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Physical Review A

Phase fromQfunction via linear amplification

W. P. Schleich, A. Bandilla, Henryk Paul

Published May 1, 1992• 72 citations
Publisher

Abstract

We relate the phase distribution of a quantum state obtained by integrating its corresponding Q function over radius (1) to a specific measurement scheme using a linear amplifier and (2) to a particular phase operator. A simple relation between the P distribution of the amplified macroscopic state and the Q function of the initial unamplified microscopic state allows us to prove the identity of the phase distributions obtained by integrating the corresponding distributions.

DOI
10.1103/physreva.45.6652

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We relate the phase distribution of a quantum state obtained by integrating its corresponding Q function over radius (1) to a specific measurement scheme using a linear amplifier and (2) to a particular phase operator. A simple relation between the P distribution of the amplified macroscopic state and the Q function of the initial unamplified microscopic state allows us to prove the identity of the phase distributions obtained by integrating the corresponding distributions.
Abstract A novel approach to the quantum mechanical description of phase properties of a one‐mode low‐intensity radiation field is presented. The procedure is based upon the correspondence between Glauber states and classical waves of fixed phase ϕ and amplitude a, and, in contrast to the conventional quantum mechanical phase concepts, allows to make contact between certain operators ℰm corresponding to the classical quantities exp {imφ} (m = ± 1, ± 2, …) and realistic measurements. In the observation, a large amplification of the microscopic field by means of a linear quantum amplifier plays a decisive role. Since the amplified field is a macroscopic one, the measurement of phase is not problematic, and hence, in particular, average values «exp {imϕ}» can be determined experimentally. The main result of our analysis is that the mean values «exp {imϕ}» are equal to the quantum mechanical expectation values for the operators ℰm with respect to the field before amplification. To obtain this result it became necessary to calculate Glauber's P function for the amplified field in case of an arbitrary state for the initial field.
Using the description of the linear phase insensitive amplification of a quantum state available in the literature we prove that to every state , produced from the initial state by such amplification, there corresponds a phase space distribution , where is the Husimi function of the initial state. The scaling parameter satisfies the relation and decreases with time while, as we have shown earlier, the scaled function is again a Husimi distribution. We prove, using these facts, that if the definition of the phase insensitive amplification is physically correct then the distribution of the phase of a state is unambiguously given as the corresponding marginal distribution obtained from its Husimi distribution represented in polar coordinates.
The experimental schemes for measuring quantum-mechanical phase properties of light suggested and partly also realized thus far, namely (i) amplification, with the help of a quantum amplifier, of the microscopic field before phase measurement, (ii) heterodyning the field with a strong local oscillator, and (iii) performing two separate homodyne measurements on the field after beam splitting, are compared from a theoretical point of view.They share the common feature that undesired noise enters the experimental setup, which makes the measurement fuzzy.It will be pointed out that all three schemes amount to measuring the Q function of the original field, and hence are fully equivalent.Since the Q function can be interpreted as a smoothed Wigner function, one may associate with the introduced noise a smoothing process in which intrinsically quantum-mechanical features displayed by finer details of the Wigner function -especially by the occurrence of negative values -are lost.As a consequence, the measured phase distribution will be broader than the "true" one based on the concept of a quantum-mechanical phase operator.In realistic experiments, the nonunit detection efficiency further deteriorates the measuring results.It will be shown that also this echect can be properly described by an (additional) smoothing process leading to a certain s-parametrized quasiprobability distribution, with a parameter s that is connected with the detection efficiency in a simple way, as the distribution that is actually measurable.
Extending recent calculations of GLASSGOLD and HOLLIDAY [3] the amplification process by means of a laser amplifier is treated theoretically in case of an arbitrary quantum mechanical state of the incident electromagnetic field. An explicit expression for GLAUBER's P‐function representing the state of the field after amplification, is found. Interpreting this function as a classical distribution function for the complex amplitude of the field, a distribution function W(φ; t) for the phase is obtained by averaging over the real amplitude. With the help of W(φ; t) the phase uncertainty (δφ)2 for the amplified field is calculated. The result approximately agrees with the expression for(δφ)2 following from the phase operator originally introduced by HEITLER if applying it to the incident field. Thus a connection between the phase operator and an experimental arrangement is established.