Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Cotangent states of the electromagnetic field: Squeezing and phase properties

Pierre Meystre, John J. Slosser, Martin Wilkens

Published May 1, 1991• 55 citations
Publisher

Abstract

We analyze the phase and squeezing properties of the cotangent states of the electromagnetic field. We calculate the phase distribution, phase variance, and number-phase uncertainty product for these states. Under appropriate conditions, the phase distribution develops oscillations resulting from the formation of states, reminiscent of macroscopic superpositions. In other cases the cotangent states are nearly minimum number-phase uncertainty states, emulating the coherent states. We also study the quadrature squeezing properties of the cotangent states and find that under a wide range of conditions they are highly squeezed.

DOI
10.1103/physreva.43.4959

Related papers

Electromagnetic field fluctuations and uncertainty relation for general quantum superpositions of coherent states

A. V. Kozlovskiĭ

Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.

Journal of Modern Optics•Jun 28, 2016•3 citations

Superposition Phases and Squeezing

Hai-Woong Lee

Quadrature variances of a radiation field depend not only on the photon number distribution in the field but also on the relative phases of the photon number probability amplitudes. Two fields with the same photon number distribution can show different degrees of squeezing if photon number states are superposed with different relative phases. It is thus possible, for example, for a radiation field with Poissonian photon statistics to exhibit squeezed quadrature fluctuations. Since different relative superposition phases in general yield different maximum and minimum values of the quadrature variances, measurement of the variances can yield information concerning the relative phases between different number states.

Journal of Modern Optics•Jun 1, 1993•3 citations

Generation of number-phase squeezed states

Matthew J. Collett

A number-phase interaction is suggested that is analogous to the usual quadrature squeezing Hamiltonian for a mode of the electromagnetic field. Applied to an initial coherent state, it produces number-phase squeezed states. For moderate squeezing in either direction the result is close to a minimum uncertainty state. Two different ${\mathrm{\ensuremath{\chi}}}^{(3)}$ approximations to the ideal interaction are found, appropriate for the different directions of squeezing. Even with the approximate interaction, the fluctuations in photon number can be reduced to less than half a photon for an arbitrarily intense field.

Physical Review Letters•May 31, 1993•21 citations

Phase properties of macroscopic superposition states

Edwin E. Hach, Christopher C. Gerry

Phase properties of states consisting of superpositions of the states mod a) and mod -a), coherent states of a single mode quantized electromagnetic field, are studied. These states are also known as macroscopic superposition states and 'Schrodinger cat' states. After first reviewing the squeezing and antibunching properties of these states, from the density operator and the phase states we derive a probability distribution for the phase. We also compare with this distributions obtained from the associated Q-function and the Wigner function.

Quantum Optics Journal of the European Optical Society Part B•Dec 1, 1993•11 citations

q-PHASE-COHERENT STATES AND THEIR SQUEEZING PROPERTIES

Lin Zhixin, YANG YA-PING, FENG WEI-GUO +1 more

Squeezing properties of q-phase-coherent states with respect to the phase quadrature operators are studied.The relations between number-phase squeezing and number-phase uncertainty are also studied in detail for a two-state system.Some new number-phase minimum uncertainty states are found.

Acta Physica Sinica•Jan 1, 1998•2 citations

Coherent and squeezed states of the radiation field

Marlan O. Scully, Muhammad Suhail Zubairy

Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.

Cambridge University Press eBooks•Sep 4, 1997•3 citations
We analyze the phase and squeezing properties of the cotangent states of the electromagnetic field. We calculate the phase distribution, phase variance, and number-phase uncertainty product for these states. Under appropriate conditions, the phase distribution develops oscillations resulting from the formation of states, reminiscent of macroscopic superpositions. In other cases the cotangent states are nearly minimum number-phase uncertainty states, emulating the coherent states. We also study the quadrature squeezing properties of the cotangent states and find that under a wide range of conditions they are highly squeezed.
Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.
Quadrature variances of a radiation field depend not only on the photon number distribution in the field but also on the relative phases of the photon number probability amplitudes. Two fields with the same photon number distribution can show different degrees of squeezing if photon number states are superposed with different relative phases. It is thus possible, for example, for a radiation field with Poissonian photon statistics to exhibit squeezed quadrature fluctuations. Since different relative superposition phases in general yield different maximum and minimum values of the quadrature variances, measurement of the variances can yield information concerning the relative phases between different number states.
A number-phase interaction is suggested that is analogous to the usual quadrature squeezing Hamiltonian for a mode of the electromagnetic field. Applied to an initial coherent state, it produces number-phase squeezed states. For moderate squeezing in either direction the result is close to a minimum uncertainty state. Two different ${\mathrm{\ensuremath{\chi}}}^{(3)}$ approximations to the ideal interaction are found, appropriate for the different directions of squeezing. Even with the approximate interaction, the fluctuations in photon number can be reduced to less than half a photon for an arbitrarily intense field.
Phase properties of states consisting of superpositions of the states mod a) and mod -a), coherent states of a single mode quantized electromagnetic field, are studied. These states are also known as macroscopic superposition states and 'Schrodinger cat' states. After first reviewing the squeezing and antibunching properties of these states, from the density operator and the phase states we derive a probability distribution for the phase. We also compare with this distributions obtained from the associated Q-function and the Wigner function.
Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.