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All-Order Theory of Continuous Quantum Noise Measurements: Application to Spin Noise Spectroscopy

D. Hägele

Published Nov 7, 2016• 0 citations
OpenAlex

Abstract

The stochastic detector output $z(t)$ of a general continuously measured quantum system is calculated using a stochastic master equation approach. The resulting continuous quantum noise formula (CQNF) is valid in all orders of the measurement strength. The CQNF provides a rigorous route to the calculation of higher order noise spectra of quantum systems with applications in spin noise spectroscopy and quantum measurement theory in general.

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The stochastic detector output z(t)z(t)z(t) of a general continuously measured quantum system is calculated using a stochastic master equation approach. The resulting continuous quantum noise formula (CQNF) is valid in all orders of the measurement strength. The CQNF provides a rigorous route to the calculation of higher order noise spectra of quantum systems with applications in spin noise spectroscopy and quantum measurement theory 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)z(t)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 AAA. 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 analysis of a continuous measurement record z(t)z(t)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)z(t)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.
The optimization of quantum control for physical qubits relies on accurate noise characterization. Probing the spectral density $S(\ensuremath{\omega})$ of semiclassical phase noise using a spin interacting with a continuous-wave (CW) resonant excitation field has recently gained attention. CW noise spectroscopy protocols have been based on the generalized Bloch equations (GBE) or the filter function formalism, assuming weak coupling to a Markovian bath. However, this standard protocol can substantially underestimate $S(\ensuremath{\omega})$ at low frequencies when the CW pulse amplitude becomes comparable to $S(\ensuremath{\omega})$. Here we derive the coherence decay function more generally by extending it to higher orders in the noise strength and discarding the Markov approximation. Numerical simulations show that this provides a more accurate description of the spin dynamics compared to a simple exponential decay, especially on short timescales. Exploiting these results, we devise a protocol that uses an experiment at a single CW pulse amplitude to extend the spectral range over which $S(\ensuremath{\omega})$ can be reliably determined to $\ensuremath{\omega}=0$.
We propose a self-contained and accessible derivation of an exact formula for the n -point correlation functions of the signal measured when continuously observing a quantum system. The expression depends on the initial quantum state and on the Stochastic Master Equation (SME) governing the dynamics. This derivation applies to both jump and diffusive evolutions and takes into account common imperfections of realistic measurement devices. We show how these correlations can be efficiently computed numerically for commonly filtered and integrated signals available in practice.
In this paper, we establish a general theoretical framework for the description of continuous quantum measurements and the statistics of the results of such measurements. The framework concerns the measurement of an arbitrary quantum system with arbitrary number of detectors under realistic assumption of instant detector reactions and white noise sources. We attend various approaches to the problem showing their equivalence. The approaches include the full counting statistics (FCS) evolution equation a for pseudo-density matrix, the drift-diffusion equation for a density matrix in the space of integrated outputs, and discrete stochastic updates. We provide the derivation of the underlying equations from a microscopic approach based on full counting statistics method, a phenomenological approach based on Lindblad construction, and interaction with auxiliary quantum systems representing the detectors. We establish the necessary conditions on the phenomenological susceptibilities and noises that guarantee the unambiguous interpretation of the measurement results and the positivity of the density matrix. Our results can be easily extended to describe various quantum feedback schemes where the manipulation decision is based on the values of detector outputs.
We derive a stochastic master equation (SME) which describes the decoherence dynamics of a system in spin environments conditioned on the measurement record. Markovian and non-Markovian nature of environment can be revealed by a spectroscopy method based on weak continuous quantum measurement. On account of that correlated environments can lead to a non-local open system which exhibits strong non-Markovian effects although the local dynamics are Markovian, the spectroscopy method can be used to demonstrate that there is correlation between two environments.