Environment-controlled Floquet-state paramagnetism

We study the response of ideal spin systems which are interacting with both a strong oscillating magnetic field, and a thermal environment, to a weak probing magnetic field. We demonstrate that even the sign of the resulting mean magnetization depends on the amplitude of the driving field and that its absolute value can be significantly larger than the equilibrium magnetization in the absence of time-periodic forcing. Since the underlying Floquet-state occupation probabilities are determined by the precise form of the system-bath coupling, future measurements of such effects have the potential to establish a particularly innovative line of research, providing information on nonequilibrium thermodynamics and giving access to quantities which usually remain hidden when probing equilibrium systems.

Figure 1

One Brillouin zone of quasienergies for a spin $J=7/2$ driven by a right-circularly polarized high-frequency magnetic field according to Eq. (7), and exposed to a static magnetic field of scaled strength ${\omega }_0/\omega =0.1$. Observe the complete quasienergy collapse at $F/\omega \approx 0.44$, indicating the principal resonance $\mathrm{\Omega }/\omega =1$. Further particularly high degeneracies appear at $F/\omega =1.2$ ($\mathrm{\Omega }/\omega =3/2$) and $F/\omega \approx 1.79$ ($\mathrm{\Omega }/\omega =2$).

Figure 2

Quasithermal expectation values (20) for $J=7/2$, ${\omega }_0/\omega =0.1$, and right-circularly polarized driving described by Eq. (7). Here ${\gamma }_y={\gamma }_z=0$; the scaled bath temperature is $k_{\mathrm{B}}T/\left(\hslash \omega \right)=1.0$ in the upper panel (a) and $k_{\mathrm{B}}T/\left(\hslash \omega \right)=0.1$ in the lower one (b). The bath density of states is taken to be constant (dotted), quadratic (dashed) according to Eq. (22), and Gaussian (full lines) with ${\omega }_{\mathrm{c}}/\omega =5.0$; see Eq. (23).

Figure 3

One Brillouin zone of quasienergies for a linearly driven spin $J=7/2$ with ${\omega }_0/\omega =0.1$. Observe how the scale of the abscissa here differs from that in Fig. 1.

Figure 4

Quasithermal expectation values (20) for $J=7/2$, ${\omega }_0/\omega =0.1$, and linearly polarized driving described by Eq. (24). Here ${\gamma }_x={\gamma }_y={\gamma }_z$; the scaled bath temperature is $k_{\mathrm{B}}T/\left(\hslash \omega \right)=1.0$ in the upper panel (a) and $k_{\mathrm{B}}T/\left(\hslash \omega \right)=0.1$ in the lower one (b). The bath density of states is taken to be constant (dotted), quadratic (dashed) according to Eq. (22), and Gaussian (full lines) with ${\omega }_{\mathrm{c}}/\omega =5.0$; see Eq. (23).

Figure 5

One Brillouin zone of quasienergies for a spin $J=7/2$ driven by a left-circularly polarized high-frequency magnetic field according to Eq. (26), and exposed to a static magnetic field of scaled strength ${\omega }_0/\omega =0.1$. High degeneracies are found at $F/\omega \approx 1.02$ ($\mathrm{\Omega }/\omega =3/2$) and $F/\omega \approx 1.67$ ($\mathrm{\Omega }/\omega =2$).

Figure 6

As Fig. 2, but for left-circularly polarized driving described by Eq. (26). Observe that the quasithermal magnetization is enhanced, so that the system is effectively cooled through the application of moderately strong fields.