A Transmon Quantum Annealer: Decomposing Many-Body Ising Constraints Into Pair Interactions

  1. Martin Leib,
  2. Peter Zoller,
  3. and Wolfgang Lechner
Adiabatic quantum computing is an analog quantum computing scheme with various applications in solving optimization problems. In the parity picture of quantum optimization, the problem
is encoded in local fields that act on qubits which are connected via local 4-body terms. We present an implementation of a parity annealer with Transmon qubits with a specifically tailored Ising interaction from Josephson ring modulators.

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.

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.

Thermal emission in the ultrastrong coupling regime

  1. A. Ridolfo,
  2. Martin Leib,
  3. S. Savasta,
  4. and M. J. Hartmann
We study thermal emission of a cavity quantum electrodynamic system in the ultrastrong-coupling regime where the atom-cavity coupling rate becomes comparable the cavity resonance
frequency. In this regime, the standard descriptions of photodetection and dissipation fail. Following an approach that was recently put forward by Ridolfo et al.[arXiv:1206.0944], we are able to calculate the emission of systems with arbitrary strength of light matter interaction, by expressing the electric field operator in the cavity-emitter dressed basis. Here we present thermal photoluminescence spectra, calculated for given temperatures and for different couplings in particular for available circuit QED parameters.

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.

Networks of nonlinear superconducting transmission line resonators

  1. Martin Leib,
  2. Frank Deppe,
  3. Achim Marx,
  4. Rudolf Gross,
  5. and Michael Hartmann
We investigate a network of coupled superconducting transmission line resonators, each of them made nonlinear with a capacitively shunted Josephson junction coupling to the odd flux
modes of the resonator. The resulting eigenmode spectrum shows anticrossings between the plasma mode of the shunted junction and the odd resonator modes. Notably, we find that the combined device can inherit the complete nonlinearity of the junction, allowing for a description as a harmonic oscillator with a Kerr nonlinearity. Using a dc SQUID instead of a single junction, the nonlinearity can be tuned between 10 kHz and 4 MHz while maintaining resonance frequencies of a few gigahertz for realistic device parameters. An array of such nonlinear resonators can be considered a scalable superconducting quantum simulator for a Bose-Hubbard Hamiltonian. The device would be capable of accessing the strongly correlated regime and be particularly well suited for investigating quantum many-body dynamics of interacting particles under the influence of drive and dissipation.

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.