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Unconventional Landau-Zener type transitions in many-body systems

Jonas Larson

Published Oct 29, 2013• 0 citations

Abstract

Institut fu¨r Theoretische Physik, Universita¨t zu K¨oln, De-50937 K¨oln, Germany(Dated: December 16, 2013)Landau-Zener transitions in a Lipkin-Meshkov-Glick model are considered, both as a single systemand in a lattice where each individual site hosts a realization of the corresponding many-bodyproblem. The single site mean-ﬁeld problem is conceptually diﬀerent from many other non-linearLandau-Zener models - it does not support any hysteresis and a new phenomenon appears whereadiabaticity totaly breaks down. More precisely, thisbreakdown occurs for isolated classical solutionspropagating on a separatrix in phase space and during the Landau-Zener sweep they get locked toa ﬁxed point which prohibits adiabatic following. Quantum ﬂuctuations tend to destabilize thesesolutions such that for moderate particle numbers the phenomenon is lost. At the quantum levelit is also shown that the non-adiabatic corrections P of the single site problem obey a power-lawdependenceon the Landau-Zenersweep velocity λ in the adiabatic regime, i.e. P ∼ λ

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Institut fu¨r Theoretische Physik, Universita¨t zu K¨oln, De-50937 K¨oln, Germany(Dated: December 16, 2013)Landau-Zener transitions in a Lipkin-Meshkov-Glick model are considered, both as a single systemand in a lattice where each individual site hosts a realization of the corresponding many-bodyproblem. The single site mean-ﬁeld problem is conceptually diﬀerent from many other non-linearLandau-Zener models - it does not support any hysteresis and a new phenomenon appears whereadiabaticity totaly breaks down. More precisely, thisbreakdown occurs for isolated classical solutionspropagating on a separatrix in phase space and during the Landau-Zener sweep they get locked toa ﬁxed point which prohibits adiabatic following. Quantum ﬂuctuations tend to destabilize thesesolutions such that for moderate particle numbers the phenomenon is lost. At the quantum levelit is also shown that the non-adiabatic corrections P of the single site problem obey a power-lawdependenceon the Landau-Zenersweep velocity λ in the adiabatic regime, i.e. P ∼ λ

We formulate and approximately solve a specific many body generalization of the Landau-Zener problem. Unlike with the single particle Landau-Zener problem, our system does not abide in the adiabatic ground state, even at very slow driving rates. The structure of the theory suggests that this finding reflects a more general phenomenon in the physics of adiabatically driven many particle systems. Our solution can be used to understand, for example, the behavior of two-level systems coupled to an electromagnetic field, as realized in cavity QED experiments.

We consider dynamics of Dicke models, with and without counterrotating terms, under slow variations of parameters which drive the system through a quantum phase transition. The model without counterrotating terms and sweeped detuning is seen in the contexts of a many-body generalization of the Landau-Zener model and the dynamical passage through a second-order quantum phase transition (QPT). Adiabaticity is destroyed when the parameter crosses a critical value. Applying semiclassical analysis based on concepts of classical adiabatic invariants and mapping to the second Painleve equation (PII), we derive a formula which accurately describes particle distributions in the Hilbert space at wide range of parameters and initial conditions of the system. We find striking universal features in the particle distributions which can be probed in an experiment on Feshbach resonance passage or a cavity QED experiment. The dynamics is found to be crucially dependent on the direction of the sweep. The model with counterrotating terms has been realized recently in an experiment with ultracold atomic gases in a cavity. Its semiclassical dynamics is described by a Hamiltonian system with two degrees of freedom. Passage through a QPT corresponds to passage through a bifurcation, and can also be described by PII (after averaging over fast variables), leading to similar universal distributions. Under certain conditions, the Dicke model is reduced to the Lipkin-Meshkov-Glick model.

By considering a quantum critical Lipkin-Meshkov-Glick model we analyze a new type of Landau-Zener transitions where the population transfer is mediated by interaction rather than from a direct diabatic coupling. For this scenario, at a mean-field level the dynamics is greatly influenced by quantum interferences. In particular, regardless of how slow the Landau-Zener sweep is, for certain parameters almost no population transfer occurs, which is in stark contrast to the regular Landau-Zener model. For moderate system sizes, this counterintuitive mean-field behaviour is not duplicated in the quantum case. This can be attributed quantum fluctuations and the fact that multi-level Landau-Zener-Stückelberg interferences have a `dephasing' effect on the above mentioned phenomenon. We also find a discrepancy between the quantum and mean-field models in terms of how the transfer probabilities scale with the sweep velocity.

The Landau-Zener-Stueckelberg (LZS) effect in a model system of interacting tunneling particles is studied numerically and analytically. Each of N tunneling particles interacts with each of the others with the same coupling J. This problem maps onto that of the LZS effect for a large spin

S=N/2.

The mean-field limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ corresponds to the classical limit $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ for the effective spin. It is shown that the ferromagnetic coupling

J>0

tends to suppress the LZS transitions. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ there is a critical value of J above which the staying probability P does not go to zero in the slow sweep limit, unlike the standard LZS effect. For

J<0

LZS transitions are boosted and

P=0

for a set of finite values of the sweep rate for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Various limiting cases such as strong and weak interaction, slow and fast sweep are considered analytically. It is shown that the mean-field approach works well for arbitrary N if the interaction J is weak.