Non-Abelian Geometric Dephasing

We study the adiabatic dynamics of degenerate quantum states induced by loop paths in a control parameter space. The latter correspond to noisy trajectories if the system is weakly coupled to environmental modes. On top of conventional dynamic dephasing, we find a universal non-Abelian geometric dephasing (NAGD) contribution and express it in terms of the non-Abelian Berry connection and curvature. We show that NAGD implies either decay or amplification of coherences as compared to the coherences when only dynamic dephasing is present. The full NAGD matrix structure can be probed through interference experiments. We outline such a detection scheme for modified Majorana braiding setups.

Figure 1

An interferometric setup for observing NAGD. For details, see main text.

Figure 2

Schematic setup for NAGD detection using noisy Majorana braiding protocols. (a) Five-star Majorana setup with tunnel couplings ${\mathrm{\Delta }}_{j=1,\dots ,5}$ between Majorana operators ${\gamma }_0$ and ${\gamma }_j$. (b) Three-star setup with ${\mathrm{\Delta }}_3={\mathrm{\Delta }}_5=0$ (solid black lines indicate ${\mathrm{\Delta }}_j>0$) and topologically protected Majorana braiding protocol [29, 41, 42]. The shown sequence implies an exchange of ${\gamma }_1$ and ${\gamma }_2$. For instance, in the first step, one starts with only ${\mathrm{\Delta }}_4>0$. The blue arrow then means that ${\mathrm{\Delta }}_4(t)$ is slowly reduced to zero while ${\mathrm{\Delta }}_1(t)$ is simultaneously ramped up. (c) Elementary steps for the NAGD detection protocol, where ${\mathrm{\Delta }}_5=0$ at all times. The gray solid line indicates an additional coupling ${\mathrm{\Delta }}_3^{\prime }\ne 0$ needed for generating NAGD. For a full description of the protocol, see main text.