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Quantum Optics Journal of the European Optical Society Part B

Phase destruction in quantum non-demolition measurement of the photon number

Henryk Paul

Published Jun 1, 1991• 21 citations
Publisher

Abstract

The decay of the off-diagonal elements of the density matrix for the field that accompanies the quantum non-demolition measurement, recently proposed by Brune et al (Phys. Rev. Lett., vol.65, p.976, 1990) of the number of photons present in a microwave cavity, is studied. Averaging over the time of flight of the atoms, a simple estimate of the number of atoms needed for a precise measurement, is found. It agrees well with the result obtained by Brune et al by numerically simulating the measurement, and, moreover, allows a determination, in a systematic manner, of the optimum values for the relevant physical parameters.

DOI
10.1088/0954-8998/3/3/003

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The decay of the off-diagonal elements of the density matrix for the field that accompanies the quantum non-demolition measurement, recently proposed by Brune et al (Phys. Rev. Lett., vol.65, p.976, 1990) of the number of photons present in a microwave cavity, is studied. Averaging over the time of flight of the atoms, a simple estimate of the number of atoms needed for a precise measurement, is found. It agrees well with the result obtained by Brune et al by numerically simulating the measurement, and, moreover, allows a determination, in a systematic manner, of the optimum values for the relevant physical parameters.
It is found by computer simulation that in the atomic beam deflection experiment proposed by Holland et al. to perform a quantum non-demolition measurement of the photon number, the rate of entropy decrease owing to the transition to a Fock state depends in a quasi-periodic way on the interaction parameter. Moreover, a simple analytic expression is derived for the decay of the off-diagonal field density matrix elements. With the help of this expression the phase destruction of the field due to the measurement process is investigated.
The decay of a single photon in a microwave cavity is shown to be retarded by interaction with a resonant two-level atom in the experimental setup recently developed by Nogues and co-workers [see G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond and S. Haroche, Nature vol 400, 239(1999)]. The effect may be interpreted in terms of the temporary removal of the photon from the cavity thereby protecting it from the effects of the environment to wich the cavity is coupled. Realistic parameters lead to a significative increase of the survival probability of the photon subsequently to the monitoring interaction.
When atoms are illuminated by an off-resonant field, the AC Stark effect will lead to phase shifts in atomic states. The phase shifts are proportional to the photon number of the off-resonant illuminating field. By measuring the atomic phase with newly developed atom-light hybrid interferometers, we can achieve quantum non-demolition measurement of the photon number of the optical field. In this paper, we analyze theoretically the performance of this QND measurement scheme by using the QND measurement criteria established by Holland et al [Phys. Rev. A 42, 2995 (1990)]. We find the quality of the QND measurement depends on the phase resolution of the atom-light hybrid interferometers. We apply this QND measurement scheme to a twin-photon state from parametric amplifier to verify the photon correlation in the twin beams. Furthermore, a sequential QND measurement procedure is analyzed for verifying the projection property of quantum measurement and for the quantum information tapping. Finally, we discuss the possibility for single-photon-number-resolving detection via QND measurement.
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.
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.