aa r X i v : . [ qu a n t - ph ] S e p Jahn-Teller systems from a cavity QED perspective
Jonas Larson
NORDITA, 106 91 Stockholm, Sweden (Dated: October 27, 2018)Jahn-Teller systems and the Jahn-Teller effect are discussed in terms of cavity QED models. Byexpressing the field modes in a quadrature representation, it is shown that certain setups of a two-level system interacting with a bimodal cavity are described by the Jahn-Teller E × ε Hamiltonian.We identify the corresponding adiabatic potential surfaces and the conical intersection. The effectsof a non-zero geometrical Berry phase, governed by encircling the conical intersection, are studiedin detail both theoretically and numerically. The numerical analysis is carried out by applying awave packet propagation method, more commonly used in molecul or chemical physics, and analyticexpressions for the characteristic time scales are presented. It is found that the collapse-revivalstructure is greatly influenced by the geometrical phase and as a consequence, the field intensitiescontain direct information about this phase. We also mention the link between the Jahn-Teller effectand the Dicke phase transition in cavity QED.
PACS numbers: 42.50.Pq, 03.65.Vf, 31.50.Gh, 71.70.Ej
I. INTRODUCTION
The
Jahn-Teller (JT) effect, due to Hermann Jahn andEdward Teller [1], states that a symmetry-breaking islikely (only exceptions are linear molecules or moleculespossessing
Kramers degeneracy points [2]) to occur when-ever there is an isolated degeneracy of electronic statesin a molecule, a so called conical intersection (CI) [3, 4].Over the years, the JT effect has gained enormous atten-tion, mainly in molecular and condensed matter physics[2, 5, 6]. A simple model system Hamiltonian possess-ing a CI, later termed E × ε , was presented by H. C.Longuet-Higgens et al. [7]. The main result of this workstates that the angular momentum quantum number ishalf integer valued rather than an integer. This phe-nomenon arises from a geometric phase , on top of thedynamical one, obtained while encircling the CI. Theadditional phase must be introduced in order to havea single valued total (electronic and vibrational) wavefunction. This was further analyzed in [8], where it wasshown that the double valuedness of the electronic wavefunction can indeed be removed by introducing a “vec-tor potential” term in the Hamiltonian. For CI models,this resembles the Aharanov-Bohm effect [9], and gaverise to the molecular Aharanov-Bohm effect and molecu-lar gauge theory [10, 11]. A deeper understanding of thisphase effect was gained with the seminal paper of M. V.Berry [12], which presents a general formalism for the ge-ometrical phase factors that an adiabatic change in theHamiltonian brings about. In the spring from this workcame several papers on the geometrical phase related toCIs, see Refs. [10, 11].The effect of the geometrical phase on physical ob-servables has since then been discussed and experimen-tally verified in several reports [13, 14]. The modulation,caused by the geometrical phase, of the wave functionhas been addressed in Ref. [15], where a dynamical wavepacket approach, like the one used in this article, is ap-plied to the E × ε JT model. Inclucion of spin-orbit cou- plings has been investigated in terms of the JT effect [16]and of wave packets [17]. Recently, other properties ofJT models, not only the E × ε , have been considered, forexample, quantum chaos [19] and ground state entangle-ment [18, 20].Although the JT effect has not, to the best of myknowledge, been discussed within cavity quantum electro-dynamics (QED), the geometrical Berry phase has beenanalyzed in the framework of cavity QED [21, 22, 23].These references study the effects induced by the vac-uum field on the geometrical phase. In other words, thetreatment of the two-level particle in the time-varyingfield is then considered on a fully quantum mechanicalfooting. The degeneracy point is not, however, identifiedas a CI in these works, and the rotating wave approxi-mation (RWA) has been applied which is not the case inthe present article. In addition, in the situation studiedhere, the geometrical phase is said to be of dynamicalcharacter as it originates from the intrinsic evolution ofthe system [11] rather than from “external” changing ofthe Hamiltonian [21, 22, 23]. Thus, the circumstancesand approaches are notably different between this workand the ones of Refs. [21, 22, 23]. Among others, weexamine the cavity QED system in a conjugate repre-sentation in which the intracavity fields are expressed intheir quadrature operators rather than the standard usedcreation and annihilation ladder operators. In this pic-ture, the link to JT systems and to CIs is revealed and,in fact, the Dicke normal-superradiant phase transition in cavity QED is seen to be related to the JT effect.The model system is a two-level quantum-dot embed-ded in a cavity and interacting with two degenerate fieldmodes. The numerical analysis is carried out using wavepacket propagation methods; an initial state of the sys-tem is let to evolve under the corresponding Hamilto-nian. The full Hamiltonian dynamics is considered andcompared to results obtained from a second Hamiltonianwhich shares the same adibatic potential surfaces (APS)but lacks a geometrical phase. We give analytical ex-pressions for the characteristic time scales, collapse andrevivals, for both systems and it is found that the revivaltime is in a sense twice as long in the case where thegeometrical phase is excluded. A consequence of this isreflected in the intracavity field intensities. Thus, mea-surements of the field intensities of the two modes give adirect indication of the geometrical phase.The outline of the paper is as follows. In Sec. II wereview some basics of JT models and CIs. Especially, inSubsec. II A we introduce the E × ε Hamiltonian and dis-cuss its APSs, while Subsec. II B derives the geometricalphase accumulated by encircling the CI, and Subsec. II Cconsiders the JT effect in general and in the E × ε Hamil-tonian in particular. The next Sec. III is devoted to ourcavity QED model and it is shown how the E × ε modeloccurs for a two-level system interacting with degeneratebimodal fields. A discussion of the corresponding JT ef-fect in cavity QED is outlined in Subsec. III B, where aparallel with the Dicke normal-superradiant phase tran-sition is drawn. Our numerical results of the cavity JTsystem are presented in Sec. IV, both analyzing the shortand long term behavior and how the geometrical phasecomes into play. Finally we summarize in Sec. V. II. THE JAHN-TELLER MODEL
Jahn-Teller systems are characterized by a degeneracypoint of coupled potential surfaces, a CI. In one dimen-sion, the simplest example is the E × β model, also called Rabi or spin-boson model [24]. It describs a spin 1/2particle coupled to a single boson mode [25]. In certainparameter regimes, a RWA can be applied in which thismodel relaxes to the one of Jaynes and Cummings [26].In one dimension, the wave packet (state of the system)cannot encircle the CI without passing through it, andtherefore there is no corresponding dynamical geometricphase [12]. In two dimensions, generalization of the E × β model leads in certain situations to the E × ε model whichwill be the subject of this section. A. The E × ε Hamiltonian
The simplest Jahn-Teller Hamiltonian with two vibra-tional degrees of freedom is the so called E × ε one, givenby [7] H JT = ˆ p x m + ˆ p y m + mω x + ˆ y )+ λ ˆ x ˆ σ x + λ ˆ y ˆ σ y . (1)Here ˆ p i and ˆ x i are momentum and position in the i di-rection of the “particle” with mass m . The ˆ σ -operatorsare the standard Pauli matrices obeying the commuta-tion relations [ˆ σ i , ˆ σ j ] = 2 ε ijk ˆ σ k (2) and with the z -eigenstates, ˆ σ z |±i = ±|±i and λ the cou-pling constant. Clearly, at the origin ˆ x = ˆ y = 0, the twopotential surfaces are degenerate. In the presence of ei-ther spin-orbit coupling [17, 27] or an external magneticfield [20, 28], an additional detuning term ∆ˆ σ z / diabaticbasis and diabatic potentials , namely; a diabatic state iswritten as Ψ( x, y ) = f ± ( x, y ) |±i for some normalizedfunction f ± ( x, y ), and the diabatic potentials, once thedetuning ∆ is included, are mω ( x + y ) / ± ∆ /
2. Beforedefining the APSs, we express the Hamiltonian in polarcoordinates x ± iy = ρ e ± iϕ (3)giving [15, 17, 27] H JT = − ¯ h m (cid:18) ∂ ∂ ˆ ρ + 1ˆ ρ ∂∂ ˆ ρ + 1ˆ ρ ∂ ∂ ˆ ϕ (cid:19) + mω ρ + ∆2 λ ˆ ρ e i ˆ ϕ λ ˆ ρ e − i ˆ ϕ − ∆2 (4)Let us introduce the unitary operator [8, 15] U = (cid:20) sin( ν ) cos( ν ) − cos( ν )e iµ sin( ν )e iµ (cid:21) , (5)where tan(2 ν ) = 2 λ ˆ ρ ∆ (6)and µ = ˆ ϕ , which diagonalizes the last term of Eq. (4).However, U does not commute with the kinetic term inEq. (4) and consequently, the transformed Hamiltonian,˜ H JT = U − H JT U , is non-diagonal. The off-diagonalterms are the non-adiabatic couplings, which usually aresmall far from the crossing. Omitting these terms definesthe adiabatic Hamiltonian H adJT = T + V ad ± + V cent + V gauge , (7)where [8, 15] T = − ¯ h m (cid:18) ∂ ∂ ˆ ρ + 1ˆ ρ ∂∂ ˆ ρ + 1ˆ ρ ∂ ∂ ˆ ϕ (cid:19) ,V ad ± = + mω ρ + ˆ σ z s(cid:18) ∆2 (cid:19) + λ ˆ ρ ,V cent = λ ∆ ω + 4 λ ˆ ρ ) ,V gauge = ω ρ p ∆ + 4 λ ˆ ρ ! (cid:18) − ∂∂ ˆ ϕ (cid:19) . (8)We will assume that the evolution takes place mainlyupon the lower APS, i.e. , ˆ σ z = −
1, but in the numer-ical simulations we consider full dynamics without anyapproximations. Close to the crossing, the non-adiabaticcouplings may have a significant impact on the dynamics[4, 29]. The term V ad ± defines the APSs, while V cent and V gauge , arising from the commutator between U and thekinetic energy operator, are centrifugal corrections [30].The last term has been labeled gauge for reasons that willbecome clear later on. We display two examples, ∆ = 0and ∆ = 0, of the APSs V ad ± in Fig. 1. For a non-zerodetuning, as pointed out, the crossing at the CI becomesavoided, with splitting amplitude ∆. The lower surfacehas the familiar sombrero shape, while the upper pos-sesses a single global minimum at x = y = 0. For largedetunings ∆, the Mexican hat structure is lost, and theminimum of the lower APS is at the origin. Especially,the radius giving the potential minima is given by [17] ρ min = s(cid:18) λω (cid:19) − (cid:18) ∆ λ (cid:19) , ω | ∆ | < λ , ω | ∆ | ≥ λ . (9) FIG. 1: Two examples of the adiabatic surfaces in the E × ε Jahn-Teller model with detuning ∆ = 0 (a) and ∆ = 0 (b). B. Berrys geometrical phase in the E × ε model The adiabatic states , defined by U , are arbitrary up toan overall phase. The phase choice in Eq. (5) is cho-sen such that the states are singled valued as ϕ is variedby 2 π . For example, the alternative obtained by multi-plying U by exp ( − iϕ/
2) implies double-valued adiabaticstates. Unitary transformation of the Hamiltonian in thissecond case leads to an adiabatic Hamiltonian lacking theterm proportional to i∂/∂ ˆ ϕ . The very last term of (7),containing the differential operator ∂/∂ ˆ ϕ , can be viewedas a vector potential. Indeed, this term can be combinedwith the canonical momenta to define a kinetic momenta.Thus, the two options of overall phase of the adiabaticstates given above result in either single-valued stateswith a vector potential present in the Hamiltonian or nosuch vector potential term but double-valued states [31].The source of a vector potential term is the cause for having a non-zero geometric phase as the system encir-cles the CI in analogous to the Aharanov-Bohm effect[17].For the system evolving along a closed loop C in pa-rameter space, the geometrical phase can be calculatedaccording to [12] γ n ( C ) = I C h n ( R ) |∇ R n ( R ) i · d R . (10)Here, | n ( R ) i is the n ’th adiabatic eigenstate and R the set of parameters. In particular, in our case | n ( R ) i = (cid:0) sin( θ ) , − cos( θ )e iϕ (cid:1) and as we consider a time-independent dynamical problem, the varying parameters R are the coordinates ρ and ϕ . Especially, we consider awave packet located at the minima of the sombrero po-tential, such that h ˆ ρ i ≈ ρ min , and ϕ is changed from 0to 2 π . For a general radius R we find the E × ε geometricphase [17] γ JT ( R ) = − π (cid:18) √ ∆ + λ R (cid:19) , (11)which for R = ρ min becomes γ JT ( ρ min ) = − π (cid:18) ω ∆ λ (cid:19) . (12)For ∆ = 0 we obtain the well known sign change of thewave function when encircling a CI, causing half integerangular momentum quantum numbers. C. The Jahn-Teller effect
Using group theoretical arguments, Jahn and Tellerproved that for almost any degeneracy (CIs) among elec-tronic states in a molecule, a symmetry-breaking is “al-lowed” which removes the degeneracy and lowers the to-tal energy of the model system ground state [1]. It turnsout that this symmetry-breaking indeed takes place inthe majority of cases with some exceptions [2, 5]. Hence,the molecule favors a distortion of its most symmetricalstate. This effect is quenched when spin-orbit couplingis taken into account, but may still exist [16]. Returningto the APSs of the E × ε Hamiltonian (7), the state withhighest symmetrical is a wave packet centered around theorigin ( x = y = 0). Loosely speaking, for vanishing de-tuning ∆, is it intuitive to expect the wave packet to slidedown the potential surfaces towards the minima of thesombrero. Semi-classically, the wave packet experiencesa non-zero force F = −∇ V ( x, y ). For large detuningshowever, we saw from Eq. (9) that the potential surfacesmay possess a single global minimum which will preventthe JT-distortion. On the other hand, for small but non-zero detuning, the sombrero structure is present for thelower APS and here quantum fluctuations will permit asymmetry breaking. Naturally, the above arguments aresemi-classical and the full evolution is quantum mechan-ical and described by the coupled system. Nonetheless,it gives some insight and intuition of the dynamics. III. JAHN-TELLER MODELS IN CAVITY QED
Spin boson models naturally occur in cavity QED.Here the boson subsystem represents a single or severalquantized modes of an intracavity field, while the spin de-grees of freedom describes either two-level atom/atoms[32] or solid state quantum-dot/dots [33]. Contrary tostandard formulations of Jahn-Teller models, here thebosons are the photons of the field rather than vibrationalphonons, and the internal structure corresponds to twodiscrete energy levels of the atom or quantum-dot. In thesingle mode case, a microscopic derivation gives the E × β Hamiltonian [35] in the assumption of dipole approxima-tion and neglecting the self-energy (see below). In mostcavity QED experiments involving atoms, the applica-tion of the RWA is justified, in which the Hamiltonianidentifies the analytically solvable Jaynes-Cummings one[26]. The APSs (or rather adiabatic potential curves) ofthe Jaynes-Cummings model contain the differential mo-mentum operator [24], and they are therefore said to beof non-potential form . Nonetheless, even if the pictureof potential surfaces is less intuitive due to the momen-tum dependence, the JC model renders a sort of general-ized CI (curve crossing). To go beyond the RWA regime,the coupling to the field must be substantially increasedcompared to atomic cavity QED setups. This is indeedthe case for solid state quantum-dots coupled to a cav-ity. In fact, the crucial parameter, coupling divided bythe two-level transition frequency, can be made severalorders of magnitude larger in condensed matter systems[33] compared to atom-cavity ones. Another possibilityto achieve ultrastrong atom-field couplings is to considerBose-Einstein condensates coupled to an intracavity field[34].
A. The model system Hamiltonian
To obtain CIs rather than curve crossings, multi-modecavities must be considered [36]. For simplicity we willassume two degenerate cavity modes such that they sharethe same frequency ˜ ω and also same coupling amplitude˜ λ to the quantum-dot. The Hamiltonian in the dipoleapproximation reads [37, 38] H cav = ¯ h ˜ ω (cid:16) ˆ a † ˆ a + ˆ b † ˆ b (cid:17) + ¯ h ˜Ω2 ˆ σ z +¯ h ˜ λ √ h (cid:0) ˆ a † + ˆ a (cid:1) (cid:0) ˆ σ + e − iφ + ˆ σ − e iφ (cid:1) + (cid:16) ˆ b † + ˆ b (cid:17) (cid:0) ˆ σ + e − iθ + ˆ σ − e iθ (cid:1) i . (13)Here ˆ a † and ˆ b † (ˆ a and ˆ b ) are creation (annihilation) oper-ators for the two field modes, φ and θ field phases, ˜Ω thequantum-dot transition frequency, and 2ˆ σ ± = ˆ σ x ± i ˆ σ y .In the following we will label the two cavity modes by a and b . Before proceeding, for brevity we introduce acharacteristic energy ¯ h ˜ ω and time scale ˜ ω − , such thatwe consider dimensionless variables λ = ˜ λ ˜ ω , Ω = ˜Ω˜ ω , τ = ˜ ωt, (14)where t is the unscaled time. In a conjugate variablerepresentation defined by the operator relationsˆ p x = i √ (cid:0) ˆ a † − ˆ a (cid:1) , ˆ x = 1 √ (cid:0) ˆ a † + ˆ a (cid:1) , ˆ p y = i √ (cid:16) ˆ b † − ˆ b (cid:17) , ˆ y = 1 √ (cid:16) ˆ b † + ˆ b (cid:17) , (15)where [ˆ x, ˆ p x ] = [ˆ y, ˆ p y ] = i , the Hamiltonian (13) takesthe form H cav = ˆ p x p y x y σ z +2 λ ˆ x (cid:2) cos( φ )ˆ σ x + sin( φ )ˆ σ y (cid:3) +2 λ ˆ y (cid:2) cos( θ )ˆ σ x + sin( θ )ˆ σ y (cid:3) . (16)For the simple example of φ = 0 and θ = π/ E × ε Hamiltonian (1). In fact,for | φ − θ | = ( j + 1 / π, j integer , (17) H cav is unitarilly equivalent with the E × ε Hamiltonian H JT by identifying Ω with ∆. In some special cases ofthe phases, the last two terms of (16) can be written as2 λ (ˆ x + ˆ y ) (cid:20) − iφ e iφ (cid:21) , for θ − φ = 2 jπ, λ (ˆ x − ˆ y ) (cid:20) − iφ e iφ (cid:21) , for θ − φ = (2 j + 1) π, (18)for some integer j . For these situations the CI is re-placed by an intersecting curve in the directions of ϕ =3 π/ , π/ ϕ = π/ , π/ V ad ± ( ρ, ϕ ) = ρ ± s(cid:18) Ω2 (cid:19) + 4 λ ρ [1+cos( φ − θ ) sin(2 ϕ )] . (19)The lower APS has two minima for angels ϕ = π/ , π/ E × ε Hamiltonian hasthree local minima in the sombrero shaped potential [15,17]. This derives from a term of the form sin(3 ϕ ) in theAPSs. Here we have a sin(2 ϕ )-dependence instead andhence the double minima structure.The single valued adiabatic states can again be writtenΨ u ( ρ, ϕ ) = (cid:20) sin( ν ) − cos( ν )e iµ (cid:21) , Ψ l ( ρ, ϕ ) = (cid:20) cos( ν )sin( ν )e iµ (cid:21) , (20)but withtan(2 ν ) = 4 λρ Ω p ϕ ) cos( φ − θ ) ,µ = tan (cid:18) cos( ϕ ) sin( φ ) + sin( ϕ ) sin( θ )cos( ϕ ) cos( φ ) + sin( ϕ ) cos( θ ) (cid:19) . (21)Encircling the CI at a radius R we find the geometricphase (12) γ cav ( R ) = − Z π cos ( α ) ∂ψ∂ϕ dϕ (cid:12)(cid:12)(cid:12)(cid:12) ρ = R . (22)The phase (22) is depicted in Fig. 2 as a function of θ and λ for fixed φ = 0 and Ω = 1 (a) and φ = 1 / λ is either − π or 0 (modulo 2 π ). The radius R is taken to be theminimum of the cylindrically symmetric case ( θ = π/ φ = 0, the greatest effect generated by the geometri-cal phase is seen to be in the symmetric case of θ = π/ φ = 0 the situation becomes more complex.Note that, according to Eq. (9), and identifying Ω with∆, λ = 0 in order to have a sombrero structure of thelower APS, which is the reason why λ is not approaching0. B. The Jahn-Teller effect in cavity QED
The Jahn-Teller effect states that, in the presence ofa CI, lowering the symmetry may be energetically fa-vorable in various systems, typically for molecules andcrystals. In terms of the E × ε Hamiltonian this is intu-itive, since a wave packet centered at the origin (the CI)will have a larger “potential energy” than a wave packetlocated at ρ min . The corresponding symmetry breakingin cavity QED implies that the system ground state con-sists of non-zero fields in the two modes. At the sametime, the quantum-dot is not entirely in its lower statebut in a superposition of its two internal states. This isrelated to a well known phenomenon in quantum optics,namely superradiance [39]. This, of course, comes aboutdue to the strong interaction between the quantum-dotand the cavity fields. For multi quantum-dot systemsand in the thermodynamic limit, where the number oftwo-level quantum-dots and the volume tend to infinitywhile the density is kept fixed, this results in a secondorder quantum phase transition between a normal (thefield in its vacuum) and a superradiant phase (a macro-scopic non-zero field) [40]. The critical coupling of thisphase transition is given by λ c = p | ∆ | ω [40], which in-deed follows from Eq. (9). FIG. 2: The geometric phase (22) as function of coupling λ and field phase θ . The radius R = ρ min , where ρ min = p λ − (Ω / λ ) is the adiabatic potential minima of the som-brero when θ − φ = π/ φ = 0 and Ω = 1 (a)and φ = 1 / However, it can be shown that for a quantum-dot inwhich the lower state is its ground state, the normal-superradiant phase transition is an artifact from neglect-ing the self-energy term from the Hamiltonian [41]. Fora single mode, and in unscaled units, this term is givenby H se = e m π ¯ hV ω ˆ x , (23)where e is the electron charge, m its mass and V theeffective mode volume. This should be compared withthe matter-field coupling λ = Ω d r π ¯ hV ω , (24)where d is the dipole moment of the transition of inter-est in the quantum-dot. It is clear that the self-energyterm H se tends to quench the Mexican hat structure,and further that λ and H sc are not fully independent.Indeed, in Ref. [41] it is demonstrated, either using theThomas-Reiche-Kuhn sum rule or simple thermodynam-ical and gauge invariance arguments, that the sombreroshape cannot be obtained for any set of physical param-eters.To circumvent this obstacle one may use a two-photonRaman type of interaction, where three levels of thequantum-dot are coupled through the cavity mode andan external classical laser field [42, 43]. In the large de-tuning limit of one of the internal levels it can be adia-batically eliminated [44] and one arrives at an effectivemodel very similar to the one above. In such procedureone introduces an additional independent parameter, thedetuning δ of the eliminated level [45]. As the detuningenters in the effective matter-field coupling parameter,but not in the self energy term, these two become in prin-ciple independent, and in particular λ can be made largein comparison with H se . The effective model, once thedetuned level has been eliminated, contains some Starkshift terms that will modify the potential surfaces, butthe CI and sombrero structure are still present. Theexternal laser fields have the advantage of being easilycontrollable in terms of system parameters such as am-plitude and phase. A system Hamiltonian suitable for re-alizing the Jahn-Teller model can be found in Ref. [22].It should be noted though, that the effective Hamilto-nian is in general time dependent, which is prevailed byimposing a RWA. This, however, induces a “momentum-dependent” potential surface, but nonetheless, the JTsymmetry breaking is still present in this approximation[40].Another possibility to surmount the problem with theself-energy, which is assumed in this paper, is to usethe fact that the two internal levels that couple to thecavity modes are normally highly excited meta stable(Rydberg) states. For these states, neither the Thomas-Reiche-Kuhn sum rule nor the thermodynamical argu-ments apply, and the symmetry breaking may still oc-cur. We therefore discard the self-energy terms as theywould only modify the frequencies of the harmonic po-tentials. In general, also for the Raman coupled model,the states involved are highly excited metastable statesand the sum rule cannot be applied in those cases ei-ther. A benefit of the Raman model, compared to a onephoton model, is the higher controllability of the systemparameters; especially the diagonal element is a detuningparameter (and not a transition frequency Ω) that can bemade small compared to the matter-field coupling. Thedrawback of an effective Raman model is that the anal-ysis is considerably less intuitive due to the RWA. Here,we therefore choose the simpler model of the two as thephysical phenomena may be more easily extracted fromit.
IV. NUMERICAL RESULTS
Contrary to molecular or solid state systems, proper-ties of the cavity fields are in comparison easily measured,for example, phoson distribution [46], field quadratures[47] and, in fact, the whole phase space distribution us-ing quantum tomography [48]. Using wave packet prop-agation methods, the dynamics of such quantities willbe studied in this section with emphasizes on the effectsemerging from the geometrical phase. As an initial statewe take a disentangled one, given in cartesian coordinatesby Ψ( x, y,
0) = ψ ( x, y,
0) 1 √ (cid:20) − (cid:21) , (25)where ψ ( x, y,
0) = 1 √ π e − (Im x ) − (Im y ) e − ( x − x − ( y − y . (26)The initial quantum-dot is a linear combination of its twointernal states with equal amplitudes, and the two fieldmodes are in Gaussian states corresponding to coherentfield states; | x / √ i and | y / √ i respectively [50]. Suchinitial states are readily prepared experimentally. Wewill further pick y = 0 and x = 2 λ such that the initialwave packet is approximately centered at the minima ofthe sombrero. Note that the initial average momentumis zero, and that Ψ( x, y,
0) is different from the adiabaticstates (20). A consequence of this is that the wave packetevolution will not be restricted to a single APS. However,the upper adiabatic state is only marginally populatedfor our particular choice of initial state and the mainphenomena studied here, the effects of the geometricalphase on the field properties, is indeed seen even thoughslight interference between the two adiabatic states oc-curs. Hence, we emphasize that the dynamics take placemainly on the lower adiabatic surface. Properties of theupper APS have been studied in Ref. [49].We restrict the analysis to the cylindrically symmetriccase, where the time evolved stateΨ( x, y, τ ) = 1 √ (cid:18) ψ e ( x, y, τ ) (cid:20) (cid:21) + ψ g ( x, y, τ ) (cid:20) (cid:21)(cid:19) , (27)will predominantly spread along the minima of the som-brero potential. As the wave packet broadens it will, aftera certain time, start to self-interfere. We may estimatethe characteristic time for this process by approximatethe inherent spreading by free evolution along the min-ima of the sombrero potential to get T in ≈ p π λ − ≈ πλ. (28)Within this time, the wave packet width has expandedover a distance 2 πρ min .From the full system state (27), we can derive the re-duced density operators for the separate constitutes ρ i ( τ ) = Tr j,k h ρ ( τ ) i , (29)where the subscripts represent, either thetwo modes a and b or the quantum-dot, and ρ ( τ ) = Ψ ∗ ( x, y, τ )Ψ( x, y, τ ). Using the reduceddensity operators we will especially study the photonstatistics and the Husimi Q -distribution [50, 51] P i ( n ) = h n | ρ i ( τ ) | n i , h ˆ n i i = X n nP i ( n ) ,Q i ( α ) = 1 π h α | ρ i ( τ ) | α i . (30)Here, | n i is the n ’th-photon Fock state, | α i a coherentstate with amplitude α and the subscript i = a, b for therespective modes. A. Dynamics on the T in time scale Discussed in Sec. II, it is the term V gauge that givesrise to a geometric phase. To correctly describe the adi-abatic evolution each term of H adJT in Eq. (7) must betaken into account, and it is not enough to study dy-namics upon the potentials V ad ± . In this subsection westudy the full dynamics using Hamiltonian (16), and wehence go beyond any adiabatic approximation. However,in order to identify the effects of the geometrical phasewe compare the results with the ones obtained by prop-agating the same initial state using the “semi-adiabatic”Hamiltonian defined as˜ H adJT = T + V ad − , (31)where T and V ad − are both given in Eq. (8). Accordingly,a wave packet evolving via the Hamiltonian ˜ H adJT aroundthe origin will not accumulate any geometrical phase.The characteristic time scale T in determines how longit takes for the particular initial state (26) to inherentlyspread out across the CI and start to self-interfere. It istherefore a measure of the collapse time. The effect ofthe geometrical phase on the probability wave functions | Ψ( x, y, τ ) | = | ψ e ( x, y, τ ) | / | ψ g ( x, y, τ ) | / x > y = 0 such that interference of the evolved wavepacket sets off at − x where the two tails of the packetfirst join. Destructive and constructive interference causenodes (vanishing probability distribution) and anti-nodes(non-vanishing probability distribution) in | Ψ( x, y, t ) | ,and the ring-shaped wave packet splits up in localizedblobs. In the case of ˜ H adJT , in which the geometric phaseis zero, an anti-node builds up at x = − x , while for γ JT ( ρ min ) = − π (as is the case of zero detuning in the E × ε model) a node is formed at x = − x . The locationof the corresponding node or anti-node depends on thevalue of γ JT ( ρ min ), and in all our examples Ω < λ suchthat γ JT ( ρ min ) ∼ − π giving a node at x ≈ − x . Thesefeatures are visible in Fig. 3 showing the numerical results of the propagated distributions | Ψ( x, y, τ ) | for three dif-ferent times τ . Full dynamics governed by Hamiltonian(16), with φ = 0 and θ = π/ τ and system parameters and then especially ρ min .Note that very similar results where presented in Refs.[15, 17]. However, the Hamiltonians used for the sim-ulations in Refs. [15, 17] are in general different fromthe one ˜ H adJT utalized here; a single surface approxima-tion [14, 52] is applied in most examples of Refs. [15, 17]while here it is only considered for the non-geometricalphase case. FIG. 3: Snapshots of the wave packet distributions | Ψ( x, y, τ ) | at times τ = T in / τ = T in / τ = T in (e) and (f) for the cases with (left) andwithout geometrical phase (right). In the last two plots, thedifference between the interference structures is clearly visi-ble. See Refs. [15, 17] for similar results. The dimensionlessparameters are Ω = 0 . λ = 3. As the initial wave packet starts to spread, a non-zerofield will begin to build up in the vacuum b mode, onthe cost of decreasing field intensity of mode a . How-ever, without the RWA, the total number of excitationsis not conserved. In Fig. 4 we display the individual pho-ton distributions P i ( n ) at a quarter of the interferencetime T in . Already at this instant has the initially emptymode a non-zero field intensity, and its photon distri-bution consists mostly of even number of photon states.This is a typical characteristic of Schr¨odinger cat states[32], and in the next subsection we will indeed show thatsuch a state is created in the system at certain times.The small but non-zero population of odd photon num-bers in the b mode is caused by non-adiabaticity; for thesemi-adiabatic Hamiltonian (31) the odd photon num-bers are never populated for the given initial state (25)with y = 0. n P i ( n ) =T in /4 ab FIG. 4: The photon distributions P a ( n ) (white) and P b ( n )(black) after a time τ = T in /
4. The dimensionless parametersare the same as in Fig. 3.
B. Dynamics beyond the T in time scale Seen in the previous subsection, for an initial local-ized state mainly located at the minima of the som-brero potential and with zero average momentum, thetime scale T in determines the collapse time; the time ittakes for the localized wave packet to spread out overits accessible phase space region. Over longer periods, arevival structure in physical quantities is expected [53],where localized bumps are formed in phase space signal-izing fractional or full revivals [54]. It has been pointedout however, that the collapse-revival characteristics arerather different in models where the RWA has been ap-plied [24, 55]. Typically, in the RWA regime the varioustime scales become long. In this work we are outside sucha regime, and we will in particular find that the revivaltime is given by a multiple of λT in and that phase spaceevolution is significantly different for the two Hamiltoni-ans (16) and (31) due to the geometrical phase.From Fig. 3 we see that after a time T in , the initialwave packet is spread throughout the minima of the som-brero potential and the self-interference causes nodes inthe probability distribution. The number of localizedbumps depends on ρ min (9), but also on the time τ ;at first, when the self-interference sets off, the numberof bumps increases to a maximum value and then thenumber begins to decrease and eventually form a singlelocalized wave packet. Full revival occurs when a singlelocalized bump is formed at the same position as the ini-tial wave packet. To study the field dynamics we use the Q -function for the two modes a and b of Eq. (30). Wewill present the two functions Q a and Q b in the sameplots for brevity, but mark them with letters a and b respectively. At the initial time τ = 0, Q a and Q b areGaussians centered at α = ( √ λ,
0) and α = (0 ,
0) re-spectively. As time evolves, the b mode builds up its intensity and the Q -function moves away from the ori-gin, while Q a at first decreases its intensity by tendingtowards the origin. However, over longer time scales, τ > T in , a swapping of energy between the two modeswill take place. This phenomenon has been discussedin our model, but only when the RWA has been imposed[38]. As our analysis concerns a regime far from the RWAone, this exchange of energy between the modes occursat very different time scales than in Ref. [38], similar towhat was found for the inversion in the JC model [24].Namely, the characteristic time scales in the parameterregimes of the RWA and without the RWA in the JCmodel can differ by orders of magnitude.From our numerical simulations we have found that lo-calization of the phase space distributions comes about atmultiples of time T frac = λT in , which hence are the char-acteristic scales for fractional revivals [53, 54]. The largerthe radius ρ min , the better resolved wave packet localiza-tions. In Fig. 5 we display examples of the Q -functions Q a and Q b (indicated in the figures by a and b ) obtainedeither from the full system Hamiltonian (16) (left) orfrom the semi-adiabatic Hamiltonian (31) (right). Thetimes are here, τ = λT in , λT in , λT in , λT in . A cleardiscrepancy is seen between the two models. For exam-ple, at τ = 2 λT in (c) and (d), mode a in the left plot (withgeometrical phase) is approximately in vacuum, while forthe semi-adiabatic system (without geometrical phase),mode b is roughly empty. At this instant, the non-emptymode is in a Schr¨odinger cat state. For τ = 4 λT in thefull system has revived; the Q -functions have evolved intoapproximate replicas of their initial states. This is trueup to an overall phase for the semi-adiabatic case, whichis typical for a half-revival [53]. From this figure we findthe revival time for a wave packet encircling the CI inthe E × ε model to be T rev = 4 λT in ≈ πλ . (32)For the semi-adiabatic model, exact revivals (in termsof restoring also the overall phase) occur at twice thistime. It should be pointed out that formula (32) hasbeen verified for a large set of different parameters.Even though the phase space distribution of a cavitymode is in principle measureable [48], the field intensityis directly regained from the cavity output field using aphoton-counter detector. Already Fig. 5 indicates thatthe average number of photons h n i i differ considerablybetween the full model and the semi-adiabatic one. Thisis verified in Fig. 6 showing the time evolution of h n a i and h n b i for both models. Judging from the field intensi-ties in this figure, the revival time of the semi-adiabaticmodel seems to be half the one of the full model, buthere the a mode is indeed not in a coherent state but ina Schr¨odinger cat. FIG. 5: Snapshots of the Husime Q -functions at times τ = λT in (a) and (b), τ = 2 λT in (c) and (d), τ = 3 λT in (e)and (f) and τ = 4 λT in (g) and (h). The left plots are theresults with geometric phase, using Hamiltonian (16), and thenon-geometric phase results, obtained by the Hamiltonian inEq. (31), are displayed in the right figures. The Q -functionof the a -mode is labeled by a in the plots, while b labels thesecond mode Q -function. The dimensionless parameters are λ = 6 and Ω = 0 . V. CONCLUSIONS
In this paper we have shown how a system of a two-level “particle” interacting with the fields of a bimodalcavity may fall in the category of JT models. By repre-senting the model Hamiltonian in terms of field quadra-ture operators, rather than boson ladder operators, weidentified its APSs and a CI. In this nomeclature, and inparticular for the multi-particle analogue (Dicke model),the JT effect of cavity QED was identified with thenormal-superradiant phase transition. The system stud-ied here was described by the well known E × ε Hamilto-nian. Knowledge from earlier research on this model, al-most exclusively in molecular or chemical and condensedmatter physics, has been applied on this cavity QEDcounterpart. Our main interest concerned the geomet-rical Berry phase reign from encircling the CI. The effectof the geometrical phase was studied by comparing phys-ical quantities, such as the field phase space distributionsand the field intensities, obtained from the evolution ofeither the E × ε Hamiltonian or the semi-adiabatic one in which no geometrical phase occurs. Clear distinctionsbetween the two models were found when the system islet to evolve over longer time periods. Energy is swappedbetween the two field modes, and this exchange is highlyaffected by the geometrical phase. From our numerical < n i > /T in < n i > (a)(b) FIG. 6: The average photon numbers h ˆ n i i for both modes a (black) and b (gray) as a function of scaled time τ /T in . Theupper plot presents the results from using the full Hamiltonian(16), which includes a geometrical phase, while the lower plotshows the results from using Hamiltonian (31). The effect ofthe geometrical phase is remarkably reflected in the two fieldintensities. The dimensionless parameters are the same as inFig. 5. results we could as well present analytical expressions forthe collapse-revival times for a wave packet encircling theCI in the E × ε model.In addition, by introducing the notion of a wave packetevolving on two coupled potential surfaces, a deeper un-derstanding of cavity QED models is obtained. Thiswork, analyzing the geometrical phase, serves as an al-ternative viewpoint of the phenomenon in comparison toprevious studies such as Refs. [21, 22]. It is indeed be-lieved that the wave packet method used here will giveeven more thorough insight into cavity QED problems,or even trapped ion systems where related CI models areexpect to occur [55]. We plan to study, using the currentapproach, the dynamics of the Dicke normal-superradiantphase transition. Another project underway is to investi-gate the “ molecular Aharanov-Bohm effect” and “molec-ular gauge theory” in terms of cavity QED models. Acknowledgments
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