Journal of Experimental and Theoretical Physics

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

A. V. Kozlovskiĭ

Published Dec 1, 2021• 0 citations

Abstract

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.

DOI: 10.1134/s1063776121110108

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Journal of Experimental and Theoretical Physics

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

A. V. Kozlovskiĭ

Published Dec 1, 2021• 0 citations

Publisher

Abstract

DOI: 10.1134/s1063776121110108

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{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.

{\mathit{n}}_{1}

{\mathit{n}}_{2}

〉, 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

{\mathit{n}}_{1}

{\mathit{n}}_{2}

\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.