Nondegenerate three-wave mixing with the Josephson ring modulator

The Josephson ring modulator (JRM) is a device, based on Josephson tunnel junctions, capable of performing nondegenerate mixing in the microwave regime without losses. The generic scattering matrix of the device is calculated by solving coupled quantum Langevin equations. Its form shows that the device can achieve quantum-limited noise performance both as an amplifier and a mixer. Fundamental limitations on simultaneous optimization of performance metrics like gain, bandwidth, and dynamic range (including the effect of pump depletion) are discussed. We also present three possible integrations of the JRM as the active medium in a different electromagnetic environment. The resulting circuits, named Josephson parametric converters (JPC), are discussed in detail, and experimental data on their dynamic range are found to be in good agreement with theoretical predictions. We also discuss future prospects and requisite optimization of JPC as a preamplifier for qubit readout applications.

Figure 1

General nondegenerate three-wave mixing device consisting of three LC oscillators coupled by a nonlinear medium, giving a trilinear term in the Hamiltonian of the form $K{\Phi }_a{\Phi }_b{\Phi }_c$, where the fluxes ${\Phi }_{a,b,c}$ are those of the inductors. Each oscillator is fed by a transmission line with characteristic impedance $R_{a,b,c}$.

Figure 2

Characteristic frequency landscape of nondegenerate three-wave mixing devices. Three separate oscillators have resonant frequencies ${\omega }_a<{\omega }_b<{\omega }_c={\omega }_a+{\omega }_b$. They are fed by transmission lines, giving them a full linewidth at half maximum ${\gamma }_a$, ${\gamma }_b$, and ${\gamma }_c$, respectively. The nonlinear coupling strength, expressed in photon amplitude language, is much smaller than these linewidths. The device can be pumped at ${\omega }_c$ and operates then as a phase-preserving amplifier with photon gain for frequencies ${\omega }_a$ and ${\omega }_b$ (top), or it can be pumped at one of the two lower frequencies ${\omega }_a$ or ${\omega }_b$ and operates then as a noiseless frequency converter or dynamical cooler, upconverting signals into oscillator at ${\omega }_c$ (bottom). In this figure, the spectral density of weak signal corresponds to thin arrows, whereas the spectral densities of pump signals corresponds to thick arrows.

Figure 3

An amplifier reaching the quantum limit must have a minimal scattering matrix, with the signal in port $a$ being reflected with amplitude gain $G^{1/2}$ while the signal in port $b$ is phase-conjugated and transmitted to port $a$ with amplitude gain ${(G-1)}^{1/2}$. This can be realized in case 1 of Fig. 2.

Figure 4

Signal flow graph for a three-wave mixing device operating in conversion without photon gain, realized in case 2 of Fig. 2. The incoming signal in port $a$ ($b$) is reflected with amplitude $r$ and transmitted with up conversion (down conversion) to port $b$ (a) with amplitude ${(1-r^2)}^{1/2}$.

Figure 5

Three-wave mixing element (see ellipse marked $K$ in Fig. 1) consisting of a loop of four nominally identical Josephson junctions threaded by a flux in the vicinity of half a flux quantum. Mutual inductances, not shown here, couple this circuit to inductances $L_a$, $L_b$, and $L_c$ of Fig. 1 via the inductances $L_X$, $L_Y$, and $L_Z$, respectively, which are much larger than the junction inductance $L_J$. The three currents $I_X$, $I_Y$, and $I_Z$ correspond to the three orthogonal modes of the structure.

Figure 6

A calculated response of the signal output power $P_{\mathrm{out}}$ vs the signal input power $P_{\mathrm{in}}$ of a typical three-wave mixing device, which exhibits a pump depletion effect. The different blue curves correspond to different $G_0$ setpoints. The definition of the other lines in the figure is given in the text. The parameters used in the calculation are ${\omega }_a/2\pi =7\mathrm{GHz}$, ${\omega }_b/2\pi =8\mathrm{GHz}$, ${\omega }_c/2\pi =15\mathrm{GHz}$, ${\gamma }_a/2\pi ={\gamma }_b/2\pi =50\mathrm{MHz}$, ${\gamma }_c/2\pi =0.6\mathrm{GHz}$, $Q_a=140$, $Q_b=160,$ $p_a=p_b=0.03$, $p_c=0.02,$ $I_0=1\mathrm{error}$, $E_J^{a,b}=E_J/\sqrt{2}=16.3K$, $P_{1\mathrm{ph}}=-128$ dBm, $G_{\mathrm{ZPF}}^{\max }=35$ dB, $P_{\mathrm{cav}}^{\max }=P_b^{\max }=-86$ dBm, and $g_3/2\pi =0.7\mathrm{MHz}$.

Figure 7

Circuit model of the microstrip resonator JPC (MRJ).

Figure 8

Optical microscope image of a microstrip resonator JPC (MRJ). The resonators denoted $a$ and $b$ are half-wave microstrip resonators, which intersect at a JRM. A zoomed-in view of the JRM, which consists of four Josephson junctions arranged in Wheatstone bridge configuration, is shown on the right. The MRJ is coupled to 50 $\Omega$ feedlines via gap capacitors.

Figure 9

Circuit model of the compact resonator JPC (CRJ).

Figure 10

Optical microscope image of a compact resonator JPC (CRJ). The device consists of four equal interdigitated capacitors denoted $C$ and two inductive elements denoted $L_a$ and $L_b$, which are realized using narrow superconducting lines of different lengths. The JRM of the device resides at the intersection of the two lines. An optical image of the JRM is shown in the inset. The CRJ is coupled to $50\mathrm{error}$ microstrip feedlines via interdigitated capacitors denoted $C_c$.

Figure 11

Circuit model of the shunted JPC (SJ).

Figure 12

Output power $P_{\mathrm{out}}$ measurement of a CRJ amplifier (device A) as a function of input power $P_{\mathrm{in}}$ measured at ${\omega }_b$. The data curves plotted in blue correspond to different $G_0$ setpoints obtained for different pump powers. The red line corresponds to 0 dB (unity gain), where $P_{\mathrm{out}}=P_{\mathrm{in}}$. The dashed black vertical line indicates the input power of 1 photon at the signal frequency per inverse dynamical bandwidth of the device at $G_0=20$ dB. The top horizontal line labeled $P_{\mathrm{cav}}^{\max }$ corresponds to the maximum circulating power in the resonator cavity given by Eq. (82). The green line corresponds to an upper limit on the device gain set by the saturation of the amplifier due to zero-point fluctuations given by Eq. (87). The dashed magenta line is a theoretical prediction for $P_{\mathrm{out}}^{\max }$, which corresponds to the maximum circulating power in the device given by Eq. (114). The measured and calculated parameters of this device (A) are listed in Table .

Figure 13

Output power $P_{\mathrm{out}}$ measurement of a MRJ amplifier (device B) as a function of input power $P_{\mathrm{in}}$ measured at ${\omega }_a$. The data curves plotted in blue correspond to different $G_0$ setpoints obtained for different pump powers. The red line corresponds to 0 dB (unity gain), where $P_{\mathrm{out}}=P_{\mathrm{in}}$. The dashed black vertical line indicates the input power of 1 photon at the signal frequency per inverse dynamical bandwidth of the device at $G_0=20$ dB. The top horizontal line labeled $P_{\mathrm{cav}}^{\max }$ corresponds to the maximum circulating power in the resonator cavity given by Eq. (82). The green line corresponds to an upper limit on the device gain set by the saturation of the amplifier due to zero-point fluctuations given by Eq. (87). The solid magenta line is a theoretical prediction for $P_{\mathrm{in}}^{\max }$ of the device and the corresponding $P_{\mathrm{out}}^{\max }$ due to pump depletion effect given by Eq. (115). The measured and calculated parameters of this device (B) are listed in Table .

Figure 14

Output power $P_{\mathrm{out}}$ measurement of another CRJ amplifier (device C, with different parameters from device A) as a function of input power $P_{\mathrm{in}}$ measured at ${\omega }_b$. The data curves plotted in blue corresponds to different $G_0$ set points obtained for different pump powers. The data curves of this device exhibit abrupt drop in the gain in the vicinity of the maximum input powers, which suggests that the device enters an unstable regime at elevated input powers. The red line corresponds to 0 dB (unity gain) where $P_{\mathrm{out}}=P_{\mathrm{in}}$. The dashed black vertical line indicates the input power of 1 photon at the signal frequency per inverse dynamical bandwidth of the device at $G_0=20$ dB. The top horizontal line labeled $P_{\mathrm{cav}}^{\max }$ corresponds to the maximum circulating power in the resonator cavity given by Eq. (82). The green line correspond to an upper limit on the device gain set by the saturation of the amplifier due to zero-point fluctuations given by Eq. (87). The solid magenta line is a theoretical prediction for $P_{\mathrm{in}}^{\max }$ of the device and the corresponding $P_{\mathrm{out}}^{\max }$ due to pump depletion effect given by Eq. (115). The measured and calculated parameters of this device (C) are listed in Table .

Figure 15

An optimized JPC response drawn in blue for large maximum input power in excess of 100 input photons at 12 $\mathrm{GHz}$ per inverse dynamical bandwidth of the device at 20 dB. The definition of the other lines shown in the figure is similar to Fig. 12. The parameters used in the calculation are ${\omega }_a/2\pi =11\mathrm{GHz}$, ${\omega }_b/2\pi =12\mathrm{GHz}$, ${\omega }_c/2\pi =23\mathrm{GHz}$, $Z_a=36\mathrm{error}$, $Z_b=33\mathrm{error}$, $L_a=0.51$ n$H$, $L_b=0.42$ n$H$, $C_a=C_b=0.4\mathrm{pF}$, $C_{C_a}=31$ f$F$, $C_{C_b}=28$ f$F$, ${\gamma }_a/2\pi ={\gamma }_b/2\pi =44.2\mathrm{MHz}$, ${\gamma }_c/2\pi =3\mathrm{GHz}$, $Q_a=249$, $Q_b=271$, $Q_c=8$, $p_a=0.028$, $p_b=0.034$, $p_c=0.02$, $I_0=30\mathrm{error}$, $E_J/\sqrt{2}=490K$, $E_J^{a,b}=49K$, $P_{1\mathrm{ph}}=-126$ dBm, $G_{\mathrm{ZPF}}^{\max }=48$ dB, $P_{\mathrm{cav}}^{\max }=P_b^{\max }=-86.3$ dBm, and $g_3/2\pi =0.6\mathrm{MHz}$.

Figure 16

The damping of a circuit by a resistance $R$ can take place in a parallel or series way, depending on whether the resistance is placed across a branch or in series with it. The Nyquist model represents the resistance by a transmission line with characteristic impedance $Z_c=R$. The noise source associated with the resistance (fluctuation-dissipation theorem) is a parallel current source in the parallel case and a series voltage source in the series case. The noise source is replaced in the Nyquist model by incoming thermal radiation whose amplitude $A^{\mathrm{in}}$ is the square root of the power flux of the radiation ($A^{\mathrm{in}}$ should not be associated to a vector potential and is rather like the square root of the length of the Poynting vector).