Static and dynamic phases of a Tonks-Girardeau gas in an optical lattice
SStatic and dynamic phases of a Tonks-Girardeau gasin an optical lattice
Mathias Mikkelsen , Thom´as Fogarty and Thomas Busch Quantum Systems Unit, OIST Graduate University, Onna, Okinawa 904-0495,JapanE-mail: [email protected]
Abstract.
We investigate the properties of a Tonks-Girardeau gas in the presenceof a one-dimensional lattice potential. Such a system is known to exhibit a pinningtransition when the lattice is commensurate with the particle density, leading to theformation of an insulating state even at infinitesimally small lattice depths. Here weexamine the properties of the gas at all lattices depths and, in addition to the staticproperties, also consider the non-adiabatic dynamics induced by the sudden motion ofthe lattice potential with a constant speed. Our work provides a continuum counterpartto the work done in discrete lattice models. a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice
1. Introduction
The versatility of modern ultracold atomic experiments has in recent years allowedfor the realisation of exotic matter wave phases borne from the competition betweendifferent physical forces acting on many-body systems [1, 2, 3]. The paradigmaticexample of this has been the observation of the superfluid-to-Mott-insulator phasetransition in ultracold gases confined in three-dimensional optical lattice potentials,which stems from the interplay between the tunnelling dynamics and the onsiteinteractions [4, 5]. In one dimensional lattices and for strong repulsive interactions ithas been shown that the phases of a system are entirely controlled by the particle fillingstatistics [6, 7, 8, 9, 10]. In this case, a commensurate particle density leads to an integerfilling of each lattice site and thus realises an insulating pinned phase where dynamics issuppressed due to vanishing long range coherence. For incommensurate particle densitieseach lattice site has a non-integer filling which has the effect of creating delocalisedmodes which can be considered as defects [11, 12]. These defects promote dynamicsas they preserve a degree of coherence and allow the system to attain superfluid-likeproperties. Quantum phase transitions in these systems can be probed by switchingthe control parameter across a critical point [13, 14, 15, 16, 17, 18]. Additionally,the transport properties of such systems can be linked to classical models of frictionwhich describe stick-slip motion between two surfaces [19, 20, 21, 22, 23], an effectwhich has been recently observed [24, 25, 26]. In fact, precise control of the competinglength scales between the two surfaces can be used to effectively control the amount offriction present[27] and can give insights into designing frictionless dynamical processesfor quantum systems.In this work we focus on describing the static and dynamical phases of an ultracoldgas in an optical lattice potential with periodic boundary conditions [28, 29, 30]. Wefocus on the Tonks-Girardeau (TG) limit in one dimension [31, 32], where the appearanceof the pinned phase can be observed for particle numbers commensurate with theunderlying lattice, such that the system becomes insulating even for very shallow latticedepths. Suddenly setting the lattice potential into motion with a constant rotation speedallows us to probe the dynamical response of the TG gas to a quench in the externalpotential and to study the relationship between between friction and commensurability.The observed behaviour is similar to a type of quantum sprocket, where the rotatinglattice will impart maximum angular momentum to the gas if the system is in theinsulating phase, while in the superfluid phase only reduced momentum transfer isobserved due to tunneling of the particles between lattice sites. Our work is related torecent investigations of driven cold atom systems in the continuum [33, 34, 35, 36, 37, 38],and in the Bose-Hubbard model [39, 40, 41, 42], which is naturally connected to thecontinuous model in the limit of tight trapping. Another recent related work hasinvestigated the localization properties of a TG gas in a continuum version of the Aubry-Andre model (using a bi-chromatic lattice) [43]. In this manuscript we explore the phasediagram from shallow to deep lattices for commensurate and incommensurate fillings tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice
2. Basic Model
We consider a one-dimensional gas of N neutral bosons of mass m which are subject toan external potential V ext ( x n ). The Hamiltonian is given by H = N (cid:88) n =1 (cid:20) − (cid:126) m ∂ ∂x n + V ext ( x n ) (cid:21) + g (cid:88) i 3. Statics of the TG gas In the following we will first discuss the ground state properties of the TG gas in thepresence of a lattice. tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice V /E r E / E r Figure 1. Lowest lying 150 single-particle eigenenergies as a function of lattice depthfor M = 50. The Hamiltonian (1) is known to possesses two distinct ground-state phases in the TGlimit, which appear as a function of the ratio of number of bosons to lattice sites F = NM . (7)For incommensurate fillings ( F (cid:54) = N , with N a positive integer) in shallow latticesthe system has superfluid-like characteristics with long-range coherence and goodconductivity due to the delocalisation of the wavefunction over many lattice sites.However for commensurate filling ( F = N ), the bosons becomes localised at individuallattice sites and the total system becomes pinned to the lattice. This pinned phasehas no coherence, behaves as an insulator and is the hard-core continuum analogue ofthe Mott-insulator phase in the Bose-Hubbard model (BHM). In the continuum TGmodel the pinning happens at infinitesimally small lattice depths for F = 1. It was firsttheoretically proposed by B¨uchler et al. [6] and has been experimentally observed innumerous cold atom experiments [7, 10].The degree of coherence is a good property to characterize the two different phasesof the system. It can be obtained from the off-diagonal elements of the reduced single-particle density matrix (RSPDM) ρ ( x, y ) = (cid:104) Ψ B | ˆ ψ † ( x ) ˆ ψ ( y ) | Ψ B (cid:105) , (8)which describes the spatial auto-correlation of a single particle, giving the probabilitythat a particle is at y immediately after it has been measured at x . For classical tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice φ i ( x ), known as the naturalorbitals, with the corresponding eigenvalues λ i giving the respective occupation numbers ρ ( x, y ) = (cid:88) i λ i φ ∗ i ( x ) φ i ( y ) , (9)where (cid:80) λ i = N . For a non-interacting Bose gas at zero temperature only the lowestlying orbital is occupied, λ = N , which corresponds to a completely coherent andsuperfluid system. Conversely, a ground state wavefunction with λ = 1 correspondsto a completely incoherent gas where the N lowest orbitals are equally occupied. Theoccupation number of the lowest lying eigenstate of the RSPDM can therefore be usedas a measure of the coherence of the system. In fact, for a TG gas in free space itis known that the lowest lying orbital has an occupation proportional to √ N , whilethe occupation of orbitals with i (cid:29) N tends to zero [49, 50]. Clearly such a stateis not strictly superfluid, however it is also not completely incoherent, as λ is of theorder √ N . It therefore does possess some superfluid properties. For simplicity, we willrefer to phases displaying finite coherence as superfluid in the rest of this manuscript.In the pinned phase, however, each particle is highly localised which destroys thecoherence between individual wells once tunneling is sufficiently suppressed. The systemis therefore reduced to N non-interacting particles with coherence λ i = 1 for the N lowestlying orbitals.Even though the coherence can be used to identify different regimes of the system, itis not easy to measure it in a direct way. However, it is closely linked to the momentumdistribution, which can be readily observed through time-of-flight experiments andcalculated from the Fourier transform of ρ ( x, y ) as n ( k ) = (cid:90) dx dyρ ( x, y ) e − ik ( x − y ) . (10)This can be recast in terms of the Fourier transform of the eigenstates of the RSPDM, (cid:101) φ i ( k ), using the same occupation numbers n ( k ) = (cid:88) i λ i (cid:101) φ i ∗ ( k ) (cid:101) φ i ( k ) . (11)It is therefore clear that the momentum distribution depends directly on the distinctcharacteristics of the coherence and long-range order of the system. The lowest eigenvalue of the RSPDM, λ , and the peak-value of the momentumdistribution, n ( k = 0), are shown in Fig. 2 as a function of the filling ratio and thelattice depth. One can immediately see that the coherence decreases quickly for unitfilling, F = 1, eventually reaching the value of λ = 1 corresponding to the pinned tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice (a) F V /E r F F V /E r (b) n ( k = ) n ( k = ) F Figure 2. (a) Coherence λ and (b) height of the central peak of the momentumdistribution n ( k = 0) as a function of the filling ratio F and the lattice depth V (inunits of E r ). The number of lattice wells is fixed at M = 50. The insets shows thecoherence and the zero momentum peak as a function of F for V = 20 E r . phase. For double filling, F = 2, a finite lattice depth is required before this transitioncan happen, as the lattice needs to be deep enough to overcome the repulsive two-bodyinteractions at each site. For incommensurate fillings a slow decrease of the coherencewith increasing lattice depth can be seen. This behavior is qualitatively mirrored inthe magnitude of the momentum distribution at k = 0 and in the deep lattice limit( V = 20 E r ), both quantities display an oscillating dependence on the filling ratio F ,as can be seen in the insets of Fig. 2. This is similar to the behaviour known for thesuperfluid fraction in the hard-core BHM [51].The distinct characteristics of the pinned phase and the superfluid phase alsomanifest themselves in the full momentum distribution. In order to characterize theincommensurate phase we show the momentum distribution for F = as a functionof the lattice depth in Fig. 3(a) and with a comparison to the commensurate case inFig. 3(b). In free space ( V = 0) the TG gas is delocalised in position space andtherefore localises around a single, central peak in momentum space (black solid linein Fig. 3(b)), whereas for V = 20 in the pinned phase ( F = 1) the gas is localisedin position space and is therefore delocalised in momentum-space (magenta line). Inthe incommensurate phase ( F = ) the momentum distribution can be seen to displayadditional back-scattering peaks at multiples of 2 k l , the number of which and theirintensity increases with the lattice depth. The gas in this phase therefore representsa momentum delocalised superfluid that exhibits some of the distinct characteristicsof both the superfluid and the pinned phase. The observed momentum distribution issimilar to the momentum distributions found in supersolids [52] and it has recently beensuggested that the incommensurate phase of the TG gas in deep lattices is similar to adefect-induced superfluid phase and can be utilized to investigate the Andreev-Lifschitz- tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice (a) V /E r k / k l -5 0 500.020.040.060.080.1 (b) n ( k ) / N k/k l l og [ n ( k ) / N ] Figure 3. (a) Normalised momentum distribution for an incommensurate state with F = as a function of lattice depth. (b) Normalised momentum distribution ina deep lattice ( V = 20 E r ) for the incommensurate ( F = , red dashed line) andthe commensurate ( F = 1, full magenta line) case. For comparison the black linecorresponds to the case where no lattice is present. Chester mechanism [53, 12].It is instructive to consider two distinct defect regimes of the incommensuratephase. If the deviation from the commensurate phase at F = 1 is microscopic, suchthat N = M ± 1, the phase becomes gapless with a quadratic dispersion relation. Onthe other hand, for a macroscopic number of defects, such as N = M ± M/ 2, the phaseis also gapless, but with a linear dispersion relation [12]. These regimes are, however,continuously connected which can be seen in Fig.4(a), where the momentum distributionas a function of F is plotted for a constant depth V = 20 E r . Far from commensuratevalues, i.e. for a macroscopic number of defects, the multiple peaks stemming fromthe delocalised superfluid are the dominant contribution to the momentum distribution(see Fig. 4(b)), but as F → N = M ± 1, the momentum distributionessentially corresponds to that of the delocalised pinned phase, but with additionalsmall peaks on top of it. Beyond unit filling the Gaussian contribution of the first N = M particles remains, while the additional particles lead to a multi-peak structureon top of it. This means that the first N = M particles are effectively filling the lowestenergy-band, while the superfluid physics happens in the second band. The multi-peak structure observed in the momentum distribution of the incommensurateTG gas is a clear indicator that off-diagonal long-range order (ODLRO) and densitylong-range order (LRO) exist in the gas. To confirm this we calculate the spatial auto- tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice F (a) k / k l -5 0 500.020.040.060.080.1 n ( k ) / N k/k l (b) -5 0 500.010.020.030.04 n ( k ) / N (c) k/k l Figure 4. (a) Normalized momentum distribution as a function of the filling factor.(b) Momentum distribution for filling factors F = 1 (magenta line), F = (dashedred line) and F = (black line). (c) Momentum distribution for filling factors F = 1(magenta line) and F = (dashed black line). All plots in this figure are for a latticedepth of V = 20 E r . correlations, which we define at a distance r as the average of the RSPDM-values at( x, y ), i.e. ρ ( r ) = 1 L (cid:90) dxdyρ ( x, y ) δ ( | x − y | − r ) . (12)This function then explicitly quantifies the long-range order of the system and wedisplay it for different regimes in Fig. 5. One can see that for no external lattice (redcurve in panel (a)), the correlations decay according to the well-known r − power lawbehaviour of a free TG gas [54]. In the presence of a lattice for incommensurate particledensity (such as F = ) an exponential decay is observed at small r followed by apower-law decay. This demonstrates quasi-off-diagonal long range order (QODLRO) inthe gas for the Hamiltonian (1) and mimics the behavior of the hard-core BHM [55].In the continuous case, however, the power-law decay has an additional, very intuitivesuper-structure: the auto-correlations are modulated with the lattice periodicity. Thiscorresponds to the fact that the atoms are preferably located at the lattice minima andtheir probability to be at a lattice maximum is significantly decreased. In fact, if theauto-correlations at these distances are going to zero, the system will enter a regimein which the tight-binding approximation works well. Once the second energy-bandis occupied a deeper lattice is required to create modulations in the auto-correlations(compare F = in (a) and (b)). For F = 1 the pinned phase leads to an exponentialdecay of correlations, with no correlations surviving at distances longer than one latticesite for V = 20. In the case of a microscopic number of defects ( F = ) the initialexponential decay is much faster than for a macroscopic number of defects ( F = ). Infact it is similar to the one found in the pinned phase, but unlike the pinned phase itshows small revivals at subsequent lattice sites. The magnitude of these revivals decaysslightly, but it is difficult to ascertain the precise nature of this small decay, as we arerestricted to a periodic 50-site lattice.Since the modulations in the correlation-function stem from the off-diagonal terms tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice -4 -4 (a) ⇢ ( r ) / N r ML (b) ⇢ ( r ) / N r ML F = F = F = 1 V = 0 F = 1 F = F = F = Figure 5. Auto-correlation functions as a function of r for different filling factors andfor a lattice depth of (a) V = 5 E r (except the red line) and (b) V = 20 E r . Thedashed lines correspond to a r − decay and are displayed for V = 0 and F = , . in the RSPDM, they relate directly to the peaks that appear in the momentumdistribution and therefore to the superfluid properties of the system. The externallattice breaks the continuious spatial symmetry and imposes discrete spatial symmetry,that is LRO (crystaline order), while the incommensurability between the number oflattice sites and particles allows for a superfluid flow giving rise to QODLRO. Thisresults in numerous peaks at multiples of 2 k l with a structure that is dependent onthe lattice depth and particle-to-site ratio F . Contrarily, in the hard-core BHM forincommensurate lattice-sites and particle numbers, the momentum distribution showsonly a single peak at k = 0 associated with superfluidity. This is due to the spatialsymmetry not being broken within the model. We note that a similar multi-peakstructure for incommensurate systems is also obtained when a super-lattice is imposedon the BHM, as this also breaks the spatial symmetry and introduces oscillations in thecorrelation function [56]. 4. Probing phases with a driven lattice Sudden perturbations are a powerful method for studying many-body effects in quantumsystems, and for TG gases notable examples that have been investigated in recentyears include the orthogonality catastrophe and decay [57, 58, 59, 60, 61, 62], persistentcurrents and probing superfluidity [33, 34, 35, 36, 37, 38, 41, 42]. Such exploration of thenon-equilibrium dynamics can be used to further characterise the different phases wehave examined in the static model, and quenches can be used to detect phase transitionsdue to the creation of non-trivial dynamics induced by large quantum fluctuations atcriticality [13, 14, 15, 16, 17].Here we quench by assuming that at t = 0 our system is in an eigenstate at a given tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice v , so that for t > V ext ( x n − vt ) = V cos ( k l ( x n − vt )). The time-dependentmany-body wavefunction after the quench can still be calculated via the Bose-Fermimapping theorem, so that we only need to evolve the initial non-rotating single particlestates according to the time-dependent Schr¨odinger equation i (cid:126) ∂ t ψ n ( x, t ) = (cid:18) − (cid:126) m ∂ ∂x + V ext ( x − vt ) (cid:19) ψ n ( x, t ) . (13)The time-dependent solutions can be found exactly in terms of the eigenfunctions ψ n ( x, 0) of the initial Hamiltonian and the eigenfunctions ψ j,q ( x ) and eigenvalues E j,q of the Hamiltonian in the co-rotating frame as [36] ψ n ( x, t ) = e iqx e − i (cid:126) q m t (cid:88) j c jn e − iE j,q t/ (cid:126) ψ j,q ( x − vt ) , (14)where q = mv (cid:126) . The eigenfunctions in the co-rotating frame are a solution to theSchr¨odinger equation E j,q ψ j,q ( x ) = (cid:18) − (cid:126) m ∂ ∂x + V ext ( x ) (cid:19) ψ j,q ( x ) , (15)obeying the twisted boundary conditions ψ j,q ( x + L ) = e − iqL ψ j,q ( x ). The coefficients c jn then contain all the information about the initial state of the system and are defined as c jn = (cid:104) ψ j,q ( x ) | e − iqx | ψ n ( x, (cid:105) = (cid:90) dx ψ ∗ j,q ( x ) e − iqx ψ n ( x, . (16)The applied rotation will lead to transport of particles with respect to the labframe and the average flow of particles can be quantified by calculating the averagemomentum K ( t ) = (cid:82) n ( k, t ) k dk . While calculating the full momentum distribution isa numerically demanding process for large particle numbers, the average momentum canbe calculated efficiently as it is related to the probability current density of the systemas K ( t ) = L (cid:82) j ( x, t ) dx . For a TG gas, the probability current density is identical tothat of the free Fermi-gas and is simply given as the sum of the single-particle currentscorresponding to the occupied energy levels j ( x, t ) = (cid:126) m Im N − (cid:88) n =0 ψ ∗ n ( x, t ) ∂ x ψ n ( x, t ) . (17)For the time-dependent wavefunction given in Eq. (14), the explicit form of the averagemomentum can therefore be calculated as K ( t ) = K + K t ( t )= N (cid:126) qmL + (cid:126) mL N − (cid:88) n =0 (cid:88) j | c jn | Im [ F jj ]+ Im (cid:34) (cid:126) mL N − (cid:88) n =0 (cid:88) j (cid:54) = k c jn c ∗ kn e − i ( E k,q − E j,q ) t/ (cid:126) F jk ( t ) (cid:35) , (18) tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice qL/⇡ 12 32 52 72 92 (a) qL/⇡ 12 32 52 72 92 (b) h K i m L ⇡ N ~ h K i m L ⇡ N ~ Figure 6. (cid:104) K (cid:105) /N as a function of q for different filling ratios and depths. In (a) theblack line corresponds to F = and V = 0 . E r , while the red line corresponds to F = and V = 20 E r . In (b) the black line corresponds to F = and V = 0 . E r ,while the red line corresponds to F = 1 and V = 0 . E r . The dashed black lineindicates (cid:104) K (cid:105) /N = 0. where F jk ( t ) = (cid:90) dxψ ∗ j,q ( x − vt ) ∂ x ψ k,q ( x − vt ) . (19)The elements F jj are independent of time in all cases we consider, as the twoeigenfunctions are the same, which means that the time-dependence just shifts theintegrand. For j (cid:54) = k , however, the two eigenfunctions are different and the time-dependence of the integrand is therefore more complicated and does affect the valueof the integral. The average momentum therefore has a time-independent part K consisting of the first two terms in Eq.(18) and a time-dependent part K t ( t ) whichconsists of the last term.The effect of the underlying many-body phase of the TG gas on the averagetransport in the system should be reflected in the average momentum and Eq. (18)allows us to explain the observed behavior in some mathematical detail. Furthermore,the same behaviour appears for a spin-polarised Fermi gas, even though its momentumdistribution is entirely different. This is because the expectation value of the momentumoperator is the same for the Fermi and TG gas, which means that the average momentumalone is insufficient for fully characterising the dynamics of the different many-bodybosonic phases. We will therefore investigate and compare the average momentum, thecoherence and the momentum distribution. It was shown by Schenke et al. [36] that persistent currents can be created in a TG gasby driving it with a rotating delta-barrier. If the driving momentum is q = ηπ/L , where tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice qL/⇡ qL/⇡ E / E r E / E r (a) (b) Figure 7. (a) Single-particle energy spectrum for PBC (odd number of particles) and(b) A-PBC (even number of particles), plotted as a function of the driving momentum q for M = 4 and V = 0 . E r . η is an integer, this yields maxima in the time-averaged average momentum per particle (cid:104) K (cid:105) /N , while outside these values it stays essentially zero. A similar resonant behaviouris also present in our system when rotating the lattice potential at small lattice depthsand for incommensurate filling (see black curve in Fig. 6), however, the momentum nolonger vanishes away from the resonance values. In fact it oscillates between positiveand negative momentum with a small amplitude. In order to simplify the discussionin this section we define the scaled driving momentum Ω = qL/π . For F = 1 thevalues of (cid:104) K (cid:105) /N around Ω = 2 η + 1 become large, while the same behavior can be seenfor F = around Ω = 2 η . This can be understood by looking at the single-particlespectrum for odd particle numbers (with PBC) and for even particle numbers (withA-PBC) as a function of q , which is plotted in Fig. 7 for M = 4 and V = 0 . E r . Notethat these numbers are chosen for visual simplicity, and the structure is exactly thesame for M = 50 and V = 0 . 02 (with a slightly smaller gap). The structure of the twospectra reflects the difference in the average momentum, that is the spectrum for oddparticle numbers close to Ω = 2 η + 1 looks similar to the one for even particle numbersclose to Ω = 2 η . The reason for this difference is that the ground-state is non-degeneratein the initial state for an odd number of particles, while it is two-fold degenerate in theinitial state for an even number of particles. A lattice with an even number of sitestherefore forces a splitting of the M, M + 1 degenerate states for odd particle numbers,but there is no splitting for even particle numbers. An odd number of sites switchesaround the behavior for odd and even particle numbers, but the explanation is analogous.For deeper lattices we see that (cid:104) K (cid:105) /N grows linearly with q even for incommensurateparticle densities. A detailed explanation of the behavior in the different regimes shownin Fig. 6 can be found in Appendix A. tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice 45 50 5500.5 (a) F V /E r N F V /E r (b) h i 45 50 55510 N h i h K i m L ~ N q h K i m L ~ N q Figure 8. (a) Time-averaged average momentum and (b) time-averaged coherence asa function of the lattice depth V and the filling ratio F . The insets shows the samenear F = 1 for a lattice depth of V = 0 . E r . These plots are based on calculationsfor M = 50 and with the time-average taken over the interval 0 to 10 t . As we are interested in using the driven lattice to probe the phases of the system,we restrict our simulations to small values of the rotation speed, namely q = 2 . π/L ,which is off resonant so that the rotation does not excite the particles into higher bands.This therefore allows for a clear distinction between the dynamics of the superfluid andinsulating phases in the system. In order to understand the nature of the bosonic phases,we investigate the coherence and the momentum distribution in addition to the averagemomentum. In Fig. 8 we show the time-averaged averagemomentum (cid:104) K (cid:105) /N and the time averaged coherence (cid:104) λ (cid:105) as a function of the fillingratio, F , and the lattice depth, V . For (cid:104) λ (cid:105) one can see qualitatively the same behavioras for the static coherence shown in Fig. 2(a), although for (cid:104) λ (cid:105) after the quench anodd-even effect is present near commensurability in shallow lattices (see the inset inFig. 8(b)). The same odd-even effect is present for (cid:104) K (cid:105) /N , as discussed in section 4.1,and for which a detailed explanation of the origin can be found in Appendix A. Theminimum in the coherence obtained for F = 1 indicates the presence of the pinnedphase and the maximum in (cid:104) K (cid:105) /N at F = 1 (and F = 2) can therefore be attributed tothis state, which restricts tunneling between the sites resulting in the gas being movedalong with the lattice. For incommensurate values of F , the relatively large degreeof coherence indicates the presence of a superfluid phase (as does the small value of (cid:104) K (cid:105) /N ), where particles can freely tunnel through the lattice as it rotates through thegas. For deep lattices ( V > E r ) we find (cid:104) K (cid:105) /N ≈ qL/π for both commensurate tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice -5 0 500.0050.010.0150.02 -5 0 500.020.040.060.080.1 (a) h n ( k ) i / N k/k l (b) k/k l h n ( k ) i / N Figure 9. Time-averaged momentum distribution (black line) after a rotationalquench for F = (a) and F = (b). The dashed black lines indicate the asymmetriesbetween positive and negative momentum occupations. and incommensurate fillings and this quantity thus no longer distinguishes between thecommensurate insulating and the incommensurate supersolid-like phases. The time-averaged coherence, however, clearly distinguishes between these phases which impliesthat the time-averaged full momentum distribution must contain information about theirdynamical properties as well. For F = 1 the pinned phase Gaussian-like momentumdistribution is slightly asymmetrical, leading to the overall transport observed, while themomentum distribution in the frictionless superfluid phase (for V (cid:28) E r and F (cid:54) = N ) hasa single, symmetrical superfluid peak centered at k = 0, even when rotation is applied.As this is difficult to visually distinguish we do not plot the time-averaged momentumdistributions in these cases. However, the time-averaged momentum distributions aftera quench for F = and F = are shown in Fig. 9. For F = , correspondingto a macroscopic number of defects, the transport is obtained due to an asymmetryin the population of the back-scattering momentum peaks, with the peaks at positivemomenta having a higher probability than the peaks at negative momenta. For F = a combination of an asymmetrical Gaussian shape (for the first 50 pinned particles) andthe asymmetrical back-scattering momentum peaks (for the remaining 20) is responsiblefor the transport. In the case of a microscopic number of defects the bulk of the transportis due to the pinned phase, but a small part is contributed from the asymmetry betweenthe population of the back-scattering momentum peaks, similar to what is seen for amacroscopic number of defects in Fig. 9. So although (cid:104) K (cid:105) /N is the same for the pinnedphase and the supersolid-like phase at different filling ratios in the deep lattice, the time-averaged momentum distributions and the type of transport is very different, reflectingthe different natures of the two many-body phases. tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice 45 50 550510 (a) F V /E r N F V /E r (b) 45 50 5500.10.2 N K m L ~ q K m L ~ q Figure 10. Standard deviation of (a) the average momentum and (b) the time-averaged coherence as a function of the lattice depth and the filling ratio. The insetsshows the same near F = 1 for a lattice depth of V = 0 . E r . These plots are based oncalculations for M = 50 and with the time-average taken over the interval 0 to 10 t . The time-averaged variables only give a partialpicture of the dynamics and a fuller understanding can be gained by considering theinstantaneous properties of the average momentum and coherence. For this we plotthe standard deviation of the average momentum and coherence in Fig. 10 (based onthe same calculation as the time averages reported in Fig. 8). These can be seen tobe essentially zero in the superfluid phase for shallow lattices (incommensurate particlenumbers), while they become large as F approaches integer values. For deeper latticesthe standard deviation of the coherence stays zero, while the standard deviation ofthe average momentum becomes larger. To understand the physics under-lying thesefluctuations in more detail, we will next consider the instantaneous properties of thecoherence and average momentum at some representative values of the depth and fillingratio.The dynamics of the coherence, λ ( t ) /N , and of the average momentum, K ( t ),after the quench are shown in the shallow lattice limit in Fig. 11. In the commensuratecase (red line) the gas exhibits collective many-body oscillations between the coherentsuperfluid and a somewhat less coherent insulating phase with a periodicity ∼ t = L/ ( M v ). The same behaviour appears for F = 2 (black line), but the frequency ofthe oscillation has roughly doubled. The average momentum in the insulating phaseis much higher than the superfluid phase, which means that the transport is verysimilar to classical stick-slip motion described by the Frenkel-Kontorova model [20].The periodicity of these oscillations is derived in Appendix A and shown to be relatedto the discrete time-symmetry of the Hamiltonian for the average momentum. This typeof collective oscillation is present for both the coherence (a many-body measure) andthe average momentum (single-particle measure), whereby the momentum dynamics tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice (a) t/t / N (b) t/t K m L ~ N q Figure 11. (a) Coherence λ /N and (b) average momentum per particle as a functionof time after a quench. The red lines correspond to V = 0 . E r and F = 1, whilethe blue lines correspond to V = 0 . E r and F = . The black lines correspond to V = 0 . E r and F = 2. is clearly a reflection of the oscillation between many-body phases as implied by thecoherence. For F = , where a macroscopic number of defects exist, time-fluctuationsin any of the parameters are essentially absent and the gas is always in the coherentsuperfluid phase which corresponds to the average momentum being small and slightlynegative, due to our choice of q (see section 4.1 and Appendix A) at all times. Theexistence of the superfluid phase thus results in essentially frictionless dynamics, inwhich particles simply tunnel through the shallow barriers without responding to them.In shallow lattices, for both commensurate fillings ( F = 1) and microscopic numbers ofdefects, there exist regular oscillations of the coherence and average momentum. As thedepth is increased, but still within the region of non-zero fluctuations of the coherence(see Fig. 10(b)), the stick-slip motion becomes less regular. The collective many-bodyfluctuations around F = 1 in shallow lattices is suggestive as this region corresponds tothe critical region for the commensurate-incommensurate pinning transition. The largefluctuations observed in the coherence, which is an order parameter that distuingushesthe pinned and superfluid phases, is therefore a manifestation of the underlyingincommensurate-commensurate transition. This aligns our results with other indicationsthat critical points and regions introduce large fluctuations in the dynamics of order-parameters [2, 63, 64, 65, 8, 15, 16].For deeper lattices the fluctuations in the coherence are small (see Fig. 10(b)) andthe instantaneous coherence is therefore not expected to contain any information thatcannot be understood from the average coherence, although we plot it for completenessin Fig. 12(a). Indeed, one can see that there are only small fluctuations and that thecoherence is close to 1 in the commensurate pinned phase, while the gas is slightly morecoherent in the presence of a microscopic number of defects ( F = ) and approaches tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice (a) / N t/t (b) t/t (c)(d) K m L ~ N q K m L ~ N q K m L ~ N q Figure 12. (a) Coherence λ /N and (b)-(d) average momentum per particle as afunction of time for V = 20 E r . The red lines correspond to F = 1, while the bluelines correspond to F = and the black lines correspond to F = . a √ N degree of coherence for a macroscopic number of defects ( F = ). This directlyreflects the results shown in Fig. 8(b). For the average momentum, however, Fig. 12(b)shows that large fluctuations are still present in deep lattices. As these fluctuations areonly present for the average momentum they do not correspond to oscillations betweenmany-body phases as was the case for the shallow lattice oscillations. In this deeplattice limit particles are transported along with the lattice on average with (cid:104) K (cid:105) = qL/π and the short-time oscillations with period ∼ π (cid:126) E M +1 − E M correspond to particles beingexcited to higher bands at each lattice site, creating on-site dynamics due to the finitewidth of the continuum lattice (see Appendix A). These types of fluctuations areclearly distinct from the fluctuations between many-body phases observed in shallowlattices. Additionally they are present for both commensurate and incommensurateparticle densities. The instantaneous average momentum is therefore not a useful wayto distinguish between the pinned phase and the supersolid-like phase in the deeplattice limit. For that information an investigation of the full momentum distributionis required. As can be understood from the coherence, however, the time fluctuationsof the full momentum distribution are not particularly interesting: the time-averagedmomentum distribution, which we discussed in section 4.2.1, already contains all therelevant information. 5. Conclusion Utilizing the Bose-Fermi mapping theorem and studying the reduced single-particledensity matrix we have investigated the phases of the Tonks-Girardeau gas ina continuum optical lattice model for commensurate and incommensurate particledensities. For shallow lattices we reproduce the well-known results, namely the existence tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice k = 0 for incommensurate particle numbers. For deeper lattices, we find thata supersolid-like phase emerges for incommensurate particle numbers. This defect-induced superfluid phase is well-characterized by the momentum distribution, whichshows a multi-peaked structure reflecting spatial modulations in the auto-correlation inthe presence of an optical lattice due to the breaking of the continuous spatial symmetry.The momentum distribution depends sensitively on the filling ratio and it is usefulto consider two distinct, but continuously connected regimes; a microscopic numberof defects where we have shown that the momentum distribution corresponds to alarge number of pinned particles, with a few superfluid particles (or holes) on topand a macroscopic number of defects, where the multi-peak structure is dominant.Additionally we have shown that dynamically rotating the lattice allows one to probethe structure of the phases. The average momentum, which measures the degreeof transport in the system, clearly reflects the shallow lattice phase diagram withmaximum transport obtained for commensurate particle densities due to the pinnedphase. Additionally a collective many-body stick-slip-like behavior is observed forparticle numbers close to commensurability where the gas oscillates between the pinnedand superfluid phases with a period T ≈ t . On the other hand, in deeper lattices,despite the supersolid-like phase retaining some coherence, the average particle transportis very similar to the pinned phase, as the particles are transported along with the latticeon average. The average momentum has short time-scale oscillations in this regimecorresponding to on-site dynamics induced by on-site particle excitations in the lattice.These oscillations are therefore very different from the many-body phase oscillationobserved for shallow lattices. Investigating the full momentum distribution, however,shows that the underlying physics of the transport is very different for commensurate andincommensurate particle densities, the former is the result of an asymmetric Gaussian-like shape, while the latter is the result of an asymmetry between the occupation of thepositive and negative back-scattering momentum side-peaks.Our approach is experimentally realizable as ring lattice potentials have beencreated using spatial light modulators [66, 67], and alternative possible setups includeoptical nanofibers [68] and rapidly moving lasers that can “paint” arbitrary optical trapshapes [69]. Another possibility is using an optical lattice in a long one-dimensional boxtrap. Theoretically, the calculation of such dynamics for a TG-gas on a lattice confinedby a square well is feasible and an obvious future extension of the work presented inthis paper. Another natural extension is to consider finite interactions, rather than theTG limit. This would allow for an investigation of the interplay between interactionsand commensurability, similar to the results presented in [29], but in the presence ofa moving lattice. Solving the dynamics of such systems is feasible for small particlenumbers, for example by utilizing the exact diagonalization scheme outlined in [70]. tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice Acknowledgments The authors thank Rashi Sachdeva for frictionless discussions. This work was supportedby the Okinawa Institute of Science and Technology Graduate University. TFacknowledges support under JSPS KAKENHI-18K13507. Appendix A. Detailed explanation of the average momentum behaviorafter a quench In this appendix we present a detailed analysis of Eq. (18) in the different regimes ofinterest, which will allow for a mathematical explanation and clarification of the observedbehavior of the average momentum. This in turn helps clarify the physical propertiesand processes responsible for this behaviour. The appendix is structured as follows: Thefirst section explains the incommensurate behavior in the shallow lattice as a function ofthe driving momentum q , by considering the free space wavefunctions analytically. Thesecond section contains a numeric investigation of the important quantities in Eq. (18)in the presence of a lattice potential. Appendix A.1. Free space analysis The incommensurate gas in shallow lattices can be qualitatively understood byevaluating the expression for the average momentum given in Eq. (18) for plane-waves,assuming that the lattice only leads to a small perturbation. In free space and withPBC (the analysis for A-PBC for even particle numbers is similar) we can thereforeassume the wavefunctions to be given by ψ l ( x, 0) = 1 √ L e i lπx/L , ψ j, Ω ( x ) = 1 √ L e i (Ω+2 j ) πx/L , (A.1)where l, j = { , ± , ± , ± . . . } . The energies in the rotating frame are therefore givenby E j = (cid:126) π mL (Ω + 2 j ) . The points Ω = κ (where κ is an integer) are special as therotational eigenstates ψ j,κ ( x ) and ψ j (cid:48) ,κ ( x ) are degenerate for j (cid:48) = − j − κ . For κ = 2 η wehave j (cid:48) = j with j = − η , therefore the lowest energy state will be unpaired, see Fig. 7(a).This means that almost all eigensfunctions of the system can be expressed alternativelyas entirely real or imaginary linear combinations of the j and j (cid:48) eigenstates of the form φ j + ,κ ( x ) = (cid:113) L cos( π ( κ + 2 j ) x/L ) and φ j − ,κ ( x ) = (cid:113) L i sin( π ( κ + 2 j ) x/L ). This thenleads to Im( F jj ( t )) = 0, which means that K ( t ) = N (cid:126) qmL + K t ( t ). The time-dependent-part of the average momentum creates oscillations around the time-independent part,which matches exactly the average visible in Fig. 6.For values Ω (cid:54) = κ we find c jl = sin( π [ − Ω − j + l ])) π ( − Ω − j + l ) , (A.2) F jk ( t ) = i [Ω + 2 k ] π/Le i π ( j − k ) vt/L δ jk , (A.3) tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice κ/ 2, we find | c jl | = | c j (cid:48) l | when j (cid:48) = 2 l − n − j , which canbe used to evaluate the average momentum analytically as K ( t ) = N (cid:126) qmL + (cid:126) mL ( N − / (cid:88) l = − ( N − / (cid:88) j | c jl | [ q + 2 πj/L ] (A.4)= 2 N (cid:126) qmL + (cid:126) mL N − / (cid:88) l = − ( N − / (cid:88) j | c jl | π [2 l − n ] (A.5)= 2 N (cid:126) qmL + − N πn (cid:126) mL = 2 N (cid:126) qmL − N (cid:126) qmL = 0 . (A.6)This corresponds to the average value we obtain for the full numerical calculationsshown in Fig. 6. For κ < Ω < κ/ (cid:12)(cid:12)(cid:12)(cid:80) ( N − / l = − ( N − / (cid:80) j | c jn | Im[ F jj ] (cid:12)(cid:12)(cid:12) < N q , whilefor κ/ < Ω < κ we find (cid:12)(cid:12)(cid:12)(cid:80) ( N − / l = − ( N − / (cid:80) j | c jn | Im[ F jj ] (cid:12)(cid:12)(cid:12) > N q . This gives rise to theregions of negative average momentum visible in in Fig. 6, in which particles rotate inthe direction opposite to the lattice. As Ω = κ is approached the average momentumtends towards zero, before the discontinuous jump to (cid:104) K ( t ) (cid:105) = N (cid:126) qmL at the point wherethe energy spectrum becomes degenerate. Appendix A.2. Numeric analysis for the lattice To explain the behavior close to commensurability and for deeper lattices, we use thenumeric wavefunctions obtained by finite difference diagonalization. The introductionof the energy gap changes the relative amplitude between the real and imaginary partsof ψ M − d, Ω ( x ) ( d = 0 , ... ), with an opposite shift in ψ M +1+ d, Ω ( x ) close to Ω = 2 η for oddparticle numbers. Similar shifts are observed close to Ω = 2 η + 1, but they are negligiblefor very small gaps ( V = 0 . E r ), although the difference becomes less pronouncedas the gap size is increased. As expected, the opposite behavior with respect to Ω isobserved for the systems with even particle numbers.In the free space analysis F jk ( t ) = 0 for j (cid:54) = k , which is still approximately trueonce the lattice is introduced, but as a consequence of the change in the wave-functionsthe terms F ( M − d )( M +1+ d ) ( t ) = F ( M +1+ d )( M − d ) ( t ) (cid:54) = 0 contribute significantly to the time-oscillating part of the average momentum. F ( M − d )( M +1+ d ) ( t ) generally oscillates with aperiodicity related to t = L/ ( M v ) as the discrete time symmetry of the system meansthat the single-particle wave functions obey the relation ψ j, Ω ( x − vt ) = ψ j, Ω ( x − v ( t + t )).The size of the d > d = 0 contribution forshallow lattices, but as the lattice depth increases so does the size of F ( M − d )( M +1+ d ) . Appendix A.2.1. Commensurable particle numbers in the shallow lattice For verysmall gaps ( V = 0 . E r ) only the d = 0 contributions matter and we will considerthis case in some detail in order to explain the behavior close to commensurability. tatic and dynamic phases of a Tonks-Girardeau gas in an optical lattice c Mn c ∗ ( M +1) n (cid:54) = 0, that is when the overlapbetween the initial state and the M and M + 1 rotating states are comparativelylarge. For A-PBC (even particle numbers) a significant overlap with the M and M + 1 rotating states is only obtained for n = { M − , M, M + 1 , M + 2 } states, with c M ( M − c ∗ ( M +1)( M − ≈ − c M ( M +2) c ∗ ( M +1)( M +2) and c MM c ∗ ( M +1) M ≈ − c M ( M +1) c ∗ ( M +1)( M +1) .There will therefore be a significant contribution to the time-dependent oscillation for N = M . For N = M + 2 and higher particle numbers the time-dependent oscillationbecomes small again as the contributions from the n = { M − , M } terms are canceledout by the contributions from the n = { M + 1 , M + 2 } terms. For PBC (odd particlenumbers) only the states n = { M − , M − , M + 2 , M + 3 } have significant overlap withthe the M and M + 1 rotating states, with c M ( M − c ∗ ( M +1)( M − ≈ − c M ( M +2) c ∗ ( M +1)( M +2) and c M ( M − c ∗ ( M +1)( M − ≈ − c M ( M +3) c ∗ ( M +1)( M +3) . This means that there will be asignificant contribution for N = M − N = M + 1. For N = M + 3 andhigher particle numbers the time-dependent oscillation becomes small again as thecontributions from n = { M − , M − } and n = { M +2 , M +3 } cancel out. The analysisfor states close to 2 M is exactly the same, that is only F M (2 M +1) ( t ) = F (2 M +1)2 M ( t ) (cid:54) = 0are important and they contribute in the same pattern. However, in this case a deeperlattice ( V ≈ . E r ) is required for significant gaps to be introduced. The large finitevalues of the average momentum for commensurate particle numbers visible in Fig. 6are therefore entirely due to a time-dependent oscillation induced when quenching tothe rotating lattice system. The time-dependence of F ( M )( M +1) ( t ) is periodic with t , while F (2 M )(2 M +1) ( t ) is periodic with t . The latter periodicity is because therelevant SP eigenfunctions from which F (2 M )(2 M +1) ( t ) is obtained are also periodic with ψ j ≥ M, Ω ( x − vt ) ≈ ψ j ≥ M, Ω ( x − v ( t + t )). In these cases the periodicity associated withthe gap energy, T = π (cid:126) E M +1 − E M , π (cid:126) E M +1 − E M is very large, as the gap is very small (seeFig. 7). This long time scale oscillation is therefore obscured by the more importantshort time scale oscillation, see Fig. 11.For slightly deeper lattices (0 . E r < V < . E r ), the d > η + 1 and Ω = 2 η also becomes smalleras the gap between the states M − d and M + 1 + d increases and the energy stateswithin each band move closer to each other. The odd-even effect therefore eventuallydisappears. Appendix A.2.2. Intermediate and deep lattices For a lattice of intermediate depth1 . E r < V < E r the finite momentum observed close to commensurability in Fig. 8stems mostly from the time-independent part of the average momentum. 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