Intertwining of lasing and superradiance under spintronic pumping

We introduce a quantum optics platform featuring the minimal ingredients for the description of a spintronically pumped magnon condensate, which we use to promote driven-dissipative phase transitions in the context of spintronics. We consider a Dicke model weakly coupled to an out-of-equilibrium bath with a tunable spin accumulation. The latter is pumped incoherently in a fashion reminiscent of experiments with magnet-metal heterostructures. The core of our analysis is the emergence of a hybrid lasing-superradiant regime that does not take place in an ordinary pumped Dicke spin ensemble, and which can be traced back to the spintronics pumping scheme. We interpret the resulting nonequilibrium phase diagram from both a quantum optics and a spintronics standpoint, supplying a conceptual bridge between the two fields. The outreach of our results concern dynamical control in magnon condensates and frequency-dependent gain media in quantum optics.

Figure 1

(a) An ensemble of spins-1/2 $\mathcal{S}$ (in blue) is coupled to a boson mode (in red) which models a magnon condensate or a cavity boson which can become macroscopically occupied for large values of the Dicke coupling, $\lambda$. It is also coupled to a spin-1/2 subsystem $\mathcal{T}$ (in green) under incoherent spin relaxation and pumping controlled by the spin accumulation $\mu$ and temperature $T$. The population inversion of $\mathcal{T}$ can induce a coherent dynamical response in $\mathcal{S}$, which is the central mechanism explored in this paper. Dissipation with strength $\kappa$ (in brown) acts on the boson mode and is shown by a red wiggle line. The model contains the essential ingredients of both quantum optics (b) and spintronics (c) platforms, as detailed in the main text. Corresponding elements in different setups are highlighted in the same color.

Figure 2

Dynamical phases resulting from the interplay of spin pumping and Dicke coupling. (a) For ${\gamma }_{\uparrow }<{\gamma }_t/2$, the usual critical coupling (${\lambda }_c\simeq 0.5$) associated to the Dicke transition separates the normal (NS) from the superradiant (SR) phase. For ${\gamma }_{\uparrow }>{\gamma }_t/2$, the normal state becomes unstable, and observables in the ensemble $\mathcal{S}$ oscillate with zero average value of ${\mathcal{S}}^x$ in region L1, and around one of the minima of the SR phase in region L2, as shown in the Bloch spheres. Inside the irregular (IR) region the motion of the collective spin covers uniformly a large part of the Bloch sphere without any structured pattern and is suggestive of chaotic behavior. (b)–(d) The absolute value of time-averaged $\langle {\mathcal{S}}^x\rangle$ (dashed line) and amplitude of its oscillations (solid line) along transition lines 1–3 in the main inset (a). Here we have chosen ${\omega }_c={\omega }_z={\omega }_z^{\prime }=1,{\gamma }_t=1, \eta ={\eta }^{\prime }=0.1,\kappa =0.06.$

Figure 3

(a) Dynamics of the photon number $n$ with parameters as in Fig. 2, with the exception of ${\omega }_z^{\prime }$, which is chosen in resonance with the upper polariton frequency (${\omega }_z^{\prime }\simeq {\mathrm{\Omega }}_{\text{U}}$). Insets show a stretched time axis. The system is prepared in the SR state and evolves with system parameters ${\gamma }_{\uparrow }$ and $\lambda$ inside region L1. The initial stage of dynamics is governed by the decaying lower polariton mode, while at stage C the upper polariton mode undergoes the dynamical instability triggered by the resonance with the incoherent subsystem; accordingly, the photon number starts to grow until it saturates around times $t\simeq 500.$ At long times, $n$ oscillates at the upper polariton frequency, see stage $D$. This is illustrative of the spin ensemble $\mathcal{T}$ acting as a frequency-dependent gain medium. (b) The effective decay rate for the photon number $n\propto exp(-{\kappa }_{eff}t)$ extracted in stage A (blue circles) and stage C (red triangles) of dynamics in (a), as a function of the frequency of the incoherent subsystem ${\omega }_z^{\prime }$. Close to the resonant frequency of the upper polariton, ${\mathrm{\Omega }}_{\text{U}}=1.63$, the effective damping ${\kappa }_{eff}=2{\kappa }_{\text{U}}$ can change signs, indicating a dynamical instability, which results in the polariton lasing. Solid lines show analytical dependence according to Eq. (4).

Figure 4

Stability analysis of the normal state. The parameters are the same as in Fig. 2. The stable normal state is indicated in white. The purple color corresponds to one real positive eigenvalue of the matrix $A$ and it indicates superradiance. The yellow region corresponds to two complex conjugated eigenvalues with positive real part and it corresponds to the lasing region. The orange region corresponds to the three positive eigenvalues. Here $\omega ={\omega }_z^{\prime }.$

Figure 5

Stability analysis with the resonance condition ${\omega }_z^{\prime }={\mathrm{\Omega }}_{\text{U}}$. The rest of the parameters are the same as in Fig. 2. The vertical boundary between normal state (white) and lasing (yellow) is given by the critical pumping rate ${\gamma }_{\uparrow }^c\approx 0.7{\gamma }_t$ [see Eq. (5)].

Figure 6

Dynamical phase diagram with Gilbert damping [see Eq. (C1)]. All parameters are as in Fig. 7 and the color code follows Fig. 2.

Figure 7

Stability analysis for model with Gilbert damping. All parameters are as in Fig. 2, $\kappa =0.04,$ and the color code follows Fig. 4.

Figure 8

Fit of the timescale $t_E\propto N^{\delta }$ with $\delta \simeq 0.5$ as a function of number of spins in the system, $N$, within the lasing region. We extract $t_E$ as the time when the ratio between second cumulants and mean-field expectation values becomes of order $\sim 0.1$.

Figure 9

(a) Amplitude of the oscillations of the $\langle {\mathcal{S}}^x\rangle$ (solid line) and absolute value of the time-averaged $\langle {\mathcal{S}}^x\rangle$ (dashed line) as a function of $\lambda .$ We choose all parameters as in Fig. 2 and fixed ${\gamma }_{\uparrow }=0.9{\gamma }_t.$ The colors are the same as in Fig. 2. Yellow, orange, and red colors correspond to L1, L2, and IR phases, respectively. (b) Dynamics of the spin $\langle \mathcal{S}\rangle$ on the Bloch sphere inside L1 and L2 regions for different values of $\lambda .$ Note that inside the L2 region, depending on the initial conditions, one of two trajectories is possible with opposite time-averaged values of ${\langle {\mathcal{S}}^x\rangle }_t=\pm s_0^x.$