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Soviet Journal of Quantum Electronics

Quantum indeterminacy relationship for optical video pulses

Volodymyr S. Ovechko

Published May 31, 1990• 0 citations
Publisher

Abstract

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.

DOI
10.1070/qe1990v020n05abeh006513

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Europhysics Letters (EPL)•May 24, 2006•7 citations
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.
When any two optical fields are superposed, a fringe pattern will be formed if the observation interval is short enough (within the coherent time) and a phase value may be extracted from the fringe and assigned as the phase difference between the fields. When measured against a common reference field, phase can be defined for an optical field. However, when the intensities of the optical fields are low or the observation time is so short that only a few photons are detected, the fringe pattern is unrecognizable due to the particle nature of light. By analyzing the fringe patterns simulated by a quantum Monte Carlo method, we find that there exist uncertainties in determining the phase value. This uncertainty is universal for any optical field with a finite photon number and decreases to zero as the photon number of the field increases to infinity or the classical limit is approached. The existence of this uncertainty in determining the phase value makes it problematic to define a quantum mechanical phase. An interference experiment with coherent state is performed to confirm the theoretical predictions. The implication of the above is that phase is a classical quantity that is associated with a large number of particles and may not have a quantum mechanical analogy.
We analyze extremal measurement error relations for time and frequency parameters of optical pulses, which are compared with the uncertainty relation. It is shown that the product of the mean-square measurement error limit for a single photon corresponds to the minimum value of ˙/2 in the quantum uncertainty relation for time-energy (coordinate-momentum) and decreases with deviation from a Gaussian pulse shape, for which the information entropy is maximal. The error relation is in the form of the uncertainty relation with the inequality reversed.
Summary form only given. Optical techniques are widely used in many areas of science and technology to make accurate measurements. The precision of these measurements suffers from limits due to the unavoidable quantum fluctuations of light. When the light is in the quasi-classical coherent state, this limit is called ”standard quantum limit” and scales as 1/√N with N the number of photons. While better scaling can be achieved with exotic quantum states, coherent light is experimentally interesting as we are able to produce it with large N. We consider light as a probe for the measurement of a parameter p that affects the medium or the length through which the light propagates. We derived the ultimate limit of sensitivity, the so-called quantum Cramér-Rao bound, for the measurement of p [1]: for u(ω, p) the normalized mean field mode of the light, the minimal fluctuation of p that can be measured is given in Fig. 1. We also showed that this ultimate limit of sensitivity can be reached experimentally in a homodyne detection scheme, where the intense local oscillator is taken in the mode wp o ∂u ∂p, called detection mode associated with p. This measurement scheme can be applied to space-time positioning experiments such as ranging or clock synchronization [2]. The basic method for space-time positioning relies on the exchange of regular light pulses between two emitters. For coherent broadband light pulses, such as the ones delivered by a mode-locked femtosecond laser, of central frequency ω0 and of spectral width Δω, the quantum limits of the measurements of the delay t are given by δtphase = 2√Nω0 1 for interferometric measurements and δttof = 2√NΔω 1 for time-of-flight measurements. However, these two methods do not reach the quantum Cramer-Rao bound; this ultimate limit is achieved by a homodyne detection with local oscillator w(ω) o iωu(ω), and is given by 1 δtmin = (1) 2√N . Δω2 + ω2 0 For usual characteristics of a Titane:Sapphire femtosecond laser and 1 ms integration time, δtmin = 10-23s. We demonstrate this limit experimentally by using pulse shaping techniques to shape the local oscillator in the detection mode. We also study the influence of the intensity and phase noise of a mode-locked femtosecond laser on this limit. This technique can also be applied in the case of ranging in a dispersive medium such as air. In this case, dispersive effects due to environmental parameters such as pressure, temperature or humidity affect the precision of measurements. Usually, this problem is tacked by measuring separately the environmental parameters or using several wavelengths, which can be difficult and require post-processing. We show that using the same technique as previously shown, the parasitic effects due to the medium in a distance measurement can be canceled by shaping properly the local oscillator [3]: we choose for the local oscillator a mode that is orthogonal to the environmental effects, so the homodyne detection is insensitive to them, at the expense of a lower sensitivity. This scheme is an all-optical and real-time measurement of distance in a dispersive medium.
We show that the width of an arbitrary function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. In the case of ergodic stochastic processes, an ensuing uncertainty relationship is then demonstrated for the product of correlation length and variance. A closely related uncertainty principle is also established for the average degree of fourth-order coherence and the spatial width of modes of bosonic quantum fields. However, it is shown that, in the case of stochastic and quantum observables, certain non-classical states with sub-Poissonian statistics, such as for example photon number squeezed states in quantum optics, can overcome the "classical" noise-resolution uncertainty limit. This uncertainty relationship, which is fundamentally different from the Heisenberg and related uncertainty principles, can define an upper limit for the information capacity of communication and imaging systems. It is expected to be useful in a variety of problems in classical and quantum optics and imaging.