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arXiv:1603.05402 · Journal of Mathematical Physics

Deterministic submanifolds and analytic solution of the quantum stochastic differential master equation describing a monitored qubit

Alain Sarlette, Pierre Rouchon

Published Jun 1, 2017• 13 citations
arXivHTMLPDF

Abstract

This paper studies the stochastic differential equation (SDE) associated with a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non-demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving surface or curve inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface or curve. Then we relate the existence in general of such deterministically evolving submanifolds to an accessibility question of control theory, which can be answered with an explicit algebraic criterion on the SDE. This allows us to show that, for a qubit, the above two sets of weak measurements are essentially the only ones featuring deterministic surfaces or curves.

DOI
10.1063/1.4984587

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This paper studies the stochastic differential equation (SDE) associated with a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non-demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving surface or curve inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface or curve. Then we relate the existence in general of such deterministically evolving submanifolds to an accessibility question of control theory, which can be answered with an explicit algebraic criterion on the SDE. This allows us to show that, for a qubit, the above two sets of weak measurements are essentially the only ones featuring deterministic surfaces or curves.
The evolution of two-level quantum systems (qubits) in the presence of continuous-time weak measurements can be described by stochastic differential equations, which is dominated by the Hamiltonian and decoherence channels. In more concrete terms, the conventional density matrix evolves according to the stochastic master equation in the Schrödinger picture. Particularly, for a monitored qubit, the deterministic submanifolds concerning the dynamical evolution of the density matrix has been previously obtained. As a complement, We have recently developed deterministic submanifolds for the backward-evolving quantum state for a monitored qubit. In this paper, in order to better understand the behavior of a monitored qubit, by using the Bloch sphere representation and the properties of Pauli matrices, we provide a closed-form expression for probability distribution over the deterministic submanifolds of the backward evolving dynamics in different scenarios. We further analyze the backward-evolving trajectories of the quantum state by taking into account the analytic solutions and the corresponding submanifolds under heterodyne/homodyne detection associated with non-demolition or fluorescence measurements.
Abstract The evolution of a quantum system subject to continuous weak measurement is described by a stochastic differential equation (SDE). It has been shown that the conventional density matrix characterized by such SDE, under experimentally relevant settings, remains within a deterministically evolving manifold of low dimension, whose associated equations can be computed explicitly. In this paper, as a complement, it is demonstrated that the effect matrix , associated to a quantum system at the time t to express its measurement during the interval , also remains confined to a deterministically evolving manifold. In particular, by using the Bloch sphere representation together with the properties of Pauli matrices (as the quantum representation of quaternions), deterministic submanifolds for the effect matrix concerning a monitored qubit are given explicitly. The qubit stays on a deterministically evolving curve in the homodyne amplitude detection scenario, and on a deterministically moving surface under heterodyne amplitude detection. This deterministic characterization should be of interest for parameter estimation and towards obtaining computationally efficient quantum filters for real‐time quantum control.
We analyze an experiment with continuous heterodyne measurement of an energy loss operator on a superconducting qubit. We have observed that in the associated mathematical model, due to the degeneracy of the diffusion operator, the resulting quantum trajectories are supported not in the entire Bloch sphere, but instead they belong to the surface of a deterministically evolving ellipsoid. This allows to analytically solve for the probability distribution of the prior-to-measurement quantum state, governed by the stochastic differential master equation.
We consider a stochastic dynamics describing the evolution of a qubit controlled by an external field and subject to continuous-time measurements. Motivated by stabilization techniques recently developed in, e.g., [10], [11], we investigate the support of the corresponding solution, which is a random variable taking values on the space of two-by-two density matrices. By making use of the Strook-Varadhan support theorem and by classical geometric control arguments we compute the support for two possible choices of the measurement and Hamiltonian operators. In one case we show that, in Bloch coordinates, the support is always contained inside an ellipsoid depending on the physical parameters of the system. More precisely, a solution starting from the ellipsoid never exit it, with probability one, and every open subset of the ellipsoid is visited with nonzero probability for some choice of the control function. In the second case the support coincides with (the interior of) the Bloch ball: every open subset of the Bloch ball is visited with nonzero probability up to suitably choosing the control function.
Quantum systems subjected to a continuous weak measurement process evolve according to stochastic differential equations. Depending on the outcomes of these stochastic measurements, the quantum state may diffuse in various directions across the state space. This note points out that, in many scenarios relevant to quantum engineering, this diffusion is effectively constrained to a low-dimensional space. Specifically, the quantum state remains confined to a low-dimensional, nonlinear manifold---often time-dependent, yet independent of the specific measurement outcomes. This note derives the corresponding low-dimensional formulations for expressing the stochastically evolving state in several prototypical cases: quantum nondemolition measurements in arbitrary dimensions; quadrature measurements of a harmonic oscillator (linear quantum system); and measurements of subsystems within multipartite quantum systems. Additionally, it introduces an algebraic criterion to determine whether such low-dimensional manifolds exist or persist when additional dynamics are present.
Suppose a single qubit is repeatedly prepared and evolved under imperfectly-controlled conditions. A drunk model represents uncontrolled interactions on each experimental trial as random or stochastic terms in the qubits Hamiltonian operator. Time evolution of states is generated by a stochastic differential equation whose sample paths evolve according to the Schrödinger equation. For models with Gaussian white noise which is independent of the qubits state, the expectation value of the solution obeys a master equation which is identical to the high-temperature limit of the Bloch equation. Drunk models predict that experimental data can appear consistent with decoherence even if qubit states evolve by unitary transformations. Examples are shown in which reversible evolution appears to cause irreversible information loss. This paradox is resolved by distinguishing between the true state of a system and the estimated state inferred from an experimental dataset.