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IL NUOVO CIMENTO VOL. 19 D, N. 5 Maggio 1997 NOTE BREVI

Canonical dressing in the multimode Jaynes-Cummings model

C. F. LO

Department of Physics, The Chinese University of Hong Kong Shatin, New Territories, Hong Kong

(ricevuto il 20 Febbraio 1997; approvato il 27 Marzo 1997)

Summary. — The construction of a series of unitary decoupling transformations,

which diagonalizes the multimode generalization of the Jaynes-Cummings model and provides us with an extremely convenient basis to gain a deeper understanding of the dressing processes present in the matter-radiation interaction, is presented. PACS 42.50 – Quantum optic.

PACS 32.90 – Other topics in atomic properties and interactions of atoms and ions with photons.

Many interesting features related to the interaction between radiation and matter can be successfully described by a rather simplified but non-trivial model proposed by Jaynes and Cummings more than three decades ago [1], which idealizes the real situation by simply concentrating on the near-resonant linear coupling between a single two-level atomic system and a quantized radiation mode (ˇ 41):

H 4v0Sz1 va†a 1e(a†S21 aS1) ,

(1)

where the radiation mode of frequency v is described by the bosonic operators a and

a†, the two atomic levels separated by an energy difference v0are represented by the

spin-half operators Sz and S6, and the atom-field coupling strength is measured by the

parameter e. Despite its simplicity, the Jaynes-Cummings (JC) model is of great significance because recent technological advances have enabled us to experimentally realize this rather idealized model [2-5] and to verify some of the theoretical predictions. Stimulated by the success of the JC model, more and more people have paid special attention to extending and generalizing the model in order to explore new quantum effects [6]. One possible generalization is the multimode version of the JC model: H 4v0Sz1

!

k Kva † k Ka k K₁

!

k Kek K(a† k KS 21 aKkS1) , (2)

where all the radiation modes are degenerate. This generalized model allows us to investigate the multimode squeezing and the effect of intermode correlation on the atomic transitions. As in the original JC model, the exact eigenstates of the system of

C.F.LO

750

this generalized case can in principle be obtained by diagonalizing the secular matrix of the total Hamiltonian [1]. These eigenstates identify a new physical unit, known as a

dressed atom, representing the combined atom-field system in the sense that its

spectrum coincides with the energy spectrum of the generalized JC Hamiltonian and its stationary states, called dressed states, are precisely the eigenstates of the coupled system. This algebraic approach of the atomic dressing, however, does not seem to offer an incisive tool to catch the dressing meaning and its role. Accordingly, in this paper we would like to present an alternative approach based upon constructing an appropriate unitary transformation which accomplishes the exact canonical dressing of a two-level atomic system by the radiation field. This canonical approach has been previously applied to the original JC model, and has proved itself to be much more effective than the algebraic approach in elucidating the physical origin of the dressing processes [7]. Furthermore, the advantages of having an explicit unitary operator which performs the required dressing of the two-level atomic system are obvious since any quantum-mechanical computation can be performed by directly transforming the operators appearing in the problem at hand, without going through the wavefunctions, which are of a very complicated structure, especially in the multimode case.

To begin with, let us first consider the unitary transformation _{R 4} exp [i

!

j , kFjkaj†ak] where F†4 F. This operator R is the generalized rotation

operator which transforms the annihilation operators in the following way: R†_a k

K_{R 4}

!

Kq(eiF)Kk, qKaKq. One should note that the total number operator N 4

!

K_ka†

k

Ka

k

K is

invariant under the rotation transformation. Thus, we have

H A_fR† HR 4v0Sz1

!

k K vaK_k†a k K1

!

k K (j *K_ka k K†S₂1 j k Ka k KS₁) , (3)

where jKk4

!

KqeKq(eiF)Kq, kK. It is not difficult to observe that

!

K_kj *

k Kj k K4

!

k Ke2 k K; in

other words, the norm is preserved under the rotation transformation. Obviously, we can always find a suitable R such that j *K_{k4 j}

k K₄

k

!

q Ke2 q K and j q K4 0 for q K cKk. As a result, all radiation modes are decoupled and the Hamiltonian HA can be written as H A 4

!

q K_cK_k vaK_q†aK_{q1 v0}S_{z1 va}† k Ka k K1 j k K(a k K†S₂1 a k KS₁) . (4)

This Hamiltonian, in fact, corresponds to a two-level atomic system coupled to a single radiation mode only. In other words, with an appropriate choice of normal modes, the two-level atomic system coupled to a dispersionless photon bath behaves like a two-level atomic system coupled to a single radiation mode. To diagonalize the Hamiltonian HA, we can simply apply the dressing unitary transformation T4 exp [u( 4 N )21 /2_(a

k

K†S₂2 aK_kS

1) ], where N 4aKk †

aK_k1 S_z1 1 O2, sin u 4 22 jK_kk NOD,

cos u 4 (v2v0) OD and D4

k

(v 2v0)21 4 jK2_k N, so that

H f T21_HA T4

!

k K vaK_k†a k K1 (v 2 D) S_z. (5)

The eigenstates of the diagonal Hamiltonian H are simply the direct products of the number states of each radiation mode and the spin states, and the corresponding

CANONICAL DRESSING IN THE MULTIMODE JAYNES-CUMMINGS MODEL 751

eigenvalues can be easily found in the form

.

`

/

`

´

E_]nK_{q(, H}4 v

!

q K nKq1 1 2

m

v 2

k

(v 2v0) 2 1 4 jK2_k(n k K1 1 )

n

, E]nKq(, I4 v

!

q Knq K₂ 1 2

m

v 2

k

(v 2v0) 2 1 4 jKk 2 nK_k

n

. (6)

The spin-dependent part in eq. (5) can be interpreted as the dressed-atom Hamiltonian.

Next, let us briefly discuss how to find explicitly the appropriate operator R for the decoupling of all bosonic modes. Since the requirement of jKk being real implies

that iF must be real and antisymmetric, the operator R can be re-written as R 4 exp [

!

m

!

n DmLKkm, k K nJk K m, k K n], where L f iF and Jk K m, k K nf(a † k K mak K n2 a † k K nak K m). It is not difficult to show that the set of operators JKkm, k

K

n(with m c n) form a closed algebra, i.e. the set of commutation relations among these operators is closed: [JKkm, k

K n, Jk K r, k K s] 4 dKkn, k K rJk K m, k K s2 dk K m, k K sJk K r, k K n1 dk K m, k K rJk K s, k K n2 dk K n, k K sJk K m, k K

r. Therefore, the operator R can be expressed in the so-called disentangled form: R 4

»

m

»

n Dmexp [uKkm, k

K nJk K m, k K n], where the uKkm, k K

n’s are related to the Lk

K

m, k

K

n’s by the well-known disentanglement procedure [8]. It is then very straightforward to show that an appropriate choice of the desired operator R is simply

R 4exp [uKk1, k K 2Jk K 1, k K 2] exp [uk K 2, k K 3Jk K 2, k K 3] R exp [uk K j, k K j 11Jk K j, k K j 11] R , where tan uKkj, k K j 114

k

!

j l 41e 2 k K l

O

ek K

j 11, and that the transformation rules of ak

K m’s can be expressed as follows: R†aKk1R 4 eKk2 bKk2 aKk11 ek K 1

!

M 21 n 42 eKkn 11 bKknbk K n 11 aKkn1 eKk1 bKkM aKkM , (7) R†aKkmR 42 bKkm 21 bKkm aKkm 211 ek K m

!

M 21 n 4m eKkn 11 bKknbk K n 11 aKkn1 eKkm bKkM aKkM, (8) where 2 GmGM and bKkm4

k

!

m n 41e 2 k K

n. Here M is the total number of radiation modes.

In summary, the main result of this paper is the construction of a series of unitary decoupling transformations, which diagonalizes the multimode generalization of the JC Hamiltonian and provides us with an extremely convenient basis to gain a deeper understanding of the dressing processes present in the matter-radiation interaction. As pointed out above, a peculiar aspect of this canonical transformation approach is represented by the possibility to evaluate explicit expressions for the dressed operators in terms of the bare ones. For instance, the dressed version of the bare spin operator Sz

is given by SAzfT21R21SzRT 4cos uSz2 sin u( 4 N )21 /2(aK_k†S_{21 a}

k

KS₁). It is obvious

that this circumstance opens, in principle, the possibility to achieve a much more physically transparent interpretation of the dressed operators than in the algebraic approach.

C.F.LO

752

* * *

This work is partially supported by the Direct Grant for Research from the Research Grants Council of the Hong Kong Government.

R E F E R E N C E S

[1] JAYNES E. T. and CUMMINGS F. W., Proc. IEEE, 51 (1963) 89.

[2] GOY P., RAIMOND J. M., GROSS M. and HAROCHE S., Phys. Rev. Lett., 50 (1983) 1903. [3] GABRIELSE G. and DEHMELT H., Phys. Rev. Lett., 55 (1985) 67.

[4] MESCHEDED., WALTHER H. and MU¨LLER G., Phys. Rev. Lett., 54 (1985) 551.

[5] HAROCHES. and RAIMONDJ. M., Advances in Atomic and Molecular Physics, edited by D. R. BATESand B. BEDERSON, Vol. 20 (Academic, New York) 1985, p. 347.

[6] SHOREB. W. and KNIGHT P. L., J. Mod. Optics, 40 (1993) 1195.

[7] CARBONARO P., COMPAGNO G. and PERSICO F., Phys. Lett. A, 73 (1979) 97. [8] WEI J. and NORMAN E., J. Math. Phys., 4 (1963) 575.