Signal-to-pump back action and self-oscillation in double-pump Josephson parametric amplifier

Archana Kamal, Adam Marblestone, and Michel Devoret

Phys. Rev. B 79, 184301 – Published 13 May 2009

Abstract

We present the theory of a Josephson parametric amplifier employing two-pump sources. Our calculations are based on input-output theory, and can easily be generalized to any coupled system involving parametric interactions. We analyze the operation of the device, taking into account the feedback introduced by the reaction of the signal and noise on the pump power, and in this framework, compute the response functions of interest—signal and idler gains, internal gain of the amplifier, and self-oscillation signal amplitude. To account for this back action between signal and pump, we adopt a mean-field approach and self-consistently explore the boundary between amplification and self-oscillation. The coincidence of bifurcation and self-oscillation thresholds reveals that the origin of coherent emission of the amplifier lies in the multiwave mixing of the noise components. Incorporation of the back action leads the system to exhibit hysteresis, dependent on parameters such as temperature and detuning from resonance. Our analysis also shows that the resonance condition itself changes in the presence of back action and this can be understood in terms of the change in plasma frequency of the junction. The potential of the double-pump amplifier for quantum-limited measurements and as a squeezer is also discussed.

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Received 3 February 2009

DOI:https://doi.org/10.1103/PhysRevB.79.184301

Authors & Affiliations

Archana Kamal^*, Adam Marblestone, and Michel Devoret^†

Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520-8284, USA

^*archana.kamal@yale.edu
^†michel.devoret@yale.edu

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Vol. 79, Iss. 18 — 1 May 2009

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Figure 1
Scattering representation of a parametric amplifier with its signal, idler, and pump ports as well as its internal coordinates $X$ and $Y$ . The bidirectional coupling between the internal coordinate at signal frequency $(X)$ and that oscillating at the pump frequency $(Y)$ is shown using the curved arrows. This coupling leads, in one direction (left arrow) to amplification of the outgoing wave signal while in the other direction (right arrow) to depletion of the outgoing pump wave. It may be noted that this process also leads to a deterministic signal of finite amplitude at the idler port as the pump also populates the outgoing idler wave along with the signal, even if there is no idler input.Reuse & Permissions
Figure 2
(Color online) Characteristic frequencies of a double-pump amplifier (DPA). The upper and lower axes represent the frequencies at the input-output port. Diagonal dashed arrows represent the parametric coupling between various frequency components. The shade of the gradient in the middle band is a schematic representation of the oscillator bandwidth $2 Γ$ . The resonant frequency of the oscillator is $ω_{0}$ . The pump frequencies are denoted by $ω_{1}$ and $ω_{2}$ , respectively, and their average $ω_{g}$ defines the band center. The reduced detuning of $ω_{0}$ from $ω_{g}$ is $Ω = (ω_{0} - ω_{g}) / Γ$ . The generic frequency of the signal and that of the corresponding idler are denoted by $ω_{S}$ and $ω_{I} = 2 ω_{g} - ω_{S}$ , respectively. The reduced detuning of the signal (idler) from $ω_{g}$ is $Δ = (ω_{S} - ω_{g}) / Γ$ ( $- Δ$ for the idler). The reduced detunings $Ω$ and $Δ$ are not indicated to scale and have been exaggerated for clarity.Reuse & Permissions
Figure 3
Top panel: Josephson nonlinear oscillator pumped with two current sources $I_{{RF}_{1}}$ and $I_{{RF}_{2}}$ . The Josephson element, denoted by a cross symbol, is used for its nonlinear inductance $[δ L_{J} = (L_{J} / 6) (L_{J}^{2} I^{2}) / {(ℏ / 2 e)}^{2} + O (I^{2})]$ which performs coherent multiwave mixing. Bottom panel represents full schematic of the DPA. The current $I_{S}$ corresponds to the input signal while $I_{N}$ represents the noise current due to the resistance $R$ , modeling the internal resistances of current sources. The symbol $Φ$ is the node flux associated with the Josephson element.Reuse & Permissions
Figure 4
(Color online) Traveling wave representation of circuit of Fig. 3. The resistance $R$ is replaced with a transmission line of characteristic impedance $Z_{c} = R$ and the voltages and currents are treated as particular linear combinations of incoming and outgoing wave amplitudes.Reuse & Permissions
Figure 5
(Color online) (a) Signal-to-signal response of the internal coordinate to the incoming signal (internal cisgain of the DPA) and (b) signal-to-idler response of the internal coordinate to the incoming signal (internal transgain of the DPA) plotted for reduced pump detuning $Ω = 2$ as a function of reduced signal detuning $Δ$ . The values $Δ = 2$ and $Δ = 0$ correspond to $ω_{S} = ω_{0}$ and $ω_{S} = ω_{g}$ , respectively. The two spectra diverge in a similar fashion as the system is driven to the bifurcation threshold power $F_{B}$ .Reuse & Permissions
Figure 6
(Color) Upper panels show gain profiles for signal $({| r |}^{2})$ and interconversion $({| s |}^{2})$ gains while lower panels show cis- $({| A |}^{2})$ and trans- $({| B |}^{2})$ gains as a function of reduced drive strength $F$ and signal detuning $Δ$ . Here the value $Ω = 5$ , which is the upper limit of the frequency range in Fig. 5, was used to display the salient features clearly. The plots for ${| r |}^{2}$ and ${| s |}^{2}$ are symmetric about the ghost frequency corresponding to $Δ = 0$ . The internal transgain of the amplifier also exhibits this symmetry. However, the cisgain is asymmetric since it knows about the relative positions of $ω_{0}$ and $ω_{g}$ .Reuse & Permissions
Figure 7
(Color online) Self-oscillation amplitude as a function of drive $F$ and reduced pump detuning $Ω$ . Panel (a) represents cuts of the surface shown in panel (b). Note that self-oscillation develops only beyond a bifurcation threshold $F_{B}$ which has a parabolalike dependence on the reduced pump detuning. The lowest bifurcation value $F_{B} = 1$ is found at zero pump detuning (i.e., $ω_{g} = ω_{0}$ ) and defines the minimum/critical bifurcation power $F_{c}$ .Reuse & Permissions
Figure 8
(Color online) (a) Phase diagram of the DPA in absence of noise at the input port. Self-oscillation occurs above the solid line. The vertical dashed line represents the condition for optimal parametric pumping. In this diagram, the bifurcation value $F_{B}$ is a single-valued function of $Ω$ . (b) Corresponding phase diagram of the Josephson bifurcation amplifier, showing the bifurcation curves for upper (blue) and lower (pink) bifurcation points. Here, there exists a critical detuning $Ω_{c} = \sqrt{3}$ below which the system does not exhibit bifurcation. Below $Ω_{c}$ , the bifurcation power is a double-valued function of $Ω$ . The curved dashed line corresponds to the location of maximum oscillation amplitude. See Fig. 18 (Appendix ) for the behavior of oscillation signal as a function of drive strength in the JBA.Reuse & Permissions
Figure 9
(Color online) Integrated noise power gain and self-oscillation amplitude, for the case of stiff pump with $Ω = 1.0$ , as a function of reduced pump power. The pump frequencies were $ω_{1} = 3 / 4 ω_{0}$ and $ω_{2} = 5 / 4 ω_{0}$ , giving $γ ≃ 0.1$ .Reuse & Permissions
Figure 10
(Color) Squeezing action of the DPA. (Upper panel) Deamplified and amplified quadrature power gain for different drive strengths, as a function of reduced signal detuning, with $Ω = 1$ . Maximum squeezing corresponds to the bifurcation threshold ( $F = F_{B}$ in Fig. 9), obtained for both peak noise power and onset of self-oscillation. (Lower panel) Squeezing $fraction = 1 - \sqrt{\frac{I}{Q}}$ increases from 0 (no squeezing) at zero drive to its maximum value 1 (perfect squeezing) at $Ω$ -dependent bifurcation threshold. Beyond the threshold, squeezing decreases.Reuse & Permissions
Figure 11
(Color online) Poles of the noise back action factor $Π$ in the complex plane of reduced signal frequency $Δ$ . They move as the drive strength is changed. The poles in panel (a) before the bifurcation threshold move toward the imaginary axis. In panel (b), they hit the imaginary axis as $F_{BA}$ takes the value $Ω$ . As $F_{BA}$ is further increased, in panel (c), they move along imaginary axis in two opposite directions. Finally, in panel (d), one pair of poles hits the real axis as $F_{BA}$ attains the bifurcation threshold.Reuse & Permissions
Figure 12
(Color online) Schematic displaying the structure of the set of Eqs. (29, 32, 35). The drive strength $F$ , self-oscillation amplitude $Ξ$ and noise back action factor $Π$ , all involve back action corrected pump strength $F_{BA}$ and are interdependent.Reuse & Permissions
Figure 13
(Color online) Maximum signal gain (dotted curve), integrated noise power gain (solid curve), self-oscillation component (dotted-dashed curve) as a function of reduced pump power for $Ω = 0.5$ and $η = 3 \times 10^{- 3}$ (values corresponding to $λ = 1 / 6$ and $Θ_{eff} = 3 %$ ). The dashed curves show the respective stiff pump responses. Back action manifests itself through a pronounced reentrant behavior and a shift of the bifurcation threshold to higher powers. The inset shows the noise back action factor $Π$ as a function of $F$ evaluated for the same system parameters. The slope represents the impedance seen looking into the pump port. This impedance acquires power dependence as bifurcation threshold is approached.Reuse & Permissions
Figure 14
(Color online) Integrated noise power gain in dB (solid curve) and self-oscillation (dotted-dashed curve) responses, at $Ω = 0.5$ , for different values of temperature (a) $η = 10^{- 3}$ (b) $η = 3 \times 10^{- 3}$ and (c) $η = 10^{- 2}$ . The noise back action factor scales linearly with temperature of the noise at the input and hence its effect becomes more pronounced at higher temperatures. Curves above reflect this through a higher fold over and increase in threshold power with temperature. The dashed curves representing the behavior of respective stiff pump response functions are included for comparison.Reuse & Permissions
Figure 15
(Color online) Variation in hysteresis due to back action in integrated noise power gain as a function of pump frequency. The direction of back bending of the response curves reverses as we move from negative to positive detuning.Reuse & Permissions
Figure 16
(Color) This plot shows the dependence of system response on pump detuning $Ω$ in the presence of back action for $η = 10^{- 3}$ . Back action manifests itself in the form of hysteresis whose range increases as we move away from a renormalized resonant frequency of the oscillator, whose value corresponds here to $Ω_{0} = - 0.03$ . This value, at which hysteresis disappears, is offset from the stiff pump lowest bifurcation value $Ω = 0.0$ . As in Fig. 14, the dashed curves representing the respective stiff pump behavior are included for comparison.Reuse & Permissions
Figure 17
(Color online) Phase diagram for the DPA with back action correction, for different values of effective temperature $Θ_{eff}$ . At each temperature, the curves corresponding to the two branches of noise back action factor $Π$ are plotted parametrically [cf. Eq. (35)]. The region between the two curves corresponds to the hysteresis of the system. The intersection of the two curves marks the locus of second-order phase-transition points (dashed line). The bifurcation threshold is seen to shift upward to higher pump strengths and toward higher values of $Ω$ with temperature.Reuse & Permissions
Figure 18
(Color online) Comparison between the steady-state oscillation amplitude of the DPA (left panel) and the JBA (right panel), at the ghost and pump frequency respectively, in absence of noise. In both the case, these frequencies correspond to the average of the signal and the idler frequencies. Unlike the DPA, JBA displays a steady-state oscillatory signal as soon as the pump is turned on. Note that the dimensionless pump power $F$ is related to the physical pump strength by different numerical factors in the two systems [see Eqs. (15, A6)].Reuse & Permissions
Figure 19
(Color online) Phase diagram of the JBA in $E − Ω$ plane, showing the bifurcation curves (dotted blue), the curve corresponding to the most efficient parametric amplification (solid green with dashed portion indicating the bifurcation regime), and the asymptote to upper bifurcation level (dotted-dashed, red). Note that the $Ω = 2 F$ line does not intersect the bifurcation curve at the lowest value of detuning $Ω = \sqrt{3}$ marking the commencement of bifurcation regime.Reuse & Permissions
Figure 20
Top panel shows a damped LC oscillator driven by an RF current source. Bottom panel shows the equivalent circuit in which the current source and its internal resistance have been replaced by a semi-infinite transmission line (input-output analog). The two cases are equivalent from the point of view of the LC oscillator if we make the identification $Z_{C} = R$ and $A^{in} = \sqrt{R} I_{b} / 2$ .Reuse & Permissions

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