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Physical Review A

Quantum-nondemolition measurement of small photon numbers and the preparation of number states

Rüdiger Schack, Arvid Breitenbach, Axel Schenzle

Published Mar 1, 1992• 14 citations
Publisher

Abstract

We analyze in detail the quantum-nondemolition (QND) scheme of Brune et al. [Phys. Rev. Lett. 65, 976 (1990)]. It is shown that the quantum treatment of the scheme based on the density-matrix formalism is fully equivalent to an information-theoretical description. In this context, it is explicitly seen that the scheme works well even with a low detection efficiency. We show further that a careful choice of the atomic velocities can significantly reduce the number of atoms needed to perform a measurement. It is also possible to use a monoenergetic atomic beam, although in this case the number of atoms needed increases. Finally we discuss how the results are modified if the atomic velocities are only known approximately.

DOI
10.1103/physreva.45.3260

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We analyze in detail the quantum-nondemolition (QND) scheme of Brune et al. [Phys. Rev. Lett. 65, 976 (1990)]. It is shown that the quantum treatment of the scheme based on the density-matrix formalism is fully equivalent to an information-theoretical description. In this context, it is explicitly seen that the scheme works well even with a low detection efficiency. We show further that a careful choice of the atomic velocities can significantly reduce the number of atoms needed to perform a measurement. It is also possible to use a monoenergetic atomic beam, although in this case the number of atoms needed increases. Finally we discuss how the results are modified if the atomic velocities are only known approximately.
We propose a scheme for realizing a quantum nondemolition (QND) measurement of a small number of optical photons. Our scheme has two stages: First, we couple a propagating light pulse with fixed photon number to a trapped cold atomic gas within a cavity, such that the pulse is stored within a collective polariton mode. Second, a small-photon-number measurement is engineered by monitoring the cavity-transmission spectrum. Since the polariton mode profile is preserved during the process of detecting the spectrum, photon-number QND measurements could be achieved by retrieving the light pulse from the polariton mode. We also discuss a method which uses QND measurements to generate small-photon Fock states.
Many so called paradoxes of quantum mechanics are clarified when the measurement equipment is treated as a quantized system. Every measurement involves nonlinear processes. Self consistent formulations of nonlinear quantum optics are relatively simple. Hence optical measurements, such as the quantum nondemolition (QND) measurement of photon number, are particularly well suited for such a treatment. It shows that the so called 'collapse of the wave function' is not needed for the interpretation of the measurement process. Coherence of the density matrix of the signal is progressively reduced with increasing accuracy of the photon number determination. If the QND measurement is incorporated into the double slit experiment, the contrast ratio of the fringes is found to decrease with increasing information on the photon number in one of the two paths.
This paper intends to clarify some issues in the theory of quantum measurement by taking advantage of the self-consistent quantum formulation of nonlinear optics. A quantum-nondemolition measurement of the photon number of an optical pulse can be performed with a nonlinear Mach-Zehnder interferometer followed by a balanced detector. The full quantum-mechanical treatment shows that the shortcut in the description of the quantum-mechanical measurement, the so-called ``collapse of the wave function,'' is not needed for a self-consistent interpretation of the measurement process. Coherence in the density matrix of the signal to be measured is progressively reduced with increasing accuracy of the photon-number determination. The quantum-nondemolition measurement is incorporated in the double-slit experiment and the contrast ratio of the fringes is found to decrease systematically with increasing information on the photon number in one of the two paths. The ``gain'' in the measurement can be made arbitrarily large so that postprocessing of the information can proceed classically.
Considering ultracold atoms traversing a high-QQQ Fabry-P\'erot cavity, we theoretically demonstrate a quantum nondemolition measurement of the photon number. This fully quantum mechanical approach may be understood utilizing concepts as effective mass and group velocity of the atom. The various photon numbers induce a splitting of the atomic wave packet, and a time-of-flight measurement of the atom thereby reveals the photon number. While repeated atomic measurements increase the efficiency of the protocol, it is shown that by considering long interaction times only a few atoms are needed to resolve the photon number with almost perfect accuracy.
We describe the optimum detection procedure to follow in order to measure by atomic interferometry the number of photons present in a cavity by using a Quantum nondemolition scheme. The method we discuss is the one recently introduced by Brune et al. [1, 2], in which the field intensity is detected by recording atomic interferences in a Ramsey type set up. A succession of two-level atoms interacts with the cavity field which is the analog of a birefringent retarding plate for the atomic wave function. The information is acquired in the form of binary digits (atoms detected in one level or the other). We discuss an optimal strategy which yields the value of the photon number with a number of atoms of the order of the base 2 logarithm of the field photon number dispersion. The procedure described here is the one we intend to use in an experiment in preparation.