Figure 1

Schematic of the different phases and transitions of master Eq. (1). In the two main phases of the system $A$ (blue) and $B$ (red)—which together cover the whole phase diagram—the system is found in a RSTSS (cf. text). While phase $A$ is characterized by normal spin-pumping behavior (large nuclear polarization in the direction of the dissipation) and a low effective temperature, phase $B$ displays anomalous spin-pumping behavior (large nuclear polarization in opposing direction to the dissipation) and high temperature. They are separated by the first-order phase boundary $b$, which is associated with a region of bistability $C$ (framed by the boundary $c$). Here a second non-Gaussian solution appears, besides the normal spin-pumping mode of $A$. The region of bistability $C$ culminates in a second-order phase transition at (${\omega }_0,{\Omega }_0$). Below this critical point the system is supercritical and no clear distinction between phases $A$ and $B$ exists. In this region a dynamical phase $D$ emerges, characterized by anomalous behavior in the approach to the steady state. For a detailed description of the different phases and transitions, see Sec. 3b.Reuse & Permissions

Figure 2

The system observables of the RSTSS solution in the thermodynamic limit show clear signatures of first- and second-order transitions ($\gamma =a$). (a) The nuclear polarization in the $z$ direction ${\langle I_z/J\rangle }_{ss}$ switches abruptly from minus to plus at the phase boundary $b$. (b) The electron polarization in the $x$ direction ${\langle S_x\rangle }_{ss}$ shows a similar discontinuous behavior along $b$. (c) The nuclear polarization in the $x$ direction changes smoothly across the phase boundary $b$. Along the segment $x$ ($\omega ={\omega }_0,\Omega <{\Omega }_0$) the nuclear field in the $x$ direction builds up linearly to cancel the external driving. (d) The electron polarization in the $z$ direction also does not show signatures of the first-order transition $b$. Along segment $x$ the electron is fully polarized in the $-z$ direction up to the second-order critical point $({\omega }_0,{\Omega }_0)$, where it changes nonanalytically (see also Fig. 6).Reuse & Permissions

Figure 3

Asymptotic decay rate (ADR, cf. text) for $\gamma =a$ within the perturbative framework. Along $b$ the ADR vanishes nonanalytically, indicating the stabilizing and destabilizing of the modes of regions $A$ and $B$, respectively. $b$ is a first-order phase boundary culminating in a second-order critical point at $({\omega }_0,{\Omega }_0)$. From here region $D$ opens, which is characterized by a nonanalyticity in the ADR at a finite value. This indicates a change in the dynamic properties of the system which cannot be detected in steady-state observables. Within $D$ the system shows an overdamped behavior in the vicinity of the steady state.Reuse & Permissions

Figure 4

Properties of the steady state CM ${\Gamma }_{ss}$ [Eq. (42)]. (a) The fluctuations $D$ are low in most parts of the phase diagram except for a small wedge around the Gaussian phase boundary. (b) Fluctuations $D$ along the line $◯I$ [green line of (a)]. The phase boundaries separate a mode with low fluctuations (enlarged in the inset), from a mode with large fluctuations. For large $\Omega$ fluctuations increase, and the system eventually approaches a fully mixed state. (c) The squeezing measure $C$ (cf. text) in the thermodynamic limit. $C$ approaches 1 at $({\omega }_0,{\Omega }_0)$, indicating diverging entanglement in the system. (d) $C$ along the line $\omega ={\omega }_0$ (solid line). The red circles indicate the the squeezing parameter $1-{\xi }_{{\widehat{e}}_y}^2=1-\sqrt{1-{(\Omega /{\Omega }_0)}^2}$ (cf. text).Reuse & Permissions

Figure 5

The ADR ($\gamma =a$) for $J=50$, 100, 150 (broken lines) in comparison with the perturbatively calculated (solid line, cf. Sec. 4b2) along $\omega ={\omega }_0$. For finite systems one finds an avoided crossing at ${\Omega }_0$. The size of the gap reduces with the system size until it closes in the thermodynamic limit (solid line). Below ${\Omega }_0$ the ADR in the thermodynamic limit is given by Eq. (52).Reuse & Permissions

Figure 6

Electron inversion $\langle S_z\rangle$ and the nuclear field in the $x$ direction $\langle I_x\rangle$ along $\omega ={\omega }_0$ in the thermodynamic limit according to the perturbative theory (circles) in comparison with the numeric values from exact diagonalization for a finite system of $J=150$ (solid lines). The perturbative theory shows excellent agreement with the numerical solutions. Further, the numerically determined electron inversion and the expectation value of the inhomogeneous nuclear operator $\langle A_x\rangle$ are displayed for a model of two inhomogeneously coupled nuclear shells ($g_1=2g_2$) of size $J_{1,2}=8$ (dashed lines) and for five inhomogeneously coupled nuclear spins (dotted lines) are displayed (discussion see Sec. 6).Reuse & Permissions

Figure 7

Effective temperature $T_{\mathrm{eff}}$ of the Gaussian mode. Temperatures $T_{\mathrm{eff}}>6$ are cut off, as the temperature diverges along the phase boundary $b$. (a) The first-order phase boundary $b$ separates the low-temperature phase $A$ from the high-temperature phase $B$. (b) $T_{\mathrm{eff}}$ along $\omega ={\omega }_0$: On segment $x$ the system is in a zero entropy state ($T_{\mathrm{eff}}=0$). Above the second-order critical point $\Omega >{\Omega }_0$ the system enters a high-temperature phase. Here the temperature rises with increasing driving strength.Reuse & Permissions

Figure 8

The ADR and the imaginary part of the respective eigenvalue ($\gamma =a$) for $J=150$ (solid lines) in comparison with the perturbatively calculated value (dots) along $◯II$. In the region where the coherent part of Eq. (25) is a squeezing Hamiltonian, the ADR (i.e., real part of the lowest Liouvillian eigenvalue pair) splits. At the same time the imaginary part of the lowest eigenvalue pair vanishes (black lines), indicating that the system is overdamped.Reuse & Permissions

Figure 9

The ADR ($\gamma =a$) for $J=50$, 100, 150 (broken lines) in comparison with the perturbatively calculated (solid line) along $◯I$. The vertical black lines indicate the asymptotic boundaries of the region of bistability. In the whole region the ADR tends to zero in the thermodynamic limit due to the appearance of a non-Gaussian stable mode. (Inset) The next-higher excitations in the spectrum for $J=150$ display equidistant splittings in regions far from the region of bistability. This is an indication for the bosonic character of the steady state, which is exploited in the perturbative approach.Reuse & Permissions

Figure 10

Complex energy of the two modes corresponding to the semiclassical solutions ${\beta }_{\pm }$ for $\gamma =a$. The solid line in the nonshaded area represents the ADR of Fig. 9 and Fig. 5, respectively. (a) Along $◯I$ ($\omega =1.5{\omega }_0$). The eigenvalues miss each other in the complex plane. The real parts cross directly. (b) $\omega ={\omega }_0$. The eigenvalues degenerate asymptotically (in both real and imaginary parts) at the critical point. This closing of the gap originates from an avoided crossing in finite systems with the relevant gap vanishing in the thermodynamic limit (see also Fig. 5).Reuse & Permissions

Figure 11

Diagonal elements $p\left(m\right)=\left.〈m\right|\rho \left|m\right.〉$ of the nuclear density matrix in the $z$ basis ($I_z\left|m\right.〉=m\left|m\right.〉$) across the region of bistability for $\omega =1.5{\omega }_0$ ($J=150$, $\gamma =a$). In the bistable region two stable modes– -the Gaussian normal spin-pumping mode (lower branch; ${\rho }_{\mathrm{lo}}$) and a non-Gaussian (upper branch; ${\rho }_{\mathrm{up}}$)—coexist. At the boundary $b$ the latter transforms into the anomalous spin-pumping mode, which is the sole stable solution above $b$. The blue dots (red diamonds) in the plane indicate the average polarization in the $z$ direction $\langle I_z\rangle$ for the lower (upper) solution.Reuse & Permissions

Figure 12

(a) The nuclear polarization in the $z$ direction ${\langle I_z/J\rangle }_{ss}$ of the RSTSS solution in the thermodynamic limit for (a) $\gamma =0.2a$ and (b) $\gamma =5a$. In the first case [(a)] the phase diagram bears strong resemblance with the case $\gamma =a$ (compare Fig. 2). In the case of large dissipation [(b)] the first-order boundary is rotated clockwise toward the line (${\omega }_0,\Omega >{\Omega }_0$) and the distinction of the phases according to their nuclear polarization in the $z$ direction becomes less prominent. Instead, other criteria like the polarization in the $y$ direction (not displayed) emerge.Reuse & Permissions