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Journal of Modern Optics

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

A. V. Kozlovskiĭ

Published Jun 28, 2016• 3 citations
Publisher

Abstract

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.

DOI
10.1080/09500340.2016.1203463

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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.
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.
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.
Quantum superposition is a fundamental principle of quantum mechanics, which provides a crucial basis to observe phenomena beyond the predictions of classical physics. For example, a quantum entangled state can exhibit stronger correlation than classically possible one. In quantum state engineering, many new quantum states can be obtained from the superposition of many known states. In recent decades, the superposition of coherent states (CSs) with the same amplitude but two different phases has been a subject of great interest. This superposition state was often called Schrodinger cat state (here, we also name it 2-headed cat state (2HCS)), which becomes an important tool to study a lot of fundamental issues. Surprisingly, some studies have extended the quantum superposition to involving more than two component coherent states. In order to produce the superposition of three photons, people have considered the superposition of coherent states with three different phases (here, we also name it 3-headed cat state (3HCS)). Furthermore, in microwave cavity quantum electrodynamics of bang-bang quantum Zeno dynamics control, people have proposed the superposition of coherent states with four different phases (here, we also name it 4-headed cat state (4HCS)). In this paper, we make a detailed investigation on the quantum statistical properties of a phase-type 3HCS. These properties include photon number distribution, average photon number, sub-Poissionian distribution, squeezing effect, and Wigner function, etc. We derive their analytical expressions and make numerical simulations for these properties. The results are compared with the counterparts of the CS, the 2HCS and the 4HCS. The conclusions are obtained as follows. 1) The CS, the 2HCS, the 3HCS and the 4HCS have k, 2k, 3k and 4k photon number components, respectively (k is an integer); 2) small difference in average photon number among these quantum states in small-amplitude range can be observed, while their average photon numbers become almost equal in large-amplitude range; 3) the CS exhibits Poisson distribution, and the 2HCS, the 3HCS and the 4HCS exhibit super-Poisson distributions in most amplitude ranges, however, sub-Poisson distribution can be seen for the 3HCS and the 4HCS in some specific amplitude ranges; 4) except for the 2HCS that may have the squeezing property, no squeezing properties can be found in the CS, the 3HCS and the 4HCS; 5) negative values can exist in the Wigner functions for the 2HCS, the 3HCS and the 4HCS, while it is not found in the CS. Similar to the 2HCS and 4HCS, the Wigner function of the 3HCS has negative component, which implies its nonclassicality. Different from the 2HCS, the 3HCS exhibits sub-Poisson photon number distribution in a certain amplitude range, it is weaker than that of the 4HCS. At the same time, no squeezing is found in the 3 or 4HCS, which is another difference from the 2HCS.
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.
Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.