Collective modes and superfluidity of a two-dimensional ultracold Bose gas
CCollective modes and superfluidity of a two-dimensional ultracold Bose gas
Vijay Pal Singh and Ludwig Mathey
1, 2 Zentrum f¨ur Optische Quantentechnologien and Institut f¨ur Laserphysik, Universit¨at Hamburg, 22761 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany (Dated: October 2, 2020)The collective modes of a quantum liquid shape and impact its properties profoundly, includ-ing its emergent phenomena such as superfluidity. Here we present how a two-dimensional Bosegas responds to a moving lattice potential. In particular we discuss how the induced heating ratedepends on the interaction strength and the temperature. This study is motivated by the recentmeasurements of Sobirey et al. arXiv:2005.07607 (2020), for which we provide a quantitative un-derstanding. Going beyond the existing measurements, we demonstrate that this probing methodallows to identify first and second sound in quantum liquids. We show that the two sound modesundergo hybridization as a function of interaction strength, which we propose to detect experimen-tally. This gives a novel insight into the two regimes of Bose gases, defined via the hierarchy ofsounds modes.
INTRODUCTION
The emergent phenomena of quantum liquids such assuperfluidity and sound modes depend on a multitude ofsystem features, such as interaction strength, dimension-ality and temperature. These two classes of phenomenaare linked in an intricate manner, as exemplified by theLandau criterion, which predicts dissipationless flow fora perturbation moving with a velocity below a criticalvelocity. This critical velocity in turn is associated withthe creation of elementary excitations [1–5]. While studyof superfluids was first motivated by the properties ofhelium-II, ultracold atoms have expanded the scope ofthe study of superfluidity by a wide range of trappablequantum liquids including superfluids having reduced di-mensionality and tunable interactions. Measurements ofthe critical velocity have been performed by perturbingultracold atom clouds with a moving laser beam [6–8] orlattice potential [9, 10].Superfluidity in two-dimensional (2D) systems is a par-ticularly intriguing case, due to their critical proper-ties, which differ from higher dimensional systems. Al-though 2D systems have no long-range order due toincreased thermal fluctuations, they can become a su-perfluid via the Berezinksii-Kosterlitz-Thouless (BKT)mechanism [11–13]. The superfluid phase is a quasi-condensate characterized by an algebraically decayingphase coherence. Ultracold atom systems provide un-precedented control and tunablity, allowing the study ofsuperfluidity in 2D. This led to the observation of paircondensation [14], phase coherence [15–17], critical veloc-ity [7, 18], and sound propagation [19, 20]. Two soundmodes were recently detected in Ref. [21]. However, theirhybridization and sublinear dissipation for small pertur-bation velocities, which we identify in this paper, havenot been detected yet.In this article, we use both classical-field dynamics andanalytical estimates to investigate the induced heatingrate as a function of the velocity of a moving lattice po- −
50 0 50 V x ( µ m) k (cid:15) ( k ) k v B R ( v ) v (mm / s) A v B NBB v B = v c C FIG. 1.
Superfluid response. ( A ) A 2D superfluid in a boxpotential is probed by moving a lattice potential with latticevector k through it, at a velocity v . ( B ) The resulting heatingof the superfluid derives from the excitation branches at k .For the weak-coupling regime, for which the velocity of thenon-Bogoliubov (NB) mode is larger than the velocity v B ofthe Bogoliubov (B) mode, the Landau criterion predicts thecritical velocity v c = v B , as indicated. ( C ) Simulated heatingrate R ( v ) for k /k c = 0 .
6, where k c = 2 . µ m − is the wavevector above which the Bogoliubov dispersion approaches aquadratic momentum dependence. The fit (red curve) yieldsa sharp increase at v c = 4 . / s, see text. The two maximaof the heating rate correspond to the two modes, where thefirst maximum is close to v B = 5 . / s. tential. The simulated heating rate shows two maximacorresponding to two sound modes, and sublinear dis-sipation at low velocities. By changing the periodicityof the lattice potential we examine the heating rate atvarying wave vector. We find phononic excitations forlow wave vectors and free-particle excitations for highwave vectors. This is in excellent agreement with Bo-goliubov theory and the measurements of Ref. [18]. Wedetermine a critical velocity from the sharp onset of dissi- a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p pation and compare it with the measurements for variousinteractions [18]. Below the critical velocity, we find thatthe heating rate has a power-law dependence in the ve-locity and its magnitude increases with the temperature,which is supported by the analytic estimate of the qua-sicondensed phase. Finally, we determine the two modevelocities from the heating rate and identify their hy-bridization dependent on interaction and temperature. RESULTSSystem and dynamical response
We simulate bosonic clouds of Li molecules confinedto 2D motion in a box potential. This geometry offerstunability of the effective interaction strength, and wasused in Ref. [18], and depicted in Fig. 1A. The systemis perturbed by a lattice potential moving at a constantvelocity v . The unperturbed system is described by theHamiltonianˆ H = (cid:90) d r (cid:104) ¯ h m ∇ ˆ ψ † ( r ) · ∇ ˆ ψ ( r ) + g ψ † ( r ) ˆ ψ † ( r ) ˆ ψ ( r ) ˆ ψ ( r ) (cid:105) . (1)ˆ ψ ( ˆ ψ † ) is the bosonic annihilation (creation) operator.The 2D interaction parameter is g = ˜ g ¯ h /m , where˜ g = √ πa s /(cid:96) z is the dimensionless interaction, m themolecular mass, a s the 3D molecular scattering length,and (cid:96) z the harmonic oscillator length in the transversedirection [22]. We consider a rectangular system withdimensions L x × L y = 100 × µ m and a density n = 1 . µ m − , comparable to the experimental param-eters [18]. We solve the dynamics using the classical-fieldmethod described in Refs. [23, 24], see also Methods.According to this methodology, we replace the operatorsˆ ψ in Eq. 1 and in the equations of motion by complexnumbers ψ . We sample the initial states from a grand-canonical ensemble with a chemical potential µ and atemperature T via a classical Metropolis algorithm. Toobtain the many-body dynamics of the system, we propa-gate the state using the classical equations of motion. Wemodel the lattice perturbation via the additional term H p = (cid:90) d r V ( r , t ) n ( r ) , (2)where n ( r ) is the density at the location r = ( x, y ). Thelattice potential V ( r , t ) is directed along the x direction: V ( x, t ) = V cos [ k ( x + vt ) / . (3) V is the strength, k = 2 π/λ l the wavevector, and v the velocity, where we define λ l as the distance be-tween two maxima of the potential. We move thispotential for a fixed perturbation time t p of 100 ms.We calculate an ensemble average of the energy change ∆ E = (cid:104) H ( t p ) (cid:105) − (cid:104) H (0) (cid:105) using Eq. 1 and ψ ( r , t ). Fromthis change of energy, we determine the heating rate∆ E/t p for various sets of parameters v , k , V , ˜ g , and T /T . Throughout this paper, we will use the tempera-ture T = 2 πn ¯ h / ( mk B D c ), with the critical phase-spacedensity D c = ln(380 / ˜ g ), as the temperature scale, seeRefs. [25, 26]. This scale gives an estimate of the crit-ical temperature T c , which is the temperature of theBKT transition. We define a dimensionless heating rate R = ¯ h ∆ E/ ( t p V N ), where N is the number of molecules.This heating rate and its velocity, temperature and in-teraction dependence are central quantities of this pa-per. As an example, in Fig. 1C we show R ( v ) for˜ g = 1 and T /T = 0 .
3. We used V /µ = 0 .
05 and k /k c = 0 . µ = gn is the mean-field energy and k c ≡ √ /ξ is the wave vector above which the Bogoli-ubov dispersion approaches a quadratic momentum de-pendence, with ξ = ¯ h/ √ mgn being the healing length.As we depict in Fig. 1C, the heating rate is small atlow velocities. Below, we comment on the velocity de-pendence in this regime in more detail. The heating rateincreases rapidly around a velocity which we refer to asthe critical velocity of the condensate. As a simple esti-mate of this velocity, we fit the heating rate below theBogoliubov velocity v B = (cid:112) gn/m = 5 . / s to thefunction f ( v ) = A max[0 , v − v c ], with A and v c as fit-ting parameters. The fit gives v c = 4 . / s, as depictedin Fig. 1C.In addition to the sudden increase of the heating rate,captured by the critical velocity, the heating rate displaystwo maxima. These two maxima derive from the two ex-citation branches of the system. In the example shownin Fig. 1C, the maximum at lower velocities is close tothe Bogoliubov estimate v B , as shown. We give furtherevidence for this interpretation below. We refer to thisbranch as the Bogoliubov (B) mode. Additionally, thereis a second maximum at a higher velocity, which we referto as the non-Bogoliubov (NB) mode. This is consistentwith an excitation spectrum sketched in Fig. 1B. Char-acterizing these two modes further is the second objectiveof this paper, in addition to the critical velocity. Analytical estimates
We present two analytical estimates for properties ofthe heating rate. The first estimate uses Bogoliubov the-ory, and the second estimate uses the quasicondensateproperties of 2D quantum gases. We derive the heatingrate perturbatively at second order in the probing term.The Bogoliubov estimate of the heating rate is [27] dEdt = 2 π ¯ h (cid:88) k ω k ( u k + v k ) N | V k | δ ( ω k − vk ) . (4)¯ hω k = (cid:112) (cid:15) k ( (cid:15) k + 2 mv ) is the Bogoliubov dispersion, u k and v k are the Bogoliubov parameters, with ( u k + v k ) =
012 0 0.5 1 v / v s k /k c
012 0 0.5 1 v / v s k /k c k /k c k /k c k /k c k /k c A Experiment A Experiment B Simulation B Simulation 00.8 C Bogoliubov C Bogoliubov
FIG. 2.
Heating rate dependence on the lattice vector. ( A ) Measurements of the response r ( v ) of Ref. [18], determinedfrom the heating of the condensate density and normalized individually for each column, as a function of k /k c and v/v s .The sound velocity v s is determined from the propagation of a density wave [27]. ( B ) Simulated heating rate R ( v ) and ( C )Bogoliubov estimate R B ( v ), for the same interaction and the same lattice parameters as in the experiment. The density n = 1 . µ m − , the interaction strength ˜ g = 1 .
6, and the temperature
T /T = 0 . v max /v B = (cid:112) k /k c ) of Eq. 5. The experimental result is primarily due to the Bogoliubov mode, reflectedin the good agreement with the simulation and the analytical estimate. v / v F ln( k F a ) v B v s,T/T =0 . v c,T/T =0 . v c,T/T =0 . v c, Experiment
FIG. 3.
Critical velocity.
Measurements of the critical ve-locity (diamonds) of Ref. [18] are compared to the simulationsof
T /T = 0 . . k F a ). k F is the Fermi wave vector, a isthe 2D scattering length, and v F is the Fermi velocity. Wealso show the propagation velocity of a density wave (circles)for T /T = 0 . v B (continuousline). (cid:15) k / ¯ hω k . (cid:15) k = ¯ h k / (2 m ) is the free-particle dispersion. V k = V δ k y ( δ k x − k + δ k x + k ) / N is the number of condensed atoms. In dimensionlessform R ≡ ¯ h ( dE/dt ) / ( N V ) and after simplifying Eq. 4we obtain R B = π
16 ¯ hk m δ [ (cid:113) v / v − v ] , (5)where v is v = ¯ hk /m . The normalized heating rate R B has a maximum at v max = ( v / v ) / . The location ofthe maximum moves to higher velocities with increasing k or v . We show v max as a red line in Fig. 2, which cap- tures the trend of the measurement and the simulation.We include the thermal damping of the phonon modesby considering a Landau-type damping Γ k = v Γ k , where v Γ is the damping velocity. This results in [27] R B = 116 v v Γ ( (cid:112) v / v − v ) + v . (6)We show this estimate in Fig. 2C.To describe the heating rate at low velocities we relatethe heating rate to the momentum distribution n k , andarrive at the expression [27] dEdt = 2 π ¯ h (cid:88) kq ( E k + q − E q ) n q | V k | δ ( vk + ω q − ω k + q ) , (7)where E k = ¯ hv B | k | is the phononic dispersion at longwavelengths. The momentum density n k follows a power-law dependence in the wave vector k as n k = n πτ r τ/ | k | τ/ − , (8)where r is the short-range cutoff of the order of ξ and τ isthe algebraic scaling exponent. τ increases monotonouslyfrom 0 to 1 as the temperature is increased from 0 to T c .In dimensionless form and for v < v B , we obtain theheating rate of a quasicondensate at low temperatures[27] R qc = π τ v v − v , (9)which scales as R qc ∝ τ v for low v and yields a rapidincrease for v close to v B . It vanishes, as τ approaches 0. α T /T − − − − R v/v B − − − − R v/v B . T . T . T . T FIG. 4.
Low-velocity dependence of the heating rate.
Simulated heating rate R ( v ) on a log-log scale for a weakperturbation and various T /T . The dashed lines are thealgebraic fits with the function f (˜ v ) = α ˜ v , where ˜ v = v/v B is the scaled velocity. The fit parameter α ( T ) is shown in theinset. R v/v B B-mode NB-mode0 . T . T . T . T . T FIG. 5.
Two sound modes. R ( v ) for ˜ g = 1 . k /k c = 0 . T /T . The transition temperature is T c /T =0 .
86, which is determined from a zero critical velocity [27].
Comparison
In this section, we compare the analytical estimateswith the simulation results and the experimental results.In Fig. 2A we show the measurements of Ref. [18]. Themeasurement was performed at ln( k F a ) = − .
8, where k F = √ πn is the Fermi wave vector and a is the 2Dscattering length. The maximum of this measured re-sponse is close to the phonon velocity for small k /k c and shifts to larger velocities with increasing k /k c . Weperform a simulation for the same system parameters,which results in the heating rate shown in Fig. 2B. Theheating rate displays the same overall dependence on thelattice wave vector k . In particular, for vanishing k the maximum of the heating rate approaches the soundvelocity. Furthermore, the simulation recovers the mea-surements for intermediate and high wave vectors. We note that for k /k c > k ap-proaches the maximum momentum set by the system dis-cretization length, so that the simulation becomes quan-titatively unreliable. In Fig. 2C we show the Bogoliubovestimate R B of Eq. 6. We set the value of v Γ /v B = 0 . R B increases with increasing k as in thesimulation in Fig. 2B, and provides a good estimate forboth the measurement and the simulation. We note thatthe simulation also displays a faint second branch abovethe Bogoliubov branch. As we elaborate below, this isdue to the non-Bogoliubov mode. This feature becomesmore pronounced and realistically detectable for highertemperatures and for an interaction strength near thehybridization interaction, as we discuss below.Next, we determine the critical velocity for the samevalue k /k F = 0 . v c for T /T = 0 . . T /T = 0 . k F a ) < ∼ − .
5, except for thedata point at ln( k F a ) = − .
2. For strong interac-tions ln( k F a ) > ∼ − .
5, the measurements are closer tothe simulations of
T /T = 0 .
1. This might suggest thatthe experimental results were obtained at temperaturesthat varied with varying interaction strengths. We alsoshow the simulation results of the sound velocity v s for T /T = 0 .
1, which is determined from the propagationof a density wave [27]. For all interactions, this soundvelocity is slightly below the Bogoliubov velocity v B andabove the results of v c . The difference between v s and v c is higher for strong interaction and high temperature,which is due to the broadening of the heating rate.Finally, we examine the low-velocity behavior of theheating rate at v/v B (cid:28)
1. In Fig. 4 we show the sim-ulated heating rate R ( v ) for k /k c = 0 . T /T . R ( v ) shows a power-law dependence at low veloci-ties, visible as a linear dependence on v on a log-log scale.More specifically, we observe a quadratic dependence on v , as supported by fitting with the function f (˜ v ) = α ˜ v ,where α is the fitting parameter. ˜ v = v/v B is the scaledvelocity. We refer to this power-law dependence as sub-linear dissipation, because the power-law dependence issublinear. We note that above the critical temperature,where the bosons form a thermal cloud, the dissipationis linear in v [28]. The quantity α ( T ) increases with thetemperature, as we show in the inset of Fig. 4. Boththe power-law dependence and α ( T ) are consistent withthe estimate R qc in Eq. 9, where α ( T ) is related to theBKT exponent τ . Since the magnitude of the heatingrate is small, it would be demanding to extract τ exper-imentally. However, in principle this result relates thelow-velocity heating rate to the quasicondensate scalingexponent. v ( mm / s ) v ( mm / s ) ˜ g BNB 1 1.5 2 2.5 3˜ g E BNB NBB 1 1.5 2 2.5 3˜ g F BNB NBB00.2 v A T/T = 0 . B T/T = 0 . C T/T = 0 . v v v s D FIG. 6.
Hybridization of the two sound modes. ( A , B , C ) R ( v ) as a function of ˜ g for T /T = 0 .
1, 0 .
3, and 0 .
6. Thehorizontal dashed line denotes v = 2 mm / s corresponding to k = 1 . µ m − . ( D , E , F ) The two mode velocities v / determinedfrom the heating rate are compared with the propagation velocity of density waves (crosses), and the Bogoliubov estimatesemploying the total density (dashed line) and the numerical superfluid density (continuous line). B denotes the Bogoliubovsound mode and NB the non-Bogoliubov mode. The vertical dashed line indicates the hybridization point, see text. Two sound modes
In this section we characterize the two sound modesthat we observe in the heating rate. In Fig. 5 we showthe simulation results of R ( v ) for ˜ g = 1 . k /k c = 0 . T /T . R ( v ) displays two maxima. The firstmaximum corresponds to the B mode and the secondmaximum to the NB mode. At low temperatures the Bmode has a higher magnitude than the NB mode. Thischanges as the temperature is increased. For high tem-peratures the NB mode has a higher magnitude than theB mode. This shift of the magnitude is consistent withthe shift of spectral weights of the two modes in the dy-namic structure factor [29]. Thus, the heating rate pro-vides direct access to the relative amplitude of two modes[30]. The first maximum disappears and turns into abroad background of the diffusive sound mode above thecritical temperature T c /T = 0 .
86, where the value of T c is identified by the critical velocity going to zero [27].Next, we analyze the interaction dependence by vary-ing ˜ g in the range 0 . − . k = 1 . µ m − . In Figs. 6(A-C) we show R ( v ) as a function of ˜ g for T /T = 0 .
1, 0 .
3, and 0 . R ( v ) displays a hybridization of the two sound modesas a function of ˜ g . This hybridization is more visible atintermediate and higher temperatures, due to the largerweight of the non-Bogoliubov mode, which we observedin Fig. 5. For low temperatures, as in Fig. 6A, this hybridization occurs at a high value of ˜ g , which shifts tolow values of ˜ g for high temperatures in Figs. 6B and6C. Similarly, the magnitude of the velocity differenceat the avoided crossing increases with increasing tem-perature. To study this hybridization we determine thetwo mode velocities by fitting R ( v ) with a single anda bi-Lorentzian function. In Figs. 6(D-F) we presentthe interaction dependence of the two velocities v / ob-tained from the fit. We indicate the hybridization pointat the interaction at which the magnitude of the heatingrate of the high-velocity ( v ) mode exceeds the magni-tude of the heating rate of the low-velocity ( v ) mode bya vertical dashed line in Figs. 6E and 6F. We compare v / with the velocity v s obtained from the propagationof a density wave [27]. We also show the Bogoliubovvelocities v B and v B ,T = (cid:112) gn s ( T ) /m determined usingeither the total density or the numerically obtained su-perfluid density n s ( T ) [27]. The lower velocity v agreeswith v s and v B ,T for all interactions and all tempera-tures. The upper velocity v agrees with v B above thehybridization point only. This suggests that for weakinteraction and low temperature the system is in thenon-hydrodynamic regime, where the lower velocity isdescribed the Bogoliubov (B) approach and the uppermode is the non-Bogoliubov (NB) mode, as we pointedout in Refs. [29, 31]. This scenario changes for strong in-teraction and high temperatures, where the system entersthe hydrodynamic regime and a hydrodynamic two-fluidmodel describes the two sound modes. Here, the uppermode propagates with the total density and the lowermode with the superfluid density [32, 33]. DISCUSSION
We have determined and discussed the heating rate ofa superfluid 2D Bose cloud by perturbing it with a mov-ing lattice potential, using classical-field simulations andanalytical estimates. This study is primarily motivatedby the experiment reported in Ref. [18]. Indeed, we find,firstly, that the signature of the Bogoliubov mode in theheating rate is well-reproduced in our study, for whichwe present a numerical result as well as an analytical es-timate. Secondly, we show that the critical velocity thatemerges in this type of stirring experiment is consistentwith the experimental findings of Ref. [18].However, our study also suggests to broaden the scopeof this type of stirring experiment. The results that wereport here, elucidate the general behavior of sound exci-tations in 2D Bose gases, in particular they give access tothe two-mode structure of the excitation spectrum, andtheir interaction and temperature dependence. We showthat the heating rate has two maxima, as a function ofits velocity and for fixed lattice wave length, correspond-ing to the two sound modes of the fluid. The relativeweight of these modes changes significantly as a func-tion of temperature, and reverts its hierarchy. Further-more, we show that the two modes undergo hybridizationas a function of the interaction strengths. At interac-tion strengths below the hybridization strength, the non-Bogoliubov mode has a higher velocity than the Bogoli-ubov mode, whereas for interaction strengths above thehybridization strengths the hierarchy is reversed, whichis consistent with the scenario described in [29]. Thisprovides a new insight into the collective modes of Bosegases, which we put forth here, and which we propose tobe tested experimentally.
METHODS
To perform numerical simulations we discretize spaceusing a lattice of size N x × N y = 200 ×
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[30] Daniel Kai Hoffmann, Vijay Pal Singh, Thomas Paintner,Manuel Jger, Wolfgang Limmer, Ludwig Mathey, andJohannes Hecker Denschlag, “Second sound in the BEC-BCS crossover,” (2020), arXiv:2003.06847.[31] Ilias M. H. Seifie, Vijay Pal Singh, and L. Mathey,“Squeezed-field path-integral description of second soundin Bose-Einstein condensates,” Phys. Rev. A , 013602(2019).[32] C. J. Pethick and H. Smith, BoseEinstein Condensationin Dilute Gases , 2nd ed. (Cambridge University Press,2008).[33] Lev Pitaevskii and Sandro Stringari,
Bose-Einstein con-densation and superfluidity , International series of mono-graphs on physics (Oxford University Press, Oxford,2016).[34] Christophe Mora and Yvan Castin, “Extension of Bo-goliubov theory to quasicondensates,” Phys. Rev. A ,053615 (2003).[35] Krzysztof Gawryluk and Miros(cid:32)law Brewczyk, “Signa-tures of a universal jump in the superfluid density of atwo-dimensional Bose gas with a finite number of parti-cles,” Phys. Rev. A , 033615 (2019). ACKNOWLEDGEMENTS
We thank Lennart Sobirey, Thomas Lompe, and Hen-ning Moritz for insightful discussions and providing uswith the experimental data. This work was supported bythe DFG in the framework of SFB 925 and the excellenceclusters ‘The Hamburg Centre for Ultrafast Imaging-EXC 1074 - project ID 194651731 and ‘Advanced Imag-ing of Matter - EXC 2056 - project ID 390715994.
SUPPLEMENTARY MATERIALS
Supplementary TextFig. S1-S4References (17, 24, 28-29, 35)
Supplemental Materials: Collective modes and superfluidity of a two-dimensionalultracold Bose gas
ANALYTIC HEATING RATE
We determine the heating rate peturbatively by considering a weak perturbation term. To second order, the heatingrate is given by [28] d (cid:104) ˆ H ( t ) (cid:105) dt = − h (cid:90) t dt (cid:104) (cid:2) ˆ H s,I ( t ) , [ ˆ H s,I ( t ) , ˆ H ] (cid:3) (cid:105) . (S1)ˆ H is the unperturbed Hamiltonian. ˆ H s,I is the perturbation term in the interaction picture. In momentum spacethe perturbation term is described as ˆ H s = (cid:88) k V k ( t )ˆ n k , (S2)where V k ( t ) is the Fourier transform of the potential V ( r , t ) and ˆ n k = (cid:80) q a † q a k + q is the Fourier transform of thedensity operator ˆ n ( r ). a k ( a † k ) is the bosonic annihilation (creation) operator. V ( r , t ) is the lattice potential directedalong the x direction: V ( x, t ) = V cos [ k ( x + vt ) / V is the strength, v the velocity, and k the wave vector.We Fourier transform V ( x, t ) and obtain V k ( t ) = V δ k y ( δ k x − k + δ k x + k ) exp( − ik x vt ) . (S3)We transform Eq. S2 to the interaction picture via ˆ H s,I ( t ) = exp( i ˆ H t ) ˆ H s ( t ) exp( − i ˆ H t ). Bogoliubov heating rate
Here we derive the heating rate for a condensate at zero temperature. We use the Bogoliubov approximation andobtain the diagonalized Hamiltonian H = (cid:88) k ¯ hω k ˆ b † k ˆ b k , (S4)where ˆ b k and ˆ b † k are the Bogoliubov operators, and ¯ hω k = (cid:112) (cid:15) k ( (cid:15) k + 2 mv ) is the Bogoliubov dispersion. (cid:15) k =¯ h k / (2 m ) is the free-particle dispersion and v B is the Bogoliubov velocity. We expand the momentum occupationaround the condensate mode as ˆ n k = √ N ( u k + v k )(ˆ b †− k +ˆ b k ), where N is the number of condensed atoms, and u k and v k are the Bogoliubov parameters, with ( u k + v k ) = (cid:15) k / ¯ hω k . This results in H s ( t ) = (cid:80) k V k ( t ) √ N ( u k + v k )(ˆ b †− k +ˆ b k ).For the interaction picture we use ˆ b k → ˆ b k exp( − iω k t ) and ˆ b † k → ˆ b † k exp( iω k t ), which yields H s,I ( t ) = (cid:88) k V k ( t ) (cid:112) N ( u k + v k )(ˆ b †− k e iω k t + ˆ b k e − iω k t ) . (S5)Using Eqs. S4 and S5 we solve the commutator in Eq. S1 and arrive at the heating rate [28] dEdt = 2 π ¯ h (cid:88) k ω k ( u k + v k ) N | V k | δ ( ω k − vk ) . (S6) V k is the time-independent part of the lattice potential in Eq. S3. Using the expression of | V k | and ω k ( u k + v k ) =¯ hk / (2 m ), we obtain dEdt = π N V m k δ [ ω k , − vk ] . (S7)The delta term in the heating rate gives an onset of dissipation at the velocity v = ω k , /k , where ω k , = k (cid:112) v / v and v = ¯ hk /m . In dimensionless form R ≡ ¯ h ( dE/dt ) / ( N V ) we have R B = π
16 ¯ hk m δ [ (cid:113) v / v − v ] . (S8)We extend this result to nonzero temperatures by including the thermal damping of the phonon modes. We consider aLandau-type damping Γ k = v Γ k , where v Γ is the damping velocity. We replace the delta distribution by a Lorentziandistribution, i.e πδ ( x ) = lim (cid:15) → (cid:15)/ ( x + (cid:15) ). This results in R B = 116 v v Γ ( (cid:112) v / v − v ) + v . (S9) Quasicondensate heating rate
To derive the heating rate for a quasicondensate, we consider a Hamiltonian of the form H = (cid:88) k E k a † k a k , (S10)where E k ≡ ¯ hω k is the excitation spectrum. For the perturbation term, we transform Eq. S2 to the interactionpicture by using a k → exp( − iω k t ) a k . This results in H s,I = (cid:88) kq V k ( t ) e i ( ω q − ω k + q ) t a † q a k + q . (S11)We now use Eqs. S10 and S11 to calculate the commutator [ H s,I ( t ) , H ], which gives[ H s,I ( t ) , H ] = (cid:88) kq ( E k + q − E q ) V k ( t ) e i ( ω q − ω k + q ) t × a † q a k + q . (S12)Using Eqs. S11 and S12 we calculate the commutator (cid:2) H s,I ( t ) , [ H s,I ( t ) , H ] (cid:3) and obtain (cid:2) H s,I ( t ) , [ H s,I ( t ) , H ] (cid:3) = − (cid:88) kq ( E k + q − E q ) | V k | cos (cid:2) ( vk + ω q − ω k + q )( t − t ) (cid:3) a † q a q . (S13)Integrating Eq. S13 over time t yields the heating rate dEdt = 2¯ h (cid:88) kq ( E k + q − E q ) | V k | sin[( vk + ω q − ω k + q ) t ]( vk + ω q − ω k + q ) (cid:104) a † q a q (cid:105) , (S14)which at long times approaches dEdt = 2 π ¯ h (cid:88) kq ( E k + q − E q ) n q | V k | δ ( vk + ω q − ω k + q ) . (S15)This result relates the heating rate to the momentum distribution n k . We consider the phononic dispersion at longwave lengths, i.e., E k = ¯ hv B | k | . The momentum distribution is given by n k = n πτ r τ/ | k | τ/ − , (S16)where n is the real-space density, r is the short-range cutoff of the order of ξ , and τ is the algebraic scaling exponent.We simplify Eq. S15 and obtain dEdt = N V h ( k r / τ/ τ ˜ v (1 − ˜ v ) τ/ − I ( φ ) , (S17)0 τ ˜ v − − − R q c ˜ v B − − − − A R qc τ = 0 . τ = 0 . τ = 0 . τ = 0 . FIG. S1.
Heating rate for a quasicondensate. ( A ) The estimate R qc of Eq. S17 as a function of the scaled velocity˜ v = v/v B and the algebraic scaling exponent τ . The magnitude of the heating rate is displayed on a log scale. ( B ) Theestimates of Eq. S17 (continuous line), Eq. S20 (dots), and R qc = πτ ˜ v /
32 (dashed line) are plotted on a log-log scale, for τ = 0 .
1, 0 .
2, 0 .
4, and 0 . with I ( φ ) = (cid:90) π dφ (1 + ˜ v − v cos φ )(˜ v − cos φ ) − (1+ τ/ . (S18)˜ v = v/v B is the scaled velocity. I ( φ ) can be solved, giving I ( φ ) = π (cid:104) v − − x (˜ v + 1) F (cid:16) − , x, , − v − (cid:17) + 3(˜ v + 1) − x (˜ v − F (cid:16) − , x, , v + 1 (cid:17) − (˜ v − − x (˜ v + 4˜ vx + 4 x − F (cid:16) , x, , − v − (cid:17) − (˜ v + 1) − x (˜ v − vx + 4 x − F (cid:16) , x, , v + 1 (cid:17)(cid:105) , (S19)where x = (1 + τ /
4) and F ( a, b, c, d ) are the hypergeometric functions. We expand I ( φ ) as I ( φ ) = 4 π ˜ v + O ( τ ). Thedimensionless heating rate R = ¯ h ( dE/dt ) / ( N V ) is R qc = π τ ˜ v − ˜ v + O ( τ ) . (S20)This low-temperature estimate scales linearly in τ , while it vanishes for τ approaching zero. At this order thedependence of k and r drops out. For low velocity the heating rate scales quadratically in v as R qc = πτ ˜ v /
32. InFig. S1(a) we show the result of Eq. S17 as a function of ˜ v and τ for k r = 2. The dissipation is nonzero below theBogoliubov velocity and increases with increasing both ˜ v and τ . This nonzero dissipation at low velocities is peculiarto 2D superfluids. In Fig. S1(b) we compare the results of Eqs. S17 and S20, which show good agreement for all ˜ v and all τ . The low-velocity sublinear behavior is also captured well by the estimate R qc = πτ ˜ v / INFLUENCE OF TEMPERATURE AND LATTICE STRENGTH ON THE CRITICAL VELOCITY
As we show in the main text, the critical velocity is smaller for high temperatures. In this section we analyze thisthermal reduction systematically by varying the temperature
T /T in the range 0 . − .
9, where T is the estimate ofthe critical temperature. We simulate the heating rate R ( v ) for n = 1 . µ m − , ˜ g = 1 .
6, and k /k c = 0 . k c = √ /ξ is determined by the healing length ξ . In Fig. S2A we show R ( v ) as a function of T /T for the lattice strength V /µ = 0 .
01. With increasing temperature, the broadening of the heating rate increases and the onset of rapidincrease occurs at a lower velocity. At low temperatures the heating rate primarily shows one maximum close to theBogoliubov velocity v B = 7 . / s. At high temperatures the heating rate shows two maxima corresponding to thetwo sound modes. We fit the heating rate below v B to the function f ( v ) = A max[0 , v − v c ], with A and v c asfitting parameters. We show the determined values of v c in Fig. S2A. The critical velocity decreases linearly withthe temperature. We fit this temperature dependence of v c with a linear function to determine the critical velocityat zero temperature v c (0) = 7 . / s, which is in excellent agreement with the Bogoliubov velocity v B . From the1
012 0.2 0.4 0.6 0.8 v / v B T /T
012 0.2 0.4 0.6 0.8 v / v B T /T V /µ V /µ AA v c BB v c FIG. S2.
Influence of temperature and lattice strength on the critical velocity. ( A ) Simulated heating rate R ( v ) asa function of T /T . The values of the critical velocity v c are shown as circles connected with a dashed line. The linear fit (redcontinuous line) is employed to determine the critical temperature T c /T = 0 .
86, for which v c is zero. ( B ) R ( v ) as a functionof V /µ for T /T = 0 . µ is the mean-field energy. The system parameters are n = 1 . µ m − and ˜ g = 1 . fit, we also extrapolate the temperature T /T = 0 .
86, for which v c is zero. We denote this temperature as the criticaltemperature. In Fig. S2B we show R ( v ) as a function of the lattice strength V /µ for T /T = 0 . µ is the mean-fieldenergy. The lattice strength introduces an additional broadening of the heating rate. This broadening is higher forhigher V /µ . We determine the value of the critical velocity v c , which is shown in Fig. S2B. The critical velocitydecreases in a non-linear fashion as a function of V /µ . DETERMINING THE SOUND VELOCITY FROM THE PROPAGATION OF A DENSITY WAVE
To determine the sound velocity we excite a density wave following the method of Refs. [17, 24]. We imprint a phasedifference on one-half of the system along x direction. This sudden imprint of the phase results in an oscillation of thephase difference between the two subsystems, ∆ φ ( t ), as shown in Fig. S3A. We fit ∆ φ ( t ) with a damped sinusoidalfunction f ( t ) = A exp( − Γ t ) sin(2 πf + φ ) to determine the amplitude A , the damping rate Γ, the frequency f , andthe phase shift φ . From the fit to the results in Fig. S3A we obtain f = 35 . / (2 π ) = 1 .
29 Hz. The soundvelocity is determined by v s = 2 f L x , where L x is the system length in the x direction. This results in v s = 7 .
16 mm / sand the damping velocity v Γ = 0 .
26 mm / s, which are v s /v B = 0 .
98 and v Γ /v B = 0 .
03, respectively. The reduction ofthe sound velocity is small for
T /T = 0 .
1, as expected. In Fig. S3B we show the spectrum of ∆ φ ( t ), which yieldsthe peak at the sound frequency and is the same as the one determined from the time evolution in Fig. S3A. DETERMINING THE SUPERFLUID DENSITY
We determine the superfluid density by calculating the current-current correlations in momentum space, see also[29, 35]. The current density j ( r ) is defined as j ( r ) = ¯ h im [ ψ ∗ ( r ) ∇ ψ ( r ) − ψ ( r ) ∇ ψ ∗ ( r )] . (S21)We choose the gradient along x/y directions and calculate the Fourier transform of the current density ( j k ) x/y in the x and y directions. This allows us to determine (cid:104) ( j ∗ k ) x ( j k ) y (cid:105) , which in the limit of k → (cid:104) ( j ∗ k ) l ( j k ) m (cid:105) = k B Tm A (cid:16) n s k l k m k + n n δ lm (cid:17) . (S22) n s and n n are the superfluid and the normal fluid density, respectively. A is the system area. By taking a cut along theline k x = k y = k/ √ k = 0 value using a linear fit, we determine the superfluid density basedon Eq. S22. In Fig. S4 we show the interaction dependence of the numerically determined superfluid density. While2 -0.0500.05 0 50 100 150 ∆ φ / π t (ms) A
012 0 20 40 60 80 | ∆ φ | f (Hz) B FIG. S3.
Sound propagation. ( A ) Time evolution of the phase difference between the two subsystems, ∆ φ ( t ), for n =1 . µ m − , T /T = 0 .
1, and ˜ g = 1 .
6. The dashed line is the fit with a damped sinusoidal function, see text. ( B ) Spectrum of∆ φ ( t ) shows a peak at the sound frequency. n s / n ˜ gT /T = 0 . T /T = 0 . T /T = 0 . FIG. S4.
Superfluid density.
Numerical superfluid fraction n s /n as a function of the interaction strength ˜ g for n = 1 . µ m − and T /T = 0 . . .6 (circles).