On the challenge of simulating dipolar contributions to spin relaxation with generalized cluster correlation expansion methods

Abstract

The study of spin decoherence is often performed by assuming that spin-phonon interactions lead to relaxation at high temperatures, and spin-spin dipolar interactions instead contribute to pure dephasing at low temperatures. This has resulted in the neglect of spin relaxation due to spin-spin dipolar interactions and its influence on decoherence at low temperatures. For a complete understanding of low temperature spin dynamics, it is then imperative to focus also on the latter mechanism. One such method which has shown great promise in the efficient calculation of central spin dynamics due to spin-spin dipolar interactions with a surrounding spin bath is the Cluster-Correlation Expansion (CCE). An extension of this method through the explicit inclusion of the central spin degrees of freedom, known as the generalized Cluster-Correlation Expansion (gCCE) is capable of simulating the transfer of energy from the central spin into the bath, and thus could have the potential to investigate spin relaxation in this setting. In this work, we show that gCCE, in its standard form, is insufficient for providing even a qualitatively accurate description of spin-spin relaxation. A full mathematical deconstruction of the underlying theory of gCCE clearly points to the origin of such a breakdown and provides a starting point for its potential future resolution.

Figures (6)

• Figure 1: Spin Relaxation. A central spin undergoes relaxation with a spin bath by exchanging energy with resonant bath spins through spin flip-flops.
• Figure 2: Spin Dephasing. Spin flip-flops between the bath spins cause fluctuations in the magnetic field experienced by the central spin, which result in continuous shifts of its Zeeman levels. Its transition frequency then broadens from a well-defined value to a spectrum of possible values.
• Figure 3: Generalized Cluster-Correlation Expansion (gCCE). The central spin (center, red) interacts with a spin bath of different spin types (red, blue and purple) through spin-spin dipolar interactions (wavy lines). In the gCCE framework the bath spins are partitioned into clusters of different sizes, shown by the pairs and triplets of bath spins with the dashed outlines. Only the red bath spins will contribute to relaxation, as they are the same spin type as the central spin.
• Figure 4: CCE Relaxation Convergence. Dynamical evolution of the spin-up state of a free electron interacting with a maximally mixed spin bath of 8 free electrons. The steady state of the dynamics either becomes unphysical or reaches an overdamped value where there is no longer any spin-up population.
• Figure 5: CCE Convergence/Divergence. Simulated dynamics for a 9 electron spin system at CCE order 2. Convergent behavior is calculated when there is no overlap between the order 2 clusters and the order 1 clusters (blue line), while the dynamics diverges once the order 2 clusters begin to overlap with the order 1 clusters (orange line).