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arXiv:1211.5905 · New Journal of Physics

Exact out-of-equilibrium central spin dynamics from integrability

Davide Fioretto, Jean-Sébastien Caux, Vladimir Gritsev

Published Apr 24, 2014• 15 citations
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Abstract

We consider a Gaudin magnet (central spin model) with a time-dependent exchange couplings. We explicitly show that the Schrödinger equation is analytically solvable in terms of generalized hypergeometric functions for particular choices of the time dependence of the coupling constants. Our method establishes a new link between this system and the Wess–Zumino–Witten model, and sheds new light on the implications of integrability in out-of-equilibrium quantum physics. As an application, a driven four-spin system is studied in detail.

DOI
10.1088/1367-2630/16/4/043024

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We consider a Gaudin magnet (central spin model) with a time-dependent exchange couplings. We explicitly show that the Schrödinger equation is analytically solvable in terms of generalized hypergeometric functions for particular choices of the time dependence of the coupling constants. Our method establishes a new link between this system and the Wess–Zumino–Witten model, and sheds new light on the implications of integrability in out-of-equilibrium quantum physics. As an application, a driven four-spin system is studied in detail.
The spin degree of freedom has become subject to one of the broadest and most intensively studied fields of research in physics. The present thesis is part of this field and deals with so-called central spin models. The thesis is devided into two parts. In the first one we investigate very basic properties of central spin models on an abstract level. We introduce a new, or rather enhanced, notion of quantum integrability. From this we derive an explicit and, to a certain extent, analytical relation between integrability and decoherence. Then we specify a model of two coupled Gaudin models with all coupling constants being equal to each other (``homogeneous''). We numerically study the spectral properties of the model and find the spectrum to exhibit systematically degenerate multiplets. We briefly discuss possible applications in solid state quantum information processing. The second part is devoted to concrete physical considerations in the context of solid state quantum information processing. Here we consider two exchange coupled electron spins confined in semiconductor quantum dots and evaluate the hyperfine induced spin dynamics for the case of a weak magnetic field applied to the electron spins by exact methods. In a first step we consider the two exchange coupled electron spins to interact with the same bath of nuclear spins via homogeneous couplings. In particular we derive a scaling law for the decoherence time. In a second step we try to turn vice into virtue by investigating to what extent it is possible to swap and entangle two separate nuclear spin baths. Surprisingly, it turns out that a nuclear swap is possible for weakly coupled nuclear spin baths of realistic size, but not for smaller ones. We indicate that under the same circumstances it should also be possible to fully entangle the baths. We close the thesis with a summary and an outlook on further important questions.
Quantum integrability has been applied to a large variety of low dimensional Hamiltonians in Quantum Field Theory, Condensed Matter Physics, and Statistical Mechanics to obtain exact expressions for the spectrum and thermodynamics of these systems. In most of these studies the coupling constants are constant in time. Here we present an exact solution of the nonstationary Schrödinger equation for the Kondo Hamiltonian with a time-dependent spin-exchange coupling J(t) J ( t ) of the form \lambda t + p(t) ± \sqrt{(\lambda t + p(t))^2 + 16/3} λ t + p ( t ) ± ( λ t + p ( t ) ) 2 + 16 / 3 , where p(t) p ( t ) is an arbitrary periodic function, under periodic boundary concditions. Unlike previously studied time-dependent integrable models, which are rooted in the classical Yang–Baxter structure and associated Knizhnik–Zamolodchikov equations, our approach is based on the quantum Knizhnik–Zamolodchikov framework and the quantum Yang–Baxter algebra. Our results broaden the domain of time-dependent integrability to a genuinely quantum class of models and provide new tools for exploring coherent nonequilibrium dynamics in strongly correlated systems.
Correlation functions in the Wess-Zumino-Novikov-Witten (WZNW) theory satisfy a system of Knizhnik-Zamolodchikov (KZ) equations, which involve constants of motion of an exactly solvable model, known as Gaudin magnet. We show that modified KZ equations, where the Gaudin operators are replaced by constants of motion of the closely related pairing Hamiltonian, give rise to a deformed WZNW model that contains terms breaking translational symmetry. This boundary WZNW model is identified and solved. The solution establishes a connection between the WZNW model and the pairing Hamiltonian in the theory of superconductivity. We also argue and demonstrate on an explicit example that our general approach can be used to derive exact solutions to a variety of dynamical systems.
Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.
Abstract We establish the most general class of spin- integrable Richardson–Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin- relaxes the usual integrability constraints, allowing for a general solution where the couplings between spins lack the usual antisymmetric properties of Richardson–Gaudin models. The full set of conserved charges are constructed explicitly and shown to satisfy a set of quadratic equations, allowing for the numerical treatment of a fully anisotropic central spin in an external magnetic field. While this approach does not provide expressions for the exact eigenstates, it allows their eigenvalues to be obtained, and expectation values of local observables can then be calculated from the Hellmann–Feynman theorem.
We investigate the connection between energy level crossings in integrable systems and their integrability, i.e. the existence of a set of non-trivial integrals of motion. In particular, we consider a general quantum Hamiltonian linear in the coupling u , H ( u ) = T + uV , and require that it has the maximum possible number of nontrivial commuting partners also linear in u . We demonstrate how this commutation requirement alone leads to: (1) an exact solution for the energy spectrum and (2) level crossings, which are always present in these Hamiltonians in violation of the Wigner–von Neumann non-crossing rule. Moreover, we construct these Hamiltonians explicitly by resolving the above commutation requirement and show their equivalence to a sector of Gaudin magnets (central spin Hamiltonians). In contrast, fewer than the maximum number of conservation laws does not guarantee level crossings.