Homodyne Detection of Temporally Resolved Quantum States

TL;DR

The paper addresses time-domain measurement of temporally varying quantum states using balanced homodyne detection by introducing a formalism that treats the state in a principal temporal mode while the detector operates in a time-bin basis. It develops an efficient algorithm to simulate continuous homodyne photocurrent under realistic conditions, enabling analysis of mode reconstruction and tomography. The study quantifies how modal mismatch, timing jitter, and phase jitter degrade marginals and Wigner-function reconstructions via the Bhattacharyya coefficient and related metrics, providing practical error budgeting. An open-source implementation accompanies the framework, offering a tool for high-fidelity quantum state estimation in continuous-variable quantum information processing.

Abstract

We present an analysis of the time domain measurement of temporally resolvable quantum states using balanced homodyne detection. Our approach outlines a formalism of detecting quantum states in arbitrary temporal modes via projection of the temporal mode onto a natural detector basis. We then present an algorithm for simulating the resultant photocurrent of continuous homodyne detection in the presence of a temporally resolved mode, and use this algorithm to explore the effects of realistic measurement errors on marginal reconstruction and quantum state tomography. A complete implementation of the method is provided through open source code on a GitHub repository.

Figures (5)

• Figure 1: A quantum state exists in a particular mode which is an element of a basis $\{f\}_k$ (right). The detector naturally measures in time-bin mode (left). In the detector basis, the state is spread over multiple time bins, as are the additional vacuum modes.
• Figure 2: (a) Example photocurrent for a single simulated trace of both vacuum and a Fock state $\ket{1}$. The single photon and vacuum cases are nearly indistinguishable for a single trace as the photon's quadrature measurement occupies a single mode within this trace. This is seen in (b) where the variance for each time bin over $5\times10^4$ traces is shown and the higher variance photon begins to appear.
• Figure 3: (a) $5\times10^4$ integrated homodyne traces, giving the corresponding quadrature sample. (b) Histogram of these quadrature samples, revealing the single-photon Fock state marginal being measured by the detector.
• Figure 4: Similarity between theoretical marginal and reconstructed marginal subject to different measurement errors. (a) Mode overlap error: A temporal mode $f(t)$ is reconstructed for various translations $f(t-t_0)$ resulting in different mode overlap. (b) Timing jitter: A temporal mode $f(t)$ is subject to random temporal shifts $f(t-\tau)$ for $\tau$ drawn from a normal distribution with standard deviation $\sigma$, averaged over $10^5$ realizations. (c) Phase jitter: The mode is always reconstructed in the correct temporal mode, but subjected to a random rotation in phase space. Note that symmetric states such as a Fock state are insensitive to these phase errors.
• Figure 5: Wigner functions of maximum likelihood quantum state reconstruction subject to measurement error. The reconstructed Wigner function approaches the vacuum state as errors become large. (a) The original cat state in a Gaussian mode with standard deviation $\sigma$, to generate the photocurrent traces. (b), (c), (d), (e) The reconstructed Wigner function subject to a timing jitter standard deviation of $0$, $\sigma$, $2\sigma$ and $5\sigma$ respectively. (f) The theoretical vacuum state Wigner function.