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Journal of Physics A Mathematical and General

Reconstructing the density matrix of a spinsthrough Stern - Gerlach measurements

Jean-Pierre Amiet, Stefan Weigert

Published Aug 7, 1998• 19 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/31/31/001

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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.
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. 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.
The problem of determining a pure state vector from measurements is investigated for a quantum spin of arbitrary length. Generically, only a finite number of wave functions is compatible with the intensities of the spin components in two different spatial directions, measured by a Stern-Gerlach apparatus. The remaining ambiguity can be resolved by one additional well-defined measurement. This method combines efficiency with simplicity: only a small number of quantities have to be measured and the experimental setup is elementary. Other approaches to determine state vectors from measurements, also known as the ``Pauli problem,'' are reviewed for both spin and particle systems.
Coherent states provide an appealing method to reconstruct efficiently a 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 hbar.
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 (4 s +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 zeros 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 .