Geometric phase in Stückelberg interferometry

We study the time evolution of a two-dimensional quantum particle exhibiting a two-band energy spectrum with two Dirac cones as, for example, in the honeycomb lattice. A force is applied such that the particle experiences two Landau-Zener transitions in succession in the vicinity of the Dirac cones. The adiabatic evolution between the two transitions leads to Stückelberg interferences, due to two possible trajectories in energy-momentum space. In addition to well-known dynamical and Stokes phases, the interference pattern reveals a geometric phase which depends on the chirality (winding number) and the mass sign associated with each Dirac cone, as well as on the type of trajectory (parallel or diagonal with respect to the two cones) in parameter space. This geometric phase reveals the coupling between the bands encoded in the structure of the wave functions. Stückelberg interferometry therefore appears as a way to access both intra- and interband geometric information.

Figure 1

(a) A Stückelberg interferometer made of two avoided crossings $(D$ and $D^{\prime })$ in an energy spectrum $E$ as a function of momentum $p$. A particle initially in the lower band is forced through the two avoided crossings that act as beam splitters. $P_f$ is the probability for the particle to end up in the upper band. This can occur through two different paths in energy-momentum space. (b) The geometric phase ${\phi }_g$ is revealed in the interference pattern $(P_f$ as a function of the distance $D{D}^{\prime })$. The dashed blue line $\left({\phi }_g=0\right)$ corresponds to trajectory (c) and the solid red line $\left({\phi }_g\ne 0\right)$ corresponds to trajectory (d). (c),(d) Double Dirac cone energy spectrum as a function of two-dimensional momentum with parallel [(c) blue arrow] or diagonal [(d) red arrow] trajectories. Chirality (shown as directed circles) and mass $M$ of each Dirac cone are also indicated (see text).

Figure 2

Bloch sphere representation of a Zeeman-like Hamiltonian $H=\vec{B}·\vec{\sigma }=E_{+}\vec{n}·\vec{\sigma }$, where the unit vector $\vec{n}$ is parametrized by spherical coordinates given by the polar angle $0\le \theta \le \pi$ and the azimuthal angle $0\le \varphi <2\pi$.

Figure 3

Similar energy spectra (a),(b) corresponding to different eigenstates (c),(d). (a),(b) Low-energy spectra featuring two gapped Dirac cones: panel (a) corresponds to the universal model with opposite chiralities [Eq. (1)]; panel (b) corresponds to the universal model with identical chiralities [Eq. (4)]. Although the two spectra look qualitatively the same, panel (b) has rotational symmetry around each Dirac point, but not panel (a). (c),(d) Plot of the relative phase $\varphi$ between ${\sigma }_x$ and ${\sigma }_y$ components of the Hamiltonian (azimuthal phase on the Bloch sphere) as a function of the momentum $(p_x,p_y)$, with Dirac points located at $D,D^{\prime }$. (c) Hamiltonian with opposite chirality (winding number $-1$ at $D$ and $+1$ at $D^{\prime })$. (d) Hamiltonian with same chirality (winding number $+1$ at $D$ and $D^{\prime })$.

Figure 4

(a) Two trajectories (parallel or diagonal) realized by applying a constant force $\vec{F}$. (b) Energy landscape as seen by the particle under acceleration (similar for both trajectories).

Figure 5

Final transition probability $P_f$ as a function of the time interval $2t_0=2\sqrt{{\Delta }_{*}}$ between the two Dirac cones for the time-dependent Hamiltonians of cases No. 1 and No. 2, as indicated. The dots corresponds to the full numerical solution. The lines correspond to the Stückelberg theory $P_f=4P_{\text{LZ}}(1-P_{\text{LZ}}){sin}^2({\phi }_S+{\phi }_{\text{dyn}}/2+{\phi }_g/2)$. In case No. 1 (dashed line) ${\phi }_g=0$, whereas ${\phi }_g=\Delta \phi =-2arctan\frac{c_y{F}_y\sqrt{{\Delta }_{*}}}{M}\approx -\pi /4$ in case No. 2 (solid line); see Sec. 5c. The parameters are $c_y{F}_y=0.14,M^2=0.{35}^2-{\left(c_y{F}_y{t}_0\right)}^2$ and $c_y{p}_y=c_y{F}_y{t}_0$ such that the gap at the avoided crossings is $2\sqrt{M^2+{\left(c_y{F}_y{t}_0\right)}^2}=0.7$ in both cases.

Figure 6

Time evolution of the Hamiltonian $H\left(t\right)=E_{+}\left(t\right)\vec{n}\left(t\right)·\vec{\sigma }$ represented by the curve traced by $\vec{n}\left(t\right)$ on the Bloch sphere from $t=t_i$ to $t=t_f$. The dot in the middle of the trajectory is at $t=0$.

Figure 7

Final transition probability $P_f$ as a function of the time interval $2t_0=2\sqrt{{\Delta }_{*}}$ between the two Dirac cones for the case of No. 2 with $M=0$. The open circles are the full numerical solution. The solid curve corresponds to Stückelberg theory without the geometric phase $P_f=4P_{\text{LZ}}(1-P_{\text{LZ}}){sin}^2({\phi }_S+{\phi }_{\text{dyn}}/2)$. The parameters are $c_y{F}_y=0.1$ with the gap at the avoided crossings given by $2c_y{F}_y{t}_0$. There is a clear $\pi$ shift.

Figure 8

Trajectories (solid line) of the Hamiltonian (17) as the dimensionless mass parameter $M/\left(c_y{F}_y\sqrt{{\Delta }_{*}}\right)$ decreases from 0.2 in (a) to 0 in (d). The dotted line is the shortest geodesic line connecting positions of the initial and the final points of the Hamiltonian curve. We see that the area of the enclosed region evolves smoothly to the value $\pi$ for $M=0$.

Figure 9

Phase shift $\Delta \phi$ as a solid angle on the Bloch sphere for the four cases in Table 5. Cases No. 1 and No. 6 is shown in red (null solid angle), cases No. 2 and No. 5 in blue (same as Fig. 8), cases No. 3 and No. 8 in green, and cases No. 4 and No. 7 in black $(\pi$ solid angle, the geodesic closure is half of a great circle).

Figure 10

(a) The analytic structure of the function $g\left(t\right)\dot{\alpha }{e}^{i\alpha \left(t\right)}$ in the complex $t$ plane. The function has four branch points [because of $\alpha \left(t\right)$] in $t_1$ to $t_4$. In addition, there are five poles at $t_1,t_2,t_3,t_4$, and $t_6$. (b) In complex $\alpha$ plane, the function $g\left(\alpha \right)e^{i\alpha }$ has five simple poles $({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$, and ${\alpha }_6)$ indicated by crosses, three of which are in the upper complex plane $({\alpha }_1,{\alpha }_4$, and ${\alpha }_6)$. The integration contour is shown in red. The point ${\alpha }_5=-{\alpha }_6$ is also indicated but is not a pole of $g\left(\alpha \right)$.

Figure 11

The analytic structure of $g\left(\alpha \right)e^{i\alpha }$ for the massless case. The function has four simple poles $({\alpha }_1,{\alpha }_2,{\alpha }_3$, and ${\alpha }_4)$ indicated by crosses, two of which are in the upper complex plane $({\alpha }_1,{\alpha }_4)$. The integration contour is shown in red.

Figure 12

The shaded region area is bounded by the two paths, $\mathcal{C}$ and ${\mathcal{C}}_g$, which are defined by the Hamiltonian trajectory $\mathcal{C}$ and the shortest geodesic ${\mathcal{C}}_g$ connecting the final and initial points.