Different Types of Integrability and Their Relation to Decoherence in Central Spin Models

We investigate the relation between integrability and decoherence in central spin models with more than one central spin. We show that there is a transition between integrability ensured by the Bethe ansatz and integrability ensured by complete sets of commuting operators. This has a significant impact on the decoherence properties of the system, suggesting that it is not necessarily integrability or nonintegrability which is related to decoherence, but rather its type or a change from integrability to nonintegrability.

Figure 1

Binary tree with $n=N_c+N=2+4=6$ leaves. In order to embed the Hamiltonian of an arbitrary central spin model with homogeneous couplings into a CSCO using the binary tree formalism, two binary trees $\alpha$, $\beta$ with $N_c$ and $N$ leaves, respectively, must be grafted together. The Hamiltonian results as given in (8).

Figure 2

Spin dynamics for $N_c=2$, $N=2N_D+1=11$, where $S_i=I_i=(1/2)$, and $B=3.441A$, $J=0.023A$. The initial state of the system is $|\Uparrow \Downarrow ⟩\otimes \prod _{i=1}^{N_D}{I}_i^{-}|0⟩$. We consider two integrable models fulfilling $A_i^1=A_i^2≕A_i$ with either $A_i\ne A_j$ (upper panel), chosen due to a nonuniform distribution, or $A_i=A_j$ (lower panel). Although in both cases $B$ is larger than any other energy scale, for homogeneous couplings the dynamics still decays, indicating decoherence.

Figure 3

Sketch of flip-flop processes in a $N_c=2$ central spin system with inhomogeneous (upper panel) and homogeneous (lower panel) couplings. For homogeneous couplings, the energy changes due to simultaneous central and nuclear spin flips can compensate each other. This is not possible for inhomogeneous couplings because the energies of flips differ for different nuclear spins.