The Boundary Time Crystal as a light source for quantum enhanced sensing beyond the Heisenberg Limit

Abstract

Modern precision measurements, such as interferometry for detecting gravitational waves, rely on the estimation of optical phases encoded in light fields. Here, we propose to exploit the collectively enhanced output field of a driven-dissipative many-body open quantum system as a light source in order to improve the precision of estimating optical phases. Pronounced temporal correlations of such output fields benefit the sensitivity of measurement protocols, which we show theoretically by employing a boundary time crystal as a light source. The fundamental bound on the precision of such estimation shows scaling with system size that surpasses the Heisenberg limit and obeys the standard quantum limit in the measurement time. This scaling can be partially harnessed by a protocol, in which the phase shifted light field is guided into an auxiliary replica system, which serves as a detector that is sensitive to non-trivial temporal correlations of the light.

Figures (11)

• Figure 1: The boundary time crystal as a light source. (a) The emission field of a boundary time crystal (BTC) composed of $N$ emitters and driven with Rabi frequency $\omega$ is used as a resource for probing an unknown sample. The sample encodes a phase $\varphi$ on the output field, which is described by the unitary $e^{-i\hat{\Lambda}\varphi}$, with the photon count operator $\hat{\Lambda}$. (b) An auxiliary, identical BTC with a tunable phase shift $\varphi'$ imprinted on its output is used as a decoder for the information encoded on the output field. (c) The output time averaged intensity $I_T$ of the source-decoder system is monitored and is sensitive to the phase difference $\Delta\varphi = \varphi-\varphi'$. For $\Delta\varphi=0$, the average intensity vanishes, rendering the decoder a perfect absorber for the phase shifted input light. Values presented are computed for $N=6$ and $\omega/\omega_\mathrm{c}=4$. (d) The output field is superposed with the field of a strong local oscillator in a homodyne detection scheme, using a low reflectivity beam splitter (LRBS) with homodyne phase $\beta$. The indicated output port is monitored, yielding the homodyne current $J_T$ which is sensitive to the phase difference $\varphi-\beta$, for which results are presented in (e) for $N=6$ and $\omega/\omega_\mathrm{c}=0.5$.
• Figure 2: Estimation error using the average homodyne current protocol. (a) Time rescaled long-time estimation error $\overline{\delta\varphi}$ for the homodyne protocol as a function of the Rabi frequency $\omega$, for $N=40$ and different values of $\varphi-\beta$. The dashed lines represent the respective Holstein-Primakoff (HP) approximated curves. (b) Derivative of the long-time averaged homodyne current $|\partial_\varphi J_T|$ and its variance $\overline{\sigma_{J_T}}^2$ as a function of the Rabi frequency $\omega/\omega_\mathrm{c}$, for $N=40$ and $\varphi-\beta = 0$.
• Figure 3: Stationary properties of the cascaded system.$N=6$ for all panels. (a) Time averaged intensity in the stationary state as a function of Rabi frequency $\omega$ and phase difference $\Delta \varphi$. (b) Purity of the stationary state as a function of Rabi frequency $\omega$ and phase difference $\Delta \varphi$. (c)-(d) Von Neumann entropy $S_\mathrm{VN}$ reduced to the source (c) and to the decoder (d) in the stationary state. For (a), (b) and (d), the red dashed line indicates the cascaded mean field transition line $\omega_\mathrm{c, casc}$ from the stationary regime (i) to an intermediate regime, where only the decoder is in a time crystal regime (ii). For (a)-(d), the red dotted line corresponds to the single system transition line $\omega_\mathrm{c}$ and marks the transition to a regime where both source and decoder display time crystal behavior (iii).
• Figure 4: Estimation error using the perfect absorber protocol. (a) Time rescaled long-time estimation error as a function of phase difference $\Delta\varphi$, for $\omega/\omega_\mathrm{c}=4$ in the cascaded time crystal regime (iii) and $\omega/\omega_\mathrm{c}=0.25$ in the cascaded stationary regime (i), and system size $N=11$. The black dashed-dotted line is the HP approximated result for $\omega/\omega_\mathrm{c} = 0.25$. (b) Estimation error as a function of system size for different values of $\Delta\varphi$, and $\omega/\omega_\mathrm{c}=4$ in the cascaded time crystal regime (iii). The green dotted line indicates the limiting values given by the quantum Fisher information (QFI) rate in the limit of $\omega/\omega_\mathrm{c}\to\infty$. Fitting a power law $\overline{\delta\varphi} = bN^{-\alpha}$ through the presented data points yields the scaling exponents $\alpha=1.222\pm0.018$ for $\Delta\varphi = 0.005$ and $\alpha=1.04\pm0.04$ for $\Delta\varphi = 0.01$. (c) Absolute derivative and variance of the time averaged intensity as a function of the phase difference $\Delta \varphi$, for different system sizes and fixed value $\omega/\omega_\mathrm{c}=4$, in the cascaded time crystal regime.
• Figure S1: Beyond Heisenberg scaling of the QFI rate. (a) Square root of the system size rescaled long-time limit QFI rate as a function of the Rabi frequency $\omega$ and for varying system sizes $N$. The dashed lines indicate the saturating value $f_{\varphi,\infty}$. (b) Square root of the system size rescaled QFI rate as a function of the system size $N$ for selected constant values of $\omega/\omega_\mathrm{c}$. For i and ii, the asymptotic behavior is determined by a power law fit yielding scaling behaviors of $f_\varphi \propto N^{\alpha_i}$ with $\alpha_{\text{i}} = 3.923 \pm 0.004$ and $\alpha_{\text{ii}} = 2.8417 \pm 0.0018$. The red dashed line indicates the saturating value $f_{\varphi,\infty}$. For both panels, numerical values are obtained, using diagonalization of Eq. (\ref{['eq:deformed_me_qfi']}) and $f_{\varphi,\infty}$ corresponds to Eq. (\ref{['eq:infinite driving qfi']}).