Linear-response theory for superradiant lasers

We theoretically study a superradiant laser, deriving both the steady-state behaviors and small-amplitude responses of the laser's atomic inversion, atomic polarization, and light field amplitude. Our minimum model for a three-level laser includes atomic population accumulating outside of the lasing transition and dynamics of the atomic population distribution causing cavity frequency tuning, as can occur in realistic experimental systems. We show that the population dynamics can act as real-time feedback to stabilize or destabilize the laser's output power, and we derive the cavity frequency tuning for a Raman laser. We extend the minimal model to describe a cold-atom Raman laser using ${}^{87}$Rb, showing that the minimal model qualitatively captures the essential features of the more complex system [Bohnet et al., Phys. Rev. Lett. 109, 253602 (2012)]. This work informs our understanding of the stability of proposed millihertz linewidth lasers based on ultranarrow optical atomic transitions and will guide the design and development of these next-generation optical frequency references.

Figure 1

Energy-level diagram of a three-level superradiant laser using the optical transition from $|e\rangle$ to $|g\rangle$. The emitted optical laser light is nearly resonant with the cavity mode (dashed lines), detuned from ${\omega }_{\mathrm{c}}$ by ${\delta }_{bcav}$. The atoms are incoherently pumped to a third state $|3\rangle$ at a rate $W$. Atoms in $|g\rangle$ also Rayleigh scatter at a rate ${\Gamma }_R$, but leaves them in state $|g\rangle$. The incoherent pumping from $|3\rangle$ to $|e\rangle$ at rate ${\Gamma }_{3e}$ completes the cycle.

Figure 2

Representing the state of the $|e\rangle$, $|g\rangle$ two-level system using a collective Bloch vector J.

Figure 3

(a) Steady-state photon flux ${\dot{\overline{M}}}_c$ vs ground-state repumping rate $W$, with a series of curves showing the effects of repumping through the additional state $|3\rangle$. The case of $r=\infty$ is the two-level model of Ref. [1] (top curve). (b) Photon flux ${\dot{\overline{M}}}_c$ vs $r$ with $W=W_{\mathrm{opt}}$. For all curves, ${\delta }_0^{\prime }=0$ and ${\Gamma }_R=0$, and the photon flux is plotted in units of $P_{2lvl}=N^{2}C\gamma /8$.

Figure 4

(a) Steady-state photon flux ${\dot{\overline{M}}}_c$ vs ground-state repumping rate $W$, with a series of curves showing the effects of detuning of the cavity resonance frequency from the emitted light frequency ${\delta }_0^{\prime }$. (b) Photon flux ${\dot{\overline{M}}}_c$ vs ${\delta }_0^{\prime }$ with $W=W_{\mathrm{opt}}\left({\delta }_0^{\prime }\right)$. The photon flux is plotted in units of $P_{2lvl}=N^{2}C\gamma /8$. For all curves, $r=\infty$ and ${\Gamma }_R=0$.

Figure 5

(a) Steady-state photon flux ${\dot{\overline{M}}}_c$ vs ground-state repumping rate $W$, with a series of curves showing the effects of decoherence in the form of Rayleigh scattering from the ground state $|g\rangle$. (b) Photon flux ${\dot{\overline{M}}}_c$ vs ${\Gamma }_R$ with $W=W_{\mathrm{opt}}\left({\Gamma }_R\right)$. The photon flux is plotted in units of $P_{2lvl}=N^{2}C\gamma /8$. For all curves, $r=\infty$ and ${\delta }_0^{\prime }=0$.

Figure 6

Output photon flux transfer function for different ground-state repumping rates, with $r=\infty$, ${\delta }^{\prime }=0$, $\alpha =0$, and ${\Gamma }_R=0$.

Figure 7

Output photon flux transfer function for different repumping ratios $r$, with $W=W_{\mathrm{opt}}$, ${\delta }^{\prime }=0$, $\alpha =0$, and ${\Gamma }_R=0$.

Figure 8

Output photon flux transfer function for different dressed cavity detuning from emitted light frequency $\delta$, with $W=W_{\mathrm{opt}}\left(\delta \right)=\frac{NC\gamma }{2(1+{\delta }^{\prime 2})}$, $r=\infty$, and ${\Gamma }_R=0$. The dashed (dot-dashed) lines show $\alpha =NC\gamma \times {10}^{-3}$ ($\alpha =2NC\gamma \times {10}^{-3}$) to demonstrate the effects of increased cavity feedback.

Figure 9

Stability plot using ${\gamma }_0$ stability condition of Eq. (28) as a function of detuning ${\delta }^{\prime }$ and the cavity shift parameter $\alpha$, assuming $N={10}^6$. The stability condition also assumes $W=W_{\mathrm{opt}}({\delta }^{\prime },{\Gamma }_R)$. The region of stability is exact for the two-level model ($r=\infty$), and a good approximation for all values of $r$. The blue region shows where the real part of all the poles of the $j_{\perp }$ solution are negative, indicating a damped return to steady-state conditions for a perturbation. The red region shows where any of the real parts of the poles become positive, making $J_{\perp }$ unstable, with no steady-state solutions.

Figure 10

Output photon flux transfer function for different Rayleigh scattering rates ${\Gamma }_R$, with $\overline{W}=W_{\mathrm{opt}}\left({\Gamma }_R\right)$, $r=\infty$, $\alpha =0$, and ${\delta }^{\prime }=0$. The dot-dashed line shows the transfer functions when $\overline{W}$ is held to $NC\gamma /2$, not varied to remain at $W_{\mathrm{opt}}$, and ${\Gamma }_R=0.3$. A dot-dashed red curve is not shown, as with ${\Gamma }_R=0.6$ and $\overline{W}=NC\gamma /2$ the maximum repumping rate threshold has been exceeded and the output photon flux is zero.

Figure 11

Response of the 2D Bloch vector to external modulation of the repumping rate. The steady-state Bloch vector, i.e., ${\overline{J}}_{\perp }$ and ${\overline{J}}_z$ from Eqs. (13) and (14), is indicated by the blue line, plotted on the axis with units of $N$, so $N/2$ is the maximum value. The ellipse is the trajectory of the Bloch vector responding to the modulation of the repumping rate $w\left(t\right)=\epsilon \mathrm{Re}\left[e^{i\omega t}\right]$, described by the small signal responses $j_{\perp }$ and $j_z$ in Eqs. (23) and (24). The parameters are $\epsilon =0.1$, $r=5$, ${\delta }^{\prime }=0$, $\alpha =0$, and ${\Gamma }_R=0$. The black arrow indicates the direction of the trajectory, starting from the blue dot at $t=0$. The values of $\omega$ are chosen to show $\omega \ll {\omega }_{\mathrm{res}}$, $\omega \approx {\omega }_{\mathrm{res}}$, and $\omega \gg {\omega }_{\mathrm{res}}$.

Figure 12

Parametric plots of the response of the three degrees of freedom $J_z$, $J_{\perp }$, and $A$, highlighting effect of cavity frequency tuning on response of atomic coherence and output light field. The solid blue line represents the steady-state atomic Bloch vector, ${\overline{J}}_z$ and ${\overline{J}}_{\perp }$, from Eqs. (13) and (14). The solid blue ellipse shows small signal response of the Bloch vector to a modulation of the repumping rate $W$, given by $j_z$ and $j_{\perp }$ from Eqs. (23) and (24). The parametric response is plotted with units of $N$, so $N/2$ is the maximum value. The red dashed line is the trajectory formed by the response of the cavity field $a$ [Eq. (27)] and $j_z$. The cavity field is plotted as a fraction of the average field, then centered on the steady-state Bloch vector to compare with the atomic response. The large red and blue dots give the response at $t=0$, and the small black dot indicates the direction of the response with respect to a modulation $W\left(t\right)=\overline{W}(1+\epsilon e^{i\omega t})$. Here $NC\gamma ={10}^4$ s${}^{-1}$, $r=5$, ${\Gamma }_R=0$, $\overline{W}=W_{\mathrm{opt}}\left({\delta }^{\prime }\right)$, $\epsilon =0.1$, and $\omega =0.02NC\gamma$, chosen to show the stable $J_{\perp }$ response. (a) When ${\delta }^{\prime }<0$, the cavity feedback can be positive, leading to larger oscillations compared to the case of no feedback ${\delta }^{\prime }=0$ shown in (b). Because of the coupling of $J_z$ to the cavity mode frequency, $A$ is not locked to the $J_{\perp }$ response, as in (b), but is anticorrelated with $J_z$. In (c), where ${\delta }^{\prime }>0$, the negative feedback reduces the response amplitudes in all quadratures. The cavity tuning again shifts the cavity amplitude response, but with the opposite phase relationship due the change in sign of the slope of the Lorentzian, so $A$ follows $J_z$. The inset shows a closeup of the response.

Figure 13

Cavity tuning stabilizing the cavity field amplitude. By changing the average repumping rate $\overline{W}$ from $W_{\mathrm{opt}}$ to $0.44\times {10}^4$ for the same parameters as Fig. 12 ($NC\gamma ={10}^4$ s${}^{-1}$, $r=5$, ${\Gamma }_R=0$, $\epsilon =0.1$, ${\delta }^{\prime }=1$, and $\omega =0.02NC\gamma$), the response of the Bloch vector (solid blue ellipse) becomes primarily perpendicular to the steady state Bloch vector (blue line). Under these conditions, the cavity field response $A$ (dashed red ellipse) has the smallest fractional deviation of the three degrees of freedom. The inset shows a closeup of the response. The large red and blue dots give the response at $t=0$, and the small black dot indicates the direction of the response with respect to a modulation $W\left(t\right)=\overline{W}(1+\epsilon e^{i\omega t})$.

Figure 14

Energy-level diagram for a superradiant laser enabled by an induced Raman transition. States $|e\rangle$ and $|g\rangle$ are two metastable states separated by a nonoptical frequency ${\omega }_{eg}$. They share an optically excited state $\left|i\right.〉$ that has a linewidth $\Gamma$. Using a Raman dressing laser, detuned from $\left|i\right.〉$ by ${\Delta }_d$, we can induced a optical decay to $|g\rangle$, which, in absence of collective effects, would proceed at a rate $\gamma =\frac{\Gamma }{4}{\left(\frac{{\Omega }_d}{\Delta }\right)}^2$. Including a single optical cavity mode, coupled to the $\left|i\right.〉$ to $|g\rangle$ transition with coupling constant $2g$, gives rise to collective emission. The cavity mode frequency is ${\omega }_c$, detuned from $\left|i\right.〉$ by ${\Delta }_c$, making the two-photon detuning ${\delta }_0={\omega }_c-({\omega }_d+{\omega }_{eg})$. To complete the laser cycle, the atoms are incoherently repumped from $|g\rangle$ to $|e\rangle$ at a rate $W$.

Figure 15

Two-step repumping process on the ${}^{87}$Rb D2 line (780 nm). The diagram is drawn showing on only positive Zeeman states, but the process is symmetric with respect to $m_f=\pm 1,\pm 2$. The desirable decay branches (solid magenta) show the most direct repumping sequence, although any particular repumping sequence could go through many ground hyperfine states due to other decay channels (dashed lines). The optically excited state on the D2 line has a linewidth ${\Gamma }_{D2}/2\pi =6.07$ MHz.

Figure 16

Lasing transition and Raman dressing scheme on the ${}^{87}$Rb D1 line (795 nm). The dressing light (straight red) and collective emission (waved blue) are a superposition of ${\sigma }^{+}$ and ${\sigma }^{-}$ polarizations because the direction of propagation of the light is along the quantization axis defined by the direction of the magnetic field at the atoms. The Raman dressing laser is detuned by $\Delta$ from the atomic transition. The bare cavity detuning is ${\delta }_0={\omega }_c-({\omega }_d+{\omega }_{hf})$. The optically excited state on the D1 line has a linewidth ${\Gamma }_{D1}/2\pi =5.75$ MHz. The effective population decay from $|F=2,m_f=0\rangle$ to $|F=1,m_f=0\rangle$ is $\gamma =\frac{{\Gamma }_{D1}}{4}{\left(\frac{{\Omega }_d}{\Delta }\right)}^2$.

Figure 17

Response for different repumping rates. (a) Light field amplitude response transfer function $T_A$ vs repumping rate $W$. (b) Resonant response $T_A\left({\omega }_{\max }\right)$ and (c) the resonant modulation frequency ${\omega }_{\max }$ as a function of repumping rate $W$. Here $NC\gamma =4\times {10}^5$ s${}^{-1}$, ${\delta }^{\prime }=\alpha =0$, and $r=0.71$.

Figure 18

Response for different repumping ratio. (a) Light field amplitude response transfer function $T_A$ vs repumping ratio $r$. (b) Resonant response $T_A\left({\omega }_{\max }\right)$ and (c) the resonant modulation frequency ${\omega }_{\max }$ as a function of repumping ratio $r$. Here $NC\gamma =4\times {10}^5$ s${}^{-1}$, ${\delta }^{\prime }=0$, and $\overline{W}=W_{\mathrm{opt}}$.

Figure 19

Response for different detunings. (a) Light field amplitude response transfer function $T_A$ vs detuning ${\delta }^{\prime }$. The transfer function is not plotted in regions of instability. (b) Resonant response $T_A\left({\omega }_{\max }\right)$ and (c) the resonant modulation frequency ${\omega }_{\max }$ as a function of ${\delta }^{\prime }$. The red shaded regions indicate parameters in which the system is unstable and no steady-state solutions exist. Here $NC\gamma =4\times {10}^5$ s${}^{-1}$, $r=0.71$, and $\overline{W}=W_{\mathrm{opt}}$. The cavity shift parameter is given by ${\vec{\alpha }}_{+}$. These parameter values reflect the conditions of the experimental system in Ref. [21].

Figure 20

Stability diagram for the full model of a superradiant Raman laser in ${}^{87}$Rb, plotting the region any of the real parts of the poles of the $j_{\perp }$ solution are positive. When any pole becomes positive, the system is unstable and has no steady-state solutions. The stability regions are shown vs the detuning of the cavity from the emission frequency ${\delta }^{\prime }$ and the detuning of the bare cavity frequency from the atomic frequency $\Delta$. The critical contour (bold) marks where the pole changes sign. The dashed line indicates the detuning $\Delta$ of experimental work (Refs. [3, 21]). For the calculation we use $NC\gamma =1\times {10}^{-4}$ s${}^{-1}$, $\overline{W}=W_{\mathrm{opt}}$, and $r=0.71$.

Figure 21

Critical stability conditions for variable values of the repumping ratio $r$. Each line shows the contour as a function of $\Delta$ and ${\delta }^{\prime }$ separating stable lasing from unstable. The unstable region is defined as any set of parameters that results in a positive value for the real part of any pole of the $J_{\perp }$ response solution. The stability conditions change as a function of the repumping ratio $r$. (a) As $r$ increases from 0, the unstable region gets smaller until it reaches some value between 0.4 and 0.5, after which (b) the unstable region grows to its asymptotic value.

Figure 22

The value of $\Delta$ for the critical contour vs $r$, assuming ${\delta }^{\prime }=-1$. Lower $\Delta$ indicates that more of the parameter space has stable, steady-state solutions. Here we assume $NC\gamma =1\times {10}^{-4}$ s${}^{-1}$ and $\overline{W}=W_{\mathrm{opt}}$.