Photodetection of Squeezed Light: a Whittaker-Shannon Analysis

TL;DR

The paper develops a nondegenerate Whittaker-Shannon decomposition to describe squeezed light with temporally localized modes, enabling a CW-limit analysis that complements traditional Schmidt-based approaches. By disentangling the multimode squeezing operator and partitioning the ket into time windows, it introduces a squeezing-strength measure |𝔟β| tied to photon-pair density and derives explicit expressions for quadrature variances, time-window coincidence probabilities, and Hong-Ou-Mandel visibilities in both CW and pulsed regimes. The approach yields general, bandlimited formulas applicable to arbitrary joint amplitudes, and reveals contrasting roles of strong vs. weak squeezing: quadrature squeezing grows with many pairs, while pair-correlations dominate in the weak regime. Overall, the work provides a framework for predicting and interpreting quantum-optical measurements in CW, multimode squeezed light, with potential extensions to a temporally local Schmidt analysis and practical relevance for quantum sensing and information protocols.

Abstract

The Whittaker-Shannon decomposition provides a temporally localized description of squeezed light, making it applicable in the CW limit and leading to a definition of squeezing strength based on the number of photon pairs at a time. We show examples of its usefulness by calculating quadrature variance in a homodyne detection scheme, coincidence detection probabilities in the continuous-wave limit, and analyzing the Hong-Ou-Mandel effect for strongly squeezed light. Quadrature uncertainty falls farther below the shot noise limit when squeezing is strong, but effects due to correlations between photon pairs are most significant with weak squeezing. Our analysis extends previous results to more general scenarios, and we leverage the Whittaker-Shannon formalism to interpret them based on the temporal properties of photon pairs.

Figures (12)

• Figure 1: a) Normalized double-Gaussian joint spectral amplitude $|\gamma(\omega_1,\omega_2)|^2$ with axes normalized by $\Omega$. b) Normalized double-Gaussian joint temporal intensity $|\overline{\gamma}(t_1,t_2)|^2$ with axes normalized by $T_p$. c) Amplitudes $r_{nm}$ of the Whittaker-Shannon decomposition of the double-Gaussian joint amplitude. Observe that $r_{nm}$ is small when more than one space away from the diagonal. The black square represents the nonzero elements of $\boldsymbol{\beta}^J$ we could take if we were interested in times close to $t_J=7\tau$; any $r_{nm}$ outside of the box is either small, or has both $n\tau$ and $m\tau$ far from $t_J$. These plots were made with $T_p/T_c=15$.
• Figure 2: Homodyne Detection Scheme. The signal of interest (for us, a multimode degenerate squeezed state, represented by $\overline{a}(t)$) are mixed on a 50:50 beam splitter with a local oscillator represented by $\overline{c}(t)$. Photodetectors produce currents $i_1(t)$ and $i_2(t)$, and we measure the differential current $i(t)=i_1(t)-i_2(t)$.
• Figure 3: Whittaker-Shannon modes $\overline{\chi}_n(t)$ for $n=4,12$ inside a time window $T\gg\tau$, they are approximately orthogonal inside the window. If $T$ is sufficiently large there will be enough modes inside the window that we can neglect edge effects.
• Figure 4: Variance of spectral squeezing ($\theta=\pi/2$) and anti-squeezing ($\theta=0$) vs. $\omega$ for the double Gaussian joint amplitude in the CW limit with $|\mathring{\beta}|=0.1$, in a time window of size $d_J=60$ centered at $t=0$. The most squeezing is obtained for $\omega=0$ at the center of the joint spectral amplitude, and squeezing is reduced as $\omega$ gets farther from the center. To maintain accuracy for $\omega$ close to $\Omega/2$ and show where the squeezing goes to zero, this calculation used an increased bandlimit of $\Omega'=2\Omega$.
• Figure 5: Squeezing (upward slopes) and anti-squeezing (downward slopes) vs. $|\mathring{\beta}|$ of the double Gaussian in a time window of size $d_J=60$ centered at $t=0$. The squeezing of the total homodyne charge measurement increases as the ratio $T_p/T_c$ gets larger. We also plot the squeezing of a spectral analysis homodyne measurement at $\omega=0$ in the CW limit; it is not quite as strong as for the total charge measurement in the CW limit, aligning with the fact that spectrum analysis is not necessarily the optimal homodyne measurement Shapiro:97. For all scenarios, the squeezing (in dB) depends linearly on $|\mathring{\beta}|$.