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Physical Review A

Longitudinal quantum interferometry

Steffen Frank Straub, Matthias Freyberger

Published May 1, 1999• 2 citations
Publisher

Abstract

We describe an interferometric method to uncover the quantum state $|\ensuremath{\psi}〉$ describing a single-mode cavity field. The information needed to retrieve the corresponding Fock coefficients is imprinted on the quantized longitudinal motion of atoms that have interacted with the cavity field. In particular, we show that a set of four atomic momentum distributions can be analyzed easily to reconstruct $|\ensuremath{\psi}〉$.

DOI
10.1103/physreva.59.3832

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We describe an interferometric method to uncover the quantum state ∣\ensuremathψ〉|\ensuremath{\psi}〉∣\ensuremathψ〉 describing a single-mode cavity field. The information needed to retrieve the corresponding Fock coefficients is imprinted on the quantized longitudinal motion of atoms that have interacted with the cavity field. In particular, we show that a set of four atomic momentum distributions can be analyzed easily to reconstruct ∣\ensuremathψ〉|\ensuremath{\psi}〉∣\ensuremathψ〉.
We present various schemes for measuring the quantum state of a single mode of the electromagnetic field. These involve measuring the photon statistics for the mode before and after an interaction with either one or two two-level atoms. The photon statistics conditioned on the final state of the atoms, for two choices of the initial set of atomic states, along with the initial photon statistics, may be used to calculate the complete quantum state in a simple manner. Alternatively, when one atom is used, two unconditioned sets of photon statistics, each after interaction with a single atom in different initial states, along with the initial photon statistics may be used to calculate the initial state in a simple manner. When the cavity is allowed to interact with just one atom, only pure cavity states which do not contain zeros in the photon distribution may be reconstructed. When two atoms are used we may reconstruct pure states which do not contain adjacent zeros in the photon distribution. Coherent states and number states are among those that may be measured with one-atom interaction, and squeezed states and ‚Schrödinger cats‘ are among those that may be measured with a two-atom interaction.
We propose a scheme for the measurement of a two-mode entangled field-state in a high-QQQ cavity. The scheme utilizes the momentum distribution spectrum in the Raman-Nath regime of a three-level atom in VVV configuration. Due to the two modes of the electromagnetic field the atom may have xxx interactions with mode AAA, and yyy interactions with mode BBB, causing a complex momentum distribution. The momentum distribution of the atom after interaction with the quantized cavity fields contains the information of the field photon statistics. We reconstruct the joint photon statistics of the entangled field with the help of recorded momentum spectrum. We also propose to reconstruct the Wigner function of a two-mode entangled field state by injecting two coherent states resonant to each mode into the cavity and then measuring the joint photon statistics of the displaced field.
We discuss a simple method to probe the wave function $\ensuremath{\psi}(x)$ describing the transverse center-of-mass motion of an atomic beam. We detect position distributions of the atoms after they have interacted with two counterpropagating laser beams. The resulting interference pattern depends crucially on the cross term ${\ensuremath{\psi}}^{*}(x)\ensuremath{\psi}(x\ensuremath{-}\ensuremath{\Delta}x)$, which finally allows us to reconstruct the phase of $\ensuremath{\psi}(x)$.