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arXiv:1407.4433 · Physical Review E

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

Oleksandr Gamayun, Oleg Valentinovich Lychkovskiy, Vadim Cheianov

Published Jul 8, 2015• 11 citations
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Abstract

In our recent paper [O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, 032132 (2014)] we studied the dynamics of a mobile impurity particle weakly interacting with the Tonks-Girardeau gas and pulled by a small external force F. Working in the regime when the thermodynamic limit is taken prior to the small force limit, we found that the Bloch oscillations of the impurity velocity are absent in the case of a light impurity. Further, we argued that for a light impurity the steady-state drift velocity V(D) remains finite in the limit F→0. These results are in contradiction with earlier works by Gangardt and co-workers [D. M. Gangardt and A. Kamenev, Phys. Rev. Lett. 102, 070402 (2009); M. Schecter, D. M. Gangardt, and A. Kamenev, Ann. Phys. (NY) 327, 639 (2012)]. One of us has conjectured [O. Lychkovskiy, Phys. Rev. A 91, 040101 (2015)] that the central assumption of these works, the adiabaticity of the dynamics, can break down in the thermodynamic limit. In the preceding Comment [M. Schecter, D. M. Gangardt, and A. Kamenev, Phys. Rev. E 92, 016101 (2015)], Schecter et al. have argued against this conjecture and in support of the existence of Bloch oscillations and the linearity of V(D)(F). They have suggested that the ground state of the impurity-fluid system is a quasibound state and that this is sufficient to ensure adiabaticity in the thermodynamic limit. Their analytical argument is based on a certain truncation of the Hilbert space of the system. We argue that extending the results and intuition based on their truncated model on the original many-body problem lacks justification.

DOI
10.1103/physreve.92.016102

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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.

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In their Comment Giraud and Combescot (GC) [Phys. Rev. A 82, 037603 (2010)] point out that the contribution to the impurity-boson distribution function ${\ensuremath{\rho}}_{\mathit{bi}}(x\ensuremath{-}y)$ of a term we dropped is not negligible, rather than being negligible in the thermodynamic limit as we had conjectured. We now agree with them, but our results for ${\ensuremath{\rho}}_{\mathit{bi}}$ are highly accurate for large impurity-boson mass ratio ${m}_{i}/m$, for all values of $x\ensuremath{-}y$ and boson-impurity coupling constant.

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We study the motion of an impurity under the action of a constant force through a one-dimensional system of weakly-interacting bosons. The interplay of the impurity-boson interaction, the boson-boson interaction, and the driving force gives rise to a rich dynamics. We focus on the influence of a finite external force. Under these far-from-equilibrium conditions, we show that in a wide range of forces, one part of the momentum transferred to the system is periodically channeled into the Bose gas through the emission of dispersive density shock waves, solitons, density waves and the creation of additional phase gradients. As a result, the impurity velocity does not increase indefinitely, but periodically oscillates in time around the drift velocity. We uncover and characterize different dynamical regimes in a wide range of the impurity-boson coupling, the impurity mass and the external force. At a sufficiently large force, the Bloch oscillations cease and the impurity exhibits an unlimited acceleration.

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In our recent paper [O. Gamayun, O. Lychkovskiy, and V. Cheianov, Phys. Rev. E 90, 032132 (2014)] we studied the dynamics of a mobile impurity particle weakly interacting with the Tonks-Girardeau gas and pulled by a small external force F. Working in the regime when the thermodynamic limit is taken prior to the small force limit, we found that the Bloch oscillations of the impurity velocity are absent in the case of a light impurity. Further, we argued that for a light impurity the steady-state drift velocity V(D) remains finite in the limit F→0. These results are in contradiction with earlier works by Gangardt and co-workers [D. M. Gangardt and A. Kamenev, Phys. Rev. Lett. 102, 070402 (2009); M. Schecter, D. M. Gangardt, and A. Kamenev, Ann. Phys. (NY) 327, 639 (2012)]. One of us has conjectured [O. Lychkovskiy, Phys. Rev. A 91, 040101 (2015)] that the central assumption of these works, the adiabaticity of the dynamics, can break down in the thermodynamic limit. In the preceding Comment [M. Schecter, D. M. Gangardt, and A. Kamenev, Phys. Rev. E 92, 016101 (2015)], Schecter et al. have argued against this conjecture and in support of the existence of Bloch oscillations and the linearity of V(D)(F). They have suggested that the ground state of the impurity-fluid system is a quasibound state and that this is sufficient to ensure adiabaticity in the thermodynamic limit. Their analytical argument is based on a certain truncation of the Hilbert space of the system. We argue that extending the results and intuition based on their truncated model on the original many-body problem lacks justification.
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.
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 study the dynamics of a mobile impurity in a quantum fluid at zero temperature. Two related settings are considered. In the first setting an impurity is injected in a fluid with some initial velocity v0{\mathbf v}_0v0​, and we are interested in its velocity at infinite time, v∞{\mathbf v}_\inftyv∞​. We derive a rigorous upper bound on ∣v0−v∞∣|{\mathbf v}_0-{\mathbf v}_\infty|∣v0​−v∞​∣ for initial velocities smaller than some critical velocity. In the limit of vanishing impurity-fluid coupling this bound can be regarded as a rigorous proof of the Landau criterion of superfluidity. In the case of a finite coupling the velocity of the impurity can drop, but not to zero; the bound quantifies the maximal possible drop. In the second setting a small constant force acts on the impurity. We identify two possible regimes of the impurity dynamics -- backscattering oscillations of the impurity velocity and saturation of the velocity without oscillations. Which regime is realized depends on the spectrum of the fluid and the mass of the impurity. In particular, backscattering oscillations are possible for sufficiently heavy impurities in fluids with a roton feature of the spectrum. We critically discuss a concept of Bloch oscillations of a mobile impurity in a one-dimensional quantum fluid introduced in [Phys. Rev. Lett. 102, 070402 (2009)] and argue that this type of oscillations is absent in the thermodynamic limit.
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.
In their Comment Giraud and Combescot (GC) [Phys. Rev. A 82, 037603 (2010)] point out that the contribution to the impurity-boson distribution function ${\ensuremath{\rho}}_{\mathit{bi}}(x\ensuremath{-}y)$ of a term we dropped is not negligible, rather than being negligible in the thermodynamic limit as we had conjectured. We now agree with them, but our results for ${\ensuremath{\rho}}_{\mathit{bi}}$ are highly accurate for large impurity-boson mass ratio mi/m{m}_{i}/mmi​/m, for all values of $x\ensuremath{-}y$ and boson-impurity coupling constant.
We study the motion of an impurity under the action of a constant force through a one-dimensional system of weakly-interacting bosons. The interplay of the impurity-boson interaction, the boson-boson interaction, and the driving force gives rise to a rich dynamics. We focus on the influence of a finite external force. Under these far-from-equilibrium conditions, we show that in a wide range of forces, one part of the momentum transferred to the system is periodically channeled into the Bose gas through the emission of dispersive density shock waves, solitons, density waves and the creation of additional phase gradients. As a result, the impurity velocity does not increase indefinitely, but periodically oscillates in time around the drift velocity. We uncover and characterize different dynamical regimes in a wide range of the impurity-boson coupling, the impurity mass and the external force. At a sufficiently large force, the Bloch oscillations cease and the impurity exhibits an unlimited acceleration.