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Journal of Experimental and Theoretical Physics

Trigonometric Phase-Difference Operators of Quantum Electromagnetic Fields

A. V. Kozlovskiĭ

Published Feb 1, 2021• 1 citation
Publisher

Abstract

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.

DOI
10.1134/s1063776121010027

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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.
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 formalism that we have previously developed for the phase difference between two quantized electromagnetic fields, which is intimately connected with the measurement process, is explored further, both theoretically and experimentally. We calculate the higher moments of the measured cosine and sine operators for certain two-mode Fock states \ensuremath{\Vert}n1{\mathit{n}}_{1}n1​,n2{\mathit{n}}_{2}n2​〉, and show how the measurement itself exerts a bias on the outcome. We find that the corresponding phase difference becomes uniform over the interval 0 to 2\ensuremath{\pi} only in the limit n1{\mathit{n}}_{1}n1​,n2{\mathit{n}}_{2}n2​\ensuremath{\rightarrow}\ensuremath{\infty}. On the other hand, the phase difference associated with the product of a coherent state \ensuremath{\Vert}v〉 with a Fock state can be random for large \ensuremath{\Vert}v\ensuremath{\Vert}, because of the availability of an infinite number of photons. Several of our theoretical predictions are compared with predictions based on the Susskind-Glogower and the Pegg-Barnett operators, and they are also tested by experiment. We find that the experimental results confirm our theory in every case, and this includes tests of the higher moments of the measured cosine operator.
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.
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.