Dynamical evolution of an inverted spin ensemble in a cavity: Inhomogeneous broadening as a stabilizing mechanism

We study the evolution of an inverted spin ensemble coupled to a cavity. While the inversion itself presents an inherent instability in the system, the inhomogeneous broadening of spin-resonance frequencies provides a stabilizing mechanism. The detailed behavior of mean values and variances of the spin components and the cavity field is accounted for under both stable and unstable conditions.

Figure 1

The physical setup under consideration. A spin ensemble is coupled to a cavity field ${\widehat{a}}_{\mathrm{c}}$, which is subjected to decay through the two mirrors with field-decay rates ${\kappa }_1$ and ${\kappa }_2$. For diagnostics purposes an external driving field $\beta$ may be applied, giving rise to reflected and transmitted fields, ${\widehat{a}}_{\mathrm{R}}$ and ${\widehat{a}}_{\mathrm{T}}$, as discussed in Sec. 5.

Figure 2

Cavity-field decay vs time when the inhomogeneous broadening is Lorentzian [dashed lines, from Eq. (15)] and Gaussian [symbols, numerical simulation]. In all cases, $g_{\mathrm{ens}}=2\Gamma$ and ${\gamma }_{\perp }=0$, while $\kappa$ is varied such that $C$ attains the values of 0.05 (tip-up triangles), 0.2 (circles), 0.5 (squares), 1 (diamonds), and 2 (tip-down triangles). The vertical axis contains $\langle {\widehat{X}}_{\mathrm{c}}\rangle =(\langle {\widehat{a}}_{\mathrm{c}}\rangle +\langle {\widehat{a}}_{\mathrm{c}}^{†}\rangle )/\sqrt{2}$. In the range $\Gamma t⩾2.5$ the slope of the Gaussian-broadening decay curves are compared to a numerically determined value (solid lines) as discussed in the text prior to Eq. (17). The solid curve through the tip-up triangles is given by Eq. (18).

Figure 3

Panels (a)–(e) show various dynamical parameters vs time with fixed $g_{\mathrm{ens}}=2\Gamma$ and varying $\kappa$, such that $C$ attains the values $0.05$ (magenta [light gray]), $0.2$ (red [gray]), $0.5$ (green [light gray]), 1 (blue [dark gray]), and 2 (black), represented in the order from the the lower to the upper sets of dashed and solid curves in all plots. Solid lines have been calculated for Gaussian inhomogeneous broadening (${\gamma }_{\perp }=0$), while the dashed lines correspond to both Lorentzian broadening (${\gamma }_{\perp }=0$) and homogeneous broadening (${\gamma }_{\perp }=\Gamma$). (a) Excess variance of ${\widehat{S}}_x$ relative to the coherent-state value of $N$. (b) Excess variance of ${\widehat{P}}_{\mathrm{c}}$ relative to the coherent-state value of $\frac{1}{2}$. (c) The mean value of $S_x$ normalized to $S_x(t=0)$. (d) The mean value of ${\widehat{P}}_{\mathrm{c}}$ in arbitrary units. (e) The relative deviation of $\langle \delta {\widehat{S}}_x^2\rangle$ from its steady-state value, $R\left(t\right)$; see the text for details. (f) The Gaussian-broadened steady-state excess variance $[\langle \delta {\widehat{S}}_x^2\left(\infty \right)\rangle /N-1]$ (circles) or $[2\langle \delta {\widehat{P}}_{\mathrm{c}}^2\left(\infty \right)\rangle -1]$ (diamonds), relative to the corresponding values from Eq. (20). The results are very similar for the different coupling strengths $g_{\mathrm{ens}}=3\Gamma$ (red [gray]), $g_{\mathrm{ens}}=4\Gamma$ (green [light gray]), and $g_{\mathrm{ens}}=5\Gamma$ (blue [gray]).

Figure 4

The real part of the transmission coefficient vs external driving frequency. The calculations assume a Lorentzian profile, ${\kappa }_1={\kappa }_2$, $\kappa =10\Gamma$, and $g_{\mathrm{ens}}$ is varied in order to obtain the $C$ values marked on the curves. Solid lines are noninverted ($p=-1$) while the dotted line for $C=0.2$ corresponds to an inverted sample ($p=1$).