A Superconducting Quantum Simulator for Topological Order and the Toric Code

  1. Mahdi Sameti,
  2. Anton Potocnik,
  3. Dan E. Browne,
  4. Andreas Wallraff,
  5. and Michael J. Hartmann
Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based
on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations and prospects for protecting stored quantum information against errors. Here we show that the Hamiltonian of the central model of this class of systems, the Toric Code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via Superconducting Quantum Interference Devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along non-contractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single qubit operations allow to create and propagate elementary excitations of the Toric Code and to verify their fractional statistics. The architecture we propose allows to implement a large variety of many-body interactions and thus provides a versatile analog quantum simulator for topological order and lattice gauge theories.

Quantum State Engineering with Circuit Electromechanical Three-Body Interactions

  1. Mehdi Abdi,
  2. Matthias Pernpeintner,
  3. Rudolf Gross,
  4. Hans Huebl,
  5. and Michael J. Hartmann
We propose a hybrid system with quantum mechanical three-body interactions between photons, phonons, and qubit excitations. These interactions take place in a circuit quantum electrodynamical
architecture with a superconducting microwave resonator coupled to a transmon qubit whose shunt capacitance is free to mechanically oscillate. We show that this system design features a three-mode polariton–mechanical mode and a nonlinear transmon–mechanical mode interaction in the strong coupling regime. Together with the strong resonator–transmon interaction, these properties provide intriguing opportunities for manipulations of this hybrid quantum system. We show, in particular, the feasibility of cooling the mechanical motion down to its ground state and preparing various nonclassical states including mechanical Fock and cat states and hybrid tripartite entangled states.

Steady-state phase diagram of a driven QED-cavity array with cross-Kerr nonlinearities

  1. Jiasen Jin,
  2. Davide Rossini,
  3. Martin Leib,
  4. Michael J. Hartmann,
  5. and Rosario Fazio
We study the properties of an array of QED-cavities coupled by nonlinear elements in the presence of photon leakage and driven by a coherent source. The main effect of the nonlinear
couplings is to provide an effective cross-Kerr interaction between nearest-neighbor cavities. Additionally correlated photon hopping between neighboring cavities arises. We provide a detailed mean-field analysis of the steady-state phase diagram as a function of the system parameters, the leakage and the external driving, and show the emergence of a number of different quantum phases. A photon crystal associated to a spatial modulation of the photon blockade appears. The steady state can also display oscillating behavior and bi-stability. In some regions the crystalline ordering may coexist with the oscillating behavior. Furthermore we study the effect of short-range quantum fluctuations by employing a cluster mean-field analysis. Focusing on the corrections to the photon crystal boundaries, we show that, apart for some quantitative differences, the cluster mean field supports the findings of the simple single-site analysis. In the last part of the paper we concentrate on the possibility to build up the class of arrays introduced here, by means of superconducting circuits of existing technology. We consider a realistic choice of the parameters for this specific implementation and discuss some properties of the steady-state phase diagram.

Synchronized Switching in a Josephson Junction Crystal

  1. Martin Leib,
  2. and Michael J. Hartmann
We consider a superconducting coplanar waveguide resonator where the central conductor is interrupted by a series of uniformly spaced Josephson junctions. The device forms an extended
medium that is optically nonlinear on the single photon level with normal modes that inherit the full nonlinearity of the junctions but are nonetheless accessible via the resonator ports. For specific plasma frequencies of the junctions a set of normal modes clusters in a narrow band and eventually become entirely degenerate. Upon increasing the intensity of a red detuned drive on these modes, we observe a sharp and synchronized switching from low occupation quantum states to high occupation classical fields, accompanied by a pronounced jump from low to high output intensity.

The Circuit Quantum Electrodynamical Josephson Interferometer

  1. Robert Jirschik,
  2. and Michael J. Hartmann
Arrays of circuit cavities offer fascinating perspectives for exploring quantum many-body systems in a driven dissipative regime where excitation losses are continuously compensated
by coherent input drives. Here we investigate a system consisting of three transmission line resonators, where the two outer ones are driven by coherent input sources and the central resonator interacts with a superconducting qubit. Whereas a low excitation number regime of such a device has been considered previously with a numerical integration, we here specifically address the high excitation density regime. This is of particular interest as intra cavity fields might undergo a transition from low excitation number quantum fields to high amplitude classical fields when increasing the input drives. We present analytical approximations to these regimes in the form of two methods. The first method is a Bogoliubov expansion in quantum fluctuations which can be understood as an approximation for weak nonlinearities. As the second method we introduce a combination of mean-field decoupling for the photon tunneling with an exact approach to a driven Kerr nonlinearity which can be understood as an approximation for low tunneling rates.

Photon solid phases in driven arrays of non-linearly coupled cavities

  1. Jiasen Jin,
  2. Davide Rossini,
  3. Rosario Fazio,
  4. Martin Leib,
  5. and Michael J. Hartmann
We introduce and study the properties of an array of QED cavities coupled by non-linear elements, in the presence of photon leakage and driven by a coherent source. The non-linear
couplings lead to photon hopping and to nearest-neighbor Kerr terms. By tuning the system parameters, the steady state of the array can exhibit a photon crystal associated to a periodic modulation of the photon blockade. In some cases the crystalline ordering may coexist with phase synchronisation. The class of cavity arrays we consider can be built with superconducting circuits of existing technology.

A Quantum Single Photon Transistor in Circuit Quantum Electrodynamics

  1. Lukas Neumeier,
  2. Martin Leib,
  3. and Michael J. Hartmann
We introduce a circuit quantum electrodynamical setup for a quantum single photon transistor. In our approach single photons propagate in two open transmission lines that are coupled
via two interacting transmon qubits. The interaction is such that photons are not exchanged between the two transmission lines but a photon in one line can completely block respectively enable the propagation of photons in the other line. High on-off ratios can be achieved for feasible experimental parameters. Our approach is inherently scalable as all photon pulses can have the same pulse shape and carrier frequency such that output signals of one transistor can be input signals for a consecutive transistor.

Many Body Physics with Coupled Transmission Line Resonators

  1. Martin Leib,
  2. and Michael J. Hartmann
We present the Josephson junction intersected superconducting transmission line resonator. In contrast to the Josephson parametric amplifier, Josephson bifurcation amplifier and Josephson
parametric converter we consider the regime of few microwave photons. We review the derivation of eigenmode frequencies and zero point fluctuations of the nonlinear transmission line resonator and the derivation of the eigenmode Kerr nonlinearities. Remarkably these nonlinearities can reach values comparable to Transmon qubits rendering the device ideal for accessing the strongly correlated regime. This is particularly interesting for investigation of quantum many-body dynamics of interacting particles under the influence of drive and dissipation. We provide current profiles for the device modes and investigate the coupling between resonators in a network of nonlinear transmission line resonators.

Photon Blockade in the Ultrastrong Coupling Regime

  1. Alessandro Ridolfo,
  2. Martin Leib,
  3. Salvatore Savasta,
  4. and Michael J. Hartmann
We explore photon coincidence counting statistics in the ultrastrong-coupling regime where the atom-cavity coupling rate becomes comparable to the cavity resonance frequency. In this
regime usual normal order correlation functions fail to describe the output photon statistics. By expressing the electric-field operator in the cavity-emitter dressed basis we are able to propose correlation functions that are valid for arbitrary degrees of light-matter interaction. Our results show that the standard photon blockade scenario is significantly modified for ultrastrong coupling. We observe parametric processes even for two-level emitters and temporal oscillations of intensity correlation functions at a frequency given by the ultrastrong photon emitter coupling. These effects can be traced back to the presence of two-photon cascade decays induced by counter-rotating interaction terms.

Bose-Hubbard dynamics of polaritons in a chain of circuit QED cavities

  1. Martin Leib,
  2. and Michael J. Hartmann
We investigate a chain of superconducting stripline resonators, each interacting with a transmon qubit, that are capacitively coupled in a row. We show that the dynamics of this system
can be described by a Bose-Hubbard Hamiltonian with attractive interactions for polaritons, superpositions of photons and qubit excitations. This setup we envisage constitutes one of the first platforms where all technological components that are needed to experimentally study chains of strongly interacting polaritons have already been realized. By driving the first stripline resonator with a microwave source and detecting the output field of the last stripline resonator one can spectroscopically probe properties of the system in the driven dissipative regime. We calculate the stationary polariton density and density-density correlations $g^{(2)}$ for the last cavity which can be measured via the output field. Our results display a transition from a coherent to a quantum field as the ratio of on site interactions to driving strength is increased.