Quench Dynamics of Three-Dimensional Disordered Bose Gases: Condensation, Superfluidity and Fingerprint of Dynamical Bose Glass
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov Quench Dynamics of Three-Dimensional Disordered Bose Gases: Condensation,Superfluidity and Fingerprint of Dynamical Bose Glass
Lei Chen, Zhaoxin Liang, ∗ Ying Hu, and Zhidong Zhang Shenyang National Laboratory for Materials Science, Institute of Metal Research,Chinese Academy of Sciences, Wenhua Road, 72, Shenyang, China Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
In an equilibrium three-dimensional (3D) disordered condensate, it’s well established that disordercan generate an amount of normal fluid equaling to of the condensate depletion. The conceptthat the superfluid is more volatile to the existence of disorder than the condensate is crucial to theunderstanding of Bose glass phase. In this Letter, we show that, by bringing a weakly disordered3D condensate to nonequilibrium regime via a quantum quench in the interaction, disorder candestroy superfluid significantly more, leading to a steady state in which the normal fluid density farexceeds of the condensate depletion. This suggests a possibility of engineering Bose Glass in thedynamic regime. As both the condensate density and superfluid density are measurable quantities,our results allow an experimental demonstration of the dramatized interplay between the disorderand interaction in the nonequilibrium scenario. PACS numbers: 05.70.Ln, 67.85.De, 61.43.-j
Interaction and disorder are two basic elements in na-ture. Their competition underlies many intriguing phe-nomena in the equilibrium physics, such as Anderson lo-calization [1] and the emergence of Bose glass phase [2].Very recently, the remarkable experimental progress ofquenching ultracold Bose gas [3] in tunable disorderedpotentials [4–6] has generated a surge of new interests instudying this old problem in the non-equilibrium regime,in which the combined effects of disorder and interactionscan become much more dramatic. In this context, whilea focus of theoretical research [8, 9] has been on howa disordered system relaxes after a variety of quantumquench, we address below the problem of how to feasi-bly illustrate the quench effects on the interplay betweenthe disorder and interaction in experimentally observablequantities.The point of this work is to revisit two fundamentaland measurable quantities, condensate density [10–13]and the superfluid density [14–17], in the new context ofquench dynamics of a disordered Bose-Einstein conden-sate (BEC) at three dimension (3D). In the equilibriumregime, the different fate of this two quantities in thepresence of disorder has been crucial for the understand-ing of the Bose glass state [2]. In the pioneering work [18]on disordered 3D BECs at their ground state, it is pointedout that the weak disorder can generate a normal fluid ρ n equaling of the condensate depletion ρ ex even atzero temperature. This ratio has been later shown to bequite generic, arising also in 3D BECs with both stronginteraction and strong disorder [19], as well as in exter-nal potentials [20]. This has led to the speculation onthe existence of Bose glass, in which superfluid vanishesbut finite condensate survives. The present work showsthat, by combining with the effect of quantum quench inthe interaction, disorder can destroy even more superfluidthan the condensate compared to the equilibrium case. In particular, a ratio ρ n /ρ ex >> / Model Hamiltonian.—
We consider a weakly interact-ing 3D Bose gas in the presence of disordered potentialsunder a quantum quench in the interaction. The corre-sponding second-quantized Hamiltonian reads [18, 21–23] H − µN = Z d r ˆΨ † ( r ) h − ~ ∇ m − µ + V dis ( r )+ 12 g ( t ) ˆΨ † ( r ) ˆΨ( r ) i ˆΨ( r ) , (1)where ˆΨ( r ) is the field operator for bosons with mass m , µ is the chemical potential, N = R d r ˆΨ † ( r ) ˆΨ ( r ) is thenumber operator and V dis ( r ) represents the disorderedpotential. The g ( t ) in Hamiltonian (1) describes thequench protocol for the interaction parameter. Specif-ically, we consider the case when the system is initiallyprepared at the ground state | Ψ(0) i of Hamiltonian (1)with g = g i labeled by H i ; then, at t = 0, the interac-tion strength is suddenly switched to g = g f such thatthe time evolution from t > H f . Accordingly, we write g ( t ) = g i [1 + Θ( t ) (˜ g − , (2)with ˜ g = g f /g i and Θ( t ) being the Heaviside function.Experimentally, the interaction quench in Eq. (2) can beachieved by using Feshbach resonance [24].For V dis ( r ) in Hamiltonian (1), we consider itsrealization [4–6, 18–23] via the random distributionof quenched impurity atoms described by V dis ( r ) = t/ τ MF ρ e x / ρ e x , e q t/ τ MF ρ e x / ρ e x , e q t/ τ MF ρ e x / ρ e x , e q t/ τ MF ρ e x / ρ e x , e q (a) (b)(c) (d) FIG. 1. (Color online) Dynamics of the quantum depletion ρ ex ( t ) after the quench. The solid curves are the quantumdepletion ρ ex ( t ) redistribution of a disordered Bose gas fol-lowing a sudden change of interactions from g i to g f via thetime. The dashed curves are the excitation fraction at thefinal interaction strength in equilibrium. The blue and redcurves correspond to the interaction- and disordered-inducedquantum depletion, whereas the black curve describes thequantum depletion due to the combined effects. The char-acteristic density and relaxation time are set by ρ ex,eq =1 / π ζ + (1 / πζ ) ˜ R and τ MF = ~ /g f ρ respectively. Pa-rameters are given for (a) ˜ R = 0 . g = 0 .
4; (b) ˜ R = 0 . g = 0 .
4; (c) ˜ R = 0 . g = 0 .
6; (d) ˜ R = 0 . g = 0 . g imp P N imp i =1 δ ( r − r i ) with g imp being the coupling con-stant of an impurity-boson pair [18], r i the randomlydistributed positions of the impurities, and N imp count-ing the number of r i . The randomness is uniformly dis-tributed and Gaussian correlated [22], such that h V i = g imp N imp /V ( V is the system’s volume) and R = 1 V h V − k V k i , (3)with V k = (1 /V ) R d r e i k · r V dis ( r ) and h ... i describing theensemble average over all possible realizations of disorderconfigurations.We shall focus on the regime of both weak interactionand weak disorder, in which Hamiltonian (1) can be welldescribed using the standard Bogoliubov approximation[18, 22, 25] and the resulting expression reads H eff ( t ) = X k =0 ( ǫ k − µ ) ˆ a † k ˆ a k + √ ρ X k =0 (cid:16) ˆ a † k V − k + ˆ a k V k (cid:17) + 12 g ( t ) ρ X k =0 (cid:16) ˆ a † k ˆ a †− k + ˆ a k ˆ a − k (cid:17) , (4)where ˆ a k (ˆ a † k ) annihilates (creates) a bosonic atom withmomentum k , and ρ = N /V is the condensate densitywith N being the number of condensed atoms. Hamil-tonian (4) describes the process when a pair of bosonic atoms with momenta { k , − k } are annihilated throughthe two body interaction (and vice versa), as well asthe process when a single particle with momenta k isscattered by the disordered potential into the condensate(and vice versa).As the system is initially prepared at the ground stateof Hamiltonian H eff (4) with g = g i , the quench from g i to g f will bring the system to the nonequilibrium. For thequadratic Hamiltonian (4), the nonequilibrium dynam-ics can be exactly described as | Φ( t ) i = Π k U k ( t ) | Φ(0) i with | Φ(0) i and | Φ( t ) i being the many-body wavefunc-tion before and after the quench, respectively, and U k ( t )represents the evolution operator for each momenta k .By noticing that Hamiltonian H eff with g = g f con-tains the operators of K ( k ) = (ˆ a † k ˆ a k + ˆ a †− k ˆ a − k ) / K + ( k ) = ˆ a † k ˆ a †− k and K − ( k ) = ˆ a k ˆ a − k which form thegenerators of SU(1,1) Lie algebra [26, 27], we can obtain U k ( t ) as U k ( t ) = e α ∗ k ( t )ˆ a † k e α k ( t )ˆ a k exp [ β ( k , t ) K ( k ) + iφ k ( t )] × exp [ β + ( k , t ) K + ( k )] exp [ β − ( k , t ) K − ( k )] . (5)Here, φ k ( t ) is a trivial phase and α k ( t ) = − √ ρ ω k V k (cid:16) | u k | + 2 u k v ∗ k + | v k | (cid:17) ,β + ( k , t ) = v ∗ k ( t ) /u ∗ k ( t ) , β − ( k , t ) = − v k ( t ) /u ∗ k ( t ) ,β ( k , t ) = − u ∗ k ( t ) , (6)are expressed in terms of the disorder potential V k andthe time-dependent Bogoliubov amplitudes u k ( t ) and v k ( t ) which are determined from (cid:18) u k ( t ) v k ( t ) (cid:19) = " cos( ω f k t ) ˆ I − i sin( ω f k t ) ω f k × (cid:18) ǫ k + g f ρ g f ρ − g f ρ − ( ǫ k + g f nρ ) (cid:19) u k (0) v k (0) (cid:19) , (7)with u k (0) ( v k (0)) = ± q(cid:2) ( ǫ k + ρ g i ) /ω i k ± (cid:3) / ω f ( i ) k = q ǫ k (cid:0) ǫ k + 2 ρ g f ( i ) (cid:1) and ǫ k = ~ k / m . Notethat | u k ( t ) | − | v k ( t ) | = 1 is always satisfied during thetime evolution. For self-consistency, hereafter we limitourselves in the regime where the time dependence of ρ can be ignored [25]. Quantum depletion after quench.–
We are now wellequipped to study the time evolution of the non-condensed fraction ρ ex ( t ) = h Φ(0) | P k ˆ a † k ˆ a k | Φ(0) i of theconsidered system, given that the initial condition that | Φ(0) i is the ground state of H i . Straightforward deriva-tion using Eq. (5) yields [28] ρ ex ( t ) = X k | υ k ( t ) | . (8)By substituting Eq. (7) into Eq. (8), we can arrive at ρ ex ( t )(3 π ζ ) − = 1 + 3 π ˜ R √ ˜ g − √ Z ∞ dk ˜ g (1 − ˜ g ) k sin (cid:16)p k ( k + 2˜ g ) t (cid:17) ( k + 2˜ g ) p k ( k + 2)+ 6 √ R (1 − ˜ g ) Z ∞ dk (cid:0) k + k + ˜ gk (cid:1) sin ( p k ( k + 2˜ g ) t ) + ˜ g (1 − ˜ g ) sin (2 p k ( k + 2˜ g ) t )( k + 2) ( k + 2˜ g ) , (9)where ζ = ~ / √ mg i ρ is the initial healing length, and wehave introduced the dimensionless parameter˜ R = ρ R / ( g i ρ ) (10)to characterise the relative disorder strength. Note that,for vanishing disorder ˜ R = 0, Equation (9) agrees ex-actly with the corresponding result in Ref. [25]; whereasfor ˜ g = 1 (no quench), the system simply remainsin the ground state | Φ(0) i with the depletion ρ ex =(3 π ζ ) h π ˜ R / i as in Ref. [18]. Therefore, thelast two terms in Eq. (9) presents the combined effectof the interaction quench ˜ g = 1 and disorder ˜ R on thecondensate depletion in the non-equilibrium regime.We are interested in the asymptotic behavior of ρ ex ( t )at the long time after the quench. To comprehensivelyreveal the roles of ˜ R and ˜ g , we consider three cases fornumerical analysis of Eq. (9), as illustrated in Fig. 1.(i) Firstly, we show how the quench strength ˜ g affectsthe asymptotic depletion. As such, we fix ˜ R = 0 andcalculate ρ ex ( t ) (blue solid line), which is compared tothe corresponding equilibrium depletion for a BEC with g f (blue dashed line). (ii) Secondly, to extract the role ofdisorder, we fix ˜ g = 0 that corresponds to a quench to anon-interacting BEC (red solid line) with disorder, andthen compare ρ ex ( t ) with the corresponding equilibriumvalue (red dashed line). (iii) Finally, the combined effectsof disorder and quenched interaction is illustrated by theblack solid curve. In all cases, we have found enhanceddepletion in the asymptotic steady state compared to thecorresponding case at zero temperature. This indicatethat the ability of disorder or interaction to deplete thecondensate is magnified in the non-equilibrium scenario.Compared to the equilibrium disordered 3D BEC, theincreased depletion in the steady state of the correspond-ing system under an interaction quench can be qual-itatively understood in terms of the Loschmidt echo L ( t ) = |h Ψ ( t ) | Ψ( t ) i| [29, 30]. The Loschmidt echohas been intensively studied recently in quenched sys-tems. The connection between the condensate depletionand the Loschmidt echo is best illustrated in the caseof ˜ R = 0 in the quadratic Hamiltonian H eff , when theLoschmidt echo can be calculated as L ( t ) = 1 / Π k | u k | [26]. By using Eq. (8) and | u k | = 1 + | υ k | , we esti-mate L ( t ) ≈ / (1 + ρ ex ( t )) (after ignoring higher orderterms in | u k | and | υ k | ). Then, building on the squarerelation L sq ( t → ∞ ) = L ad ( t → ∞ ) established in Ref.[26, 31, 32], which connects the steady-state Loschmidt echo in a sudden quench ( L sq ) to that of an adiabaticinteraction change ( L ad ), we can estimate ρ sqex ( t → ∞ ) =(1 + ρ adex ) −
1. To see how this formula fits, we inputthe adiabatic value ρ eqex ( t ) /ρ ex,eq = 0 .
196 (obtained fromthe dashed blue line in Fig. 1 a) and calculate the sud-den quench depletion as ρ sqex ( t → ∞ ) /ρ ex,eq = 0 .
43, whichagrees fairly well with the numerical results in the steadystate (solid blue line in Fig. 1 a). g f /g i ρ n / ρ e x FIG. 2. (Color online) Quench enhanced ratio between nor-mal density ρ n in Eq. (12) and quantum depletion ρ ex ( t →∞ ) in Eq. (9) via g f /g i . The dashed line corresponds to thevalue of ρ n /ρ ex = 4 / Superfluid depletion after quench.–
The superfluidcomponent and the normal fluid component can beclearly distinguished from their response to a slow ro-tation: normal fluid rotates but superfluid does not.This is the essential concept behind typical experimentalschemes to measure the superfluid density in an atomicBose gas [14–16]. While the techniques to generate rota-tions in a Bose gas differ in various schemes, the keyquantity measured boils down to the current-currentresponse function χ i,j ( r t, r ′ t ′ ) = h [ J i ( r , t ) , J j ( r ′ , t ′ )] i ( i, j = x, y, z ) with J i ( r , t ) = ~ / (2 i )[ ˆΨ † ( r ) ∇ ˆΨ( r ) − ˆΨ( r ) ∇ ˆΨ † ( r )] being the current density of system and h ... i averaged with the initial ground state | Ψ(0) i . Partic-ularly, the superfluid density ρ s corresponds to the re-sponse to the irrotational (longitudinal) part of the per-turbation; whereas, the normal fluid density ρ n describesthe transverse response, in an isotropic translationallyinvariant system, we have mχ ij ( q → , ω ) = ρ s q i q j q + ρ n δ ij . (11)Based on the current experimental approaches to mea-sure the superfluid response, we have calculated ρ n = χ zz ( q → , ω ) with J z q = ~ / (2 m ) P k ( k z + q z / a † k ˆ a k + q for the steady state of the considered BEC after the in-teraction quench (as the system is isotropic, we are freeto choose the slow rotation around the z axis ). Then,by using Eqs. (5) and (7) [28], we derive the normal fluiddensity in the steady state of the quenched system as ρ n (3 π ζ ) − = 2 π ˜ R p ˜ g " (7˜ g − F ( , ; 1; (˜ g − ˜ g )(˜ g − / ˜ g − F ( − , ; 1; (˜ g − ˜ g )(˜ g − / ˜ g , (12)with F ( a , b ; c ; x ) being the hypergeometric function[33].Now, with both the condensate depletion and the nor-mal fluid density at hand for the steady state of theconsidered system, we plot ρ n /ρ ex as a function of thequench strength ˜ g = g f /g i , as illustrated in Fig. 2. Toset a reference point, we have shown that in the limit˜ g → ρ n /ρ ex = 4 / ρ n /ρ ex , the stronger the quench is, the higher valueof ρ n /ρ ex is found, which can even approach 2. More-over, we have found that the ratio does not depend onthe individual absolute value of g f and g i , but rather ontheir relative strength ˜ g = g f /g i . Figure 2 shows thatthe ability of disorder to deplete more superfluid thanthe condensate is remarkably amplified when combinedwith the quench effect, which presents the major result ofthis work. In principle, the value of ρ s /ρ can be furthersuppressed by repeating the process of sudden quenchin the interaction: quench from g i to g f , holding time t and then adiabatically change interaction from g f to g i , then quench interaction n times (bang-bang proto-col). It’s highly expected that the disordered BEC canbe quenched into the regime of ρ s /ρ ≪ Discussion and Conclusion.—
We have shown that,by quenching in the interaction of a 3D BEC in disor-dered potential, the system can relax to a steady state where the ratio ρ n /ρ ex > / ρ ex and normalfluid density ρ n . In typical experiments, the-state-of-art t/ τ MF g ( ) δ ( t ) − ρ / | g ( ) δ ( t → ∞ ) | −5 t/ τ MF ( g ( ) δ ( t ) − g ( ) δ ( )) / ρ ζ − δ / ζ ( g ( ) δ − ρ ) π / ρ ζ − FIG. 3. (Color online) Quench dynamics of density-densitycorrelation function, which develops in an oscillatory mannerand rapidly saturate at times t − δ/c > τ MF with δ being theseparation between points, δ = | r − r ′ | . The top inset: long-time density correlation function via time. The bottom inset:long-time behavior of density-density correlation function viathe separation between points. The parameters are given as˜ δ = δ/ζ = 4, ˜ g = 0 .
6, and ˜ R = 0 . higher-resolution imaging techniques allow one to probethe time-dependent quantum depletion ρ ex [13, 34, 35],while the experimental schemes reported in Refs. [14–16] can be used to measure ρ n in both equilibrium andnonequilibrium regimes. One concern that may arise isrelated to how fast the considered system can relax tothe steady state. Figure 1 indicates the rapid relaxationof the one-body correlation. Let us further analyze thequench dynamics of the two-body correlation function, aquantity directly relevant for the the Bragg spectroscopy[36, 37] that has been a routine technique for studyingexcitations of a Bose gas. The two-body correlation func-tion is defined as g (2) ( t ) = P q e i q · ( r − r ′ ) h ρ q ( t ) ρ − q ( t ) i with ρ q ( t ) = P k ˆ a † k + q ( t ) ˆ a k ( t ) being the density oper-ator. Following similar procedures [25], we calculate thecorrelation function from Eq. (5) and obtain g (2) δ ( t ) = ρ + ρ π ˜ δζ Z kdk sin (cid:16) √ k ˜ δ (cid:17) h k √ k + 2 − g − k sin (cid:16)p k ( k + 2˜ g ) t (cid:17) √ k + 2 ( k + 2˜ g ) − i + 4 ρ π ˜ δζ ˜ R Z dk k sin (cid:16) √ k ˜ δ (cid:17) ( k + 2) ( k + 2˜ g ) (cid:26) cos (cid:16)p k ( k + 2˜ g ) t (cid:17) + k + 2 k + 2˜ g sin (cid:16)p k ( k + 2˜ g ) t (cid:17)(cid:27) , (13)with the dimensionless parameter ˜ δ = δ/ζ for δ = | r − r ′ | .Again, for vanishing disorder ˜ R = 0, Equation (9) agreeswith Ref. [25]; whereas for ˜ g = 1, our result is consistentwith Ref. [18]. The time evolution of g (2) is presented inFig. 3, which shows a rapid relaxation to a finite valueon a time scale t ∼ τ MF . Both the rapid relaxation ofone-body matrix in Fig. 1 and the two-body correlationfunction in Fig. 3 suggest that, after the 3D disorderedBEC is brought out of equilibrium by a quantum quenchin the interaction, it relaxes to a steady state on a time-scale within the experimental reach.We thank Biao Wu and Li You for stimulating dis-cussions. This work is supported by the NSF of China(Grant Nos. 11004200 and 11274315). Y. H. also ac-knowledges support by the Austrian Science Fund (FWF)through SFB F40 FOQUS. ∗ Corresponding author: E-mail: [email protected][1] P. W. Anderson, Phys. Rev. , 1492 (1958).[2] J. A. Hertz, L. Fleishman, and P. W. Anderson, Phys.Rev. Lett. , 942 (1979); T. Giamarchi and H. J. Schulz,Phys. Rev. B , 325 (1988); K. G. Singh and D. S.Rokhsar, Phys. Rev. B , 9013 (1994); D. S. Fisher,and M. P. A. Fisher, Phys. Rev. Lett. , 1847 (1988);M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D.S. Fisher, Phys. Rev. B , 546 (1989); D. K. K. Lee andJ. M. F. Gunn, J. Phys. Condens. Matter , 7753 (1990);S. Giorini, L. Pitaevskii, and S. Stringari, Phys. Rev. B , 12938 (1994); M. Kobayashi and M. Tsubota, Phys.Rev. B , 174516 (2002).[3] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[4] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P.Lugan, D. Cl´ement, L. Sanchez-Palencia, P. Bouyer andA. Aspect, Nature (London) , 891 (2011).[5] G. Roati, Chiara D’Errico, L. Fallani, M. Fattori, C. Fort,M. Zaccanti, G. Modugno, M. Modugno, and M. Ingus-cio, Nature (London) , 895 (2008).[6] L. Sanchez-Palencia and M. Lewenstein, Nat. Phys. , 87(2010).[7] S. Krinner, D. Stadler, J. Meineke, J. P. Brantut, and T.Esslinger, Phys. Rev. Lett. , 100601 (2013).[8] M. Tavora, A. Rosch, and A. Mitra, Phys. Rev. Lett. , 010601 (2014).[9] C. D’Errico, E. Lucioni, L. Tanzi, L. Gori, G. Roux, andI. P. McCulloch, Phys. Rev. Lett. , 095301 (2014).[10] D. Pines and P. Nozi´eres, The theory of quantum liquids (Benjamin, New York, 1966), Vol. I; P. Nozi´eres and D.Pines,
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