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Modeling of the lossy Josephson parametric amplifier

Waldemar Kaiser, Michael Haider, Johannes A. Russer, P. Russer, Christian Jirauschek

Published Jan 1, 2016• 0 citations
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Abstract

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).

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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.

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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.

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With a large portfolio of elemental quantum components, superconducting quantum circuits have contributed to dramatic advances in microwave quantum optics. Of these elements, quantum-limited parametric amplifiers have proven to be essential for low noise readout of quantum systems whose energy range is intrinsically low (tens of µeV). They are also used to generate non classical states of light that can be a resource for quantum enhanced detection. Superconducting parametric amplifiers, like quantum bits, typically utilize a Josephson junction as a source of magnetically tunable and dissipation-free nonlinearity. The magnetic control is not an industry standard for devices and starts already to be an issue in large scale circuits. In recent years, efforts have been made to introduce semiconductor weak links as electrically tunable nonlinear elements, with demonstrations of microwave resonators and quantum bits using semiconductor nanowires, a two dimensional electron gas, carbon nanotubes and graphene. However, given the challenge of balancing nonlinearity, dissipation, participation, and energy scale, parametric amplifiers have not yet been implemented with a semiconductor weak link. The work presented in this PhD thesis demonstrates the design, fabrication and performances of a parametric amplifier leveraging a graphene Josephson junction.The graphene is encapsulated in between hBN flakes in order to improve its quality. Using such high quality junctions, we demonstrate amplification with 20 dB gain which is the milestone for overcoming noise of the classical amplifiers in the rest of the amplification chain. The Josephson parametric amplifier being based on a resonant structure (around 6 GHz) suffers from the gain bandwidth product limiting the amplification to a few MegaHertz frequency range. The use of a side gate to electrically tune the graphene gives control of the Josephson junction inductance and thus the resonant frequency of the amplifier. We demonstrate a near 1 GHz tunability of the amplification frequency with the use of a side gate which is as good as the tunability offered by using a magnetic flux on a SQUID for similar amplifier designs. Moreover, graphene has shown to exhibit nonlinear loss under a strong microwave irradiation arising because of dynamics in the Andreev states. Dissipation is known to add intrinsic noise on quantum amplifiers and could be a problem for quantum limited parametric amplification. We demonstrate that despite the presence of nonlinear loss, graphene based Josephson parametric amplifiers can reach near quantum limited amplification. A two photon loss model was used to describe the behaviour of the device but we show that a more complex model is required in order to take into account the nonlinear dissipation and the non-sinusoidal current phase relation. We also studied the dynamic range and showed that the graphene based Josephson parametric amplifier can reach a 1 dB compression point of -123 dBm which is as good as single tunnel junctions based parametric amplifiers. Our results expand the toolset for electrically tunable superconducting quantum circuits and offer new opportunities for the development of quantum technologies such as quantum computing, quantum sensing and fundamental science.

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Quantum noise in Josephson-junction parametric amplifiers

L. S. Kuzmin, K. K. Likharev, V. V. Migulin +1 more

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.

IEEE Transactions on Magnetics•May 1, 1983•9 citations
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).
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.
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.
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.
With a large portfolio of elemental quantum components, superconducting quantum circuits have contributed to dramatic advances in microwave quantum optics. Of these elements, quantum-limited parametric amplifiers have proven to be essential for low noise readout of quantum systems whose energy range is intrinsically low (tens of µeV). They are also used to generate non classical states of light that can be a resource for quantum enhanced detection. Superconducting parametric amplifiers, like quantum bits, typically utilize a Josephson junction as a source of magnetically tunable and dissipation-free nonlinearity. The magnetic control is not an industry standard for devices and starts already to be an issue in large scale circuits. In recent years, efforts have been made to introduce semiconductor weak links as electrically tunable nonlinear elements, with demonstrations of microwave resonators and quantum bits using semiconductor nanowires, a two dimensional electron gas, carbon nanotubes and graphene. However, given the challenge of balancing nonlinearity, dissipation, participation, and energy scale, parametric amplifiers have not yet been implemented with a semiconductor weak link. The work presented in this PhD thesis demonstrates the design, fabrication and performances of a parametric amplifier leveraging a graphene Josephson junction.The graphene is encapsulated in between hBN flakes in order to improve its quality. Using such high quality junctions, we demonstrate amplification with 20 dB gain which is the milestone for overcoming noise of the classical amplifiers in the rest of the amplification chain. The Josephson parametric amplifier being based on a resonant structure (around 6 GHz) suffers from the gain bandwidth product limiting the amplification to a few MegaHertz frequency range. The use of a side gate to electrically tune the graphene gives control of the Josephson junction inductance and thus the resonant frequency of the amplifier. We demonstrate a near 1 GHz tunability of the amplification frequency with the use of a side gate which is as good as the tunability offered by using a magnetic flux on a SQUID for similar amplifier designs. Moreover, graphene has shown to exhibit nonlinear loss under a strong microwave irradiation arising because of dynamics in the Andreev states. Dissipation is known to add intrinsic noise on quantum amplifiers and could be a problem for quantum limited parametric amplification. We demonstrate that despite the presence of nonlinear loss, graphene based Josephson parametric amplifiers can reach near quantum limited amplification. A two photon loss model was used to describe the behaviour of the device but we show that a more complex model is required in order to take into account the nonlinear dissipation and the non-sinusoidal current phase relation. We also studied the dynamic range and showed that the graphene based Josephson parametric amplifier can reach a 1 dB compression point of -123 dBm which is as good as single tunnel junctions based parametric amplifiers. Our results expand the toolset for electrically tunable superconducting quantum circuits and offer new opportunities for the development of quantum technologies such as quantum computing, quantum sensing and fundamental science.
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.