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Fortschritte der Physik

Phase of a Microscopic Electromagnetic Field and Its Measurement

Henryk Paul

Published Jan 1, 1974• 116 citations
Publisher

Abstract

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.

DOI
10.1002/prop.19740221104

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Laser‐Verstärker und Phasenunschärfe

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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.

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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.

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Abstract In a preceding paper [1] the phase uncertainty of an electromagnetic wave amplified by an (ideal) laser amplifier has been evaluated. Similar calculations are now performed for the amplitude and the intensity of the radiation. A special result is the conclusion that a field with minimum intensity fluctuations after amplification corresponds uniquely to a signal with a sharp photon number. The same holds approximately for the amplitude, the approximation being exactly the same as in [1] which allowed us to identify the phase uncertainty after amplification with the expression following formally from the phase operator applied to the incident field.

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Phase fromQfunction via linear amplification

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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.

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Phase properties of the quantized single-mode electromagnetic field

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The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.

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Strong linear amplification of quantum fields

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Abstract Going ahead with investigations started by ideas of H. Paul it is examined how different quantum properties in the initial field state show up in the strongly amplified output field. As examples of extreme quantum fields the number state and the recently introduced phase optimized state are used. For the linear laser amplifier the inevitable noise is fully taken into account. These results are compared with those for a coherent input field. The characteristic differences in the macroscopic output fields allow unique conclusions onto the initial field. For the calculations the positive definite Glauber's P‐representation after strong amplification is applied.

Annalen der Physik•Jan 1, 1992•5 citations
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.
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.
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.
Abstract In a preceding paper [1] the phase uncertainty of an electromagnetic wave amplified by an (ideal) laser amplifier has been evaluated. Similar calculations are now performed for the amplitude and the intensity of the radiation. A special result is the conclusion that a field with minimum intensity fluctuations after amplification corresponds uniquely to a signal with a sharp photon number. The same holds approximately for the amplitude, the approximation being exactly the same as in [1] which allowed us to identify the phase uncertainty after amplification with the expression following formally from the phase operator applied to the incident field.
The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.
Abstract Going ahead with investigations started by ideas of H. Paul it is examined how different quantum properties in the initial field state show up in the strongly amplified output field. As examples of extreme quantum fields the number state and the recently introduced phase optimized state are used. For the linear laser amplifier the inevitable noise is fully taken into account. These results are compared with those for a coherent input field. The characteristic differences in the macroscopic output fields allow unique conclusions onto the initial field. For the calculations the positive definite Glauber's P‐representation after strong amplification is applied.