Quantum Phase Transition and Berry Phase in an Extended Dicke Model
QQuantum Phase Transition and Berry Phase in an Extended DickeModel
C. A. Estrada Guerra , , J. Mahecha-G´omez and J. G. Hirsch Abstract — We investigate quantum phase transitions,quantum criticality, and Berry phase for the ground stateof an ensemble of non-interacting two-level atoms embeddedin a non-linear optical medium, coupled to a single-modequantized electromagnetic field. The optical medium is pumpedexternally through a classical electric field, so that there isa degenerate parametric amplification effect, which stronglymodifies the field dynamics without affecting the atomic sector.Through a semiclassical description the different phases ofthis extended Dicke model are described. The quantum phasetransition is characterized with the expectation values ofsome observables of the system as well as the Berry phaseand its first derivative, where such quantities serve as orderparameters. It is remarkable that the model allows the controlof the quantum criticality through a suitable choice of theparameters of the non-linear optical medium, which couldmake possible the use of a low intensity laser to access thesuperradiant region experimentally.
I. INTRODUCTIONThermal phase transitions occur in many physical systems.They are observed as changes in macroscopic properties,quite often discontinuous, when certain thermodynamicparameters of the system change. Unlike thermal phasetransitions, which happen at finite temperature, quantumphase transition (QPT) occurs at T = 0 , where quantumfluctuations survive and are determined by the Heisenberguncertainty principle. They are characterized by suddenchanges in some order parameters (or their derivatives) whenexternal parameters are varied. QPT is a topic of currentinterest in the areas of chaos, quantum optics, condensedmatter, among others [1]–[4].The Dicke model [5] describes the interaction betweena single-mode electromagnetic field contained in an opticalcavity and an ensemble of N non-interacting identical atoms.The model is known in the literature of quantum optics andcondensed matter due to superradiance; the emission processthat interferes constructively and with an energy densityproportional to N [6]. This system shows a second-orderthermal phase transition, which occurs at finite temperature[7, 8]. Furthermore, for T = 0 , the ground state of thesystem becomes degenerate with the first excited state,when the atom-field interaction reaches its critical value,leading the system from a normal to a superradiant phase.In the thermodynamic limit, this level crossover becomesa real QPT, exhibiting a discontinuity in the derivatives of Universidad de Antioquia, Instituto de F´ısica, Facultad de CienciasExactas y Naturales, Calle 70 No. 52-21, Medell´ın, Colombia. Email:[email protected] Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma deM´exico, Apartado Postal 70-543, C.P. 04510 CDMX, Mexico. the ground state energy, the average number of photonsand of excited atoms [9]. QPT have been characterizedin several extended Dicke models, adding the interactionbetween the atoms [10]–[14], nonlinear light interactions[15, 16], atom-optomechanical systems [17], and others.These studies generated interesting theoretical advances, butthe experimental observation of the superradiant phase ina cavity-atom system remains challenging. It is due to thestrong coupling that it is required between the atoms andphotons, which must be of the order of the atomic andfield frequencies [18]. Although the no-go theorem [19]–[22]prohibits the transition to the superradiant phase, severalexperimental results have shown that it is possible to reachthat phase [23]–[25].From a different perspective, Berry showed that forHamiltonians which depend on a set of parameters thatvary cyclically and adiabatically over time, the associatedwave function acquires a phase factor of geometric nature,in addition to the dynamical phase due to temporal evolution[26]. Therefore, since the ground state of many-body systemsshows crossover or avoided crossover between the groundstate and the first excited state, due to the variation of anexternal parameter of the system, the geometric phase andits derivative with respect to the external parameter can detectthese irregularities [27]. The behavior of the geometric phasein the thermodynamic limit has been studied in the XY spinmodel [28], the Lipkin-Meshkov-Glick model [29], the Dickemodel [12, 30, 31], and experimentally in the Heisenberg XY model [32] and the Zak phase in topological Bloch bands[33].In this work, we consider an extended model based onan ensemble of non-interacting two-level atoms, which areembedded in a nonlinear optical material pumped by aclassical field. Using semi-classical analysis, exact in thethermodynamic limit, it is shown that the critical valueof the atom-field interaction can be noticeable reduced bythe presence of the the non-linear terms, in comparisonwith the one needed in the standard Dicke model to reachthe ultra-strong coupling regime. We also show that theexpectation values of the field operators change, while theexpectation values of atomic operators are not affected bynon-linear terms. Finally, we derive the geometric phase ofthe ground state induced by the change in the quantized field,showing its usefulness in detecting the QPT, as well as thescaling behavior in the vicinity of the separatrix betweenphases.The structure of the article is as follows. In section 2,we give an introduction to the Dicke model, showing their a r X i v : . [ qu a n t - ph ] J un ain characteristics. Afterwards, we present the extendedmodel. The semiclassical description is given in section 3,where we obtain the classical Hamiltonian, the classificationof the fixed points, and the expectation values of both theatomic and photonic operators. In section 4 we deducethe expression of the Berry phase for the ground state, itsderivative with respect to the atom-field coupling parameterand its scaling behavior. Finally, the conclusions of ourresults are given in section 5.II. M ODEL H AMILTONIAN
The Dicke model describes an ensemble of N non-interacting identical two-level atoms with atomictransition frequency ω , interacting with a single-moderadiation field with frequency ω f inside of a high finesseoptical cavity. To obtain the Dicke model Hamiltonian,the following assumptions are made: ( i ) the dipoleapproximation (long-wavelength limit, where the fieldwavelength is much greater than the size of region in whichthe atoms are confined). ( ii ) Only two atomic levels thatinteract with the electromagnetic field are considered. ( iii ) The quantum state of lowest energy is metastable so thatwe can neglect decays towards other atomic states. In thisway, the Dicke model Hamiltonian is given by (taking (cid:126) = 1 henceforth) [9] ˆ H D = ω f ˆ a † ˆ a + ω ˆ J z + γ √ N (cid:0) ˆ a † + ˆ a (cid:1) (cid:16) ˆ J + + ˆ J − (cid:17) , (1)where γ is the coupling constant for the atom-fieldinteraction, ˆ a † and ˆ a are the creation and annihilationoperators for the single-mode of the cavity, respectively,and satisfy the commutation rule (cid:2) ˆ a, ˆ a † (cid:3) = 1 . The atomicensemble is described through the pseudo-spin collectiveoperators ˆ J = (cid:80) Nk =1 ˆ j k , with ˆ j k = ˆ σ k / is the kthcomponent of the pseudo-spin operator for the atom k , whichsatisfy the SU(2) commutation relations [ ˆ J z , ˆ J ± ] = ± ˆ J ± and [ ˆ J + , ˆ J − ] = 2 ˆ J z , where ˆ J z , ˆ J ± are the atomic relativepopulation and the atomic transition operators, respectively.The total spin quantum number is selected to be j = N/ ,corresponding to the subspace which includes the groundstate of the system and is completely symmetric.In the Dicke model there are two types of phasetransitions: ( i ) a thermal second-order phase transition foundby Hepp and Lieb [7], and mathematically described byWang and Hioe [8]. When γ > √ ω f ω , there is a thermalphase transition for a temperature T c . For T > T c , thesystem is in a normal phase where there are no atomic orphotonic excitations, while for T < T c the system reaches thesuperradiant phase. ( ii ) At T = 0 , there is a second-orderQPT which occurs at the critical point γ Dc = √ ω f ω / .The system is in the normal phase, where the ground stateis non-degenerate, and there are no atomic or photonicexcitations for γ ≤ γ Dc . For γ > γ Dc , the system is in thesuperradiant phase, where the symmetry is broken, causingdegeneration in the ground state, and the photons and theatomic ensemble have macroscopic occupations [9].In this work, we study an extended Dicke model (EDM),where a non-linear optical material is within a high-finesse optical cavity. It contains a quantized field mode and itis pumped with an electromagnetic field with frequency ω f , described in the parametric approximation [34]. Thissystem is modeled through the inclusion of two termscontaining non-linear operators, which correspond to the realand imaginary parts of the square of the field amplitude,introduced by Hillery [35]. In this way, a nonlinear effectknown as degenerate parametric amplification (DPA) isproduced within the cavity. The Hamiltonian for EDM is ˆ H ED = ω f ˆ a † ˆ a + ω ˆ J z + γ √ N (cid:0) ˆ a † + ˆ a (cid:1) (cid:16) ˆ J + + ˆ J − (cid:17) + K (cid:0) ˆ a † + ˆ a (cid:1) + i K (cid:0) ˆ a † − ˆ a (cid:1) , (2)with K and K are the coupling of real and imaginaryparts of the squared amplitude, respectively. Furthermore,the EDM Hamiltonian retains the same symmetry propertiesas Dicke model. When K = K = 0 , we recover the DickeHamiltonian. Some properties for the case of K = 0 havebeen studied in [15]. When there is no interaction betweenthe atoms and the field ( γ = 0) , the Hamiltonian of thesystem is equivalent to a degenerate parametric amplifierwith an extra term given by the energy of the atoms.Parity symmetry is characterized by the unitarytransformation ˆ U (Φ) = e i ΦˆΛ , with ˆΛ = ˆ a † ˆ a + ˆ J z + (cid:113) ˆ J + 1 / − / being the operator representing thetotal number of excitations, and its respective eigenvaluesgiven by Λ = n + m + j . The number of photons isrepresented by n , and m + j is the number of atomicexcitations. This transformation acts on the atomic andphotonic operators in the form ˆ U ˆ J + ˆ U † = e − i Φ ˆ J + and ˆ U ˆ a ˆ U † = e i Φ ˆ a . Thus, the transformed Dicke Hamiltonian is ˆ U ˆ H ED ˆ U † = ω f ˆ a † ˆ a + ω ˆ J z + γ √ N (cid:16) ˆ a † ˆ J − + ˆ a ˆ J + (cid:17) + γ √ N (cid:16) e − i Φ ˆ a † ˆ J + + e i Φ ˆ a ˆ J − (cid:17) + K (cid:0) e − i Φ ˆ a † + e i Φ ˆ a (cid:1) + i K (cid:0) e − i Φ ˆ a † − e i Φ ˆ a (cid:1) . (3)The Hamiltonian is invariant under the action of ˆ U (Φ) for Φ = 0 , π , and the invariant group is given by C = { , e iπ ˆΛ } .Two projection operators emerge, ˆ P ± = ( ± e iπ ˆΛ ) , thatclassify the eigenvalues of ˆΛ into even (+) and odd ( − ) .It implies that parity is a conserved quantity of the EDM, (cid:104) ˆ H D , ˆ U ( π ) (cid:105) = 0 . States belonging to each parity subspacesare not mixed by the Hamiltonian with states in the othersubspace [36].III. S EMICLASSICAL A NALYSIS
The semiclassical version of the EDM Hamiltonian isobtained employing coherent states [37, 38], both for theatomic sector (Bloch coherent states) and for the photonicector (Glauber states), given, respectively, by | z (cid:105) = 1(1 + | z | ) j e z ˆ J + | j, − j (cid:105) , (4) | α (cid:105) = e −| α | / e α ˆ a † | (cid:105) , (5)where z and α are complex parameters. α depends on thecanonical variables for the electromagnetic field, q and p ,defined as α = ( q + ip ) / √ . And z = tan( θ/ e iφ , wherethe angle θ measures the zenith angle with respect to the − ˆ z axis such that j z = − j cos θ , and φ is the azimuthal angle.The canonical variables ( q, p ) represent the classical analogof the electromagnetic field quadratures. In this form, thesemiclassical Hamiltonian is E = H ( q, p, φ, θ ) = (cid:104) z | ⊗ (cid:104) α | ˆ H ED | α (cid:105) ⊗ | z (cid:105) = ω f q + p ) − ω j cos θ + 2 (cid:112) j γ q sin θ cos φ + K q − p ) + K q p. (6)The Hamilton’s equations are given by ˙ q = ∂ H ∂p = ω f p − K p + K q, (7) ˙ p = − ∂ H ∂q = − ω f q − (cid:112) jγ sin θ cos φ − K q − K p, (8) ˙ φ = ∂ H ∂θ = ω j sin θ + 2 (cid:112) jγq cos θ cos φ, (9) ˙ θ = − ∂ H ∂φ = 2 (cid:112) jγq sin θ sin φ. (10)In order to find the equilibrium points, we calculate ∇H ( q c , p c , φ c , θ c ) = 0 . The resulting critical pointsare ( q c , p c , φ c , θ c ) = (0 , , φ, and ( q c , p c , φ c , θ c ) =(0 , , φ, π ) for any value of the coupling constant γ . Thevalue θ c = 0 corresponds to the South Pole of the Blochsphere. It is stable for γ ≤ γ c and unstable for γ > γ c .The point θ c = π represents the North Pole, which isunstable for any value of γ . In both of them the valueof φ c is not well defined. Two other critical points appearfor φ c = 0 with ( q c , p c , , arccos[( γ c /γ ) ]) and φ c = π with ( − q c , − p c , π, arccos[( γ c /γ ) ]) . They are stableequilibrium points, which can only exist for γ ≥ γ c . Thequantities q c , p c and γ c are given by q c = − ω √ j Γ2 γ c (cid:112) − Γ − , (11) p c = − K ω f − K q c , (12) γ c = 12 (cid:115) ω ( ω f − K − K ) ω f − K , (13)with Γ = γ/γ c and γ c is the critical value of the atom-fieldcoupling. Enforcing γ c to be real restricts the values ofthe ( K , K ) to be in the region K < ω f and K < (cid:113) ω f − K . As one the goals of this work is to find situationsin which γ c < γ Dc , the case K > ω f will no be considered. It is worth to mention that, when γ = 0 , there is a richsemi-classical dynamics of the Hamiltonian 2 which can beunveiled using SU(1,1) coherent states [39].When K < ω f , it is clear from Eq. 13 that along thecircular line K + K = ω f the critical value is null: γ c = 0 . The ground state undergoes a second-order phasetransition and it becomes degenerate with the first excitedstate. When K + K ≥ ω f the system is a the normalphase, characterized by the absence (in average) of excitedatoms and photons within the cavity. When K + K < ω f the system is in the superradiant phase, with a macroscopicpopulation of photons and excited atoms inside the cavity[40, 41]. It is interesting to notice that having non-linearmaterials inside the cavity which can fulfill this condition,the system will always be in the superradiant phase, for anypositive value of the coupling parameter γ , however small itcould be. This represents an alternative way to achieve thestrong coupling limit.Figure 1 shows γ c as function of the non-linear parameters,at the resonance condition, ω f = ω = 1 , in the region ≤ K , K ≤ . Fig. 1:
Atom-field coupling constant γ c as a function of K and K , with resonance condition ω f = ω = 1 . The ground state scaled energy for each phases is (cid:15) = E ω j = (cid:26) − , for γ ≤ γ c , − (cid:0) Γ + 1 / Γ (cid:1) , for γ > γ c . (14)The functional dependence of the ground state energy onthe coupling parameter γ is the same as in the standard Dickemodel [42]. The influence of the non-linear terms K , K ishidden inside the critical parameter γ c , given in Eq. 13.Another way to visualize the QPT is by expressing theHamiltonian in terms of the variables θ c and φ c , that is, (cid:15) = − cos θ c − Γ θ c cos φ c . (15)In Figure 2, we plot the contour plots of the ground statescaled energy showing the changes in the surface for threedifferent values of Γ . In (a), we have Γ = 0 . , for which itis observed that θ = 0 is the minimum stable of the energy (cid:15) , and it belongs to the normal phase. In (b), Γ = 1 . , andthe equilibrium point is still stable, but with a stable regiongreater than the previous case. In (c), for Γ = 2 . , the energyof the system is doubly degenerated, for φ = 0 and φ = π ,and a saddle point appears. The system is in the superradiantphase. ig. 2: Contours of the energy surfaces for Eq. (15) with
Γ =0 . , Γ = 1 . and Γ = 2 . . In (a) and (b), the system is in thenormal phase, and there is one stable minimum energy. In (c), adegeneration associated with symmetry breaking appears for θ =0 , π , and the system is in the superradiant phase. The expectation values of the operators ˆ q, ˆ p, ˆ a † ˆ a and ˆ J z serve as order parameters and characterize the quantumphases of the system. These quantities are calculated inthe same way as the energy surface, using the coherentstates. Table I shows these expressions and their respectivefluctuations. Values in the normal phase can be obtainedtaking Γ → . Operator Mean Value Fluctuation (cid:104) ˆ q (cid:105) − ω √ j Γ2 γ c √ − Γ − (cid:104) ˆ p (cid:105) − K ω f − K (cid:104) ˆ q (cid:105) (cid:104) ˆ a † ˆ a (cid:105) ( ω f − K ) + K ω f − K ) (cid:104) ˆ q (cid:105) ω f − K ) + K ω f − K ) (cid:104) ˆ q (cid:105) (cid:104) ˆ J x (cid:105) j √ − Γ − j Γ − (cid:104) ˆ J y (cid:105) j (cid:104) ˆ J z (cid:105) − j Γ − j (1 − Γ − ) TABLE I:
Expectation values of photonic and atomic operators inthe superradiant phase with their respective fluctuations.
Their dependence on the nonlinear parameters K and K is shown in Figure 3, for different values of ( K , K ) . Noticethat, when K + K approach ω f , the value of γ c goes tozero, Γ = γ/γ c grows without limits and tends to divergefor any finite value of γ . In this case the expectation of q c , Eq. 11, becomes approximately proportional to Γ , andalso p c , Eq. 12, and the average number of photons in thecavity. This is the most significative effect of the presence ofthe non-linear material inside the cavity. To account for thiseffect and allow a comparison, the values for K = K = 0 . (black line) have been multiplied by − . In the normalphase the operators (cid:104) ˆ q (cid:105) , (cid:104) ˆ p (cid:105) and (cid:104) ˆ a † ˆ a (cid:105) have null expectationvalues.In Figure 3(a), we plot (cid:104) ˆ q (cid:105) / ( ω √ j ) for different values of K and K . In the superradiant phase, the plots for K = K = 0 (Dicke model - blue line) and ( K , K ) = (0 . , . (green line) are equals since they have the same value of γ c = 0 . . For ( K , K ) = (0 . , (red line), with γ c = 0 . ,the (absolute) values are larger than those of Dicke model.For ( K , K ) = (0 . , . (black line), with γ c = 0 . , theexpectation values are increased by two order of magnitudesdue to the presence of the non-linear terms. In Figure 3(b),we plot (cid:104) ˆ p (cid:105) / ( ω √ j ) . In the superradiant phase, for K = Fig. 3: (cid:104) ˆ q (cid:105) / ( ω √ j ) and (cid:104) ˆ p (cid:105) / ( ω √ j ) , (cid:104) ˆ a † ˆ a (cid:105) / ( ω j ) and (cid:104) ˆ J z (cid:105) /j as afunction of Γ in the normal and superradiant phases for the groundstate with different values of K and K . The plots for the values K = K = 0 . are multiplied for − . We take ω f = 1 . K = 0 and ( K , K ) = (0 . , , the value of the quadrature p is zero, since it is proportional to K , like in Dicke model[43]. For ( K , K ) = (0 . , . , the value does not vanish,since K is different from zero, even having the same value γ c as Dicke model. When ( K , K ) = (0 . , . , we find amuch higher values than in the previous case since we havea smaller value of γ c .The mean photon number (MPN) (cid:104) ˆ a † ˆ a (cid:105) / ( ω j ) , is shownin Figure 3(c), where the values are compared with the Dickemodel, blue line. For γ c > . (red line), we obtain lowervalues than those obtained in the Dicke model. For γ c = 0 . (green line), the values are higher due to the contributionof (cid:104) ˆ p c (cid:105) . For γ c < . (black line), much larger valuesare obtained since we have a smaller γ c . For the chosenvalues for K and K , the MPN has been multiplied by avalue of − , to allow a visual comparison with the Dickemodel. As mentioned above, these high values of MPN arecaused by the pumping of the non-linear medium, wherethe effect of the parametric amplification is obtained. It isa great experimental advantage since it could be possibleto achieve the quantum phase transition with far smalleratom-field couplings than in Dicke model. In this sense, themanipulation of the parameters ( K , K ) allows to reachdifferent values for both the critical coupling γ c and theMPN, depending on the experimental needs. Finally, themean value of the population inversion operator is shownin the Figure 3(d). In the normal phase all atoms are, onaverage, in their ground state and therefore (cid:104) ˆ J z (cid:105) = − .For Γ > , we see that a macroscopic atomic populationappears in the system, increasing with respect to Γ up to (cid:104) ˆ J z (cid:105) = 0 , which coincides with the standard Dicke modelbecause the inclusion of nonlinear terms does not affect thetomic subsystem.IV. B ERRY P HASE I NDUCED BY THE C AVITY
Our goal in this section is to investigate the Berryphase induced by the cavity field and its connection to theQPT present in the extended model, at the thermodynamiclimit. The Berry phase is a quantum phase of topologicalorigin acquired, in addition to the dynamical phase, by theeigenstates of a Hamiltonian which are varied cyclicallyand adiabatically along a closed path C in the parameterspace of the system. Along a quantum phase transition anon-analyticity appears in the geometric phase of the groundstate [27]. The geometric phase can be found when a familyof Hamiltonians is generated through the application of theunitary transformation on ˆ H ED . This is done by making anadiabatic rotation of the system around the z -axis, throughof the unitary transformation given by U ( β ) = e − iβ ˆ a † ˆ a , andadiabatically varying the angle β from to π , forminga closed path C in the parameter space. The transformedextended Dicke Hamiltonian is ˆ H ( β ) = U ( β ) ˆ H D U † ( β )= ω f ˆ a † ˆ a + ω ˆ J z + γ √ N (ˆ a † e − iβ + ˆ ae iβ )( ˆ J + + ˆ J − )+ K a † e − iβ + ˆ a e iβ )+ i K a † e − iβ − ˆ a e iβ ) . (16)The effect of the transformation on the Hamiltonian is theaddition of a phase on the creation and annihilation operators,such that ˆ a † → ˆ a † e − iβ and ˆ a → ˆ ae iβ . The transformedGlauber coherent state for the Eq. 5 is | α ( β ) (cid:105) = e −| α | / e α ˆ a † e − iβ | (cid:105) , (17)The Berry phase for the ground state is written as λ = i (cid:90) π (cid:28) ψ ( β ) (cid:12)(cid:12)(cid:12)(cid:12) ddβ (cid:12)(cid:12)(cid:12)(cid:12) ψ ( β ) (cid:29) dβ =2 π (cid:104) ˆ a † ˆ a (cid:105) , (18)where | ψ ( β ) (cid:105) = | α ( β ) (cid:105) ⊗ | z (cid:105) is the transformed groundstate of the system. This result exhibits the proportionalitybetween the Berry phase and the average number of photons,it has also been found in [31]. Therefore, the geometricphase, in the thermodynamic limit, is λ ω N = (cid:40) ≤ , π Γ (1 − Γ − )8 γ c (cid:104) K ( ω − K ) (cid:105) Γ > . (19)In the normal phase, the geometric phase is zero. In thesuperradiant phase and for K = K = 0 Eq. 19 reproducethe result for the Dicke model reported in [12].The first-order derivative of the Berry phase with respectto Γ , for each phase, is given ∂λ / ( ω N ) ∂ Γ = (cid:40) ≤ , π Γ(1+Γ − )4 γ c (cid:104) K ( ω f − K ) (cid:105) Γ > . (20) Fig. 4:
The scaled Berry phase γ / ( ω N ) for the ground stateversus the Γ . In the inset is shown the first-order derivative as afunction of Γ . The values for ( K = K = 0 . have been dividedby 10, and ω f = 1 . In the Fig. (4), we show the scaled Berry phase and itsfirst-order derivative as a function of Γ , for different valuesof ( K , K ) . In the normal phase, the Berry phase is zerofor any value of ( K , K ) . For Γ > , we see that theBerry phase increases as Γ increases, in the same way asthe average number of photons due to the proportionalitygiven in Eq. 19. The inset shows the first-order derivativewith respect to Γ , with a non-analyticity in its derivative atthe critical point Γ = 1 . The values for the correspondingplots for K , K = 0 . (black line), are divided by a factorof . In this way, the Berry phase acts as an indicator ofquantum phase transitions.In the thermodynamic limit, the scaling behavior of theBerry phase for the ground state, close to the critical value, γ c (Γ = 1) , is given by γ N (Γ →
1) = πω γ c (cid:20) K ( ω f − K ) (cid:21) | Γ − | . (21)The first order derivative with respect to Γ diverges linearlywith N , in the vicinity of Γ = 1 , in the form lim N →∞ dγ d Γ (cid:12)(cid:12)(cid:12)(cid:12) Γ → = πω γ c (cid:20) K ( ω f − K ) (cid:21) N. (22)V. C ONCLUSIONS
We have analyzed an extended model of the Dicke model,including two non-linear terms that represent the real andimaginary part of the square of the field amplitude. Througha semiclassical analysis, in the thermodynamic limit, westudied the ground state energy, which exhibits a quantumphase transition, whose dependence in the Hamiltonianparameters was analized in detail. Furthermore, our modelshows a degenerate parametric amplification effect, whichis revealed in the expectation values of the photon fieldoperators and has a strong sensitivity to the values of thenon-linear parameters, without affecting the atomic sector. Ithould be underlined that this effect can be relevant, sinceit could allow the experimental access to the ultra-strongcoupling regime, reducing the required intensity of theatom-field coupling parameter.On the other hand, the Berry phase shows a non-analyticityin the ground state for the critical value of the atom-fieldcoupling, confirming the Berry phase as an indicatorof quantum criticality. Furthermore, we observe that thegeometric phase scales linearly with N (number of atoms) inthe vicinity of the QPT. The geometric phase induced by thecavity field is proportional to the MPN. As a consequence,the high MPN values could help to detected experimentallythe Berry phase. R EFERENCES[1] S. Sachdev, “Quantum phase transitions,”
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