Balanced homodyne detectors in quantum field theory

By examining balanced homodyne defectors (BHDs) from the general quantum-field-theoretical point of view we derive a number of generalizations of the standard analysis of their response. In particular, by allowing for interactions restricted in time (by a smooth function) we present general expressions for BHD output (in which the usual, simplifying limits can but need not be taken). Moreover, we point out the need for nonstrictly monochromatic local oscillators (i.e., the need for “pulsed” ones) in order to have well-defined quantum-field-theory (QFT) observables (the products of which, e.g., have finite vacuum expectation values). Furthermore, we show how the analysis of the detectors generalize to situations with BHDs in waveguides, Casimir cavities, or other time-independent but inhomogeneous $ϵ$ and $\mu$. The general treatment also allows us to comment on important QFT features of the detector observables such as locality (i.e., commuting for causally separated measurements) and non-null vacuum response. This leads to the conclusion that balanced homodyne-type detectors (and not single photodiodelike ones) are the appropriate tools in testing intriguing QFT predictions such as negative (subvacuum) energy densities. Finally, by recalling results on field-autocorrelation functions for QFT in Casimir cavities we show that interesting effects (large reductions of fluctuations) are to be expected if a version of BHDs were to be placed in such cavities.

Figure 1

Balanced homodyne detector with a local oscillator. The linearly polarized signal field $S$ (if present) is “blended” with a coherent state (LO), which is polarized orthogonally to $S$, on the polarizing beam splitter (PBS1). The half wave plate (HWP) reflects the planes of polarization with respect to its optical axis, thereby inducing $\pi ∕4$ shift of the plane of polarization of the signal field $S$. The subsequent PBS2 separates the two orthogonally polarized signals, which are detected at the photodiodes $Dx$ and $Dy$. The charge collected at $V$ provides a measure of the expectation value of the observable $J$ (and of its higher moments). Note that the setup is arranged in such a way, that if $S$ happened to be a monochromatic coherent state, it would be phase-matched to the LO at the point $x$, but shifted in phase by $\pi$ at the point $y$.

Figure 2

The normalized difference $[{\sigma }_G(\omega ,\underset{̱}{\mathbf{x}})-{\sigma }_{\Omega }(\omega ,\underset{̱}{\mathbf{x}})]∕{\sigma }_{\Omega }(\omega ,\underset{̱}{\mathbf{x}})$ between vacuum and Casimir spectral densities for $\omega ∊[0.4,4.5]$. The black color corresponds to negative values (subvacuum spectral densities). For $\omega L<2\pi$ Casimir spectral density vanishes, ${\sigma }_G(\omega ,\underset{̱}{\mathbf{x}})=0$, for all $\underset{̱}{\mathbf{x}}$. Discontinuities appear at $\omega L=2n\pi$.

Figure 3

Frequency dependence of the suppression of fluctuations in the dB scale, $10{\log }_{10}[{\sigma }_G(\omega ,\underset{̱}{\mathbf{x}})∕{\sigma }_{\Omega }(\omega ,\underset{̱}{\mathbf{x}})]$, for the ground state in the Casimir geometry. Plotted curves correspond to $x=5.2$ (solid) and $x=7.5$ (dashed).