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Annalen der Physik

Laser‐Verstärker und Phasenunschärfe

A. Bandilla, Henryk Paul

Published Jan 1, 1969• 76 citations
Publisher

Abstract

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.

DOI
10.1002/andp.19694780704

Related papers

Amplituden‐ und Intensitätsschwankungen nach Verstärkung mit einem Laserverstärker

A. Bandilla, Henryk Paul

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.

Annalen der Physik•Jan 1, 1970•6 citations

Phase of a Microscopic Electromagnetic Field and Its Measurement

Henryk Paul

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.

Fortschritte der Physik•Jan 1, 1974•116 citations

Evolution of the quasiprobability distribution function of the radiation from a nonlinear optical amplifier

A. V. Kozlovskiĭ, А. Н. Ораевский, P. N. Lebedev

A study is carried out of the evolution of the classical phase portrait of the density operator of nonlinearly amplified light. A single-mode optical amplifier is considered, with quantumstatistical fluctuations of the two-level active medium and the light being amplified. The Heisenberg-Langevin equations for the field operators are obtained by adiabatic elimination of the atomic variables in the van der Pol approximation. The Fokker-Planck equation for the P-function, corresponding to the Heisenberg-Langevin equations, was solved numerically using the polynomial-expansion method. The initial quasiprobability distribution function corresponded to a pure coherent state. The Wigner quasiprobability distribution was calculated with the help of an integral transform of the P-function. Such quantumstatistical parameters of the amplifier radiation as the mean photon number and its variance, the Mandel parameter, and the mean field and the variance of the phase-amplitude quadratures of the field were calculated. O 1996 American Institute of Physics. [S1063-7761(96)00405-21

Journal of Experimental and Theoretical Physics•May 1, 1996•0 citations

Realistic Measurement of Phase

Henryk Paul, Ulf Leonhardt

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.

Acta Physica Polonica A•Jul 1, 1994•3 citations

Quantum phase and phase dependent measurements

W. P. Schleich, Stephen Mark Barnett

The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.

Physica Scripta•Jan 1, 1993•74 citations

Quantum Theory of Parametric Amplification. I

B. R. Mollow, R. J. Glauber

The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the $P$ representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and $n$-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative $P$ representation for the density operator for a single mode of oscillation.

Physical Review•Aug 25, 1967•537 citations
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.
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.
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.
A study is carried out of the evolution of the classical phase portrait of the density operator of nonlinearly amplified light. A single-mode optical amplifier is considered, with quantumstatistical fluctuations of the two-level active medium and the light being amplified. The Heisenberg-Langevin equations for the field operators are obtained by adiabatic elimination of the atomic variables in the van der Pol approximation. The Fokker-Planck equation for the P-function, corresponding to the Heisenberg-Langevin equations, was solved numerically using the polynomial-expansion method. The initial quasiprobability distribution function corresponded to a pure coherent state. The Wigner quasiprobability distribution was calculated with the help of an integral transform of the P-function. Such quantumstatistical parameters of the amplifier radiation as the mean photon number and its variance, the Mandel parameter, and the mean field and the variance of the phase-amplitude quadratures of the field were calculated. O 1996 American Institute of Physics. [S1063-7761(96)00405-21
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.
The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.
The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the PPP representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and nnn-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative PPP representation for the density operator for a single mode of oscillation.