Spectroscopic transitions in a two-level atom and supersymmetry

Volume 145, number 4

PHYSICS LETTERS A

9 April 1990

SPECTROSCOPIC TRANSITIONS IN A TWO-LEVEL ATOM AND SUPERSYMMETRY C.J. LEE’ Department ofChemistry, Princeton University, Princeton, NJ 08544-1009, USA Received 1 June 1989; revised manuscript received 13 February 1990; accepted forpublication 14 February 1990 Communicated by B. Fricke

The Jaynes—Cummings model for a two-level atom interacting an electromagnetic field is analyzed in terms of supersymmetry quantum mechanics. Supersymmetry arguments are utilized to calculate eigenvalues and eigenstates of the model and to discuss its correspondence to the classical model of Lorentz. Extension to arbitrary rn-photon processes is discussed.

The simplest nontrivial model of spectroscopic processes involves a two-level atom (TLA) interacting with a single electromagnetic field mode. In the dipole and the rotating wave approximations, this model is solvable exactly and is known as the Jaynes— Cummings model (JCM) [1]. Despite its oversimplified nature, JCM is a basic model for many optical phenomena and a number of interesting features of the model has been reported. In this Letter the dynamics of the atom—photon interaction will be examined from another perspective with JCM as a prototype example. The first observation we make is that TLA—photon is a manybody system of bosonic and fermionic quasi-partides. Naturally, then, it is useful to employ supersymmetry (SUSY) theory, which has emerged as a powerful tool for analysis of many quantum mechanical problems. As a demonstration of the usefulness of this approach we calculate eigenvalues and eigenfunctions of JCM directly using susy arguments. JCM is a quantum electrodynamics model, and as such it would be interesting to see what it corresponds to in the classical limit. We elucidate the correspondence also with the help of symmetry considerations. Finally, extension ofthis approach to mphoton processes will be discussed, Consider the two linear energy-conserving processes shown in fig. 1. The Hamiltonian correspondPresent address: Chemical Analysis Laboratory, Korea Institute of Science and Technology, Seoul, Korea.

1

1

(a)

(b)

Fig. I. In diagram (a) a photon is absorbed and the transition l—~2occurs. The reverse process is shown in diagram (b). All processes proceed from left to right.

ing to these processes is JCM, which may be written as (h = 1) 2

~ W~cfcJ+Wbtb+~K(bc~c, +b~c1c2) .

(1)

In the above, w1 and w are energies of the atomic level j> and photons (y-quanta), respectively, and K is the atom—field coupling parameter. c,t (c1) and bt (b) are creation (annihilation) operators for electrons in Ii> and the photons, respectively. The zero of the atomic energy is chosen such that w1 +w2=O, with the energy difference w0 (~ w1 > 0). Then w1 = ~w0and w2= ~w0,and the atomic states may be represented by —

—

2>

=

0375-9601/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)

(~)

I +>

I>

=

(~)

I —>

.

(2) 177

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PHYSICS LETTERS A

It is assumed that n+ 1 photons exist in the field associated with the lower state I 1>. Thus the basis set for JCM is { In+ 1, >, In, + >}~where In, ±> In> ® I ±>. All subsequent discussions will be confined within the space spanned by this basis set. The operators c and b satisfy the equal-time commutation and anticommutation rules —

9 April 1990

a property of angular momentum j= Therefore, the transition operators can be realized by 2 x 2 Pauli matrices, /0 ~ /0 1 ~ ~= (,~ -‘~=~ . (11) ~.

~)

{c,, c~}=511, {c1, ç}={c~,c~}=0,

After some algebra, eq. (1) reduces to tb+ ~ic(fth+btf) —~w H=w0ff+wb 0.

[b, bt] = 1,

Thus with the introduction of even symmetry gen-

[b

[b, b] = (bt, bt] =0,

[b c~]_~ [bt, c]

C] =

=

[bt, c~]=0

(3)

eratorsfandft eq. (1) is written entirely in terms of bilinear operators. Note, however, that now H contains generators of odd symmetry.

(4)

These odd symmetry generators transform a boson into a fermion, and vice versa. To be more explicit, consider operators Q and Qt as defined by

(5)

Qt=V~wfth, Q_—~Twb~f.

The number operators are given by ñ~=c~tc,, ñ=b~b,

and the transition operators [2] are defined by f=cIc2,

ft=c~c,.

(12)

(13)

These operators connect particles of the same spin, and hence are even symmetry generators in the sense of Haag, Lopuszañski, and Sohnius [3]. It is not difficult to show that the Hermitian operators t+ñ. f t,f1 (6)

Obviously the state I~B,flF>, where n~and n,~-are eigenvalues of n~=bb and nF=ftf respectively, transforms under Qt and Q as

satisfy the familiar angular momentum commutation rules:

Inordertoseehowthebasisset{In+l,

.f1=W

2~j(ft_J), R=~[f (1,m,n=l,2,3).

[fi,f,n]~(imnf,,

2

1=

CJCJ

=

1

transforms under

(8)

FtF

I

—

>~

~l) /

(1 =

\

)

(14)

—>, In, +>~

Q and Q note that

lo\

(7)

In general, solutions to eq. (7) have (21+ 1 )-dimensional representations with j=0, ~, 1 However, in a TLA no transitions are allowed if both atomic levels are either empty or filled, so only the subspace with ~

QInB,nF>xInB+l,nF—l> QtInH,nF>xlnB—l,nF+l>.

o\Io\

~o o)

~

/

7

~

1) 1~

) ~)

10 =

0 ~I 7i\ 1 (\)•

n~I + > =f~f(\, = (~ = (15) 0 0 0 0 So the original basis set { n + 1, —>‘ I n, + > } corresponds to {InB=n+ 1, nF=O>, In~=n,~F= 1 >} in the boson—fermion space. It follows that

1

is of our interest. Using eqs. (3), (5), and (8) one can easily verify the following commutation and anticommutations rules: {f,ft} 1, {f,J}{ft,f~}0, (9a) [b,f]

=

[b,ft]

=

[b~,f]

=

[bt,ftl =0.

(9b)

Thus fcan be regarded as a fermion operator. Moreover, as a consequence of eq. (9a) the operators in eq. (6) satisfy {J,f,n}=~c5t,,~

178

(1,m=l,2,or3),

(10)

QIn,+>xln+l,—>, Qt n + 1,

—

> x n, +> .

(16)

.

The statistics-changing generators of symmetry Q and Qt are generators of SUSY. A SUSY Hamiltonian is constructed from H ={Qt Q} SS

Q as (17)

‘

The Hamiltonian for the SUSY generator given by eq. (13) is then

Q (Q~)

Volume 145, number 4

~

PHYSICS LETTERS A

=w{f~b,btj} =

9 April 1990

total spin of a state by Therefore, the supercharge behaves as a spinorial tensor. This can further be made explicit by normalizing operators Q and Q~as ~.

w (b ~b+f VL

(18)

which is identical to eq. (12) when K—*0 and at resonance, w=w0. The SUSY structure may be made

q=

—~—,

q~

more explicit by rewriting eq. (18) as [4] H~.=w[ (btb+ ~ ) + (fff

Then q and qt are quantized according to Fermi— Dirac statistics: {q, qt}1J, {q, q}{qt, qt}0 . (27)

1)]

=W[(flB+~)+(flF—fl],

n,~=O,l

~,

(26)

=

(19)

or

In fact, it can be shown that in the space spanned by the basis set { n, +>, I n + 1, > } q~and q behave as the raising and the lowering operators 1+ = 1, ± i12 of a spin-i [7]. The third component 13 is easily obtamed as /3 = [1 1] = ~ (28) —

= ~w({b~, b} + [f~t])~ (20) which contains both the anticommutator (for the bosonic sector) and the commutator (for the fermionic sector). Also, it is easy to show that

[Q, Hss]

=

(21)

[Qf, H~~]0 .

Eqs. (17) and (21) constitute SUSY algebra. In the above SUSY quantum mechanical model there are two kinds of “excitations”: fermionic and bosonic excitations. The total excitation number operator [5] is the sum of each excitation number operator: (22)

N=nB+nF=btb+ftf,

.

We now calculate eigenvalues and eigenstates of JCM. In the interaction picture defined by the transformation U= exp ( ~iwt) exp (

—

iH~~t),

(29)

the Hamiltonian reduces to a simple form: H,=~w [1~ (qt+q) (30)

which differs from H~5only by a factor of w. Thus

where ~tw (~w0—w) is the detuning. One can immediately realize that H, can be diagonalized by the

[Ic’,H~~] =0 ,

“rotation”

(23)

and the eigenvalue of this conserved quantity is
I NI n~,flF>

=



==n+l, or N= n~+ ~F

(24)

R (0) = exp ( 2i0/2) , (31) where tan 20= (sw) ‘K(n+ 1)~2. Then the eigenvalue equation becomes —

~,I~>—weT~Ith>~oethIm>,

th=±~, (32) and the transformed quantity where We= [(L~w)is defined by A’= RARt. The eigenstates of H, are 2+K2(n+l)]~2

= fl +

I

.

(25)

The invariance of Q is guaranteed by eq. (21) and Q is called the supercharge. It is well known that in a two-dimensional isotropic oscillator the angular momentum operatordefinedbyL= —i(b1b

Ith÷> =R(O) In, +> =cos0ln, +>+sin0ln+l,

—>

2—b~b,)

serves as a charge [6]. By analogy, one is interested in relating the supercharge with angular momentum. For the isotropic two-dimensional harmonic oscillator the angular momentum is given in terms of two boson modes. However, the supercharge connects a boson mode with a fermion mode, and changes the

Ith_ > =R(Ofln+l, —> =cos0 n+1, —>—sin0 In, +>

.

(33)

Furthermore, because I-Is, is invariant under the supergauge transformation R(9), the eigenstates I th±> (th ±= ±fl must also be eigenstates of ~ and the 179

Volume 145, number 4

PHYSICS LETTERS A

eigenvalue of H~.is of course (n+ 1) w. Therefore. the eigenvalues of JCM are (n+ ~) w±~ The same eigenvalues and eigenstates can be found in the litcrature [1,8]. According to Witten [9], a SUSY quantum mechanical model in d= 1 can be constructed from the supercharges given by

9 April 1990

ilar to (bare) Einstein phonons. The weak boson-fermion interaction causes the transformation of one particle into another. In the classical model of Lorentz [11] the atom interacting with light is taken as a one-dimensional charged oscillator of the electron—ion pair: (p2+rn2w~x2)

.

Q±=

[p~idW(x)/dx]

(34)

~.

V

where p= satisfying ‘c

~— —

—

id/dx and ~ are Grassmann variables

1

‘c

~0

‘c

1~±’

‘

—

~ ‘

~

(35\ —~

.

When the potential is that of a harmonic oscillator (massm=l) FJ”(x)=~wx2,theSUSYharmonicoscillator term H~.results. However, no choice of W(x) yields supercharges which give the JCM Hamiltonian. In this model SUSY is broken explicitly by the non-vanishing perturbation I’ ( H— ~ such that [V

]0

(

— —

or

t)

(36’/

for some Q, expressing the fact that V is not invariant under all SUSY transformations. Putting this positively, however, eq. (36) implies that V should transform like a function of Q1. the generator of the invariance group of the SUSY harmonic oscillator. For strong interactions between elementary particles the dynamical form of the interactions is not known, and Gell-Mann [10] gave a similar argument to deduce the transformation properties of the SU (3) symmetry-breaking interactions. The dynamical form of V in JCM is given a priori and it is straightforward to express it in terms of Q,. Considering the supercharges as the “normal coordinates” of the SUSY harmonic oscillator, therefore, one may think of JCM as a model analogous to weakly excited (T—~0)lattice vibrations in a crystal. To a first approximation the energy levels are determined by the SUSY harmonic oscillator, consisting of free bosonic and fermionic oscillators. The coupling between the two types of oscillators can be regarded as a small anharmonicity term given in terms of the normal coordinates, and this 0B’ causes between the suflF> transitions and In perpartner states I 8 ±1, nF±1>. In an alternative picture, in the absence of interaction the model describes a system of freely moving quasi-partides. nF fermionic and n~bosonic excitations sim180

“alo~=

(37)

From eq. (12) we see that the simple substitution of classical variables x and p with the usual creation and annihilation operators of a (bosonic) harmonic oscillator would not give the correct quantum mechanical analog. Rather, it is given by the (onemode) /èrinioni’c oscillator H —W ~ 38) ~toni

—

—

2

with the Grassmann variables fandf ~.Note that the correspondence would not be obvious if the two original fermionic oscillators w~)c 1(/ 1. 2) were used. As a summary, we have shown that JCM may be thought of as a model describing either a system of weakly interacting Bose- and Fermi-type excitations, or vibrations in a weakly excited superoscillator, which in the classical limit corresponds to the Lorentz model of an atom interacting with light. The eigenvalue spectrum of JCM was obtained with very little calculations by utilizing SUSY arguments. The same eigenvalue spectrum can also be obtained by canonical dressing with a transformation similar to eq. (31). (However, because of the Bose—Fermi symmetry eq. (31) is not a usual canonical transformation.) The dressed-atom technique has been successfully employed to two-photon processes as we!! [12]. A common problem with the canonical transformation technique is that in general it is difficult to find a necessary transformation itself for the exact diagonalization of a Hamiltonian. The above SUSY considerations, however, provide a specific recipe for the eigenvalue problems of not just one- and twophoton Hamiltonians but any rn-photon Hamiltonian whenever the atom—photon interaction can be written in terms of the Q-type supercharges. The procedure is (1) construct pseudospin-~operators from the supercharges, and (2) utilize properties of angular momentum to diagonalize Hamiltonians. The complete details for the calculation of the exact

Volume 145, number 4

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eigenvalue spectra of rn-photon Hamiltonians with interaction of the type V=~(KBtf+,c’I’Bft) with B=B(b~, b~) (i, 1=1, 2, will be published elsewhere [13]. ...),

9 April 1990

[4] L.E. Gendenshtein and IV. Knve, Soy. Phys. Usp. 28 (1985) 645; J. Fuchs, Ann. Phys. (NY) 165 (1985) 285. [5] L. Allen and J.H. Eberly, Optical resonance and two-level atoms (Dover, New York, 1987).

161 A. Messiah, Quantum

This was supported by the National Science Foundation under grant no. CHE-87 19545. References [1] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51(1963) 89. [2] J.H. Eberly, in: Foundations of radiation theory and quantum electrodynamics, ed. A.O. Barut (Plenum, New York, 1980) pp. 23—35. [31R. Haag, J.T. Lopuszañski and M. Sohnius, NucI. Phys. B 88 (1975) 257.

mechanics, Vol. 1 (North-Holland, Amsterdam, 1961). [7IC.J.Lee,Chem.Phys.Lett. 155 (1989) 399. [8] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1973). [9lE.Witten,Nucl.Phys.B 188 (1981) 513. [10] M. Gell-Mann, Caltech Rep. CTSL-20 (1961); Phys. Rev. 125 (1962) 1067. [11] H.A. Lorentz, The theory ofelectrons, 2nd Ed. (Dover, New York, 1952). [l2]G. Compagno and F. Persico, in: Coherence and quantum optics V, eds. L. Mande! and E. Wolf (Plenum, New York, 1984) p. 1117. [13] C.J. Lee, Pseudospin-~technique for rn-photon processes in a two-level atom, Phys. Rev. A, to be published.

181