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arXiv:1212.2070 · Annalen der Physik

Circuit QED lattices: Towards quantum simulation with superconducting circuits

Sebastian Schmidt, Jens Koch

Published Apr 26, 2013• 312 citations
arXivHTMLPDF

Abstract

Abstract The Jaynes‐Cummings model describes the coupling between photons and a single two‐level atom in a simplified representation of light‐matter interactions. In circuit QED, this model is implemented by combining microwave resonators and superconducting qubits on a microchip with unprecedented experimental control. Arranging qubits and resonators in the form of a lattice realizes a new kind of Hubbard model, the Jaynes‐Cummings‐Hubbard model, in which the elementary excitations are polariton quasi‐particles. Due to the genuine openness of photonic systems, circuit QED lattices offer the possibility to study the intricate interplay of collective behavior, strong correlations and non‐equilibrium physics. Thus, turning circuit QED into an architecture for quantum simulation, i.e., using a well‐controlled system to mimic the intricate quantum behavior of another system too daunting for a theorist to tackle head‐on, is an exciting idea which has served as theorists’ playground for a while and is now also starting to catch on in experiments. This review gives a summary of the most recent theoretical proposals and experimental efforts.

DOI
10.1002/andp.201200261

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The Jaynes-Cummings model describes the coupling between photons and a single two-level atom.In circuit quantum electrodynamics, this model is implemented by combining microwave resonators and superconducting qubits with unprecedented experimental control.This can be used for quantum simulation, i.e., using a well-controlled system to mimic the intricate quantum behavior of another system too daunting for a theorist to tackle head-on.

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The field of cavity quantum electrodynamics (cavity QED) studies the interaction between light and matter on a fundamental level: a single atom interacts with a single photon. If the atom-photon coupling is larger than any dissipative effects, the system enters the strong-coupling limit. A peculiarity of this regime is the possibility to form coherent superpositions of light and matter excitations - a kind of 'molecule' consisting of an atomic and a photonic contribution. The novel research field of circuit QED extends cavity QED concepts to solid-state based system. Here, a superconducting quantum bit is coupled to an on-chip superconducting one-dimensional waveguide resonator. Owing to the small mode-volume of the resonant cavity, the large dipole moment of the 'artificial atom' and the enormous engineering potential inherent to superconducting quantum circuits, remarkable atom-photon coupling strengths can be realized. This thesis describes the theoretical framework, the development of fabrication techniques and the implementation of experimental characterization techniques for superconducting quantum circuits for circuit QED applications. In particular, we study the interaction between superconducting flux quantum bits and high-quality coplanar waveguide resonators in the strong-coupling limit. Furthermore, we report on the first experimental realization of a circuit QED system operating in the ultrastrong-coupling regime, where the atom-photon coupling rate reaches a considerable fraction of the relevant system frequencies. In these experiments we could observe phenomena that can not be explained within the renowned Jaynes-Cummings model.

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Abstract The Jaynes‐Cummings model describes the coupling between photons and a single two‐level atom in a simplified representation of light‐matter interactions. In circuit QED, this model is implemented by combining microwave resonators and superconducting qubits on a microchip with unprecedented experimental control. Arranging qubits and resonators in the form of a lattice realizes a new kind of Hubbard model, the Jaynes‐Cummings‐Hubbard model, in which the elementary excitations are polariton quasi‐particles. Due to the genuine openness of photonic systems, circuit QED lattices offer the possibility to study the intricate interplay of collective behavior, strong correlations and non‐equilibrium physics. Thus, turning circuit QED into an architecture for quantum simulation, i.e., using a well‐controlled system to mimic the intricate quantum behavior of another system too daunting for a theorist to tackle head‐on, is an exciting idea which has served as theorists’ playground for a while and is now also starting to catch on in experiments. This review gives a summary of the most recent theoretical proposals and experimental efforts.
Recent progress in Cavity QED (CQED) has brought forth the possibility of wiring up individual CQED systems into lattices to investigate collective behavior and correlated phases of strongly coupled light and matter [1-9]. Such Lattice CQED systems operate on polaritonic quasi-particles that are hybrids of light and matter in a controllable proportion, combining long-range coherence of photons and strong interactions typically displayed by massive particles. In this talk, I will review recent work on proposed quantum simulation of Hubbard models with Lattice CQED systems, which are akin to the celebrated Bose-Hubbard model extensively studied with cold atomic gases in optical lattices and Josephson junction arrays. A crucial feature of Lattice CQED systems that I will focus on is their non-equilibrium nature. Unavoidable photon loss coupled with the ease of feeding in additional photons through continuous external driving renders such lattices open quantum systems. Under the influence of drive and dissipation, Lattice CQED systems typically display steady-state phases thereby providing a platform for realizing novel, non-equilibrium phases of light and matter that may be very dierent from those realized with condensed matter systems.
The field of cavity quantum electrodynamics (cavity QED) studies the interaction between light and matter on a fundamental level: a single atom interacts with a single photon. If the atom-photon coupling is larger than any dissipative effects, the system enters the strong-coupling limit. A peculiarity of this regime is the possibility to form coherent superpositions of light and matter excitations - a kind of 'molecule' consisting of an atomic and a photonic contribution. The novel research field of circuit QED extends cavity QED concepts to solid-state based system. Here, a superconducting quantum bit is coupled to an on-chip superconducting one-dimensional waveguide resonator. Owing to the small mode-volume of the resonant cavity, the large dipole moment of the 'artificial atom' and the enormous engineering potential inherent to superconducting quantum circuits, remarkable atom-photon coupling strengths can be realized. This thesis describes the theoretical framework, the development of fabrication techniques and the implementation of experimental characterization techniques for superconducting quantum circuits for circuit QED applications. In particular, we study the interaction between superconducting flux quantum bits and high-quality coplanar waveguide resonators in the strong-coupling limit. Furthermore, we report on the first experimental realization of a circuit QED system operating in the ultrastrong-coupling regime, where the atom-photon coupling rate reaches a considerable fraction of the relevant system frequencies. In these experiments we could observe phenomena that can not be explained within the renowned Jaynes-Cummings model.
Gate-based digital quantum simulations offer an exciting new paradigm for studying the many-body physics of strongly correlated systems. In this context, electron-phonon models are challenging for qubit-only quantum simulators, as bosonic degrees of freedom require costly finite-dimensional encodings. Here, we elaborate on an alternative approach based on a digital-analog circuit QED architecture, where fermions are encoded in transmon qubits while bosons are represented directly by microwave resonators. The central building block of this framework is a qubit-resonator Rabi gate that emulates strong electron-phonon coupling and can be implemented through a sequence of resonant Jaynes-Cummings gates interleaved with layers of single-qubit rotations. Using this Rabi gate as the fundamental unitary operation, we construct quantum circuits for the Hubbard-Holstein and Yukawa-Sachdev-Ye-Kitaev models, which describe, respectively, strongly correlated electrons coupled to phonons and phonon-mediated interactions among Majorana fermions. We further demonstrate how nonclassical phonon physics and signatures of quantum chaos in these models can be probed through circuit simulations, and develop measurement and variational protocols tailored to near-term superconducting quantum hardware.
In the last decade, the field of Quantum Optics has been enriched with a completely new platform: superconducting quantum circuits, also known as circuit-QED. This platform builds on the original designs of superconducting qubits, that is few-level systems made of superconducting islands and Josephson junctions, but adds to it a new ingredient, microwave photons. In this talk I will review circuit-QED with a focus on two complementary lines of research: ultrastrong light-matter coupling and strongly correlated propagating photons. The first topic deals with novel designs of superconducting qubits, engineering their coupling to the microwave radiation field to reach Rabi frequencies that approach the qubit and photon frequencies. In this regime, known as ”ultrastrong coupling”, the rotating wave approximation from Quantum Optics is no longer valid, the vacuum may spontaneously polarize, producing photons that get bound to the qubit, and novel physics is expected to emerge. I will review recent experiments on this topic, together with various theoretical proposals to tame this ultrastrong coupling. The second topic revolves around the notion of ”propagating” photons in these circuits. Most experiments in circuit-QED so far adopted the typical configuration of artificial atoms (qubits) interacting with artificial cavities (LC resonators). This proved useful in order to reach the strong coupling regime, demonstrating Rabi oscillations and developing a variety of protocols for quantum information, tomograhy, entanglement, etc. However, recent experiments have shown that it is possible to also work with propagating photons, replacing the LC resonators with open transmission lines, and studying the scattering properties of those artificial atoms. In this context we have worked on developing new technology to detect and control these photons, engineering single-photon detectors, quantum metamaterials and photonic crystals, or even new nonlinear media based on coupled cavities.
We review recent developments regarding the quantum dynamics and many-body physics with light, in superconducting circuits and Josephson analogues, by analogy with atomic physics. We start with quantum impurity models addressing dissipative and driven systems. Both theorists and experimentalists are making efforts towards the characterization of these non-equilibrium quantum systems. We show how Josephson junction systems can implement the equivalent of the Kondo effect with microwave photons. The Kondo effect can be characterized by a renormalized light frequency and a peak in the Rayleigh elastic transmission of a photon. We also address the physics of hybrid systems comprising mesoscopic quantum dot devices coupled with an electromagnetic resonator. Then, we discuss extensions to Quantum Electrodynamics (QED) Networks allowing one to engineer the Jaynes–Cummings lattice and Rabi lattice models through the presence of superconducting qubits in the cavities. This opens the door to novel many-body physics with light out of equilibrium, in relation with the Mott–superfluid transition observed with ultra-cold atoms in optical lattices. Then, we summarize recent theoretical predictions for realizing topological phases with light. Synthetic gauge fields and spin–orbit couplings have been successfully implemented in quantum materials and with ultra-cold atoms in optical lattices — using time-dependent Floquet perturbations periodic in time, for example — as well as in photonic lattice systems. Finally, we discuss the Josephson effect related to Bose–Hubbard models in ladder and two-dimensional geometries, producing phase coherence and Meissner currents. The Bose–Hubbard model is related to the Jaynes–Cummings lattice model in the large detuning limit between light and matter (the superconducting qubits). In the presence of synthetic gauge fields, we show that Meissner currents subsist in an insulating Mott phase.