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Quantum Electronics and Laser Science Conference

Density matrix reconstruction of atoms with large angular momentum

Gerd Klose, Greg Smith, Poul Jessen

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

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.

Related papers

Density matrix reconstruction of a large angular momentum

Gerd Klose

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.

•Jan 1, 2001•1 citation

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We present a filtered backprojection algorithm for reconstructing the Wigner function of a system of large angular momentum j from Stern–Gerlach-type measurements. Our method is advantageous over the full determination of the density matrix in that it is insensitive to experimental fluctuations in j , and allows for a natural elimination of high-frequency noise in the Wigner function by taking into account the experimental uncertainties in the determination of j , its projection m and the quantization axis orientation. No data binning and no arbitrary smoothing parameters are necessary in this reconstruction. Using recently published data (Riedel et al 2010 Nature 464 1170), we reconstruct the Wigner function of a spin-squeezed state of a Bose–Einstein condensate of about 1250 atoms, demonstrating that measurements along quantization axes lying in a single plane are sufficient for performing this tomographic reconstruction. Our method does not guarantee positivity of the reconstructed density matrix in the presence of experimental noise, which is a general limitation of backprojection algorithms.

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Reconstructing the density matrix of a spinsthrough Stern-Gerlach measurements: II

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The density matrix of a spin s is fixed uniquely if the probabilites to obtain the value s upon measuring . are known for 4 s ( s +1) appropriately chosen directions in space. These numbers are just the expectation values of the density operator in coherent spin states, and they are easily obtained in an experiment carried out with a Stern-Gerlach apparatus. Furthermore, the experimental data is non-redundant, and it can be inverted explicitly which allows for a parametrization of the statistical operator by the 4 s ( s +1) positive parameters.

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Quantum-state reconstruction of atoms in optical lattices

Sascha Wallentowitz, A. Zucchetti, W. Vogel +1 more

Summary form only given. The preparation of the quantum state of physical systems and its diagnostics by quantum-state reconstruction has been a subject of large progress in recent years. Various experiments have successfully demonstrated the feasibility of reconstructing the quantum state of optical fields, molecular vibrations, trapped ions, and matter waves. The impressive success in laser cooling of neutral atoms opens new possibilities for preparing quantum states of ensembles of atoms in optical lattices and also quantum computing. In this context, the diagnostics of the quantum state, would provide a very useful tool for studying the accuracy of prepared quantum states or exploring decoherence mechanisms. However, to our knowledge, a reconstruction method for optical lattices that accounts for typical features of optical lattice potentials such as anharmonicity and band structure, is not yet available. We propose such a method, that we think is relatively simple to implement, since only phase shifts of the lattice laser beams and measurements of the band populations in the lattice potential are needed. These techniques are feasible by use of electro/acousto-optical modulators and by measuring the momentum distribution of the atoms after adiabatically turning off the lattice lasers.

Quantum Electronics and Laser Science Conference•Jan 1, 2000•0 citations
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.
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.
The Pauli problem is solved for a spin of length s prepared in an arbitrary (unnormalized) mixed state which has free real parameters. The reconstruction of its density operator is possible if one knows the probabilities of the spin components along each of directions in space. These probabilties are directly accessible through measurements performed with a Stern - Gerlach apparatus. A multipole expansion of the density operator establishes the link between the matrix elements of and the measured intensities.
We present a filtered backprojection algorithm for reconstructing the Wigner function of a system of large angular momentum j from Stern–Gerlach-type measurements. Our method is advantageous over the full determination of the density matrix in that it is insensitive to experimental fluctuations in j , and allows for a natural elimination of high-frequency noise in the Wigner function by taking into account the experimental uncertainties in the determination of j , its projection m and the quantization axis orientation. No data binning and no arbitrary smoothing parameters are necessary in this reconstruction. Using recently published data (Riedel et al 2010 Nature 464 1170), we reconstruct the Wigner function of a spin-squeezed state of a Bose–Einstein condensate of about 1250 atoms, demonstrating that measurements along quantization axes lying in a single plane are sufficient for performing this tomographic reconstruction. Our method does not guarantee positivity of the reconstructed density matrix in the presence of experimental noise, which is a general limitation of backprojection algorithms.
The density matrix of a spin s is fixed uniquely if the probabilites to obtain the value s upon measuring . are known for 4 s ( s +1) appropriately chosen directions in space. These numbers are just the expectation values of the density operator in coherent spin states, and they are easily obtained in an experiment carried out with a Stern-Gerlach apparatus. Furthermore, the experimental data is non-redundant, and it can be inverted explicitly which allows for a parametrization of the statistical operator by the 4 s ( s +1) positive parameters.
Summary form only given. The preparation of the quantum state of physical systems and its diagnostics by quantum-state reconstruction has been a subject of large progress in recent years. Various experiments have successfully demonstrated the feasibility of reconstructing the quantum state of optical fields, molecular vibrations, trapped ions, and matter waves. The impressive success in laser cooling of neutral atoms opens new possibilities for preparing quantum states of ensembles of atoms in optical lattices and also quantum computing. In this context, the diagnostics of the quantum state, would provide a very useful tool for studying the accuracy of prepared quantum states or exploring decoherence mechanisms. However, to our knowledge, a reconstruction method for optical lattices that accounts for typical features of optical lattice potentials such as anharmonicity and band structure, is not yet available. We propose such a method, that we think is relatively simple to implement, since only phase shifts of the lattice laser beams and measurements of the band populations in the lattice potential are needed. These techniques are feasible by use of electro/acousto-optical modulators and by measuring the momentum distribution of the atoms after adiabatically turning off the lattice lasers.