Random-time sampling of a quantum system is introduced as a new approach to continuous quantum measurements with prospects for ultra-low-perturbation measurements. Random sampling is, e.g., naturally realized in an optical spin noise experiment when weak probe-laser light exhibits random single-photon events in the detector. We show that a direct evaluation of these detector time-traces yields power spectra that are equivalent to those of the usual continuous measurement regime. Surprisingly, this holds true even for average sampling rates much lower than the typical frequency range of the measured quantum dynamics. The third-order quantum polyspectrum (bispectrum) also contains the same information as its continuous counterpart. Many applications of random-time sampling are envisioned for high-resolution spectroscopy, circuit quantum electrodynamics, quantum sensing, and quantum measurements in general.
Random-time sampling of a quantum system is introduced as a new approach to continuous quantum measurements with prospects for ultra-low-perturbation measurements. Random sampling is, e.g., naturally realized in an optical spin noise experiment when weak probe-laser light exhibits random single-photon events in the detector. We show that a direct evaluation of these detector time-traces yields power spectra that are equivalent to those of the usual continuous measurement regime. Surprisingly, this holds true even for average sampling rates much lower than the typical frequency range of the measured quantum dynamics. The third-order quantum polyspectrum (bispectrum) also contains the same information as its continuous counterpart. Many applications of random-time sampling are envisioned for high-resolution spectroscopy, circuit quantum electrodynamics, quantum sensing, and quantum measurements in general.
The analysis of a continuous measurement record z(t) poses a fundamental challenge in quantum measurement theory. Different approaches were used in the past as records can, e.g., exhibit predominantly Gaussian noise, telegraph noise, or clicks at random times. The last case may appear as photon clicks in an optical spin-noise measurement at very low probe laser power. Here we show that such random-time quantum measurements can, similarly to the first two cases, be analyzed in terms of higher-order temporal correlations of the detector output z(t) and be related to the Liouvillian of the measured quantum system. Our analysis in terms of up to fourth-order spectra (quantum polyspectra) shows that this type of spectra reveals the same valuable information as previously studied higher-order spectra in the case of the usual continuous quantum measurements. Surprisingly, broadband system dynamics is revealed even for deliberately low average measurement rates. Many applications are envisioned in high-resolution spectroscopy, single-photon microscopy, circuit quantum electrodynamics, quantum sensing, and quantum measurements, in general.
We present general quantum mechanical expressions for higher-order moments, cumulants, and spectra of continuously measured quantum systems with applications in spin noise spectroscopy, quantum transport, and measurement theory in general. Starting from the so-called stochastic master equation of continuous measurement theory, we find that the leading orders of the fluctuating detector output z(t) with respect to the measurement strength $\ensuremath{\beta}$ are a white shot noise background, a constant measurement offset, and the leading order quantum noise of the measured operator A. Starting from quantum expressions for the multitime moments $\ensuremath{\langle}z({t}_{n})\ensuremath{\cdots}z({t}_{1})\ensuremath{\rangle}$ we derive three- and four-time cumulants that are valid in all orders of $\ensuremath{\beta}$ covering the full regime between the weak- and strong-measurement limit (Zeno limit). Intriguingly, quantum expressions for the cumulants were found that exhibit the same simple structure as those for the moments after introduction of only a slightly modified system propagator. Very compact expressions for the cumulant-based third- and fourth-order spectra (bispectrum and trispectrum) follow naturally. We illustrate the usefulness of higher-order spectra by treating a real-world two-spin system with strong hyperfine interaction. Moreover, spin noise spectroscopy is shown to have the potential for investigating the transition from weak measurements to the famous quantum Zeno regime for realistic probe laser intensities.
The theory of continuous quantum measurement allows to reconstruct the state ρt of a system from a continuous stochastic measurement record It. However, this truly continuous-time signal It is never available in practice. In experiments, one generally has access to its digitization, i.e., to a series of time averages Ik over finite intervals of duration Δt. In this letter, we take this digitization seriously and define ρˉn as the best Bayesian estimate of the quantum state given (only) a digitized record (I1,…,In). We show that ρˉn+1 can be computed recursively from In+1 and ρˉn using an exact formula. The latter can be evaluated numerically exactly, or used as the basis for a perturbative expansion into successive powers of Δt. This allows reconstructing quantum trajectories in regimes of coarse Δt where existing methods fail, estimating parameters at fixed Δt without bias, and directly sampling digitized quantum trajectories with schemes of arbitrarily high order.
Abstract A new method for efficient, high-quality randomness extraction is presented. The method relies on quantum processes such as the emission of single photons and their subsequent detection, where each detection event has an associated detection time. We show that, by utilising the number of possibilities to arrange a timed quantum measurement sequence, the efficiency can be increased many-fold compared to current methods. The method delivers fundamentally uniform randomness and therefore does not need any post processing or de-biasing.
Measurements with randomly chosen settings determine many important properties of quantum states without the need for a shared reference frame or calibration. They naturally emerge in the context of quantum communication and quantum computing when dealing with noisy environments, and allow the estimation of properties of complex quantum systems in an easy and efficient manner. In this review, we present the advancements made in utilising randomised measurements in various scenarios of quantum information science . We describe how to detect and characterise different forms of entanglement, including genuine multipartite entanglement and bound entanglement. Bell inequalities are discussed to be typically violated even with randomised measurements, especially for a growing number of particles and settings. Furthermore, we also present an overview on the estimation of non-linear functions of quantum states and shadow tomography from randomised measurements. Throughout the review, we complement the description of theoretical ideas by explaining key experiments.
The blinking statistics of quantum emitters and their corresponding Markov models play an important role in high-resolution microscopy of biological samples as well as in nano-optoelectronics and many other fields of science and engineering. Current methods for analyzing the blinking statistics like the full counting statistics and the Viterbi algorithm break down for low photon rates. We present an evaluation scheme that eliminates the need for both a minimum photon flux and the usual binning of photon events which limits the measurement bandwidth. Our approach is based on higher-order spectra of the measurement record which we model within the recently introduced method of quantum polyspectra from the theory of continuous quantum measurements. By virtue of this approach we can determine on- and off-switching rates of a semiconductor quantum dot at light levels 1000 times lower than in a standard experiment and 20 times lower those than achieved with a scheme from full counting statistics. Thus, a very powerful high-bandwidth approach to the parameter-learning task of single-photon hidden Markov models is established with applications in many fields of science.