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arXiv:1804.03726 · Physical review. B./Physical review. B

Necessary and sufficient condition for quantum adiabaticity in a driven one-dimensional impurity-fluid system

Oleg Valentinovich Lychkovskiy, Oleksandr Gamayun, Vadim Cheianov

Published Jul 19, 2018• 13 citations
arXivHTMLPDF

Abstract

We study under what conditions the quantum adiabaticity is maintained in a closed many-body system consisting of a one-dimensional fluid and an impurity particle dragged through the latter by an external force. We employ an effective theory describing the low-energy sector of the system to derive the time dependence of the adiabaticity figure of merit: the adiabatic fidelity. We find that in order to maintain adiabaticity in a large system the external force ${F}_{N}$ should vanish with the system size $N$ as $1/N$ or faster. This improves the necessary adiabatic condition ${F}_{N}=O(1/lnN)$ obtained for this system previously [O. Lychkovskiy et al., in Fourth International Conference on Quantum Technologies (ICQT-2017), AIP Conf. Proc. No. 1936 (AIP, New York, 2018), p. 020024]. The experimental implications of this result and its relation to the quasi-Bloch oscillations of the impurity are discussed.

DOI
10.1103/physrevb.98.024307

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We study under what conditions the quantum adiabaticity is maintained in a closed many-body system consisting of a one-dimensional fluid and an impurity particle dragged through the latter by an external force. We employ an effective theory describing the low-energy sector of the system to derive the time dependence of the adiabaticity figure of merit: the adiabatic fidelity. We find that in order to maintain adiabaticity in a large system the external force FN{F}_{N}FN​ should vanish with the system size NNN as 1/N1/N1/N or faster. This improves the necessary adiabatic condition FN=O(1/lnN){F}_{N}=O(1/lnN)FN​=O(1/lnN) obtained for this system previously [O. Lychkovskiy et al., in Fourth International Conference on Quantum Technologies (ICQT-2017), AIP Conf. Proc. No. 1936 (AIP, New York, 2018), p. 020024]. The experimental implications of this result and its relation to the quasi-Bloch oscillations of the impurity are discussed.
The quantum adiabatic theorem states that a driven system can be kept arbitrarily close to the instantaneous eigenstate of its Hamiltonian if the latter varies in time slowly enough. When it comes to applying the adiabatic theorem in practice, the key question to be answered is how slow slowly enough is. This question can be an intricate one, especially for many-body systems, where the limits of slow driving and large system size may not commute. Recently we have shown how the quantum adiabaticity in many-body systems is related to the generalized orthogonality catastrophe [arXiv 1611.00663, to appear in Phys. Rev. Lett.]. We have proven a rigorous inequality relating these two phenomena and applied it to establish conditions for the quantized transport in the topological Thouless pump. In the present contribution we (i) review these developments and (ii) apply the inequality to establish the conditions for adiabaticity in a one-dimensional system consisting of a quantum fluid and an impurity particle pulled through the fluid by an external force. The latter analysis is vital for the correct quantitative description of the phenomenon of quasi-Bloch oscillations in a one-dimensional translation invariant impurity-fluid system.
The evolution of a driven quantum system is said to be adiabatic whenever the state of the system stays close to an instantaneous eigenstate of its time-dependent Hamiltonian. The celebrated quantum adiabatic theorem ensures that such pure state adiabaticity can be maintained with arbitrary accuracy, provided one chooses a small enough driving rate. Here, we extend the notion of quantum adiabaticity to closed quantum systems initially prepared at finite temperature. In this case adiabaticity implies that the (mixed) state of the system stays close to a quasi-Gibbs state diagonal in the basis of the instantaneous eigenstates of the Hamiltonian. We prove a sufficient condition for the finite temperature adiabaticity. Remarkably, it turns out that the finite temperature adiabaticity can be more robust than the pure state adiabaticity with respect to increasing the system size. This can be the case for one-body systems with large Hilbert spaces, such as a particle in a large box, as well as for certain many-body systems. In particular, we present an example of a driven many-body system where, in the thermodynamic limit, the finite temperature adiabaticity is maintained, while the pure state adiabaticity breaks down. On the other hand, for generic many-body systems the scaling of the finite temperature adiabatic condition with the system size is exponential, analogously to pure state adiabatic conditions.
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 study the dynamics of a mobile impurity in a quantum fluid at zero temperature. Two related settings are considered. In the first setting, the impurity is injected in the fluid with some initial velocity v0{\mathbf{v}}_{0}v0​, and we are interested in its velocity at infinite time, ${\mathbf{v}}_{\ensuremath{\infty}}$. We derive a rigorous upper bound on $|{\mathbf{v}}_{0}\ensuremath{-}{\mathbf{v}}_{\ensuremath{\infty}}|$ for initial velocities smaller than the generalized critical velocity. In the limit of vanishing impurity-fluid coupling, this bound amounts to ${\mathbf{v}}_{\ensuremath{\infty}}={\mathbf{v}}_{0}$, which 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 is exerted upon the impurity. We argue that two distinct dynamical regimes exist---backscattering oscillations of the impurity velocity and saturation of the velocity without oscillations. For fluids with vcL=vs{v}_{c\mathrm{L}}={v}_{s}vcL​=vs​ (where vcL{v}_{c\mathrm{L}}vcL​ and vs{v}_{s}vs​ are the Landau critical velocity and sound velocity, respectively), the latter regime is realized. For fluids with vcL<vs{v}_{c\mathrm{L}}<{v}_{s}vcL​<vs​, both regimes are possible. Which regime is realized in this case depends on the mass of the impurity, a nonequilibrium quantum phase transition occurring at some critical mass. Our results are equally valid in one, two, and three dimensions.
Establishing quantitative adiabaticity criteria at finite temperature remains substantially less developed than in the pure-state setting, even though realistic quantum systems are never at absolute zero. Here, by combining a mixed-state quantum speed limit with mixed-state fidelity susceptibility within the Liouville-space formulation of quantum mechanics, we derive rigorous bounds on the Hilbert-Schmidt fidelity between mixed states. Focusing on protocols that drive an initial Gibbs state toward a quasi-Gibbs target, these bounds yield an explicit threshold driving rate for the onset of nonadiabaticity. For a broad class of local Hamiltonians in gapped phases, we show that, in the thermodynamic limit, the threshold driving rate factorizes into a system-size contribution that recovers the zero-temperature scaling and a universal temperature-dependent factor. The latter is exponentially close to unity at low temperature, whereas at high temperature it is linear in temperature. We verify the predicted scaling in several spin-1/2 chains by obtaining closed-form expressions for the threshold driving rate. Our results provide a practical and largely model-independent criterion for finite-temperature adiabaticity in driven many-body systems under closed-system unitary evolution.
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.