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Physical Review Letters

Direct Measurement of the Wigner Function of a One-Photon Fock State in a Cavity

Patrice Bertet, Alexia Auffèves, Paolo Maioli, S. Osnaghi, Tristan Meunier, M. Brune, J. M. Raimond, S. Haroche

Published Oct 28, 2002• 344 citations
Publisher

Abstract

We have measured the complete Wigner function W of the vacuum and of a single-photon state for a field stored in a high-Q cavity. This experiment implements the direct Lutterbach and Davidovich method [L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997)]] and is based on the dispersive interaction of a single circular Rydberg atom with the cavity field. The nonclassical nature of the single-photon field is exhibited by a region of negative W values. Extensions to other nonclassical cavity field states are discussed.

DOI
10.1103/physrevlett.89.200402

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We have measured the complete Wigner function W of the vacuum and of a single-photon state for a field stored in a high-Q cavity. This experiment implements the direct Lutterbach and Davidovich method [L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997)]] and is based on the dispersive interaction of a single circular Rydberg atom with the cavity field. The nonclassical nature of the single-photon field is exhibited by a region of negative W values. Extensions to other nonclassical cavity field states are discussed.
Following a proposal by two of us [L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997)], we have measured the Wigner function at the origin of phase space for a single photon field. Its value is negative, exhibiting the nonclassical nature of this state. The experiment is based on the absorption-free detection of the microwave field stored in a superconducting cavity [G. Nogues et al., Nature (London) 400, 239 (1999)]. Extension to a measurement of the Wigner function over the complete phase space is discussed.
The photon number parity is a binary observable playing an important role in the description of the non-classical features of light. In microwave experiments, this observable can be measured by using a single Rydberg atom evolving in a two dimension space and interacting dispersively with a field stored in one or two cavities. The procedure, based on atomic interferometry, maps the field parity (+ 1 or − 1) onto the states of the two-level atom. By performing repeated measurements of the field parity, it is possible to observe field quantum jumps, to reconstruct the complete Wigner function of an arbitrary field and to study how it evolves due to decoherence.
A quantum-nondemolition method to measure the number of photons stored in a high-Q cavity, introduced by Brune et al. [Phys. Rev. Lett. 65, 976 (1990)], is described in detail. It is based on the detection of the dispersive phase shift produced by the field on the wave function of nonresonant atoms crossing the cavity. This shift can be measured by atomic interferometry, using the Ramsey separated-oscillatory-field method. The information acquired by detecting a sequence of atoms modifies the field step by step, until it eventually collapses into a Fock state. At the same time, the field phase undergoes a diffusive process as a result of the back action of the measurement on the photon-number conjugate variable. Once a Fock state has been generated, its evolution under weak perturbation can be continuously monitored, revealing quantum jumps between various photon numbers. When applied to an initial coherent field, the intermediate steps of the measuring sequence produce quantum superpositions of classical fields, known as ``Schr\"odinger cat states.'' Ways to prepare and detect these states in a cavity subjected to a weak relaxation process are discussed. The effects analyzed in this article could realistically be observed by using circular Rydberg atoms and very high-Q superconducting microwave cavities. The possibility of photon ``manipulation'' through nonresonant atom-field interactions opens a domain in cavity QED studies.
We present the experimental realisation of a cavity QED atom interferometer based on two consecutive resonant interactions of a circular Rydberg atom with the field mode of a high Q microwave resonator that is initially prepared in the vacuum state. The interferometer scheme is analogous to the Ramsey method with the classical field replaced by the quantum field stored in the cavity mode. We demonstrate the possibility to use this technique to probe, by dispersive interaction, the field stored in a second mode of the resonator.
We propose a scheme for the reconstruction of a cavity field state. In the scheme the cavity field is first displaced by a microwave source, and then is placed in front of one slit of the two-slit apparatus. Ground state atoms pass through the apparatus, hitting upon the screen far away from the two-slit apparatus. The atom-field interaction is dispersive. The contrast of interference fringes is directly related to the Wigner function for the field state. The scheme can be easily generalized to measure the Wigner function of an entangled state of two spatially separated single-mode cavities.
We measure complete and continuous Wigner functions of a two-level cesium atom in both a nearly pure state and highly mixed states. We apply the method [T. Tilma et al., Phys. Rev. Lett. 117, 180401 (2016)] of strictly constructing continuous Wigner functions for qubit or spin systems. We find that the Wigner function of all pure states of a qubit has negative regions and the negativity completely vanishes when the purity of an arbitrary mixed state is less than 23\frac{2}{3}32​. We experimentally demonstrate these findings using a single cesium atom confined in an optical dipole trap, which undergoes a nearly pure dephasing process.