Post-processed estimation of quantum state trajectories

TL;DR

This work demonstrates that post-processing with future information can significantly sharpen quantum trajectory reconstruction for continuously measured systems, by extending quantum state smoothing to linear Gaussian quantum (LGQ) dynamics. The authors develop a practical LGQ smoothing formalism, apply it to a strongly monitored optomechanical resonator, and show that smoothed trajectories are closer to the true long-time-limit state and to the ideal true state than real-time filtered estimates. A key finding is that quantum smoothing yields a purer, inherently stochastic trajectory, in contrast to the smooth but potentially unphysical trajectories from classical smoothing; this nondifferentiable quantum behavior contrasts with classical expectations and is supported by autocorrelation analyses. The study also validates the framework experimentally, including noise-injection tests that reveal smoothing mitigates measurement inefficiency and enhances trajectory accuracy, with implications for quantum sensing, control, and error correction.

Abstract

Weak quantum measurements enable real-time tracking and control of dynamical quantum systems, producing quantum trajectories -- evolutions of the quantum state of the system conditioned on measurement outcomes. For classical systems, the accuracy of trajectories can be improved by incorporating future information, a procedure known as smoothing. Here we apply this concept to quantum systems, generalising a formalism of quantum state smoothing for an observer monitoring a quantum system exposed to environmental decoherence, a scenario important for many quantum information protocols. This allows future data to be incorporated when reconstructing the trajectories of quantum states. We experimentally demonstrate that smoothing improves accuracy using a continuously measured nanomechanical resonator, showing that the method compensates for both gaps in the measurement record and inaccessible environments. We further observe a key predicted departure from classical smoothing: quantum noise renders the trajectories nondifferentiable. These results establish that future information can enhance quantum trajectory reconstruction, with potential applications across quantum sensing, control, and error correction.

Figures (10)

• Figure 1: Post-processed quantum trajectories in experiment. a, A high-quality mechanical resonator, formed by the out-of-plane mode of a hierarchical arrangement of stressed silicon nitride strings, is in contact with a cryogenic thermal bath. It is also in contact with an optical bath, via a photonic crystal cavity coupled with rate $g$. A fraction $\eta$ (yellow) of the optical bath is observed, and the acquired photocurrent is processed to reconstruct the resonator's quantum trajectory. Trajectories obtained from real-time (blue) and post-processed (red) analysis of a sample measurement record diffuse in the rotating phase space spanned by mechanical quadratures $X_{1,2}$. Their means start at distant points (crosses), while their final values coincide. At intermediate time $t$, variances of the inferred Gaussian quantum states are indicated (circles), illustrating that the post-processed state $\rho_{_{\text{S}}}(t)$ is purer, with smaller variance, than the real-time filtered state $\rho_{_{\text{F}}}(t)$. b, Optical image of the large optomechanical device, with vertically stacked optical cavity and tapered waveguide for fibre coupling. A scanning electron micrograph (inset) shows the cavity's and waveguide mirror's photonic crystal design.
• Figure 1: Experimental setup. a, Control and measurement circuitry. IM, intensity modulator (electro-optic). FG, function generator. SA, spectrum analyzer. oscillo, oscilloscope. PI, proportional--integral controller. The $\phi$-labelled green box indicates a fibre stretcher. An on-resonance laser beam (red) interacts with the optomechanical device located inside a dilution refrigerator. The mechanical displacement is imprinted on the phase quadrature of the back-reflected beam, allowing the resonator’s motion to be monitored via homodyne detection. The measurement data is recorded by the oscilloscope. To stabilise the device, weak feedback is applied using a detuned laser (orange) and IM 2. IM 1 and the SA are used only for characterisation. b, Holding structure. The chip, with the optomechanical device fabricated on top, is placed upside-down in a holder and mounted onto a stack of three piezo-positioners (pos. x, y, z) to provide three-axis motorised control. The tapered fibre is fixed into place with adjustable angle to optimise alignment. A thermal anchor is sandwiched between chip holder and positioning stack, while a thermometer provides temperature readings. The full structure is placed on the bottom of the dilution refrigerator, over a series of windows that allow optical access for alignment.
• Figure 2: Cold optical and mechanical characterisation. a, Cavity reflectivity $|\mathcal{R}|^2$ is measured (circles) as the drive laser wavelength $\lambda_0 = 2\pi c / \omega_{\text{L}}$ is stepped over the optical resonance. The theoretical function (line) is fit to the measurements with linewidth $\kappa/2\pi = 11.50$ GHz. b, Overcoupling test. The DC output $V_\text{dc} \sim \operatorname{Re}[\mathcal{R}\exp{(i\omega_{\text{L}} L/c)}]$ of the balanced detector is measured as the laser wavelength is quickly swept without locking the interferometer phase --- $L$ is the optical path difference between the two arms. As here $\exp{(i\omega_{\text{L}} L/c)}\approx1$ at $\omega_{\text{L}}=\omega_{\text{c}}$, the negative sign of $V_\text{dc}$ at the resonance wavelength indicates that the cavity is overcoupled. c, Mechanical spectrum around the resonance frequency $\Omega/2\pi=1.04$ MHz (with feedback off). d, Mechanical ring-down. An exponential decay is fitted with intrinsic energy decay rate $\Gamma = 2\pi\times(11.5\,$mHz).
• Figure 3: Cold characterisation of single-photon optomechanical coupling rate. a-c, Mechanical spectra of the fundamental mode ($\Omega_\text{int}$) as the laser detuning $\Delta$ is varied, with constant input power $P_\text{in}$ and feedback laser off. For low power $P_\text{in}=2$ nW (a), the typical detuning-asymmetric spring shift behaviour is absent. Instead, the mechanical frequency $\Omega_\text{eff}$ tracks the cavity photon number $\bar{n}_\text{cav}$, indicating an absorption-induced thermal shift. For intermediate power $P_\text{in}=60$ nW (b), the optical spring effect appears on top of the thermal shift. For higher power $P_\text{in}=300$ nW (c), high-amplitude self-oscillations occur at both positive and negative detunings. d, Mechanical resonance frequencies $\Omega_\text{eff}$ extracted from mechanical spectra measured on resonance ($\Delta = 0$) for a range of photon numbers $\bar{n}_\text{cav}$. A thermal shift is observed at lower cavity intensities that saturates once $\bar{n}_\text{cav}\gtrsim7$. An exponential saturation curve is fitted for reference. e-h, Mechanical spring shift spectra for constant cavity photon number $\bar{n}_\text{cav}$, measured around the thermally shifted mechanical frequency $\Omega$ with feedback laser off. At the lowest $\bar{n}_\text{cav} = 0.5$, an inverted spring shift stemming from photothermal back-action is observed. The shift is stronger before condensed gas boil-off (e) than after (f). At intermediate $\bar{n}_\text{cav} = 2.3$ (g), radiation pressure and photothermal back-action tend to cancel each other, leading to a small spring effect. At a higher $\bar{n}_\text{cav} = 7.0$ (h), where the thermal shift in panel d is saturated, a strong optical spring with the typical detuning dependence is observed. i, Radiation-pressure-induced spring is fitted to the mechanical frequencies extracted from panel h. j, Maximal squared spring shift $\Delta \Omega_\text{max}^2 = \max_\Delta\{|\Omega_\text{eff}^2 - \Omega^2|\}$ obtained from measuring at the three detunings $\Delta = 0, \pm \kappa/2$ with high values of $\bar{n}_\text{cav}$. Error bars represent the difference between the measured values of $\Delta \Omega_\text{max}^2$ with $\Delta = +\kappa/2$ and $\Delta = -\kappa/2$, while the data points are found by averaging those values. For radiation-pressure-induced spring, $\Delta \Omega_\text{max}^2=4\Omega g_0^2/\kappa\times\bar{n}_\text{cav}$. The reliable linear fit (with a zero y-intercept) gives the single-photon coupling rate as $g_0/2\pi=159$ kHz.
• Figure 4: Fluctuations of the mean value traces. The quadrature "velocity" $({\rm d}\langle\hat{X}_j\rangle_\text{C}^{\eta\downarrow}/{\rm d} t)$ autocorrelation function (VACF) for filtering (blue), quantum state smoothing (red) and classical smoothing (green). Its decay, indicated by the dashed line, is markedly slower for classical smoothing. The function is first averaged over $j$ and across the ensemble of measurement records in the noise-injection experiment, and then normalized to its maximum value.