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 1050-2947/99/60͑3͒/1817͑7͒/$15.00PRA 60 1817 ©1999 The American Physical Society 1818 CHUMAKOV, FRANK, AND WOLF PRA 60 ␹ 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 nonlin- earity 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 .

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