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Hamiltonian Formulations for Lossy Nonlinear Quantum Circuits

Christian Jirauschek, P. Russer

Published Jul 11, 2012• 7 citations
OpenAlex

Abstract

Hamiltonian formulations for lossy nonlinear quantum circuits are investigated. As an example for a nonlinear quantum circuit a DC biased Josephson junction connected to a resonator is treated and a circuit quantum electrodynamic representation is introduced. The DC bias is chosen such that a strong interaction of the nonlinear Josephson junction with two resonant modes occurs. The DC-biased nonlinear Josephson junction acts as a frequency converter and power exchange occurs between the two considered resonator modes and also between the resonator modes and the DC power supply. The thermal losses in the resonator are described by a heat bath and treated by the Langevin approach.

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Hamiltonian formulations for lossy nonlinear quantum circuits are investigated. As an example for a nonlinear quantum circuit a DC biased Josephson junction connected to a resonator is treated and a circuit quantum electrodynamic representation is introduced. The DC bias is chosen such that a strong interaction of the nonlinear Josephson junction with two resonant modes occurs. The DC-biased nonlinear Josephson junction acts as a frequency converter and power exchange occurs between the two considered resonator modes and also between the resonator modes and the DC power supply. The thermal losses in the resonator are described by a heat bath and treated by the Langevin approach.
Summary Recent research in superconducting quantum circuits operating close to the quantum limit results in the need of a quantum mechanical treatment of losses. Of special interest is the dynamic behaviour of an open quantum system. As an example, a negative‐resistance Josephson parametric amplifier is treated. The DC bias voltage is chosen such that a strong interaction between the Josephson junction and the two resonant circuits, the signal and the idler circuit, is achieved. Power exchange occurs between the two considered resonator modes and also between the resonator modes and the DC power supply. Losses in the resonators are modeled by the quantum Langevin method, which describes the losses by coupling the resonators to a heat bath representing a photon gas in thermal equilibrium. The derived dynamic behaviour does not provide signal energy saturation, like classically expected for parametric amplifiers. Introducing a phenomenological multi‐photon coupling approach, saturation of the amplified signal is ensured. The time evolution of the signal and noise energy is calculated and numerically evaluated for a specific example in cases of both, the quantum Langevin method with and without the phenomenological multi‐photon coupling approach. Copyright © 2017 John Wiley & Sons, Ltd.
Abstract. In this work, we derive the dynamics of the lossy DC pumped non-degenerate Josephson parametric amplifier (DCPJPA). The main element in a DCPJPA is the superconducting Josephson junction. The DC bias generates the AC Josephson current varying the nonlinear inductance of the junction. By this way the Josephson junction acts as the pump oscillator as well as the time varying reactance of the parametric amplifier. In quantum-limited amplification, losses and noise have an increased impact on the characteristics of an amplifier. We outline the classical model of the lossy DCPJPA and derive the available noise power spectral densities. A classical treatment is not capable of including properties like spontaneous emission which is mandatory in case of amplification at the quantum limit. Thus, we derive a quantum mechanical model of the lossy DCPJPA. Thermal losses are modeled by the quantum Langevin approach, by coupling the quantized system to a photon heat bath in thermodynamic equilibrium. The mode occupation in the bath follows the Bose-Einstein statistics. Based on the second quantization formalism, we derive the Heisenberg equations of motion of both resonator modes. We assume the dynamics of the system to follow the Markovian approximation, i.e. the system only depends on its actual state and is memory-free. We explicitly compute the time evolution of the contributions to the signal mode energy and give numeric examples based on different damping and coupling constants. Our analytic results show, that this model is capable of including thermal noise into the description of the DC pumped non-degenerate Josephson parametric amplifier.
Union Radio-Scientifique Internationale International Union of Radio Science U.R.S.I. Landesausschuss in der Bundesrepublik Deutschland e.V. Kleinheubacher Tagung 2013 Miltenberg • Germany • 23. - 25. Septembe r U.R.S.I. Landesausschuss in der Bundesrepublik Deutschland e.V. Kleinheubacher Tagung 2016 Miltenberg • Germany • 26. – 28. September 2016 14:10 – 14:30 Uhr: KH2016-D-10 Waldemar Kaiser, Michael Haider, Johannes Russer, Peter H. Russer, Christian Jirauschek (Technische Universitat Munchen, Germany) Modeling of the lossy Josephson parametric amplifier Recent advances in superconducting nanoelectr onic circuits exhibit the potential for future electronic applications. Experimental and theoretical research on superconducting quantum circuits based on the Josephson effect have recently gained more and more attention due to developments in quantum information theory and quantum metrology (Russe r and Russer, 2011). The Josephson effect is observed, if two superconducting materials ar e weakly coupled across a tunnel barrier or a narrow conducting bridge. Electronic devices based on the Josephson effect allow the generation, detection, mixing and parametric amplification of hi gh-frequency signals. Mor eover, Josephson devices are nanometre-sized low power devices and show hi gh sensitivity. So far, Josephson parametric amplifiers have been treated classically as well as quantum mechanically. The existing quantum mechanical models do not include loss contributions (Russer and Russer, 2012). In this work the lossy negative- resistance Josephson parametric am plifier is analyzed. Losses are considered using the quantum Langevin method. Hereby , the resonator circuits, i.e. the signal and the idler circuit, are coupled to a heat bath, represented by a photon gas in thermal equilibrium. In this way, also temperature dependencies are taken into account. In our model, the heat bath consists of harmonic oscillators, inducing fluctuations in the resonators a nd causing damping of the signal energy (Jirauschek and Russer, 2012). The DC biased Josephson junction is causing a strong interaction between the idler and the signal modes. Thus, power exchange occurs between the signal and the idler mode, as well as between the DC power supply and the resonator ci rcuits. The energy conversion in the DC-pumped Josephson parametric amplifier follows the Manley-Row e equations, with an additional DC term (Russer, 1971). The time evolution of the signal e nergy and the noise contributions ar e derived based on the Heisenberg equations of motion. The rotating wave approximati on is applied in order to simplify the Josephson coupling Hamiltonian.We assume the heat bath noise cont ributions to follow the statistics of a Markovian process. We have explicitly computed the time evolut ion of the various energy contributions for different initial settings. The signal photon number in therma l equilibrium is determined by the photon occupation number of the heat bath, given by Boltzmann statistics, and the damping constants. Our analytic results show, that this model is capable of including th ermal noise into the description of the Josephson parametric amplifier. For low damping constants, the signal energy is amplified exponentially as shown in previous publications without loss considerations (Russer and Russer, 2012).
In RF nanoelectronics and mesoscopic electronics quantum effects play a role on the circuit and system level. In this work a quantum electrodynamics based circuit theory is introduced that allows to establish quantum mechanical models of distributed circuits. Based on the Hamiltonian description of the Foster equivalent circuits a quantization of the equivalent circuits is performed. The quantum mechanical interaction of the modes via nonlinear elements is discussed. A DC-pumped Josephson parametric amplifier is treated quantum mechanically.
We present a description of the spatial evolution of quantized discrete-mode operators along a lossy nonlinear transmission line. The nonlinearity is formed by hundereds or even thousands of Josephson junctions which are placed periodically along a microwave transmission line. Dissipation is added to the system Hamiltonian by coupling the nonlinear transmission line to an Ohmic bath. Using the Hamiltonian of the open quantum system, Heisenberg equations of motion for the discrete mode operators can be derived in terms of quantum Langevin equations. The temporal equations of motion are then translated to the spatial domain to investigate the performance of a nonlinear four-wave-mixing process, while signals propagate along the transmission line.