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Quantum Electronics

Phase—photon-number uncertainty relation for the field of optical pulses

Natalia Korolkova, Anatolii S Chirkin

Published Apr 30, 1998• 5 citations
Publisher

Abstract

An analysis is made of the phase—photon-number uncertainty relation for multimode electromagnetic fields (wave packets). It is shown that the variance of fluctuations of the investigated quantities depends strongly on the relation between the pulse duration τ p and the recording time T . If T ≪ τ p , the uncertainty relation is similar to that applicable to a single mode, and the phase and photon number variances follow adiabatically the changes in the pulse envelope.

DOI
10.1070/qe1998v028n04abeh001201

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An analysis is made of the phase—photon-number uncertainty relation for multimode electromagnetic fields (wave packets). It is shown that the variance of fluctuations of the investigated quantities depends strongly on the relation between the pulse duration τ p and the recording time T . If T ≪ τ p , the uncertainty relation is similar to that applicable to a single mode, and the phase and photon number variances follow adiabatically the changes in the pulse envelope.
The uncertainty principle applied to electromagnetic fields implies that the field phase and photon number cannot simultaneously have arbitrarily narrow distributions. Here we describe an experiment which determines, for the first time, the variances of phase and photon number for a field containing small numbers of photons. The measurements allow a test of the uncertainty relation for the phase and photon number of a mode of the electromagnetic field when the field mode is in a very weak coherent state. The phase of the field is described by the Pegg-Barnett phase operator formalism. [1] The measurements are accomplished by experimentally determining the quantum state of the field by using optical homodyne tomography. [2] This method of state characterization requires that an ensemble of many measurements be performed. [3] In this case, the relevant interpretation of the uncertainty relation is that the state of the field does not have well defined phase and photon number.
A theoretical investigation is made of the conditions under which an optical video pulse of femtosecond duration can be represented using classical optics. It is shown that the minimum value of the indeterminacy (uncertainty) parameter E τ 0 ℏ −1 ( E is the energy and τ 0 is the duration of a pulse) depends on the dimensionless quantity ω max τ 0 (ω max is the upper frequency limit which determines the precision of reconstruction of the shape of an optical video pulse) and on the average number of photons in a mode ⟨ N min ⟩. The latter parameter is related to indeterminacy of the mode phase and is a consequence of the quantum nature of light.
We have experimentally determined all of the quantities involved in the uncertainty relation for the phase and photon number of a mode of the electromagnetic field when the field mode is in a coherent state of small average photon number. This is accomplished by determining the quantum state of the field using optical homodyne tomography, which uses measured distributions of electric-field quadrature amplitude to determine the Wigner function and hence the density matrix. The measured state is then used to calculate the uncertainty product for the number and phase, as well as the expectation value of the commutator of the number and phase operators. The experimental results agree with the quantum-mechanical predictions. We also present measured phase- and photon-number distributions for these weak coherent states, as well as their measured complex wave functions.
Abstract Two complementary properties, such as frequency and duration of a pulse, cannot be precisely measured simultaneously. The lower bound on the time-bandwidth product, known also as uncertainty relation, plays a very important role in quantum theory. In this work, we consider single-photon pulses with arbitrary temporal and spectral profiles, and derive a lower bound for the single-photon time-bandwidth product. We investigate how this bound depends on the modal purity, the single-mode fidelity, and the effective number of modes. Our results can be used to solve the inverse problem, namely, to estimate characteristics of the single-photon pulse based on the time-bandwidth product obtained from temporal and spectral measurements of the pulse.