Macroscopic Quantum Superposition States and Wigner Functions on the Sphere

PHYSICAL REVIEW A VOLUME 60, NUMBER 3 SEPTEMBER 1999

Finite Kerr medium: Macroscopic quantum superposition states and Wigner functions on the sphere

Sergey M. Chumakov,1 Alejandro Frank,2 and Kurt Bernardo Wolf3 1Departamento de Fı´sica, Universidad de Guadalajara, Corregidora 500, 44420 Guadalajara, Jalisco, Mexico 2Instituto de Ciencias Nucleares and Centro de Ciencias Fı´sicas, Universidad Nacional AutońomadeMe´xico, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, Mexico 3Centro de Ciencias Fı´sicas Universidad Nacional AutońomadeMe´xico, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, Mexico ͑Received 30 September 1997; revised manuscript received 20 November 1998͒ We propose a spin model which exhibits the main properties of the Kerr medium. The description of the model uses a recently generalized quasiprobability distribution ͑Wigner function͒ for the SU͑2͒ group. This function is naturally defined on the sphere, which plays the role of phase space for the spin system. Our model leads to macroscopic quantum superposition states on the sphere. ͓S1050-2947͑99͒06908-5͔ PACS number͑s͒: 03.65.Bz, 42.50.Fx

I. INTRODUCTION ͑ ͒ ͓ ͔ ϱ distribution on the SU 2 group 6 . In the limit l˜ , when the su͑2͒ algebra of spin operators contracts to the To stress the intriguing implications of the quantum su- Heisenberg-Weyl algebra of boson operators, the sphere perposition of states which are macroscopically different, opens to the phase plane and the model coincides with the Schro¨dinger ͓1͔ proposed quantum states with the following quantum harmonic oscillator or, through a renormalization, structure: with the common Kerr medium. After a brief review of the Kerr dynamics in Sec. II we ␣͉living cat͘ϩ␤͉dead cat͘, ͑1͒ formulate our model in Sec. III. Section IV describes the Wigner function on the SU͑2͒ group; we discuss the evolu- where the two components are superposed with amplitudes ␣ tion of the spin-coherent states into cat states in Sec. V. and ␤. These have become known as Schro¨dinger cat states. Possible physical applications are briefly discussed in the It is certainly difficult to obtain and characterize states ͑1͒ in concluding section ͑VI͒. Schro¨dinger’s original setting, so Yurke and Stoler ͓2͔ proposed instead to take a single mode of a radiation field, and II. KERR MEDIUM superpose its coherent states with macroscopically different parameters. Similar ideas were formulated earlier by The Kerr Hamiltonian ͑2͒ characterizes certain general Bialynicka-Birula ͓3͔, and we also point out the odd and properties of quantum dynamics. It can be used to generate even coherent states discussed by Dodonov, Malkin, and both quadrature ͓7͔ and amplitude squeezing ͓8͔ for short Man’ko in Ref. ͓4͔. times, when the initial wave packet spreads in phase and It was shown in Ref. ͓2͔ that a superposition of coherent revolves around the origin in the phase plane ͓9͔. When the states with macroscopically different phases can be gener- phase spread exceeds 2␲, the front of the wave packet inter- ated in the course of the evolution of a single-field mode in a faces with its tail. This self-interference is a quantum feature Kerr medium with the Hamiltonian ͓5͔ and has no classical counterpart ͓10͔. At some time instants, the self-interference leads to standing waves which corre- ϭ␻ ϩ␹ 2 ͑ ͒ HKerr n n , 2 spond to Schro¨dinger cat states. A similar phenomenon also appears in other examples of quantum nonlinear evolution, where បϭ1. Here the operators a, a†, and nϭa†a describe a such as the Jaynes-Cummings model ͓11͔, where it is called single-field mode of frequency ␻, and ␹ is the Kerr constant. fractional revivals; the Dicke model ͓12͔; Rydberg wave While the photon number distribution is clearly unchanged packets ͓13͔; and particles in one-dimensional anharmonic by Hamiltonian ͑2͒, it still leads to a nontrivial phase evolu- potentials ͓14,15͔. tion. It is convenient to ignore the fast rotations of the phase In this paper, we propose a spin model which exhibits the plane generated by the linear term ␻n in Eq. ͑2͒, and to main properties of the Kerr medium. The model is formu- introduce the time scale ␶ϭ␹t. The nonlinear part n2 of ϭ ͑ ͒ lated in terms of spin operators Si , i 1, 2, and 3, acting on Hamiltonian 2 has an integer spectrum, and which is a the (2lϩ1)-dimensional Hilbert space of a particle of spin l. specific feature of the Kerr medium. It is distinct from the The natural phase space is a sphere of radius l; the spin- Jaynes-Cummings and Dicke models, where the effective- coherent states are thus represented by spots on the sphere. field Hamiltonian ͑for special initial conditions ͓16͔͒ is writ- Here our model leads to Schro¨dinger cat states on the sphere, ten as Hϳͱnϩ1/2 ͓17͔. The latter will also lead to phase i.e., superpositions of several spots located ‘‘far’’ from each spread and self-interference, but the standing waves are not other. The description of the model and its dynamics will be so well pronounced as for the Kerr medium. Due to the in- made using a recently generalized Wigner quasiprobability teger spectrum of Hamiltonian ͑2͒, it is easy to prove that at

␹ ϭ␲ ͑ ͒ϭ i␻t͑2lϩ1͒/2l Ϫi␻͑t/l͒N ϭ ͒ † ͒ϭ times t K/M where K and M are mutually prime inte- Sϩ͑t e e Sϩ ͓SϪ͑t ͔ , S3͑t const, gers͒ the wave function is a superposition of M copies of the ͓ ͔ initial state, placed equidistant along the circle ¯nϭconst ͓18͔ in complete analogy with the Kerr medium 7,10 .Wecan ͑see also Ref. ͓19͔͒. When Mϭ1, the initial state is repro- write this solution in the forms duced up to a global phase factor. ␻ S ͑t͒ϭei t/2l͓cos͑⍀t͒S Ϫsin͑⍀t͒S ͔, To describe a state ⌿ of a quantum system, it is clearest 1 1 2 ͑7͒ to draw the picture of the corresponding Wigner quasiprob- ␻ S ͑t͒ϭei t/2l͓sin͑⍀t͒S ϩcos͑⍀t͒S ͔, ability distribution W(⌿͉p,q), where p and q are classical 2 1 2 ͓ ͔ coordinates of phase space 20 . Generalized coherent states where ⍀ϭ␻(1ϪN/l) is an angular frequency which de- ͓ ͔ have Gaussian Wigner functions that are positive 21 . pends on the excitation number operator N. The time evolu- Schro¨dinger cat states cannot be produced from a single tion of Sជ (t) is a rotation around the z axis, but the precession Gaussian by linear dynamics. Quantum nonlinear dynamics frequency depends on the latitude on the sphere. introduces oscillations into the Wigner functions on phase ϱ ͑ ͒ space ͓9͔, with regions where the Wigner function takes In the limit l˜ , Hamiltonian 6 becomes the usual negative values. For a two-component cat state, these oscil- quantum harmonic-oscillator Hamiltonian, i.e., the nonlinearity disappears. We can also modify Eq. ͑6͒ to obtain the lations are localized near the midpoint between the centers of ␻ the Gaussians ͑i.e., the ‘‘smile of the cat’’ ͓22͔͒. optical Kerr Hamiltonian in this limit, replacing /2l by a free parameter ␹. Indeed,

III. ANALOG OF THE KERR HAMILTONIAN FOR SPIN ϭ␻͑ ϩ 1 ͒Ϫ␹ 2 ͑ ͒ H N 2 N 8 SYSTEMS is the common Kerr Hamiltonian ͑2͒ as l ϱ. Note, that in It is well known that the Heisenberg-Weyl algebra can be ͓ ͔ ˜ obtained by contraction from the su͑2͒ algebra. For this pur- molecular physics 23 , this model corresponds to a diatomic pose we consider the su͑2͒ commutation relations molecule approximated by a Morse potential. It was recently shown ͓24͔ that the effective Hamiltonian of type ͑8͒ de- ϭϮ ϭ ϭ Ϯ scribes the interaction of the collective atomic system with ͓S3 ,SϮ͔ SϮ , ͓Sϩ ,SϪ͔ 2S3 , SϮ S1 iS2 , the off-resonant quantum radiation field in a dispersive cav- and, in a definite unitary irreducible representation l,we ity. build the operators IV. WIGNER FUNCTION ON SU„2… SϪ Sϩ ϭ †ϭ ϭ ϩ ͑ ͒ b ͱ , b ͱ , N S3 l. 3 In this section we define the Wigner function on the group 2l 2l SU͑2͒ introduced in Refs. ͓22͔ and ͓6͔. We do this in a The su͑2͒ algebra in this representation is thus written as mathematically careful manner to avoid confusion with other different distribution functions on the sphere that have been 1 used for spin systems ͓25͔. We stress that with this definition ͓N,b†͔ϭb†, ͓N,b͔ϭb, ͓b,b†͔ϭ1Ϫ N. ͑4͒ we shall have full SU͑2͒ covariance, that the generalized l overlap formula holds, and that the reconstruction of states In the limit l ϱ these operators have the commutation re- and density matrices can be accomplished, just as in quan- ˜ ͓ ͔ lations of the oscillator algebra, with b and b† being the tum mechanics with the usual Wigner function 26 . ជ ϭ͕ ͖d usual boson operators. We denote by S Si iϭ1 the row vector of spin operators Let us keep l finite, however, and consider the oscillator- closing into the su͑2͒ algebra. The generic SU͑2͒ group ele- like Hamiltonian ment in polar parametrization ͑indicated by brackets͒ is given by a 3 vector yជ of length ␰ϭ͉yជ͉ and unit direction uជ ␻ ϭyជ/␰ϭ(sin ␪ sin ␾,sin ␪ cos ␾,cos ␪), Hϭ ͑bb†ϩb†b͒. ͑5͒ 2 ͓ ជ ͔ϭ ͑Ϫ ជ ជ ͒ϭ ͓Ϫ ␰͑ ϩ ϩ ͔͒ g y exp iS•y exp i S1u1 S2u2 S3u3 . In terms of the spin-l operators ͓Eq. ͑3͔͒, this is written ͑9͒

␻ ␻ 1 1 This is a rotation by the angle ␰ around the axis uជ , with 0 ϭ ͑ ϩ ͒ϭ ͑ 2Ϫ 2͒ϭ␻ͩ ϩ Ϫ 2 ͪ H SϩSϪ SϪSϩ S S N N , 4l 2l 3 2 2l р␰р2␲,0р␪р␲,0р␾Ͻ2␲; the group manifold is the 3 S ͑6͒ sphere 3 . The Euler angles are also often used, writing an arbitrary SU͑2͒ group element in the form where ␻ is a constant. The analogy of Eq. ͑6͒ with the usual ␣ ␤ ␥͒ϭ ␣ ͒ ␤ ͒ ␥ ͒ ͑ ͒ Kerr Hamiltonian ͑2͒ is evident, since the operator of the g͑ , , exp͑i S3 exp͑i S1 exp͑i S3 . 10 spin excitation number N has a non-negative integer spectrum kϭ0,1,2,... . The difference between the photon The connection between polar and Euler parameters is well number operator n and N is that the spectrum of the latter is known ͑see, e.g., Ref. ͓27͔͒. р ជ ϭ bounded from above: k 2l. The three generators S (S1 ,S2 ,S3) transform under the Using the commutation relation f (N)SϩϭSϩ f (Nϩ1) we action of the SU͑2͒ group as the components of a row vector itH ϪitH obtain Sϩ(t)ϭe Sϩe , and from here the solution of with the 3ϫ3 orthogonal matrices R of the adjoint represen- the Heisenberg equations of motion is tation PRA 60 FINITE KERR MEDIUM: MACROSCOPIC QUANTUM . . . 1819

Ϫ ϭ ␰ ͓ ͔ g͓yជ ͔Sជ͑g͓yជ ͔͒ 1ϭSជ R͓yជ ͔. ͑11͒ The case w wHaar( ) was considered in Ref. 6 , where the corresponding W-matrix elements were obtained in terms of ϭ ͓ ជ ͔ l ជ Applying the group transformation g g y to an arbitrary the common D-matrix elements Dm,mЈ(y) ជ ជ ͑ ͒ ជ ជ Ϫ1 ϭ͗l,m͉exp(Ϫiyជ xជ)͉l,mЈ͘, of angular momentum theory ͓31͔, operator S•x from the su 2 algebra, we have g(S•x)g • ϭSជ xជЈ, where xជЈϭRxជ. Column vectors xជ belong to the also but independently associated with the name of Wigner • 3 ͑see the Appendix͒. It was shown in Ref. ͓6͔ that the radial three-dimensional real space R , conjugate to the vector space of the Lie algebra, and carry the coadjoint representa- projection of the Wigner function tion of the group ͓28͔, xជЈϭRxជ. This space xជ෈R3 we will call the metaphase-space of the classical dynamical system. W͑␹͒ϭ ͵ duជ W͑⌿͉␹uជ ͒, ͉xជ͉ϭ␹ S Acting with all group transformations g͓yជ ͔ on an arbi- 2 ជ trary fixed vector x0 in the coadjoint representation, we obtain its group orbit. This orbit has a symplectic structure and has its maximum value over the sphere for radius ␰ is the common phase space ͓29,30͔. When we choose a defi- Ϸͱl(lϩ1). The figures in this paper are values of the nite irreducible representation of spin l, the group acts on the Wigner functions plotted for this radius. We now list the Hilbert space of states ͉⌿͘ of the corresponding quantum three most relevant properties of the Wigner function: cova- ͗⌽͉⌿͘ϭ͚l ␾ ␺ system with the scalar product mϭϪl m* m .An riance, overlap formula, and density-matrix reconstruction. arbitrary pure spin state ͉⌿͘ is represented by a column vec- Covariance: From the form of integral ͑12͒ and the defi- ⌿ ϭϪ Ϫ ϩ ͑ ͒ tor with components m , m l, l 1,...,l. For sim- nition of the adjoint representation 11 , it follows that the plicity we use the same notation for the group element g and Wigner operator ͑12͒ satisfies the SU͑2͒ covariance: its unitary representation image ͓Eq. ͑9͔͒. ͓ ជ ͔W͑ជ ͒͑ ͓ ជ ͔͒Ϫ1ϭW ͑ ជ ͒ជ ͑ ͒ We now define the Wigner operator as an operator-valued g y x g y „R y x…. 16 function on metaphase space, This means that the linear quantum dynamics can be com- pletely described by the corresponding transformation in W ͑ជ ͒ϭ ͵ ជ ͑␰͒ ͓ ͑ជϪ ជ ͒ ជ ͔ w x dy w exp i x S y SU͑2͒ • classical metaphase space. Covariance holds for any choice of the function w(␰) in Eq. ͑12͒. ϭ ͵ ជ ͑␰͒ ͑ ជ ជ ͒ ͓ ជ ͔ ͑ ͒ Overlap formula: Consider two states of the same spin l, dy w exp ix y g y . 12 ͉⌿͘ ͉⌽͘ ⌿͉ជ SU͑2͒ • and , with corresponding Wigner functions Ww( x) and W (⌽͉xជ), for two different weight functions w(␰) and ជϭ ␰ϭ͉ជ͉ ␰ w¯ Here dy dy1dy2dy2 , y , w( ) is a weight function, w˜ (␰). We shall prove that the overlap of the two states can and the integral is over the group manifold. be calculated by integration over the metaphase space of the ͓ ͔ In Ref. 6 we considered the Haar invariant measure over product of their Wigner functions, i.e., ͑ ͒ ␰ ϭ 1 ␰Ϫ2 2 1␰ the group for integral 12 , wHaar( ) 2 sin 2 . While this is an obvious choice, it is not the only one; see below. 2lϩ1 ͵ ជ ͑⌿͉ជ ͒ ͑⌽͉ជ ͒ϭͦ͗⌿͉⌽ͦ͘2 ͑ ͒ dx W x W¯ x , 17 We shall drop the subscript w from the Wigner operator, ␲5 3 w w 16 R until more than one weight is used. For a given value of l, the Wigner operator is reduced to the if and only if the two weight functions are conjugate in the (2lϩ1)ϫ(2lϩ1)-dimensional Hermitian matrix following sense: ʈ l ជ ʈ Wm,mЈ(x) . The Wigner ͑quasiprobability distribution͒ function is the 1 ␰ w͑␰͒w˜ ͑␰͒ϭw ͑␰͒ϭ sin2 . ͑18͒ matrix element of the Wigner operator between the states ͉⌿͘ Haar 2␰2 2 and ͉⌽͘ with definite l,

l This is the counterpart of the corresponding well-known ͑⌿ ⌽͉ជ ͒ϭ͗⌿͉W͑ជ ͉͒⌽͘ϭ ͚ ⌿* l ͑ជ ͒⌽ property of the common Wigner function on the Heisenberg- W , x x mWm,mЈ x mЈ . ͓ ͔ m,mЈϭϪl Weyl group 20 , where the Haar weight function is 1. ͑13͒ The integral on the left-hand side of Eq. ͑17͒ can be un- derstood as a mean value of an operator ␴ˆ which acts on the Most often this is used for ⌿ϭ⌽; then we indicate direct product of two Hilbert spaces of the same value of l, W(⌿,⌿͉xជ)byW(⌿͉xជ), and we suppress l when convenient. When instead of pure states we have a density matrix ͵ ជ ͑⌿͉ ͒ ͑⌽͉ជ ͒ϭ ͗⌿͉ ͗⌽͉␴͉⌿͘ ͉⌽͘ ␳, the Wigner function is dx Ww xˆ Ww˜ x 1 2 ˆ 1 2 , ͑ ͒ l 19 W͑␳͉xជ ͒ϭtr͕W͑xជ ͒␳͖ϭ ͚ Wl ͑xជ ͒␳ . ͑14͒ m,mЈ mЈ,m where m,mЈϭϪl The W-matrix elements are ␴ϭ ͵ ជ W ͑⌿͉ជ ͒ W ͑⌽͉ជ ͒ ͑ ͒ ˆ dx x ˜ x . 20 3 w w 2␲ R Wl ͑xជ ͒ϭ ͵ w͑␰͒␰2d␰ ͵ dvជ exp͑i␰vជ xជ ͒Dl ͑␰vជ ͒, m,mЈ S • m,mЈ ␴ 0 2 We will show now that ˆ is proportional to an exchange ͑15͒ operator. Integral ͑20͒ is found by replacing the deﬁnition of 1820 CHUMAKOV, FRANK, AND WOLF PRA 60

the Wigner operator ͑12͒ into Eq. ͑20͒; this can be integrated 2lϩ1 ͵ ជ ˜ ͑ជ ͒ ͑⌿͉ជ ͒ over xជ yielding a Dirac ␦. Due to relation ͑18͒ between the 5 dx Wn,m x W x 16␲ 3 weight functions, we have R 2lϩ1 ϭ ͗ f ͉ ͗⌿͉␴ˆ ͉f ͘ ͉⌿͘ ␴ϭ͑ ␲͒3 ͵ ͓ ជ ͔ ͑ ជ ជ ͒ ͑Ϫ ជ ជ ͒ 16␲5 1 n 2 m 1 2 ˆ 2 dg y exp iy S1 exp iy S2 , SU͑2͒ • • ͑ ͒ ϭ ͉⌿ ⌿͉ ϭ␳ 21 1͗ f n ͘12͗ f m͘2 n,m . ͓ ជ ͔ϭ 1 2 ␰ ␰ ␾ ␪ where dg y 2 sin ( /2)d d d cos is the invariant ͕͉ ͖͘l ͑Haar͒ measure, and the integral is over the group manifold. For mixed states, consider the basis rn nϭϪl where the ͉ ͉ Ϫ р р ␳ϭ͚ ͉ ͗͘ ͉ ␳͉ជ Let ͕ m͘1 mЈ͘2 , l m,mЈ l͖ be an orthonormal basis in density matrix is diagonal: cn rn rn . Then W( x) ϭ͚ ͉ជ the tensor product of the two copies of the Hilbert space. cnW(rn x), and we have Consider the matrix elements of Eq. ͑21͒, 2lϩ1 l ͵ ជ ˜ ͑ជ ͒ ͑␳͉ជ ͒ ͗ ͉ ͗ Ј͉␴͉ ͘ ͉ Ј͘ ϭ͑ ␲͒3 ͵ ͑ ͒ 5 dx Wn,m x W x 1 m 2 m ˆ n 1 n 2 2 dg Dmn g 16␲ 3 SU͑2͒ R ϫ l ͑ Ϫ1͒ 2lϩ1 DmЈnЈ g . ϭ ͚ c ͵ dxជ W˜ ͑xជ ͒W͑r ͉xជ ͒ϭ␳ . ␲5 k 3 n,m k n,m 16 k R l Ϫ1 ϭ͓ l ͔ From unitarity we have DmЈnЈ(g ) DnЈ,mЈ(g) *. Be- cause of the well-known property of orthogonality of the We note that the SU͑2͒ Wigner function provides the D-matrix elements, we ﬁnd SU͑2͒ Q function used by Agarwal, Puri, and Singh in Ref. ͓24͔. Indeed, the overlap of the w Wigner function of a state ͗ ͉ ͗ Ј͉␴͉ ͘ ͉ Ј͘ ϭ͑ ␲͒3 ͵ l ͑ ͒ ͉⌿͘ with the w˜ Wigner function of a coherent states ͉␣,␤͘ on 1 m 2 m ˆ n 1 n 2 2 dg Dmn g SU͑2͒ the sphere ͑see below͒ is Q(␣,␤)ϭ͉͗␣␤͉⌿͉͘, the non- ͓ ͔ 16␲5 negative function studied in Ref. 24 . ϫ͓Dl ͑g͔͒*ϭ ␦ ␦ . nЈ,mЈ 2lϩ1 mnЈ nmЈ V. EVOLUTION OF COHERENT STATES ON THE ␴ Thus the operator ˆ exchanges the states of the ﬁrst and the SPHERE second Hilbert spaces,

16␲5 The above concepts will be now applied to describe the ␴͉⌿͘ ͉⌽͘ ϭ ͉⌽͘ ͉⌿͘ ͑ ͒ evolution in a Kerr medium of spin-coherent states. ˆ 1 2 ϩ 1 2 . 22 2l 1 Spin-coherent states. We now particularize the spin-l model to a system of 2l two-level atoms prepared initially in Comparing this result with Eq. ͑19͒, we ﬁnd Eq. ͑17͒. a state symmetric under permutations of atoms. If the inter- Clearly, if we choose the ‘‘self-conjugate’’ weight func- action Hamiltonian also has this symmetry, then there are tion 2lϩ1 accessible states ͉mϭkϪl͘,0рkр2l, called the ͓ ͔ Ϫ 1 ␰ Dicke states 32 ; here 2l k is the number of atoms in the ␰͒ϭ ␰͒ϭ ͑ ͒ w0͑ w˜ 0͑ sin , 23 ground state, and k is the number of atoms in the excited &␰ 2 state. The simplest examples of spin-coherent states ͓33,30͔, are then the overlap formula holds for the same w Wigner func- the states ͉0͑͘spin down͒ and ͉2l͘ ͑spin up͒, both of which tion. Another interesting choice of the weight function is have uncertainties ⌬S ϭ0 and ⌬S ϭ⌬S ϭͱl/2. All the ␰ ϭ ϭ 3 1 2 w( ) w˜ Haar 1. other spin-coherent states ͉␣,␤͘ can be produced by rotation Reconstruction of the state: Given the Wigner function from the state ͉0͘,as ␳͉ជ ϭ W ជ ␳ W( x) tr͕ w(x) ͖, we can reconstruct the density matrix ␳ ͕͉ ͖͘l of the corresponding state of the system. Let f n nϭϪl be ␣ ␤ ␥ ͑␣ ␤ ␥͉͒ ϭ i S3 i S1 i S3͉ an orthonormal basis in the (2lϩ1)-dimensional Hilbert g , , 0͘ e e e 0͘ ˜ Ϫil␥ i␣S i␤S Ϫil␥ space. Denote by Wn,m the matrix elements of the ϭe e 3e 1͉0͘ϭe ͉␣,␤͘, ͑25͒ ͉ w˜ -conjugate Wigner operator between the states f n͘ and ͉f ͘, m where we have used Eq. ͑10͒. The rotation angle ␥ leads to a phase factor which may be disregarded. Mathematically this W˜ ͑xជ ͒ϭ͗ f ͉W˜ ͑xជ ͉͒f ͘. n,m n w m corresponds to the reduction from SU͑2͒ to the factor space ͑ ͒ ͑ ͒ϭS ͓ ͔ The elements of the density matrix in this basis are given by SU 2 /SU 1 2 30 . The general form of spin-coherent ␣ ␤ states is thus ͉␣,␤͘ϭei S3ei S1͉0͘. The manifold of spin- 2lϩ1 coherent states is phase space ͓30͔, and is a sphere of radius ␳ ϭ ͵ dxជ W˜ ͑xជ ͒W͑␳͉xជ ͒. ͑24͒ n,m ␲5 3 n,m l ͑called the Bloch sphere in quantum optics and the Poin- 16 R care´ sphere in polarization optics ͓35͔͒. The components of S ͑ ͒ Indeed, for the pure states ␳ϭ͉⌿͗͘⌿͉, from Eqs. ͑20͒ and spin-coherent states in the eigenbasis of 3 are the SU 2 ͑22͒ it follows that d-matrix elements ͓27͔ PRA 60 FINITE KERR MEDIUM: MACROSCOPIC QUANTUM . . . 1821

and keeps the probability distribution of the states ͉m͘. The phase ␻tk leads to the rigid rotation of the sphere around the vertical axis; this will be extracted so that we consider only the nonlinear phase shift t␹k2. In analogy with the usual Kerr medium, as time evolves we can expect phase spread and squeezing along some direction different from the 3-axis. Indeed, this is shown in Fig. 1͑b͒. Slanted squeezing was reported previously for this Hamiltonian by Kitagawa and Ueda ͓36͔. For longer times, self-interference appears. This is shown in Figs. 1͑c͒ and 1͑d͒, for triple and double resonances, respectively, at times ␹tϭ␲/3 and ␲/2. We now proceed to prove that the latter are true Schro¨dinger cat states ͓24͔ and find the amplitudes of the components in the cat states. Schro¨dinger cats on the sphere. The figures suggest that at resonance times ␹tϭ␲K/M ͑with K and M mutually prime; we take Kϭ1 for simplicity͒, the initial coherent state unfolds into a sum of M coherent states placed equidistant around the equator. Indeed, the nonlinear phase in Eq. ͑27͒ is produced by the unitary operator exp(i␹tN2), where N is the level number operator defined in Eq. ͑3͒. At times ␹tϭ␲/M, for M even, the eigenvalues exp(i␲k2/M) of the operator exp(i␹tN2) are periodic in k with period M ͑invariant under the substitution ϩ k˜k M). For M odd, we consider instead the operator FIG. 1. Time evolution of the Wigner function on the sphere exp͓i␹tN(NϪ1)͔ times an oscillator phase ͑linear in N͒.As ϭ governed by a nonlinear finite Kerr Hamiltonian, for l 8. The plots for the common Kerr medium ͓19,18͔, this periodicity in N have the following line code: , visible grid, positive, and zero- allows us to use the finite Fourier transform basis of phases level lines; , invisible grid and positive-level lines; ͕ Ϫ1/2 Ϫ2␲isk/M͖MϪ1 ͓ ͔ • • • M e sϭ0 . Thus we expand 34 , , visible negative- and zero-level lines; ¯ , invisible negative-level lines. The times are ͑a͒ ␹tϭ0 ͑initial coherent state͒, ␲ MϪ1 ei /4 ͑b͒ ␹tϭ0.15, ͑c͒ ␹tϭ␲/3, and ͑d͒ ␹tϭ␲/2. i␲N2/Mϭ Ϫ2␲isN/M e ͚ f se , ͱM sϭ0 ͗m͉␣,␤͘ϭeim͑␣Ϫ␲/2͒dl ͑␤͒ϭeim͑␣Ϫ␲/2͒ m,0 ϭ Ϫi␲s2/M ͑ ͒ f s e , M even, 28 ͑2l͒! 1/2 ␤ ␤ lϩm lϪm ϫͩ ͪ cos sin . ␲ MϪ1 ͑lϪm͒!͑lϩm͒! 2 2 ei /4 i␲N͑NϪ1͒/Mϭ Ϫ2␲isN/M e ͚ f se , ͑26͒ ͱM sϭ0

ϭ Ϫi␲͑sϪ1/2͒2/M ͑ ͒ The Wigner function of the spin-coherent state ͉␣, ␤͘ is a f s e , M odd. 29 round, concentrated blob on this sphere, centered at the point The evolution operator at cat times is therefore a sum of ͑␣, ␤͒. The radius of the Wigner blob on the sphere is ͱl/2, ␲ rotations e2 isN/M by angles 2␲s/M, sϭ0,1,..., MϪ1, whereas the radius of the sphere is l. The classical limit for acting on the initial coherent state. The wave function is thus the spin system occurs when the dimension of the represen- ϩ a sum of M coherent states with these rotation angles and tation, 2l 1, grows without bound and thus the direction of ¨ ជ with the amplitudes f s i.e., an M-component Schrodinger cat the vector S becomes sharper. Hence the classical limit of a state. As in the case of the Heisenberg-Weyl algebra ͑the spin system is a magnetic needle, i.e., a compass. Wigner function on the plane͒, the Wigner function on the The evolution of the initial spin coherent state under the sphere shows interference fringes between different compo- ͓ ͑ ͔͒ ͑ ͒ ͑ ͒ finite Kerr Hamiltonian Eq. 6 is shown in Figs. 1 a –1 f nents of the cat state ͑the ‘‘smile of the cat’’͒; see Figs. 1͑c͒ for lϭ8. The level plots of the Wigner function with w(␰) and 1͑d͒. ϭ ␰ ͉ជ͉ wHaar( ) for the value of the radial coordinate x ϭͱ ϩ l(l 1) are shown for various subsequent times. Let us VI. CONCLUSIONS analyze these figures. ͑ ͒ ϭ We start in Fig. 1 a at t 0 with an eigenstate of S1 of We have used a recently defined Wigner function on eigenvalue l; this is a coherent state. As time evolves, the SU͑2͒ to understand spin-coherent states and their evolution ␣ ͑ points of phase space move only in geographical longi- under a physically realistic Kerr Hamiltonian. This spin sys- ͒ ␤ ͑ ͒ tude, or phase , keeping colatitude constant. This evolu- tem is realized in quantum optics as a collective atomic sys- ͉ ϭ Ϫ ͘ tion multiplies eigenstates m k l by phases tem ͓32,12͔, as a pair of radiation field modes of two different polarizations ͓35͔, and also as an atomic interferometer Ϫ ͓␻͑ ϩ 1 ͒Ϫ␹ 2͔ ͑ ͒ exp͕ it k 2 k ͖, 27 ͓37,38͔. The effective nonlinearity in the case of a collective 1822 CHUMAKOV, FRANK, AND WOLF PRA 60

atomic system arises due to the dynamical Stark shift ͓39,40͔ it follows that the Wigner operator at arbitrary point xជ ͑and it has been proposed to generate the atomic Schro¨dinger ϭ␩(sin ␪ sin ␾,sin ␪ cos ␾,cos ␪) can be obtained by the ro- ͓ ͔͒ cat states in Ref. 41 or due to the interaction with radiation tation of the Wigner operator at the North pole, W(␩kជ), kជ field in a dispersive cavity out of resonance with the atomic ϭ(0,0,1), transition frequency; see Ref. ͓24͔, where the equivalent ͑ Ϫ ␾ Ϫ ␪ ␪ ␾ model was described in terms of the atomic Q function the W͑xជ ͒ϭe i Sze i SxW͑␩kជ ͒ei Sxei Sz. diagonal matrix element of the atomic density matrix between the spin-coherent states͒. In an atomic interferometer In turn, it is easy to prove that at the Wigner operator is the nonlinearity may be created by a nonlinear active ele- l ϭ␦ diagonal at the North pole, Wm,mЈ m,mЈWm . The quanti- ment such as a Coulomb coupler ͓42͔. ␩ ties Wm( ) are the eigenvalues of the Wigner operator; they In wave optics our model describes the propagation of a depend only on ␩ϭ͉xជ͉ and can be written in the form finite number of transverse modes in a waveguide ͓6͔.An l ideal ͑Gaussian͒ waveguide corresponds to the infinite equi- 1 ͑␩͒ϭ ␲ ͵ ␪͉ l ͑ ␪͉͒2 ͑␩ ␪Ϫ ͒ distant spectrum and usually is described by a harmonic os- Wm 2 ͚ d cos dmk cos F cos k , kϭϪl Ϫ1 cillator. In real waveguides with a finite number of modes, the spectrum is always different from the equidistant one, 2␲ ␰ which leads to the effective nonlinearity. Finally, in molecu- F͑y͒ϭ ͵ d␰␰2w͑␰͒ei y. lar physics ͓23͔, our model corresponds to a diatomic mol- 0 ecule approximated by a Morse potential. where, for the case w(␰)ϭ(1/2␰2)sin2(␰/2),

2l ACKNOWLEDGMENTS ͑Ϫ1͒ sin͑2␲y͒ Ϫ1 2 Ϫ1 F͑y͒ϭ ͫ ϩ ϩ ͬ, 8 yϩ1 y yϪ1 We thank our colleagues, Dr. Natig M. Atakishiyev, Dr. Andrey B. Klimov, Dr. Valery P. Karassiev, and Dr. Andrey and for the case w(␰)ϭ(1/␰)sin(␰/2), N. Leznov, for interesting discussions. This work is a result of the DGAPA-UNAM Project Nos. I104198 and IN101997 1 1 F͑y͒ϭ͑Ϫ1͒2lͭ ␲ei2␲yͩ Ϫ ͪ and European Community Project No. CI1-CT94-0072. One yϩ1/2 yϪ1/2 of us ͑S.Ch.͒ would like to acknowledge Project No. i2␲y 2l 3927P-E of CONACYT. ͑e ϩ͑Ϫ1͒ ͒ 1 1 ϩ ͩ Ϫ ͪͮ . 2i ͑yϪ1/2͒2 ͑yϩ1/2͒2 APPENDIX Note that k takes integer or half-integer values depending on Here we give a convenient formula for the calculation of whether l is an integer or half-integer; this leads to the factor the Wigner function for a spin system. From the covariance (Ϫ1)2l in the last equation.

͓1͔ E. Schro¨dinger, Naturwissenschaften 23, 807 ͑1935͒. C. A. Holmes, ibid. 44, 4704 ͑1991͒. ͓2͔ B. Yurke and D. Stoler, Phys. Rev. Lett. 57,13͑1986͒. ͓11͔ I. Sh. Averbukh and N. F. Perel’man, Phys. Lett. A 139, 449 ͓3͔ Z. Białynicka-Biruła, Phys. Rev. 173, 1207 ͑1968͒. ͑1989͒. ͓4͔ V. V. Dodonov, L. A. Malkin, and V. I. Man’ko, Physica ͓12͔ S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, ͑Amsterdam͒ 72, 597 ͑1974͒. Opt. Commun. 118, 529 ͑1995͒; Phys. Rev. A 49, 4972 ͓5͔ Usually the Kerr nonlinearity is introduced in a slightly differ- ͑1994͒. ent way: as n2Ϫnϭ(a†)2a2. This definition introduces a small ͓13͔ J. Parker and C. R. Stroud, Phys. Rev. Lett. 56, 716 ͑1986͒;T. additional detuning, which nevertheless can be canceled with a Bluhm, V. A. Kostelecky´, and B. Tudose, Phys. Rev. A 55, redefinition of the parameters. 819 ͑1997͒. ͓6͔ K. B. Wolf, N. M. Atakishiyev, and S. M. Chumakov, in Pho- ͓14͔ P. Kasperkowitz and M. Peev, Phys. Rev. Lett. 75, 990 ͑1995͒. tonic Quantum Computing, Proceedings of SPIE, Vol. 3076 ͓15͔ P. Stifter, C. Leichtle, W. P. Schleich, and J. Marklof, Z. ͑SPIE, Bellingham, WA, 1997͒, pp. 196–206; N. M. Atakish- Naturforsch., A: Phys. Sci. 52, 377 ͑1997͒; F. Großmann, iyev, S. M. Chumakov, and K. B. Wolf, Wigner distribution J.-M. Rost, and W. P. Schleich, J. Phys. A 30, L277 ͑1997͒. for finite systems ͑unpublished͒. ͓16͔ J. Gea-Banacloche, Phys. Rev. A 44, 5913 ͑1991͒. ͓7͔ R. Tanas´,inCoherence and Quantum Optics V, edited by L. ͓17͔ S. M. Chumakov, A. B. Klimov, and C. Saavedra, Phys. Rev. Mandel and E. Wolf ͑Plenum, New York, 1984͒, p. 645. A 52, 3153 ͑1995͒. ͓8͔ M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974 ͓18͔ Ts. Gantsog and R. Tanas´, Quantum Opt. A 3,33͑1991͒; ͑1986͒. Phys. Rev. A 44, 2086 ͑1991͒. ͓9͔ N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera and K. B. ͓19͔ K. Tara, G. S. Agarwal, and S. Chaturvedi, Phys. Rev. A 47, Wolf, Phys. Lett. A 215, 128 ͑1996͒; A. L. Rivera, N. M. 5024 ͑1993͒. Atakishiyev, S. M. Chumakov, and K. B. Wolf, Phys. Rev. A ͓20͔ E. Wigner, Phys. Rev. 40, 749 ͑1932͒; M. Hillery, R. F. 55, 876 ͑1997͒. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, ͓10͔ G. J. Milburn, Phys. Rev. A 33, 674 ͑1986͒; G. J. Milburn and 121 ͑1984͒; H.-W. Lee, ibid. 259, 147 ͑1995͒. PRA 60 FINITE KERR MEDIUM: MACROSCOPIC QUANTUM . . . 1823

͓21͔ E. A. Akhundova, V. V. Dodonov and V. I. Man’ko, Theor. ͓29͔ A. A. Kirillov, Elements of the Representation Theory Math. Phys. 60, 413 ͑1984͒. ͑Springer, Berlin, 1970͒. ͓22͔ K. B. Wolf, Opt. Commun. 132, 343 ͑1996͒. ͓30͔ A. Perelomov, Generalized Coherent States and Their Appli- ͓23͔ A. Frank and P. Vansacker, Algebraic Methods in Molecular cations ͑Springer-Verlag, Heidelberg, 1986͒. and Nuclear Structure ͑Wiley Interscience, New York, 1994͒. ͓31͔ N. Ya. Vilenkin and A. U. Klimyk, Representations of Lie ͓24͔ G. S. Agarwal, R. R. Puri, and R. P. Singh, Phys. Rev. A 56, Groups and Special Functions ͑Kluwer, Dordrecht, 1991͒, 2249 ͑1997͒. Vol. 1. ͓25͔ R. L. Stratonovich, Zh. Eksp. Teor. Fiz. 58, 1884 ͑1970͓͒Sov. ͓32͔ R. H. Dicke, Phys. Rev. 93,99͑1954͒. Phys. JETP 31, 1012 ͑1970͒; G. S. Agarwal, Phys. Rev. A 24, ͓33͔ J. M. Radcliffe, J. Phys. A 4, 313 ͑1971͒; F. T. Arecchi, E. 2889 ͑1981͒; R. Gilmore, in The Physics of Phase Space, ed- Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 ited by Y. S. Kim and W. W. Zachary, Lecture Notes in Phys- ͑1972͒; G. J. Glauber and F. Haake, ibid. 13, 357 ͑1976͒. ics Vol. 278 ͑Springer, Berlin, 1987͒, p. 211; J. P. Dowling, G. ͓34͔ A. B. Klimov and S. M. Chumakov, Phys. Lett. A 235,7 S. Agarwal, and W. P. Schleich, Phys. Rev. A 49, 4101 ͑1994͒. ͑1997͒. ͓26͔ The definition given by G. S. Agarwal ͓Phys. Rev. A 24, 2889 ͓35͔ V. P. Karassiov and V. I. Puzyrevsky, J. Sov. Laser Res. 10, ͑1981͔͒ in terms of SU͑2͒ irreducible tensor operators also 229 ͑1989͒; V. P. Karassiov, J. Phys. A 26, 4345 ͑1993͒; Phys. provides covariance and overlap relations. Our definition Lett. A 190, 1387 ͑1994͒; J. Lehner, U. Leonhardt, and H. seems to be more convenient to work with, and the correspon- Paul, Phys. Rev. A 53, 2727 ͑1996͒; A. P. Alodjants, S. M. dence to the common Wigner function is more clear. The re- Arakelian, and A. S. Chirkin, Quantum Semiclassic. Opt. 9, lation between our construction and the Agarwal definition as 311 ͑1997͒. well as the contractions to the Heisenberg-Weyl group ͑the ͓36͔ M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 ͑1993͒. common Wigner function͒ and to the Euclidean group ͓Wigner ͓37͔ B. Yurke, Phys. Rev. Lett. 56, 1515 ͑1986͒; B. Yurke, S. Ł. function on the cylinder; see L. M. Nieto, N. M. Atakishiyev, McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 ͑1986͒. S. M. Chumakov, and K. B. Wolf, J. Phys. A 31, 3875 ͑1998͔͒ ͓38͔ D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, will be considered elsewhere. Phys. Rev. A 50,67͑1994͒. ͓27͔ L. C. Biedenharn and J. D. Louck, Angular Momentum in ͓39͔ M. J. Holland, D. F. Walls, and P. Zoller, Phys. Rev. Lett. 67, Quantum Physics, Encyclopedia of Mathematics and its Appli- 1716 ͑1991͒. cations, edited by G.-C. Rota ͑Addison-Wesley, Reading, MA, ͓40͔ A. B. Klimov, J. Negro, and S. M. Chumakov ͑unpublished͒. 1981͒. ͓41͔ C. C. Gerry and R. Grobe, Phys. Rev. A 56, 2390 ͑1997͒. ͓28͔ F. A. Berezin, Commun. Math. Phys. 40, 153 ͑1975͒. ͓42͔ M. Kitagawa and M. Ueda, Phys. Rev. Lett. 67, 1852 ͑1991͒.