Enhanced multi-parameter metrology in dissipative Rydberg atom time crystals

Abstract

The pursuit of unprecedented sensitivity in quantum enhanced metrology has spurred interest in non-equilibrium quantum phases of matter and their symmetry breaking. In particular, criticality-enhanced metrology through time-translation symmetry breaking in many-body systems, a distinct paradigm compared to spatial symmetry breaking, is a field still in its infancy. Here, we have investigated the enhanced sensing at the boundary of a continuous time-crystal (CTC) phase in a driven Rydberg atomic gas. By mapping the full phase diagram, we identify the parameter-dependent phase boundary where the time-translation symmetry is broken. This allows us to use a single setup for measuring multiple parameters, in particular frequency and amplitude of a microwave field. By increasing the microwave field amplitude, we first observe a phase transition from a thermal phase to a CTC phase, followed by a second transition into a distinct CTC state, characterized by a different oscillation frequency. Furthermore, we reveal the precise relationship between the CTC phase boundary and the scanning rate, displaying enhanced precision beyond the Standard Quantum Limit. This work not only provides a promising paradigm rooted in the critical properties of time crystals, but also advances a method for multi-parameter sensing in non-equilibrium quantum phases.

Figures (5)

• Figure 1: Experimental diagram and the criticality enhanced metrology model. (a) Energy level diagram of model. The level structure consists of the atomic ground state $\ket{g}$and three Rydberg states $\ket{R_1}$, $\ket{R_2}$ and $\ket{R_3}$. (b) Schematic diagram of the experimental setup. The experiment employs a three-photon Rydberg excitation scheme. The probe field propagates in the opposite direction to the dressing field and the coupling field, passing through the atomic vapor cell and finally being received by a photo-detector. Electrodes are used to radiate RF electric field, while antennas are used to generate microwave electric field. (c) The criticality enhanced metrology model. As the microwave amplitude (or its frequency) is varied, the system undergoes a phase transition from a thermal equilibrium phase to a time crystal phase. Near the critical point of the phase transition, the system exhibits higher sensitivity to external perturbations, which can be exploited for enhanced metrology. (c2) and (c4) represent the measured probe transmission in time and frequency domain in the thermal equilibrium phase. (c3) and (c5) correspond to the cases in the time crystal phase.
• Figure 2: Measured phase diagrams. (a) Transmission spectrum obtained by sweeping the microwave frequency $f_{\rm{MW}}$ from 6.405 to 6.605 GHz. The coupling of the microwave field to the Rydberg states induces a pronounced splitting of the transmission resonance into two distinct peaks, accompanied by coherent spectral oscillations that signal the emergence of the time crystal phase. (b) Spectrum recorded at a fixed microwave frequency $f_{\rm{MW}} = 6.455$ GHz and field amplitude $E_{\rm{MW}} = 9.33~ \rm{mV/cm}$. (c) Normalised probe transmission as a function of $f_{\rm{MW}}$ under conditions of fixed detuning, $\Delta=2\pi \times 23.4~\rm{MHz}$, and fixed microwave amplitude, $E_{\rm{MW}} = 9.33~ \rm{mV/cm}$. (d) Evolution of the transmission spectrum with increasing microwave field amplitude $E_{\rm{MW}}$ (from 1.66 to 16.58 mV/cm) under resonant drive ($f_{\rm{MW}} = 6.505$ GHz). The peak splitting widens progressively with $E_{\rm{MW}}$, and the spectral region associated with the time crystal phase also bifurcates. (e) Spectrum corresponding to $E_{\rm{MW}} = 9.33~\rm{mV/cm}$. (f) Phase transition observed by varying $E_{\rm{MW}}$ while maintaining a constant detuning of $\Delta=-2\pi\times37.9$ MHz, demonstrating a crossover from a thermal to a time crystal phase. The colour bar represents the normalised probe transmission intensity. Grey-shaded regions in all panels denote the parameter space where the time crystal phase is stabilised.
• Figure 3: Cascaded phase transitions with scanning the microwave amplitude. (a) The measured transmission $I_{\rm{T}}$ with $E_{\rm{MW}}$ varying from 0.7 mV/cm to 3.06 mV/cm when the RF field is turned off. The system exhibits no evidence of time crystal phase. (b) The recorded transmission $I_{\rm{T}}$ when the RF field is turned on, the system undergoes phase transition from the no-continuous-time-crystalline (no-CTC) phase to the CTC-1 phase and finally to the CTC-2 phase by increasing $E_{\rm{MW}}$. (c)-(e) represent the time-domain response in different phases, where (c) corresponds to no-CTC phase with $E_{\rm{MW}}=1.1$ mV/cm, (d) corresponds to the CTC-1 phase with $E_{\rm{MW}}=1.8$ mV/cm, and (e) corresponds to the CTC-2 phase with $E_{\rm{MW}}=2.8$ mV/cm, respectively. (f) Critical scaling in phase transitions. We extracted the relative oscillation amplitude of the transmission spectrum, as shown by the blue and red data points. The blue and red lines correspond to the critical scaling from the no-CTC phase to the CTC-1 phase and from the CTC-1 phase to the CTC-2 phase, respectively. The fit function is $\Delta I_{\rm{T}} = A/(1 + e^{B(E_{\rm{MW}} - E_0)}) + C$, with $A=-0.41, B=59.63, E_0= 1.44, C=0.46$ (blue line) and $A=-0.52, B=3.31, E_0= 2.47, C=0.58$ (red line).
• Figure 4: Multi-parameter enhanced sensing. (a) Measured transmission difference with $f_{\rm{MW}}$ sweeping from 6.465 GHz to 6.525 GHz and $E_{\rm{MW}}$ from 0 to 2.63 mV/cm over 5 ms. Color scale indicates the probe transmission intensity $I_{\rm{T}}$. The transmission difference $\Delta I_{\rm{T}}$ is obtained by subtracting a fitted baseline to isolate microwave-induced variations. The CTC phase, identified as the region of sharp spikes, is outlined by a gray dashed line representing the quadratic fit $E_{\rm{MW}}=a_0 f_{\rm{MW}}^2-b_0 f_{\rm{MW}}+c_0$ to its envelope, with $a_0 = 1.93\times 10^{3}$, $b_0 = 2.50\times 10^{4}$, $c_0 = 8.13\times 10^{4}$. The optical detuning $\Delta$ is fixed at approximately $2\pi\times4~\rm{MHz}$ for all measurements. (b) Measured transmission difference $\Delta I_{\rm{T}}$ at fixed microwave frequencies $f_{\rm{MW}}$ = 6.510 GHz (top), 6.505 GHz (middle), and 6.500 GHz (bottom). (c) Phase transition boundaries obtained at microwave electric field scanning rates $\nu_E$ of 0.59 $\rm{mV~cm^{-1}~ms^{-1}}$ (red squares), 1.77 $\rm{mV~cm^{-1}~ms^{-1}}$ (blue diamonds), and 4.13 $\rm{mV~cm^{-1}~ms^{-1}}$ (purple hexagons), respectively. All three datasets are fitted with parabolic functions $E_{\rm{MW}}=a_0 f_{\rm{MW}}^{2}+b_0 f_{\rm{MW}}+c_0$, where the coefficients $a_0$ and $b_0$ are fixed to the values obtained from the global fit in (a), and the red, blue, and purple curves correspond to $c_0=81328.41$, $81328.78$, and $81329.11$, respectively. (d) Evolution of the transmission difference $\Delta I_{\rm{T}}$ with the scanning rate $\nu_E$ at a frequency of $f_{\rm{MW}}=6.505~\rm{GHz}$, as shown for values of 0.59 (top), 1.77 (middle) and 2.06 (bottom) in units of $\rm{mV~cm^{-1}~ms^{-1}}$.
• Figure 5: Criticality-enhanced sensing and beyond SQL. (a) Measured phase diagram with varying scanning rates $\nu_{E}$. Color scale indicates the probe transmission intensity $I_{\rm{T}}$. The variation in $\nu_{E}$ shifts the critical point in time domain, forming a phase boundary (white circles). (b) The measured critical points as a function of $\nu_E$ with different $f_{\rm{MW}}$, resulting in distinguishable boundaries. The fit function is $t=A_{\rm{CTC}}/\nu_E + B_{\rm{CTC}}/\sqrt{\nu_E}$, yielding $A_{\rm{CTC}}=1.50$, $B_{\rm{CTC}}=-0.59$ for $f_{\rm{MW}}=6.505$ GHz; $A_{\rm{CTC}}=1.34$, $B_{\rm{CTC}}=0.26$ for $f_{\rm{MW}}=6.485$ GHz; $A_{\rm{CTC}}=1.43$, $B_{\rm{CTC}}=0.35$ for $f_{\rm{MW}}=6.480$ GHz; and $A_{\rm{CTC}}=1.38$, $B_{\rm{CTC}}=0.97$ for $f_{\rm{MW}}=6.475$ GHz. (c) shows the critical amplitude $E_c$ with error $\delta E_c$ within three independent measurements. (d) Measured system response with the RF field off. We fix the position of the contour points (white circles) at a level of -0.4, and defined as the critical amplitude $E_c$ for the case of thermal phase. (e) is the recorded distribution of contour points with $f_{\rm{MW}}=6.485$ GHz and $f_{\rm{MW}}=6.505$ GHz. The fit function is $t=A_{\rm{Th}}/\nu_E + B_{\rm{Th}}/\sqrt{\nu_E}$; the $95\%$ prediction bands are shown as red shading ($f_{\rm{MW}}=6.505~\rm{GHz}$, $A_{\rm{Th}}=1.29$, $B_{\rm{Th}}=-0.03$) and yellow shading ($f_{\rm{MW}}=6.485~\rm{GHz}$, $A_{\rm{Th}}=0.79$, $B_{\rm{Th}}=1.61$). (f) Comparison of measured error $\delta E_c$ between the thermal phase contour points (blue diamonds) and the CTC phase critical points (red circles) at $f_{\rm{MW}}=6.5~\rm{GHz}$. For the thermal system, the error $\delta E_c$ increases with $\nu_E$ and is fitted by $\delta E_c=a_{\rm{Th}}\sqrt{\nu_E}+b_{\rm{Th}}$ (blue solid curve, $a_{\rm{Th}}=0.51$, $b_{\rm{Th}}=-0.13$), with the blue band indicating the $95\%$ prediction interval. In contrast, for the CTC phase, $\delta E_c$ remains nearly constant and consistently lower, as indicated by its $95\%$ prediction band (red shading). Error bars denote the standard deviation from three independent measurements.