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Physical Review A

Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields

Christopher C. Gerry, Adil Benmoussa, Richard A. Campos

Published Nov 29, 2005• 49 citations
Publisher

Abstract

The parity of photonic number states is known to be an important observable for quantized electromagnetic fields with applications to quantum information processing and to Heisenberg-limited measurement of phase shifts in quantum interferometry performed with maximally entangled states and with twin number states. In this paper we describe an approach to the quantum nondemolition measurement of parity for quantized optical fields. The method proposed involves the use of a cross-Kerr interaction where we assume a large Kerr nonlinearity is available through the techniques of electromagnetically induced transparency. Our proposed method does not require the measurement of photon number but rather measures parity directly. The method not only allows for the quantum nondemolition measurement of parity but also allows for the von Neumann projection of parity eigenstates from an arbitrary field state. The generation and detection of higher-order parity eigenstates is also discussed. Losses from dissipation and the effects of detector efficiency are considered.

DOI
10.1103/physreva.72.053818

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The parity of photonic number states is known to be an important observable for quantized electromagnetic fields with applications to quantum information processing and to Heisenberg-limited measurement of phase shifts in quantum interferometry performed with maximally entangled states and with twin number states. In this paper we describe an approach to the quantum nondemolition measurement of parity for quantized optical fields. The method proposed involves the use of a cross-Kerr interaction where we assume a large Kerr nonlinearity is available through the techniques of electromagnetically induced transparency. Our proposed method does not require the measurement of photon number but rather measures parity directly. The method not only allows for the quantum nondemolition measurement of parity but also allows for the von Neumann projection of parity eigenstates from an arbitrary field state. The generation and detection of higher-order parity eigenstates is also discussed. Losses from dissipation and the effects of detector efficiency are considered.
We review the use of parity measurements to obtain Heisenberg-limited phase shift measurements from interferometry performed with maximally path entangled two mode states containing N photons, contrasting this measure with the method of coincidence detection which requires a number of such detections that scales with N. For large N, coincidence measurements will become impractical. In contrast, the parity method we propose is independent of N. In addition, with the parity approach, one can use an entanglement of a coherent state and the vacuum to approach the Heisensberg limit of phase resolution without the requirement of first producing a Fock state of large photon number. We also show in a very simple way, using Maximally entangled states and parity measurements that it is possible to go beyond the Heisenberg limit if the phase shifts are generated by a nonlinear Kerr-like interaction.
The generation of an entangled coherent state is one of the most important ingredients of quantum information processing using coherent states. Recently, numerous schemes to achieve this task have been proposed. In order to generate travelling-wave entangled coherent states, cross-phase-modulation, optimized by optical Kerr effect enhancement in a dense medium in an electromagnetically induced transparency (EIT) regime, seems to be very promising. In this scenario, we propose a fully quantized model of a double-EIT scheme recently proposed [D. Petrosyan and G. Kurizki, Phys. Rev. A 65, 33 833 (2002)]: the quantization step is performed adopting a fully Hamiltonian approach. This allows us to write effective equations of motion for two interacting quantum fields of light that show how the dynamics of one field depends on the photon-number operator of the other. The preparation of a Schr\"odinger cat state, which is a superposition of two distinct coherent states, is briefly exposed. This is based on nonlinear interaction via double EIT of two light fields (initially prepared in coherent states) and on a detection step performed using a 50:50 beam splitter and two photodetectors. In order to show the entanglement of an entangled coherent state, we suggest to measure the joint quadrature variance of the field. We show that the entangled coherent states satisfy the sufficient condition for entanglement based on quadrature variance measurement. We also show how robust our scheme is against a low detection efficiency of homodyne detectors.
The role of decoherence of an electromagnetic field in the process of quantum nondemolition measurement of the number of photons using a nonlinear Mach-Zehnder interferometer is investigated. This decoherence is caused by distributed losses in a Kerr medium. A long interaction time between the field and the medium is required to achieve a high accuracy in the photon number measurement. The losses in the Kerr medium accompanying the resonance four-wave interaction between the measured and the probe fields lead to a measurement error close to unity at γτ ≫1 (γ is the rate of losses in the medium, and τ is the time of interaction between the field and the medium); consequently, nondemolition quantum measurement turns out to be impossible in the scheme considered. Under these conditions, an increase in the intensity of the probe field does not result in achievement of the required measurement accuracy.
Photon number states are assigned a parity of +1 if their photon number is even and a parity of −1 if odd. The parity operator, which is minus one to the power of the photon number operator, is a Hermitian operator and thus a quantum mechanical observable although it has no classical analogue, the concept being meaningless in the context of classical light waves. In this paper we review work on the application of the parity operator to the problem of quantum metrology for the detection of small phase shifts with quantum optical interferometry using highly entangled field states such as the so-called N00N states, and states obtained by injecting twin Fock states into a beam splitter. With such states and with the performance of parity measurements on one of the output beams of the interferometer, one can breach the standard quantum limit, or shot-noise limit, of sensitivity down to the Heisenberg limit, the greatest degree of phase sensitivity allowed by quantum mechanics for linear phase shifts. Heisenberg limit sensitivities are expected to eventually play an important role in attempts to detect gravitational waves in interferometric detection systems such as LIGO and VIRGO.
We propose a technique for quantum nondemolition (QND) measurement and heralded preparation of Fock states by dynamics of electromagnetically induced transparency (EIT). An atomic ensemble trapped in an optical cavity is driven by two external continuous-wave classical fields to form EIT in steady state. As soon as a weak coherent field is injected into the cavity, the EIT system departs from steady state, falls into transient state dynamics by the dispersive coupling between cavity injected photons and atoms. Because the imaginary part of time-dependent linear susceptibility Im[X(t)] of the atomic medium explicitly depends on the number n of photons during the process of transient state dynamics, the measurement on the change of transmission of the probe field can be used for QND measurement of small photon number, and thus create the photon Fock states in particular single-photon state in a heralded way.