Continuous Real-Time Tracking of a Quantum Phase Below the Standard Quantum Limit

We propose a scheme for continuously measuring the evolving quantum phase of a collective spin composed of $N$ pseudospins. Quantum nondemolition measurements of a lossy cavity mode interacting with an atomic ensemble are used to directly probe the phase of the collective atomic spin without converting it into a population difference. Unlike traditional Ramsey measurement sequences, our scheme allows for real-time tracking of time-varying signals. As a bonus, spin-squeezed states develop naturally, providing real-time phase estimation significantly more precise than the standard quantum limit of $\mathrm{\Delta }{\varphi }_{\mathrm{SQL}}=1/\sqrt{N}\mathrm{rad}$.

Figure 1

Schematic and working principle. (a) Two lasers drive a collection of atoms to interact with a cavity mode. The relative phase $\varphi (t)$ can be continuously tracked by homodyne detection of the field leaking out. (b) Cavity-assisted Raman transitions: The red (blue) pathway leads to the emission of a cavity photon accompanied by a spin flip $|\downarrow ⟩\to |\uparrow ⟩$ ($|\uparrow ⟩\to |\downarrow ⟩$). (c) Hierarchy of frequencies. (d) Classical Bloch vector picture: The red and blue pathways set up balanced, opposing superradiance pathways that lead to a coherent cancellation of the intracavity field when the Bloch vector (green) is along the $y$ axis ($\varphi =0$). When the Bloch vector has a small $x$ component ($\varphi \ne 0$), the intracavity field from the two pathways add constructively, giving rise to nonzero output field.

Figure 2

Real-time continuous tracking of a time-varying phase. (a) A single experimental run: A squeezed state is prepared during $[-50T_0,0]$, with the initial measured phase ${\varphi }_0^{(m)}$ (blue triangle) varying in each run. Subsequently, a phase modulation ${\varphi }^{(a)}(t)=15\mathrm{mrad}\times \sin (t/40T_0)$ (black line) is applied, e.g., using a time-varying magnetic field. The blue, solid (red, hollow) markers are estimates ${\varphi }_{\mathrm{SSS}}^{(m)}$ (${\varphi }_{\mathrm{CSS}}^{(m)}$) of the phase using the measured photocurrent in windows of duration $8T_0$ that account for (do not account for) the initial offset ${\varphi }_0^{(m)}$. The gray shaded region indicates the $1\sigma$ SQL tolerance for this applied signal. Representative Bloch spheres for $t\le 0$ indicate the state before and after the state preparation stage. For $t>0$, Bloch spheres indicate the deflection of the spin as a result of the phase modulation (black dots on the spheres indicate the zero phase reference), as well as the equivalent spin state used for the respective estimates ${\varphi }_{\mathrm{CSS}}^{(m)}$, ${\varphi }_{\mathrm{SSS}}^{(m)}$. (b) Histogram of phase errors ${\varphi }_{\mathrm{SSS}}^{(m)}-{\varphi }^{(a)}$ (blue) and ${\varphi }_{\mathrm{CSS}}^{(m)}-{\varphi }^{(a)}$ (red) over 2048 runs in one particular measurement window [$48T_0$, $56T_0$]. (c) Single-run precision gain of the estimates ${\varphi }_{\mathrm{SSS}}^{(\mathrm{m})}$ relative to the SQL at different window centers $t$. Here, $\mathrm{\Delta }{\varphi }_{\mathrm{SSS}}^2$ is the variance of Gaussian fits to histograms such as the blue histogram in (b). Decoherence results in decreased gain over time.

Figure 3

(a) A sudden jump in the phase with amplitude ${\varphi }_J=40\mathrm{mrad}$ at $T_J=50T_0$ is tracked in the same run using moving windows of durations $T_W=2T_0$ (red) and $T_W=20T_0$ (blue), showing the faster response of the shorter window. (b) Protocol to estimate ${\varphi }_J$. (c) Histograms, over 2048 runs, of ${\varphi }_J^{(m)}$ for $T_W=2T_0$ (red) and $T_W=20T_0$ (blue), demonstrating the greater precision of the longer window. For $T_W=2T_0$, $W_2$ was offset by a small time $0.2T_0$ to allow transients on timescales of ${\kappa }^{-1}$ to decay. (d) Gain in precision over a CSS in the Ramsey mode as the duration of $W_1$ and $W_2$ is varied, for fixed $C{\gamma }_{\mathrm{sc}}$ and different values of $C$. Analytic results (lines) calculated using Eqs. (7) and (8) are in excellent agreement with simulations (markers).