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

Optics and Spectroscopy

Uncertainty Relations for the Number of Photons and the Phase Operator of an Electromagnetic Field for Phase Quantum Superpositions of Coherent States

A. V. Kozlovskiĭ

Published Mar 1, 2020• 2 citations
Publisher

Abstract

Abstract We examine various uncertainty relations: (electromagnetic-field phase (the so-called “Pegg–Barnett operator”))–(number of photons), (trigonometric functions of the phase)–(number of photons), and (trigonometric functions of the phase)–(field phase). An electromagnetic field in quantum states of phase superpositions of coherent states, which are general superpositions of two coherent states with identical moduli but arbitrary phases, is considered. The rigorous uncertainty relation (Cauchy inequality) and the soft uncertainty relation (Heisenberg inequality) are examined and compared.

DOI
10.1134/s0030400x2003011x

Related papers

Uncertainty Relation for Trigonometrical Phase-Difference Operators of Quantum Electromagnetic Fields

A. V. Kozlovskiĭ

Abstract The uncertainty relations for the phase-difference operators of two electromagnetic fields, which have been proposed earlier in [10], are obtained and analyzed. The uncertainty relations for the phase-difference cosine and sine operators, as well as for the operators of the sum of the photon number and phase-difference operators for two fields, are investigated. The Fock and coherent quantum states of the fields, as well as general states of quantum superpositions of coherent field states and states of the Schrödinger’s cat of the fields, are considered. The rigorous uncertainty relation (Cauchy–Schwartz inequality) and the Heisenberg uncertainty relation for these operators and quantum field states are analyzed. The differences between the rigorous uncertainty relations and the Heisenberg uncertainty relations for the trigonometric phase-difference operators are demonstrated using examples of considered field states. It is shown that the rigorous uncertainty relations and the Heisenberg uncertainty relations are qualitatively different for coherent states as well as the states of quantum superpositions, but coincide in the case of the Fock states of the fields.

Journal of Experimental and Theoretical Physics•Dec 1, 2021•0 citations

Fluctuations and a rigorous uncertainty relation of trigonometric operators of the phase and the number of photons of an electromagnetic field for general quantum superpositions of coherent states

A. V. Kozlovskiĭ

The quantum-statistical properties of states of an electromagnetic field of general superpositions of coherent states of the form of N α,β(α⌤+e iξ β⌤ are investigated. Formulas for the fluctuations (variances) of Hermitian trigonometric phase field operators Ŝ ≡ côs φ, Ĉ ≡ sîn φ (the so-called “Susskind–Glogower operators”) are found. Expressions for the rigorous uncertainty relations (Cauchy inequalities) for operators of the number of photons and trigonometric phase operators, as well as for operators Ŝ and Ĉ, are found and analyzed. The states of amplitude $${N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i\varphi }}\rangle + {e^{i\xi }}\left| {{{\sqrt {{n_\beta }e} }^{i\varphi }}\rangle } \right.} \right.} \right)$$ , φ = φα = φβ, and phase $${N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i{\varphi _\alpha }}}\rangle + {e^{i\xi }}\left| {{{\sqrt {ne} }^{i{\varphi _\beta }}}\rangle } \right.} \right.} \right)$$ , n = n α = n β, superpositions of coherent states are considered separately. The types of quantum superpositions of meso- and macroscales (n α, n β » 1) are found for which the sines and/or cosines of the phase of the field can be measured accurately, since, under certain conditions, the quantum fluctuations of these quantities are close to zero. A simultaneous accurate measurement of cosφ and sinφ is possible for amplitude superpositions, while an accurate measurement of one of these trigonometric phase functions is possible in the case of certain phase superpositions. Amplitude superpositions of coherent states with a vacuum state are quantum states of the field with a “maximum” level of the quantum uncertainty both in the case of a mesoscopic scale and in the case of a macroscopic scale of the field with an average number of photons n α/β ≈ 0, n β/α » 1.

Optics and Spectroscopy•May 25, 2017•2 citations

A rigorous uncertainty relation (Cauchy Inequality) for an electromagnetic field in quantum superpositions of coherent states and in a two-photon coherent state

A. V. Kozlovskiĭ

It is shown that the Heisenberg uncertainty relation (or soft uncertainty relation) determined by the commutation properties of operators of electromagnetic field quadratures differs significantly from the Robertson–Schrödinger uncertainty relation (or rigorous uncertainty relation) determined by the quantum correlation properties of field quadratures. In the case of field quantum states, for which mutually noncommuting field operators are quantum-statistically independent or their quantum central correlation moment is zero, the rigorous uncertainty relation makes it possible to measure simultaneously and exactly the observables corresponding to both operators or measure exactly the observable of one of the operators at a finite measurement uncertainty for the other observable. The significant difference between the rigorous and soft uncertainty relations for quantum superpositions of coherent states and the two-photon coherent state of electromagnetic field (which is a state with minimum uncertainty, according to the rigorous uncertainty relation) is analyzed.

Optics and Spectroscopy•Apr 1, 2016•4 citations

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

Trigonometric Phase-Difference Operators of Quantum Electromagnetic Fields

A. V. Kozlovskiĭ

Quantum-mechanical operators of phase difference between two electromagnetic fields are proposed and their properties are analyzed. The Hermitian phase-difference operators are determined using the operators of interference of the two fields, which appears upon their mixing on a beam splitter. The results of calculations for the measured mean values and dispersions (fluctuations) of the trigonometric functions of the phase-difference operators, which follow from the Pegg–Barnett theory, are compared with the corresponding values for the proposed operators. Calculations and comparison are carried out for the Fock, coherent, and squeezed quantum states of electromagnetic fields. Analysis is performed for quantum microscopic fields with mean values $$\hat {n}$$ ~ 1 of the number of photons, which are currently used in quantum technologies. The quantitative agreement between two different theories is demonstrated for the Fock and squeezed states of the fields; in the case of coherent states, an appreciable quantitative difference in the predictions of the theories is observed.

Journal of Experimental and Theoretical Physics•Feb 1, 2021•1 citation

Fluctuations of the hermitian phase operator of electromagnetic field for quantum phase superpositions of coherent states

A. V. Kozlovskiĭ

Quantum-statistical properties of the phase of electromagnetic field in the case of phase superpositions of coherent states of the type Nα,β,ξneiϕα+eiξneiϕβ have been studied. Formulas for the average values and fluctuations (variances) of Hermitian phase operator ϕˆθ of the electromagnetic field (so-called Pegg–Barnett optical phase operator) in the case of general superposition of coherent states are obtained and analysed. The states of phase superpositions of the ‘Schrödinger cat’ type Nα,−α,ξ(|α⟩+eiξ|−α⟩),ϕβ=ϕα+π, and superpositions of coherent states of the type Nα,α∗,ξ(|α⟩+eiξ|α∗⟩),ϕβ=−ϕα, are considered separately.

Journal of Modern Optics•Nov 29, 2018•3 citations
Abstract We examine various uncertainty relations: (electromagnetic-field phase (the so-called “Pegg–Barnett operator”))–(number of photons), (trigonometric functions of the phase)–(number of photons), and (trigonometric functions of the phase)–(field phase). An electromagnetic field in quantum states of phase superpositions of coherent states, which are general superpositions of two coherent states with identical moduli but arbitrary phases, is considered. The rigorous uncertainty relation (Cauchy inequality) and the soft uncertainty relation (Heisenberg inequality) are examined and compared.
Abstract The uncertainty relations for the phase-difference operators of two electromagnetic fields, which have been proposed earlier in [10], are obtained and analyzed. The uncertainty relations for the phase-difference cosine and sine operators, as well as for the operators of the sum of the photon number and phase-difference operators for two fields, are investigated. The Fock and coherent quantum states of the fields, as well as general states of quantum superpositions of coherent field states and states of the Schrödinger’s cat of the fields, are considered. The rigorous uncertainty relation (Cauchy–Schwartz inequality) and the Heisenberg uncertainty relation for these operators and quantum field states are analyzed. The differences between the rigorous uncertainty relations and the Heisenberg uncertainty relations for the trigonometric phase-difference operators are demonstrated using examples of considered field states. It is shown that the rigorous uncertainty relations and the Heisenberg uncertainty relations are qualitatively different for coherent states as well as the states of quantum superpositions, but coincide in the case of the Fock states of the fields.
The quantum-statistical properties of states of an electromagnetic field of general superpositions of coherent states of the form of N α,β(α⌤+e iξ β⌤ are investigated. Formulas for the fluctuations (variances) of Hermitian trigonometric phase field operators Ŝ ≡ côs φ, Ĉ ≡ sîn φ (the so-called “Susskind–Glogower operators”) are found. Expressions for the rigorous uncertainty relations (Cauchy inequalities) for operators of the number of photons and trigonometric phase operators, as well as for operators Ŝ and Ĉ, are found and analyzed. The states of amplitude Nα,β(∣neiφ⟩+eiξ∣nβeiφ⟩){N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i\varphi }}\rangle + {e^{i\xi }}\left| {{{\sqrt {{n_\beta }e} }^{i\varphi }}\rangle } \right.} \right.} \right)Nα,β​(​ne​iφ⟩+eiξ​nβ​e​iφ⟩) , φ = φα = φβ, and phase Nα,β(∣neiφα⟩+eiξ∣neiφβ⟩){N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i{\varphi _\alpha }}}\rangle + {e^{i\xi }}\left| {{{\sqrt {ne} }^{i{\varphi _\beta }}}\rangle } \right.} \right.} \right)Nα,β​(​ne​iφα​⟩+eiξ​ne​iφβ​⟩) , n = n α = n β, superpositions of coherent states are considered separately. The types of quantum superpositions of meso- and macroscales (n α, n β » 1) are found for which the sines and/or cosines of the phase of the field can be measured accurately, since, under certain conditions, the quantum fluctuations of these quantities are close to zero. A simultaneous accurate measurement of cosφ and sinφ is possible for amplitude superpositions, while an accurate measurement of one of these trigonometric phase functions is possible in the case of certain phase superpositions. Amplitude superpositions of coherent states with a vacuum state are quantum states of the field with a “maximum” level of the quantum uncertainty both in the case of a mesoscopic scale and in the case of a macroscopic scale of the field with an average number of photons n α/β ≈ 0, n β/α » 1.
It is shown that the Heisenberg uncertainty relation (or soft uncertainty relation) determined by the commutation properties of operators of electromagnetic field quadratures differs significantly from the Robertson–Schrödinger uncertainty relation (or rigorous uncertainty relation) determined by the quantum correlation properties of field quadratures. In the case of field quantum states, for which mutually noncommuting field operators are quantum-statistically independent or their quantum central correlation moment is zero, the rigorous uncertainty relation makes it possible to measure simultaneously and exactly the observables corresponding to both operators or measure exactly the observable of one of the operators at a finite measurement uncertainty for the other observable. The significant difference between the rigorous and soft uncertainty relations for quantum superpositions of coherent states and the two-photon coherent state of electromagnetic field (which is a state with minimum uncertainty, according to the rigorous uncertainty relation) is analyzed.
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.
Quantum-mechanical operators of phase difference between two electromagnetic fields are proposed and their properties are analyzed. The Hermitian phase-difference operators are determined using the operators of interference of the two fields, which appears upon their mixing on a beam splitter. The results of calculations for the measured mean values and dispersions (fluctuations) of the trigonometric functions of the phase-difference operators, which follow from the Pegg–Barnett theory, are compared with the corresponding values for the proposed operators. Calculations and comparison are carried out for the Fock, coherent, and squeezed quantum states of electromagnetic fields. Analysis is performed for quantum microscopic fields with mean values n^\hat {n}n^ ~ 1 of the number of photons, which are currently used in quantum technologies. The quantitative agreement between two different theories is demonstrated for the Fock and squeezed states of the fields; in the case of coherent states, an appreciable quantitative difference in the predictions of the theories is observed.
Quantum-statistical properties of the phase of electromagnetic field in the case of phase superpositions of coherent states of the type Nα,β,ξneiϕα+eiξneiϕβ have been studied. Formulas for the average values and fluctuations (variances) of Hermitian phase operator ϕˆθ of the electromagnetic field (so-called Pegg–Barnett optical phase operator) in the case of general superposition of coherent states are obtained and analysed. The states of phase superpositions of the ‘Schrödinger cat’ type Nα,−α,ξ(|α⟩+eiξ|−α⟩),ϕβ=ϕα+π, and superpositions of coherent states of the type Nα,α∗,ξ(|α⟩+eiξ|α∗⟩),ϕβ=−ϕα, are considered separately.