Bistable Mott-insulator to superfluid phase transition in cavity optomechanics
W. Chen, K. Zhang, D. S. Goldbaum, M. Bhattacharya, P. Meystre
aa r X i v : . [ qu a n t - ph ] M a y Bistable Mott-insulator to superfluid phase transition in cavity optomechanics
W. Chen , K. Zhang , D. S. Goldbaum , M. Bhattacharya , and P. Meystre B2 Institute, Department of Physics and College of Optical Sciences, The University of Arizona, Tucson, AZ 85721, USA State Key Laboratory of Precision Spectroscopy, Department of Physics, East China Normal University, Shanghai 200062, China
We study the many-body state of ultracold bosons in a bistable optical lattice potential in anoptomechanical resonator in the weak-coupling limit. New physics arises as a result of bistability anddiscontinuous jumps in the cavity field. Of particular interest is the situation where the optical cavityis engineered so that a single input beam can result in two radically different stable ground statesfor the intracavity gas: superfluid and Mott-insulator. Furthermore, the system we describe can beused as an adjustable template for investigating the coupling between cavity fields, nanomechanicalsystems operating in the quantum regime, and ultracold atomic gases.
PACS numbers: 42.50Pq, 37.30+i, 37.10.Jk, 05.30.Jp
Recent years have witnessed a remarkable convergenceof interests in atomic, molecular and optical physics, con-densed matter physics, and nanoscience. Specific ex-amples include the use of ultracold atomic and molecu-lar systems as quantum simulators of solid-state systems[1, 2], the demonstration of the analog of cavity QED ef-fects with superconducting boxes [3], and the laser cool-ing of nanoscale cantilevers [4], leading to the emergingfield of cavity optomechanics.The central element of most cavity optomechanical sys-tems consists of a Fabry-P´erot type cavity with one end-mirror vibrating about its equilibrium position under theeffect of radiation pressure. These devices can exhibit op-tical bistability, that is, the light transmitted through thecavity can take two distinct intensity values for a givenincident intensity [5].In this letter we show that optical bistability can leadto fascinating new effects in the dynamics of an ultracoldsample of bosonic atoms trapped inside such resonators.In particular, at the simplest level of weak coupling andclassical mirror motion we predict a bistable quantumphase transition between a Mott-insulator (MI) state anda superfluid (SF) state of the many-atom system. Inthe more general case where these approximations areremoved, this system opens the way to the exploration ofa completely new regime of interaction between light, ul-tracold atoms and quantum mechanical nanostructures.We note at the outset that clearly, a bistable transi-tion between a MI and a SF does not require the use ofa cavity optomechanical system: any arrangement pro-ducing optical bistability would work just as well. How-ever, it is expected that it will soon be possible to ef-ficiently laser-cool one or more modes of vibration ofmoving nanoscale cantilevers or mirrors to their quan-tum mechanical ground state. An added advantage ofthe optomechanical cavity setup is its ability to serve asa diagnostic: the reflected or transmitted fraction of lightdriving the cavity has been shown to contain informationabout atomic [6, 7, 8] and mirror [4] dynamics. This iswhat makes the coupling of ultracold atoms to optome-chanical systems so promising. One main purpose of thisnote is to demonstrate that these studies are rapidly be- coming experimentally viable.We recall that the MI to SF transition can occur whenan ultracold gas of bosonic atoms is trapped by an op-tical lattice in the tight-binding regime [9]. The groundstate properties of the system are largely determined bythe relative strength of the interwell tunneling energy J ,and the intrawell pair-interaction energy U . When tun-neling dominates, the ground state tends to be SF. Inthe opposite case, the ground state tends to be a MI,characterized by a fixed atom number at each site.Consider then an ultracold gas of bosonic atomstrapped in the optical lattice provided by the standingoptical wave inside an optical cavity exhibiting bistabil-ity. In the lower intensity branch the optical lattice isshallow, so that interwell tunneling dominates and themany-atom ground state is SF. In the upper branch amuch deeper optical lattice suppresses tunneling, and themany-atom ground state is a MI. The state of the atomicsystem is therefore bistable, with a SF or a MI beingformed for the same incident light field, depending onthe history of the system. In the following we describe ascenario where this effect can be observed for realizableparameters in optomechanical resonators.Our study complements recent experiments on coldatomic gases in optical cavities with fixed ends. In eachsystem two dynamical quantities are strongly coupled,necessitating a self-consistent, and generally nonlinear,description of their time evolution. Slama et al. [10] stud-ied the gain mechanisms behind superradiant Rayleighscattering and collective atomic recoil lasing by investi-gating a ring cavity system. Two separate groups inves-tigated optomechanical systems, where collective excita-tions of the confined gas played the role of the mechanicaloscillator. Brennecke et al. [8] demonstrated a couplingbetween a density modulated Bose-Einstein condensate(BEC) and the cavity field, where the phase space evo-lution was mapped onto that of a harmonically confined,cavity-coupled mechanical oscillator. Gupta et al. [11]and Murch et al. [12] found that cavity-field coupling toa collective center-of-mass-motion excitation of the con-fined gas, resulted in oscillatory displacement of the gas.Our work has an especially close correspondence withthat of Larson et al. [13]. Like us, they investigate a coldgas of bosonic atoms trapped by a bistable optical lattice.However, in contrast to our study, their cavity had fixedends, and the bistability results from the strong couplingbetween the cavity field and the atomic gas. Accordingly,their system is modeled by a Bose-Hubbard Hamiltoniancharacterized by the parameters J , U , and chemical po-tential µ calculated self-consistently with the many-bodyatomic state. This self-consistent dependence results ina radically different ground-state phase diagram than forthe Bose-Hubbard Hamiltonian describing our system.Furthermore, our system has an additional dynamicalcomponent – the movable end-mirror, and as alreadymentioned it can be used to investigate couplings be-tween these three dynamical components when one, twoor all of them operate in the quantum regime. In thisletter we focus on the new physics in the weak couplingregime and for classical mirror motion. Understandingthis limit is an important first step in the study of themore complicated regimes that can be realized in oursetup.On the microscopic level, the lattice potential resultsfrom the coupling between the intracavity field and anatomic resonance with frequencies, ω and ω a , respec-tively. As already mentioned we investigate the weak-coupling limit defined by N g / | ∆ | ≪ κ , where N isthe total number of atoms, g is the atom-field cou-pling strength, κ is the cavity’s natural line-width, and∆ = ω − ω a is the atom-field detuning [14]. In this limitthe intracavity field has no significant dependence on theintracavity atomic population. We thus explain the gen-eration of the intracavity optical lattice potential by us-ing the theory of an empty cavity [15].We briefly derive the necessary results from the one-dimensional equilibrium theory of the Fabry-Perot cavityshown in figure 1. The cavity consists of two mirrors, onefixed along x = 0 and the other harmonically confinedabout x = L . Each mirror has complex transmissionand reflection coefficients t and r , where | r | = 0 .
99 and | t | + | r | = 1. We only consider internal reflections whereone may assume a π -phase shift, and thus we replace thecomplex r defined above with − r , where the new r ispositive and real. The phase of t has no bearing on ourresults.A driving laser field E in of frequency ω is incident on,and directed normal to, the outer surface of the fixed mir-ror. This configuration allows a one-dimensional treat-ment. For a fixed cavity length, L , we follow the dis-cussion of Loudon [16] to determine the transmitted in-tensity, I trans , exiting the cavity. The right-moving in-tracavity field at x = 0, E R , is determined by solv-ing E R = tE in − rE L under the equilibrium condition E L = − r exp[ i kL ] E R , where E L is the left-moving cav-ity field at x = 0 and k = ω/c is the wavenumberof the light [20]. The resulting transmitted intensity, L(cid:13) ξI in I trans x Figure 1: Fabry-P´erot cavity of length L with left-end mir-ror fixed along x = 0 and right-end mirror oscillating about x = L , where L = L + ξ . The input- and transmitted-light intensities are labeled I in and I trans , respectively. Theintracavity intensity at resonance ( I in = I trans ) is representedschematically by the sine-squared wave drawn inside the cav-ity. In this letter, L ∼ ∼ ∼ I trans . I trans = | E trans | , is I trans = I in F π sin ( kL ) , (1)where E trans = tE R , I in = | E in | , and F = πr/ (cid:0) − r (cid:1) is the cavity finesse.Small mirror displacements due to the intracavity ra-diation pressure are given by ξ = ηI trans , where η = AM Ω c rπ F , A is the cross-sectional area of the input laserbeam, M is the mass of the moveable mirror and Ω itsoscillation frequency. Substituting L = L + ξ into equa-tion (1) results in a nonlinear equation for I trans whichis multistable with respect to I in . We concentrate on thephysics near cavity resonances, where kL = nπ , with n apositive integer. For small displacements from resonance,the governing equation is approximately cubic in I trans ,and predicts radiation pressure bistability [5, 15].It follows that the intracavity field intensity, I cav ( x ) = | E R ( x ) + E L ( x ) | , is also bistable, and leads to a bistableoptical lattice potential for the atoms (see Figure 2), V OL ( x ) = V osc sin [ k ( L − x )] + V L , (2)where V osc = F π αI trans , V L = (1 − r )(1+ r ) αI trans , α = (cid:0) πc Γ (cid:1) / (cid:0) ω a ∆ (cid:1) , and Γ is the natural linewidth of theatomic resonance. The microscopic origin of the propor-tionality constant α is the AC-Stark shift of the single-atom ground state. (We ignore V L in the following sinceit is tiny compared to all relevant energies.) The positionof the individual lattice wells is bistable as well, since amirror displacement, ξ , displaces each optical lattice wellby ξ in the same direction. However, we consider a regimewhere ξ/ ( π/k ) ∼ − , and thus we ignore this effect.We consider a gas of ultracold bosonic atoms trappedin the one-dimensional optical lattice potential V OL ( x ).In the weak-coupling limit the atomic state does not alter I in [ E re /α ] V o s c [ E r e ] DA BC
Figure 2: (Color Online) Bistability of the intracavity opticallattice depth, V osc , with respect to the input light intensity I in . The bistability curve is drawn with respect to the unitlessquantities αI in /E re and V osc /E re , where α = V OL ( x ) /I cav ( x )and E re = ` ~ k ´ / (2 m ) is the recoil energy. The detuningfrom resonance is quantified by φ = − . π , and k η =0 . π α/E re , where φ = mod π [ kL ]. The curves AB andDC are the lower and upper branches of V osc in the bistableregion. The dashed green line connecting D and B marks theunstable lattice depths. The dashed gray lines DA and BCmark discontinuous jumps in the lattice height. the cavity field. Thus the atomic state is described bythe Hamiltonianˆ H = Z dx ˆ ψ † ( x ) (cid:18) − ~ m d dx + V OL ( x ) + g n ( x ) (cid:19) ˆ ψ ( x ) , (3)where ˆ ψ † ( x ) , ˆ ψ ( x ) are bosonic field operators, ˆ n ( x ) is thecorresponding number operator, m is the atomic massand g is the two-body interaction.We are interested in the SF–MI transition, where themany-atom system is accurately described by a tight-binding approximation that results in 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 − − µ X i ˆ n i , (4)where ˆ a † i (ˆ a i ) is the bosonic creation (annihilation) op-erator for site i , ˆ n i = ˆ a † i ˆ a i , and the subscript h i, j i de-notes a sum over nearest neighbor hopping moves. Thetunneling matrix element is J , U is the pair interactionenergy, and µ is the chemical potential. The param-eters J and U are calculated by expanding the bosonfield operators in a basis of lowest band Wannier states,ˆ ψ ( x ) = P i ˆ a i w ( x − x i ), and then evaluating the perti-nent integrals [1].The ground state of the many-body system describedby equation (4) is largely determined by the value of J/U ,which depends on the intensity, wavelength and detuningfrom atomic resonance of the intracavity standing wavefield. Of particular interest to us is the bistable regime ofthe optical lattice potential where the many-body groundstate corresponding to the lower branch of the potential isa SF, while the ground state corresponding to the upperbranch is a MI. Figure 3 summarizes key features of the system for acavity length L = 1 mm and a moving end mirror of mass M = 10 mg and oscillation frequency Ω = 2 π × (25 Hz).The cavity is loaded with a Bose-Einstein condensate ofabout 1000 sodium-23 atoms. We use an input laser ofwavelength λ = 985 nm to generate the intracavity opti-cal lattice potential. The optical lattice consists of about2000 sites, however we neglect the effects of direct atom-mirror interactions by assuming that only ∼ µ/U andlog (2 J/U ) [17], [21]. For
J/U smaller than this bound-ary the ground state is the single-particle MI, otherwisethe ground state is SF. This diagram is overlayed by aplot of log (2 J/U ) versus I in for the many-body sys-tem described above, the logarithmic scale reflecting theexponential dependence of tunneling on intensity. Thelower and upper branches are labeled with points { A,B } and { D,C } , respectively. These labeled points correspondto the lattice depths labeled in Fig. 2. For I in just abovezero, the lattice potential is too shallow for the system tobe described by a single-band tight-binding limit. Thusassigning a value of J/U is meaningless there. How-ever, by adiabatically increasing the intensity of the inputlaser, the condensate settles into a single band of the op-tical lattice potential. For high enough lattice intensity,the system enters the tight-binding limit. However, atthe low-intensity edge of the bistable region, labeled Ain Fig. 3, the system is very near the single-band tight-binding regime. That is, at point A a treatment withEq. (4) is appropriate for our present purpose, but afuture in-depth calculation will require including higherband effects. At point B, in contrast, the system is safelyin the single-band limit, and is accurately described byEq. (4). It should be noted that the semi-log bistabil-ity plot has no direct correspondence to µ in Fig. 3. Wemerely specify that the system is prepared so that the up-per branch of the bistability region lies inside the Mott-lobe, while the lower branch corresponds to a SF groundstate.For input intensities between the points A, D ( I in ∼ .
86 mW for our choice of parameters) and B,C ( I in ∼ .
62 mW), the system is bistable. During an initial adi-abatic intensity increase, the system first resides in thelower branch, where the ground state is SF. Above pointB there is only a single stable state, in the upper branch ofintracavity intensity and lower branch of interwell tunnel-ing. At that point the atoms experience a much strongerlattice confinement, with a discontinuous phase transi-tion to a MI.The time scale over which this transition occurs is de-termined by the longest of the interwell tunneling times τ ∼ ~ /J and the switching time of the intracavity field.In most cases, the intracavity field reaches a new steady- -3-2-1010.0 0.1 0.2 l og ( J / U ) µ/U I in [ E re /α ] AD CB
Figure 3: (Color Online) Bistability of the many-body groundstate. Thick black line: mean-field SF–MI phase boundarywith respect to µ/U and log (2 J/U )). The ground stateis the single-particle MI inside the lobe and a SF outside.The phase plot is overlayed with the intersite tunneling bista-bility curve, log (2 J/U ) vs. I in curve. The (DC) branchcorresponds to MI ground states, while the (AB) branch cor-responds to SF ground states. The labeled values here cor-respond to the lattice depths labeled in Fig. 2. The dashedgreen line indicates unstable solutions, and the arrows BCand DA indicate the discontinuous jumps between differentbranches. state value following an abrupt change in the incidentfield after a time of the order of the inverse cavity decayrate κ ≃ c | t | /L [18]. However, for intensities switchedfrom below point B to a value just above it the systemundergoes a critical slowing down [19], with a large de-lay before the field switches from the lower to the upperbranch. The resulting possibility to vary the switchingtime of the light field compared to the tunneling time provides an important tool to investigate a variety of dy-namical phenomena. Most optical lattice experiments areperformed using adiabatic tuning of the lattice height inan attempt to keep the system in its ground state. Af-ter sweeping through the discontinuity, though, we ex-pect that in general the many-body state will be excitedabove the ground state corresponding to the optical lat-tice potential. The nature of this excited state and itsrelaxation pathways are a subject of current research. Al-ternatively, applying a time-dependent incident field suchthat the system oscillates about the discontinuity pro-vides an additional tool to probe non-equilibrium prop-erties, and perhaps induce coupling between MI and SFground states.Similar considerations hold when initially preparing astable state in the upper branch, and then decreasing I in past point D, the optical lattice magnitude discontinu-ously jumps to its lower branch value at point A.In general, the setup that we described can be used toinvestigate the dynamics of coupled cold atomic gases,cavity fields and nanomechanical dynamics. We consid-ered explicitly the weak-coupling limit where the cou-pling between the cavity-field and the movable mirror re-sults in a bistable optical lattice potential for the atoms.We have discussed how such a cavity plus cold-atom sys-tem can be engineered so that SF and MI phases arebistable ground states for the cold-atom gas. Futurework will extend these considerations to the situationwhere the mirror motion is quantized, and discuss in de-tail the dynamics of the coupled system of light, ultracoldatoms and quantized nanostructure both in the weak andthe strong-coupling regime. With these considerations inmind an important first step is to construct an experi-mentally viable template, where the basic physics of eachconstituent system (cold gas, cavity-field, moving mirror)is well understood, and one can tune the couplings. Thesetup presented above is ideal for this purpose.This work is supported in part by the US Office ofNaval Research, by the National Science Foundation, andby the US Army Research Office. [1] D. Jaksch et al. , Phys. Rev. Lett. , 3108 (1998).[2] D. Jaksch and P. Zoller, Ann. Phys. , 52 (2005).[3] R. J. Schoelkopf and S. M. Girvin, Nature , 664(2008).[4] C. H. Metzger and K. Karral, Nature , 1002 (2004).[5] A. Dorsel et al. , Phys. Rev. Lett. ,023812 (2007).[7] I. B. Mekhov, C. Maschler, and H. Ritsch, Nature Phys. , 319 (2007).[8] F. Brennecke, S. Ritter, T. Donner, and T. Esslinger,Science , 235 (2008).[9] Greiner et al. , Nature , 39 (2002).[10] Slama et al. , Phys. Rev. Lett. , 053603 (2007); S. Slama et al. , Phys. Rev. A , 063620 (2007).[11] S. Gupta, K. Moore, K. Murch, and D. Stamper-Kurn,Phys. Rev. Lett. , 213601 (2007).[12] K. Murch, K. Moore, S. Gupta, and M. Stamper-Kurn,Nature Physics , 561 (2008).[13] J. Larson, B. Damski, G. Morigi, and M. Lewenstein,Phys. Rev. Lett. , 050401 (2008).[14] P. Horak, S. Barnett, and H. Ritsch, Phys. Rev. A ,033609 (2000).[15] P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes,J. Opt. Soc. Am. B , 1830 (1985).[16] R. Loudon, The Quantum Theory of Light (Oxford Sci-ence Publications, 2003), 3rd ed.[17] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D.
S. Fisher, Phys. Rev. A , 033609 (2000).[18] P. Meystre, Optics Commun. , 147 (1978).[19] R. Bonifacio and P. Meystre, Optics Commun.29