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Density matrix reconstruction of a large angular momentum

Gerd Klose

Published Jan 1, 2001• 1 citation
OpenAlex

Abstract

A complete description of the quantum state of a physical system is the fundamental knowledge necessary to statistically predict the outcome of measurements. In turning this statement around, Wolfgang Pauli raised already in 1933 the question, whether an unknown quantum state could be uniquely determined by appropriate measurements--a problem that has gained new relevance in recent years. In order to harness the prospects of quantum computing, secure communication, teleportation, and the like, the development of techniques to accurately control and measure quantum states has now become a matter of practical as well as fundamental interest. However, there is no general answer to Pauli's very basic question, and quantum state reconstruction algorithms have been developed and experimentally demonstrated only for a few systems so far. This thesis presents a novel experimental method to measure the unknown and generally mixed quantum state for an angular momentum of arbitrary magnitude. The (2F + 1) x (2F + 1) density matrix describing the quantum state is hereby completely determined from a set of Stern-Gerlach measurements with (4F + 1) different orientations of the quantization axis. This protocol is implemented for laser cooled Cesium atoms in the 6S₁/₂(F = 4) hyperfine ground state manifold, and is applied to a number of test states prepared by optical pumping and Larmor precession. A comparison of the input and the measured states shows successful reconstructions with fidelities of about 0.95.

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A complete description of the quantum state of a physical system is the fundamental knowledge necessary to statistically predict the outcome of measurements. In turning this statement around, Wolfgang Pauli raised already in 1933 the question, whether an unknown quantum state could be uniquely determined by appropriate measurements--a problem that has gained new relevance in recent years. In order to harness the prospects of quantum computing, secure communication, teleportation, and the like, the development of techniques to accurately control and measure quantum states has now become a matter of practical as well as fundamental interest. However, there is no general answer to Pauli's very basic question, and quantum state reconstruction algorithms have been developed and experimentally demonstrated only for a few systems so far. This thesis presents a novel experimental method to measure the unknown and generally mixed quantum state for an angular momentum of arbitrary magnitude. The (2F + 1) x (2F + 1) density matrix describing the quantum state is hereby completely determined from a set of Stern-Gerlach measurements with (4F + 1) different orientations of the quantization axis. This protocol is implemented for laser cooled Cesium atoms in the 6S₁/₂(F = 4) hyperfine ground state manifold, and is applied to a number of test states prepared by optical pumping and Larmor precession. A comparison of the input and the measured states shows successful reconstructions with fidelities of about 0.95.
We demonstrate a general method to measure the quantum state of an angular momentum of arbitrary magnitude. The (2F+1)x(2F+1) density matrix is completely determined from a set of Stern-Gerlach measurements with (4F+1) different orientations of the quantization axis. We implement the protocol for laser cooled Cesium atoms in the 6S(1/2)(F = 4) hyperfine ground state and apply it to a variety of test states prepared by optical pumping and Larmor precession. A comparison of input and measured states shows typical reconstruction fidelities F >/~0.95.
Summary form only given. Determining the unknown state of a quantum system from measurements of observables, a question first posed by Pauli, is a nontrivial task, which does not always permit a solution. The few examples where this inverse problem was experimentally solved include light fields, molecular vibrations, the motion of trapped ions and free atomic waves. We describe a method to reconstruct the complete density matrix of a generally mixed spin state of trapped neutral atoms. Compared to spin- 1/2 systems, the reconstruction of the spin density matrix for a large angular momentum is a much more challenging task. In our system with cesium atoms in the hyperfine ground state, a reconstruction of the 9/spl times/9 density matrix requires 80 real numbers to be experimentally determined. This can be achieved by measuring the 2F+1=9 populations along 4F+1=17 carefully chosen orientations of a Stern-Gerlach apparatus. A numerical simulation of the reconstruction shows this method to be robust against expected experimental uncertainties. We report on an experiment in progress to implement this measurement protocol in a standard laser cooling and trapping arrangement of neutral cesium atoms.
One of the fundamental problems of experimental quantum physics is the determination of the state of a system, which provides its accurate description and contains all the available information about it. The standard solution is quantum state tomography, a technique that employs a large number of measurements of complementary observables to reconstruct a general density operator. Recently, new protocols have been proposed which allow to obtain the matrix elements one by one with fewer and less complicated observations by exploiting so-called weak measurements, that perturb the system less than ideal projections at the cost of being less precise. This thesis presents an evolution of these schemes that eschews the weakness of the measurement, thus gaining in the accuracy of the results and in practical feasibility. We realized an experiment that uses these methods to reconstruct the density operator of the polarization state of light in single photons regime.
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.
Coherent states provide an appealing method to reconstruct efficiently the pure state of a quantum mechanical spin s. A Stern-Gerlach apparatus is used to measure (4s+1) expectations of projection operators on appropriate coherent states in the unknown state. These measurements are compatible with a finite number of states which can be distinguished, in the generic case, by measuring one more probability. In addition, the present technique shows that the zeroes of a Husimi distribution do have an operational meaning: they can be identified directly by measurements with a Stern-Gerlach apparatus. This result comes down to saying that it is possible to resolve experimentally structures in quantum phase-space which are smaller than ¯h. To determine the unknown state of a quantum system poses a problem which is interesting from a theoretical, an experimental, and a conceptual point of view. To set up a successful approach to state reconstruction (see [1] and [2] for reviews) means to identify a collection of observables, a quorum, such that their expectation values contain
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.