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arXiv:1805.07863 · Physical review. D/Physical review. D.

Maximal subvacuum effects: A single mode example

Anastasia Korolov, Lawrence H. Ford

Published Aug 28, 2018• 2 citations
arXivHTMLPDF

Abstract

We discuss an example of a subvacuum effect, where a quantum expectation value is below the vacuum level, and is hence negative. The example is the time average of the mean squared electric field in a nonclassical state where one mode is excited. We give some specific examples of such states, and discuss the lower bound on the squared field or its time average. We show when a lower bound can be obtained by diagonalization of the squared electric field operator, and calculate this bound. We also discuss the case of an instant time mean squared electric field, when the operator cannot be diagonalized. In this case, a lower bound still exists but is attained only by the limit of a sequence of quantum states. In general, the optimum lower bound on the mean squared electric field is minus one-half of the mean squared electric field in a one-photon state. This provides a convenient estimate of the subvacuum effect, and may be useful for attempts to experimentally measure this effect.

DOI
10.1103/physrevd.98.036020

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We discuss an example of a subvacuum effect, where a quantum expectation value is below the vacuum level, and is hence negative. The example is the time average of the mean squared electric field in a nonclassical state where one mode is excited. We give some specific examples of such states, and discuss the lower bound on the squared field or its time average. We show when a lower bound can be obtained by diagonalization of the squared electric field operator, and calculate this bound. We also discuss the case of an instant time mean squared electric field, when the operator cannot be diagonalized. In this case, a lower bound still exists but is attained only by the limit of a sequence of quantum states. In general, the optimum lower bound on the mean squared electric field is minus one-half of the mean squared electric field in a one-photon state. This provides a convenient estimate of the subvacuum effect, and may be useful for attempts to experimentally measure this effect.
Subvacuum effects arise in quantum field theory when a classically positive quantity, such as the local energy density, acquires a negative renormalized expectation value. Here we investigate the case of states of the quantized electromagnetic field with negative mean-squared electric field and their effects on the propagation of light pulses in a nonlinear dielectric material with a nonzero third-order susceptibility. We identify two distinct signatures of the subvacuum effect in this situation. The first is an increase in the speed of the pulse, which is analogous to the superluminal light propagation in gravity which can arise from negative energy density. This increase in speed leads to a phase shift which might be large enough to observe. The second effect is a change in the frequency and power spectra of the pulse. We identify a specific measure of the modified spectra which can signal the presence of a negative mean-squared electric field. These ideas are implemented in the particular example of a wave guide filled with a nonlinear dielectric material.
The general expressions for the time development of the ensemble averages of E^- and E^-E^+ are found for N two level molecules (TLMs) interacting with a single mode electromagnetic field. The TLMs are assumed to be at equivalent mode positions in the field. In the main body of the paper, the general results are then simplified by assuming that all TLMs are in the up state at time 0. Examples of spontaneous emission behavior including both Super Radiant behavior and emission suppression are shown in Appendix E while stimulated emission with all TLMs initially in the lower state is examined in Appendix F. Both resonant and non resonant cases are considered in the main body. Analytic results are presented for low numbers of TLMs. In the appendices, it will be shown that for the mean photon number large in comparison to the number of TLMs, that the Average Field Approximation (AFA) is a useful representation with much simpler analytic form. The validity of this approximation will also be examined. In the main paper, a comparison between the exact results and results using the AFA will be presented for a small number of TLMs and a large average photon number. It will be found that the results are in general remarkably similar but for non resonance differ somewhat at a finer level of detail. Of interest is that the ensemble averages for N TLMs is similar to the ensemble average for a single TLM multiplied by the number of TLMs which is somewhat disappointing but is believed to be the consequence of assuming that all TLMs are at equivalent mode positions and large mean photon number. Some results for smaller photon number will also be presented
Quantifying nonclassicality of a bosonic mode is an important but challenging task in quantum optics. Recently, a nonclassicality measure based on the concept of operational resource theory has been proposed [W. Ge, K. Jacobs, S. Asiri, M. Foss-Feig, and M. S. Zubairy, Phys. Rev. Res. 2, 023400 (2020)], which shows several crucial properties as a resource for quantum metrology. Here we apply the measure to evaluate and categorize different classes of nonclassical states. We discover a class of states that can achieve the maximum nonclassicality in the asymptotic limit of a large mean number of excitations. These states can provide the same sensitivity in place of a squeezed vacuum in a sensing scheme, e.g., the laser interferometer gravitational-wave observatory (LIGO) experiments. We also discover that the nonclassicality of certain states can be greatly improved with a single-photon addition. In addition, we explore some examples on how to evaluate analytically the nonclassicality of mixed states, which are in general difficult to calculate due to the convex roof nature of the measure. Our results will be useful for preparing and utilizing nonclassical states for applications in precision-sensing tasks, such as quadrature sensing and phase sensing in a Mach-Zehnder interferometer.
We calculate spatial localization and phase properties of spontaneously emitted photons. Our model is simple yet fully quantized: the emitting atom is a two-level atom located in a one-dimensional multimode optical cavity. Although the photon state vector does not have a position-space representation, the expectation value of the square of the electric field operator (intensity) is spatially localized and this pattern shifts at the speed of light. The emitted photon exhibits classical-like phase properties in the intensity expectation value when it “interferes with itself” after reflection. The phase properties of the emitted radiation are also evident at times sufficiently long for the radiation to have returned to the emitting atom before the decay of excited atomic state is complete.
The isolation usually encountered in optics between the part of a field that is of interest and its source motivates the consideration of a radiation mode weakly coupled to a quantum-mechanical source. After the introduction of some refinements into the quantum mechanics of a damped radiation mode, the field is expressed as the sum of two parts, one due to the source (the source field) and the other due to the loss mechanism (the "vacuum" field). The characteristic function for the field is calculated up to second order in perturbation theory. This function is then compared with the characteristic function for the field in the presence of a classical source. A method is exhibited by which a classical source can be found such that the two characteristic functions are identical when averaged over a half cycle. In particular, the two sources yield the same expectation values for the instantaneous amplitude and energy of the field. The description of the equivalent classical source must be given in statistical terms, in general, and requires only a knowledge of 〈S(0)(t)〉〈{S}^{(0)}(t)〉〈S(0)(t)〉 and 〈S(0)(t1)S(0)(t2)〉〈{S}^{(0)}({t}_{1}){S}^{(0)}({t}_{2})〉〈S(0)(t1​)S(0)(t2​)〉, where S(0){S}^{(0)}S(0) is the dipole-moment operator of the quantum-mechanical source unperturbed by the mode under consideration (but otherwise arbitrarily complex, with the possibility of strong coupling to other modes). The theory is illustrated by a consideration of several simple sources---a two-level system, a harmonic oscillator, and a blackbody---for which equivalent classical sources are found. The two-time correlation functions for the field obtained with the two types of sources are compared and are shown to be the same up to first order in $\ensuremath{\xi}\ensuremath{\tau}$, where $\ensuremath{\tau}$ is the difference between the two times and $\ensuremath{\xi}$ is the inverse of the field relaxation time; the physical meaning of the second-order difference in the correlation functions is discussed. A limiting process, in which both the coupling to the source and the damping become small, is suggested as a method of adapting the results to free fields, but it is pointed out that for discussion of a single mode, a free field is physically less satisfactory than a damped field. It is concluded that, within a reasonable approximation scheme, the source field may be described classically (the "vacuum" field furnishing all the necessary quantum-mechanical properties of the total field).