--- title: Nondegenerate Josephson Mixers url: https://www.emergentmind.com/topics/nondegenerate-josephson-mixers-jms type: topic --- # Nondegenerate Josephson Mixers Nondegenerate Josephson mixers (JMs) are Josephson-based microwave devices in which the useful modes are distinct in frequency and, in many implementations, also distinct in physical channel. In the canonical superconducting realization, a Josephson ring modulator (JRM) provides a dissipationless three-wave nonlinearity that couples signal, idler, and pump modes, enabling phase-preserving amplification, noiseless frequency conversion, and two-mode squeezing near the quantum limit [1208.3142][1212.2231]. The term is also used more broadly for Josephson circuits that mix distinct local-oscillator, signal, idler, or sideband frequencies in microwave engineering, including flux-driven parametric amplifiers, flux-controlled bridge modulators, heterodyne high-\(T_c\) junction mixers, and Josephson-photonics circuits with two nondegenerate resonators [1610.07987][1609.09041][1405.5998][2203.15373]. Across these realizations, the central theme is frequency-selective parametric coupling mediated by a Josephson nonlinearity without relying on normal-metal dissipation in the active element. ## 1. Definition and scope of the term In the strict JRM/JPC usage, “nondegenerate” means that signal and idler occupy two different resonant modes, commonly labeled \(a\) and \(b\), while a third mode or drive furnishes the pump. In amplifier mode the pump is applied at the sum frequency, \(f_p=f_a+f_b\), so that one pump photon is converted into a signal-idler pair; in conversion mode the pump is applied at the difference frequency, \(f_p=f_b-f_a\), so that photons are coherently exchanged between the two modes without gain [1208.3142][1212.2231]. In the continuous-variable entangler realization, the same nondegeneracy is emphasized as two different frequencies propagating in different physical channels, which is what allows the entangled outputs to be spatially separated [2501.05537]. A broader operational usage appears in microwave modulation and sideband physics. The double-balanced Josephson bridge developed for qubit control mixes a microwave carrier in the \(6\text{--}10\ \text{GHz}\) band with a lower-frequency control signal from DC to \(850\ \text{MHz}\), so it functions as a nondegenerate Josephson mixer in the microwave-engineering sense even though the paper presents it primarily as a modulator [1610.07987]. Likewise, a flux-driven JPA becomes a nondegenerate Josephson mixer when a pump at \(2\omega_0\) amplifies a signal at \(\omega_{\mathrm s}=\omega_{\mathrm{pump}}/2+\delta\omega\) and generates an idler at \(\omega_{\mathrm i}=\omega_{\mathrm{pump}}/2-\delta\omega\) [1609.09041]. In heterodyne high-\(T_c\) mixers, the nondegenerate regime means a strong LO at \(f_{\mathrm{LO}}\), a weaker signal at \(f_s\), and an intermediate frequency \(f_{\mathrm{IF}}=|f_{\mathrm{LO}}-f_s|\) [1405.5998][1203.1734]. This terminological spread matters because not all nondegenerate Josephson mixers share the same Hamiltonian, port structure, or operating objective. Some are three-wave amplifiers or converters, some are heterodyne detectors described by RSJ three-port theory, and some are modulation elements for cryogenic control. A plausible implication is that “nondegenerate Josephson mixer” is best understood as a functional class rather than a single circuit topology. ## 2. Core nonlinear physics and scattering descriptions The JRM-based mixer is built around a ring of four Josephson junctions biased near half a flux quantum so that even-order nonlinearities are suppressed and a dominant cubic interaction remains [1208.3142]. In one formulation the mode Hamiltonian is $$ \frac{H_0^{\mathrm{RWA}}}{\hbar} = \omega_a a^\dagger a+\omega_b b^\dagger b+\omega_c c^\dagger c + g_3\left(a^\dagger b^\dagger c + a b c^\dagger\right), $$ while a compact lumped-element implementation writes the mixing term as $$ H_{mix}=-E_J\sin(\Phi/4\varphi_0)\varphi_a\varphi_b\varphi_p. $$ Both forms encode the same central fact: the Josephson nonlinearity furnishes a three-wave interaction among distinct modes [1208.3142][1503.08185]. Under a stiff pump, amplifier-mode operation produces a Bogoliubov scattering relation. On resonance, the generic nondegenerate JRM/JPC scattering matrix reduces to $$ \begin{pmatrix} \cosh \tau_0 & -i e^{-i\phi}\sinh\tau_0\\ i e^{i\phi}\sinh\tau_0 & \cosh\tau_0 \end{pmatrix}, $$ with power gain $$ G_0 = (\cosh\tau_0)^2 = \left(\frac{1+|\rho|^2}{1-|\rho|^2}\right)^2. $$ Because phase-preserving amplification requires idler participation, the device reaches the Caves limit of at least half a photon of added noise referred to the input [1208.3142]. Experimental realizations close to this regime include the widely tunable shunted device, whose noise analysis with a tunnel junction source gave an apparent added noise \(N_{add}=2.8\) quanta for the full measurement chain, with \(0.5\) quanta attributed to unavoidable idler-port vacuum noise [1202.1315], and later coupled-mode mixers with inferred \(n_{\rm add}\approx 0.5\text{--}0.6\) [2508.06636]. In conversion mode, the same three-wave interaction becomes unitary rather than amplifying. The Josephson parametric converter pumped at the difference frequency behaves as a frequency-domain beam splitter with $$ |r|^2+|t|^2=1, $$ and at full conversion the measured losses are below \(10^{-2}\) [1212.2231]. The corresponding on-resonance unitary matrix may be written as $$ \begin{pmatrix} \cos\tau_0 & e^{-i\phi}\sin\tau_0\\ e^{i\phi}\sin\tau_0 & \cos\tau_0 \end{pmatrix}, $$ or, in the JPC conversion notation, $$ \begin{pmatrix} \cos\tau_0 & i e^{-i\varphi_p}\sin\tau_0\\ i e^{i\varphi_p}\sin\tau_0 & \cos\tau_0 \end{pmatrix}. $$ The pump phase enters with opposite sign for opposite propagation directions, which is the microscopic origin of later Josephson-mixer nonreciprocity [1212.2231][1702.01149]. ## 3. Principal circuit families and realizations The most established JM architecture is the Josephson parametric converter, in which a JRM is embedded between two resonant modes. Early devices used half-wave microstrip resonators, while the widely tunable variant replaced the bare ring by “four Josephson junctions shunted by a cross of four linear inductances,” allowing the phase configuration to remain unique over a wide flux range while preserving a strong cubic interaction [1202.1315]. That device demonstrated more than \(400\ \mathrm{MHz}\) of center-frequency tunability with reproducible gain \(>20\ \mathrm{dB}\), a \(3.2\ \mathrm{MHz}\) dynamical bandwidth at \(20\ \mathrm{dB}\), and \(1\ \mathrm{dB}\) compression points from \(-133\ \mathrm{dBm}\) to \(-118\ \mathrm{dBm}\) [1202.1315]. A separate line of development pursued compactness and impedance engineering. The lumped-element JM of 2015 made the \(a\) and \(b\) resonators from the JRM itself shunted by plate capacitors and galvanically connected them to \(50\,\Omega\) ports, reaching \(50\ \mathrm{MHz}\) bandwidth at \(20\ \mathrm{dB}\) gain, up to \(30\ \mathrm{dB}\) maximum gain, quantum efficiency \(\eta \approx 0.7\), and a \(1\ \mathrm{dB}\) compression point of \(-112\ \mathrm{dBm}\) at the JRM input [1503.08185]. A hybrid-less and coil-less JPC then replaced off-chip \(180^\circ\) hybrids and magnetic coils by an on-chip three-port power divider and an on-chip flux line with mutual inductance about \(0.6\ \text{pH}\), while retaining gains exceeding \(20\ \text{dB}\) and enabling time-multiplexed amplification through \(2\ \mu\text{s}\) flux pulses [1609.00401]. The term also covers Josephson circuits whose function is modulation rather than direct signal-idler translation. The Josephson junction microwave modulators for qubit control use a flux-tunable double-balanced bridge embedded in a \(4\)-pole Chebyshev band-pass filter with center frequency \(8\ \text{GHz}\) and target bandwidth \(4\ \text{GHz}\). They operate in the \(6\text{--}10\ \text{GHz}\) band, provide DC–\(850\ \text{MHz}\) IF bandwidth, achieve greater than \(25\ \text{dB}\) LO/RF isolation, and exhibit input saturation powers in excess of \(1\ \text{nW}\) [1610.07987]. The paper explicitly distinguishes this from the JPC-style frequency-conversion emphasis, but confirms that the circuit is a nondegenerate Josephson mixer in the practical operational sense. Outside the JRM/JPC lineage, high-\(T_c\) and SQUID-based realizations broaden the concept further. Ion-irradiated YBa\(_2\)Cu\(_3\)O\(_7\) nano-junctions embedded in wideband spiral THz antennas realize nondegenerate heterodyne mixing described by RSJ and three-port models, with operation studied from \(20\ \text{GHz}\) to \(420\ \text{GHz}\), characteristic frequencies up to about \(85\ \text{GHz}\) in one study and \(75\ \text{GHz}\) in another, and conversion efficiencies from about \(0.1\%\) at \(20\ \text{GHz}\) down to about \(0.01\%\) at \(140\ \text{GHz}\) in one dataset, or about \(0.3\%\) at \(20\ \text{GHz}\) down to about \(0.02\%\) at \(140\ \text{GHz}\) in another [1405.5998][1203.1734]. Flux-driven JPAs terminated by dc-SQUIDs furnish yet another form: their finite-screening two-dimensional SQUID potential controls a nondegenerate signal-idler response, so overcoupled devices act as amplifiers whereas undercoupled devices can produce deamplification up to \(-30\ \text{dB}\) [1609.09041]. ## 4. Nonreciprocity, gyration, and directional signal routing One major extension of the nondegenerate JM is from amplification and conversion to active nonreciprocal microwave elements. The Josephson-mixer gyrator couples two nominally identical JPCs through their idler ports and operates each in noiseless frequency-conversion mode. On resonance, the transmission phases obey $$ \angle S_{21} = \varphi_d + \varphi, \qquad \angle S_{12} = \varphi_d - \varphi, $$ with \(\varphi=\varphi_1-\varphi_2\), so $$ \angle S_{21}-\angle S_{12}=2\varphi. $$ At \(\varphi=\pi/2\), the phase difference is \(\pi\), which realizes the gyrator condition [1702.01149]. Experimentally, the reflections \(S_{11}\) and \(S_{22}\) showed dips of about \(-15\ \text{dB}\), the transmissions \(S_{21}\) and \(S_{12}\) peaked at about \(-4.6\ \text{dB}\), and the phase difference was approximately \(180^\circ\). The paper interprets the pump-phase gradient as an artificial gauge potential for microwave photons and describes the effect as a photonic Aharonov–Bohm analogue [1702.01149]. The same synthetic-gauge concept underlies the Multi-Path Interferometric Josephson Isolator, which combines two identical nondegenerate JPC mixers through a \(90^\circ\) hybrid, an idler transmission line, and internal matched loads. In the ideal symmetric \(50{:}50\) operating point with \(\varphi=-\pi/2\), the theory gives \(S_{12}=0\), \(S_{11}=S_{22}=0\), and \(|S_{21}|=2\sqrt{2}/3\), corresponding to about \(0.5\ \text{dB}\) insertion loss [1810.07234]. The realized device showed more than \(20\ \text{dB}\) isolation, about \(1.8\ \text{dB}\) insertion loss, about \(11\ \text{MHz}\) dynamical bandwidth, and a \(1\ \text{dB}\) isolation-compression point around \(-108\ \text{dBm}\); when inserted into a qubit measurement chain it provided more than \(20\ \text{dB}\) of protection against amplified noise reflected off the Josephson amplifier [1810.07234]. Directional amplification can be synthesized in an analogous two-JPC interferometer. The Multi-Path Interferometric Josephson Directional Amplifier uses same-frequency pumps with relative phase \(\varphi=\varphi_1-\varphi_2\), with maximum forward gain near \(\varphi=\pi/2\). At its main working point, the measured forward gain was about \(19.1\ \text{dB}\), the backward gain about \(2.5\ \text{dB}\), the high-gain dynamical bandwidth about \(7\ \text{MHz}\), and the fit yielded \(n_{\mathrm{add}}\approx 1/2\), consistent with near-quantum-limited phase-preserving amplification [1710.02521]. A common misconception is to equate all Josephson “directionality” with genuine nonreciprocity. The one-wavelength reflection amplifier with integrated directionality employs two SQUID-based Josephson junction oscillators, a branch-line coupler, and tailored impedance engineering; it achieves \(20\ \text{dB}\) gain over \(220\ \text{MHz}\), predicts a factor-of-four increase in dynamic range, and refers a total photon shot noise of maximally \(1.3\) photons per second per Hertz of bandwidth to the input, but its routing is reciprocal and arises from passive interference rather than from broken reciprocity [1706.09300]. This distinction is central in the JM literature. ## 5. Continuous-variable entanglement and nonclassical radiation sources Nondegenerate JMs are also sources and processors of nonclassical microwave fields. When pumped at the sum frequency \(f_p=f_a+f_b\), the JM can act as a nondegenerate two-mode squeezer with input-output relations $$ a_{\rm out}=\cosh(r)\,a_{\rm in}+e^{i\varphi_p}\sinh(r)\,b_{\rm in}^\dagger, $$ $$ b^\dagger_{\rm out}=\cosh(r)\,b^\dagger_{\rm in}+e^{-i\varphi_p}\sinh(r)\,a_{\rm in}, $$ so vacuum input yields a two-mode squeezed state [2501.05537]. In a recent microwave continuous-variable platform, a JM flux-tuned to \(f_a=7.231\ \text{GHz}\) and \(f_b=9.695\ \text{GHz}\), pumped at \(f_p=16.926\ \text{GHz}\), produced maximum squeezing of about \(9.2\ \text{dB}\), logarithmic negativity \(E_N=1.5\), entanglement of formation \(E_F\approx 1.25\), and entanglement bit rate about \(70\) Mebit/s [2501.05537]. The same work used two nondegenerate JMs for teleportation and three for entanglement swapping. In the teleportation setup, a second JM at Alice acted in the high-gain limit as a which-path information eraser, and vacuum-state teleportation reached fidelity \(F_q=0.73\), exceeding the \(50\%\) classical limit; coherent-state fidelities were reported in the range \(0.62\text{--}0.73\), with the ceiling mainly limited by intermediate losses [2501.05537]. In entanglement swapping, the remote noninteracting modes held by Alice and Bob acquired a maximum measured logarithmic negativity \(E_N=0.53\), while turning off the first entangler reduced the arrangement to a classical feedforward process for which \(E_N\to 0\) [2501.05537]. A different nonclassical-source paradigm appears in Josephson photonics. A dc voltage-biased Josephson junction in series with a charge qubit and two nondegenerate resonators realizes the resonance condition $$ \omega_J = \delta + \omega_1 + \omega_2, $$ so a Cooper-pair tunneling event excites the qubit and creates one photon in each resonator [2203.15373]. Because the qubit is a two-level system, subsequent tunneling is blocked until relaxation occurs, producing photon-pair blockade. In the regime \(\kappa\gg\gamma\), the predicted signatures are \(g_{11}^{(2)}(0)\rightarrow 0\), \(g_{22}^{(2)}(0)\rightarrow 0\), \(g_{12}^{(2)}(0)\gg 1\), and \(g_{12,12}^{(2)}(0)\ll 1\), with an emission rate of order MHz for representative parameters [2203.15373]. This suggests a route from nondegenerate Josephson mixing to bright antibunched two-mode microwave sources. ## 6. Performance limits, tradeoffs, and current directions The theory of JRM-based JMs identifies hard tradeoffs among gain, bandwidth, and dynamic range. In the high-gain limit the amplifier bandwidth scales as $$ B = 2\gamma\,G_0^{-1/2} = \frac{2\gamma_a\gamma_b}{\gamma_a+\gamma_b}\,G_0^{-1/2}, $$ so the gain-bandwidth product is approximately constant [1208.3142]. Pump depletion further constrains usable input power, with $$ P_{\mathrm{in}}^{\max}\propto G_0^{-3/2}, $$ and compact-lumped analyses express related requirements through inequalities such as \(p_a Q_a p_b Q_b>\Xi\) and the signal-power bound \(p_kG\left(P_{in}/(2\pi\Gamma_k)+hf\right)\Xi'