aa r X i v : . [ qu a n t - ph ] M a y Non-Equilibrium Quantum Phases of Two-Atom Dicke Model
Aranya B. Bhattacherjee
School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067, India
In this paper, we investigate the non-equilibrium quantum phases of the two-atom Dicke model,which can be realized in a two species Bose-Einstein condensate interacting with a single light modein an optical cavity. Apart from the usual non-equilibrium normal and inverted phases, a non-equilibrium mixed phase is possible which is a combination of normal and inverted phase. A new kindof quantum phase transition is predicted from non-superradiant mixed phase to the superradiantphase which can be achieved by tuning the two different atom-photon couplings. We also show thata quantum phase transition from the non-superradiant mixed phase to the superradiant phase isforbidden for certain values of the two atom-photon coupling strengths.
Keywords:
Non-Equilibrium Dicke Model, Quantum Phase Transition.
PACS numbers: 37.30.+i,42.50.Pq
I. INTRODUCTION
The interaction of a collection of atoms with a radia-tion field has always been an important topic in quantumoptics. The Dicke model (DM) which describes interac-tion of N identical two level atoms with a single radia-tion field mode, established the importance of collectiveeffects of atom-field interaction, where the intensity ofthe spontaneously emitted light is proportional to N rather than N [1]. The spatial dimensions of the en-semble of atoms are smaller than the wavelength of theradiation field. As a result, all the atoms experience thesame field and this gives rise to the collective and co-operative interaction between light and matter. The DMexhibits a second-order quantum phase transition (QPT)from a non-superradiant normal phase to a superradiantphase when the atom-field coupling constant exceeds acertain critical value [2–5]. The experimental observationof the QPT predicted in the DM required that the col-lective atom-photon coupling strength to be of the sameorder of magnitude as the energy separation between thetwo atomic levels. In conventional atom-cavity setup thiscondition was impossible to satisfy until it was observedrecently in a trapped Bose-Einstein condensate (BEC) inan optical cavity [6–9]. In the BEC setup, the two spinstates of the original DM are the two momentum states ofthe BEC which are controlled by the atomic recoil energyand Raman pumping schemes. This approach is similarto a novel scheme proposed by Dimer et. al.[10]. An im-portant aspect of these experimental developments is thepossibility to explore exotic phases mediated by the cav-ity field. The superradiance phase transition in a BECis accompanied by self-organization of the atoms into achecker board pattern [6–9, 11].Interesting equilibrium and non-equilibrium phaseshave been predicted in the DM with BEC [12, 13], in-cluding crystallization and frustration [14], as well asspin glass phase [15–18]. Multimode DM has also beenexplored recently, revealing interesting physics such asAbelian and non-Abelian gauge potentials [19], spin-orbitinduced anomalous Hall effect [20], and prediction of the Nambu-Goldstone mode [21]. An interesting extensionof the BEC Dicke model is the optomechanical Dickemodel which has been proposed for detection of weakforces [22, 23]. In the present paper, we investigate thenon-equilibrium properties of the two-atom Dicke model,which can be realized by two species BEC in an opticalcavity. Apart from the usual non-equilibrium normal andinverted phases, the dynamical phase diagram reveals anew kind of non-equilibrium mixed phase. This givesrise to a new quantum phase transition from the mixedphase to the superradiant phase by manipulation of thetwo distinct atom-photon coupling strengths. In addi-tion, we show that a quantum phase transition from thenon-superradiant phase to the superradiant phase is notallowed for certain values of the atom-photon couplingstrengths of the two set of atoms. II. THE MODEL
Figure 1: (color online)Experimental setup showing two setsof cold atoms (blue and green) in an optical cavity with trans-verse pumping. The two sets of atoms have different atom-photon coupling strengths which depends on their position inthe cavity. On increasing the transverse pump intensity, onetype of atoms can reach the critical point earlier.
We consider two different ensembles of N and N atoms coupled simultaneously to the quantized field of anoptical cavity mode (Fig.1). The two sets of atoms havetransition frequencies ω and ω while the frequency ofthe cavity mode is ω c . The cavity is pumped by an trans-verse external laser with frequency ω p . The light-mattercoupling strengths for the two sets of atoms are λ and λ . These coupling strengths λ and λ can be written as λ i = λ i Ω P / ω p − ω i ) ( i = 1 , λ i is the single atom-cavity mode coupling while Ω P is the transverse pumpbeam Rabi frequency. The detuning ( ω p − ω i ) is consid-ered to be large so as to avoid spontaneous emission. Theeffective Hamiltonian of the system takes the form of atwo-atom Dicke model with H = ~ ω J z + ~ ω J z + ~ ω c a † a (1)+ ~ λ √ N ( J + J − ) (cid:0) a + a † (cid:1) + ~ λ √ N ( J + J − ) (cid:0) a + a † (cid:1) , where ~J i = ( J ix , J iy , J iz ) is the effective collective spinof length N i / J i ± = J ix ± iJ iy .We now discuss the non-equilibrium dynamics arisingfrom the above two-atom Dicke model.The semi-classicalequations of motion for the system are given by˙ J z = iλ √ N (cid:0) a † + a (cid:1) ( J − − J ) , (2)˙ J z = iλ √ N (cid:0) a † + a (cid:1) ( J − − J ) , (3)˙ J − = − iω J − + 2 iλ √ N (cid:0) a † + a (cid:1) J z , (4)˙ J − = − iω J − + 2 iλ √ N (cid:0) a † + a (cid:1) J z , (5)˙ a = − ( κ + iω c ) a − iλ √ N ( J + J − ) (6) − iλ √ N ( J + J − ) . Here κ is the decay rate of the cavity photons. Inaddition the magnitude of pseudo-angular momentum isconserved, J z + | J − | = N , (7) J z + | J − | = N . (8) The long time steady state solutions from the equa-tions of motion can be studied with ˙ ~J i = 0( i = 1 ,
2) and˙ a = 0. These fixed point solutions can be stable or un-stable. Separating a = a + ia , J ± i = J ix ± J iy ( i = 1 , κa − ω c a = 0 , (9) κa + ω c a = − λ √ N J x − λ √ N J x , (10) ω J y = 0 , (11) ω J x = 4 λ √ N a J z , (12) ω J y = 0 , (13) ω J x = 4 λ √ N a J z . (14)An analysis of these equations leads us to four typesof steady states, namely ( a = 0 , J z = ± N / , J z = ± N / a = 0 , J z = − N / , J z = − N /
2) is the normal phase while ( a = 0 , J z = N / , J z = N /
2) is the inverted phase. The states( a = 0 , J z = − N / , J z = N /
2) and ( a = 0 , J z = N / , J z = − N /
2) are called mixed phases. As weshall show later that these mixed phases generate in-teresting non-equilibrium phase diagrams. The criticalcoupling strength corresponding to the onset of superra-diance starting from the normal, inverted or mixed phaseis obtained by putting ~J i = (0 , , ± N i /
2) ( i = 1 , J z = − N J z = − N N ormal P hase ) : λ c = s ( κ + ω ) ω ω − λ ω ω , (15) λ c = s ( κ + ω ) ω ω − λ ω ω , (16) J z = N J z = N Inverted P hase ) : λ c = − s ( κ + ω ) ω ω + λ ω ω , (17) λ c = − s ( κ + ω ) ω ω + λ ω ω , (18) J z = − N J z = N M ixed P hase
1) : λ c = s ( κ + ω ) ω ω + λ ω ω , (19) λ c = − s ( κ + ω ) ω ω − λ ω ω , (20) J z = N J z = − N M ixed P hase
2) : λ c = − s ( κ + ω ) ω ω − λ ω ω , (21) λ c = s ( κ + ω ) ω ω + λ ω ω . (22) Λ Λ a Λ Λ b Figure 2: (color online)Dynamical phase diagrams of the sta-ble roots corresponding to the normal phase (Eqns.27) in the( λ , λ ) plane. The parameters used are ω /κ = ω /κ = ω c /κ .The white region is the non-superradiant normal phase whilethe superradiant phase is indicated by the contours. Thedarker region in the contour corresponds to low superradiancewhile light region corresponds to high superradiance. The leftplot is the ω + root while the right plot is the ω − root. These set of expressions reveals one interesting pointthat the critical coupling strength for one set of atomsdepends on the coupling strength of the other set of atoms. For a given set of J z and J z , Eqns. 15-22 deter-mines the boundary between the nonsuperradiant (nor-mal/inverted/mixed) and superradiant phase. A triv-ial manipulation of Eqns. 9-14 leads us to the followingequation for J z = − N / J x = 0, J x (cid:18) ω (cid:0) κ + ω c (cid:1) + 8 λ N ω c J z (cid:19) = 0 . (23)Now there are two possibilities depending on whether J x = 0 and J z = ± N / J x = 0 and J z = − N ω ( κ + ω c ) / λ ω c . The first condition impliesthat both the set of atoms are in the non-superradiantphase. The second solution corresponds to the casewhere the first set of atoms are in the non-superradiantnormal phase while the second set of atoms are in thesuperradiant phase. Substituting the second expres-sion for J z from above in the expression for λ c = q ω ( κ + ω c )4 ω c + λ ω N ω J z ,one obtains λ c = 0. This im-plies that by keeping one coupling strength arbitrarilylow, one could enter the superradiant phase by manip-ulating the second coupling strength alone. This pointwould be more clear when we discuss the dynamical phasediagrams in the next section. III. DYNAMICAL PHASE DIAGRAMS
In this section, we explore the fluctuation dynamicsabove the steady state (fixed points). In particular, wewill consider the instability of the normal ( ↓↓ ), inverted( ↑↑ )and mixed phases ( ↑↓ or ↓↑ ). To this end, we write a = a + δa , J i − = J i − + δJ i − ( i = 1 , a = 0, J i − = 0 and J iz = ± N i / i = 1 , δa = − ( κ + iω c ) δa − iλ √ N ( δJ + δJ − ) (24) − i iλ √ N ( δJ + δJ − )˙ δJ − = − iω δJ − + 2 iλ √ N (cid:0) δa † + δa (cid:1) J z , (25)˙ δJ − = − iω δJ − + 2 iλ √ N (cid:0) δa † + δa (cid:1) J z . (26)We now write δa = Ae − iηt + B ∗ e iη ∗ t , δJ − = Ce − iηt + D ∗ e iη ∗ t and δJ − = Ee − iηt + F ∗ e iη ∗ t and equate coeffi-cients with the same time dependence to obtain algebraicequations for A , B , C , D , E and F . The correspondingself consistency equations yields a quadratic equation for ω , whose roots characterize the possible instabilities for η = 0. These instabilities describe the boundaries in the Λ Λ a Λ Λ b Figure 3: (color online)Dynamical phase diagrams of the sta-ble roots corresponding to the mixed phase 1 (Eqns.29) in the( λ , λ ) plane. The parameters used are ω /κ = ω /κ = ω c /κ .The left plot is the ω + root while the right plot is the ω − root.The white region is the non-superradiant mixed phase 1 whilethe superradiant phase is indicated by the contours. emerging dynamical phase diagrams. In particular, thevarious boundaries between exponentially growing anddecaying fluctuations are given as:Normal Phase: ω ± = 2 (cid:18) λ ω + λ ω (cid:19) ± s (cid:18) λ ω + λ ω (cid:19) − κ , (27)Inverted Phase: ω ± = − (cid:18) λ ω + λ ω (cid:19) ± s (cid:18) λ ω + λ ω (cid:19) − κ , (28)Mixed Phase 1: ω ± = 2 (cid:18) λ ω − λ ω (cid:19) ± s (cid:18) λ ω − λ ω (cid:19) − κ , (29)Mixed Phase 2: ω ± = − (cid:18) λ ω − λ ω (cid:19) ± s (cid:18) λ ω − λ ω (cid:19) − κ , (30)Note that the ”inverted phase” is the inversion of the”normal phase” around ω = 0 boundary while ”mixedphase 2” is the mirror inversion of ”mixed phase 1”.The contour plot of the stable roots ω ± of Eqns.27 asa function of λ and λ for the normal phase is shownin Fig.2. Fig.2(a) shows the boundary separating thenon-superradiant phase and the superradiant phase forthe ω + root. This boundary is the curve that joins λ c (with λ = 0) and λ c (with λ = 0). Below this bound-ary is the non-superradiant normal phase while above this curve is the superradiant phase. As we move alongthe y -axis ( λ = 0), we reach the superradiant phase at λ c = ( κ + ω ) ω / ω . This analysis agrees with oursteady state analysis of the previous section. Thus alongthe x or the y axis, the system behaves as if only oneset of atoms are present. In any other direction, both setof atoms contribute to the dynamics. Note that whiteregion is the non-superradiant normal phase while thesuperradiant phase is indicated by the contours. Thedarker region in the contour corresponds to low superra-diance while light region corresponds to high superradi-ance. Fig2b shows the plot of ω − root. The combinationof ω + and ω − determine the complete boundary betweenthe non-superradiant phase and superradiant phase. Thephase diagrams of the inverted phase (not shown) is themirror inversion of the normal phase. In a similar man-ner, one can determine the dynamical phase diagramsfor the mixed phases. In fig.3(a) and 3(b), we demon-strate this for the mixed phase 1. A new kind of dynam-ical phase diagram emerges for the mixed phase. Thephase diagram now splits into two distinct superradiantregimes separated by the non-superradiant phase. Thetwo critical points λ c ( λ = 0) and λ c ( λ = 0) alongthe x and y axis are still the same. In Fig.3a, ω + root isshown and on moving along the x axis (increasing λ ), weencounter the usual superradiant phase with increasingenergy. On the other hand, moving along the y axis (in-creasing λ ), we get a superradiant phase of constant lowenergy. There are regions in the phase diagram, whereeven when λ > λ c and λ > λ c , the system stays inthe non–superradiant phase. Interestingly for λ = λ ,the superradiant phase can never be reached. Infact theenergy landscape in the ( λ , λ ) plane gives an impres-sion of anti-crossing of energy levels. Fig.3(b) shows theplot of ω − root whose behavior is opposite to that of the ω + root. A superradiant phase with decreasing energy isencountered along the y axis while a constant high en-ergy phase is encountered along the x axis. Note that ifwe choose ω = ω , then the energy plots of Fig.2 andFig.3 becomes asymmetric (figure not shown).The current predictions can be tested in an experimentsimilar to that of Ref.[6] but with two species condensate.In addition one has to look into the long duration of theseexperiments beyond the 10 ms time scale as noted inRef.[12] IV. CONCLUSIONS
In summary, we have investigated the non-equilibriumquantum phases of a two-atom Dicke model, which isrealized in a collection two set of cold atoms couplingsimultaneously to a single quantized cavity mode. Withinthe framework of the non-equilibrium two-atom Dickemodel, we reveal a rich and new set of phase diagrams.We have shown the existence of a new kind of quantumphase transition from the non-superradiant mixed phase(where one set of atoms are in the normal phase whilethe other set is in the inverted phase) to the superradiantphase. In addition, we have demonstrated that in thequantum phase diagram of the mixed phase, there areregions where the superradiant phase cannot exist even ifthe light-matter coupling constants of both set of atomsare above the critical value. These predictions can berealized in a two species cold atoms in an optical cavity.
V. ACKNOWLEDGEMENTS
A. Bhattacherjee acknowledges financial support fromthe Department of Science and Technology, New Delhi forfinancial assistance vide grant SR/S2/LOP-0034/2010. [1] R.H. Dicke, Phys. Rev. , 99 (1954).[2] K. Hepp and E.H. Lieb, Ann. Phys.(NY), ,360 (1973).[3] Y.K. Wang and F. T. Hioes, Phys. Rev. A, , 831 (1973).[4] F.T. Hioes, Phys. Rev. A, , 1440 (1973).[5] Clive Emary and Tobias Brandes, Phys. Rev. E ,066203 (2003).[6] K. Baumann, C. Guerlin, F. Brennecke and T. Esslinger,Nature (London), , 1301 (2010).[7] K. Baumann, R. Mottl, F. Brennecke and T. Esslinger,Phys. Rev. Lett. , 140402 (2011).[8] F. Brennecke, R. Mottl, K. Baumann, R. Landig, T. Don-ner and T. Esslinger, Proc. natl. Acad. Sci.USA, ,11763 (2013).[9] H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger,Rev. Mod. Phys. , 553 (2013).[10] F. Dimer, B.Estienne, A. S. Parkins and H.J. Carmichael,Phys. Rev. A, , 013804 (2007).[11] D. Nagy, G. Konya, G. Szirmai and P. Domokos, Phys.Rev. Lett. , 130401 (2010).[12] M.J. Bhaseen, J. Mayoh, B.D. Simons and J. Keeling,Phys. Rev. A, , 013817 (2012).[13] Ni Liu, Jinling Lian, Jie Ma, Liantuan Xiao, Gang Chen, J.-Q. Liang and Suotang Jia, Phys. Rev. A, , 033601(2011).[14] S. Gopalakrishnan, B.L. Lev and P.M. Goldbart, Nat.Phys. , 845 (2009).[15] S. Gopalakrishnan, B.L. Lev and P.M. Goldbart, Phys.Rev. Lett., , 277201 (2011).[16] P. Strack, S. Sachdev, Phys. Rev. Lett. , 277202(2011).[17] M. Buchhold, P. Strack, S. Sachdev and S. Diehl, Phys.Rev. A, , 063622 (2013).[18] A. Andreanov and M. M¨uller, Phys. Rev. Lett., ,177201 (2012).[19] J. Larson and S. Levin, Phys. rev. Lett. , 013602(2009).[20] J. Larson, Phys. Rev. A, , 051803 (2010).[21] Jingtao Fan, Zhiwei Yang, Yuanwei Zhang, Jie Ma, GangChen and Suotang Jia, Phys. Rev. A, , 023812 (2014).[22] Neha Aggarwal, Sonam Mahajan and A. Bhattacherjee,Jour. of Mod. Opt., , 1263 (2013).[23] Priyanka Verma, A. Bhattacherjee and Man Mohan,Canad. J. Phys.90