Dynamical Phases and Quantum Correlations in an Emitter-Waveguide System with Feedback

We investigate the creation and control of emergent collective behavior and quantum correlations using feedback in an emitter-waveguide system using a minimal model. Employing homodyne detection of photons emitted from a laser-driven emitter ensemble into the modes of a waveguide allows for the generation of intricate dynamical phases. In particular, we show the emergence of a time-crystal phase, the transition to which is controlled by the feedback strength. Feedback enables furthermore the control of many-body quantum correlations, which become manifest in spin squeezing in the emitter ensemble. Developing a theory for the dynamics of fluctuation operators we discuss how the feedback strength controls the squeezing and investigate its temporal dynamics and dependence on system size. The largely analytical results allow to quantify spin squeezing and fluctuations in the limit of large number of emitters, revealing critical scaling of the squeezing close to the transition to the time crystal. Our study corroborates the potential of integrated emitter-waveguide systems—which feature highly controllable photon emission channels—for the exploration of collective quantum phenomena and the generation of resources, such as squeezed states, for quantum enhanced metrology.

Figure 1

Measurement-based feedback. A chain of $N$ emitters (modeled as two-level systems) in the vicinity of a waveguide is driven resonantly with Rabi frequency $\mathrm{\Omega }$ by an external laser with wave vector perpendicular to the chain. Photons with wavelength $\lambda$ are emitted into the right- and left-propagating modes of the waveguide with equal emission rate ${\gamma }_R={\gamma }_L$. When the separation between nearest neighbors, $a$, becomes commensurate with $\lambda$, the emission can be described as a collective process, where the emitters act as a single macroscopic spin of length $N/2$. The measurement of the light emitted into the right-propagating mode via homodyne detection is fed back as a modulation of the laser field with strength $g$.

Figure 2

Phase diagram and mean-field vs exact numerics. (a) The mean-field diagram shows a time crystal and a stationary phase where $⟨m_z{⟩}_{\mathrm{ss}}$ assumes a nonzero value. (b) Cut along the line $g=\frac{1}{2}$. The stationary value of $⟨m_z⟩$ converges to the mean-field prediction (dashed line) as the number of emitters, $N$, increases. The transition to the time-crystal phase takes place at $\mathrm{\Omega }=\frac{1}{2}\mathrm{\Gamma }$. As shown in the inset, at the critical point the spectral gap $\mathrm{\Lambda }$ of the master operator [Eq. (2)] scales as $\propto N^{-(1/2)}$. In the time-crystal phase the gap decreases as $N^{-1}$, while it is constant outside. (c) Time dependence of $⟨m_z⟩$. Inside the time-crystal phase $⟨m_z⟩$ exhibits damped oscillations whose damping rate decreases proportional to $N^{-1}$, according to the scaling of the spectral gap. Outside the time-crystal phase the system reaches a stationary state.

Figure 3

Spin squeezing. (a) Long-time behavior of the spin squeezing $\xi$ in the thermodynamic limit [Eq. (10)], computed at $\mathrm{\Gamma }t=100$. In the time-crystal phase, $\xi >1$ and increases indefinitely [see the corresponding inset in (b)]. In the stationary phase, for $g>-\frac{1}{2}$, the emitter ensemble is spin squeezed. In this region, the smallest values of $\xi$ are reached at the transition line. In the stationary phase, for $g<-\frac{1}{2}$, $\xi$ approaches a finite value larger than 1. (b) The dashed line shows the behavior of $\xi$ in the thermodynamic limit [Eq. (10)] as a function of $\mathrm{\Omega }/\mathrm{\Gamma }$, for $g=\frac{1}{2}$. For $\mathrm{\Omega }/\mathrm{\Gamma }>\frac{1}{2}$, the spin squeezing parameter increases indefinitely. Solid lines show a comparison with the spin squeezing parameter for finite-size systems [Eq. (9)], computed with the stationary state of Eq. (2). The insets show $\xi$ as a function of $\mathrm{\Gamma }t$ for the different regimes, in the thermodynamic limit. For $\mathrm{\Omega }/\mathrm{\Gamma }<\frac{1}{2}$ the stationary state of the emitters becomes quantum correlated ($\xi <1$). For $\mathrm{\Omega }/\mathrm{\Gamma }>\frac{1}{2}$, $\xi$ grows indefinitely and features persistent oscillations. On the critical line, $\xi$ tends to zero with a power-law decay $\xi \propto t^{-1}$. (c) Comparison between exact analytical results in the thermodynamic limit and finite-size numerics for the entry ${\mathrm{\Sigma }}_{xx}$ of the covariance matrix.