Critical phenomena and nonlinear dynamics in a spin ensemble strongly coupled to a cavity. I. Semiclassical approach

We present a theoretical study of the nonlinear dynamics and stationary states of an inhomogeneously broadened spin ensemble coupled to a single-mode cavity driven by an external drive with constant amplitude. Assuming a sizable number of constituents within the ensemble allows us to use a semiclassical approach and to formally reduce the theoretical description to the Maxwell-Bloch equations for the cavity and spin amplitudes. We explore the critical slowing-down effect, quench dynamics, and asymptotic behavior of the system near a steady-state dissipative phase transition accompanied by a bistability effect. Some of our theoretical findings have recently been successfully verified in a specific experimental realization based on a spin ensemble of negatively charged nitrogen-vacancy centers in diamond strongly coupled to a single-mode microwave cavity [see A. Angerer et al., Sci. Adv. 3, e1701626 (2017)].

Figure 1

(a), (b) Cavity probability amplitude $a\left(t\right)$ vs time $t$ (in units of $1/{\gamma }_{\parallel }$) under the action of an external drive with constant amplitude $\eta /\kappa =0.054,0.11,0.19,0.24$ (from light to dark green), which is switched on at $t=0$. We present the numerical results (a) on a linear timescale and (b) on a logarithmic time scale to cover a much longer time interval. (c) Stationary solutions for the cavity amplitude $a_0^2$ as a function of the driving amplitude $\eta$ (log-log scale). Solid and dashed curves are stable and unstable solutions, respectively, represented by a node and a saddle. The two points at which they meet are saddle-node bifurcations. Hysteresis behavior of $a_0^2$ under smooth sweeping of the amplitude $\eta$ through the critical region is indicated by two arrows. Green symbols are stationary solutions to which the system eventually settles for the values of $\eta$ from (a) and (b) [the colors of the symbols in (c) correspond to those of the curves in (a) and (b)]. (a)–(c) $\eta$ is normalized such that $\eta =1$ lies exactly at the middle between the upper and lower saddle-node (SN) bifurcations occurring at, respectively, ${\eta }_{\mathrm{SN}}^{\mathrm{up}}=0.955$ and ${\eta }_{\mathrm{SN}}^{\mathrm{down}}=1.045$ (solid circles).

Figure 2

Pairs of critical values for the (a) driving amplitudes ${\eta }_{\mathrm{SN}}$ and (b) corresponding cavity amplitudes squared ${\left(a_0^{SN}\right)}^2$ at which SN bifurcations occur for Gaussian, $q$-Gaussian, and Lorentzian distributions (labeled by G, qG, and L in both panels). Solid circles in (a) are the upper and lower SN bifurcations from Fig. 1 and for the $q$-Gaussian distribution the same parameters are used as in Fig. 1.

Figure 3

The $z$ component of the central spin operator expectation value, ${\sigma }_c^z\left(t\right)$, [(a) in log-log scale] and the cavity probability amplitude $a\left(t\right)$ [(b) in log-lin scale] vs time $t$ (in units of $1/{\gamma }_{\parallel }$) under the action of an external drive with constant amplitude, $\eta \left(t\right)=\eta \mathrm{\Theta }\left(t\right)$, where $\mathrm{\Theta }\left(t\right)$ is the Heaviside step function and $\eta =\text{const}$. The values for $\eta$ are chosen to be the same as those shown by symbols in Fig. 1. Solid lines: full numerical solutions. Dashed lines: numerical solution of Eqs. (10) and (11) obtained under adiabatic elimination of ${\sigma }_k^{-}\left(t\right)$ and $a\left(t\right)$.

Figure 4

Long transient quench dynamics in the vicinity of the (a),(b) upper and (c),(d) lower SN bifurcations shown in Fig. 1. (a),(c) Cavity probability amplitude squared $a^2\left(t\right)$ and (b),(d) the $z$ component of the central spin operator expectation value, ${\sigma }_c^z\left(t\right)$, vs time $t$ (in units of $1/{\gamma }_{\parallel }$) under the action of an external drive with constant amplitude, $\eta \left(t\right)=\eta \mathrm{\Theta }\left(t\right)$, where $\mathrm{\Theta }\left(t\right)$ is the Heaviside step function and $\eta =\text{const}$. The values for $\eta$ are chosen slightly below (above) the upper (lower) SN bifurcation. As an initial condition, a stationary solution on (a), (b) the upper branch or (c), (d) the lower branch is chosen, which lies far away from the critical region. The closer the value of $\eta$ to the corresponding critical value ${\eta }_{\mathrm{SN}}^{\mathrm{up}}$ or ${\eta }_{\mathrm{SN}}^{\mathrm{down}}$, the longer the transient time (transient time increases from light to dark curves in all panels). Gray regions designate a gap in values of $|a_0{|}^2$, where no stable stationary solution exists (see Fig. 1).

Figure 5

Phase portraits and scalings for the quench dynamics near SN bifurcations. (a) Phase trajectories in the $[d{a}^2\left(t\right)/dt,a^2\left(t\right)]$ plane (log-log scale) for an initial condition chosen as a stationary state lying on the upper branch of the S-shaped bistability curve in Fig. 1. The driving value $\eta$ is then abruptly decreased to a value located slightly below the one at which the upper SN bifurcation occurs, ${\eta }_c\equiv {\eta }_{\mathrm{SN}}^{\mathrm{up}}=0.955$ (in the normalization of Fig. 1). For the purpose of demonstration, a few values in the proximity of the bifurcation point are taken. (b) The time $T-T_0$ (measured in units of $1/{\gamma }_{\parallel }$) needed to pass the slowing-down region [gray area in (a)] vs ${log}_{10}\left(\right|\eta -{\eta }_c\left|\right)$ is displayed by empty symbols. Dashed curve: the algebraic fit $T-T_0=\beta {|\eta -{\eta }_c|}^{-\alpha }$, with the exponent $\alpha =0.53$, $T_0=-6.25$, and $\beta =1.16$. (c), (d) Corresponding visualizations of the quench dynamics near the lower SN bifurcation at ${\eta }_c\equiv {\eta }_{\mathrm{SN}}^{\mathrm{down}}=1.045$. A stationary solution located at the lower branch is now used as an initial condition and $\eta$ is increased to a value located slightly above this bifurcation point. Dashed curve: the same algebraic fit as above with the exponent $\alpha =0.52$, $T_0=-74.7$, and $\beta =2.54$. Arrows in (a) and (c) indicate the system's evolution during the course of time. The darker a curve's color in (a) and (c), the closer the value of $\eta$ to the corresponding critical value ${\eta }_{\mathrm{SN}}^{\mathrm{up}}$ or ${\eta }_{\mathrm{SN}}^{\mathrm{down}}$.

Figure 6

(a) Stationary solutions for the cavity amplitude $a_0^2$ vs driving amplitude $\eta$ for different values of the coupling strength, $\mathrm{\Omega }/2\pi =7,8,8.84,12$ MHz (curves from left to right). The very right curve coincides with the one depicted in Fig. 1. (b) The decay rate $\zeta$ (in units of ${\gamma }_{\parallel }$) towards the corresponding stationary states displayed in (a) [see Eq. (15)]. Symbols in (b) are the minima of $\zeta$ connected with the corresponding stationary states [symbols in (a)] by dashed lines. $\eta$ is normalized as in Fig. 1.