Quantum entangled supercorrelated states in the Jaynes–Cummings model

16 August 1999

Physics Letters A 259 Ž1999. 285–290 www.elsevier.nlrlocaterphysleta

Quantum entangled supercorrelated states in the Jaynes–Cummings model A.K. Rajagopal a b

a,)

, K.L. Jensen a , F.W. Cummings

b,1

NaÕal Research Laboratory, Washington D.C. 50378-5320, USA UniÕersity of California, RiÕerside, (Emeritus), CA 92521, USA

Received 15 March 1999; received in revised form 1 June 1999; accepted 11 June 1999 Communicated by A.R. Bishop

Abstract The regions of independent quantum states, maximally classically correlated states, and purely quantum entangled Žsupercorrelated. states described in a recent formulation of quantum information theory by Cerf and Adami are explored here numerically in the parameter space of the well-known exactly soluble Jaynes–Cummings model for equilibrium and nonequilibrium time-dependent ensembles. q 1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 03.65.-w; 05.30.-d; 89.70.q c

The well-known exact solution of the Jaynes– Cummings ŽJC. model w1,2x 2 describing the interaction of a single-mode radiation field with a two-level atom is the basis for a vast array of the current experiments on foundations of quantum mechanics involving entangled states w3x, new ideas of quantum optics w4x, and novel device structures such as micromaser w5x, single-atom laser, etc. w6x. It is also a generic model of the interaction between two dissimilar quantum systems ŽBose field and two-level-system, ŽTLS... Thus this model contains all the subtle

)

Corresponding author. Tel: 202-767-1373; fax: 202-767-0546 or 4290; e-mail: [email protected] 1 Present address: 2365 Virginia St. a4, Berkeley, CA 947091338. 2 See also wAnn. Phys. 11 Ž1963. 411x where H. Paul also reports an exact solution to the JC model but did not explore in full the ramifications of his solution.

features of quantum entanglement albeit dependent on the interaction strength. A new quantum information theory based on the density matrix of entangled systems of Cerf and Adami w7x ŽCA. mathematically classified Ž1. independent quantum states, Ž2. maximally classically correlated states, and Ž3. a new classically forbidden and therefore purely quantum entangled states called ‘supercorrelated’ states. This is because they found that the conditional entropies of quantum entangled system can be negative, a feature missed in an earlier theory w8x. CA have subsequently explored many further consequences of this discovery w9x. The purpose of this paper is to employ the formal CA theory to elucidate these features numerically in the parameter space of ensembles constructed with the JC model, thus providing an explicit display of the features that CA discovered. We use the exact eigen-solutions of the JC model for constructing two

0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 4 0 4 - 1

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ensembles: Ža. thermal Žequilibrium. ensemble at a temperature T, where the JC Hamiltonian may be considered as a generic model ŽBose-TLS. system and Žb. a time-dependent ensemble of quantum optical interest w8,10x. Two simple models of the radiation often used in the literature, Ži. blackbody-like and Žii. pure coherent source w10,4x are considered in Žb.. For a discussion of ensembles, see w11x.. For each of these models, we deduce the individual partial density matrices of the radiation and the atomic fields from the system density matrix. This provides us with explicit expressions to examine numerically the negative conditional entropy and information contained therein, as a function of the model parameters, a feature not considered in w7x. For these ensembles, we find the three regions of states described by CA. Unlike CA, we exhibit them in the parameter space of the model and the two ensembles, the temperature and time-scale of the system. In the notations of w2x, the Hamiltonian of the JC model concerns a two-level atom ŽTLS. interacting with a single given mode of quantized radiation Žboson. field of a given frequency, v , is described in terms of the usual creation, aˆ†, and destruction, a, ˆ operators of the boson field; the TLS is represented by the z-component of the Pauli matrix operator with the energy separation of the two atomic levels v 0 , and their mutual interaction is expressed in the rotating wave approximation: HAq R s " v aˆ†aˆ q

" v0 2

Here s s 1,2 labels the entangled states which in their bare condition are the ground Ž g . and the excited Ž e . states, and n, the states of the boson field. Also V Ž n, s . s v Ž n q 1r2. q 3 y 2 s . l n ,tan un s k Ž n q 1 . rwŽ Dv .r2 q l n x, Ž Dv . s Ž v y v 0 .,

(

(

`

rAq R s

ž

< w Ž n,1 . : w Ž n,1 . ² w Ž n,1 . <

Ý ns0

q< w Ž n,2 . : w Ž n,2 . ² w Ž n,2 . < q <0, g : w Ž 0 . ²0, g < .

`

Ý Ž < n q 1: ² n q 1 <. Ž < g : ² g <. ns0

/

Ž 1.

= Ž w Ž n,1 . cos 2un q w Ž n,2 . sin2un .

Here k is the dipole-interaction strength between the radiation and the atom or in the generic model, the boson-TLS interaction. Exact solutions of this interacting system are w1,2x

q w Ž 0 . Ž <0: ²0 < . Ž < g : ² g < .

HAq R < w Ž n, s . : s " V Ž n, s . < w Ž n, s . : ,

= Ž w Ž n,1 . sin2un q w Ž n,2 . cos 2un .

Ž s s 1,2 .

q

`

q

Ý Ž < n: ² n < . Ž < e : ² e < . ns0

`

HAq R <0, g : s y" w v 0r2 x <0, g : ,

Ý Ž w Ž n,1 . y w Ž n,2 . . cos un sin un ns0

= Ž n q 1: ² n < . Ž < g : ² e < .

< w Ž n,1 . : s cos un < n q 1, g : q sin un < n,e : , < w Ž n,2 . : s ysin un , g : q cos un < n,e : .

Ž 3.

Here w Ž0. s wexpŽ b " v 0r2.xrZAqR ,w Ž n, s . s wexp y b " V Ž n, s .xrZAq R , and ZAqR is the system partition function. Here b s Ž k B T .y1 where T is the temperature. In terms of the boson and TLS states using Eq. Ž2., it is

rAq R s sˆz q " k aˆ†sˆyq aˆ sˆq .

2

and l n s Ž Dvr2 . q k 2 Ž n q 1 . . The angles un are measures of the entanglement; un s 0 in the noninteracting case Žno entanglement. and un s pr4 for all n at resonance, Dv s 0. For the sake of simplicity, the numerical results reported in this paper are for the resonant case. In Eq. Ž2., the eigenstates in terms of the complete set of states of the boson field and the TLS are also given. Ža. Equilibrium ensemble: The density matrix constructed by using the maximum entropy principle subject to given average total energy of the system w2,10x, is thus found to be:

Ž 2.

The entangled states < w Ž n, s .:4 , Ž s s 1,2;n s 0,1,2 . . . . and <0, g : are orthonormal and complete.

q Ž < n: ² n q 1 < . Ž < e :² g < . .

Ž 4.

The third sum in the above represents the entanglement as well as ‘decoherence’ features of the

A.K. Rajagopal et al.r Physics Letters A 259 (1999) 285–290

interacting system. We obtain the ‘marginal’ density matrix

rA s Tr R rAqR s fŽy. < g : ² g < q fŽq. < e : ² e < .

Ž 5. Here fŽy . s w Ž0. q Ý`ms 0 Ž w Ž m,1.cos 2um q w Ž m,2.sin2um ., and fŽq. s 1 y fŽy. are the occupation probabilities of the g and e states. Similarly, the ‘marginal’ density matrix of the boson is

287

interesting aspects of the entropy of the radiation field. There are two radiation models exhibiting different interesting behaviors w10,4x. The initial state is specified in the form where the photons come from a single mode of radiation considered in w10x and the atom is in its excited Ž e . state:

rAq R Ž 0 . s rA Ž 0 . m r R Ž 0 . , rA Ž 0 . s < e : ² e < , r R Ž 0 . s Ý p Ž n . < < n: ² n < . n

Ž 8.

`

r R s TrA rAqR s

Ý

pn < n : ² n < .

Ž 6.

ns0

Here p 0 s w Ž0. q w Ž0,1.sin2u 0 q w Ž0,2.cos 2u 0 , and for n s 1,2,3, . . . , pn s w Ž n,1.sin2un q Ž n,2.cos 2un s w Ž n y 1,1. cos 2 uny1 q w Ž n y 1, 2.sin2uny 1. From Eqs. Ž2., Ž5. and Ž6., we compute SAq R s yTrŽ rAqR ln rAqR ., SA s yTrA Ž rA ln rA ., andSR s yTr RŽ r R ln r R .. Henceforth we focus on the resonant case v s v 0 and the dimensionless variables of this ensemble are Ž b " v .y1 and Ž krv .. Žb. Non-equilibrium ensemble: We consider a unitary time-evolution ŽLiouville–von Neumann. of an initially prescribed density matrix of the system, rAq RŽ0. w1,8,10x, given by rAqRŽ t . s † Ž . where UAq R Ž t . s UAq R Ž t . rAq R Ž0.UAq R t expŽyitHAq R r" .. In terms of the eigenstates given in Eq. Ž2.,

rAq R Ž t . s <0, g : ²0, g < rAq R Ž 0 . <0, g : ²0, g < q <0, g : Ý ²0, g < rAq R Ž 0 . < w Ž m, sX . : m, sX

= exp Ž it Ž v 0r2 q V Ž m, sX . . . =² w Ž m, sX . < q h.c.q

Ý

< w Ž n, s . :

n, s ; m , sX

rAq R Ž t . s Ý < w Ž n,1 . : p Ž n . sin2un² w Ž n,1 . < n

q < w Ž n,2 . : p Ž n . cos 2un² w Ž n,2 . < q 2 < w Ž n,1 . : p Ž n .  sin un cos un =cos t Ž V Ž n,1 . y V Ž n,2 . . =² w Ž n,2 . <

4 Ž 9.

It should be emphasized that the preparation of the initial state as in Eq. Ž8. is nontrivial both conceptually and experimentally. However the above two models have served as guides and we use them here in the same spirit. In contrast to the equilibrium density matrix Eq. Ž3., the above expression contains only entanglement effects Ždependent on k only.. From this the ‘marginal’ density matrices are

rA Ž t . s < g : wg Ž t . ² g < q < e : we Ž t . ² e < ,

=² w Ž n, s . < rAq R Ž 0 . < w Ž m, sX . :

r R Ž t . s Ý < n: Pn Ž t . ² n < ,

Ž 10 .

n

= w exp Ž y it Ž V Ž n, s . yV Ž m, sX . . . x ² w Ž m, sX . < .

Model Ži.: Blackbody-like source pŽ n. s Ž N . n Ž1 q N .yŽ1 qn., N s Ý n npŽ n. s mean number of photons. Model Žii.: A single mode pure coherent source w10x for which the photon number distribution is Poissonian, pc Ž n. s Ž N . n expŽyN .rn!, with N and rAŽ0. are as in model Ži.. Note that pc Ž n. has a peak at n s N whereas pŽ n. of model Ži. is monotonic, but both vanish for large n. Using Eqs. Ž7. and Ž8. we obtain

Ž 7.

For simplicity of presentation, we now employ two special initial density matrices, first considered in w10x and later used in w8x to examine

where wg Ž t . s Ý n pŽ n.WnŽ t .,we Ž t . s 1 y w Ž t . and PnŽ t . s pŽ n.Ž1 y WnŽ t .. q pŽ n y 1.Wny1Ž t ., with WnŽ t . s Žsin2 2 un .sin2 t k Ž n q 1 . . The total system entropy in both of these cases is time-independent and solely determined by the initial state density

ž (

/

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matrix, and is just the entropy of the initial radiation field, SRŽ t s 0.: SAq R Ž t . s SAqR Ž t s 0 . s y N ln N y Ž N q 1 . ln Ž N q 1 .

Ž model Ž i . .

Ž 11a.

SAq R Ž t . s SAqR Ž t s 0 . s y N ln N y N y Ý pc Ž n . ln Ž n! . n

Ž model Ž ii. . Ž 11b. The counterparts of Eqs. Ž5. and Ž6. are calculated using Eq. Ž10. for the two models. For the resonant case the dimensionless parameters for these two ensembles are k trp N and N. The quantum conditional entropies express the residual information in the atomic and radiation systems respectively while retaining the quantum phase information and they are S Ž A q R
ž

' /

S Ž A q R
Ž 12 .

As shown formally by CA, for quantum entangled subsystems these can be nonmonotonic and may even be negative unlike their counterparts in classical conditional entropies which are non-negative. And finally the quantum mutual entropy is SŽA:R. s SA q SR y SAqR . As in the classical case, SŽA:R. G 0 but can be F 2 minw SA ,SR x. When A and R are classically maximally correlated, the classical upper bound SŽA:R. s min w SA ,SR x is saturated. The range between classical and quantum upper bounds corresponds to pure quantum entanglement and is called the state of supercorrelation w7x. This is the new feature found by CA and missed entirely in the earlier works w8x, probably because of the choice of the parameters chosen in their calculations, as will be shown here. In the present work, we will numerically explore these features as a function of the parameters of the model and the ensembles chosen, by focussing our attention on SŽA q R
Fig. 1. Ž SAq R y SR .rSA versus Ž b " v .y1 for three values of the scaled dimensionless coupling parameter, Ž k r v . s 0.5,2.5, and 5. The inset exhibits the regions of cross over from positive to negative values of Ž SAq R y SR .rSA .

™ 0 the system approaches its ground state, thus Ž SAq R y SR .rSA ™ 1 for all Ž krv ., as expected. For Ž krv . s 0.5,Ž SAq R y SR .rSA remains positive whereas for Ž krv . s 2.5 and 5, negative regions Žpure quantum entangled supercorrelated states. appear at finite temperatures. The region between the values 1 and zero of Ž SAq R y SR .rSA , represents maximally correlated states. For Ž krv . greater than 1, the interaction part of the Hamiltonian dominates over the TLS part. The eigenvalues V Ž n,2. then become negative for increasing values of n, as Ž krv . increases Žfor the examples considered here, V Ž n,2. is negative for Ž krv . s 0.5,n s 0; for Ž krv . s 2.5,n s 0,1, . . . 5; and for Ž krv . s 5,n s 0,1, . . . 25. and so at low temperatures, the corresponding weights dominate the density matrix, reminiscent of chaotic behavior of the system as Ž krv . increases. This is analogous to the result of Furuya et al. w12x, displaying chaotic features in a semiclassical version of the JC model, due to entanglement effects. We must caution that this ‘chaos’ analogy needs clarification because the origins of these two features are distinctly different. In the present analysis, the chaotic nature resides in the mixed global equilibrium Gibbs quantum density matrix. And, Furuya et al. w12x, treat a globally pure state in a non-equilibrium time-dependent ensemble where the field and the atom are initially prepared in coherent states Ža semi-classical feature. and the origin of chaos is in the interaction Hamiltonian with the non-rotating wave terms, not included in the

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present analysis. If our result has to be applicable to the radiation-atom system, such strong dipole-interaction strength can only be achieved if the cavity-size in the quantum optics experiment is of nanometer dimensions and atoms are cooled to very low temperatures as in w13x! In Fig. 2, we display Ž SAq R y SR .rSA versus Ž k trp N . for case Žb., model Ži. for three values of N. We observe for N s 1 Žcorresponding to a quantum state as in the single atom laser case w6x. Ž SAq R y SR .rSA is negative indicating the appearance of supercorrelated states for many values of Ž k trp N . whereas for N s 50, Žas for high mean photon number limit w8x, where one has quasiclassical states. these supercorrelation states do not appear except in a very small region for low Ž k trp N . values where quantum features manifest themselves because of symmetry requirements as shown in the foregoing calculations in this limit. Indeed, from Eq. Ž10. calculating the weights of the atomic states and the

'

'

'

'

Fig. 3. Ž SAq R y SR .rSA versus Ž k trp N . for pure coherent source for three values of N s1, 5, and 50. Note that the vertical scales are different for the three different cases. The insets exhibit the cross over from negative to positive values of Ž SAq R y SR .rSA for small times. Unlike in Fig. 2, we have oscillations for N s 50.

photon states for small Ž k t ., wg Ž t . ( Ž k t . 2 Ž N q 1., and PnŽ t . ( pŽ n. y Ž k t . 2 wŽ n q 1. pŽ n. y npŽ n.x, which are in conformity with the unitary time evolution and concomitant time reversal symmetry. From these it follows that Ž SAq R y SR .rSA for Ž k trp N . near zero has the form,  Ý n Ž n q 1 . p Ž n . ln Ž p Ž n . rp Ž n q 1 . . 4 ŽŽ N q 1. wlnŽ k 2 t 2 . y 1 qln Ž N q 1 . which goes to 0y Žthe numerator is positive for the two radiation models considered here.. ŽThis feature was missed in w8x; also, in their presentations of their calculations they used parameters which were not suitable for discovering the supercorrelated states.. This small time behavior is not perceptible in the figs. displayed. For N less than or equal to 5 the supercorrelated states occur but appear to be spread out; they are more spread out for N greater than 5, with a decrease in the number of supercorrelated regions as well as exhibit compression of the oscillations. There is no completely disentangled state as in case Ža. by virtue of its construction.

'

'

Fig. 2. Ž SAq R y SR .rSA versus Ž k trp N . for blackbody-like source for three values of N s1, 5, and 50. Note that the vertical scales are different for the three different cases. The insets exhibit the cross over from negative to positive values of Ž SAq R y SR .rSA for small times.

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A.K. Rajagopal et al.r Physics Letters A 259 (1999) 285–290

In Fig. 3, we similarly display the results for model Žii. of case Žb., for the same three values of N. Here again N s 1 displays large number of supercorrelated regions whereas they decrease for N s 5. For N s 50, we have a very small negative region of Ž SAq R y SR .rSA near zero time-scale as mentioned above, but there is a new feature - oscillations Žand revivals. for longer times. Similar oscillations and revivals were found and discussed in w3,4,8x. The positive regions of Ž AAq R y SR .rSA correspond to the maximally classically correlated states while we have no approach to the region of disentangled states, as expected. This is a manifestation of the difference between the two radiation models mentioned earlier. In conclusion, we have shown here by exploring numerically in the parameter space of the exactly soluble JC model that the quantum supercorrelated behavior where the conditional entropy exhibits negative values in both the equilibrium Žthermal. and nonequilibrium, Žtime-dependent. ensembles in the corresponding parameter space, clearly demonstrating the entanglement effects contained in the JC model. We believe that by this explicit example, we have elucidated the formal results of CA concerning the existence of supercorrelated states in the quantum entangled systems. As noted here, for the JC model their presence depends on the ensemble considered and the interaction strength between the two subsystems. In view of the recent advances in single-atom quantum optics, especially as we approach nanometric cavity Že.g., quantum wells. sizes, we hope these fully quantum correlated states will be explored experimentally in the near future. It may be worth pointing out that the quantum entanglement in the JC model arises from interaction between the two subsystems and therefore depends on the interaction strength between the two, in contrast to quantum entanglement discussed in the context of quantum information theory which is traditionally based on Bell-type bipartite spin-singlet states, for example. For a recent review see w14x. In this case, for the pure state density matrix associated with a single Bell state, the conditional entropy is trivially seen to be negative and hence the Bell state is supercorrelated in the CA sense. A different type of entropic analysis

based on Jaynes maximum entropy principle for this system has been given recently by one of us w15x.

Acknowledgements We wish to dedicate this paper to the memory of Professor Edwin T. Jaynes. Thanks are due to Dr. R.W. Rendell for drawing our attention to Ref. w12x. We thank the referee for his perceptive comments and suggestions for improving the presentation of this work. A.K.R. and K.L.J. thank the Office of Naval research for partial support of their work.

References w1x E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 Ž1963. 89. w2x W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1990, pp. 323–328. w3x S. Haroche, Phys. Today 51 Ž1998. 36. w4x B.W. Shore, P.L. Knight, in: W.T. Grandy Jr., P.W. Milonni ŽEds.., Physics and Probability - Essays in honor of Edwin T. Jaynes, Cambridge Univ. Press, Cambridge, UK, 1993, p. 15. w5x H. Walther, in: W.T. Grandy Jr., P.W. Milonni ŽEds.., Physics and Probability - Essays in honor of Edwin T. Jaynes, Cambridge Univ. Press, Cambridge, UK, 1993, p. 33. w6x M.S. Feld, Kyungwon An, Scientific American, 57 ŽJuly 1998.; Kyungwon An, R.R. Dasari, M.S. Feld, in: Atomic and Quantum Optics: High Precision Measurements, SPIE Proceedings Series 2799 Ž1996. 14; see also G. Raithel et al., in: P.R. Berman ŽEd.., Cavity Quantum Electrodynamics, Academic Press, New York, 1994, p. 57. w7x N.J. Cerf, C. Adami, Phys. Rev. Lett. 79 Ž1997. 5194. w8x S.J.D. Phoenix, P.L. Knight, Phys. Rev. A 44 Ž1991. 6023. w9x N.J. Cerf, C. Adami, Phys. Rev. A 55 Ž1997. 3371; A 56 Ž1997. 1721, 3470; Physica D 120 Ž1998. 62. w10x F.W. Cummings, Phys. Rev. 140 Ž1965. A1051. w11x J.R. Klauder, E.C.G. Sudarshan, Fundamentals of Quantum Optics, W.A. Benjamin, Inc., New York, 1968. w12x K. Furuya, M.C. Nemes, G.Q. Pellegrino, Phys. Rev. Lett. 80 Ž1998. 5524. w13x C.J. Hood, M.S. Chapman, T.W. Lynn, H.J. Kimble, Phys. Rev. Lett. 80 Ž1998. 4157. w14x P. Horodecki, R. Horodecki, M. Horodecki, Acta Phys. Slovaca 48 Ž1998. 141. w15x A.K. Rajagopal, quant-phr9903083, submitted for publication; R. Horodecki, M. Horodecki, P. Horodecki, Phys. Rev. A 59 Ž1999. 1799.