M.Sc Thesis
M.Sc Student Tsabary Amit Dissipation in Linear and Nonlinear Models of Quantum Systems Department of Physics Professor Emeritus Joseph Avron

Abstract

In reality, all quantum systems are in fact open systems, that undergo dissipative evolution. One of the main tools in describing the evolution of open quantum systems is the Lindblad equation, which is a general Markovian master equation which is completely positive and trace preserving.

The motivation to this work arises from the SERF effect that occurs in atomic vapor, in which the relaxation caused by interatomic collisions, decreases with the collision rate. Although the effect was explained theoretically in the past, it is difficult to draw an elementary understanding of this counterintuitive effect, as the description of the atomic system is complicated. This motivated us to search for similar effects in a simpler setting, where the effect could be easily understood. The effect also motivated us to study relaxation in different collision-inspired models, which in general are nonlinear.

We start with a linear Lindblad equation that arises from a stochastic term in the Hamiltonian, and gives a discrete set of relaxation rates. We present a simple example of a system, where certain relaxation rates decrease with the noise power, instead of increasing, a result which resembles the SERF effect. We give an interpretation that identifies this result with the quantum Zeno effect. We also show similar phenomenon occurs in general, for degenerate noise operators.

We then investigate nonlinear Lindblad equations, which have not been previously subject to much research. These equations follow from mean field equations describing interacting many body dynamics at the limit of infinitely many particles. These equations can arise, for example, from collisions between particles.

We first discuss a nonlinear equation that describes collision-induced dephasing. The solution to this equation exhibits a continuum of exponential dephasing rates. This result is fundamentally different from the relaxation in linear equations, which takes place only through a discrete set of rates.

Next, we discuss a model for qubits that undergo pairwise decay, where excited qubits are only allowed to decay to the ground state in pairs, and not separately as in usual spontaneous decay. The Lindblad equation that describes an ensemble of such qubits is solved using two different approaches. In the first approach (mean field approach), we take the number of qubits to infinity, and obtain a nonlinear equation for a single qubit. The solution to this equation exhibits a polynomial decay law, which is essentially different from the exponential decay law that arises from linear equations. In the second approach, given certain initial data, the complexity of the full Lindblad equation reduces from exponential to linear (in the qubit number). Using this, we obtain the solution for a finite number of qubits. We show agreement between the solutions arising from the two approaches, upon taking the number of qubits to infinity.