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 ${p}_{0}$ 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 ${p}_{0}$ 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 time evolution of the momentum of an impurity atom injected into a degenerate Tonks-Girardeau gas. We establish that given an initial momentum p0 the impurity relaxes to a steady state with a nonvanishing momentum p\ensuremath∞. 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∞ on p0 remains nontrivial even in the limit of vanishing interaction between the impurity and host particles. In this limit p\ensuremath∞(p0) is found explicitly.
Consider an impurity particle injected in a degenerate one-dimensional gas of noninteracting fermions (or, equivalently, Tonks-Girardeau bosons) with some initial momentum p0. We examine the infinite-time value of the momentum of the impurity, ${p}_{\ensuremath{\infty}}$, as a function of p0. 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.
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 large-time asymptotical behavior of a mobile impurity immersed in a degenerate Tonks-Girardeau gas. We derive a correct weak-coupling kinetic equation valid for N arbitrary ratio of masses of gas and impurity particles. When gas particles are either lighter or heavier than the impurity we find that our theory is equivalent to the Boltzmann theory with the collision integral calculated via the Fermi Golden Rule. On the contrary, in the equal-mass case, Fermi Golden Rule treatment gives false results due to not accounting for multiple coherent scattering events. The latter are treated by the resummation of ladder diagrams, which leads to a new kinetic equation. The asymptotic momentum of the impurity produced from this equation coincides with the result obtained by means of the Bethe ansatz.
We consider a mobile impurity particle injected into a one-dimensional quantum gas. The time evolution of the system strongly depends on whether the mass of the impurity and the masses of the host particles are equal or not. For equal masses, the model is Bethe Ansatz solvable, but for unequal masses, the model is no longer integrable and the Bethe Ansatz technique breaks down. We construct a controllable numerical method of computing the spectrum of the model with a finite number of host particles, based on exact diagonalization of the Hamiltonian in the truncated basis of the Bethe Ansatz states. We illustrate our approach on a few-body system of 5+1 particles, and trace the evolution of the spectrum depending on the mass ratio of the impurity and the host particles.
The nonequilibrium dynamics of an impurity immersed with a finite velocity into a one-dimensional system of weakly interacting bosons is studied within the framework of the time-dependent Gross-Pitaevskii equation. We uncover and characterize different regimes of relaxation dynamics. We find that the final impurity velocity remains constant in a large interval of sufficiently big and realistic initial velocities. The underlying physical mechanism is the emission of the dispersive density shock waves that carry away the excess of the initial impurity momentum, while locally the system remains in the same stationary state. In contrast, a heavy impurity with the same coupling constant relaxes differently and the regime of constant final velocity disappears. Furthermore, a fast heavy impurity exhibits damped velocity oscillations in time before reaching a stationary state. This process is accompanied by the oscillations of the local depletion of the boson density around the impurity, until their positions coincide and they continue the motion together. Decreasing the impurity-boson coupling or increasing the strength of repulsion between bosons, the oscillations get amplified. In the case of a heavy impurity with the mass bigger than the critical one, the ground-state energy as a function of momentum exhibits cusps and metastable branches. We show that they manifest themselves by a soliton emission, a considerable slowing down of the relaxation, and a change of the impurity direction of motion with respect to the initial one.
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.