Non-Abelian Berry phase for open quantum systems: Topological protection versus geometric dephasing

We provide a detailed theoretical analysis of the adiabatic evolution of degenerate open quantum systems, where the dynamics is induced by time-dependent fluctuating loop paths in control parameter space. For weak system-bath coupling, the fluctuations around a deterministic base path obey Gaussian statistics, where we assume that the quantum adiabatic theorem is also satisfied for fluctuating paths. We show that universal non-Abelian geometric dephasing (NAGD) contributions are contained in the fluctuation-averaged evolution operator. This operator plays a key role in all experimental protocols proposed in this work for the detection of NAGD. In particular, we formulate interference measurements providing full access to NAGD. Apart from a factor due to the dynamic phase and its fluctuations, the averaged evolution operator contains the averaged Berry matrix. A polar decomposition of this nonunitary matrix into the product of a unitary, $V$, and a positive-semidefinite Hermitian matrix, $R$, reveals the physics of the NAGD contributions. Unlike the conventional dynamic dephasing rate, the NAGD eigenrates encoded by $R$ depend on the loop orientation sense and, in particular, change sign under a reversal of the direction. A negative rate then implies amplification of coherences as compared to the ones when only dynamic dephasing is present. The non-Abelian character of geometric dephasing is rooted in the noncommutativity of $V$ and $R$ and causes smoking-gun signatures in interference experiments. We also clarify why systems subject to topological protection do not show geometric dephasing. Without full protection, however, geometric dephasing can arise and has to be taken into account. As concrete application, we propose spin-echo NAGD detection protocols for modified Majorana braiding setups.

Figure 1

Different mechanisms of topological protection. (a) The Berry curvature (orange) is localized at isolated points in parameter space. Fluctuations of the parameter trajectory (green) do not change the Berry matrix as long as the trajectory stays in the region of vanishing Berry curvature. This scenario is typical for the topological protection of anyon braiding. (b) Same as before but the Berry curvature remains finite in the relevant parameter space. The fluctuating trajectory (blue) feels the Berry curvature and thus the Berry matrix is sensitive to fluctuations. Hence there is no topological protection in this case. (c) Also for this example—corresponding to Majorana braiding setups with control parameters ${\mathrm{\Delta }}_j/\mathcal{E}$, see Sec. 5—the Berry curvature does not vanish. However, for system-specific reasons, it may happen that physically realizable fluctuations cannot deform certain trajectories (green). This results in another topological protection mechanism. However, more general trajectories do exhibit fluctuations (blue), and the corresponding Berry matrices are then unprotected. As explained in the main text, topologically protected systems do not exhibit NAGD.

Figure 2

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

Figure 3

The essence of two-block interference. The adiabatic evolution changes the system state $|\psi \rangle$ by the action of the Berry matrix ${\mathcal{U}}_B$ which acts differently in the two blocks (red, blue). The measured operator $M$ can have block-diagonal and block-off-diagonal components. The block-diagonal components (red, blue) are not sensitive to energy fluctuations and thus to NAGD. The block-off-diagonal component (green) probes the interference between the two blocks and is sensitive to NAGD.

Figure 4

Starlike Majorana setups and protocols for NAGD detection. (a) Five-star Majorana setup with tunnel couplings ${\mathrm{\Delta }}_{j=1,...,5}$ between Majorana operators ${\gamma }_0$ and ${\gamma }_j$, see Eq. (5.1). (b) Three-star setup with ${\mathrm{\Delta }}_3={\mathrm{\Delta }}_5=0$ (dashed black lines indicate ${\mathrm{\Delta }}_j\ne 0$). This is a poor man's non-Abelian setup—while the Hamiltonian has a degeneracy, all components of the Berry connection (and of the Berry curvature) commute with each other. (c) Five-star Majorana setup used for the NAGD detection protocol with ${\mathrm{\Delta }}_5=0$. This represents a truly non-Abelian setup. (d) Topologically protected Majorana braiding protocol in the three-star setup, cf. Ref. [23, 42, 43]. Solid black lines indicate the nonzero couplings, ${\mathrm{\Delta }}_j\left(t\right)\ne 0$, during the respective time interval. The shown sequence leads to an exchange of ${\gamma }_1$ and ${\gamma }_2$. In the first step, one starts with ${\mathrm{\Delta }}_4>0$ and all other ${\mathrm{\Delta }}_j=0$. The blue arrow indicates that ${\mathrm{\Delta }}_4\left(t\right)$ is slowly reduced to zero while ${\mathrm{\Delta }}_1\left(t\right)$ is simultaneously ramped up, and similarly for the other steps. (e) Elementary steps for the NAGD detection protocol in the poor man's non-Abelian case using the three-star setup in panel (b). The gray solid line indicates an additional constant coupling ${\mathrm{\Delta }}_4^{\prime }\ne 0$ which breaks topological protection and implies NAGD. For a full description of this protocol, see Sec. 5d. (f) Same as in panel (e) but for the truly non-Abelian case, where ${\mathrm{\Delta }}_3^{\prime }\ne 0$ denotes a constant additional coupling (gray) present during the respective time intervals. For details, see Sec. 5e.

Figure 5

Schematic setup for a possible implementation of the Majorana five-star device. Six long Majorana nanowires (green) are connected to a common floating superconductor and thus form a superconducting island with charging energy $E_C$. The outer MBSs described by the ${\gamma }_j^{\prime }$ operators are spectator modes; i.e., their presence does not affect our protocols. By arranging six Majorana nanowires ($j=0,...,5$) as shown, the five Majorana operators ${\gamma }_{j=1,...,5}$ are tunnel-coupled only to the central Majorana operator ${\gamma }_0$. One can thereby realize the setup in Fig. 4.