Exact Solutions of an Extended Dicke Model

Physics Letters A 341 (2005) 94–100 www.elsevier.com/locate/pla

Exact solutions of an extended Dicke model

Feng Pan a,b,∗,TaoWanga,JingPana, Yong-Fan Li a, J.P. Draayer b

a Department of Physics, Liaoning Normal University, Dalian 116029, PR China b Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA Received 1 February 2005; received in revised form 19 April 2005; accepted 4 May 2005 Available online 11 May 2005 Communicated by P.R. Holland

Abstract Exact solutions of the Dicke Hamiltonian plus a dipole–dipole interaction among N two-level atoms are derived based on an algebraic Bethe ansatz. As an example application, the role of the dipole–dipole interaction in ground-state-atom condensates of a system is explored.  2005 Elsevier B.V. All rights reserved.

PACS: 42.50.Ct; 42.50.Dv; 42.50.Hz; 42.50.Fx

Keywords: Extended Dicke model; Dipole–dipole interaction; Exact solution

A description of the interaction of a collection of atoms with a radiation field is a model problem in matter–field interaction phenomena that is of interest in many areas of physics. But because it is generally impossible to obtain exact solutions for this problem, approximations are normally adopted. The most common approximation assumes that the radiation field is quasi-monochromatic and the atoms are identical with no direct interactions between or among themselves. The Dicke model [1,2], which is the natural consequence of such an assumption, describes the interaction of N identical two-level atoms with a single-mode field in a perfect cavity. The spatial dimensions of the system are smaller than the wavelength of the field so that all the atoms experience the same field. The dipole–dipole interaction between atoms is neglected in the model, an assumption that is justified if the atomic wave functions do not overlap. Many aspects of this model have been studied, such as the dynamical properties of quantum entanglement and decoherence [3–5], squeezing [6–8], quantum phase transitions [9], and chaotic properties [10–13]. Recently an exact solution of the model in the electric dipole and rotating-wave approximations, including its extension to the multi-level case, has been reported [14].

* Corresponding author. E-mail address: [email protected] (F. Pan).

The Hamiltonian of the Dicke model in the electric dipole and rotating-wave approximations can be written as ˆ + ˆ † ˆ ˆ HD = ωa a + ω0S0 + g a S− + aS+ , (1) where ω and ω0 are frequencies of a single bosonic mode and the splitting of N two-level atoms, respectively, a and a† are annihilation and creation operators of the ﬁeld mode, respectively, satisfying the usual bosonic commutation relation a,a† = 1, [a,a]= a†,a† = 0, (2) ˆ = ˆ = ˆ = S0 j S0(j) is the operator of atomic inversion, and S+ j S+(j) (S− j S−(j)) is the collective atomic raising (lowering) operator. Sµ (µ = 0, +, −) satisfy the SU(2) Lie algebraic relations ˆ ˆ ˆ ˆ ˆ ˆ [S0, S±]=±S±, [S+, S−]=2S0. (3) It has been shown in [6,15,16] that dipole–dipole interaction between atoms plays a prominent role when atoms are close to each other at a distance that is much smaller than the resonance wavelength as may be the case when low modes of the cavity are involved. In such cases, the dipole–dipole interaction will be active and the effect should be taken into consideration. Spontaneous decay of two atoms in an over-damped cavity with dipole–dipole interaction was studied in [15]. Dissipation-induced stationary entanglement of two-level atoms with a dipole– dipole interaction was investigated in [16]. Recently, angular dependence of the dipole–dipole interaction in an approximately one-dimensional sample of Rydberg atoms has also been reported [17]. In what follows, it will be shown that the Dicke Hamiltonian (1) plus a dipole–dipole interaction among N atoms with ˆ ˆ ˆ ˆ N † † H = H + g S+(j)S−(j ) =− g + (ω + g )S + (ω − ω )a a + ω a + λS+ (a + λS−), (4) D 2 0 0 j=j where λ and ω are real parameters that satisfy g = λω and g = λg, is also algebraically solvable based on a Bethe ansatz used for ﬁnding exact solutions of SU(2) spin interaction [18–20], interacting boson systems [21,22], and ˆ ˆ † the original Dicke model [14]. In this case the atomic inversion plus the photon number operator Λ = S0 + a a ˆ2 ˆ ˆ ˆ ˆ and the total pseudo-spin operator S = S+S− + S0(S0 − 1) commute with the Hamiltonian (4), namely [Λ,ˆ Hˆ ]=0, Sˆ2, Hˆ = 0. (5) Therefore, Λˆ and Sˆ2 are two conserved quantities. Hence, the atomic inversion plus the photon number Λ and pseudo-spin S are good quantum numbers to be used to label the corresponding wave function of the system. It can be proven that the Bethe ansatz eigenvectors used for diagonalizing Hamiltonian (4) can be written as | ; =− + = (ζ ) (ζ ) ··· (ζ ) | =− ; = ζ Λ S k J+ x1 J+ x2 J+ xk SMS S n 0 , (6) where |SMS =−S; n = 0 is the SU(2) lowest-weight and photon vacuum state satisfying ˆ a|SMS =−S; n = 0=S−|SMS =−S; n = 0=0 (7) and (ζ ) + λ ˆ J+ x = a + S+, (8) i − (ζ ) 1 λxi (ζ ) = in which xi (i 1, 2,...,k) are spectral parameters that are to be determined, and ζ is an additional quantum number that is used to distinguish different eigenvectors with the same quantum number Λ. To prove that the Bethe ansatz eigenvectors given in (6) can indeed be used to solve the problem, one may directly apply Hamiltonian (4) to (6) to establish the corresponding eigen-equation with ˆ | ; =− + = (ζ )| ; =− + H ζ Λ S k Ek ζ Λ S k . (9) 96 F. Pan et al. / Physics Letters A 341 (2005) 94–100

Then, by using the identity

λS+ 1 = J+(x) − J+(0) , (10) 1 − λx λx and the following commutation relations: 1 † 1 S ,J+(x) = J+(x) − J+(0) , a a,J+(x) = J+(x) − J+(x) − J+(0) , (11) 0 λx λx

2 2 2λ S0 2λ J−(0), J+(x) = 1 − , J−(0), J+(x) ,J+(y) =− J+(x) − J+(y) , (12) 1 − λx x − y where + J+(0) = a + λS+,J−(0) = a + λS−, (13) the eigen-equation (9) becomes

E(ζ )|ζ ; Λ =−S + k k k N 1 1 = − g + (ω − ω )k − (ω + g )S + (ω + g − ω + ω ) |ζ ; Λ =−S + k 2 0 0 λ (ζ ) i=1 xi k (ω0 + g + ω − ω ) 1 (ζ ) (ζ ) (ζ ) (ζ ) − J+(0) J+ x ···J+ x J+ x ···J+ x |SM =−S; n = 0 λ (ζ ) 1 i−1 i+1 k S i=1 xi

k 2 (ζ ) (ζ ) 2λ S (ζ ) (ζ ) − ω J+( ) J+ x ···J+ x − J+ x ···J+ x |SM =−S; n = 0 1 i−1 (ζ ) 1 i+1 k S 0 = λx − 1 i 1 i − (ζ ) ··· (ζ ) (ζ ) ··· (ζ ) (ζ ) ··· (ζ ) ω J+(0) J+ x1 J+ xi−1 J+ xi+1 J+ xj−1 J+ xj+1 J+ xk i=j 2 2λ (ζ ) (ζ ) × J+ x − J+ x |SMS =−S; n = 0. (14) (ζ ) − (ζ ) i j xi xj Clearly, Eq. (14) consists of two parts. One is proportional to the original Bethe ansatz eigenvector (6), which gives the corresponding eigenvalue with k N 1 1 E(ζ ) =− g + k(ω − ω ) − S(ω + g ) + (ω + g − ω + ω ) , (15) k 2 0 λ 0 (ζ ) i=1 xi while the second part contains k terms not having the initial Bethe ansatz form. The cancellation of these terms { (ζ )} imposes special conditions on the c-numbers xi , which are commonly called the Bethe ansatz equations, which (ζ ) = leads to the solution when the spectral parameters xi (i 1, 2,...,k) satisfy the following set of equations:

2 2 1 2λ S 2λ g (ω0 + g − ω + ω ) + g − 1 − = 0 (16) (ζ ) (ζ ) − (ζ ) − (ζ ) xi λxi 1 j(=i) xi xj for i = 1, 2,...,k. It can be seen from (9) that the quantum number k = Λ + S. While the additional quantum number ζ labels the ζ th set of roots of Eq. (16). Since the dipole–dipole interaction is related to geometric configuration of atoms, the overall strength g only provides with an approximation to realistic situations. The dipole–dipole interaction strength will be effectively F. Pan et al. / Physics Letters A 341 (2005) 94–100 97 changed for different kinds of atoms since the strength is related to the resonant wave frequency and proportional to the spontaneous emission rate of atoms in free space. For given set of atoms, to change the dipole–dipole interaction strength g may be achieved, for example, by varying the distance among atoms. As an example application of the exact solutions, we consider a large N = 5000 collection of identical two-level atoms coupled with a single- mode field in a perfect cavity with frequency ω = ω0 = 10 and g = 0.1. With increasing dipole–dipole interaction strength g, which can be achieved by varying the parameter λ = g/ω and keeping g = 0.1 unchanged, we study how the dipole–dipole interaction affects the system’s behavior. In Fig. 1, the occupation number of atoms in excited and unexcited states, respectively, and that of photons of the field for the ground state of the system with k = 2500 and 2S = N = 5000 are shown. Since only state with Λ = 0 is considered in this example, k = S gives the total number of excited atoms plus number of photons. Ground state occupation probability of photons rp and that of excited atoms rex are defined by

= † = − rp a a g/k, rex 1 rp, (17) † respectively, where a ag stands for the expectation value of the number of photons calculated by using the normalized wave function given by (6) for the ground state determined by (16). While the quantities rex and rg are deﬁned by = = − rex krex/N, rg 1 rex, (18) in which rex provides the percentage of excited atoms in the ground state with respect to the total number of atoms N, and rg gives that of atoms remaining in the unexcited state. In Fig. 1, the left panel provides the ground state occupation probabilities of photons rp, and excited atoms rex as functions of the strength of the dipole–dipole interaction g, respectively, with respect to the total number of photons and excited atoms k. While the right panel shows the ground state occupation probabilities of excited atoms rex, and unexcited atoms rg as functions of the strength of the dipole–dipole interaction g, respectively, with respect to the total number of atoms N. As can be seen from Fig. 1, there are rp = 50% of photons and rex = 50% excited atoms in the ground state of the system when the dipole–dipole interaction is weak and negligible in comparison to the coupling strength between atoms = = and the ﬁeld. Therefore, there are rex 25% of the atoms in the excited state in this case, while rg 75% of the atoms remain in the unexcited state. With increasing dipole–dipole interaction strength g, Fig. 1 shows that the number of excited atoms becomes fewer and fewer. And if g is increased beyond a certain value, almost all atoms remain in the unexcited state, and the photons become spectators. The conclusion is that the ground state of the

Fig. 1. Occupation probabilities for the ground state of the N = 5000 atom system with a single-mode ﬁeld in a perfect cavity with frequency ω = ω0 = 10 and g = 0.1 as a function of the strength of the dipole–dipole interaction g . In the left panel, the upper and lower curves correspond to the occupation probability of photons rp and that of excited atoms rex, respectively. In the right panel, the upper and lower curves represent the occupation probability of unexcited atoms rg and that of excited atoms rex, respectively. 98 F. Pan et al. / Physics Letters A 341 (2005) 94–100

Fig. 2. The ground state entanglement measure of the system η Fig. 3. Energy (in arbitrary unit) of the first excited state of the as a function of the strength of the dipole–dipole interaction g . N = 5000 system as a function of the strength of the dipole–dipole in- The other parameters of the theory are the same as those used teraction g . The other parameters of the theory are the same as those in Fig. 1. used in Fig. 1. system will become a ground-state-atom condensate when g reaches a critical point, even if the radiation field is present. This phenomenon is due mainly to competition between the dipole–dipole interaction and the field–atom coupling. When the field–atom coupling is strong, the interaction of atoms with the field is dominant, which leads to strong entanglement between the atoms and the field. When the dipole–dipole interaction dominates, the field– atom interaction is reduced and ultimately rendered negligible because there are a large number of atoms with large pseudo-spin S so that the dipole–dipole interaction term dominates. In order to show the ground state entanglement quantitatively, we adopt the entanglement measure for any pure bipartite system defined by

=− =− η Tr(ρf logk+1 ρf) Tr(ρa logk+1 ρa), (19) where ρf and ρa are the reduced density matrix obtained by taking partial trace over the atoms and the ﬁeld, respectively. We use the logarithm to the base k + 1 instead of base 2 to ensure that the maximal measure is normalized to 1. Fig. 2 shows the ground state entanglement measure of the system as a function of the strength of the dipole–dipole interaction g. In accordance with the results shown in Fig. 1, it can be seen from Fig. 2 that the ground state entanglement measure η indeed gradually decreases with increasing of g. However, the change of η is slower than those of the occupation probabilities, which indicate that the phase transition in this case is a second order one. Fig. 3 shows the energy gap between the ﬁrst excited state and the ground state of the system as a function of the strength of the dipole–dipole interaction g. It is obvious that the energy gap increases steadily with the increasing g, which shows the system indeed undergoes the second order phase transition from the photon-atom entangled phase to the ground-state-atom condensed phase when g increases. The above model extended to including N non-identical two-level atoms, with each level splitting given by ωj , can also be solved exactly using the same procedure. The Hamiltonian is

N ˆ + ˆ † ˆ ˆ ˆ ˆ H = ωa a + ωj S0(j) + g a S− + aS+ + g S+(j)S−(j ) j=1 j=j N N ˆ ˆ † ˆ ˆ =− g + (ω − ω )Λ + (ω + g + ω − ω)S (j) + ω a + λS+ (a + λS−) (20) 2 j 0 j=1 F. Pan et al. / Physics Letters A 341 (2005) 94–100 99 with the same parameterization used for Eq. (4). In this case, the total pseudo-spin is no longer conserved. The eigenvectors of (20) with Λ = k − N/2 can be written as

N (ζ ) (ζ ) (ζ ) 1 1 1 1 1 1 |ζ ; Λ = k − N/2=A+ x A+ x ···A+ x − ; − ; ...; − ; n = 0 , (21) 1 2 k 2 2 2 2 2 2 where the lowest weight state satisﬁes

N N

1 1 1 1 1 1 ˆ 1 1 1 1 1 1 a − ; − ; ...; − ; n = 0 = S−(j) − ; − ; ...; − ; n = 0 = 0 (22) 2 2 2 2 2 2 2 2 2 2 2 2 = (ζ ) for j 1, 2,...,N. The operators A+(xi ) used in (21) are N ˆ (ζ ) + λS+(j) A+ x = a + . (23) i − + + − (ζ ) j=1 1 (ωj g ω ω)xi Then, the eigen-energies of the system with Λ = k − N/2 can be written as

N k 1 E(ζ ) =−Ng + (ω − ω )k − ω / + k j 2 (ζ ) (24) j=1 i=1 xi with the following set of Bethe equations: N 2 2 1 λ 2λ ω + ω − 1 − = 0 (25) (ζ ) + + − (ζ ) − (ζ ) − (ζ ) xp j=1 (ωj g ω ω)xp 1 q(=p) xp xq for p = 1, 2,...,k. In summary, exact solutions of the Dicke Hamiltonian plus a dipole–dipole interaction among N two-level atoms are derived based on an algebraic Bethe ansatz. The results should be helpful in understanding the effect of the dipole–dipole interaction among N atoms and the competition with the atom–field interaction, especially when atoms are close to each other at a distance much smaller then the resonance wavelength so low modes of the cavity are involved. As has been demonstrated, the role of the dipole–dipole interaction term in the system becomes important when a large number of atoms are coupled with a single-mode cavity even with comparatively weaker dipole–dipole coupling among atoms. The exact results show that the system will become the ground-state-atom condensate when the dipole–dipole coupling strength reaches the critical point. In such cases, one cannot treat the dipole–dipole interaction terms perturbatively. It is well know that non-perturbative and non-linear phenomena play dominant roles in such quantum phase transitional situation and atom–field entanglement. The results shown may be helpful in preparing atom–field entangled states by using cavity QED with large number of atoms close enough to each other with low cavity modes.

Acknowledgements

Support from the US National Science Foundation (0140300), the Southeastern Universities Research Associa- tion, the Natural Science Foundation of China (10175031), the Natural Science Foundation of Liaoning Province (2001101053), the Education Department of Liaoning Province (202122024), and the LSU–LNNU joint research program (C164063) is acknowledged. 100 F. Pan et al. / Physics Letters A 341 (2005) 94–100

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