Session Y29: Focus Session: Quantum Optics with Superconducting Circuits: Nonlinearity and Itinerant Microwave Photons

Microwave Photon Counter Based on Josephson Junctions

We describe progress in the development of a microwave photon counter based on current biased Josephson junctions; absorption of a single photon causes the junction to switch to the voltage state, producing a large and easily measured classical signal. We have combined multiple junctions with a broad-band, on-chip microwave beam splitter to realize a multiplexed microwave photon detector. We discuss application of the Josephson microwave counter to the study of full counting statistics of the microwave emission from various mesoscopic conductors, and we describe alternative biasing schemes to enable operation in more traditional photon counting modes.   

Quantum efficiency of a single microwave photon detector based on a current-biased Josephson junction

We analyze the quantum efficiency of a single microwave photon detector based on a current-biased Josephson junction. We consider the Jaynes-Cummings Hamiltonian to describe coupling between the photon field and the junction. We then take into account coupling of the junction and the resonator to the environment. Numerically solving the equation of motion of the density matrix of the resonator-junction system, we compute the quantum efficiency of the photon detector as a function of detection time, bias current and the junction's decay time. For current-biased Josephson junctions, the efficiency to detect a single photon with frequency in the microwave regime is around 50\%. Our results also indicate that a highly efficient single microwave photon detector is feasible for a moderate improvement in the junction's decay time.   

Backaction of Microwave Photon Detection by a Strongly Coupled Josephson Junction

We analyze the functionality of on-chip Josephson junctions as single microwave photon detectors, as has been demonstrated recently in Chen, {\it et al.}, arXiv:1011.4329. The Josephson junction device, which we refer to as a Josephson Photomultiplier (JPM), acts as a nearly perfect binary detectors of microwave photons by undergoing an observable switching event when there are one or more photons in an incident cavity. We analyze the backaction of this switching event on the state of incident light, including the energy dissipation and dephasing affecting an imperfect JPM. This analysis improves the efficiency and fidelity with which a JPM reconstructs the state of light in an incident transmission line `cavity', which are commonly used to store and transfer quantum states in implementations of circuit-QED.   

Cooperative effects for Qubits in a Transmission Line: Theory

Strong extinction of the transmitted power in a 1D transmission line coupled to an artificial atom has recently been achieved [1]. In contrast to the 3D case, large extinctions are made possible by the strong light-matter coupling occurring because of reduced dimensionality. Motivated by this, here we consider the situation where multiple artificial atoms (ie transmon qubits) are coupled to the 1D line. Following the work of Lehmberg for the 3D case [2], we obtain a master equation describing the dynamics of an arbitrary number of qubits coupled to the line. This master equation reveals interaction between the qubits mediated by the line. Using the input-output formalism, the model is compared to experimental results for multiple qubits coupled to the 1D line. [1] O. Astafiev et al., Science 327, 840 (2010) [2] R. H. Lehmberg. Phys. Rev. A 2, 883 (1970).   

Cooperative Effects for Qubits in a Transmission Line: Experiment

The interaction probability between freely propagating photons and atoms or atom-like systems is greatly enhanced in one dimension. Thus a system of many atoms coupled to a one-dimensional continuum of electromagnetic modes is expected to reveal many interesting phenomena - the photons emitted by an atom can be absorbed by other atoms and coherently interact with propagating modes of the continuum. We implement three superconducting qubits coupled strongly to an open transmission line to investigate such light-matter interactions in one dimension. We characterize our system by scattering radiation off the qubits and measuring the transmitted and reflected field. For low driving powers, a single qubit reflects nearly all incident radiation resonant with its transition frequency [1]. When two qubits are tuned such that their emitted radiation has a wavelength close to twice the distance between the qubits we observe interference effects in the reflection and transmission spectra. At high driving powers resonance fluorescence is measured for single qubits and multiple qubits in resonance. These results present first steps towards investigating cooperative effects for multiple qubits in open one-dimensional space.\\[4pt] [1] O.~Astafiev {\em et~al.}, Science, {\bf327}, 840 (2010)   

Hong-Ou-Mandel Interference in Circuit QED Experiments

The Hong-Ou-Mandel (HOM) effect is a quantum interference effect whereby two indistinguishable photons incident at either side of a balanced beam splitter will be detected together at one output port or the other, but never with one photon at each output port. Such experiments have long been performed in the optical domain, but recent developments have raised the possibility of performing such experiments in the microwave domain, using linear amplifiers and quadrature amplitude detectors instead of photon counting [Bozyigit \emph{et al.}, Nat. Phys. \textbf{7}, 154-158 (2010)]. Here we determine the signature of HOM interference in a system consisting of two independent circuit QED systems out-coupled into an on-chip microwave beam splitter. We have calculated the beam splitter output intensity auto- and cross-correlations for both trains of pulsed Lorentzian photons, and continuously-driven sources based on photon blockade. The HOM interference is manifest as antibunching in the output intensity cross-correlation. Controllable distinguishability may be introduced via a time delay in the pulsed case, or via a frequency offset in the continuously-driven case. The frequency offset leads to a quantum beat effect. Preliminary experimental results will be discussed.   

Generating distributable and unconditional entanglement on-chip at microwave frequencies

Entanglement is a critical requirement for quantum teleportation protocols. In a strategy to generate entanglement between two separate microwave lines, we integrate two Josephson Parametric Amplifiers (JPAs) and a quadrature hybrid onto a single chip making the entangler circuit. When two squeezed states created by the JPAs are combined on the hybrid (microwave beam splitter), the two output modes are entangled. We observe entanglement using a quantum efficient two channel quadrature measurement device. In our initial tests, the degree of entanglement has been limited by undesirable coupling among the elements of our entangler circuit. We present our investigation of the undesired coupling along with design strategies to reduce it.   

Quantifying distributable and unconditional entanglement at microwave frequencies

Unconditional and distributable entanglement can be created by combining a squeezed state and a vacuum state at a beam splitter. We create a single integrated circuit designed to pursue this strategy at microwave frequencies. The squeezed state is created with a Josephson Parametric Amplifier and then combined with a vacuum state in a hybrid (microwave beam splitter) producing entanglement of the output modes. In this talk, we will present the measurement and quantification of entanglement between separate microwave transmission lines. We quantify the entanglement and estimate the fidelity when applying this entangled state as a quantum teleportation channel.   

Microwave photonics and Josephson junction arrays

We present an architecture for a photonic crystal in the microwave regime based on superconducting transmission lines interrupted by Josephson junctions. A study of the scattering properties of a single junction in the line shows that the junction behaves as a perfect mirror when the photon frequency matches the Josephson plasma frequency. We generalize our calculations to periodic arrangements of junctions, demonstrating that they can be used for tunable band engineering, forming what we call a quantum circuit crystal. As a relevant application, we discuss the creation of stationary entanglement between two superconducting qubits interacting through a disordered media.   

Modeling of a Flying Microwave Qubit

We investigate the transfer efficiency of sending a flying microwave qubit through a transmission line from one resonator to another resonator. Our model is based on the current technology of coupling superconducting phase qubits to microwave resonators and transmission lines. Analytical as well as numerical results are presented. Procedural imperfections have been modeled, including weak detuning, imperfect timing, and deviations from the ideal time dependence of the coupling modulation. The effects of multiple reflections within the transmission line and energy dissipation in the system are also considered.   

A Four-wave Mixing Toolbox For Photon State Manipulation in Superconducting Resonators

We present a circuit scheme to generate quantum operations on superconducting resonators by engineering effective interaction Hamiltonians. We show that both the linear Bogoliubov transformations, including the beam-splitter operation, the squeezing operation, and the phase shifter, and the nonlinear interactions such as the cross-Kerr interaction can be realized with one single circuit. Our circuit is composed of two superconducting qubits coupled with each other to form a quantum four-level system. Each qubit interacts directly with one superconducting resonator. We exploit the four-wave mixing (FWM) approach and use the circuit as a toolbox to generate the above-mentioned quantum operations by controlling circuit parameters with external sources. Using numerical simulations to study the error rates, we show that the transformations can be realized with high fidelity. Arbitrary quantum operations on the microwave photons can be realized by combining these effective interactions.   

Dephasing and Kerr type interaction effects in circuit quantum electrodynamics

There has been recently a significant advance in obtaining high quality factor resonators in superconducting circuit architectures. The reduction of the resonator line width motivates us to consider subtle Kerr type interaction effects in small clusters of cavities and transmon type qubits. The Kerr interaction leads to entanglement of cavities, which in the transient regime is manifested in collapse-revival dynamics. For longer time scales, the interaction of the system with its environment becomes important and we discuss how the entangled states are modified. The signal of this steady-state Kerr interaction is a multi-photon port-to-port scattering process which can be observed in homodyne measurements or in a spectral analysis (correlations). We discuss the relevance of these effects to the challenge of building quantum memories.   

Dynamic quantum Kerr effect in circuit quantum electrodynamics

In the dispersive regime of circuit quantum electrodynamics (QED), where the qubit and resonator frequencies differ slightly, photons in the resonator exhibit induced frequency and phase shifts. The qubit-state dependent phase shift is usually measured by monitoring the resonator transmission spectrum at fixed qubit-resonator detuning. In this static scheme, the phase shift can only be monitored in the far-detuned, linear dispersion regime, in order to avoid measurement-induced demolition of the quantum state. By using a dynamic procedure to adiabatically drive the qubit frequency, here we are able to explore the dispersive interaction over a much broader range, and we further monitor the interaction using resonator Wigner tomography. Exotic non-linear effects on different photon states, e.g., Fock states, coherent states and Schrodinger cat states, are thereby directly revealed. Correspondingly, we demonstrate a quantum Kerr effect in the dynamic framework in circuit QED.   

Shaping an Itinerant Quantum Field into a Multimode Squeezed Vacuum by Dissipation

This work shows how to create tunable continuous sources of single and multimode squeezed light by controlling single emitters coupled to propagating modes of the EM field. Our work builds on recent experiments that implement the main tools of cavity Quantum Electrodynamics (QED) using superconducting qubits coupled to microwave transmission lines, as well as quantum dots coupled to microcavity photons, or plasmons. The main results of this letter, presented in sequential order are: A multicolor driving of an artificial atom modifies its coupling to the EM field, inducing sidebands. Combining the sidebands with an auxiliary bath, a single qubit may cool a quantum field in a single mode cavity to a squeezed vacuum. If instead of a cavity, the driven qubit is placed in a waveguide, the high energy modes play the role of a dissipative bath and the result is tunable multimode squeezing of the propagating quantum field. Through the manuscript we will also discuss implementations, measurement schemes and further outlook.   

Effects of nonlinearity in the emission spectrum of the driven nonlinear oscillator

Motivated by many experiments on nonlinear driven systems realized using superconducting circuits we investigate the properties of a coherently driven nonlinear resonator. By using Josephson junctions in superconducting circuits, strong nonlinearities can be engineered, which lead to a relatively low number of photons in the resonator and the appearance of pronounced nonlinear effects. Based on a master equation approach, which takes into account the quantum nature of noise, we determine the emission spectrum and observe for typical circuit QED parameters, in addition to the primary side-peaks, second-order peaks not predicted by a linearized theory.