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Journal of Optics B Quantum and Semiclassical Optics

Coherent states and the reconstruction of pure spin states

Jean-Pierre Amiet, Stefan Weigert

Published Aug 13, 1999• 32 citations
Publisher

Abstract

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 .

DOI
10.1088/1464-4266/1/5/101

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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 .
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 (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
The expectation values of a Hermitian operator in (2 s +1) 2 specific coherent states of a spin are known to determine the operator unambiguously. As shown here, (almost) any other set of (2 s +1) 2 coherent state projectors also provide a basis for self-adjoint operators. This is proved by considering the determinant of the Gram matrix associated with the coherent state projectors as a Hamiltonian of a fictitious classical spin system. The result guarantees that (almost) any experimentally desirable choice of directions is appropriate for reconstructing the state of a quantum spin by means of a Stern–Gerlach apparatus.
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.
State reconstruction for quantum spins is reviewed. Emphasis is on non-tomographic approaches which are based on measurements performed with a Stern-Gerlach apparatus. Two consequences of successfully implemented state reconstruction are pointed out. First, it allows one to determine experimentally the expectation value of an arbitrary operator without a device measuring it. Second, state reconstruction suggests a reformulation of Schroedinger's equation in terms of expectation values only, without explicit reference to a wave function or a density operator.