Gauge invariance and resonant transitions

Volume 114A, number 4

PHYSICS LETTERS

17 February 1986

G A U G E INVARIANCE A N D R E S O N A N T T R A N S I T I O N S Kuo-Ho Y A N G Physics Department, St Ambrose College, Davenport, 1A-52803, USA Received 24 June 1985; accepted in revised form 3 December 1985

It is shown that resonant transitions can be used to demonstrate experimentally the effects of the requirement of gauge invariance in quantum mechanics.

In his pioneering work on the measurement of the fine structure spectrum of the hydrogen atom, Lamb [1 ] also discovered that the two interactions A(O,t)'p and r. E(O,t) predicted different lineshapes of resonance curves in "the usual interpretation of probability amplitudes", and that his experimental data favored the r . E interaction. (In "the usual interpretation" the probability amplitudes are constructed from the wavefunction and the eigenfunctions of the "unperturbed" hamiltonian. It will be referred to as the conventional formulation, or CF, in this note.) Ever since then, a great deal of effort [ 2 - 8 ] has been devoted to the search for the underlying reasons for Lamb's observation. It is now understood [3-8] that the differences in the physical predictions betweenA .p and r °E are caused by the fact that the CF is not gauge invariant. Indeed, it has been shown [ 3 - 8 ] that there is no difference between these two interactions in the predictions of the gauge invariant formulation (GIF) that always derives the r ° E results in any arbitrary gauge under the dipole approximation of the fields, E(r, t) E(0, t) and B(r, t) ~ O. Because of this coincidence, a test between the CF and the GIF in the dipole approximation becomes a test of the choices of gauge in the CF. Past investigations [ 1,2,7-9] have firmly established the direct relationship between the gauge invariance or dependence of theoretical predictions and the lineshapes of resonance curves. All these works have shown that it is in the off-resonance portion of resonance curves that the effects of gauge invariance or dependence manifest themselves. But, so far no 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

example has been presented in which the theoretically predicted properties of resonant transitions are affected by the requirement of gauge invariance. It is our purpose here to mention one such example. We will follow closely the theory of three-level laser spectroscopy (e.g., refs. [ 10,11 ]), using the well-known Autler-Townes effect (e.g. ref. [10]) caused by atomic states strongly coupled in near resonance with an intense external radiation field. For simplicity, we shall use the one-particle system in our presentation. We consider a nonrelativistic, spinless electron of mass m and charge e in the presence of an internal atomic field - VV0(r), an intense radiation field E(r, t) and B(r, t) with angular frequency co, and a weak radiation field E'(r, t) and B'(r, t) with angular frequency co'. To eliminate the Doppler shifts, we assume either the atom is at rest or the two radiations are directed perpendicularly at the atomic beam. The wavelengths of both radiation fields are assumed to be long compared to the sizes of the bound states under investigation so that the dipole approximation is sufficient to describe the interaction of both fields with the particle. We then choose the potentialsA(r, t) and • (r, t) for the strong field E cos(cot), and A'(r, t) and ~'(r, t) for the weak field E'cos(co't), to be of the form:

A(r, t) = - ( t x c / ~ ) E sin(cot), • (r, t) = - ( 1 - a)r" E cos(cot),

(1)

A'(r, t) = -(a'c/co') E' sin(co't), ¢p'(r, t) = - ( 1 - ~')r. I=' cos(co't),

(2) 175

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where a and a t are two arbitrary real constants, the so-called gauge parameters. The Schr6dinger equation is now ihaq,/at = H ~O = ( H 0 + ] i V + h V t + fiV2)xP,

(3)

where H 0 = p 2 / 2 m + e V 0 is the usual "unperturbed" hamiltonian,

h V 2 = (e2/2mc2)(A + A')2 ,

(4)

V = ecb/h - e A . p / m c h - ½(V+e it°t + V e - i t ° t ) ,

(5)

V+_ = - ( e E / h ) . [r - a(r + p / i m w ) ] ,

(6)

and similarly for the first-order operators V' and V" for the weak field. We will use {heal} and {41(r)} to denote the eigenvalues and the orthonormal and complete set of eigenfunctions of the "unperturbed" hamiltonian H 0. Since both vector potentials are position independent, they do not cause transitions between any two different states o f H 0. Hence, the term ~tV2 in eq. (5) can be eliminated by the definition of the expansion coefficients (e.g. refs. [8,12]) t

cl(t)=(4il~(t)}exp(ioait

+if

v2(t')dt' ) .

(7)

17 F e b r u a r y 1 9 8 6

idcl/d t = l V . c 2 eiat ,

(11)

where A = co - oo21 , A' = w' -- w32 , V = (~21V-141), V ' = (431V'-142), and * denotes the complex conjugate. We now solve eqs. ( 9 ) - ( 1 1 ) in the following manner, after Geltman [13] (see also ref. [10] in the calculations of density matrix elements). We first drop the last term of eq. (10) and then use this approximated equation in conjunction with eq. (11) to solve for Cl(t ) and c2(t ) satisfying the initial conditions Cl(0) = 1 and c2(0 ) = 0. This approximation is justified since states 1 and 2 are coupled to the strong field and V' represents the weak-field coupling between states 2 and 3. Then, we use the solution for c2(t ) obtained in this manner to solve for c3(t ) perturbatively from eq. (9). If we do this, we first get c2(t ) = --(V/21a)e-iZxt/2(e iut/2 -- e-iut/2) ,

(12)

e l ( t ) = e iAt/2 ( [(/a - A)/2/.t] eiu t/2 + [(ju + A)/2/a] e - i u t t 2 ) ,

(13)

where the Rabi flopping frequency is /~(a) = [A2 + IV(~)1211/2 ,

These coefficients then satisfy the equation of motion V(c0 = [1 - ~A/6o]<42t-er" E//tl41> •

idci/dt = ~ (4 i IV(t) + vt(t)14k)Ck exp(iwikt) , k

(8)

where colk = w I - w k. In the following, we present two cases where resonant transitions can be used to demonstrate the effects of gauge invariance b y investigating the gauge dependent results of the CF. First case. We now assume that there exist three states, 1,2 and 3, o f H 0 such that states 1 and 2 are in near resonance with the strong field, and states 2 and 3 are in near resonance with the weak field. Each state is assumed to have a definite parity. The initial conditions are Cl(0 ) = 1 and cl(O) = 0 for] q= 1. For simplicity, we assume that w 1 < 602 < 603 . If we are not interested in the effects of all non-resonant terms, then the equations of motion for the amplitudes of these states can be approximated using the rotatingwave approximation (RWA): idc 3/dt = ~-V tc 2 e - iA't,

idc2/d t = :1VCle-iAt + ~1 ,v, , * v3~oia't , 176

(9) (10)

(14)

If we use eq. (12) into eq. (9), the perturbative solution for c3(t ) is c3(t ) = (iVV'/4/~) [exp(:lS+t) 1. • ~1 6 +t)/ :8 1 + sm( 1.

- exp(:,f_t) sin({6_t)/~8_],

where 8± =

w32 -

t 60 -

(15)

1 1 ~ A +- ~/.t.

The above solution suggests that the weak field will "see" two ac Stark-shifted resonance frequencies (e.g. ref. [10l), ~±(o 0 = 0932 - ~-A -+ ½(A 2 + [V(a)12) 1/2 .

(16)

Furthermore, the rates of resonant transitions at these two frequencies are R ~ (or, ott) = 27r ] V(a)/2p.(cO 12 I~ V t(ot') 128 (w' - ~± ) , where

(17)

V'(ct') = [1 - a'At/w'](431--er " E'/h 142).

(18)

Here, for convenience of our discussions we have dis-

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played the dependence of all important quantities on the gauge parameters a and a'. There are several experimentally measurable quantities that can be used to check the effects of the choices of gauge in the CF. The first quantities are the ac Stark-shifted frequencies ~2±(a) "seen" by the weak field, as is clear from eqs. (14) and (16). There is, however, one drawback here. It is that measurement of these frequencies cannot tell the difference between the A . p (a = 1) and the r . E (ct = 0) interactions at the exact resonance frequency 60 = w21. The second quantities are the resonant transition rates R± at these two frequencies induced by the weak field. Since, by design, the field coupling states 1 and 2 is a strong one, ~2± can never reduce to 6032" Hence, the rates of resonant transitions at ~2± can be used to distinguish the A . p (a' = 1) from the r . E (a' = 0) interaction. To see this point more clearly, we investigate the ratios: IV'(~' = 0,t~' = ~2+)/V'(~' = 0,60' = ~ 2 ) 1 2 = 1 , (19)

IV'(d = I, 60' = 5 2 + ) / v ' ( ~ ' = I~

/~2+12 ,

= 1,60' = ~2_)1 2

(20)

which are valid for an arbitrary detuning A of the strong field. Eq. (19) tells us that the r - E interaction in the CF predicts the same rates (or peaks) of resonant transitions at these two ac Stark-shifted frequencies. On the other hand, eq. (20) indicates that the A . p interaction in the CF predicts that the rate (or peak) of resonant transition at the lower frequency ~2_ is larger than that at the higher frequency ~2+. At this moment, we recall that the GIF predicts results which are completely independent of the gauge parameters a and a' and which happen to coincide with the above r . E results of the CF for both fields (a = a ' = 0).

Second case. We now consider a second arrangement for which we will present a numerical study. We assume that states 1 and 2 are coupled to the strong field, but the weak field now couples states 1 and 3. The initial conditions are still the same as previously described. The energy levels are assumed to be 603 < 601 < 602. Again, Cl(t ) and c2(t ) are solved as before, and c3(t ) is now solved perturbatively from the equation: idc3/d t = ~V t , el eiA't ,

17 February 1986

V ' = (q~31g+l¢l) = [1 - a'A'/60'] (¢3 I - e r . E '/~ I¢1>-

(22) If we use eq. (13) into (21) and solve for c 3 perturbatively, then 1.

c3(t ) = --~lV

t

X ([(~ -- ~)/2U] exp(½i6"t) sln(~ " 1 6_t)/~ , ~ 6_, +

[0z + A)/2U] e x p ( ~ 11.6 + t' ) sm(~6+t)]~6+), • 1 ' 1 ,

(23)

where 6+ = 60' - ~2'÷ and ~2~(ot) = 6013 - 1A + 1 [A2 + IV(or)12 ] 1/2.

(24)

The rates of resonant transitions are therefore R'±(a, tx') = 21r I~-V'l 2 I ~ -+ A)/2~12 6(60 ' -- ~2'±) ---N;(a, a ' ) 6(60' - [2'±).

(25)

At the exact resonance frequency for the strong field, A = 0, the various values of the amplitudesN~ have the same relations as in eqs. (19) and (20): U+(o~' = 0, 60' = [2+, A = 0)/

N ' ( a ' = 0, 60' = [2'_, A = 0) = 1, N ;t( a t = 1,

60t

(26)

t = ~2+, A = 0)1

N'_(a'=1,60 ' = a ' _ , ~ = 0) = 1~2_/~2+1a=0. , , 2

(27)

Here, the gauge parameter ~ for the strong field does not enter the above results because of the condition A = 0. Since eqs. (26) and (27) are similar to eqs. (19) and (20), all of our previous discussions apply here. Numerical study. One major difficulty in the design of an experiment to test the validity of eq. (26) (or (19)) against that of eq. (27) (or (20)) is to find a system such that the field strength is strong enough so that the ratio in eq. (27) (or (20)) is appreciably different from unity while, at the same time, the entire physical process is not dominated by multiphoton processes * 1. So far at least, we have found one such system. It consists of the 2 2S1/2, 2 2P3/2 and 2 2P1/2 states of the hydrogen atom, which are labelled states 1,2 and 3 in that order for the second case considered. (The presence of spin does not affect our results here.) The two energy differences are: 6021/2rr ~ 9880 MHz

(21)

where the RWA has been used, A' = 60' -- 6013, and

*1 The referee's suggestion to look into this point is gratefully acknowledged. 177

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(0.3297 cm - 1 ) and ~o13/2~r ~ 1060 MHz (0.0354 c m - 1 , the Lamb shift). The nearest energy level (n = 3) is about 4.57 X 108 MHz (15233 cm - 1 ) away, which eliminates the worry of multiphoton processes. For a linearly polarized strong field, IV(A = 0)1/27r = E X (x/'6aolel/h) = E × [3.13 MHz/(V/cm)],

(28)

where E is the amplitude of the strong field and a 0 is the Bohr radius. If we choose E = 32 V/cm, then I V(A = 0)I/2n ~ 100 MHz (which is the natural decay width (FWHM) of the 2P-states). The corresponding beam intensity is 1.36 W/cm 2, and the ratio in eq. (27) is then 1~2' /~2+[2__0 ~ (1010/1110) 2 = 0.83. In conclusion, we stress that the GIF should always be used to calculate transition probabilities and other physical quantities, since any observable, i.e., any physically measurable quantity, must be independent of gauge. Nevertheless, it is instructive to investigate the gauge dependent predictions o f the CF and to see how these predictions vary with gauge parameters. As this note and previous investigations have shown clearly, the requirement of gauge invariance affects not only the calculated off-resonance results (lineshapes) of resonance curves but also the calculated rates (or peaks) of resonant transitions.

178

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The author is very grateful to Professor T. Fulton, Professor W.C. Stwalley and Professor D.H. Kobe for many helpful discussions and suggestions.

References [1] W.E. Lamb Jr., Phys. Rev. 85 (1952) 259. [2] E.A. Power and S. Zienau, Philos. Trans. R. Soc. A251 (1959) 427. [3] K.-H. Yang, Ann. Phys. (NY) 101 (1976) 62; J. Phys. A15 (1982) 437. [4] D.H. Kobe and A.L. Smirl, Am. J. Phys. 46 (1978) 624. [5 ] D. Park, Classical dynamics and its quantum analogues (Springer, Berlin, 1979) pp. 146-150. [6] C. Leubner and P. Zoller, J. Phys. B13 (1980) 3613. [7] D. Lee and A.C. Albrecht, J. Chem. Phys. 78 (1983) 3382. [8] R.R. Schlicher, W. Becket, J. Bergou and M.O. Scully, in: Quantum electrodynamics and quantum optics, ed. A.O. Barut (Plenum, New York, 1984) p. 405. [9] D.H. Kobe, Phys. Rev. Lett. 40 (1978) 538. [ 10] S. Stenholm, Foundations of laser spectroscopy (Wiley, New York, 1984) ch. 4. [11 ] V.S. Letokhov and V.P. Chebatayev, Springer series in optical sciences, Vol. 4. Nonlinear laser spectroscopy (Springer, Berlin, 1977). [12] H.M. Worlock, in: Laser handbook, Vol. 2, eds. F.T. Arecchi and E.O. Schulz-DuBois (North-Holland, Amsterdam, 1972) p. 1323. [13] S. Geltman, J. Phys. B13 (1980) 115.