Noisy Spins and the Richardson-Gaudin Model

We study a system of spins (qubits) coupled to a common noisy environment, each precessing at its own frequency. The correlated noise experienced by the spins implies long-lived correlations that relax only due to the differing frequencies. We use a mapping to a non-Hermitian integrable Richardson-Gaudin model to find the exact spectrum of the quantum master equation in the high-temperature limit and, hence, determine the decay rate. Our solution can be used to evaluate the effect of inhomogeneous splittings on a system of qubits coupled to a common bath.

Figure 1

Spectrum of the Liouvillian [Eq. (10)] for $n=6$ spin correlations for the case of $\mathrm{\Omega }=1$, $g_{+}=800$, $g_0=0$, and ${\omega }_i\sim \mathrm{Uni}(-0.2,0.2)$ obtained by exact diagonalization. The inset is a magnified view of a split multiplet of 15 states near zero. The spectrum is symmetric with respect to the real axis due to the $PT$ symmetry of the Liouvillian.

Figure 2

Bethe root distribution corresponding to the ${\mathsf{S}}_{\mathrm{tot}}^z=0$ eigenstate descended from the maximal ${\mathsf{S}}_{\mathrm{tot}}$ state of the ${\omega }_i=0$ model ($n=20$). The curves of different color correspond to different values of $1/g_{+}$ (increasing from left to right), and the ${\omega }_i$ are shown as red circles along the imaginary axis. One can see that ${\mathsf{S}}_{\mathrm{tot}}$ is maximum by noting that all ${\mu }_i$ go to infinity as $1/g_{+}$ vanishes, and so from Eq. (17) the state is derived from $|{\chi }^{-}⟩$ simply by raising ${\mathsf{S}}_{\mathrm{tot}}^z$ to zero.

Figure 3

Comparison of the decay rate (dominant Liouvillian eigenvalue) of $n$-spin correlations for $\mathrm{\Omega }=-(n+3)$ and ${\omega }_j=j{\mathrm{\Delta }}_y$ (where ${\mathrm{\Delta }}_y=2$) evaluated by (i) exact diagonalization for $n=8$, (ii) exact solution at $n=8$ and 60, and (iii) exact solution for $n\to \infty$. The inset shows the string state formed by the Bethe roots (blue squares), found by numerical solution of Eq. (18) for $n=20$ and $g_{+}\to \infty$; as in Fig. 2, the ${\omega }_i$ are represented by red circles. The effect of finite $g_{+}$ is to push the Bethe roots in the negative real direction.

Figure 4

Typical motion of the singlet eigenvalues as the ${\omega }_i$ are smoothly translated (parameterized by $\mathrm{\Delta }\omega$), revealing the presence of level crossings.