Relaxation of a high-energy quasiparticle in a one-dimensional Bose gas
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Relaxation of a high-energy quasiparticle in a one-dimensional Bose gas
Shina Tan, Michael Pustilnik, and Leonid I. Glazman Department of Physics, Yale University, New Haven, CT 06520, USA School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
We evaluate the relaxation rate of high-energy quasiparticles in a weakly-interacting one-dimensional Bose gas. Unlike in higher dimensions, the rate is a nonmonotonic function of tem-perature, with a maximum at the crossover to the state of suppressed density fluctuations. At themaximum, the relaxation rate may significantly exceed its zero-temperature value. We also find thedependence of the differential inelastic scattering rate on the transferred energy. This rate yieldsinformation about temperature dependence of local pair correlations.
PACS numbers: 03.75.Kk,05.30.Jp
Recent experiments with ultracold atomic gases [1, 2]renewed the interest in fundamental properties of the el-ementary excitations in interacting Bose systems.A three-dimensional (3D) Bose gas undergoes theBose-Einstein condensation (BEC) phase transition ata sufficiently low temperature [3]. The transition af-fects dramatically the spectrum of elementary excitations(quasiparticles) of the system. In the Bose-condensedphase, the quasiparticles obey the Bogoliubov dispersionrelation ǫ q = sq p q/ ms ) which interpolates be-tween a phonon-like linear spectrum at small momenta(here s is sound velocity and m is each boson’s mass) anda free-particle-like spectrum at large momenta [3].The BEC transition affects strongly the lifetime of low-energy quasiparticles. The relaxation rate Γ q of quasipar-ticles in the phonon part of the spectrum is very sensitiveto both their momenta q [4] and temperature T [5–7],Γ q ∝ max { ( ε q ) , ε q T } . However, the relaxation rate ofhigh-energy quasiparticles is dominated by collisions withlarge momentum transfer, does not depend on either q or T , and thus is not sensitive to BEC transition [4]. Someof these long-standing predictions have been recently ver-ified experimentally, see [1, 2] for a review.Unlike its 3D counterpart, the one-dimensional (1D)interacting Bose gas turns at low temperatures to a qua-sicondensate in which the long-range order is destroyedby quantum fluctuations [3, 6], and the BEC transitionturns to a crossover. Yet, despite this difference, the spec-trum of elementary excitations in 1D is still describedvery well by the Bogoliubov dispersion relation [8].However, the quasiparticle lifetime in 1D is very dif-ferent from that in higher dimensions and is not aswell understood. The reason is that, due to the con-straints imposed by the energy and momentum conser-vation, two-particle collisions do not lead to a relaxationin 1D. At the same time, realizations of 1D Bose systemswith cold atoms confined in tight atomic waveguides [1]are described rather well [9] by a model of bosons withzero-range repulsive interaction (the Lieb-Liniger model),which is integrable [8, 10]. In this model, the redistri-bution of the momenta between particles in a collision,and, therefore, relaxation, are absent [10]. Such apparent lack of relaxation was recently demonstrated experimen-tally [11] (see the discussion below).The leading corrections to the Lieb-Liniger model havethe form of a local three-particle interaction term [12, 13],which breaks the integrability and brings about thequasiparticle relaxation. In this Letter, we study relax-ation of a particle with a large momentum. This problemwas considered recently in [13], where the inelastic relax-ation rate due to three-particle collisions was evaluatedin the approximation that neglects two-body repulsion.The results of Ref. [13] suggest that, very much like in3D, the relaxation rate of high-energy quasiparticles is in-dependent of momentum and temperature. However, inthe present Letter, we demonstrate that, in a dramaticdeparture from the behavior in higher dimensions, therelaxation rate in 1D depends strongly on temperatureeven at large momenta. It has a pronounced peak at thecrossover to the quasicondensate state.We evaluate the differential and the total relaxationrates. Both can be inferred from observations of collidingclouds of cold atoms [11, 14].To describe the relaxation in a weakly-interacting 1DBose gas, we consider the simplest Hamiltonian H = H + V, (1)where H = Z dx ψ † ( x ) (cid:18) − m d dx (cid:19) ψ ( x ) + c Z dx : ρ ( x ) : (2)describes 1D bosons with a repulsive contact interaction(hereinafter we set k B = ~ = 1), and V = − α m Z dx : ρ ( x ) : (3)represents the leading integrability-breaking perturba-tion [12, 13]. In Eqs. (2) and (3), ρ ( x ) = ψ † ( x ) ψ ( x )is the local density operator and the colons denote thenormal ordering. The strength of the interaction [repre-sented by the second term in Eq. (2)] is characterized [8]by the dimensionless parameter γ = mc/n , where n isthe 1D concentration. A finite three-particle scatteringamplitude appears already in the first order in α ≪ q (we assume that q >
0) and kinetic energy ξ q = q / m , which is large compared to both temperature T and a typical interaction energy per particle ω s , ξ q ≫ max { T, ω s } , ω s = ms / . (4)In the limit of a weak interaction γ ≪
1, which we con-sider from now on, the sound velocity s in Eq. (4) is givenby [8] s = ( n/m ) √ γ . The condition (4) ensures that theparticle is added to an almost empty single-particle state: f q = h ψ † q ψ q i ≪ α the differential rateof inelastic scattering is given by σ q ( ω ) = α πm Z q/ −∞ dp δ (cid:0) ω − ξ q + ξ q − p (cid:1) G ( p, ω ) , (5)where G ( p, ω ) = R dxdt e iωt − ipx G ( x, t ) is the Fouriertransform of the correlation function G ( x, t ) = (cid:10) : ρ ( x, t ) : : ρ (0 ,
0) : (cid:11) , (6)which should be evaluated for the Lieb-Liniger modelEq. (2). In writing Eq. (5) we took into account thekinematic constraint p < q/ σ q ( ω ) vanishes for ω > ξ q /
9. In terms of σ q ( ω ), thetotal relaxation rate is given byΓ q = Z dω σ q ( ω ) . (7)The differential rate (5) at large energy transfer ω isdetermined by the behavior of G ( x, t ) at t → G ( x, t ) = 2 n g Ψ , Ψ( x, t ) = (cid:16) m πit (cid:17) / e imx / t . (8)Here n g = h : ρ (0 , i is the probability of finding twobosons at point x = 0 at time t = 0, and Ψ( x, t ) isthe solution of the single-particle Schr¨odinger equationwith the initial condition Ψ( x,
0) = δ ( x ). Interactionsdo not affect the time evolution in Eq. (8) as long as | t | ≪ min { /ω s , /T } . Instead, the dependence on tem-perature and on the interaction strength enters Eq. (8)via the normalized local pair correlation g . For the Lieb-Liniger model this quantity can be evaluated exactly [17].For a weak interaction g increases monotonically with T from g = 1 at T ≪ T s to g = 2 at T ≫ T s [17], wherewe introduced a characteristic temperature scale T s = p ω s T = ns ; (9)here T = 2 n /m is the quantum degeneracy tempera-ture. (Note that ω s ≪ T s ≪ T for a weak interaction.) Substitution of Eq. (8) into Eq. (5) yields the differen-tial rate at large positive energy transfer [18] σ q ( ω ) = α T g π p ξ q ω (cid:20) p − ω/ξ q (1 − ω/ξ q )(1 + 3 p − ω/ξ q ) (cid:21) / . (10)Eq. (10) is applicable at max { ω s , T } ≪ ω < ξ q /
9. Awayfrom the upper end of this interval, at ω ≪ ξ q , Eq. (10)reduces to σ q ( ω ) = α T g π p ξ q ω . (11)To further analyze σ q ( ω ) at | ω | ≪ ξ q , we note thatin this range of ω the momenta p contributing to theintegral in Eq. (5) are small, | p | ≪ q , and it simplifies to σ q ( ω ) = α πmq G (0 , ω ) . (12)It follows from the properties of G ( p, ω ) that the differ-ential rate (12) satisfies the detailed balance condition σ q ( − ω ) = e − ω/T σ q ( ω ) . (13)Eq. (13) implies that while σ q ( ω ) ≈ σ q ( − ω ) at smallenergy transfers | ω | ≪ T , the differential rate is expo-nentially small at large negative ω .To gain further understanding of the differential rateat small momentum transfer, we consider first the regimeof relatively high temperatures T ≫ T s , when the inter-action in Eq. (2) can be neglected (except for very tinyenergy transfers, see below). The correlation function inEq. (12) is then easily evaluated resulting in σ q ( ω ) = α π mq Z Y i =1 dk i f k f k ( f k + 1)( f k + 1) (14) × δ ( k + k − k − k ) δ ( ξ k + ξ k − ξ k − ξ k + ω ) , where f k is the Bose distribution. (Eq. (14) can also bederived by using Fermi’s Golden Rule.)At T s ≪ T ≪ T the chemical potential is given by µ = − µ , µ = T /T ≪ T. (15)At | ω | ≪ T the differential rate is dominated by pro-cesses in which both the initial and the final states of thetwo low-energy particles involved in a collision belong tothe part of the spectrum with high occupation numbers: f k i ≈ f k i + 1 ≈ T / ( ξ k i + µ ) ≫
1. Evaluation of Eq. (14)with this approximation results in σ q ( ω ) = α ( T /ξ q ) / ( T /T ) F (cid:0) | ω | /µ (cid:1) , | ω | ≪ T. (16)The analytical expression for the function F ( z ) is some-what cumbersome. It is a monotonic function normal-ized as R ∞ F ( z ) dz = 1 /
8, with a power-law behavior at z ≫ F ( z ) = (2 √ /π ) z − / , and a logarithmic asymp-tote F ( z ) = (5 / π ) ln(8 e − / /z ) at z → ω → k ≈ k ≈ k ≈ k in the integral over momentain Eq. (14), and is an artifact of the free-boson approx-imation. The probability of scattering two bosons withclose momenta k ≈ k in the initial state is suppressedat | k − k | ≪ mc . (There is a similar suppression for thefinal states k , .) The logarithmic divergence in σ q ( ω ) isthus regularized at | ω | . ω s /T for T ≫ T s .At T s ≪ T ≪ T and ω ≫ µ , the main contributionto the integral in Eq. (14) comes from | k , | . √ mµ and | k , | ∼ √ mω ≫ | k , | . Neglecting k , and ξ k , in thearguments of the delta-functions in Eq. (14), we find σ q ( ω ) = α T π p ξ q | ω | g (cid:0) − e − ω/ T (cid:1) , | ω | ≫ µ , (17)where g = 2 as appropriate for T ≫ T s . Eq. (17) ex-trapolates between Eqs. (11) and (16).With lowering the temperature, Eqs. (14)-(16) becomeinadequate when µ ( T ) is of the order of the interactionenergy per particle ω s , i.e., at T ∼ T s . At T ≪ T s ,however, the Bogoliubov approximation for the local den-sity operator becomes applicable [19]. In this approxima-tion, excitations of a 1D Bose liquid are essentially freephonons (Bogoliubov quasiparticles), described by theHamiltonian H B = P k ε k b † k b k with ε k = p ξ k ( ξ k + 4 ω s ).In terms of phonons, the density operator has the form ρ ( x ) = n + P k =0 ( nξ k /Lε k ) / (cid:0) b k + b †− k (cid:1) e ikx , where L isthe size of the system. Using this representation, evalu-ation of Eq. (12) is straightforward and yields σ q ( ω ) = α T π p ω s ξ q (cid:16) ω/ω s − e − ω/ T (cid:17) (18) × (cid:2) ω/ ω s ) (cid:3) − / n (cid:2) ω/ ω s ) (cid:3) / o − / . At | ω | ≫ ω s , Eq. (18) reduces to Eq. (17) with g = 1appropriate for T ≪ T s . In fact, Eq. (17) is valid at any T ≪ T , provided that the energy transfer falls withinthe range max { µ , ω s } ≪ | ω | ≪ ξ q . For positive ω inthis range, the validity of Eq. (17) is due to the fact thatthe interaction has negligible effect [16] on the final statesof the colliding particles ( ξ k ≈ ξ k ≈ ω/ ≫ ω s ). Theapplicability of Eq. (17) for negative ω in the above rangethen follows from Eq. (13).We show the typical plots of the differential relaxationrate σ q ( ω ) at T ≫ T s and T ≪ T s in Fig. 1.We turn now to the evaluation of the total relaxationrate, Eq. (7). There are two contributions to the integralover ω in (7): Γ q = Γ ∞ + e Γ q . (19)The first contribution, Γ ∞ , comes from the high-energy“tail” of σ q ( ω ), see Eqs. (10) and (11). This contribution (cid:99) (cid:99)
50 50 (cid:110) ( T s / (cid:122) q ) / (cid:125) q ( (cid:4) ) (cid:4) / (cid:4) s T À T s T ¿ T s FIG. 1: The differential inelastic scattering rate σ q ( ω ) atdifferent temperatures (only | ω | ≪ ξ q domain is shown). Thetwo plots correspond to Eqs. (14) and (18), evaluated at T = 4 T s and T = T s /
4, respectively, with T s = T / is independent of q and is given by [18]Γ ∞ = α T g √ . (20)Note that, unlike in higher dimensions, Γ ∞ depends ontemperature via g ( T ), see the discussion above.The second contribution in Eq. (19), e Γ q ∝ ξ − / q ,comes from the processes with a small energy transfer | ω | . max { T, ω s } . Using Eq. (16), we find e Γ q = α T (cid:18) T s ξ q (cid:19) / (cid:18) T T s (cid:19) / T s T , T s ≪ T ≪ T . (21)At lower temperatures we obtain, with the help ofEq. (18), e Γ q = α T (cid:18) T s ξ q (cid:19) / (cid:18) T T s (cid:19) / (cid:18) TT s (cid:19) , T s T ≪ T ≪ T s . (22)Comparison with Eq. (20) shows that for not too largeenergies, ξ q ≪ T s ( T /T s ) , the small momentum transfercontribution e Γ q dominates the relaxation rate (19) in abroad temperature interval T s (cid:18) ξ q T s (cid:19) / (cid:18) T s T (cid:19) / ≪ T ≪ T s (cid:18) T s ξ q (cid:19) / (cid:18) T T s (cid:19) / , (23)which includes T = T s . At some temperature T max ∼ T s within this interval, the relaxation rate reaches its peakvalue Γ max = Γ q ( T max ), see Fig. 2. By extrapolatingthe asymptotes (21) and (22) to the region T ∼ T s andfinding their intersection, we estimate T max ≈ . T s , andΓ max ≈ . α T ( T s /ξ q ) / ( T /T s ) / . (24)The actual values of T max and Γ max may differ fromthe above estimates only by numerical factors; findingthese values would require a systematic description ofthe crossover regime T ∼ T s .We now discuss briefly the feasibility of observing re-laxation by inelastic collisions in a system of cold atomsconfined in a cylindrical trap. In this case the effective T (cid:99) (cid:20) T max (cid:99) max (cid:99) q ( T ) FIG. 2: Sketch of the temperature dependence of the totalinelastic relaxation rate. The dependence is nonmonotonic,with a maximum at T max ∼ T s The dashed lines indicate thehigh- and the low-temperature asymptotes Eqs. (19)-(22).
Hamiltonian (1)-(3) can be derived explicitly, by projec-tion onto the lowest subband of transverse quantization.For a model in which the interaction in 3D is describedby a pseudopotential V D ( r ) = 4 π ( a/m ) δ ( r ), where a isthe s -wave scattering length [3], and with the amplitudeof radial zero-point motion a r = ( mω r ) − / ≫ a (here ω r is the trap frequency), one finds [9, 13, 20] γ = 2 a/na r , α = 18 ln(4 / a/a r ) . (25)The main limitation arises due to 3-body recombina-tion processes [21], absent in our model. The correspond-ing rate is Γ R = βn g /a r [21], where g = h : ρ : i /n .Using Eqs. (20) and (25), we find [18]Γ ∞ / Γ R = ηg /g , η = 10 . a / ( mβ ) . (26)For Rb ( a = 5 . β = 3 × − cm / s [22]), we have η ≈
20. For a weak to a moderately strong interaction, γ .
1, the ratio g /g in Eq. (26) is of the order of 1 atall T , and Γ ∞ / Γ R ∼ γ ≪ T s /T ≪ T ≪ T , see Eqs.(21) and (22), disappears for large γ ). For the peak valueof Γ q [see Eq. (24)], we findΓ max / Γ R ∼ . η ( T s /ξ q ) / γ − / , (27)which for a fixed ratio ξ q /T s diverges in the limit γ → q is reached at T ∼ T s . The con-dition of the observability of the inelastic relaxation,Γ max ≫ Γ R , and the condition for the high-energy quasiparticle to be outside the quasicondensate yet wellwithin the lowest subband of transverse quantization, T s ≪ ξ q ≪ ω r , can be satisfied simultaneously. Forexample, for Rb the trap frequency ω r / π = 15 kHzand concentration n = 7 µ m − correspond to γ = 0 . T s = 120 nK. For ξ q /T s = ω r /ξ q = 2 .
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