Reservoir-assisted energy migration through multiple spin domains

The transfer of energy through a network of nodes is fundamental to how both nature and current technology operate. Traditionally, we think of the nodes in a network being coupled to channels that connect them, in which energy is passed from node to channel to node until it reaches its targeted site. Here we introduce an alternate approach to this, where our channels are replaced by collective environments (or, actually, reservoirs) which interact with pairs of nodes. We show how energy initially located at a specific node can arrive at a target node—even though that environment may be at zero temperature. Further, we show that such a migration occurs on much faster timescales than the damping rate associated with a single spin coupled to the reservoir. Our approach shows the power of being able to tailor both the system and environment and the symmetries associated with them to provide new directions for future quantum technologies.

Figure 1

(a) Illustration of a dissipatively coupled chain of spin domains. Each domain $D_i$ contains $N_i$ identical spin-1/2 particles. All spins have an energy $\hslash {\omega }_0$ with associated frequency ${\omega }_0/2\pi$. Two neighboring domains $D_i$ and $D_{i+1}$ are interacting via a common (zero-temperature) reservoir mediated by the dissipation rate ${\gamma }_i$. (b) Schematic representation of the effective excitation transfer from the first domain (with $N_1=3$ spins and total angular momentum $j_1=3/2$ and $1/2$) to the second domain (with $N_2=2$ spins and total angular momentum $j_2=1$ and 0): initially, $D_1$ is fully excited while $D_2$ is in the ground state. This corresponds to the (partial) excitation of states with different total angular momentum (bottom row). Due to the collective decay, which preserves the total angular momentum, the system relaxes down the ladders and reaches a steady state where $D_1$ and $D_2$ are locally not in their respective ground states. This results in excitations arising in the second domain—even though it was initially in its ground state.

Figure 2

Normalized collective spin relaxation $\langle J_i^z\rangle /N_i$ with dynamics governed by the master equation (3). (a) Two spin domains with $N_1=40$ (solid magenta), $N_2=2$ (dashed blue); (b) three domains with $N_1=12$ (solid magenta), $N_2=6$ (dashed blue), $N_3=2$ (dotted gold); (c) four domains with $N_1=8$ (solid magenta), $N_2=6$ (dashed blue), $N_3=5$ (dotted gold), and $N_4=1$ (dot-dashed mint green).

Figure 3

Normalized collective spin relaxation $\langle J_i^z\rangle /N_i$ for three domains using mean-field dynamics. (a) $N_1={10}^6$ (solid magenta), $N_2={10}^4$ (dashed blue), and $N_3={10}^2$ (dotted gold). (b) $N_1={10}^8$ (solid magenta), $N_2={10}^5$ (dashed blue), and $N_3={10}^2$ (dotted gold).

Figure 4

Collective spin relaxation $\langle J_i^z\rangle /N_i$ for an initial state of the first domain with $\langle J_1^z\rangle (t=0)=0$ showing that the relative population of the later domains never exceeds $\langle J_{2,3}^z\rangle =0$. (a) The master equation approach with $N_1=12$ (solid magenta), $N_2=6$ (dashed blue), and $N_3=2$ (dotted gold). (b) The mean-field approach with $N_1={10}^6$ (solid magenta), $N_2={10}^4$ (dashed blue), and $N_3={10}^2$ (dotted gold).