Dynamics of a bistable Mott insulator to superfluid phase transition in cavity optomechanics
aa r X i v : . [ qu a n t - ph ] S e p Dynamics of a bistable Mott insulator to superfluid phase transition in cavityoptomechanics
K. Zhang
State Key Laboratory of Precision Spectroscopy, Dept. of Physics, East China Normal University, Shanghai 200062, China
W. Chen, P. Meystre
B2 Institute, Dept. of Physics and College of Optical Sciences, The University of Arizona, Tucson, AZ 85721, USA
Abstract
We study the dynamics of the many-body state of ultracold bosons trapped in a bistable optical lattice in an optome-chanical resonator controlled by a time-dependent input field. We focus on the dynamics of the many-body systemfollowing discontinuous jumps of the intracavity field. We identify experimentally realizable parameters for the bistablequantum phase transition between Mott insulator and superfluid.
Key words: optomechanics, phase transition, BEC
PACS:
1. Introduction
The coupling of coherent optical systems to microme-chanical devices, combined with breakthroughs in nanofab-rication and in ultracold science, has opened up an excit-ing new field of research, cavity optomechanics. Severalgroups have now demonstrated very significant cooling ofthe vibrational motion of a broad range of moving mir-rors, from nanoscale cantilevers to LIGO-class mirrors [1],and there is every reason to believe that the quantum me-chanical ground state of motion of these systems will soonbe achieved. In a parallel development, ultracold gasesas well as Bose-Einstein condensates placed inside opticalresonators have been shown to behave under appropriateconditions much like moving mirrors [2, 3]. Following thesedevelopments, cavity optomechanics is rapidly becoming avery active sub-field of fundamental and applied researchat the boundary between AMO physics, condensed matterphysics, and nanoscience.Cavity optomechanics presents considerable promiseboth in opening the way to address fundamental ques-tions related to pushing quantum mechanics toward in-creasingly macroscopic systems, but also in applicationsthat span a variety of areas from quantum detection tothe coherent control of microscopic atomic and molecu-lar systems and/or of nanoscale devices. Specific exam-ples include nanomechanical cantilevers coupled to a Bose-Einstein condensate [4] or to dipolar molecules [6], to a sin-gle atom [7], oscillating mirrors coupled to atomic vapors[8], magnetized resonator tip to color center of diamond[9]. Cold atoms can probe the state of the nanostructure,or conversely, the optomechanical cavity setup can serve as a diagnostic tool for the quantum state of atoms trappedinside the resonator [11, 12]. On a more applied side suchsystems will enable the development of ultrasensitive forcesensors and may find applications in quantum informationprocessing technology.In recent work we studied the many-body state of ul-tracold bosons in a bistable optical lattice potential inan optomechanical resonator. We considered explicitlythe weak-coupling limit where the coupling between thecavity-field and the movable mirror results in a bistableoptical lattice potential for the atoms, and showed howsuch a cavity plus cold-atom system can be engineered sothat a superfluid and a Mott insulator phase are bistableground states for the ultracold atomic gas [13].The present paper extends this study to consider thedynamics of the transition between these two states asthe optical potential undergoes a bistable loop. A time-dependent variational principle combined with a Gutzwilleransatz are introduced to calculate the evolution of thequantum many-body state as a super-Gaussian incidentlight pulse [14] switches the intracavity optical field. Wediscuss conditions under which this process can proceedadiabatically, and identify experimental parameters thatwould permit to test our predictions.Section II reviews our model of a quantum-degenerateBose gas confined by the standing wave generated in aFabry-P´erot cavity with one movable end mirror, and brieflyreviews its steady-state properties. Section III introducesan effective potential description of the system and com-ments on the conditions required for an adiabatic evolutionof the atoms. It then turns to the time evolution of thesuperfluid order parameter during the bistable superfluid
Preprint submitted to Elsevier November 3, 2018 (cid:13)
L(cid:13) c(cid:13)
E(cid:13) p(cid:13)
E(cid:13) t(cid:13)
E(cid:13) q(cid:13)
Figure 1: A Fabry-P´erot cavity with a movable mirror displacedfrom its equilibrium position by q . E p , E c and E t are the pump,cavity and transmitted light amplitudes, respectively. to Mott insulator transition. Finally, Section IV is a con-clusion and outlook.
2. Model
We consider a Fabry-P´erot cavity with one fixed endmirror and the other one, of mass M , mounted on a springof frequency Ω and damping rate γ m . An optical field E p of frequency ω p is incident on the fixed mirror. Thereflectivity of both mirrors is R , with R ≃ E c . In the absence of that field the movablemirror is at its equilibrium position q = 0 and the cavitylength is L , see Fig. 1.A sample of ultracold bosonic two-level atoms withtransition frequency ω a is loaded inside the optical latticeformed by the standing wave, and the atoms are assumedto interact with the light field in the weak coupling regime N g / | ∆ a | ≪ κ c , where N is the total number of atoms, g is the single-photon atom-field coupling strength, κ c isthe cavity decay rate, and ∆ a = ω p − ω a is the atom-field detuning [15]. In this limit the intracavity field hasno significant dependence on the atomic distribution andwe can investigate the dynamics of the intracavity field byusing the theory of an empty cavity. In this paper we de-scribe both the light field and the movable mirror as clas-sical objects, while the center-of-mass motion of the atomsis quantum mechanical. The extension to mirrors coolednear their quantum mechanical ground state of vibrationwill be considered in future work.Ignoring the retardation effects due to the propagationof the light field back and forth inside the cavity, the intra-cavity field is governed by the familiar equation of motion[16, 17] dE c dt = − ( κ c − i ∆ c ) E c + 2 κ c √ T E p (1)where ∆ c = ω p − ω c is the cavity-pump detuning and T is the transmittivity of the mirrors, with R + T = 1 inthe absence of mirror absorption. The movable mirror is driven by radiation pressure, d qdt + γ m dqdt + Ω q = F RP M (2)where F RP ≃ Aǫ | E c | / A being the cross-sectional area of the laser beam.Equations (1) and (2) are coupled through the cavity-pump detuning since the cavity frequency is q -dependent, ω c = n cπL + q ≃ ncπL (1 − q/L ) , (3)where n is the integer associated with the mode closest tothe laser frequency in the single-mode theory consideredhere. The approximate equality assumes that the mir-ror displacement due to radiation pressure is small com-pared to both L and to the cavity mode wavelength λ c =2 πc/ω c .It is convenient at this point to introduce dimensionlessunits and to scale times to Ω − , lengths to λ p / p E r /cαǫ , where E r is the pho-ton recoil energy and α = 3 πc γ a / (2 ω a ∆ a ), with γ a thenatural linewidth of the atomic resonance. Equations (1)and (2) reduce then to the dimensionless form dXdτ + (cid:20) − i πT ( δ + ξ ) (cid:21) κX = 2 κ √ T Y (4) d ξdτ + γ dξdτ + ξ = β | X | (5)where τ is the (dimensionless) time, X and Y the in-tracavity and incident field amplitudes, ξ mirror’s dis-placement, κ and γ the intracavity optical field and mov-ing mirror damping rates, and δ the offset from the cav-ity resonance in the absence of radiation. Finally, β =( AE r ) / ( λ p M Ω cα ).The quantum-degenerate atomic sample, assumed tobe at zero temperature is trapped in the optical latticealong cavity axis, its transverse motion being strongly con-fined by a strong transverse potential. Its Hamiltonian canbe then approximated as the one-dimensional formˆ H = Z dx ˆΨ † ( x ) (cid:18) − ~ m d dx + V sin ( k p x ) + g ˆΨ † ˆΨ (cid:19) ˆΨ( x )where ˆΨ( x ) is the Schr¨odinger field operator, g is the two-body interaction coefficient, m is the atomic mass and V = αcǫ | E c | / | X | E r is the depth of the opticalpotential. In the tight-binding approximation this many-body Hamiltonian can be simplified to a single-band Bose-Hubbard Hamiltonianˆ H BH = − J X h i,j i ˆ a † i ˆ a j + U X i ˆ n i (ˆ n i −
1) (6)where ˆ a i is the bosonic annihilation operator for site i , ˆ n i is the corresponding number operator, the subscript h i, j i labeling nearest neighbor pairs. Finally J is the inter-site2 .03 0.04 0.05 0.0601020 Scaled Input Intensity È Y S ca l e d I n t r aca v it y I n t e n s it y È X s È Y b È Y a Figure 2: Bistable dimensionless intracavity intensity | X s | as a func-tion of the dimensionless pump intensity | Y | for T = 0 .
01, initialoffset δ = − . β = 0 . tunneling matrix element and U is the pair interactionenergy. The parameters J and U can be evaluated by ex-panding the field operators on the Wannier basis of thelowest Bloch band, and then evaluating the pertinent in-tegrals [18].Setting the time-derivatives to zero yields the steady-state solution X s = 2 Y / √ T − i π ( δ + ξ s ) /T ,ξ s = β | X s | . (7)Eliminating the equilibrium displacement ξ s in these equa-tions results in a transcendental equation for the intracav-ity field intensity | X s | , | X s | | Y | = 4 /T π ( δ + β | X s | ) /T . (8)This equation is known to have a bistable solution for anappropriate choice of parameters, see Fig. 2. The upperand lower branch with positive slope are generally stable,while the dashed branch with negative slope is unstable[19]. It is clear from Eq. (7) that the steady-state dis-placement of the movable mirror has the same bistableproperty as the intracavity field intensity. As is well known, the ground state of the atomic systemis determined by the ratio
J/U which can be controlled byvarying the lattice depth V . For a shallow potential, in-terwell tunneling dominates and the many-body groundstate is a superfluid, while for a deep enough potential on-site interactions dominate and the atoms enter the Mott-insulator phase with integral filling factor [18]. The dot-ted line in Fig. 2 indicates the critical depth V at which All calculations presented in this paper are for Na and for λ p =985nm, M = 0.078g, Ω = 2 π × R = 0 . the superfluid-Mott insulator phase transition occurs, forsodium atoms trapped by a laser of wavelength 985nm. Ofparticular interest is the bistable region where the shallowlower branch corresponds to a superfluid phase and thedeeper upper branch to a Mott insulator phase. The nextsection discusses the dynamics of the transition betweenthese two phases as the optical field is varied along thebistable loop.
3. Dynamics
Three important time scales are relevant in understand-ing the dynamics of the system: the characteristic timeΩ − of the mirror, the cavity buildup time κ − , and theinterwell tunneling time ~ /J of the atoms. In the weakcoupling limit the dynamics of the cavity system (the lightfield plus the movable mirror) is determined by the formertwo. Furthermore, in the case of a bad cavity the dynamicsof the light field is much faster than that of the movablemirror, κ − ≪ Ω − , so that it can adiabatically follow theinstantaneous displacement of the movable mirror. Notethat this approximation neglects the non-adiabatic effectsthat lead to optical damping, see Ref. [20] and referencestherein.This paper treats the motion of the moving mirror clas-sically as it is not concerned with the dynamics of mirrorcooling. We therefore ignore optical damping in the fol-lowing, and adiabatically eliminate the optical field in theEq. (5), resulting in the nonlinear oscillator equation d ξdτ + γ dξdτ + ξ = β | Y | /T π ( δ + ξ ) /T . (9)To further analyze the dynamics of this nonlinear os-cillator we introduce the effective potential [21] V ( ξ ) = Z ξξ ξ ′ dξ ′ − Z ξξ β | Y | /T π ( δ + ξ ′ ) /T dξ ′ (10)where the first term corresponds to the restoring force onthe movable mirror and the second to the radiation pres-sure force. Figure 3 shows that potential for three valuesof the input intensity. For a weak input field, Fig. 3(a) thepotential has a single minimum, but increasing it sees theappearance of a second local minimum, Fig. 3(b), indica-tive of bistability. Further increasing the input intensitypast the bistable region the initial minimum disappears asexpected, see Fig. 3(c).For concreteness we assume in the following that theoptical field incident on the Fabry-P´erot is a super-Gaussianpulse, Fig. 4(a). Since the intracavity field is assumed tofollow the motion of the mirror adiabatically, this readilyyields the time-dependent optical potential | X ( τ ) | = 4 | Y ( τ ) | /T π ( δ + ξ ( τ )) /T (11)Figure 4(b) illustrates the dynamics of the intracav-ity intensity as the incident intensity is varied to switch3 Ξ H Λ p (cid:144) L H - L H a L Ξ H Λ p (cid:144) L - - H - L H b L Ξ H Λ p (cid:144) L - - H - L H c L Figure 3: Potential V ( ξ ) for (a) input intensity | Y | = 0 .
03; (b) | Y | = 0 . | Y | = 0 .
06. Other parameters are the same as inFig. 2. All variables are dimensionless.
40 80 120 1600.04 È Y b È Y a Τ H W - L I npu t I n t e n s it y È Y H a L
40 80 120 16051015 Τ H W - L I n t r aca v it y I n t e n s it y È X H b L Figure 4: (a) Input intensity | Y | as a function of dimensionlesstime τ . The two dashed lines label the critical intensities at thebeginning and the end of the bistable region. (b) Evolution of theintracavity intensity | X | controlled by | Y | . Here γ = 0 . the system from the lower to the upper bistability branchand back. In this example | Y ( τ ) | is initially switched toa value only slightly above the critical value | Y a | wherethe first local minimum of the effective potential degener-ates into a plateau, hence it takes a relatively long time toswitch to its new steady-state value. Under appropriatecircumstances this critical slowing down [22] can be ex-ploited to allow the atoms to adiabatically follow the po-tential changes in the superfluid region, see below. Follow-ing that slow evolution stage, | X ( τ ) | grows nearly expo-nentially as the mirror falls into the newly formed potentialwell and completes its transition to the upper branch. Forthe switch back to the lower branch shown in this exam-ple, the lower value of | Y ( τ ) | is chosen to be significantlybelow | Y b | , so as to reduce the effects of critical slowingdown. As expected, | X ( τ ) | then switches much faster tothe lower branch.The time dependence of the potential | X ( τ ) | results inWannier functions that are likewise time-dependent, andhence also in time-dependent tunneling rate J ( τ ) and on-site interaction U ( τ ). In order for the many-body groundstate of the atoms to follow adiabatically the changes inpotential, two requirements must be fulfilled. The first oneis that the variation of the potential depth must be slowenough to prevent the occurrence of inter-band excitations,that is the atoms must remain in the first Bloch band at alltimes. This condition is usually easy to satisfy for atomswith quasi-momentum q ≃ | ddt V /E r | ≪ ω r . We note that for the experimental parameters consideredhere this condition is satisfied for even the fastest tran-sients in Fig. 4(b): The maximum value reached by | ddt V /E r | is 2 Ω which is much less than 16 ω r . We therefore assumein the following that that inter-band excitations are negli-gible, and evaluate J ( t ) and U ( t ) with Wannier functionsof the first band.In addition to this single-particle adiabaticity condi-tion, we also need to consider the time scale associatedwith many-body effects. As the potential depth varies intime, the atoms need a sufficient amount of time to tunneland redistribute themselves across the lattice, and henceto settle in their new ground state — think specifically of atransition between a Mott insulator and a superfluid. Al-though this is not a significant consideration if the atomsremain in the Mott insulator phase, which is quite insen-sitive to inter-well rearrangement, it is more important inthe superfluid region, due to the variation of the tunnel-ing rate J ( t )[26, 25]. That is the reason why the criticalslowing down can be useful in controlling the variation ofthe potential in the superfluid region. Whether or notthe second adiabaticity condition is satisfied can be deter-mined from the time-dependent order parameter which wecalculate next.4he evolution of the atomic ground state can be ob-tained from the time-dependent variational principle [27,28], δ h G | i ~ ∂∂t − ˆ H BH ( t ) | G i = 0 , (12)where the Bose-Hubbard Hamiltonian ˆ H BH ( t ) depends ontime via the coefficients J ( t ) and U ( t ), and | G i is takento be given by the time-dependent mean-field Gutzwilleransatz | G i = N l Y i ∞ X n =0 f ( i ) n ( t ) | n i ! . (13)Here N l is the number of lattice sites, | n i are Fock states,and f ( i ) n ( t ) are probability amplitudes that preserves thenormalization of the wave function. We assume that thelattice is uniform with unit filling factor and insert theGutzwiller ansatz into Eq. (12). This yields the set ofcoupled nonlinear equations for the amplitudes f n ( t ) i ~ ∂∂t f n = − J ( t ) (cid:0) √ nf n − h a i + (cid:10) a † (cid:11) √ n + 1 f n +1 (cid:1) + U ( t )2 n ( n + 1) f n (14)where h a i = P ∞ n =0 √ n + 1 f ∗ n f n +1 is the atomic superfluidorder parameter. Our numerical results are for an initialoptical potential depth of 5 E r , for which the tight bindingapproximation is valid and the many-body ground stateis superfluid. Figure 5(a) shows the ratio 2 J ( t ) /U ( t ) asa function of time. The transition between the super-fluid and Mott insulator phases is clearly shown by theabsolute value of the superfluid order parameter |h a i| plot-ted in Fig. 5(b). As the ratio 2 J/U decreases below thecritical value 0 .
17, corresponding to the critical potentialdepth V = 10 . E r for sodium atoms, the order parame-ter rapidly drops to zero, indicating that the atoms enterthe Mott insulator phase. When 2 J/U rises back abovethat critical value, the order parameter recovers a non-zero value, indicative of the return of the atoms to thesuperfluid phase.The small oscillations of the order parameter in theMott-insulator region are due to collapses and revivals ofthe condensate with period T = h/U [29, 30]. The oscilla-tions following the return of the system to the superfluidregion were first predicted in Ref. [31]. They result froma rapid transition of the system from being a Mott insu-lator to a superfluid, and can be suppressed via a slowervariation of potential depth. The small amplitude of theseoscillations, less than 5 percent of the initial value of theorder parameter, indicates that the second adiabatic con-dition (associated with tunneling) is well satisfied in thisexample.For comparison Fig. 5(c) shows the time-dependent su-perfluid order parameter for a situation here non-adiabaticeffects are more apparent. In this case the mass and vibra-tion frequency of the mirror have been changed to 0.031gand 2 π ×
40 80 120 160 - - - Τ H W - L L og H J (cid:144) U L H a L
40 80 120 16000.51 Τ H W - L O r d e r P a r a m e t e r È < a ` > È H b L
40 80 120 16000.51 Τ H W - L O r d e r P a r a m e t e r È < a ` > È H c L Figure 5: (a) 2 J ( t ) /U ( t ) as a function of the dimensionless time τ . The dotted line indicates the critical value 2 J/U = 0 .
17, corre-sponding to the potential depth 10 . E r for our parameters, for thesuperfluid and to Mott insulator phase transition. (b) Dynamicsof the superfluid order parameter |h a i| during the transition fromthe superfluid to the Mott insulator and back to the superfluid.Here λ p = 985 nm and ω r = E r / ~ = 2 π × . Na. Themass and vibration frequency of the mirror are M = 0 .
078 g andΩ = 2 π ×
10 Hz, respectively. In real units, the full cycle through thebistable loop takes 2 . M = 0 .
031 g andΩ = 2 π ×
50 Hz δξ which result in turn in fluctuations for the intra-cavity field strength the lattice potential depth δV . Inorder to prevent this effect we request that δV ≪ V min ,where V min is the minimum value of the potential depthduring the switching process. This inequality gives a mir-ror temperature limit of T ≪ K.
4. Conclusion
We have analyzed the dynamics of a quantum-degenerateultracold sample of bosonic atoms trapped in the opticallattice of a bistable optomechanical system. The varia-tion of intracavity optical lattice depth results in a bistablequantum phase transition between a superfluid and a Mott-insulator ground state. We considered the concrete case ofa super-Gaussian incident light pulse to drive the atomicsystem around the bistable loop, and found that the crit-ical slowing down of the optomechanical system can beuseful to prevent the excitation of the atoms in a po-tential ramping-up stage. Our numerical results indicatethat the bistable superfluid to Mott-insulator phase tran-sition should be realizable for realistic experimental pa-rameters. Future work will extend these studies to thestrong-coupling regime, where the optical field depends onthe motion of the cold atoms as well as the mirror. Wewill also investigate to which extent the mirror can serveas a measurement device to observe the dynamics of aquantum phase transition nondestructively, in particularin cases where it is cooled to near its quantum mechani-cal ground state of vibration, Additional goals include thestudy of the optically induced quantum correlations andentanglement between the ultracold atomic system and themirror, the quantum control of the mirror motion by theatoms, and conversely of the atoms by the nanomechani-cal system, and finally the development of novel types ofquantum sensors.
Acknowledgements
This paper is dedicated to the memory of KrzysztofW´odkiewicz, excellent physicist and even better friend,with whom it has been great fun to push the frontiers ofquantum optics over many years and in many places, fromthe old Max Planck Institute for Quantum Optics ”theorycontainer” to the American Southwest. We thank M. Bhattacharya and D. Goldbaum for use-ful discussions. This work is supported in part by the USOffice of Naval Research, by the National Science Founda-tion, and by the US Army Research Office.
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