Extracting information from a superradiant burst using simple measurements

Abstract

It is well known that superradiant decay of an ensemble of $N$ spins generates a complex non-classical state of light. Here, we consider the information content of a superradiant burst of photons: how is information encoded in the initial spin state distributed among the emitted photons, and can it be extracted using simple measurements? Despite the complexity of the photonic burst state, we show that a simple homodyne measurement combined with an optimized filter and linear estimator recovers the $N$-scaling of the quantum Fisher information of the initial spin state (including cases exhibiting $N^2$ Heisenberg scaling). Even more surprising, the temporal mode with optimal information content contains a vanishing fraction of the total emitted photons in the large-$N$ limit, suggesting an effective compressing of information. Our results and setup represent a new way to perform cavity based readout of solid-state spin ensembles that allows one to utilize resonant spin-photon interactions.

Figures (5)

• Figure 1: a) Generic setup and initial state: spin ensemble (with collective angular momentum operators $\hat{J}_\alpha$) in a cavity, where the spins have been infinitesimally perturbed from the fully-excited state by a parameter-dependent unitary $\hat{U}_\varphi$. In the picture, the rotation angle ${\varphi} \simeq 0$ has been enhanced for clarity. This initial spin state has a quantum Fisher information $\mathcal{F}_\varphi$. b) Long-time state: the spin ensemble decays superradiantly, with a $\varphi$-dependent photonic burst state generated in multiple temporal modes of the cavity's input-output waveguide. We show that for a variety of cases, a simple homodyne measurement that targets a single temporal mode of the output field can recover a constant fraction of $\mathcal{F}_\varphi$ in the large-$N$ limit, even in cases where one has Heisenberg scaling.
• Figure 2: a) Mode profiles $w(t)$ for the maximum-photons mode (Eq. \ref{['eq:most_populated_mode']}), the maximum-signal mode (Eq. \ref{['eq:most_sensitive']}) and the maximum-SNR mode (Eq. \ref{['eq:snr_optimizer']}). Results are shown for pure superradiant decay dynamics (i.e., $\chi = 0$ in Eq. (\ref{['eq:lindbladian_decay']})), in which case these modes are real and independent of the generator ${\hat{G}}$ (up to an overall time-independent phase). We take $N=512$ (large enough so the modes shapes have their asymptotic large-$N$ form). Each curve is labeled by the fractional photon number occupation of each mode $\bar{p}_u = \bar{n}_u / N$; for the maximum-SNR mode this ratio vanishes in the large-$N$ limit. The vertical line indicates $t=t_\text{SR} \simeq \frac{\log N}{\Gamma N}$, the time associated with the peak intensity of the superradiant burst. b) For a simple Ramsey encoding of a rotation angle, as described in Eq. \ref{['eq:initial_state_separable']}, we evaluate the ratio of the SNR to the quantum Fisher information (which has the value $N$) as a function of $N$. For each mode shape this ratio converges to a constant as $N \rightarrow \infty$; in particular, for the maximum-SNR mode (Eq. \ref{['eq:snr_optimizer']}), the ratio converges to $0.5$. The inset shows convergence to this value as a power law, $\frac{1}{2} - \frac{\text{SNR}}{N} \simeq 0.30 \cdot N^{-0.66}$. The linestyles and linecolors label the three modes the same way as a). The maximum-photons and maximum-signal modes converge to a SNR lower than the SNR of $w_\text{SNR}(t)$, as expected.
• Figure 3: a) Homodyne SNR scaled by $N$ for the three temporal modes $w_{\text{SNR}}(t)$ (solid line), $w_{\text{pop}}(t)$ (short dashed line), $w_{\text{sig}}(t)$ (long dashed line) as a function of the spin-spin interaction $\chi$ produced by a detuned cavity; we consider the unentangled Ramsey-protocol initial state in Eq. (\ref{['eq:initial_state_separable']}). For large values of $\chi$, there is a large difference between the SNR of the maximum-SNR mode and the naive maximum-photons mode. The opacity of the lines corresponds to different values of $N$ ($N=32, 64, 128$ for increasing opacity). Note that apart from an overall common scale factor, these curves are the same for any generator ${\hat{G}}$, and hence the figure also applies to the twist-untwist protocol which achieves Heisenberg scaling. b) We plot the fractional occupation number in Eq. \ref{['eq:ratio_num_photons']}, for the maximum SNR mode in Eq. \ref{['eq:snr_optimizer']}, for an ensemble of spins coupled to a detuned cavity. We fit the power law $r=a N^{b}$ on the results of the numerical computations, finding a consistent result for the exponent $b \simeq -0.37$ as the detuning is varied, and therefore the OAT term $\chi$ changes. The detuning through the OAT term has the effect of making the compression of the information in the few photons of the maximum-SNR mode even more evident.
• Figure 4: Decomposition of the maximum SNR mode in Eq. \ref{['eq:snr_optimizer']} in a signal mode (Eq. \ref{['eq:most_sensitive']}), and a zero signal mode, for a small number $N=16$.
• Figure 5: a) The picture represents the population of the most populated mode $w_{\text{pop}}(t)$ for $N=16$, the inset is its Wigner function, see Eq. \ref{['eq:most_populated_mode']}. b) Population and Wigner function of the mode that optimizes the SNR, see Eq. \ref{['eq:snr_optimizer']}. Contrary to the $w_{\text{pop}}(t)$, $w_\text{SNR}(t)$ has no Wigner negativity. We observe that the uncertainty in the number of photons in the mode $w_{\text{SNR}}(t)$ is larger then the uncertainty for the most populated mode. c) Wigner function of the derivative of the state of the optimal mode (used to compute the QFI). d) Probability density (and it's derivative) of the integrated homodyne current. The density function is not normalized for better visualization in comparison with its derivative.