Relaxation to Negative Temperatures in Double Domain Systems

The engineering of quantum systems and their environments has led to our ability now to design composite or complex systems with the properties one desires. In fact, this allows us to couple two or more distinct systems to the same environment where potentially unusual behavior and dynamics can be exhibited. In this Letter we investigate the relaxation of two giant spins or collective spin ensembles individually coupled to the same reservoir. We find that, depending on the configuration of the two individual spin ensembles, the steady state of the composite system does not necessarily reach the ground state of the individual systems, unlike what one would expect for independent environments. Further, when the size of one individual spin ensemble is much larger than the second, collective relaxation can drive the second system to an excited steady state even when it starts in the ground state; that is, the second spin ensemble relaxes towards a negative-temperature steady state.

Figure 1

Schematic representation of a double spin subdomain system coupled to a single reservoir. The double spin subdomain system may be formed from spin ensembles or giant spins. Here we denote the first spin subdomain $D_1$ containing $N_1$ spins shown as red arrows, while the second spin domain $D_2$ contains $N_2$ spins represented by blue arrows. In both subdomains, each spin couples with the bosonic reservoir (at a temperature $T$) with coupling constant $g$.

Figure 2

Plot of the normalized collective spin relaxations for balanced number configurations. (a) $N_1=N_2=10$ and (b) $N_1=N_2=100$. The blue (red) curves are for $T=0\mathrm{K}$ (400 mK). The solid (dotted) curves represent $⟨S_1^z⟩$ ($⟨S_2^z⟩$) under the initial condition $|\mathrm{SDAP}⟩$, while the dashed curves are the relaxations under the condition $|\mathrm{SDP}⟩$. We have chosen ${\omega }_{\mathrm{s}}/2\pi =10\mathrm{GHz}$ with $\gamma =0.01\mathrm{Hz}$.

Figure 3

Plot of the relaxation time ${\tau }_N$ for the initial configuration $|\mathrm{SDAP}⟩$ as a function of subdomain size $N$ for zero temperature (blue solid line) and finite (400 mK) temperature (red dotted line). Curve fitting indicates a functional form for ${\tau }_N\sim a/N+b$ with $a=422.33$, $b=1.62$ for $T=0$ and $a=365.64$, $b=1.43$ for $T>0$. The $1/N$ dependence is a clear signature of superradiant decay [19].

Figure 4

Plot of the collective spin relaxations for various unbalanced number configurations ($N_1>N_2$) at zero temperature. The configuration considered are (a) $N_1=2$, $N_2=1$ and (b) $N_1=5$, $N_2=1$. The red solid curves, blue dotted curves, and green dashed curves represent $⟨S_1^z⟩$, $⟨S_2^z⟩$, and $⟨S^z⟩$, respectively. $S_{1,\min }^z$ ($S_{2,\max }^z$) are the steady-state solution for $⟨S_{1(2)}^z⟩$.

Figure 5

Plot of the normalized collective spin relaxations $⟨S_1^z⟩$ (red solid line) and $⟨S_2^z⟩$ (blue dot line) for the unbalanced configuration at zero temperature. Here we take $N_1={10}^4$ and $N_2={10}^2$.