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Some Remarks on Quantum Tomography in Laser Cooling

Artur Czerwiński

Published Apr 16, 2015• 1 citation
OpenAlex

Abstract

In this article we take into consideration the evolution model of 3-level quantum systems known as laser cooling. The evolution in this model is given by an equation which is a special case of the general Kossakowski-Lindblad master equation. The explicit knowledge about the evolution makes it possible to apply the stroboscopic tomography to laser cooling. In this article we present some remarks concerning the minimal number of observables and moments of measurement for quantum tomography in laser cooling.

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•Jul 2, 2015•0 citations
In this article we take into consideration the evolution model of 3-level quantum systems known as laser cooling. The evolution in this model is given by an equation which is a special case of the general Kossakowski-Lindblad master equation. The explicit knowledge about the evolution makes it possible to apply the stroboscopic tomography to laser cooling. In this article we present some remarks concerning the minimal number of observables and moments of measurement for quantum tomography in laser cooling.
In this article we propose a dynamic quantum state tomography model for qutrits subject to laser cooling. We prove that one can reduce the number of distinct measurement setups required for state reconstruction by employing the stroboscopic approach. The results are in line with current advances in quantum tomography where there is a strong tendency to investigate the optimal criteria for state reconstruction. We believe that the stroboscopic approach can be considered an efficient tool for density matrix identification since it allows to determine the minimal number of distinct observables needed for quantum state tomography.
Abstract The research presented in this article concerns the stroboscopic approach to quantum tomography, which is an area of science where quantum physics and linear algebra overlap. In this article we introduce the algebraic structure of the parametric-dependent quantum channels for 2-level and 3-level systems such that the generator of evolution corresponding with the Kraus operators has no degenerate eigenvalues. In such cases the index of cyclicity of the generator is equal to 1, which physically means that there exists one observable the measurement of which performed a sufficient number of times at distinct instants provides enough data to reconstruct the initial density matrix and, consequently, the trajectory of the state. The necessary conditions for the parameters and relations between them are introduced. The results presented in this paper seem to have considerable potential applications in experiments due to the fact that one can perform quantum tomography by conducting only one kind of measurement. Therefore, the analyzed evolution models can be considered optimal in the context of quantum tomography. Finally, we introduce some remarks concerning optimal evolution models in the case of n -dimensional Hilbert space.
In this paper, we analyze selected evolution models of N-level open quantum systems to find the minimal number of observables (Hermitian operators) such that their expectation values at some time instants determine the accurate representation of the quantum system. The assumption that lies at the foundation of this approach to quantum tomography claims that time evolution of an open quantum system can be expressed by the Kossakowski–Lindblad equation of the form ρ˙=Lρ\dot{\rho } = \mathbb {L} \rho ρ˙​=Lρ , which is the most general type of Markovian and time-homogeneous master equation which preserves trace and positivity. We consider the cases when the generator of evolution can be presented by means of two or more double commutators. Determining the minimal number of observables required for quantum tomography can be the first step towards optimal tomography models for N-level quantum systems.
In the paper we discuss possible applications of the so-called stroboscopic tomography (stroboscopic observability) to selected decoherence models of 2-level quantum systems. The main assumption behind our reasoning claims that the time evolution of the analyzed system is given by a master equation of the form $\dot {\rho } = \mathbb {L} \rho $ and the macroscopic information about the system is provided by the mean values m i (t j ) = T r(Q i ρ(t j )) of certain observables $\{Q_{i}\}_{i=1}^{r} $ measured at different time instants {tj}j=1p\{t_{j}\}_{j=1}^{p}{tj​}j=1p​ . The goal of the stroboscopic tomography is to establish the optimal criteria for observability of a quantum system, i.e. minimal value of r and p as well as the properties of the observables $\{Q_{i}\}_{i=1}^{r} $ .
In this article we propose a new approach to quantum measurement in reference to the stroboscopic tomography. Generally, in the stroboscopic approach it is assumed that the information about the quantum system is encoded in the mean values of certain Hermitian operators Q1,...,QrQ_1, ..., Q_rQ1​,...,Qr​ and each of them can be measured more than once. The main goal of the stroboscopic tomography is to determine when one can reconstruct the initial density matrix ρ(0)ρ(0)ρ(0) on the basis of the measurement results ⟨Qi⟩tj\langle Q_i \rangle_{t_j}⟨Qi​⟩tj​​. In this paper we propose to treat every complex matrix as a measurable operator. This generalized approach to quantum measurement may bring some improvement into the models of stroboscopic tomography.
In this article we propose a stroboscopic approach to complex vector reconstruction in the context of quantum tomography. There are two underlying assumptions behind our reasoning. The first one claims that the evolution of a d-level pure quantum system is given by the Schr\odinger equation with a time-independent Hamiltonian and the other states that the knowledge about the quantum state is provided from projective measurements, also called intensity measurements. The problem of quantum state reconstruction is connected with the notion known as phase retrieval - recovering a complex vector from modulus of inner product with frame vectors. We believe that the stroboscopic approach can significantly improve the effectiveness of the vector reconstruction as it aims to decrease the number of distinct projectors by taking advantage of the knowledge about the evolution. General conditions and observations are applied to a specific unitary model.