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arXiv:1006.5183 · Physica Scripta

Reconstruction of Hamiltonians from given time evolutions

Julia Bernatska, ANTONINO MESSINA

Published Dec 7, 2011• 2 citations
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Abstract

In this paper we propose a systematic method to solve the inverse dynamical problem for a quantum system governed by the von Neumann equation: to find a class of Hamiltonians reproducing a prescribed time evolution of a pure or mixed state of the system. Our approach exploits the equivalence between an action of the group of evolution operators over the state space and an adjoint action of the unitary group over Hermitian matrices. The method is illustrated by two examples involving a pure and a mixed state.

DOI
10.1088/0031-8949/85/01/015001

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In this paper we propose a systematic method to solve the inverse dynamical problem for a quantum system governed by the von Neumann equation: to find a class of Hamiltonians reproducing a prescribed time evolution of a pure or mixed state of the system. Our approach exploits the equivalence between an action of the group of evolution operators over the state space and an adjoint action of the unitary group over Hermitian matrices. The method is illustrated by two examples involving a pure and a mixed state.
A simple systematic way of obtaining analytically solvable Hamiltonians for quantum two-level systems is presented. In this method, a time-dependent Hamiltonian and the resulting unitary evolution operator are connected through an arbitrary function of time, furnishing us with new analytically solvable cases. The method is surprisingly simple, direct, and transparent and is applicable to a wide class of two-level Hamiltonians with no involved constraint on the input function. A few examples illustrate how the method leads to simple solvable Hamiltonians and dynamics.
We present a general framework to approach the problem of finding time-independent dynamics generating target unitary evolutions. More specifically, given a target unitary gate G over a set of qubits, and a parametrized Hamiltonian of the form H ( λ ) = ∑ i λ i σ i with σi Hermitian operators over the qubits, we want to find a set of values λ0 such that exp ( i H ( λ 0 ) ) = G. We show that this problem is equivalently stated as a set of conditions over the spectrum of H ( λ ), reducing the problem to an inverse eigenvalue problem. We show how to solve this problem in some physically relevant instances, like for example to find time-independent dynamics implementing Toffoli and Fredkin gates without the need for ancillary gates or effective evolutions.
Abstract Time‐driven quantum systems are important in many different fields of physics as cold atoms, solid state, optics, etc. Many of their properties are encoded in the time evolution operator or the effective Hamiltonian. Finding these operators usually requires very complex calculations that often involve some approximations. To perform this task, a systematic scheme that can be cast in the form of a symbolic computational algorithm is presented. It is suitable for periodic and non‐periodic potentials and, for convoluted systems, can also be adapted to yield numerical solutions. The method exploits the structure of the associated Lie group and a decomposition of the evolution operator on each group generator. To illustrate the use of the method, five examples are provided: harmonic oscillator with time‐dependent frequency (Paul trap), modulated optical lattice, time‐driven quantum oscillator, a step‐wise driving of a free particle, and the non‐periodic Caldirola‐Kanai Hamiltonian. To the extent of the authors' knowledge, whereas the exact form of Paul trap's evolution operator is well known, its effective Hamiltonian was until now unknown. The remaining four examples accurately reproduce previous results.
We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators H(t)H(t)H(t) that generate a real phase in their time-evolution. This involves the use of invariant operators IPH(t)I_{PH}(t)IPH​(t) that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, H(t)H(t)H(t) is not generally quasi-Hermitian and does not define an observable of the system but IPH(t)I_{PH}(t)IPH​(t) obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.