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Universal off-diagonal long-range-order behavior for a trapped Tonks-Girardeau gas
Abstract
The scaling of the largest eigenvalue λ0 of the one-body density matrix of a system with respect to its particle number N defines an exponent C and a coefficient B via the asymptotic relation λ0∼BNC. The case C=1 corresponds to off-diagonal long-range order. For a one-dimensional homogeneous Tonks-Girardeau gas, a well-known result also confirmed by bosonization gives instead C=1/2. Here we investigate the inhomogeneous case, initially addressing the behavior of C in the presence of a general external trapping potential V. We argue that the value C=1/2 characterizes the hard-core system independently of the nature of the potential V. We then define the exponents γ and β, which describe the scaling of the peak of the momentum distribution with N and the natural orbital corresponding to λ0, respectively, and we derive the scaling relation γ+2β=C. Taking as a specific case the power-law potential V(x)∝x2n, we give analytical formulas for γ and β as functions of n. Analytical predictions for the coefficient B are also obtained. These formulas are derived by exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.
... [7,31]. Such studies were first extended to the ground state of 1D HCBs in harmonic traps [32], and more recently to generic inhomogeneous ground-state configurations [33,34] using an inhomogeneous Luttinger liquid approach [35][36][37]. Only recently has it become feasible to derive fully analytical expressions for the equal-time one-particle density matrix (OPDM) in generic potentials when the system is out of equilibrium [38], thanks to a novel quantum hydrodynamic approach. ...
... However, in a 1D with charge conservation, C is always strictly smaller than one because of the Mermin-Wagner theorem. Nevertheless, as long as C is close enough to one, it is customary to say that the system is a quasicondensate-a phase which in modern literature extends all the way up to C = 1/2 [14,32,34,63,83]. In contrast, in non-interacting fermion systems, the fermion OPDM would give C = 0 because of Pauli exclusion and, expectedly, there is no quasicondensate. ...
... i.e. C = 1/2, as previously noted in [14,32,34,63,83]. We have confirmed this scaling numerically using our formula for the OPDM, see figure 7. We find that, for fixed ratio t/N, the largest eigenvalue scales as λ 0 A √ N R + B for large N R , with an amplitude A that saturates for large t/N. ...
Quasicondensation in one dimension is known to occur for equilibrium systems of hard-core bosons (HCBs) at zero temperature. This phenomenon arises due to the off-diagonal long-range order in the ground state, characterized by a power-law decay of the one-particle density matrix g1(x,y)∼|x−y|−1/2—a well-known outcome of Luttinger liquid theory. Remarkably, HCBs, when allowed to freely expand from an initial product state (i.e. characterized by initial zero correlation), exhibit quasicondensation and demonstrate the emergence of off-diagonal long-range order during nonequilibrium dynamics. This phenomenon has been substantiated by numerical and experimental investigations in the early 2000s. In this work, we revisit the dynamical quasicondensation of HCBs, providing a fully analytical treatment of the issue. In particular, we derive an exact asymptotic formula for the equal-time one-particle density matrix by borrowing ideas from the framework of quantum Generalized Hydrodynamics. Our findings elucidate the phenomenology of quasicondensation and of dynamical fermionization occurring at different stages of the time evolution, as well as the crossover between the two.
... In order to distinguish the superfluid from the pinned phase and map out the phase diagram of the system, we calculate the coherence (also known as the condensate fraction) of the immersed component, C = (max n λ n )/N . It characterizes the off-diagonal long-range order [48][49][50] and it is defined in terms of the largest eigenvalue λ n of the reduced single-particle density matrix (RSPDM), obtained from ...
... This is a consequence of the RSPDM becoming mixed when interactions are finite which therefore reduces the coherence of the state and signifies the presence of quantum correlations between the particles. In fact, it is worth noting that in the Tonks-Girardeau limit λ 0 ∼ N as g → ∞ [48,51] which still characterizes a superfluidlike phase that possesses some long range order. The coherence in the TG limit therefore scales as C → 1/ N , and while this is distinct from a fully incoherent fermionic state with C → 1/N , this is no longer true in the thermodynamic limit where both values vanish. ...
We study the recently introduced self-pinning transition [Phys. Rev. Lett. 128, 053401 (2022)] in a quasi-one-dimensional two-component quantum gas in the case where the component immersed into the Bose-Einstein condensate has a finite intraspecies interaction strength. As a result of the matter-wave backaction, the fermionization in the limit of infinite intraspecies repulsion occurs via a first-order phase transition to the self-pinned state, which is in contrast to the asymptotic behavior in static trapping potentials. The system also exhibits an additional superfluid state for the immersed component if the interspecies interaction is able to overcome the intraspecies repulsion. We approximate the superfluid state in an analytical model and derive an expression for the phase transition line that coincides with well-known phase separation criteria in binary Bose systems. The full phase diagram of the system is mapped out numerically for the case of two and three atoms in the immersed component.
... In order to distinguish the superfluid from the pinned phase and map out the phase diagram of the system, we calculate the coherence C = (max n λ n )/N of the immersed component. It characterizes the off-diagonal long-range order [44][45][46] and it is defined in terms of the largest eigenvalue λ n of the reduced single-particle density matrix (RSPDM), obtained from ...
... This is a consequence of the RSPDM becoming mixed when interactions are finite which therefore reduces the coherence of the state and signifies the presence of quantum correlations between the particles. In fact, it is worth noting that in the Tonks-Girardeau limit λ 0 ∼ √ N as g → ∞ [44,47] which still characterizes a superfluid-like phase that possesses some long range order. The coherence in the TG limit therefore scales as C → 1/ √ N , and while this is distinct from a fully incoherent fermionic state with C → 1/N , this is no longer true in the thermodynamic limit where both values vanish. ...
We study the recently introduced self-pinning transition [Phys. Rev. Lett. 128, 053401 (2022)] in a quasi-one-dimensional two-component quantum gas in the case where the component immersed into the Bose-Einstein condensate has a finite intraspecies interaction strength. As a result of the matter-wave backaction, the fermionization in the limit of infinite intraspecies repulsion occurs via a first-order phase transition to the self-pinned state, which is in contrast to the asymptotic behavior in static trapping potentials. The system also exhibits an additional superfluid state for the immersed component if the interspecies interaction is able to overcome the intraspecies repulsion. We present an analytical model that includes the superfluid state and derive an approximation to the transition line in the phase diagram. The full phase diagram of the system is mapped out numerically for the case of N=2 and N=3 atoms in the immersed component.
... This regime has been experimentally achieved with ultracold atoms [3-6] allowing to study correlation and many-body effects [7][8][9][10][11][12]. Thanks to the possibility of describing the TG manybody wavefunction by an exact solution, several facets have been deeply investigated: one-body density matrix [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], momentum distribution [16,23,25,26,[31][32][33][34][35][36][37][38], and non-equilibrium properties [35,36,[39][40][41][42][43][44][45][46]. ...
... This regime has been experimentally achieved with ultracold atoms [3][4][5][6] allowing to study correlation and many-body effects [7][8][9][10][11][12]. Thanks to the possibility of describing the TG manybody wavefunction by an exact solution, several facets have been deeply investigated: one-body density matrix [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], momentum distribution [16,23,25,26,[31][32][33][34][35][36][37][38], and non-equilibrium properties [35,36,[39][40][41][42][43][44][45][46]. ...
The single-particle spectral function of a strongly correlated system is an essential ingredient to describe its dynamics and transport properties. We develop a general method to calculate the exact spectral function of a strongly interacting one-dimensional Bose gas in the Tonks-Girardeau regime, valid for any type of confining potential, and apply it to bosons on a lattice to obtain the full spectral function, at all energy and momentum scales. We find that it displays three main singularity lines. The first two can be identified as the analogs of Lieb-I and Lieb-II modes of a uniform fluid; the third one, instead, is specifically due to the presence of the lattice. We show that the spectral function displays a power-law behaviour close to the Lieb-I and Lieb-II singularities, as predicted by the non-linear Luttinger liquid description, and obtain the exact exponents. In particular, the Lieb-II mode shows a divergence in the spectral function, differently from what happens in the dynamical structure factor, thus providing a route to probe it in experiments with ultracold atoms.
... This model, even though it describes the physics of strongly interacting particles, is solvable due to the existence of a Bose-Fermi mapping theorem [26,27], which also implies that the fermionic counterpart is exactly solvable. Since the coherences in the bosonic TG case are √ N times larger than in the fermionic case [28][29][30], where N is the number of particles, these models offer insight into two interesting limits. ...
... In general, the scaling of θ 0 is determined by the large distance behavior of the RSPDM and is therefore a good quantifier of the presence of off-diagonal long-range order [29]. Therefore, in the following, we will adhere to the conventional use of θ 0 to quantify the coherence of the TG gas [30,57,58]. 3. (a) Instantaneous speed v(t ) after a quench from λ i = 1 to λ f = 8 for N = 50 particles, for the TG gas (green dots) and Fermi gas (red dashed line). ...
We discuss the effects of many-body coherence on the speed of evolution of ultracold atomic gases and the relation to quantum speed limits. Our approach is focused on two related systems, spinless fermions and the bosonic Tonks-Girardeau gas, which possess equivalent density dynamics but very different coherence properties. To illustrate the effect of the coherence on the dynamics, we consider squeezing an anharmonic potential which confines the particles and find that the speed of the evolution exhibits subtle but fundamental differences between the two systems. Furthermore, we explore the difference in the driven dynamics by implementing a shortcut to adiabaticity designed to reduce spurious excitations. We show that collisions between the strongly interacting bosons can lead to changes in the coherence which result in different evolution speeds and therefore different fidelities of the final states.
... This model, even though it describes the physics of strongly interacting particles, is solvable due to the existence of a Bose-Fermi mapping theorem [26,27], which also implies that the fermionic counterpart is exactly solvable. Since the coherences in the bosonic TG case are √ N times larger than in the fermionic case [28][29][30], where N is the number of particles, these models offer insight into two interesting limits. ...
... In general the scaling of θ 0 is determined by the large distance behaviour of the RSPDM and is therefore a good quantifier of the presence of off-diagonal long range order [29]. Therefore, in the following we will adhere to the conventional use of θ 0 to quantify the coherence of the TG gas [30,56,57]. ...
We discuss the effects of many-body coherence on the quantum speed limit in ultracold atomic gases. Our approach is focused on two related systems, spinless fermions and the bosonic Tonks-Girardeau gas, which possess equivalent density dynamics but very different coherence properties. To illustrate the effect of the coherence on the dynamics we consider squeezing an anharmonic potential which confines the particles and find that the quantum speed limit exhibits subtle, but fundamental, differences between the atomic species. Furthermore, we explore the difference in the driven dynamics by implementing a shortcut to adiabaticity designed to reduce spurious excitations. We show that collisions between the strongly interacting bosons can lead to changes in the coherence which results in larger speed limits.
... The problem we will treat is the one of a gas of hard-core bosons, also known as the TonksGirardeau gas, in a time-dependent harmonic potential V (x, t). This problem is well-known to be exactly solvable [34][35][36], and our goal is to use it to illustrate our approach, which extends recent works by others and by ourselves and our collaborators [1,[37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. We will see that we recover some known results about equal-time correlations, and we uncover new ones, including results for correlation functions at different time. ...
... Refs. [1,[37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]), by considering a truly dynamical situation: a breathing gas of hard core bosons at zero temperature. In particular, we have found new formulas for the 1/N ∼ ħ h → 0 asymptotics of 2n-point functions of boson creation/annihilation operators, and also for fermionic observables for which we provided numerical checks. ...
The recent results of [J. Dubail, J.-M. Stéphan, J. Viti, P. Calabrese, Scipost Phys. 2, 002 (2017)], which aim at providing access to large scale correlation functions of inhomogeneous critical one-dimensional quantum systems —e.g. a gas of hard core bosons in a trapping potential— are extended to a dynamical situation: a breathing gas in a time-dependent harmonic trap. Hard core bosons in a time-dependent harmonic potential are well known to be exactly solvable, and can thus be used as a benchmark for the approach. An extensive discussion of the approach and of its relation with classical and quantum hydrodynamics in one dimension is given, and new formulas for correlation functions, not easily obtainable by other methods, are derived. In particular, a remarkable formula for the large scale asymptotics of the bosonic \boldsymbol{n}𝐧 -particle function \boldsymbol{\left} is obtained. Numerical checks of the approach are carried out for the fermionic two-point function —easier to access numerically in the microscopic model than the bosonic one— with perfect agreement.
... In the TG gas, the repulsive boson is a reminiscent of non-interacting fermions due to the dynamic interaction, allowing a quite interesting and subtle correspondence between them. This Bose-Fermi map [13] offers a reliable access to the correlated properties of TG gases either of continuum or on a lattice, such as the DSF, one-body density matrix, one-particle dynamical correlation function, and SF etc [14][15][16][17][18][19][20]. A little bit away from the TG limit, the generalization of this map works for strongly interacting Bose gases as well, the DSF of ground state and finite temperatures of the Lieb-Liniger model are derived based on a pseudopotential Hamiltonian [21,22]. ...
The dynamical structure factor (DSF) represents a measure of dynamical density-density correlations in a quantum many-body system.
Due to the complexity of many-body correlations and quantum fluctuations in a system of an infinitely large Hilbert space, such kind of dynamical correlations often impose a big theoretical challenge. For one dimensional (1D) quantum many-body systems, qualitative predictions of dynamical response functions are usually carried out by using the Tomonaga-Luttinger liquid (TLL) theory. In this scenario, a precise evaluation of the DSF for a 1D quantum system with arbitrary interaction strength remains a formidable task. In this paper, we use the form factor approach based on algebraic Bethe ansatz theory to calculate precisely the DSF of Lieb-Liniger model with an arbitrary interaction strength at a large scale of particle number. We find that the DSF for a system as large as 2000 particles enables us to depict precisely its line-shape from which the power-law singularity with corresponding exponents in the vicinities of spectral thresholds naturally emerge. It should be noted that, the advantage of our algorithm promises an access to the threshold behavior of dynamical correlation functions, further confirming the validity of nonlinear TLL theory besides Kitanine {\em et. al.} 2012 {\em J. Stat. Mech.} P09001. Finally we discuss a comparison of results with the results from the ABACUS method by J.-S. Caux 2009 {\em J. Math. Phys.} {\bf 50} 095214 as well as from the strongly coupling expansion by Brand and Cherny 2005 {\em Phys. Rev.} A {\bf 72} 033619.
... where a 0 is an effective diameter of particles. Studying the properties of reduced density matrices [73][74][75][76] one concludes that the order indices are ...
The review is devoted to two important quantities characterizing many-body systems, order indices and the measure of entanglement production. Order indices describe the type of order distinguishing statistical systems. Contrary to the order parameters characterizing systems in the thermodynamic limit and describing long-range order, the order indices are applicable to finite systems and classify all types of orders, including long-range, mid-range, and short-range orders. The measure of entanglement production quantifies the amount of entanglement produced in a many-partite system by a quantum operation. Despite that the notions of order indices and entanglement production seem to be quite different, there is an intimate relation between them, which is emphasized in the review.
... where a 0 is an effective diameter of particles. Studying the properties of reduced density matrices [73][74][75][76] one concludes that the order indices are ...
The review is devoted to two important quantities characterizing many-body systems, order indices and the measure of entanglement production. Order indices describe the type of order distinguishing statistical systems. Contrary to the order parameters characterizing systems in the thermodynamic limit and describing long-range order, the order indices are applicable to finite systems and classify all types of orders, including long-range, mid-range, and short-range orders. The measure of entanglement production quantifies the amount of entanglement produced in a many-partite system by a quantum operation. Despite that the notions of order indices and entanglement production seem to be quite different, there is an intimate relation between them, which is emphasized in the review.
... Dans cette thèse, on se propose de développer plus en avant la théorie du liquide de Luttinger inhomogène à partir des travaux récents de Dubail et al. [50] qui visent à décrire les systèmes quantiques inhomogènes à une dimension par des théories conformes (CFT) dans un espace courbe. Ces travaux ont depuis donné lieu à une série d'articles par des collègues et collaborateurs [50][51][52][53][54][55][56][57][58][59][60][61][62][63]. Tous les résultats présentés ici sont essentiellement extraits de trois articles publiés au cours des trois dernières années [1][2][3]. ...
Si les systèmes quantiques à une dimension ont longtemps été vus comme de simples modèle-jouets, bon nombre sont à présent réalisés dans les expériences d’atomes ultra-froids. Dans ces expériences, le potentiel de confinement du gaz induit nécessairement une inhomogénéité spatiale. Cette inhomogénéité brise l'invariance par translation qui joue un rôle clé dans les solutions analytiques, notamment celle de l'Ansatz de Bethe. On propose dans cette thèse de développer une théorie des champs effective à même de caractériser ces gaz quantiques inhomogènes, en généralisant la théorie du liquide de Luttinger. Dans ces conditions la métrique de l'action effective est courbe. Sous une hypothèse de séparation des échelles, les paramètres de l'action peuvent néanmoins être fixés par les solutions de l'Ansatz de Bethe. Le problème peut alors se ramener au cas d'un espace plat en faisant appel aux théories conformes. On est ainsi amené à résoudre le champ libre gaussien inhomogène, qui donne accès à toutes les fonctions de corrélations du modèle considéré. Dans cette thèse, on s'intéresse plus particulièrement au modèle de Lieb-Liniger. Les résultats obtenus sont comparés au système simulé par DMRG.
... This is a semiclassical approximation in which the potential couples directly to the energy density, allowing for an easy application of our methods. Similar methods were also used in [58] to study the density matrix of a 1D Tonks-Girardeau gas. Once we have this metric, we are implicitly using this approximation, and are no longer dealing directly with the nonrelativistic fermionic system. ...
A holographic dual description of inhomogeneous systems is discussed. Notably, finite temperature results for the entanglement entropy in both the rainbow chain and the sine-square deformation model are obtained holographically by choosing appropriate foliations of the Bañados– Teitelboim–Zanelli spacetime. Other inhomogeneous theories are also discussed. The entanglement entropy results are verified numerically, indicating that a wide variety of inhomogeneous field theory phenomenology can be seen in different slicings of asymptotically AdS3 spacetimes.
In one-dimensional (1D) quantum gases, the momentum distribution (MD) of the atoms is a standard experimental observable, routinely measured in various experimental setups. The MD is sensitive to correlations, and it is notoriously hard to compute theoretically for large numbers of atoms N, which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large N, at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix 〈Ψ̂†(x)Ψ̂(x′)〉 of the out-of-equilibrium TG gas in the N→∞ limit, valid on distances |x−x′| much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well, which mimics a “quantum Newton cradle” setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.
The single-particle spectral function of a strongly correlated system is an essential ingredient to describe its dynamics and transport properties. We develop a method to evaluate exactly the spectral function for a gas of one-dimensional bosons with infinitely strong repulsions valid for any type of external confinement. Focusing on the case of a lattice confinement, we find that the spectral function displays three main singularity lines. One of them is due uniquely to lattice effects, while the two others correspond to the Lieb-I and Lieb-II modes occurring in a uniform fluid. Differently from the dynamical structure factor, in the spectral function the Lieb-II mode shows a divergence, thus providing a route to probe such mode in experiments with ultracold atoms.
Characterizing the scaling with the total particle number (N) of the largest eigenvalue of the one-body density matrix (λ0) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting λ0∼NC0, then C0=1 corresponds in ODLRO. The intermediate case, 0<C0<1, corresponds in translational invariant systems to the power-law decaying of (nonconnected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in C0 (and in the corresponding quantities Ck≠0 for excited natural orbitals) exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions in presence of short-range repulsive potentials. We show that Ck≠0=0 in the thermodynamic limit. In one dimension it is C0=0 for nonvanishing temperature, while in three dimensions it is C0=1 (C0=0) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to D=2, studying the XY and the Villain models, and the weakly interacting Bose gas. The universal value of C0 near the Berezinskii-Kosterlitz-Thouless temperature TBKT is 7/8. The dependence of C0 on temperatures between T=0 (at which C0=1) and TBKT is studied in the different models. An estimate for the (nonperturbative) parameter ξ entering the equation of state of the two-dimensional Bose gases is obtained using low-temperature expansions and compared with the Monte Carlo result. We finally discuss a “double jump” behavior for C0, and correspondingly of the anomalous dimension η, right below TBKT in the limit of vanishing interactions.
We provide a thorough characterisation of the zero-temperature one-particle density matrix of trapped interacting anyonic gases in one dimension, exploiting recent advances in the field theory description of spatially inhomogeneous quantum systems. We first revisit homogeneous anyonic gases with point-wise interactions. In the harmonic Luttinger liquid expansion of the one-particle density matrix for finite interaction strength, the non-universal field amplitudes were not yet known. We extract them from the Bethe Ansatz formula for the field form factors, providing an exact asymptotic expansion of this correlation function, thus extending the available results in the Tonks-Girardeau limit. Next, we analyse trapped gases with non-trivial density profiles. By applying recent analytic and numerical techniques for inhomogeneous Luttinger liquids,we provide exact expansions for the one-particle density matrix. We present our results for different confining potentials, highlighting the main differences with respect to bosonic gases.
Conformal field theories in curved backgrounds have been used to describe inhomogeneous one-dimensional systems, such as quantum gases in trapping potentials and non-equilibrium spin chains. This approach provided, in a elegant and simple fashion, non-trivial analytic predictions for quantities, such as the entanglement entropy, that are not accessible through other methods. Here, we generalise this approach to low-lying excited states, focusing on the entanglement and relative entropies in an inhomogeneous free-fermionic system. Our most important finding is that the universal scaling function characterising these entanglement measurements is the same as the one for homogeneous systems, but expressed in terms of a different variable. This new scaling variable is a non-trivial function of the subsystem length and system's inhomogeneity that is easily written in terms of the curved metric. We test our predictions against exact numerical calculations in the free Fermi gas trapped by a harmonic potential, finding perfect agreement.
A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue of the one-body-density matrix scales as , where N is the total number of particles. Putting to define the scaling exponent , then corresponds to ODLRO and to the single-particle occupation of the density matrix orbitals. When , can be used to quantify deviations from ODLRO. In this paper we study the exponent in a variety of one-dimensional bosonic and anyonic quantum systems. For the 1D Lieb-Liniger Bose gas we find that for small interactions is close to 1, implying a mesoscopic condensation, i.e. a value of the "condensate" fraction appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms). 1D anyons provide the possibility to fully interpolate between and 0. The behaviour of for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.
Motivated by the calculation of correlation functions in inhomogeneous
one-dimensional (1d) quantum systems, the 2d Inhomogeneous Gaussian Free Field
(IGFF) is studied and solved. The IGFF is defined in a domain equipped with a conformal class of metrics and with a
real positive coupling constant by the
action . All
correlations functions of the IGFF are expressible in terms of the Green's
functions of generalized Poisson operators that are familiar from 2d
electrostatics in media with spatially varying dielectric constants.
This formalism is then applied to the study of ground state correlations of
the Lieb-Liniger gas trapped in an external potential V(x). Relations with
previous works on inhomogeneous Luttinger liquids are discussed. The main
innovation here is in the identification of local observables in
the microscopic model with their field theory counterparts , etc., which involve non-universal coefficients that
themselves depend on position --- a fact that, to the best of our knowledge,
was overlooked in previous works on correlation functions of inhomogeneous
Luttinger liquids ---, and that can be calculated thanks to Bethe Ansatz form
factors formulae available for the homogeneous Lieb-Liniger model. Combining
those position-dependent coefficients with the correlation functions of the
IGFF, ground state correlation functions of the trapped gas are obtained.
Numerical checks from DMRG are provided for density-density correlations and
for the one-particle density matrix, showing excellent agreement.
We study Tan’s contact, i.e. the coefficient of the high-momentum tails of the momentum distribution at leading order, for an interacting one-dimensional Bose gas subjected to a harmonic confinement. Using a strong-coupling systematic expansion of the ground-state energy of the homogeneous system stemming from the Bethe-Ansatz solution, together with the local-density approximation, we obtain the strong-coupling expansion for Tan’s contact of the harmonically trapped gas. Also, we use a very accurate conjecture for the ground-state energy of the homogeneous system to obtain an approximate expression for Tan’s contact for arbitrary interaction strength, thus estimating the accuracy of the strong-coupling expansion. Our results are relevant for ongoing experiments with ultracold atomic gases.
We explore the ground-state properties of cold atomic gases focusing on the cases of noninteracting fermions and hard-core (Tonks-Girardeau) bosons, trapped by the combination of two potentials (bichromatic lattice) with incommensurate periods. For such systems, two limiting cases have been thoroughly established. In the tight-binding limit, the single-particle states in the lowest occupied band show a localization transition, as the strength of the second potential is increased above a certain threshold. In the continuous limit, when the tight-binding approximation does not hold, a mobility edge is found, instead, whose position in energy depends upon the strength of the second potential. Here, we study how the crossover from the discrete to the continuum behavior occurs, and prove that signatures of the localization transition and mobility edge clearly appear in the generic many-body properties of the systems. Specifically, we evaluate the momentum distribution, which is a routinely measured quantity in experiments with cold atoms, and demonstrate that, even in the presence of strong boson-boson interactions (infinite in the Tonks-Girardeau limit), the single-particle mobility edge can be observed in the ground-state properties.
The one-particle density matrix of the one-dimensional Tonks-Girardeau gas
with inhomogeneous density profile is calculated, thanks to a recent
observation that relates this system to a two-dimensional conformal field
theory in curved space. The result is asymptotically exact in the limit of
large particle density and small density variation, and holds for arbitrary
trapping potentials. In the particular case of a harmonic trap, we recover a
formula obtained by Forrester et al. [Phys. Rev. A 67, 043607 (2003)] from a
different method.
This monograph introduces the reader to basic notions of integrable techniques for one-dimensional quantum systems. In a pedagogical way, a few examples of exactly solvable models are worked out to go from the coordinate approach to the Algebraic Bethe Ansatz, with some discussion on the finite temperature thermodynamics. The aim is to provide the instruments to approach more advanced books or to allow for a critical reading of research articles and the extraction of useful information from them. We describe the solution of the anisotropic XY spin chain; of the Lieb-Liniger model of bosons with contact interaction at zero and finite temperature; and of the XXZ spin chain, first in the coordinate and then in the algebraic approach. To establish the connection between the latter and the solution of two dimensional classical models, we also introduce and solve the 6-vertex model. Finally, the low energy physics of these integrable models is mapped into the corresponding conformal field theory. Through its style and the choice of topics, this book tries to touch all fundamental ideas behind integrability and is meant for students and researchers interested either in an introduction to later delve in the advance aspects of Bethe Ansatz or in an overview of the topic for broadening their culture.
Conformal field theory (CFT) has been extremely successful in describing
large-scale universal effects in one-dimensional (1D) systems at quantum
critical points. Unfortunately, its applicability in condensed matter physics
has been limited to situations in which the bulk is uniform because CFT
describes low-energy excitations around some energy scale, taken to be constant
throughout the system. However, in many experimental contexts, such as quantum
gases in trapping potentials and in several out-of-equilibrium situations,
systems are strongly inhomogeneous. We show here that the powerful CFT methods
can be extended to deal with such 1D situations, providing a few concrete
examples for non-interacting Fermi gases. The system's inhomogeneity enters the
field theory action through parameters that vary with position; in particular,
the metric itself varies, resulting in a CFT in curved space. This approach
allows us to derive exact formulas for entanglement entropies which were not
known by other means.
Using anyon-fermion mapping method, we investigate the ground state properties of hard-core anyons confined in a one-dimensional harmonic trap. The concise analytical formula of the reduced one-body density matrix are obtained. Basing on the formula, we evaluated the momentum distribution, the natural orbitals and their occupation distributions for different statistical parameters. The occupation and occupation fraction of the lowest natural orbital versus anyon number are also displayed. It is shown that the ground state properties of anyons interplay between Bosons and Fermions continuously. We can expect that the hard-core anyons of larger statistical parameter exhibit the similar properties to the hard-core Bosons although anyon system satisfy specific fractional statistics.
We consider the Tonks-Girardeau gas subject to a random external potential.
If the disorder is such that the underlying one-particle Hamiltonian displays
localization (which is known to be generically the case), we show that there is
exponential decay of correlations in the many-body eigenstates. Moreover, there
is no Bose-Einstein condensation and no superfluidity, even at zero
temperature.
We investigate the superfluid-insulator transition of one-dimensional
interacting Bosons in both deep and shallow periodic potentials. We compare a
theoretical analysis based on Monte-Carlo simulations in continuum space and
Luttinger liquid approach with experiments on ultracold atoms with tunable
interactions and optical lattice depth. Experiments and theory are in excellent
agreement. It provides a quantitative determination of the critical parameter
for the Mott transition and defines the regime of validity of widely used
approximate models, namely the Bose-Hubbard and sine-Gordon models
We study the dynamical depinning following a sudden turn off of an optical
lattice for a gas of impenetrable bosons in a tight atomic waveguide. We use a
Bose-Fermi mapping to infer the exact quantum dynamical evolution. At long
times, in the thermodynamic limit, we observe the approach to a non-equilibrium
steady state, characterized by the absence of quasi-long-range order and a
reduced visibility in the momentum distribution. Similar features are found in
a finite-size system, where we obtain a quasi-steady state at sufficiently long
times before the revival time.
We provide evidence in support of a recent proposal by Astrakharchik et al for the existence
of a super Tonks–Girardeau gas-like state in the attractive interaction regime of
quasi-one-dimensional Bose gases. We show that the super Tonks–Girardeau gas-like state
corresponds to a highly excited Bethe state in the integrable interacting Bose gas for which
the bosons acquire hard-core behaviour. The gas-like state properties vary smoothly
throughout a wide range from strong repulsion to strong attraction. There is an additional
stable gas-like phase in this regime in which the bosons form two-body bound states
behaving like hard-core bosons.
We study the non-equilibrium dynamics of a Tonks-Girardeau gas released from
a parabolic trap to a circle. We present the exact analytic solution of the
many body dynamics and prove that, for large times and in a properly defined
thermodynamic limit, the reduced density matrix of any finite subsystem
converges to a generalized Gibbs ensemble. The equilibration mechanism is
expected to be the same for all one-dimensional systems.
We determine the finite-temperature momentum distribution of a strongly interacting 1D Bose gas in the Tonks-Girardeau (impenetrable-boson) limit under harmonic confinement and explore its universal properties associated to the scale invariance of the model. We show that, at difference from the unitary Fermi gas in three dimensions, the weight of its large-momentum tails-given by Tan's contact-increases with temperature and calculate the high-temperature universal second contact coefficient using a virial expansion.
We consider a 1D Bose gas with attractive interactions in an
out-of-equilibrium highly excited state containing no bound states. We show
that relaxation processes in the gas are suppressed, making the system
metastable on long timescales. We compute dynamical correlation functions,
revealing the structure of excitations, an enhancement of umklapp correlations
and new branches due to intermediate bound states. These features give a clear
indication of the attractive regime and can be probed experimentally. We
observe that, despite its out-of-equilibrium nature, the system displays
critical behaviour: correlation functions are characterised by asymptotic
power-law decay described by the Luttinger liquid framework.
We study the pinning quantum phase transition in a Tonks-Girardeau gas, both
in equilibrium and out-of-equilibrium, using the ground state fidelity and the
Loschmidt echo as diagnostic tools. The ground state fidelity (GSF) will have a
dramatic decrease when the atomic density approaches the commensurate density
of one particle per lattice well. This decrease is a signature of the pinning
transition from the Tonks to the Mott insulating phase. We study the
applicability of the fidelity for diagnosing the pinning transition in
experimentally realistic scenarios. Our results are in excellent agreement with
recent experimental work. In addition, we explore the out of equilibrium
dynamics of the gas following a sudden quench with a lattice potential. We find
all properties of the ground state fidelity are reflected in the Loschmidt echo
dynamics i.e., in the non equilibrium dynamics of the Tonks-Girardeau gas
initiated by a sudden quench of the lattice potential.
This book gives a self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations. Parts and chapter titles are as follows: fundamentals - ordinary differential equations, difference equations; local analysis - approximate solution of linear differential equations, approximate solution of nonlinear differential equations, approximate solution of difference equations, asymptotic expansion of integrals; perturbation methods - perturbation series, summation series; and global analysis - boundary layer theory, WKB theory, multiple-scale analysis. An appendix of useful formulas is included. 147 figures, 43 tables. (RWR)
We review the physics of one-dimensional interacting bosonic systems.
Beginning with results from exactly solvable models and computational
approaches, we introduce the concept of bosonic Tomonaga-Luttinger Liquids
relevant for one-dimension, and compare it with Bose-Einstein condensates
existing in dimensions higher than one. We discuss the effects of various
perturbations on the Tomonaga-Luttinger liquid state as well as extensions to
multicomponent and out of equilibrium situations. Finally, we review the
experimental systems that can be described in terms of models of interacting
bosons in one dimension.
We study the local correlations in the super Tonks-Girardeau gas, a highly
excited, strongly correlated state obtained in quasi one-dimensional Bose gases
by tuning the scattering length to large negative values using a
confinement-induced resonance. Exploiting a connection with a relativistic
field theory, we obtain results for the two-body and three-body local
correlators at zero and finite temperature. At zero temperature our result for
the three-body correlator agrees with the extension of the results of Cheianov
et al. [Phys. Rev. A 73, 051604(R) (2006)], obtained for the ground-state of
the repulsive Lieb-Liniger gas, to the super Tonks-Girardeau state. At finite
temperature we obtain that the three-body correlator has a weak dependence on
the temperature up to the degeneracy temperature. We also find that for
temperatures larger than the degeneracy temperature the values of the
three-body correlator for the super Tonks-Girardeau gas and the corresponding
repulsive Lieb-Liniger gas are rather similar even for relatively small
couplings.
We theoretically demonstrate features of Anderson localization in the
Tonks-Girardeau gas confined in one-dimensional (1D) potentials with controlled
disorder. That is, we investigate the evolution of the single particle density
and correlations of a Tonks-Girardeau wave packet in such disordered
potentials. The wave packet is initially trapped, the trap is suddenly turned
off, and after some time the system evolves into a localized steady state due
to Anderson localization. The density tails of the steady state decay
exponentially, while the coherence in these tails increases. The latter
phenomenon corresponds to the same effect found in incoherent optical solitons.
We study the quantum (zero-temperature) critical behaviors of confined
particle systems described by the one-dimensional (1D) Bose-Hubbard model in
the presence of a confining potential, at the Mott insulator to superfluid
transitions, and within the gapless superfluid phase. Specifically, we consider
the hard-core limit of the model, which allows us to study the effects of the
confining potential by exact and very accurate numerical results. We analyze
the quantum critical behaviors in the large trap-size limit within the
framework of the trap-size scaling (TSS) theory, which introduces a new trap
exponent theta to describe the dependence on the trap size. This study is
relevant for experiments of confined quasi 1D cold atom systems in optical
lattices. At the low-density Mott transition TSS can be shown analytically
within the spinless fermion representation of the hard-core limit. The
trap-size dependence turns out to be more subtle in the other critical regions,
when the corresponding homogeneous system has a nonzero filling f, showing an
infinite number of level crossings of the lowest states when increasing the
trap size. At the n=1 Mott transition this gives rise to a modulated TSS: the
TSS is still controlled by the trap-size exponent theta, but it gets modulated
by periodic functions of the trap size. Modulations of the asymptotic power-law
behavior is also found in the gapless superfluid region, with additional
multiscaling behaviors.
We develop a trap-size scaling theory for trapped particle systems at quantum
transitions. As a theoretical laboratory, we consider a quantum XY chain in an
external transverse field acting as a trap for the spinless fermions of its
quadratic Hamiltonian representation. We discuss trap-size scaling at the Mott
insulator to superfluid transition in the Bose-Hubbard model. We present exact
and accurate numerical results for the XY chain and for the low-density Mott
transition in the hard-core limit of the one-dimensional Bose-Hubbard model.
Our results are relevant for systems of cold atomic gases in optical lattices.
Quantum many-body systems can have phase transitions even at zero temperature; fluctuations arising from Heisenberg's uncertainty principle, as opposed to thermal effects, drive the system from one phase to another. Typically, during the transition the relative strength of two competing terms in the system's Hamiltonian changes across a finite critical value. A well-known example is the Mott-Hubbard quantum phase transition from a superfluid to an insulating phase, which has been observed for weakly interacting bosonic atomic gases. However, for strongly interacting quantum systems confined to lower-dimensional geometry, a novel type of quantum phase transition may be induced and driven by an arbitrarily weak perturbation to the Hamiltonian. Here we observe such an effect--the sine-Gordon quantum phase transition from a superfluid Luttinger liquid to a Mott insulator--in a one-dimensional quantum gas of bosonic caesium atoms with tunable interactions. For sufficiently strong interactions, the transition is induced by adding an arbitrarily weak optical lattice commensurate with the atomic granularity, which leads to immediate pinning of the atoms. We map out the phase diagram and find that our measurements in the strongly interacting regime agree well with a quantum field description based on the exactly solvable sine-Gordon model. We trace the phase boundary all the way to the weakly interacting regime, where we find good agreement with the predictions of the one-dimensional Bose-Hubbard model. Our results open up the experimental study of quantum phase transitions, criticality and transport phenomena beyond Hubbard-type models in the context of ultracold gases.
Ultracold atomic physics offers myriad possibilities to study strongly
correlated many-body systems in lower dimensions. Typically, only ground state
phases are accessible. Using a tunable quantum gas of bosonic cesium atoms, we
realize and control in one dimensional geometry a highly excited quantum phase
that is stabilized in the presence of attractive interactions by maintaining
and strengthening quantum correlations across a confinement-induced resonance.
We diagnose the crossover from repulsive to attractive interactions in terms of
the stiffness and the energy of the system. Our results open up the
experimental study of metastable excited many-body phases with strong
correlations and their dynamical properties.
We study the scaling properties of critical particle systems confined by a potential. Using renormalization-group arguments, we show that their critical behavior can be cast in the form of a trap-size scaling, resembling finite-size scaling theory, with a nontrivial trap critical exponent theta, which describes how the correlation length xi scales with the trap size l, i.e., xi approximately l;{theta} at T_{c}. theta depends on the universality class of the transition, the power law of the confining potential, and on the way it is coupled to the critical modes. We present numerical results for two-dimensional lattice gas (Ising) models with various types of harmonic traps, which support the trap-size scaling scenario.
Strongly correlated quantum systems are among the most intriguing and fundamental systems in physics. One such example is the Tonks-Girardeau gas, proposed about 40 years ago, but until now lacking experimental realization; in such a gas, the repulsive interactions between bosonic particles confined to one dimension dominate the physics of the system. In order to minimize their mutual repulsion, the bosons are prevented from occupying the same position in space. This mimics the Pauli exclusion principle for fermions, causing the bosonic particles to exhibit fermionic properties. However, such bosons do not exhibit completely ideal fermionic (or bosonic) quantum behaviour; for example, this is reflected in their characteristic momentum distribution. Here we report the preparation of a Tonks-Girardeau gas of ultracold rubidium atoms held in a two-dimensional optical lattice formed by two orthogonal standing waves. The addition of a third, shallower lattice potential along the long axis of the quantum gases allows us to enter the Tonks-Girardeau regime by increasing the atoms' effective mass and thereby enhancing the role of interactions. We make a theoretical prediction of the momentum distribution based on an approach in which trapped bosons acquire fermionic properties, finding that it agrees closely with the measured distribution.
We consider a homogeneous 1D Bose gas with contact interactions and a large attractive coupling constant. This system can be realized in tight waveguides by exploiting a confinement induced resonance of the effective 1D scattering amplitude. By using the diffusion Monte Carlo method we show that, for small densities, the gaslike state is well described by a gas of hard rods. The critical density for cluster formation is estimated using the variational Monte Carlo method. The behavior of the correlation functions and of the frequency of the lowest breathing mode for harmonically trapped systems shows that the gas is more strongly correlated than in the Tonks-Girardeau regime.
The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Gordon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians. The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results. The book will be essential reading for all mathematical physicists working in field theory and statistical physics.
This volume introduces the basic concepts of Bose–Einstein condensation and superfluidity. It makes special reference to the physics of ultracold atomic gases; an area in which enormous experimental and theoretical progress has been achieved in the last twenty years. Various theoretical approaches to describing the physics of interacting bosons and of interacting Fermi gases, giving rise to bosonic pairs and hence to condensation, are discussed in detail, both in uniform and harmonically trapped configurations. Special focus is given to the comparison between theory and experiment, concerning various equilibrium, dynamic, thermodynamic, and superfluid properties of these novel systems. The volume also includes discussions of ultracold gases in dimensions, quantum mixtures, and long-range dipolar interactions.
Integral equations form an important class of problems, arising frequently in engineering, and in mathematical and scientific analysis. This textbook provides a readable account of techniques for their numerical solution. The authors devote their attention primarily to efficient techniques using high order approximations, taking particular account of situations where singularities are present. The classes of problems which arise frequently in practice, Fredholm of the first and second kind and eigenvalue problems, are dealt with in depth. Volterra equations, although attractive to treat theoretically, arise less often in practical problems and so have been given less emphasis. Some knowledge of numerical methods and linear algebra is assumed, but the book includes introductory sections on numerical quadrature and function space concepts. This book should serve as a valuable text for final year undergraduate or postgraduate courses, and as an introduction or reference work for practising computational mathematicians, scientists and engineers.
We show that the contact parameter of N harmonically trapped interacting one-dimensional bosons at zero temperature can be analytically and accurately obtained by a simple rescaling of the exact two-boson solution, and that N-body effects can be almost factorized. The small deviations observed between our analytical results and density matrix renormalization group (DMRG) calculations are more pronounced when the interaction energy is maximal (i.e., at intermediate interaction strengths) but they remain bounded by the large-N local-density approximation obtained from the Lieb-Liniger equation of state stemming from the Bethe ansatz. The rescaled two-body solution is so close to the exact ones, that is possible, within a simple expression interpolating the rescaled two-boson result to the local density, to obtain N-boson contact and ground-state energy functions in very good agreement with DMRG calculations. Our results suggest a change of paradigm in the study of interacting quantum systems, giving to the contact parameter a more fundamental role than energy.
We study the superfluid response, the energetic and structural properties of a one-dimensional ultracold Bose gas in an optical lattice of arbitrary strength. We use the Bose-Fermi mapping in the limit of infinitely large repulsive interaction and the diffusion Monte Carlo method in the case of finite interaction. For slightly incommensurate fillings we find a superfluid behavior, which is discussed in terms of vacancies and interstitials. It is shown that both the excitation spectrum and static structure factor are different for the cases of microscopic and macroscopic fractions of defects. This system provides an extremely well-controlled model for studying defect-induced superfluidity.
We study the one-body reduced density matrix of a system of N one-dimensional impenetrable anyons trapped by a harmonic potential. To this purpose we extend two methods developed to tackle related problems, namely the determinant approach and the replica method. While the former is the basis for exact numerical computations at finite N, the latter has the advantage of providing an analytic asymptotic expansion for large N. We show that the first few terms of such expansion are sufficient to reproduce the numerical results to an excellent accuracy even for relatively small N, thus demonstrating the effectiveness of the replica method.
We investigate one-dimensional Bose gas with -interaction in optical
lattices at zero temperature by means of the (exact) diffusion Monte Carlo
algorithm. We compare the results obtained from the fundamental continuous
model, with those obtained from the discrete Bose-Hubbard model, using exact
diagonalization a number of approximate approaches. We map out the complete
phase diagram of the continuous model and determine the regions of
applicability of the lattice (discrete) Bose-Hubbard model. We calculate
various physical quantities characterizing the systems and demonstrate that the
quantum Sine-Gordon model used for shallow lattices is inaccurate.
The mathematical description of B.E. (Bose-Einstein) condensation is generalized so as to be applicable to a system of interacting particles. B.E. condensation is said to be present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity. Some transformations facilitating the practical use of this definition are given. An argument based on first principles is given, indicating that liquid belium II in equilibrium shows B.E. condensation. For absolute zero, the argument is based on properties of the ground-state wave function derived from the assumption that there is no "long-range configurational order." A crude estimate indicates that roughly 8% of the atoms are "condensed" (note that the fraction of condensed particles need not be identified with ρsρ). Conversely, it is shown why one would not expect B.E. condensation in a solid. For finite temperatures Feynman's theory of the lambda-transition is applied: Feynman's approximations are shown to imply that our criterion of B.E. condensation is satisfied below the lambda-transition but not above it.
Girardeau has shown that an exact analytical formula may be given for the ground‐state wave‐function of a system of one‐dimensional impenetrable bosons. Starting with this formula, we give a mathematically rigorous analysis leading to the determination of major features of the momentum distribution in the limit of an infinitely large system.
We introduce a novel notion, order indices of reduced density matrices, characterizing different types of order, long-range as well as mid-range, occurring in statistical systems. The order indices define the behaviour of the largest eigenvalues of density matrices in the thermodynamic limit. These indices are especially useful for describing mid-range order, where they are directly connected with the large-distance asymptotic forms of off-diagonal density matrices. Several examples illustrate the idea.
A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, gamma. When gamma is small, Bogoliubov's perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of gamma and show that it is analytic for all gamma, except gamma=0. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum-not one as is ordinarily supposed.
The ground state of a one-dimensional (1D) quantum gas of dipoles oriented perpendicular to the longitudinal axis, with a strong 1/x^{3} repulsive potential, is studied at low 1D densities n. Near contact the dependence of the many-body wave function on the separation x_{jℓ} of two particles reduces to a two-body wave function Ψ_{rel}(x_{jℓ}). Immediately after a sudden rotation of the dipoles so that they are parallel to the longitudinal axis, this wave function will still be that of the repulsive potential, but since the potential is now that of the attractive potential, it will not be stationary. It is shown that as nd^{2}→0 the rate of change of this wave function approaches zero. It follows that for small values of nd^{2}, this state is metastable and is an analog of the super Tonks-Girardeau state of bosons with a strong zero-range attraction. The dipolar system is equivalent to a spinor Fermi gas with spin z components σ_{↑}=⊥ (perpendicular to the longitudinal axis) and σ_{↓}=∥ (parallel to the longitudinal axis). A Fermi-Fermi mapping from spinor to spinless Fermi gas followed by the standard 1960 Fermi-Bose mapping reduces the Fermi system to a Bose gas. Potential experiments realizing the sudden spin rotation with ultracold dipolar gases are discussed, and a few salient properties of these states are accurately evaluated by a Monte Carlo method.
A rigorous one‐one correspondence is established between one‐dimensional systems of bosons and of spinless fermions. This correspondence holds irrespective of the nature of the interparticle interactions, subject only to the restriction that the interaction have an impenetrable core. It is shown that the Bose and Fermi eigenfunctions are related by ψB=ψFA, where A(x1 … xn) is +1 or −1 according as the order pq … r, when the particle coordinates xj are arranged in the order xp<xq< … <xr, is an even or an odd permutation of 1 … n. The energy spectra of the two systems are identical, as are all configurational probability distributions, but the momentum distributions are quite different. The general theory is illustrated by application to the special case of impenetrable point particles; the one‐one correspondence between bosons with this particular interaction and completely noninteracting fermions leads to a rigorous solution of this many‐boson problem.
Using the exact N -particle ground state wave function for a one-dimensional gas of hard-core bosons in a harmonic trap we develop an algorithm to compute the reduced single-particle density matrix and corresponding momentum distribution. Accurate numerical results are presented for up to N = 8 particles, and the momentum distributions are compared to a recent analytic approximation.
We study low-temperature properties of boson density matrices when macroscopic occupation of quantum energy levels can occur.
An important novelty of our approach is the definition of order indices that characterize the behaviour of largest eigenvalues
of density matrices in the thermodynamic limit. Continuous variation of the order indices gives rise to a whole spectrum of
possible order types: long-range order, mid-range order and short-range order. A relation between order indices is found.
As an illustration of the general scheme a new model exemplifying mid-range order is constructed.
The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the
theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single
particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation
of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé
VI transcendent in Σ-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles.
For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant
form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases
an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the
ground state occupations.
DOI:https://doi.org/10.1103/RevModPhys.34.694
Preface; 1. Introduction; 2. The non-interacting Bose gas; 3. Atomic
properties; 4. Trapping and cooling of atoms; 5. Interactions between
atoms; 6. Theory of the condensed state; 7. Dynamics of the condensate;
8. Microscopic theory of the Bose gas; 9. Rotating condensates; 10.
Superfluidity; 11. Trapped clouds at non-zero temperature; 12. Mixtures
and spinor condensates; 13. Interference and correlations; 14. Optical
lattices; 15. Lower dimensions; 16. Fermions; 17. From atoms to
molecules; Appendix; Index.
In this paper, we investigate the ground-state properties of a bosonic
Tonks-Girardeau gas confined in a one-dimensional periodic potential. The
single-particle reduced-density matrix is computed numerically for systems up
to N=41 bosons. Then we are able to study the scaling behavior of the
occupation numbers of the most dominant orbital for both commensurate and
non-commensurate cases, which correspond to an insulating phase and a
conducting phase, respectively. We find that, in the commensurate case, the
fractional occupation decays as with the particle number,
while in the non-commensurate case. So there is no BEC
in both cases. The study of zero momentum peaks shows that in
commensurate case, which implies that all bonons are localized in the
insulating phase.
A mapping theorem leading to exact many-body dynamics of impenetrable bosons in one dimension reveals dark and gray solitonlike structures in a toroidal trap which is phase imprinted. On long time scales revivals appear that are beyond the usual mean-field theory.
A nonzero temperature generalization of the Fermi-Bose mapping theorem is used to study the exact quantum statistical dynamics of a one-dimensional gas of impenetrable bosons on a ring. We investigate the interference produced when an initially trapped gas localized on one side of the ring is released, split via an optical-dipole grating, and recombined on the other side of the ring. Nonzero temperature is shown not to be a limitation to obtaining high visibility fringes.
We report the observation of a one-dimensional (1D) Tonks-Girardeau (TG) gas of bosons moving freely in 1D. Although TG gas
bosons are strongly interacting, they behave very much like noninteracting fermions. We enter the TG regime with cold rubidium-87
atoms by trapping them with a combination of two light traps. By changing the trap intensities, and hence the atomic interaction
strength, the atoms can be made to act either like a Bose-Einstein condensate or like a TG gas. We measure the total 1D energy
and the length of the gas. With no free parameters and over a wide range of coupling strengths, our data fit the exact solution
for the ground state of a 1D Bose gas.