Jan Carl Budich

Topological quantum matter with ultracold gases in optical lattices

Using optical lattices to trap ultracold atoms provides a powerful platform for probing topological phases, analogues to those found in condensed matter. But as these systems are highly tunable, they could be used to engineer even more exotic phases.

Majorana Bound States and Nonlocal Spin Correlations in a Quantum Wire on an Unconventional Superconductor

We study theoretically the proximity effect of a one-dimensional metallic quantum wire (in the absence of spin-orbit interaction) lying on top of an unconventional superconductor. Three different material classes are considered as a substrate: (i) a chiral superconductor in class D with broken time-reversal symmetry and a class DIII superconductor (ii) with and (iii) without a nontrivial Z(2) number. Interestingly, we find degenerate zero energy Majorana bound states at both ends of the wire for all three cases. They are unstable against spin-orbit interaction in case (i), while they are topologically protected by time-reversal symmetry in cases (ii) and (iii). Remarkably, we show that nonlocal spin correlations between the two ends of the wire can be simply controlled by a gate potential in our setup.

Observation of dynamical vortices after quenches in a system with topology

Topological phases constitute an exotic form of matter characterized by non-local properties rather than local order parameters1. The paradigmatic Haldane model on a hexagonal lattice features such topological phases distinguished by an integer topological invariant known as the first Chern number2. Recently, the identification of non-equilibrium signatures of topology in the dynamics of such systems has attracted particular attention3–6. Here, we experimentally study the dynamical evolution of the wavefunction using time- and momentum-resolved full state tomography for spin-polarized fermionic atoms in driven optical lattices7. We observe the appearance, movement and annihilation of dynamical vortices in momentum space after sudden quenches close to the topological phase transition. These dynamical vortices can be interpreted as dynamical Fisher zeros of the Loschmidt amplitude8, which signal a so-called dynamical phase transition9,10. Our results pave the way to a deeper understanding of the connection between topological phases and non-equilibrium dynamics.Non-equilibrium signatures of topology—the appearance, movement and annihilation of vortices in a cold-atom system—are identified, showing that topological phase can emerge dynamically from a non-topological state.Phase transitions are a fundamental concept in science describing diverse phenomena ranging from, e.g., the freezing of water to Bose-Einstein condensation. While the concept is well-established in equilibrium, similarly fundamental concepts for systems far from equilibrium are just being explored, such as the recently introduced dynamical phase transition (DPT). Here we report on the first observation of a DPT in the dynamics of a fermionic many-body state after a quench between two lattice Hamiltonians. With time-resolved state tomography in a system of ultracold atoms in optical lattices, we obtain full access to the evolution of the wave function. We observe the appearance, movement, and annihilation of vortices in reciprocal space. We identify their number as a dynamical topological order parameter, which suddenly changes its value at the critical times of the DPT. Our observation of a DPT is an important step towards a more comprehensive understanding of non-equilibrium dynamics in general.

Failure of protection of Majorana based qubits against decoherence

Qubit realizations based on Majorana bound states have been considered promising candidates for quantum information processing which is inherently inert to decoherence. We put the underlying general arguments leading to this conjecture to the test from an open quantum system perspective. It turns out that, from a fundamental point of view, the Majorana qubit is as susceptible to decoherence as any local paradigm of a qubit.

Dynamical topological order parameters far from equilibrium

We introduce a topological quantum number -- coined dynamical topological order parameter (DTOP) -- that is dynamically defined in the real-time evolution of a quantum many-body system and represented by a momentum space winding number of the Pancharatnam geometric phase. Our construction goes conceptually beyond the standard notion of topological invariants characterizing the wave-function of a system, which are constants of motion under coherent time evolution. In particular, we show that the DTOP can change its integer value at discrete times where so called dynamical quantum phase transitions occur, thus serving as a dynamical analog of an order parameter. Interestingly, studying quantum quenches in one-dimensional two-banded Bogoliubov de Gennes models, we find that the DTOP is capable of resolving if the topology of the system Hamiltonian has changed over the quench. Furthermore, we investigate the relation of the DTOP to the dynamics of the string order parameter that characterizes the topology of such systems in thermal equilibrium.

Group theoretical and topological analysis of the quantum spin Hall effect in silicene

Silicene consists of a monolayer of silicon atoms in a buckled honeycomb structure. It was recently discovered that the symmetry of such a system allows for interesting Rashba spin?orbit effects. A perpendicular electric field is able to couple to the sublattice pseudospin, making it possible to electrically tune and close the band gap. Therefore, external electric fields may generate a topological phase transition from a topological insulator to a normal insulator (or semimetal) and vice versa. The contribution of the present paper to the study of silicene is twofold. Firstly, we perform a group theoretical analysis to systematically construct the Hamiltonian in the vicinity of the K points of the Brillouin zone and find an additional, electric field induced spin?orbit term, that is allowed by symmetry. Subsequently, we identify a tight-binding model that corresponds to the group theoretically derived Hamiltonian near the K points. Secondly, we start from this tight-binding model to analyze the topological phase diagram of silicene by an explicit calculation of the topological invariant of the band structure. To this end, we calculate the topological invariant of the honeycomb lattice in a manifestly gauge invariant way which allows us to include Sz symmetry breaking terms?like Rashba spin?orbit interaction?into the topological analysis. Interestingly, we find that the interplay of a Rashba and an intrinsic spin?orbit term can generate a non-trivial quantum spin Hall phase in silicene. This is in sharp contrast to the more extensively studied honeycomb system graphene where Rashba spin?orbit interaction is known to compete with the quantum spin Hall effect in a detrimental way.

Topology of density matrices

We investigate topological properties of density matrices motivated by the question to what extent phenomena like topological insulators and superconductors can be generalized to mixed states in the framework of open quantum systems. The notion of geometric phases has been extended from pure to mixed states by Uhlmann in [Rep. Math. Phys. 24, 229 (1986)], where an emergent gauge theory over the density matrices based on their pure-state representation in a larger Hilbert space has been reported. However, since the uniquely defined square root

Dynamical topological quantum phase transitions for mixed states

\sqrt{\rho}

From the adiabatic theorem of quantum mechanics to topological states of matter

of a density matrix

Quantum Hall physics with cold atoms in cylindrical optical lattices

\rho