Inverted spectroscopy and interferometry for quantum-state reconstruction of systems with SU(2) symmetry

The last few years were marked by an outburst of research devoted to the problem of reconstruction of quantum states for various physical systems (see, e.g., Ref. [?] for an extensive list of the literature on the subject). The problem, as stated already in the fifties by Fano [?] and Pauli [?], is to determine the density matrix ρ from information obtained by a set of measurements performed on an ensemble of identically prepared systems. Significant theoretical and experimental progress has been achieved during the last decade in the reconstruction of quantum states of the light field [?]. Also, numerous works were devoted to reconstruction methods for other physical systems. Most recently, a general theory of quantum-state reconstruction for physical systems with Lie-group symmetries was developed [?]. In the present work we consider state-reconstruction methods for some quantum systems possessing SU(2) symmetry. The principal procedure for the reconstruction of spin states was recently presented by Agarwal [?]. A similar approach was also proposed by Dodonov and Man’ko [?], while the basic idea underlying this method goes back to the pioneering work by Royer [?]. In brief, one applies a phase-space displacement [specifically, a rotation in the SU(2) case] to the initial quantum state and then measures the probability to find the displaced system in a specific state (the so-called “quantum ruler” state). Repeating this procedure with identically prepared systems for many phase-space points [many rotation angles in the SU(2) case], one determines a function on the phase space (the so-called operational phase-space probability distribution [?,?,?]). In particular, by measuring the population of the ground state, one obtains the so-called Q function. The information contained in the operational phase-space probability distribution is sufficient to completely reconstruct the unknown density matrix of the initial quantum state. A general grouptheoretical description of this method and some examples, including SU(2), are presented in Ref. [?]. The aim of the present paper is to study how the general state-reconstruction procedure outlined above can be implemented in practice for a number of specific physical systems with SU(2) symmetry. Three systems are considered: a collection of two-level atoms, a two-mode quantized radiation field with a fixed total number of photons, and a single laser-cooled ion in a two-dimensional harmonic trap with a fixed total number of vibrational quanta. We show that a simple rearrangement of conventional spectroscopic and interferometric schemes enables one to measure unknown quantum states of these systems.