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Exact Solutions of an Extended Dicke Model

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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). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.05.005 F. Pan et al. / Physics Letters A 341 (2005) 94–100 95 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.
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