33 research outputs found
Geometry of quantum observables and thermodynamics of small systems
The concept of ergodicity---the convergence of the temporal averages of
observables to their ensemble averages---is the cornerstone of thermodynamics.
The transition from a predictable, integrable behavior to ergodicity is one of
the most difficult physical phenomena to treat; the celebrated KAM theorem is
the prime example. This Letter is founded on the observation that for many
classical and quantum observables, the sum of the ensemble variance of the
temporal average and the ensemble average of temporal variance remains constant
across the integrability-ergodicity transition.
We show that this property induces a particular geometry of quantum
observables---Frobenius (also known as Hilbert-Schmidt) one---that naturally
encodes all the phenomena associated with the emergence of ergodicity: the
Eigenstate Thermalization effect, the decrease in the inverse participation
ratio, and the disappearance of the integrals of motion. As an application, we
use this geometry to solve a known problem of optimization of the set of
conserved quantities---regardless of whether it comes from symmetries or from
finite-size effects---to be incorporated in an extended thermodynamical theory
of integrable, near-integrable, or mesoscopic systems
Three-dimensional Gross-Pitaevskii solitary waves in optical lattices: stabilization using the artificial quartic kinetic energy induced by lattice shaking
In this Letter, we show that a three-dimensional Bose-Einstein solitary wave
can become stable if the dispersion law is changed from quadratic to quartic.
We suggest a way to realize the quartic dispersion, using shaken optical
lattices. Estimates show that the resulting solitary waves can occupy as little
as -th of the Brillouin zone in each of the three directions and
contain as many as atoms, thus representing a \textit{fully
mobile} macroscopic three-dimensional object.Comment: 8 pages, 1 figure, accepted in Phys. Lett.
Triangular Gross-Pitaevskii breathers and Damski-Chandrasekhar shock waves
The recently proposed map [arXiv:2011.01415] between the hydrodynamic
equations governing the two-dimensional triangular cold-bosonic breathers
[Phys. Rev. X 9, 021035 (2019)] and the high-density zero-temperature
triangular free-fermionic clouds, both trapped harmonically, perfectly explains
the former phenomenon but leaves uninterpreted the nature of the initial
() singularity. This singularity is a density discontinuity that leads, in
the bosonic case, to an infinite force at the cloud edge. The map itself
becomes invalid at times . A similar singularity appears at ,
where is the period of the harmonic trap, with the Fermi-Bose map becoming
invalid at . Here, we first map -- using the scale invariance of the
problem -- the trapped motion to an untrapped one. Then we show that in the new
representation, the solution [arXiv:2011.01415] becomes, along a ray in the
direction normal to one of the three edges of the initial cloud, a freely
propagating one-dimensional shock wave of a class proposed by Damski in [Phys.
Rev. A 69, 043610 (2004)]. There, for a broad class of initial conditions, the
one-dimensional hydrodynamic equations can be mapped to the inviscid Burgers'
equation, a nonlinear transport equation. More specifically, under the Damski
map, the singularity of the original problem becomes, verbatim, the
initial condition for the wave catastrophe solution found by Chandrasekhar in
1943 [Ballistic Research Laboratory Report No. 423 (1943)]. At , our
interpretation ceases to exist: at this instance, all three effectively
one-dimensional shock waves emanating from each of the three sides of the
initial triangle collide at the origin, and the 2D-1D correspondence between
the solution of [arXiv:2011.01415] and the Damski-Chandrasekhar shock wave
becomes invalid.Comment: 13 pages, 2 figures. Submission to SciPos
Threshold for Chaos and Thermalization in One-Dimensional Mean-Field Bose-Hubbard Model
We study the threshold for chaos and its relation to thermalization in the 1D
mean-field Bose-Hubbard model, which in particular describes atoms in optical
lattices. We identify the threshold for chaos, which is finite in the
thermodynamic limit, and show that it is indeed a precursor of thermalization.
Far above the threshold, the state of the system after relaxation is governed
by the usual laws of statistical mechanics.Comment: 4 pages, 3 figures, published versio
Coherent matter waves emerging from Mott-insulators
We study the formation of (quasi-)coherent matter waves emerging from a Mott
insulator for strongly interacting bosons on a one-dimensional lattice. It has
been shown previously that a quasi-condensate emerges at momentum k=\pi/2a,
where a is the lattice constant, in the limit of infinitely strong repulsion
(hard-core bosons). Here we show that this phenomenon persists for all values
of the repulsive interaction that lead to a Mott insulator at a commensurate
filling. The non-equilibrium dynamics of hard-core bosons is treated exactly by
means of a Jordan-Wigner transformation, and the generic case is studied using
a time-dependent density matrix renormalization group technique. Different
methods for controlling the emerging matter wave are discussed.Comment: 20 pages, 11 figures. Published versio
Short-Distance Correlation Properties of the Lieb-Liniger System and Momentum Distributions of Trapped One-Dimensional Atomic Gases
We derive exact closed form expressions for the first few terms of the
short-distance Taylor expansion of the one-body correlation function of the
Lieb-Liniger gas. As an intermediate result we obtain the high-p asymptotics of
the momentum distribution of both free and harmonically trapped atoms and show
that it obeys a universal 1/p^4 law for_all_ values of the interaction
strength. We discuss the ways to observe the predicted momentum distributions
experimentally, regarding them as a sensitive identifier for the
Tonks-Girardeau regime of strong correlations.Comment: A significant misprint in the definition of c_2 is corrected. 5
pages, 1 EPS figure, references update
Bosons in cigar-shape traps: Thomas-Fermi regime, Tonks-Girardeau regime, and between
We present a quantitative analysis of the experimental accessibility of the
Tonks-Girardeau gas in the current day experiments with cigar-trapped alkalis.
For this purpose we derive, using a Bethe anzats generated local equation of
state, a set of hydrostatic equations describing one-dimensional
delta-interacting Bose gases trapped in a harmonic potential. The resulting
solutions cover the_entire range_ of atomic densities.Comment: 4 pages, 4 figure
The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive
-function interaction can be rigorously derived via a scaling limit
from a dilute three-dimensional Bose gas with arbitrary repulsive interaction
potential of finite scattering length. For this purpose, we prove bounds on
both the eigenvalues and corresponding eigenfunctions of three-dimensional
bosons in strongly elongated traps and relate them to the corresponding
quantities in the Lieb-Liniger model. In particular, if both the scattering
length and the radius of the cylindrical trap go to zero, the
Lieb-Liniger model with coupling constant is derived. Our bounds
are uniform in in the whole parameter range , and apply
to the Hamiltonian for three-dimensional bosons in a spectral window of size
above the ground state energy.Comment: LaTeX2e, 19 page
Bose-Fermi variational theory of the BEC-Tonks crossover
A number-conserving hybrid Bose-Fermi variational theory is developed and
applied to investigation of the BEC-Tonks gas crossover in toroidal and long
cylindrical traps of high aspect ratio, where strong many-body correlations and
condensate depletion occur.Comment: 4 pages RevTeX including 2 figures, uses epsfig. Submitted to Phys.
Rev. Let