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.
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.
Superconducting quantum circuits exhibit an extraordinary potential for future electronic applications. The most important element in superconducting quantum circuits is the Josephson junction, built by nanofabricated superconducting electrodes separated by a thin insulating layer. Superconducting devices allow mixing and parametric amplification up to the terahertz range under low energy consumption and ultra-low noise. High frequency signals at low temperatures exhibit a photon energy exceeding the thermal energy, resulting in the need of a quantum mechanical treatment of the electromagnetic field at millimeter-wave frequencies. In this work, we investigate the dynamic behaviour of a negative-resistance, dissipative DC biased Josephson parametric amplifier. The Langevin theory is used for modeling of the dissipation in the resonant circuits. We investigate Markovian dynamics for modeling the dissipative Josephson parametric amplifier in a correct manner, neglecting memory effects. We derive the equations of motions of the resonator operators and numerically evaluate the time evolution of the signal and noise energies. Applying a phenomenological multi-photon coupling approach, a correction of the Markovian assumption is achieved providing an expected saturation in the dynamical behaviour of the circuit.
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).
Crossover between thermal and quantum noise in the externally-pumped Josephson-junction parametric amplifiers is theoretically followed. Two different approaches are used for the analysis : i. Josephson junction is replaced with its RSJ model with the equilibrium source of fluctuations; ii. microscopic theory of the tunnel junctions is used, which takes into account not only thermal and quantum but also shot noise. Because of inconvenience of the noise temperature TNfor the amplifier sensitivity characterization in the quantum limit, we express our results in terms of the figure ΘNwhich is essentially the amplifier output noise energy, reduced to its input (in the classical limit,\Theta_{N}=k_{B}T_{N}). For the nondegenerate amplifier, the minimum value of ΘN. equals hω/2 and is due to quantum noise (zero-point oscillations) in the idle-frequency resonator of the amplifier. For the degenerate amplifier, sensitive to only one quadrature component of the input signal, ΘNcan be made much less than the above "quantum limit" even at operation temperatures much higher thanh\omega/k_{B}, if the Josephson junction characteristic frequency ωcis high enough,h\omega_{c}\gsimk_{B}T. Our analysis of the experimental situatlon shows that the values\Theta_{N}<h\omega/2can be achieved using two-dimensional arrays of a large number (N\simeq100) single-tunnel-junction interferometers with low inductances, inductively coupled to a microwave cavity.
Linear parametric amplification is a key operation in information processing. Our interest here is quantum-limited parametric amplification, i.e., amplification of quantum signals while adding the minimum amount of noise allowed by quantum mechanics, which is essential for any viable implementation of quantum information processing. We describe parametric amplifiers based on the dispersive nonlinearity of Josephson junctions driven with appropriate tones playing the role of pumps. We discuss two defining characteristics in the architecture of these amplifiers: the number of modes occupied by the signal, idler, and pump waves and the number of independent ports through which these waves enter into the circuit. The scattering properties of these amplifiers are also reviewed. The main focus of this work are the computations of the dynamic range and phase-space distributions of the fluctuations of the modes of the amplifiers.
The Josephson junction acts as a nonlinear lossless inductor and can be applied for mixing and parametric amplification in the microwave region. Since an alternating current can be generated in the junction by an applied dc voltage, a dc power flow must be taken into consideration. Energy relations for the Josephson junction are derived which are similar to the Manley-Rowe equations but with an additional term for the dc power. These equations show the possibility of realizing dc-pumped parametric amplifiers.