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arXiv:quant-ph/9904054 · Journal of Optics B Quantum and Semiclassical Optics

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

Constantin Brif, A. Mann

Published May 11, 2000• 19 citations
arXivHTMLPDF

Abstract

We consider how the conventional spectroscopic and interferometric schemes can be rearranged to serve for reconstructing quantum states of physical systems possessing SU (2) symmetry. The discussed systems include 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. In the proposed rearrangement, the standard spectroscopic and interferometric experiments are inverted. Usually one measures an unknown frequency or phase shift using a system prepared in a known quantum state. Our aim is just the inverse, i.e., to use a well-calibrated apparatus with known transformation parameters to measure unknown quantum states.

DOI
10.1088/1464-4266/2/3/305

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We consider how the conventional spectroscopic and interferometric schemes can be rearranged to serve for reconstructing quantum states of physical systems possessing SU (2) symmetry. The discussed systems include 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. In the proposed rearrangement, the standard spectroscopic and interferometric experiments are inverted. Usually one measures an unknown frequency or phase shift using a system prepared in a known quantum state. Our aim is just the inverse, i.e., to use a well-calibrated apparatus with known transformation parameters to measure unknown quantum states.
The quantum nondemolition measurement is applied to a two-dimensional (2D) trapped-ion model in which two laser beams drive the corresponding vibrational motions and are carrier resonant with the two-level system of the ion. The information about the ionic vibrational energy can be detected by the occupation probability of the internal electronic level. The substantial difference of the 2D model from the one-dimensional one is that two orthogonal beams have a fixed phase shift instead of statistical independence. As a result, the atomic Rabi oscillation is involved in the coherent superposition of two sub-Rabi oscillations induced by the corresponding driving beams. This means that, in the bidimensional case, the phase difference between the two laser beams plays the role of an adjustable external parameter which allows us to optimize the measurement scheme itself, in terms of precision and sensitivity. As in the one-dimensional case, our proposal leads to interesting applications as, for example, the cooling and preparation of arbitrary bimodal Fock states. The model can be considered as a cooling method for the trapped ion from the vibrational thermal state to the vacuum state. Due to the coherent superposition of two sub-Rabi oscillations, the Rabi frequency degeneration and offset may occur in this model. This provides the possibility of generating various entangled superposition of Fock states, the pair coherent state, and entangled pair coherent states.
We show how group symmetries can be used to reconstruct quantum states. The method we propose is presented in the context of the two-mode SU(1,1) states of the radiation field. In our scheme for SU(1,1) states, the input field passes through a nondegenerate parametric amplifier and one measures the probability of finding the output state with a certain number (usually zero) of photons in each mode. The density matrix in the Fock basis is retrieved from the measured data by the least-squares method after singular value decomposition of the design matrix followed by Tikhonov regularization. Several illustrative examples involving the reconstruction of a pair coherent state, a Perelomov coherent state, and a coherent superposition of pair coherent states are considered.
A scheme for preparing vibrational SU(2) states of motion in a two-dimensional ion trap is described. These anticorrelated two-mode states are formally equivalent to the output photon states of a lossless SU(2) interferometer with number-state inputs. Nontrivial statistics such as the binomial distribution and the discrete ``arcsine'' distribution can be generated in the vibrational states of trapped ions, and detected by measuring the population inversion of the ion driven by a laser field along a specific direction. \textcopyright{} 1996 The American Physical Society.
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.
Fourier transform spectroscopy with classical interferometry corresponds to the measurement of a single-photon intensity spectrum from the viewpoint of the particle nature of light. In contrast, the Fourier transform of two-photon quantum interference patterns provides the intensity spectrum of the two photons as a function of the sum- or difference frequency of the constituent photons. This unique feature of quantum interferometric spectroscopy offers a different type of spectral information from the classical measurement and may prove useful for nonlinear spectroscopy with two-photon emission. Here, we report the experimental demonstration of two-photon quantum interference of biphotons emitted via biexcitons in the semiconductor CuCl. Besides applying Fourier transform to quantum interference patterns, we reconstruct the intensity spectrum of the biexciton luminescence in the two-photon sum- or difference frequency. We discuss the connection between the reconstructed spectra and exciton states in CuCl as well as the capability of quantum interferometry in solid-state spectroscopy.
Optical quantum metrology uses specially-prepared states of light to probe sensitive samples. Choosing a suitable probe state allows one to extract more information per probe photon about an unknown parameter than is possible using any classical technique. In effect, these techniques improve the sensitivity of the measurement whenever the number of probe photons is constrained in some way. This thesis presents experimental and theoretical results on the use of pairs of time-frequency entangled photons in precision measurements. It begins with an overview of necessary background knowledge about quantum optics and quantum information. This is followed by a theoretical discussion on the estimation of a single delay using two-photon interference. We show that constraints on the probe photon's bandwidth complicate the measurement problem, requiring suitable optimization of the two-photon quantum state to avoid losing precision. We then consider the use of two-photon interference for linear spectroscopic measurements. Linear spectroscopy relates the absorption and phase spectrum of a sample to unknown electronic parameters. We describe a proof-of-principle experiment demonstrating that two-photon interferometric spectroscopy can be used for simultaneously measurement of both absorption and phase. This is followed by an information-theoretic discussion of the precision of this technique. For low losses, we show that the method provides phase sensitivity similar to classical interferometry while approaching the optimal absorption precision of any possible measurement. If the absorption and phase values are both related to a single unknown parameter, then our technique provides improved sensitivity to this parameter compared to both classical measurements and other quantum techniques in the presence of realistic levels of loss. We conclude with a discussion of quantum-optical coherence tomography, which is a two-photon interferometric method for determining the material structure of a sample. We present an ongoing experiment which aims to apply the method to tissue-like samples, where the quantum measurement is limited by a poor signal-to-noise ratio. Preliminary results demonstrate that this noise can be largely reduced by suitable frequency-domain filtering. This allows us to precisely determine the position of a scattering center using a few hundred detected photon pairs despite the presence of large amounts of noise. We follow this with a theoretical discussion of signal artifacts which are known to complicate the measured interferogram. We show these artifacts are due to a distinct interferometric effect, which can be distinguished from the signal of interest using the relative time delay between detected photons. We conclude with a brief discussion on possible techniques for artifact removal.