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arXiv:1711.05547 · AIP conference proceedings

Quantum many-body adiabaticity, topological Thouless pump and driven impurity in a one-dimensional quantum fluid

Oleg Lychkovskiy, Oleksandr Gamayun, Vadim Cheianov

Published Jan 1, 2018• 5 citations
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Abstract

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.

DOI
10.1063/1.5025462

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•Mar 28, 2014•2 citations
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.
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 adiabatic theorem is a fundamental result in quantum mechanics, which states that a system can be kept arbitrarily close to the instantaneous ground state of its Hamiltonian if the latter varies in time slowly enough. The theorem has an impressive record of applications ranging from foundations of quantum field theory to computational molecular dynamics. In light of this success it is remarkable that a practicable quantitative understanding of what "slowly enough" means is limited to a modest set of systems mostly having a small Hilbert space. Here we show how this gap can be bridged for a broad natural class of physical systems, namely, many-body systems where a small move in the parameter space induces an orthogonality catastrophe. In this class, the conditions for adiabaticity are derived from the scaling properties of the parameter-dependent ground state without a reference to the excitation spectrum. This finding constitutes a major simplification of a complex problem, which otherwise requires solving nonautonomous time evolution in a large Hilbert space.
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 provide two inequalities for estimating adiabatic fidelity in terms of two other more handily calculated quantities, i.e., generalized orthogonality catastrophe and quantum speed limit. As a result of considering a two-dimensional subspace spanned by the initial ground state and its orthogonal complement, our method leads to stronger bounds on adiabatic fidelity than those previously obtained. One of the two inequalities is nearly sharp when the system size is large, as illustrated using a driven Rice-Mele model, which represents a broad class of quantum many-body systems whose overlap of different instantaneous ground states exhibits orthogonality catastrophe.
Thouless pumping is a celebrated topological effect that enables quantized particle transport and holds promise for applications in quantum technologies. However, the standard adiabatic pumping theory dictate that the Hamiltonian must be varied continuously and slowly throughout the whole pumping process. In this work, we show that Thouless pumping can be realized by using only discrete sampling of the Hamiltonian, removing the conventional constraint of continuous and slow change of control. Specifically, we select particular parameter points of the Hamiltonian for conventional continuous quantum adiabatic evolution, and then apply the Hamiltonian at these parameter points which induces a geodesic evolution for the pseudospin with a given quasimomentum. This realizes an adiabatic evolution for the Thouless pumping according to a necessary and sufficient condition for quantum adiabatic evolution.
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.