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arXiv:1308.6260 · Physical Review A

Perpetual motion of a mobile impurity in a one-dimensional quantum gas

Oleg Valentinovich Lychkovskiy

Published Mar 13, 2014• 22 citations
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Abstract

Consider an impurity particle injected in a degenerate one-dimensional gas of noninteracting fermions (or, equivalently, Tonks-Girardeau bosons) with some initial momentum ${p}_{0}$. We examine the infinite-time value of the momentum of the impurity, ${p}_{\ensuremath{\infty}}$, as a function of ${p}_{0}$. A lower bound on $|{p}_{\ensuremath{\infty}}({p}_{0})|$ is derived under fairly general conditions. The derivation, based on the existence of the lower edge of the spectrum of the host gas, does not resort to any approximations. The existence of such bound implies the perpetual motion of the impurity in a one-dimensional gas of noninteracting fermions or Tonks-Girardeau bosons at zero temperature. The bound admits an especially simple and useful form when the interaction between the impurity and host particles is everywhere repulsive.

DOI
10.1103/physreva.89.033619

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Comment on “Kinetic theory for a mobile impurity in a degenerate Tonks-Girardeau gas”

Michael Schecter, Dimitri M. Gangardt, Alex Kamenev

In a recent paper Gamayun et al. [O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, 032132 (2014)] studied the dynamics of a mobile impurity weakly coupled to a one-dimensional Tonks-Girardeau gas of strongly interacting bosons. Employing the Boltzmann equation approach, they, in particular, arrived at the following conclusions: (i) a light impurity, being accelerated by a constant force F, does not exhibit Bloch oscillations, which were predicted and studied by Gangardt and co-workers [D. M. Gangardt and A. Kamenev, Phys. Rev. Lett. 102, 070402 (2009); M. Schecter, D. M. Gangardt, and A. Kamanev, Ann. Phys. (N.Y.) 327, 639 (2012)]; (ii) a heavy impurity does undergo Bloch oscillations, accompanied by a drift with the velocity v(D)∝√[F]. In this Comment we argue that result (i) is an artifact of the classical Boltzmann approximation. The latter misses the formation of the quasibound state between the impurity and a hole. Its dispersion relation E(b)(P,ρ) is a smooth periodic function of momentum P with the period 2k(F)=2πℏρ, where ρ is a density of the host gas. Being accelerated by a small force, such a bound-state exhibits Bloch oscillations superimposed with the drift velocity v(D)=μF. The mobility μ may be expressed exactly [M. Schecter et al., Ann. Phys. (N.Y.) 327, 639 (2012)] in terms of E(b)(P,ρ). Result (ii), while not valid at exponentially small forces, indeed reflects an interesting intermediate-force behavior.

arXiv:1404.4366•Physical Review E•Jul 8, 2015•14 citations

Time and temperature-dependent correlation function of an impurity in one-dimensional Fermi and Tonks–Girardeau gases as a Fredholm determinant

Oleksandr Gamayun, A. G. Pronko, Mikhail B. Zvonarev

We investigate a free one-dimensional spinless Fermi gas, and the Tonks–Girardeau gas interacting with a single impurity particle of equal mass. We obtain a Fredholm determinant representation for the time-dependent correlation function of the impurity particle. This representation is valid for an arbitrary temperature and an arbitrary repulsive or attractive impurity-gas δ -function interaction potential. It includes, as particular cases, the representations obtained for zero temperature and arbitrary repulsion in (Gamayun et al 2015 Nucl. Phys. B 892 83–104 ), and for arbitrary temperature and infinite repulsion in (Izergin and Pronko 1998 Nucl. Phys. B 520 594–632 ).

arXiv:1511.05922•New Journal of Physics•Apr 8, 2016•39 citations
Consider an impurity particle injected in a degenerate one-dimensional gas of noninteracting fermions (or, equivalently, Tonks-Girardeau bosons) with some initial momentum p0{p}_{0}p0​. We examine the infinite-time value of the momentum of the impurity, p\ensuremath∞{p}_{\ensuremath{\infty}}p\ensuremath∞​, as a function of p0{p}_{0}p0​. A lower bound on ∣p\ensuremath∞(p0)∣|{p}_{\ensuremath{\infty}}({p}_{0})|∣p\ensuremath∞​(p0​)∣ is derived under fairly general conditions. The derivation, based on the existence of the lower edge of the spectrum of the host gas, does not resort to any approximations. The existence of such bound implies the perpetual motion of the impurity in a one-dimensional gas of noninteracting fermions or Tonks-Girardeau bosons at zero temperature. The bound admits an especially simple and useful form when the interaction between the impurity and host particles is everywhere repulsive.
A kinetic theory describing the motion of an impurity particle in a degenerate Tonks-Girardeau gas is presented. The theory is based on the one-dimensional Boltzmann equation. An iterative procedure for solving this equation is proposed, leading to the exact solution in a number of special cases and to an approximate solution with the explicitly specified precision in a general case. Previously we reported that the impurity reaches a nonthermal steady state, characterized by an impurity momentum p(∞) depending on its initial momentum p(0) [E. Burovski, V. Cheianov, O. Gamayun, and O. Lychkovskiy, Phys. Rev. A 89, 041601(R) (2014)]. In the present paper the detailed derivation of p(∞)(p(0)) is provided. We also study the motion of an impurity under the action of a constant force F. It is demonstrated that if the impurity is heavier than the host particles, m(i)>m(h), damped oscillations of the impurity momentum develop, while in the opposite case, m(i)<m(h), oscillations are absent. The steady-state momentum as a function of the applied force is determined. In the limit of weak force it is found to be force independent for a light impurity and proportional to √[F] for a heavy impurity.
We investigate the time evolution of the momentum of an impurity atom injected into a degenerate Tonks-Girardeau gas. We establish that given an initial momentum p0{p}_{0}p0​ the impurity relaxes to a steady state with a nonvanishing momentum ${p}_{\ensuremath{\infty}}.$ The nature of the steady state is found to depend drastically on whether the masses of the impurity and the host are equal. This is due to multiple coherent scattering processes leading to a resonant interaction between the impurity and the host in the case of equal masses. The dependence of ${p}_{\ensuremath{\infty}}$ on p0{p}_{0}p0​ remains nontrivial even in the limit of vanishing interaction between the impurity and host particles. In this limit ${p}_{\ensuremath{\infty}}({p}_{0})$ is found explicitly.
We investigate the momentum distribution function of a single distinguishable impurity particle which formed a polaron state in a gas of either free fermions or Tonks-Girardeau bosons in one spatial dimension. We obtain a Fredholm determinant representation of the distribution function for the Bethe ansatz solvable model of an impurity-gas \delta δ -function interaction potential at zero temperature, in both repulsive and attractive regimes. We deduce from this representation the fourth power decay at a large momentum, and a weakly divergent (quasi-condensate) peak at a finite momentum. We also demonstrate that the momentum distribution function in the limiting case of infinitely strong interaction can be expressed through a correlation function of the one-dimensional impenetrable anyons.
In a recent paper Gamayun et al. [O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, 032132 (2014)] studied the dynamics of a mobile impurity weakly coupled to a one-dimensional Tonks-Girardeau gas of strongly interacting bosons. Employing the Boltzmann equation approach, they, in particular, arrived at the following conclusions: (i) a light impurity, being accelerated by a constant force F, does not exhibit Bloch oscillations, which were predicted and studied by Gangardt and co-workers [D. M. Gangardt and A. Kamenev, Phys. Rev. Lett. 102, 070402 (2009); M. Schecter, D. M. Gangardt, and A. Kamanev, Ann. Phys. (N.Y.) 327, 639 (2012)]; (ii) a heavy impurity does undergo Bloch oscillations, accompanied by a drift with the velocity v(D)∝√[F]. In this Comment we argue that result (i) is an artifact of the classical Boltzmann approximation. The latter misses the formation of the quasibound state between the impurity and a hole. Its dispersion relation E(b)(P,ρ) is a smooth periodic function of momentum P with the period 2k(F)=2πℏρ, where ρ is a density of the host gas. Being accelerated by a small force, such a bound-state exhibits Bloch oscillations superimposed with the drift velocity v(D)=μF. The mobility μ may be expressed exactly [M. Schecter et al., Ann. Phys. (N.Y.) 327, 639 (2012)] in terms of E(b)(P,ρ). Result (ii), while not valid at exponentially small forces, indeed reflects an interesting intermediate-force behavior.
We investigate a free one-dimensional spinless Fermi gas, and the Tonks–Girardeau gas interacting with a single impurity particle of equal mass. We obtain a Fredholm determinant representation for the time-dependent correlation function of the impurity particle. This representation is valid for an arbitrary temperature and an arbitrary repulsive or attractive impurity-gas δ -function interaction potential. It includes, as particular cases, the representations obtained for zero temperature and arbitrary repulsion in (Gamayun et al 2015 Nucl. Phys. B 892 83–104 ), and for arbitrary temperature and infinite repulsion in (Izergin and Pronko 1998 Nucl. Phys. B 520 594–632 ).