratory, HARC, The Woodlands,
TX 77381, USA.
0030.4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00161-1
144
C.H. Keitel et al. /Optics
Communications 118 (1595) 143-153
in a V-configuration, when driven coherently on both allowed atomic transitions, can produce fluorescence spectra which are characterized by sub-natural linewidths [ 61, a prediction that was confirmed experimentally by Gauthier et al. [ 71. The underlying physical mechanism of the line narrowing can also be understood fairly readily with the help of the dressed states of the coupled atom-field system [6,8]. When one places an atom into a high Q-resonator or a photon band-gap material, for example, spontaneous emission can be suppressed or enhanced relative to the rate of decay in free space [ 9- 111. From a theoretical point of view this situation can be modeled by a two-level atom coupled to the nearest resonator mode (the other modes of the cavity can be assumed to be quite removed in frequency), which in turn interacts with the modes of the surrounding environment, as discussed, for example in Ref. [ 121. Dynamical suppression of spontaneous emission has been predicted also for a driven two-level atom in a cavity field as a result of the finite response time of the environment [ 131. This effect, which is connected with non-Markovian features, was tested experimentally quite recently [ 141 and interpreted theoretically as a consequence of the renormalization of the cavity-induced spontaneous emission by the strong driving field [ 151. Other decay studies have been published, which indicate that the traditional Born-Markov assumption of essentially unperturbed source evolution needs careful attention. Some approaches to this problem have involved a modification of the Scully-Lamb laser theory with the inclusion of atom-field correlation effects during the atomic relaxation process [ 16 3 ; others have raised the possibility of inconsistencies in the approach of the Jaynes-Cummings model to thermal equilibrium if the effects of atom-field interactions are ignored in the description of the field damping process [ 171. More recently a generalized set of Maxwell-Bloch equations has been proposed to describe semiclassically the limit of strong atom-field coupling [ 181; in common with the present quantum mechanical treatment, the analysis by Kocharovskaya and collaborators [ 181 applies the Born-Markov approximation and predicts a sensitive dependence of the relaxation terms in the master equation of driven atomic systems on the modal density of the surrounding environment. For the most part, until now, the common approach to the description of resonance fluorescence has involved consideration of the interaction between the atom and the coherent driving field separately from the process of atomic decay, the latter part of the problem being modeled by the conventional interaction of the bare two-level atom with a suitable reservoir. The few studies that have taken into account the atom-field coupling together with the atomic damping process, however, have investigated the parameter domain beyond the Born-Markov approximation and have stressed the importance of the response of the environment [ 13-161. In this paper we show that interesting effects arise already in the domain in which the Born-Markov approximation is justified, i.e. we carry out this approximation, and, as shown schematically in Fig. 1, we consider the coupling to infinitely many modes of the reservoir only after taking into account the dressing operation of the atom. The order in which these two interactions are handled in the calculation is not an obvious matter, and surely the traditional approach and the one developed in this paper are not equivalent because of the approximate nature of the irreversible part of the calculation, based as it is on the Weisskopf-Wigner approach. From a conceptual point of view it appears more correct to apply the Weisskopf-Wigner approximation to the degrees of freedom of the strongly interacting subsystems, in this case the atom and the driving field regarded as a single entity. In fact, as it turns out, some details of the resonance fluorescence spectrum are sensitive to the order in which the calculation is carried out. The difference between the final results of the traditional approach and the one adopted in this work is especially evident in the argument of the density of modes which enters as a multiplicative factor in all the spontaneous emission rates: in the traditional treatment the density of modes is evaluated at the atomic resonance frequency, while in our approach it must be calculated at the various dressed states resonances. In free space, this difference is not very substantial, given the slowly varying character of the vacuum density of states, but in a high-Q resonator it can be quite large, In order to emphasize this point, we introduce the notion of “tailored vacuum” to indicate that the magnitude of the vacuum fluctuations at a particular frequency can be modified with a suitable design of the surrounding environment. In this paper we construct the Heisenberg equations of motion for the operators that describe the interaction of the combined atom-driving field system with an infinite collection of reservoir modes, and derive decoupled
C.H. Keitel et al. /Optics Conventional
Communications
118 (1995) 143-153
145
treatment:
laser mode 6.1, -
Correct treatment:
/I-
reservoir
Fig. 1. Usually the damping rates which are added phenomenologically to the equation of motion of the density matrix elements are due to the interaction of the bare atom with the reservoir in spite of the presence of the additional driving field. Here we consider the coupling of the reservoir to the atom-driving field system and perform the usual Weisskopf-Wigner approximation after the interaction of the atom with the driving field has been taken into account. The box indicates which subsystems in each approach are considered first in a pre-diagonalization: atom-reservoir or atom-laser field.
_
Ib.n+l>
-
la,n-l>
Fig. 2. The left hand side displays the bare two-level atoms with the upper level a and the lower level b and n denotes the photon number of the driving field. The right hand side shows the corresponding dressed states.
equations for the dressed coherence operators. In the specific case of a driven two-level system, we perform the Weisskopf-Wagner approximation for the corresponding dressed states operators and obtain the linewidths and weights of the various spectral components as functions of the density of modes at the dressed states resonances. The deviation from the traditional Mallow spectrum is here put forward in a simple analytic form in terms of the modal density of the environment and graphically for several realistic examples. Finally parameters are pointed out which seem suitable to demonstrate this effect experimentally. The Hamiltonian HO describing the subsystem consisting of a two-level atom with upper level a and lower level b and the coherent driving field mode with frequency OL and the photon number n can be written in terms of the dressed states as
with
Il,n) =cosela,n)
+sinf?lb,n+
l),
12,n) = -sinO)a,n)
+cosB(b,n
+ l),
(2)
and
G.n=noL*iti12,
{+fori=l,-fori=2},
012 = (LP + S2)j,
tan(20) = O/s.
(3)
146
C.H. Keitel et al. /Optics
Communications 118 (1995) 143-153
Here 0 denotes the Rabi frequency of the driving field and 6 = tiOb - WL the detuning between the atomic transition and the driving field frequency. Furthermore, we have assumed the laser field to be intense and the corresponding photon number to obey Poissonian statistics. As a consequence the width of the photon number distribution is small compared to the mean number of photons, yet still large as compared to unity, so that we can neglect the n-dependence of the Rabi frequency over a wide range around the average [ 4,5]. Thus, the dressed states energy level diagram can be viewed as consisting of periodically repeating doublets in this range; in Fig. 2 we have displayed the bare states on the left and the corresponding dressed states on the right for one particular value of the driving field photon number. The atomic dipole operator ,&,which is the source of the radiated field and whose time dependence determines the shape of the spectrum, can be represented conveniently, if somewhat clumsily, as a superposition of dressed states projectors as follows
iiL=~~li,nj~i,nlPl~,m)(~,nl=~[~,1I1, n- l)(l,nl +p22p,nj,m +p21)2,n
l)(Znl
+p12(1,n-
1)(294
n
i,n -
l)(Lnll
(4)
+h.C.,
with PII =
-CL22
= COS 6 Sin 8 p&,
CL12 =
-sin28 &&b,
P21 = C0S28 ,U&,
(5)
and where i,j E { 1,2}, m,n E { 1,2, . ..} and &b = (bl,k]u) is the atomic dipole matrix element that couples the upper level a to the lower level b. The advantage of this representation, as shown more explicitly below, is that in the secular approximation, particular lines of the fluorescence spectrum can be associated with excellent approximation to particular dressed states projectors; thus, for each sideband only one projector is relevant (if we ignore the sum over the field photon number), while for the central component of the spectrum two projectors oscillate at this frequency and must be considered simultaneously. The spectrum is governed by the two-time correlation function of the raising and lowering contributions of the dipole operator. Following the regression theorem, this reduces to a sum of one-time dipole operator expectation values with complex two-time weighting factors which are identical to those that govern the evolution of the dipole operator itself. In the dressed states basis these summands give rise to well separated spectral peaks if the driving field is sufficiently large. As a consequence, the one-time weighting factors that distinguish the time evolution of the dipole correlation function from that of the dipole operator itself play a central role in assigning the intensities of the various peaks but virtually no role in setting the linewidths of the spectral components. Thus, for example, the summand (2,n - l)(l, II I in the dipole operator of Eq. (4) leads to a spectral peak at OL + ~12 and its damping rate determines the linewidth. Our interest in the following is the calculation of the relaxation rates of these dressed coherences. The complete Hamiltonian H which includes the interaction of the dipole operator with the vacuum modes, whose annihilation operators are denoted by {ak}, takes the form H=~o+Gl,,
(6)
with
n - l)(l,nl +g22kp,IZ - l)Q,nl
H WC n +g12k]l,n
k -
I)(294
+g21k[%n
-
l)(l,nll
+h.C.
+ xbk&k,
(7)
k
where the coupling COllStZiMS gijk ZiR defined as gijk = pij&k/h and 8k = ( fUl)k/kO)f denotes the so-called electric field per photon, V is the quantization volume and Wk is the frequency of the kth vacuum mode.
-7-
C.H. Keifel et al. /Optics Commu~catiu~
I18 ff995j
143-153
147
The time dependence of the dressed population and coherence operators l’Ii,n;j,m= Ii, n>(j, ml is governed by the Heisenberg equations of motion $fli.n:j,nt = i [ Hv fli,n;j,ml~
(8)
with analogous equations describing the evolution of the field operators u&(t). For the coherence operator K7~.n;2,,r-~(i),for example, we obtain $L,:*,.-1 + i
= i(wL -t
~2)n1,~;2,~-1
Ca:lgldh-b2,n-l
+ g2ikni,n-k2,n-1
-
g22kJ11,n;2,n
-
g2uXi,n~~,nl
k
In the next step we integrate the Heisenberg equations for the vacuum field operators ak( t), insert the resulting expressions into Eq. (9), and assume a continuous density of modes by replacing the sum over the vacuum modes by an integral over their density D( wk) (in free space equal to Vw$‘(c3$)), i.e. +CC r_:
gijkgnrlk
k
---f
J --cc
+oO
dmk D(~k)gijkgntlk
=
J --oo
dwk D(@k)&jPdt
(10)
where r)( Ok) = D( Ok) ( &k/h)2 which is eq.tal to oi/( ko&c3) in free space. We assume the temperature of the reservoir to be absolute zero and thus remove terms containing the operators a&O) and CZ~ (0) by taking their vacuum exp~tation value. This step reduces the number of con~~butions that govern the equation of motion of 171,,;2,,_1 [Eq. (9) ] to 32, in addition to the free evolution term i( OJ_f ~12)ZIl,~;2,~__1;a typical contribution has the form I
al T=-
dwkB(Wk) s
0
.Ic 0
d7
exp[iMt-
7~ll~~~~j~~~,~~~,~-~~7~ffi,~--1~2,~-~~~~.
(11)
m
In general, this equation is still too complicated to be handled analytically. As usual in dressed states computations, a major simplification can be achieved in the secular limit, in which the Rabi frequency of the driving field is much larger than all the decay constants of the system. In this case I7i,n;j,moscillates mainly because of its free evolution frequency Oij + (n - M)OL with ~021= -4~12 and 011 = 022 = 0, and we can neglect terms in its equation of motion whose oscillation frequency differs by at least the Rabi frequency of the driving field. If we adopt tilda superscripts to denote the slowiy varying operators, the term T in Eq. ( 11) can then be written in the form co T = -_Im112exp[i(wL
I exp[i(wk
+ wdtl
-
wL)(t
-
711
m
(12) The integrals over Ok and T contribute significantly only for Ok -+ WI. and 7 -+ t, and the sum reduces to its element m = n because I?r~,m;~,m_~(t)~~,n_-1;2,n_~(t)= Il,m)(l,m - l\(t) Il,n - 1)(2,?I - 1\(t) = (1, rn) (2, n - 11(t) S,,,, [ 191. For this assumption, usually refered to as the Weisskopf Wigner approximation,
C.H. Keitel et al. /Optics
148
Communications 118 (199-V 143~153
to be valid, the modal density needs to vary slowly on the scale of the respective dressed states transition linewid~, however not on the scale of the Rabi frequency of the driving field. The sample term in Rq. (12) is one of six out of the complete set of thirty-two terms that oscillate with the frequency WL f ~1.5 the other terms have frequencies that differ by at least 012, and can be neglected Furthermore, we can assume (nt,,~,,+r) = (flr,,+~:& in view of the strong driving field assumption, so that, finally, we obtain the following equation for the damping of the dressed coherence
( $IL2,n-I) =
ti(wL
&
=~[~(~~)(l~~~~2
+ ~12)
-
~51(J71,~;2,~-1),
+ llu2212--~WPI&~))
=?rll.tab1*[4c0s28Sin281S(~~)
(13)
~-~~(wL+w,~)ILL~,/~~~(wL
-l-cos48b(w~-l-~12)
Jrsin”&B(wr,-ut2)].
-wz)JP~~~~I
(14)
As already pointed out we only need to know the time dependence of the dressed coherence IJI,~;~,,+~ in order to calculate the linewid~ of the outer sideband in the secular approximation. Eq. (13) can be integrated at once for this purpose. The Fourier transform of the exponential function gives a Lorentzian line centered at the frequency WL -t- 0~12with L, as the corresponding half width at half maximum. In free space with M D(w,~) we obtain the well-known linewidth of a driven two-level %D(WL-t012)MD(WL--012) b(@L) system for the outer sideband, i.e. LS = ~~(~~~)1~~*~2( 1 -I- 2cos28sin28) [47. Thus, the results in Ref. 141, which assume a negligible frequency dependence of the modal density, can be generalized by multiplying the dressed states matrix elements by the density of modes at the corresponding dressed states resonance frequency rather than the atomic resonance frequency. In an environment with a strong frequency dependence such as a high-Q resonator the situation changes signific~tly in such a way that we cannot assume a negligible frequency dependence of the density of modes. Furthermore we should note, that the coupling constant of the atom to such a cavity must be rather small because we have assumed an infinite response time for the enviro~ent and this rules out dynamical effects due to the interaction with the resonator. Thus, effects such as vacuum Rabi splitting 1201 are not expected in this treatment. However, from studies of spontaneous emission in cavities [ 9- II], the possibility of suppressed and enhanced emission is well known and its influence on resonance fluorescence should not come as a surprise. The features of the resonance fluorescence in a cavity are much richer, however. For example we can envisage ch~ging signific~tly the fluorescence spectrum by slightly altering the frequency or the intensity of the coherent driving field rather than the cavity resonance. The calculation for the other sideband is completely analogous and leads to the identical linewidth. For the central com~nent of the spectrum around the laser driving frequency WL, we have an extra implication due to the fact that two dressed coherences IJl,n;t,n_t and D2,n;2,n-t contribute, because both oscillate at the same frequency OL. In the semiclassical limit of a strong driving field, such that (fl~,n;i,n-t) M (Qn+t;& M exp( iWt.t) p(n) (fli;,) with normalization for the photon number distribution C,p(n) = 1 and (ntii> being the semiclassical population of the two semiclassical dressed states 1 and 2, we have a significant number of cancellations of terms and find, in an analogous way to the calculation for the sideband:
with
C.H. Keitel et al. /Oprics
149
Communicarions 118 (1995) 143-153
where we have included the definition of ~11 and ~22 for later purposes. The differential equations ( 15) represent a coupled system of first order differential equations in the unknown variables (nl,nn;,,_l) and (172,n;2,n_l) which can be solved to give the central component of the dipole operator (fiC) that oscillates at the frequency WL: ,=11I l,n-
(G)=(C[
l)(l,nl+~22(2,+
1)(2,4
= Pll
= PII ~((17,..;,,n-,)
+h.c.l >
n
Y12 - Y2’ exp(iW) ( Yl2 + Y2l
+
Kexp
C-(792
+
+
exp(iwLt)
7v2l)t]
-
(nz,n;2,n-1))
+c.c.
” c.c.,
(17)
>
where the integration constant K can be fixed to match the initial condition of the atom. We note that the Fourier transform of the two summands of the above expression leads to two fundamentally different structures. The first term whose time dependence is assigned by exp( i@Lt) only yields a delta-function component at the frequency CIQ_which is usually known as coherent scattering, or Rayleigh peak. The linewidth of this peak, which, ideally, is vanishingly small in the case of negligible fluctuations of the driving laser field, is unaffected by the environment. The second contribution in Eq. (171, i.e. the one whose time dependence has a damped sinusoidal behaviour generates a spectral component centered at WL with half width at half maximum Lc given by L =
Y12 +
y21 = 2d
l/w212mwL
= 2rr(E*.ub\2[sin4BD(w L -
w12)
012) +
+
cos4e
Ip2,
12mwL
D(OL
+
+ Wl2)
w2)
1
I,
(18)
which again agrees with the result in Ref. [ 41 for D( OL) M D( OL + ~12) z D ( wL - ~12) M D ( uab). This spectral component together with the two sidebands has a non vanishing linewidth, and is usually refered to as the incoherent contribution to the spectrum. As a very interesting feature in the strong field regime we find that the linewidth of the central component depends only on the density of modes at the dressed states resonances corresponding to the two sidebands. After the discussion of the linewidths we now turn our attention to the influence of the modal density on the intensities of the three incoherent lines and the coherent scattering peak. For this purpose we consider the equations of motion of the dressed populations. These can be derived following the same procedure already described in connection with the dressed coherences, and their form is given explicitly by (&I:l,n)
= _(Yll
+ Y21) (~1,“;l.“)
+ Yll(~l,n+l;l,n+l)
+ Y12(~2,*+1;2,“+1)~
(fi2,n:2.n)
= -(Y22
+ Y12) (fl2.n;2.n)
+ Y22(172,n+1;2,n+l)
+ Y2l (~l,“+l;l,“+l)~
(19)
with yii given in Eq. ( 16). These equations can be interpreted as the description of the various spontaneous emission processes between dressed states: from 11, n + 1) to 11, n) at the rate 711, from 12, n + 1) to 12, n) at the rate ~22, from ) 1, n + 1) to 12,n) at the rate ~21 and from 12,n + 1) to 11, n) at the rate ~12. In the semiclassical limit the decay from (i, n + 1) to Ii, n) occurs at the same rate as the decay from Ii, n) to (i, n - 1) and thus it has no effect on (ZZi,n;i,n), as a consequence the dressed populations are only affected by the density of modes at the transition frequencies of the Rabi sidebands 3 . Thus, in the semiclassical limit (ni,n;;,n) M (ni,,+,;i,n+l) x p(n)(ni;i) with (Z7l;l) + (D23) = 1 we can easily calculate the steady state values of the dressed populations with the results (nl:l(00))
=
Y12 Y2l + Yl2’
(n2;2(0))
=
Y21 Y21 + Y12’
3 A similar argument applies for the coherences (Lli,n;i.n_l) which can be related to the dressed populations via the relationship exp(ioLt) (Ili.n;i,n) in the semiclassical limit. Thus, we also understand that the linewidths of the central component x of the spectrum are only dependent on the density of modes at the resonance transitions of the Rabi sidebands. tnr,n;i,n-I)
150
C.H. Keitel et al. /Optics
Commrrnications I18 (1995) 143-153
Following Cohen-T~noudji and Reynaud [4] (see also Ref. 1211) the number of photons emitted on the transitions WL and WL f ~12 are equal to the total number of transitions between the corresponding dressed states. Thus, the stationary weights of the peak corresponding to the transition from Ii, n + I) to Ij, n) are given by the product of the upper level population (L’i,n;i,n(co)), the transition rate yji and a proportionality factor r that depends on the measurement time (assumed large compared to the inverse of all decay rates of the system), the detector efficiency and the number of atoms. For the sideband ffuorescence weights W, therefore, we obtain
(21) We note that the intensities of both sidebands are always equal and depend on the product of the density of modes on the resonance transitions of both Rabi sidebands. As a consequence the fluorescence intensity of both Rabi sidebands may be quenched or greatly reduced just because the modal density is very small at one sideband frequency, even if it happens to be large at the other. It may be surprising, at first glance, to have a symmetrical spectrum when the modal density is ~y~e~c~. The dressed states decay rate does indeed increase on the transition with larger modal density (Eq. ( 16)) and therefore enhance the generated fluorescence intensity on this transition frequency, this, however, is just compensated for by a reduced upper dressed population on this transition (see Eqs. (20), (21)). For the intensity of the central component, which still contains both its coherent and incoherent contribution, we find WWL)
=~(~~;(~~;l(~))
+~22(J72;2(00)))
=wl
(22)
~~~~~~~~~~~~~~~~~~~~~~~~
coherent component of scattered intensity WC&,is obtained from the absolute value squared of the steady state dipole operator Eq. ( 17), which only contributes at the central component and appears as the summand with the undamped time dependence exp( iwtt) only. The Fourier transform of the correlation function corresponding to this term gives us as expected the S-function in frequency and the integral over frequency the corresponding intensity:
The
=
sin48b(q--12) 2rr7cos28 sin2Bb( ot_) 1~~~~1~ sin48B(w_-w2)
2
-cos~~~~(w~+w~~) +~0~~6Ls(0~+0,~)
>
(23)
*
In the limit in which the driving Rabi frequency is much larger than the detuning of the field from resonance, i.e. when 8 = 71.14,and if the modal density is weakly frequency dependent, we find from Eq. (23) that the coherent scattered intensity becomes virtually zero. From the above expressions, however, we also see that this is not the case in a tailored vacuum, where B( wt. - ~12) differs from b ( wt_+ ~12). This is a clear consequence of the influence of the environment and should be observable experimentally. In order to obtain the incoherent contribution of the scattered spectrum at the central frequency Winc(#~) we must substract the coherently scattered intensity from the total fluorescence at this frequency, by taking advantage of Eqs. (22) and (23). The result is
C.H. Keitel etal. /Optics
Conmunications
151
I18 (1995) 143-153
W
mu-3
fmw==Y
Fig. 3. Resonance fluorescence spectrum in the secular limit in {a) ordinary vacuum b W and (b) a tailored vacuum with B(ot_) = 2 B, and L)(wL i: 042) = 0.5 6, and (c) a tailored vacuum with b(wL) = 0.5& and bfo~ f 012) = 2 d, and (d) a tailored vacuum with Ij(wLf = 0.5 B, and @WI_ - wra) = 2 B MCand ii{ wt, + 012) = 2/ 13 6,. The modal density is indicated in a dashed line and is displayed in units of the modal density of vacuum b W. The frequency axis is relative to the laser frequency OL and scaled in terms of the decay rate of the two level system in ordinary vacuum y = 27rj~L,42bW(w,b), the measurement time was chosen such that r = loo-’ and the Rabi frequency of the driving field Q is assumed huge compared to the detuning S of the driving field to resonance, such that @= ?r/4. Y12Y21
wnc(@L)
= W(@L)
- Wcoh(@L)
=47'11 (Yl2 fY21)2
=
sin%cos%b(q
~~mw_)Ipab12
1sin48B(WL-c0i2)
- ~12) D(wL + wt2)
(24)
+cos~~D(w~+o~~)]~'
which agrees with the result presented in Ref. [4] when L)(wL) M D(WL t 0~12)M D(WL - 012) x D(w@~) as do all the previous weights and linewidths. Now we are in the position to write down the entire spectrum S(W) in the secular limit:
s(w)
=wcoh(@L)~(~-~L)
+ ~nc(6&,>
(w_
U7.r oL12 +L2 C
with all indicated parameters derived in previous equations and 6(0 - WL) reffering to the S-function centered around the driving laser frequency wt. In order to emphasize the influence of the modal density on the fluorescence spectrum we have displayed in Fig. 3a the traditional Mollow spectrum in vacuum with a virtually uniform density of modes, and compared it with situations where the environment causes a change in the modal density by a small factor over a frequency range of the order of the Rabi frequency of the driving laser field. Thus, we see in Fig. 3b that the central line can strongly dominate the sidebands in the case of an enhanced
C.H. Keitel et al. /Optics Communications 118 (1595) 143-153
152
modal density at the laser frequency or, viceversa, the sidebands can become more intense than the central line when the modal density of both sideband frequencies is increased with respect to that at the center of the spectrum (Fig. 3~). In the case of an asymmetric density of modes with respect to the laser frequency WL, as in Fig. 3d, we note that the spectrum remains symmetric and that the coherent scattering reappears even in the secular and resonant limit. We finally would like to point out various expressions which show most clearly the effect of the frequency dependence of the modal density and which seem most favorable to test experimentally. Those seem to be the ratios RL, Rw, and RH of the linewidths, weights and maximal heights of the central incoherent spectral component compared to those of the sideband peaks. In the simple case when the driving Rabi frequency .f2 is much larger than the detuning S of the driving field from resonance, i.e. when 19= 7r/4, we obtain
L
2
~(~L--l2)/2+B(~L+~,2)/2
26(0~)/3
RL = Z
= 3
R
Wnc(WL)
= w
W(WL
&z-=3 Rw RL
f
w12)
+ ~(WL - ~12)/6+ =2
_
D(WL + ~12)/6’
RWL)
D(wL-w~~)/~+~(wL+w,~)/~'
~CWL)@&~L)/~+&~L-
w12)/6+
(B(oL--w12)/2+~(~L+w12)/2)2
~(oL+
~12)/6)
(26c) '
where we have separated the free space from the possibly substantial modal correction factor. In conclusion we have presented a novel treatment of resonance fluorescence in the Born-Markov approximation by direct coupling of the reservoir to the dressed states. We have also stressed the sensitive influence of the density of modes with the possibility of suppression and enhancement of the fluorescence. For the particular example of the driven two-level system we have put forward explicit expressions for the linewidths and weights of the spectral components which show the rather drastic role of the environment and could give a clear comparison for an experimental observation. The support of the Office of Naval Research, The Welch foundation and the Texas Advanced Research Project, the European Community and the U.K. Engineering and Physical Sciences Research Council is gratefully acknowledged. One of us (CHK) would also like to acknowledge Dr. M. Fleischhauer, Dr. M.B. Plenio, and Dr. S.Y. Zhu for helpful discussions and in particular Dr. 0. Kocharovskaya for valuable suggestions in the initial phase of this work.
References [ I] V. Weisskopf and E. Wigner, Z. Phys. 63 ( 1930) 54; 65 ( 1931) 18. [2] B.R. Mollow, Phys. Rev. 188 (1969) 1969; A 2 (1970) 76; for the experimental observation of this effect see: W. Hartig and H. Walther, Appl. Phys. 1 ( 1973) 171; F. Schuda, CR. Stroud Jr. and M. Hercher, J. Phys. B 7 (1974) L198; F.Y. Wu. R.E. Grove and S. Ezekiel, Phys. Rev. Lett. 35 (1975) 1426; R.E. Grove, EY. Wu and S. Ezekiel, Phys. Rev. A 15 (1975) 227; H. Walther, Proc. Second Spectroscopy Conf., Megeve, France, 1974, eds. S. Haroche, J.C. Pebay-Peyroula, T.W. HBnsch and S.E. Harris (Springer, Berlin, 1975) p. 358. [3] S. Haroche, Ann. Phys. (Paris) 6 (1971) 189, 327. [4] C. Cohen-Tannoudji and S. Reynaud, Multiphoton Processes, Proc. Intern. Conf., University of Rochester, Rochester, June 1977, eds. J. Eberly and P. Lambropoulos (Wiley, New York, 1978) p. 103; for a more detailled study see also, C. Cohen-Tannoudji, Frontiers in Laser Spectroscopy, L-es Houches Summer School 1975, Session 27, eds. R. Balian, S. Haroche and S. Liberman (North-Holland, Amsterdam, 1976) pp. 5-104. [ 5 ] C. Cohen-Thtmoudji and S. Reynaud, J. Phys. B 10 (1977) 345, 365, 2311.
C. H. Keitel et al. /Optics 161 17) [ 81 [9 ] [ 101
Communications
118 (1995) 143-153
L.M. Narducci, M.O. Scully, G.-L. Oppo, P Ru and J.R. Tredicce, Phys. Rev. A 42 (1990) 1630. D.J. Gauthier, Y. Zhu and T.W. Mossberg, Phys. Rev. Lett. 66 ( 1991) 2460. C.H. Keitel, L.M. Narducci and M.O. Scully, Appl. Phys. B 60 (1995) 153. E.M. Purcell, Phys. Rev. 69 (1946) 681. D. Kleppner, Phys. Rev. Lett. 47 (1981) 232. ( I I 1 S. Haroche and D. Kleppner, Phys. Today 42 ( 1989) 24. [ 121 PL. Knight, Frontiers in Quantum Optics, eds. S. Sarkar and E.R. Pike (Adam Hilger, Bristol, 1986); B. Piraux, R. Bhatt and PL. Knight, Phys. Rev. A 41 (1990) 6296. [ 131 M. Lewenstein, T.W. Mossberg and R.J. Glauber, Phys. Rev. Lett. 59 (1987) 775; M. Lewenstein and T.W. Mossberg, Phys. Rev. A 37 ( 1988) 2048. I 141 W. Lange and H. Walther, Phys. Rev. A 48 (1993) 4551. [ 151 G.S. Agarwal, W. Lange and H. Walther, Phys. Rev. A 48 (1993) 4555; M. Elk and P. Lambropoulos, Phys. Rev. A 50 (1994) 1490. [ 161 H.J. Carmichael and D.F. Walls, Phys. Rev. A 9 (1974) 2686; J. Phys. A 6 (1973) 1552. [ 171 J.D. Cresser, J. Mod. Optics 39 (1992) 2187. I 18 1 0. Kocharovskaya, S.-Y. Zhu, M.O. Scully, P Mandel and Y.V. Radeonychev, Phys. Rev. A 49 ( 1994) 4928. [ 191 M. Sargent Ill, M.O. Scully and W.E. Lamb Jr., Laser Physics (Addison-Wesley, Reading, MA, 1974). 1201 M.G. Raizen, R.J. Thompson, R.J. Brecha, H.J. Kimble and H.J. Carmichael, Phys. Rev. Lett. 63 (1989) 240. 121 I P. Meystre and M. Sargent, Elements of Quantum Optics (Springer, Berlin, 1990) Sect. 15.3.
153