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Physics Letters A

Boundary-conditioned spatial redistribution of electric-field vacuum fluctuations in a mirror–beam-splitter system

Sun-Hyun Youn

Published Aug 17, 2026• 0 citations
Publisher

Abstract

We analyze electric-field vacuum fluctuations in a beam-splitter system with one port terminated by a mirror. The aim is not to propose passive generation of a squeezed quantum state, but to clarify how Maxwell boundary conditions modify the spatial mode function on which vacuum-field operators act. The mirror-side vacuum contribution acquires a standing-wave structure, so its local electric-field variance is suppressed at electric-field nodes and enhanced at antinodes. Throughout this work, the reference quantum noise level (QNL) means the local variance contributed by an unconstrained traveling-wave vacuum mode having the same spectral, polarization, transverse-mode, and field-normalization conventions as the mirror-conditioned mode. A “sub-QNL” statement therefore refers only to a specified local, mode-resolved electric-field contribution, and not to the total noise of an arbitrary detector or to a globally squeezed freely propagating state. Using semiclassical and quantum-operator descriptions, we derive the position-dependent variance at the beam-splitter outputs and show that the effect is local and mode-selective, rather than a nonlinear transformation of the quantum state. For an ideal point detector and a perfect standing wave, the position-dependent standing-wave component is suppressed by 100% at a node relative to its antinode value, although the total measured variance retains spatially uniform contributions from the remaining modes. We provide explicit suppression and signal-to-noise expressions and discuss their degradation by finite visibility and spatial averaging. We also discuss the conditions required for observing this redistribution and its relation to feedback-based noise reduction and the standard quantum limit.

DOI
10.1016/j.physleta.2026.132091

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We analyze electric-field vacuum fluctuations in a beam-splitter system with one port terminated by a mirror. The aim is not to propose passive generation of a squeezed quantum state, but to clarify how Maxwell boundary conditions modify the spatial mode function on which vacuum-field operators act. The mirror-side vacuum contribution acquires a standing-wave structure, so its local electric-field variance is suppressed at electric-field nodes and enhanced at antinodes. Throughout this work, the reference quantum noise level (QNL) means the local variance contributed by an unconstrained traveling-wave vacuum mode having the same spectral, polarization, transverse-mode, and field-normalization conventions as the mirror-conditioned mode. A “sub-QNL” statement therefore refers only to a specified local, mode-resolved electric-field contribution, and not to the total noise of an arbitrary detector or to a globally squeezed freely propagating state. Using semiclassical and quantum-operator descriptions, we derive the position-dependent variance at the beam-splitter outputs and show that the effect is local and mode-selective, rather than a nonlinear transformation of the quantum state. For an ideal point detector and a perfect standing wave, the position-dependent standing-wave component is suppressed by 100% at a node relative to its antinode value, although the total measured variance retains spatially uniform contributions from the remaining modes. We provide explicit suppression and signal-to-noise expressions and discuss their degradation by finite visibility and spatial averaging. We also discuss the conditions required for observing this redistribution and its relation to feedback-based noise reduction and the standard quantum limit.
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Electromagnetic fields of the vacuum mode near a conducting mirror are modified with respect to those in free space, with their amplitudes having a sinusoidal spatial dependence from the mirror. Therefore if we combine this spatially amplitude-modulated vacuum field mode and intense coherent light with a beam splitter, we may detect this fluctuation of the vacuum mode in a homodyne detection scheme. It will give a new method to produce squeezed states of light with a single mirror placed close to an unused port of a beam splitter. We show that the amplitude fluctuation of the combined light can be reduced by a factor of 2 below that of the coherent light. We also discuss the limitations due to the finite line width of the laser and the effective absorption length of the photodiodes.
The generation of arbitrary single-mode quantum states from the vacuum by alternate coherent displacement and photon adding as well as the measurement of the overlap of a signal with an arbitrarily chosen quantum state are studied. With regard to implementations, the transformation of the quantum state of a traveling optical field at an array of beam splitters is considered, using conditional measurement. Allowing for arbitrary quantum states of both the input reference modes and the output reference modes on which the measurements are performed, the setup is described within the concept of two-port non-unitary transformation, and the overall non-unitary transformation operator is derived. It is shown to be a product of operators, where each operator is assigned to one of the beam splitters and can be expressed in terms of an s-ordered operator product, with s being determined by the beam splitter transmittance or reflectance. As an example we discuss the generation of and overlap measurement with Schroedinger-cat-like states.
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Abstract In the absence of a signal field, vacuum entering through the empty beam splitter port is considered to be the sole contributor to the output noise of conventional two-port homodyne detection. We study a modified configuration that alters the input coefficient of vacuum, predicting an output noise less than that of the conventional configuration. Measurements, however, reveal identical output noise profiles for both the configurations. We explain the observations in terms of the incident field noise alone, and suggest that vacuum does not contribute to homodyne noise or shot-noise. We extend our results to the measurement of squeezed light, with non-ideal detectors.