Phase Transitions in Electron Spin Resonance Under Continuous Microwave Driving

We study an ensemble of strongly coupled electrons under continuous microwave irradiation interacting with a dissipative environment, a problem of relevance to the creation of highly polarized nonequilibrium states in nuclear magnetic resonance. We analyze the stationary states of the dynamics, described within a Lindblad master equation framework, at the mean-field approximation level. This approach allows us to identify steady-state phase transitions between phases of high and low polarization controlled by the distribution of disordered electronic interactions. We compare the mean-field predictions to numerically exact simulations of small systems and find good agreement. Our study highlights the possibility of observing collective phenomena, such as metastable states, phase transitions, and critical behavior, in appropriately designed paramagnetic systems. These phenomena occur in a low-temperature regime which is not theoretically tractable by conventional methods, e.g., the spin-temperature approach.

Figure 1

(a) Steady-state polarization spectra ${\overline{p}}_z(\mathrm{\Delta })$ obtained by the mean-field formula (6) (solid lines) and the numerically exact solution (dashed lines) for $N=4$, $D=10\mathrm{MHz}$, $R_2={10}^6{\mathrm{s}}^{-1}$, and different temperature and microwave parameters: $p=0.11$, ${\omega }_1=75\mathrm{kHz}$, $R_1={10}^3{\mathrm{s}}^{-1}$ (red); $p=0.55$, ${\omega }_1=12\mathrm{kHz}$, $R_1=10{\mathrm{s}}^{-1}$ (green); $p=0.99$, ${\omega }_1=7\mathrm{kHz}$, $R_1=1{\mathrm{s}}^{-1}$ (blue). (b) Phase diagram obtained from Eq. (6) in the $(a,b)$ plane. The diagram features regions of unique (brown) and multiple (gray) solutions and displays a (cusp) critical point $G$ at $p=0.99$, ${\omega }_1=R_2={10}^5$, and $R_1=1{\mathrm{s}}^{-1}$ (for $N=150$ electrons). (c) Structure of the solutions along the cut $b=3.75$ ($D=6.3\mathrm{MHz}$) through the region with a multistable region featuring three solutions. (d) Phase diagram obtained from Eq. (7) in the $(a^{\prime },b^{\prime })$ plane featuring regions of unique and multiple solutions similar to that in panel (b) and a critical point $G^{\prime }$ belonging to the same universality class as $G$ (see text for details). The dark gray region illustrates the impact of disorder in the frequency offsets ${\mathrm{\Delta }}_k$ (inhomogeneous broadening) on the multistability region. The strength of the inhomogeneous broadening is parameterized by $c$ (see SM [26] for details).

Figure 2

Numerical simulations and fluctuations. All results in this figure are produced for parameters ${\omega }_1={10}^5\mathrm{Hz}$, $R_1=1{\mathrm{s}}^{-1}$, $R_2={10}^5{\mathrm{s}}^{-1}$, $p=0.99$, and $N=8$, and averaged over 10 disorder realizations. (a)–(c) Discrete approximations of the probability density (dark shaded area) for the observable $P_z$ for three sets of parameters, such that $\int \pi (P_z)d{P}_z=1$ over the range shown. The light-colored curves represent the densities for some individual disorders, divided by the number of disorder realizations considered so that their addition (rather than their average) would equal the full probability density. This is done to better represent the contribution each disorder realization makes to the distribution. Panel (b) shows a strongly broadened distribution signaling enhanced fluctuations. This is consistent with the presence of metastable states that are expected from the mean-field analysis. (d) The variance of the time integrated observable $P_z$ for varying $b^{\prime }$, with the fixed $a^{\prime }$ value indicated by the legend in the top right.