A compact design for the Josephson mixer: the lumped element circuit

We present a compact and efficient design in terms of gain, bandwidth and dynamical range for the Josephson mixer, the superconducting circuit performing three-wave mixing at microwave frequencies. In an all lumped-element based circuit with galvanically coupled ports, we demonstrate non degenerate amplification for microwave signals over a bandwidth up to 50 MHz for a power gain of 20 dB. The quantum efficiency of the mixer is shown to be about 70$\%$ and its saturation power reaches $-112$ dBm.

We present an optimal design in terms of gain, bandwidth and dynamical range for the Josephson mixer, the superconducting circuit performing three-wave mixing at microwave frequencies. In a compact all lumpedelement based circuit with galvanically coupled ports, we demonstrate non degenerate amplification for microwave signals over a bandwidth up to 50 MHz for a power gain of 20 dB. The quantum efficiency of the mixer is shown to be about 70% and its dynamical range reaches 5 quanta per inverse dynamical bandwidth. Analog processing of microwave signals has recently entered the quantum regime owing to the developments of superconducting circuits. Quantum limited amplifiers that are based on the non-linearity provided by Josephson junctions have been developed in various designs [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] . Non-degenerate three-wave mixing, a key operation, is realized by the Josephson ring modulator (JRM), which is a ring of four identical Josephson junctions 20,21 . This element is at the core of several tools able to generate and manipulate quantum microwave modes such as phase preserving amplifiers [22][23][24] , non-local entanglement generators 25 , frequency converters 26 , quantum memories 27 or circulators 28,29 . In all previous implementations of the Josephson mixer, the JRM was embedded at the crossing of two distributed or lumped resonators, which puts a constraint on bandwidth and dynamical range that can be detrimental to quantum operations. In this letter, we discuss the origin of this constraint and how to optimize the figures of merit of the Josephson mixer. These ideas are put in practice on an experiment in which a phase preserving amplifier is solely built out of a JRM that is shunted with lumped plate capacitors. Compared to previous implementations, we report an order of magnitude increase of its dynamical bandwidth, up to 50 MHz at a power gain of 20 dB while keeping the dynamical range as high as 5 photons per inverse bandwidth.
In a Josephson mixer, the JRM couples three independent fluxes ϕ a , ϕ b and ϕ p (Fig. 1a), through the lowest order coupling term where E J is the Josephson energy of each junction, ϕ 0 = /2e the reduced flux quantum and Φ the flux threading the ring 20,21 . Three wave mixing occurs by embedding the ring in resonant circuits (Fig. 1b) so that each flux can be expressed as ϕ k ∝ (k +k † ) wherek is the canonical annihilation operator of a microwave mode of characteristic impedance Z k and resonance frequency f k . Although the Josephson mixer can be used in various ways 20-22,24-28 , we will focus here on the amplificaa) corresponding author: francois.mallet@lpa.ens.fr tion regime in order to describe the figures of merit on a concrete case. The signals sent towards a and b modes are amplified in reflection in a phase preserving manner by driving the mode p out of resonance at the frequency f a +f b 20 . Three main specifications matter in analog processing of quantum microwave signals. First, the power gain G of the amplifier needs to be large enough so that the quantum noise at the input of the amplifier dominates all other noise sources on the detection setup. Typically 20 dB is enough if a cryogenic HEMT is used as a second stage of amplification 30 . Second, the time correlations of the quantum signals should be dominated by the system of interest and not by the Josephson mixer. This requires to have as large a dynamical frequency bandwidth Γ as possible. Finally, the maximum input power P max in that does not affect the gain by more than 1 dB needs to be large enough to avoid any limitation on the amplitude of the quantum signals.
Optimizing these three parameters for a practical amplifier has been at the center of recent experimental works in various geometries [12][13][14]16 . Recently Josephson Parametric Amplifiers have reached up to 20 dB gain over a 100 MHz bandwidth in an all lumped-element based design 14,16 while 5 dB gain over 2 GHz bandwidth has been reported in a TiN traveling wave parametric amplifiers 13 . The constraints on the parameters of the Josephson mixer are similar in origin to those of its degenerate cousin, the Josephson Parametric Amplifier 31 , but with some differences 21 . First, there is an upper bound on the energy that is stored in the p pump mode, originating from the small flux ϕ p assumption in the three-wave mixing term H mix . Therefore, in order to allow pump powers to reach the onset of parametric oscillations, at which large gain G develops, one has to ensure that 21 where Ξ is a number depending on the exact geometry of the mixer. Ξ = 8 will be used in the following 32 . In this expression, Q k is the quality factor of mode k, defined as the k mode quantifies the ratio of the total energy in this mode that is actually stored across the JRM junctions. For a serial coupling, it is given by ex is the effective extra inductance of the resonator defining the k mode and L J = ϕ 2 0 /E J is the inductance of a single junction of the JRM (Fig. 1d). Second, there is an upper bound on the power spectral density of the amplified signals coming from the the small flux ϕ a,b assumption in the three-wave mixing term H mix . Indeed, neither the input signal power P in , nor the vacuum noise should be amplified beyond a fraction of the Josephson energy E J . For large gain, this condition can be approximated as where Ξ is a number of order 1 and E J is the Josephson energy E J = ϕ 2 0 /L J . The two above constraints (1) and (2) indicate that increasing both P max in and Γ for a given gain G requires to increase the participation ratios p k and the Josephson junction energy E J . However, these two figures are related since p k is a decreasing function of E J . To minimize the detrimental influence of an increasing E J on p k , it seems that the optimal design is when all the inductive parts of the resonators originate from the Josephson junction themselves so that L ex = 0 (Fig. 1c).
It is enlightening to represent graphically these constraints in the parameter phase spaces. In figure 1e, shaded areas delimitate, for two different values of L ex , the allowed values of the quality factor Q and Josephson inductance L J for a typical quantum limited amplifier operating at a frequency f = 7 GHz with a 20 dB power gain, a saturating input power P max in = −110 dBm. Note that for the sake of clarity the two a and b modes have been set to identical parameters. As can be seen in the figure, lower Q (larger bandwidth) can be obtained only by lowering L ex . Conversely, Fig. 1f shows the maximal allowed extra inductance L max ex as a function of saturating power P max in for several desired quality factors Q. From these curves, one can deduce which maximal extra inductance L max ex can be used for a given bandwidth. If L ex comes from the geometrical inductance of some wires, their length is of the order of L ex /µ 0 , which is represented on the right axis of Fig. 1f. From these considerations, one also determines the maximal spatial extension of a Josephson mixer to ensure a given bandwidth.
Our implementation of the optimal design (Fig. 1c) is presented in Fig. 2. The a and b mode resonators are composed of the JRM that is shunted by a cross of plate capacitors. The circuit is fabricated on a 500 µm thick Si chip covered with a 300 nm layer of SiO 2 on top. In a first step, a Ti(5nm)/Al(30nm) common counter electrode for all the plate capacitors is fabricated using standard ebeam lithography (dark yellow in Fig. 2b). It spreads all over the surface underneath the rest of the circuit, except for a hole in the center and a thin stripe (brown in Fig. 2b) allowing to flux bias the circuit without the constraints imposed by the Meissner effect. The whole chip is then covered with 200 nm of amorphous dielectric silicon nitride by plasma-enhanced chemical vapor deposition. Finally, the second metallic plate of the capacitors (area 285 µm × 86 µm for the a resonator and 140 µm × 86 µm for the b resonator) and the Josephson junctions (area 4.2 µm × 1 µm) are fabricated by double angle deposition of 100 nm and 120 nm of aluminum with an intermediate oxydation step (Fig. 2c). The circuit is then placed in a copper box enclosed in a Cryoperm magnetic shielding box anchored at base temperature of a dilution refrigerator (T dil 50 mK). Two 180 o hybrid couplers address separately the differential a and b modes through their ∆ ports as well as the pump mode c through one of the Σ ports. The resonators have characteristic impedances smaller than 10 Ω and are galvanically connected to the 50 Ω ports of the device so as to maximize bandwidth, only limited by impedance mismatch. Owing to the large coupling between the differential modes and the input/output ports, the gain is sensitive to the frequency dependence of the impedance 16 . In order to probe the characteristics of the Josephson mixer alone without carefully engineering the impedance of the environment, we connect a 6 dB attenuator on the ∆ ports of the hybrid couplers. A coil allows to control the flux threading the JRM loop to tune the mixing term in H mix via its current I coil .
The effect of the flux bias can also be seen (Fig. 3a) on the resonance frequencies f a and f b which depend in an hysteretic manner of the flux with a period 4φ 0 20 . This hysteretic behavior could be removed by inserting additional inductances in the JRM in order to extend the static bandwidth of the amplifier 24 . However, this comes at the expense of lowering participation ratios, which we aim at maximizing, and becomes less useful with a large dynamical bandwidth. In this device, we observe a frequency dependence on I coil , which is not perfectly periodic. This observed non-linear dependence of Φ ext on I coil may originate form vortex dynamics in the large superconducting capacitor plate that is buried under the silicon nitride. For each resonance frequency, it was possible to measure the quality factor (inset of  = 5.7 GHz, which are close to the measured resonance frequencies at Φ ext = 0 (Fig. 3a). From there, one can estimate the participation ratios to be p a = 25% and p b = 35%. Note that similar values for the participation ratios can be obtained by fitting directly the flux dependence of the resonance frequency (see supplementary material of Ref. 27 ). Finally, one can fit the stray inductances to best recover the curve Q(f ) and find L a stray = 75 pH, L b stray = 51 pH. The Josephson mixer can be used as an amplifier by setting the flux φ 1 slightly lower than 2φ 0 = h/e and driving the c pump mode at the frequency f p = 12.26 GHz, which is close to f a + f b 20 . The gain was measured in reflection on both amplifier ports as a function of frequency (Fig. 3b) for various values of the pump power P p at the fixed flux φ 1 . As the pump power rises towards the parametric oscillation threshold, the gain increases on both ports up to 30 dB. Conversely, the operating bandwidth decreases. This curve demonstrates a bandwidth of 50 MHz at 20 dB, which is an order of magnitude higher than using previous implementations of the Josephson mixer [22][23][24] . The amplifier added noise is evaluated by amplification of zero point fluctuations in a and b modes. A spectrum analyzer measures the spectral power density coming from a out and b out as a function of P p . No signal is sent into a in and b in . The difference between spectral densities while the amplifier is ON and OFF is given by where G is the gain of the amplifier and G LN A is the total gain of the output lines. Here the modes are assumed to be in the vacuum state and G 1. In this case the added noise S add can be related to the quantum efficiency η of the mixer 25 by S add = (1 − η)/2η. Determination of η requires thus separate measurements of the amplifier gain G, as those of Fig. 3, and fine calibrations of the attenuations and gains of the input and output lines connecting the mixer to the detectors. We obtain this calibration by measuring the shift in frequency as a function of power sent into a in and b in . It provides, by comparison with numerical calculations of the circuit shown in Fig. 1 (d), a precise calibration of the attenuation of the input line between a in and the mixer. The gain G LN A between the Josephson mixer and a out is then deduced from the total transmission between a in and a out , the pump being turned off. Note that we also observe a clear frequency shift in the gain measurements of Fig. 3 (b) while changing pump power due to higher order cross-Kerr 33 terms that are proportional to P p a † a and P p b † b. Figure 4(a) presents the measured η as a function G. It indicates an efficiency of 0.7, up to 33 dB of gain above which the amplifier enters the parametric oscillation regime where η is near 0.2.
The last important specification of an amplifier is its dynamical range. It is characterized by the 1 dB compression point P max in of the amplifier. In Josephson parametric devices, such as the Josephson mixer, this saturation can be caused either by depletion of the pump (i.e. the gain is so large that the pump cannot refill quickly enough to feed the amplifier), or by reaching a large enough number of photons such that higher order non-linearities cannot be neglected. Using the calibration of the input power P in on port b above, we used a vector network analyzer to measure the output power P out as a function of input power P in (Fig. 4b) at 6.76 GHz (center frequency when G = 20 dB) for various pump powers following Ref. 21 . At low input powers P in < −120 dBm, the gain goes from 0 to 25 dB for increasing pump power P p without depending on P in . For the pump power corresponding to G = 20 dB at low input power, the amplifier behaves linearly for low power until it reaches the 1 dB compression point at −112 dBm at the JRM input. At 6.76 GHz and for a bandwidth of 50 MHz, this power corresponds to 4.5 photons per bandwidth. Above this threshold the gain drops and finally saturates. This dynamical range is large enough not only for performing qubit readout 23 but also for amplifying vacuum squeezed states 25 .
In conclusion, we have discussed an optimal and compact design for the Josephson mixer and applied these principles to demonstrate phase preserving quantum limited amplification. The resulting device operates with gains reaching 30 dB within 0.4 photons of the quantum limit of noise and a saturation power of −112 dBm, which is promising for analog information processing of quantum signals, directional amplification and on-chip circulators 29 . These specifications do not hinder the dynamical bandwidth of the mixer, which reaches 50 MHz at G = 20 dB. Such device is suited for fast operation on superconducting qubit, which are necessary to the improvement of the efficiency of quantum feedback 36,37 , multiplexing several qubits 35 or more generally quantum error correction schemes.