Synthetic Topological Matter

We propose a new protocol for engineering quantum many-body Hamiltonians with enhanced symmetries. The protocol is based on repeated pulsed application of a set of unitary operators $X_i$, with $X_i^2 = 1$, (which can be generalizedto $X^n = 1, n>2$) in a self-similar-in-time (``polyfractal”) manner. For local initial Hamiltonians, the protocol can simultaneously implement multiple global and local symmetries, with the accuracy improving superpolynomially with the fastest drive period. The effective Hamiltonian remains local and avoids heating over time scales that are stretched-exponentially long in the drive frequency. Such Floquet engineering can be used to realize novel quantum models, or in the case when two or more global symmetries $X_i$ anti-commute, engender a degenerate many-body spectrum that can be used to encode topological qubits controlled precisely by the same $X_i$. We also discuss how our protocol can be implemented robustly using a setup based on Majorana fermions. Networks of such Majoranas can be used to implement novel symmetry-protected topological phases, lattice gauge theories, among others.

Two-dimensional topological insulators (TIs) host gapless helical edge states that are predicted to support a quantized two-terminal conductance. Quantization is protected by time-reversal symmetry, which forbids elastic backscattering. Paradoxically, the current-carrying state itself breaks the time-reversal symmetry that protects it. Here we show that the combination of electron-electron interactions and momentum-dependent spin-polarization in helical edge states gives rise to feedback through which an applied current opens a gap in the edge state dispersion, thereby breaking the protection against elastic backscattering. Current-induced gap opening is manifested via a nonlinear contribution to the system's $I-V$ characteristic, which persists down to zero temperature. We discuss prospects for realizations in recently discovered large bulk band gap TIs, and an analogous current-induced gap opening mechanism for the surface states of three-dimensional TIs.

Directly observing anyons, quasiparticles with fractional exchange statistics, is a major challenge of contemporary physics, with practical implications for quantum computation. I will present a newly-developed protocol for preparing anyons in an optical cavity built by carefully aligning a set of high-quality mirrors. I will explain how one can drive the cavity with lasers to inject photons one by one, building up a fractional quantum Hall state. Additional “pinning” lasers are used to create anyonic “quasihole” excitations and move them around one another. The resulting phases are measured interferometrically. I will discuss the challenges of implementing the protocol by analyzing the adiabaticity and coherence constraints.

Topologically quantized Hall conductance in two dimensional electron system in presence of a uniform magnetic field and periodic scalar potential forms the backbone of our understanding of current topological condensed matter systems. We show with the example of graphene that system with periodic magnetic modulation ( that can be lithographically fabricated) shows topological quantisation of Hall conductivity is a combination of Integer Quantum Hall Effect and Anamolous Quantum Hall Effect proposed by Haldane. The consequence of this type of topological quantisation will be discussed. Ref. 1.Manisha Arora and Sankalpa Ghosh, PHYSICAL REVIEW B 98, 155425 (2018) 2. Manisha Arora and Sankalpa Ghosh ( in preparation)

Superconducting (SC) heterostructures offer a huge flexibility in creating complex circuits operated in the quantum regime. Some of the recent advances in implementing hardware for quantum information processing have been achieved using SC qubits. Most of these circuits use standard Josephson tunnel contacts, because they can be fabricated in large numbers and in a controlled fashion. There have been proposals to use a non-standard Josephson junction that contains one or a few modes to realize a qubit, in which the computational basis is formed by so-called Andreev bound states [1]. These levels have subsequently been detected spectroscopically [2] and even a coherent manipulation of the Andreev level qubit was demonstrated experimentally [3]. Multiterminal Josephson junctions can sustain topologically nontrivial states by the Andreev levels formed in such a device [4]. A first signature of the topological transition was subsequently identified experimentally [5]. We analyze a microscopic model that realizes the topological transition with theoretically expected Weyl nodes. In presence of time-dependent ac-oscillation of the phase differences applied at the contacts, we demonstrate the mapping to a spin-1/2 in an effective time-dependent magnetic field and show how to extract the topological properties spectroscopically. For this purpose, we suggest that circularly or linearly polarized light is topologically equivalent to driving two SC phases with controlled amplitudes and defined phase lag. This paves the way to measure topological properties like the Chern number directly using microwaves. [1] N. M. Chtchelkatchev and Y. V. Nazarov, Phys. Rev. Lett. 90, 226806 (2003); A. Zazunov et al., Phys. Rev. Lett. 90, 087003 (2003). [2] L. Bretheau, C. O. Girit, H. Pothier, D. Esteve, and C. Urbina, Nature 499, 312 (2013). [3] C. Janvier et al., Science 349, 1199 (2015). [4] R.-P. Riwar, M. Houzet, J. S. Meyer, and Y. V. Nazarov, Nat. Comm. 7, 11167 (2016). [5] E. Strambini et al., Nat. Nano. 11, 1055 (2016).

The Niu-Thouless-Wu (NTW) formula demonstrates the quantization of Hall conductance in the presence of disorders or interactions by defining the many-body version of the Chern number. In this poster, we show numerically that the integration in the NTW formula is indeed unnecessary if the system size and the excitation gap are sufficiently large. The Berry curvature is already effectively quantized with the exponential accuracy with respect to the system size. The absence of integration is particularly useful in the interacting many-body problems for a sufficiently large system size. We also discuss the unimportance of the integration in the vicinity of a quantum phase transition by using concrete models with a vanishing excitation gap. Ref: arXiv:1808.10248

The momentum and spin of charge carriers in the topological insulators are constrained to be perpendicular due to strong spin-orbit coupling. Sb2Te3 is one of the topological insulator materials with a bulk band gap of 0.28 eV and simple surface states consisting of a single Dirac cone in the band gap. We have synthesized single crystalline Sb2Te3 nanowires using low pressure catalytic chemical vapor deposition, via vapor-liquid-solid growth mechanism. Two levels of aligned E-beam lithography were used to pattern non-magnetic outer Au leads and two magnetic tunnel junction inner leads on individual Sb2Te3 nanowires. The tunnel junction leads consist of a free Py (Ni80Fe20) layer, whose magnetization determines the magnitude and direction of spin current injected into the Sb2Te3 nanowire. Measurements of the device resistance between the two Au leads reveal that the Au/Sb2Te3 has ohmic contacts. The two-point resistance measured through the topological channel as a function of magnetic field shown exhibits positive magneto-resistance, originating from weak anti-localization of carriers in the Sb2Te3 nanowire induced by spin-orbit interaction. The weak anti-localization signal serves as evidence of a strong impact of spin orbit interaction on transport in the Sb2Te3 nanowire system. Furthermore, we have also measured a non-local spin valve signal in Sb2Te3 nanowire channels. The symmetry of this non-local spin valve (NLSV) signal is dramatically different from that of a NLSV with a channel that lacks spin-momentum locking (such as graphene). Two parallel states of the injector and detector magnetic moments give rise to different non-local voltage values, which is never observed in conventional NLSVs. This unusual symmetry is a clear signature of the spin-momentum locking in the Sb2Te3 nanowire topological surface state.

We demonstrate that the non-Abelian nature of Moore-Read quasiholes can be experimentally probed without braiding or exchanging them. By applying the adiabatic theorem to the case of rigid rotations of the anyons, we establish a general relation between the statistical properties of these excitations and the angular momentum expectation value taken on the many-body state. After that, we use simple mathematical equivalences and Monte Carlo calculations to prove that the braiding phase of Moore-Read quasiholes, and the existence of multiple fusion channels, can be obtained in two different ways: by measuring (i) the mean square radius of the system or (ii) the depletions the quasiholes create in its bulk density. Consisting basically in density measurements, our protocols can be easily applied to both ultracold atoms and quantum fluids of light. Generalizations of this idea to lattice systems as well as to the simpler case of Abelian quasiholes are also discussed.

Ultracold atoms in optical lattices provide a versatile tool to study the effects of strong magnetic fields on lattice systems, which are not easily accessible in a conventional solid state setting. In this work we examine the ground state properties of the bosonic two-leg ladder lattice under the influence of an artificial staggered magnetic flux. We calculate the order parameters of the arising superfluid phases and provide the full phase diagram, which is obtained analytically through a perturbative mean-field approach and numerically through a Cluster-mean-field theory. Finally, we investigate the experimental signatures of the superfluid given by the chiral currents, which show a vortex, anti-vortex configuration.

Floquet engineering is a widely applicable method to realize novel effectively static Hamiltonians via driving a quantum system. Several experiments have successfully demonstrated Floquet Hamiltonians with non-interacting ultracold atoms. Yet, the time scales for which this effective Hamiltonian properly describes the dynamics of a driven strongly-interacting many-body state have not been explored. For long times, the system is expected to heat up due to continuous energy absorption from the drive. We experimentally study these aspects in the driven Fermi-Hubbard model using strongly-interacting ultracold fermions in a driven three-dimensional optical lattice. First, the dynamics of the engineered Floquet state is compared to the one of an equivalent static many-body state. Our observables show that these dynamics coincide for up to several hundreds of driving cycles. This time scale is ultimately limited by Floquet heating and consequently atom loss. Second, non-equilibrium dynamical mean field theory is used for an experiment-theory comparison in regimes where no static analogue is available. When the driving frequency is close to the interaction energy, double occupancies are created via resonant tunneling processes. These novel hopping mechanisms are studied in the effective static description. Good agreement between our theoretical and experimental methods prove the validity of the Floquet Hamiltonian description. Our results establish that the driven Fermi-Hubbard model can be implemented on realistic experimental time scales and in future work could be used to explore dynamical gauge fields.

The coupling between gauge and matter fields is a key concept in many models of high-energy and condensed matter physics. In these models, the gauge fields are dynamical quantum degrees of freedom, i.e. they are influenced by the spatial configuration and motion of the matter field. It has been proposed to quantum simulate this coupling mechanism, ultimately aiming at emulating lattice gauge theories. However, existing methods for generating gauge fields in ultracold atoms in optical lattices lack the back-action from the atoms. In this experiment we realize the fundamental ingredient for a density-dependent gauge field by engineering non-trivial Peierls phases that depend on the site occupation of fermions in a Hubbard dimer. Our method relies on breaking time-reversal symmetry by driving the optical lattice simultaneously at two frequencies, at resonance with the on-site interaction. In addition, a constant energy offset between the two sites of the dimer allows us to single out one tunnelling process of which we characterise both the amplitude and the associated Peierls phase. In future experiments, more elaborate driving schemes, in which certain gauge invariances are imposed, can be implemented to perform quantum simulations of intractable problems in lattice gauge theories such as quantum electro- and chromodynamics.

Topological effects are being studied in the context of superconducting circuits, where quantized transconductance has been predicted in multi-terminal superconducting weak links[1]. We propose a simple topological circuit containing three conventional Josephson tunnel junctions in the charge regime. The energy spectrum of the system shows Weyl singularities which are robust to disorder. These degeneracies are associated with non-zero Chern numbers, which lead to a quantized current across one of the junction. We then discuss an experiment to detect the Weyl singularities of the circuit with spectroscopy. We also establish the relationship between our circuit and the Cooper pair pump[2]. We conclude with an experiment allowing to measure the Chern number and thus reveal the topological nature of the circuit. [1] Riwar, R.-P. et al. Multi-terminal Josephson junctions as topological matter. Nat. Commun. 7:11167 (2016). [2] R Leone, L Lévy and P Lafarge. The Cooper Pair Pump as a Quantized Current Source. Phys. Rev. Lett. 100, 117001 (2008)

We consider a system of a square lattice of impurities immersed in a two-dimensional p-wave superconductor and study its properties in the limit of Shiba states coherence length reaching impurity lattice constant. Such a system forms an exotic band structure with extremely high Chern numbers [1-3]. We develop an effective model Hamiltonian capturing essential properties of the considered system, allowing studies of topological properties of the system and its emergent structures of gap closing points in the limit of vanishing densities of the impurities. We indicate non-trivial differences in topological properties between zero-density limit of the impurities and clean p-wave superconductor. [1] Joel Röntynen and Teemu Ojanen, Phys. Rev. Lett. 114, 236803 (2015) [2] Joel Röntynen and Teemu Ojanen, Phys. Rev. B 93, 094521 (2016) [3] Lukas Kimme and Timo Hyart, Phys. Rev. B 93, 035134 (2016)

We propose a method by which the quantization of the Hall conductance can be directly measured in the transport of a one-dimensional atomic gas. Our approach builds on two main ingredients: (1) a constriction optical potential, which generates a mesoscopic channel connected to two reservoirs, and (2) a time-periodic modulation of the channel, specifically designed to generate motion along an additional synthetic dimension. This fictitious dimension is spanned by the harmonic-oscillator modes associated with the tightly-confined channel, and hence, the corresponding “lattice sites” are intimately related to the energy of the system. We analyze the quantum transport properties of this hybrid two-dimensional system, highlighting the appealing features offered by the synthetic dimension. In particular, we demonstrate how the energetic nature of the synthetic dimension, combined with the quasi-energy spectrum of the periodically-driven channel, allows for the direct and unambiguous observation of the quantized Hall effect in a two-reservoir geometry. Our work illustrates how topological properties of matter can be accessed in a minimal one-dimensional setup, with direct and practical experimental consequences

We study ultracold fermionic ytterbium in topologically non-trivial optical lattices. This element provides two internal degrees of freedom which can be used to create synthetic ladders. A long-lived metastable clock state and the ground state can be used as a two-site dimension. Additionally, the SU(6) symmetry in $^{173}$Yb ensures that the six $m_F$ states can be used as a synthetic dimension. We present progress studying the influence of interactions in these synthetic dimensions.

We present a topological characterization of time-periodically driven two-band models in 2+1 dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. Moreover, after being cast as a Hopf insulator, not only the Chern numbers, but also the winding numbers of the Floquet topological insulator become accessible in experiments as linking numbers. We exploit this description to propose a feasible scheme for measuring the complete set of their Floquet topological invariants in optical lattices.

We study non-Hermitian systems of cavity arrays, in which the topological phases stem from the interplay of coherent and dissipative interaction. Thus, in contrast to the canonical examples known from condensed matter physics, such as the SSH model, in this case the dissipation plays a crucial part in the emergence of topologically non-trivial phases. We demonstrate that this will give rise to stable topologically amplified edge states, leading to possible applications as non-reciprocal travelling wave amplifier. The desired interaction between cavities can for instance be realised in optomechanical setups.

Topology is a topic of great importance in condensed matter systems, covering the quantum Hall effect, topological insulators and topological superconductivity. However, the microscopic origin of these effects is hard to observe in solids. Therefore, numerous approaches have been taken in order to engineer synthetic topological matter to explore this exciting field in more detail. In our experiment, we use ultracold fermionic potassium atoms in a driven honeycomb lattice in order to realize the Haldane model. By performing a state tomography protocol, we can measure the Berry curvature of the lowest band, which also allows to calculate the corresponding Chern number. A second tool we implemented is the study of circular dichroism. This technique probes the difference in the response of a system to circular lattice shaking of one chirality as compared to the other chirality. Specifically, the Chern number of the lowest band is given by the frequency integrated difference of the excitation rates, resulting in a quantized response.

Static synthetic magnetic fields give rise to phenomena including the Lorentz force and the quantum Hall effect even for neutral particles, and they have by now been implemented in a variety of physical systems. Moving towards fully dynamical synthetic gauge fields allows, in addition, for backaction of the particles’ motion onto the field. If this results in a time-dependent vector potential, conventional electromagnetism predicts the generation of an electric field. Here we show that synthetic electric fields for photons arise self-consistently due to the classical nonlinear dynamics in driven optomechanical arrays giving rise to unidirectional transport of light. The next natural step is to consider the synthetic gauge fields in the quantum regime, where quantum fluctuations cannot be neglected. We identify a single parameter controlling the strength of quantum fluctuations enabling us to investigate the classical-to-quantum crossover. We show that the generation of synthetic electric fields is robust against noise and it leads to unidirectional transport of photons also in the quantum regime, albeit with reduced isolation ratio. Our study opens the path for dynamical gauge fields in the quantum regime based on optomechanical arrays.

In the presence of strong random scattering the behavior of particles with a particle-hole symmetric spectrum is fundamentally different from Anderson localization of particles in a single band: A random gap creates geometric states rather than confining the particles to an area of the size of the localization length. These states are subject to propagation, which reflects the robustness of edge modes even in the presence of very strong randomness. The properties are discussed on a general basis and applied to disordered photonic crystals with spectral degeneracies in the form of Dirac nodes.