Implementation of low-loss superinductances for quantum circuits

The emerging field of quantum electronics exploiting large fluctuations of the superconducting phase is limited by the practical challenge of engineering an electromagnetic environment which suppresses simultaneously the quantum fluctuations of charge and the random low-frequency fluctuations of offset charges. The small value of the fine structure constant $\alpha =1/137$ entails a fundamental asymmetry between flux and charge quantum fluctuations, strongly favoring the latter. To illustrate this, let us consider the simplest case of a dissipationless LC oscillator, where the charge $Q$ on the capacitor plates and the generalized flux $\Phi$ across the inductor are conjugate variables. The ratio between quantum fluctuations of charge, $\delta q=\delta Q/\left(2e\right)$, and flux, $\delta \phi =\delta \Phi /{\Phi }_0$, in the ground state of the oscillator is given by $\delta \phi /\delta q=Z_0/R_Q$. Here $Z_0=\sqrt{L/C}$ is the characteristic impedance of the oscillator and $R_Q=h/(2e{)}^2=6.5k\Omega$ is the superconducting resistance quantum. Using only geometrical inductors and capacitors, the characteristic impedance of the oscillator $Z_0$ can not exceed the vacuum impedance $Z_{vac}=\sqrt{{\mu }_0/{ϵ}_0}$, thus imposing quantum fluctuations of charge at least an order of magnitude larger than flux fluctuations: $\delta \phi /\delta q<Z_{vac}/R_Q=8\alpha$.

In order to exceed the vacuum impedance and suppress quantum charge fluctuations, circuit elements with extremely high impedance are required. On chip resistors and long chains of Josephson junctions (JJs) in the dissipative regime have already been used to provide high impedance environments KUZMIN1991 (); Lotkhov2003 (); Corlevi2006 (). However, these Ohmic components do not help shunting charge offsets and, being dissipative, they tend to destroy the quantum coherence of the devices. In order to simultaneously suppress quantum charge fluctuations and offsets, we need a circuit element which possesses three key attributes: high impedance at frequencies of interest, perfect conduction at DC and extremely low dissipation. These attributes define the so-called “superinductance” superinductance_Kitaev ().

In this Letter, we report the successful implementation and detailed characterization of superinductances using the large kinetic inductance of arrays of Josephson junctions (see Fig 1(a)), following the proof-of-concept shown in the fluxonium circuit Manucharyan2009 (). This first superinductance implementation suffered from coherent quantum phases-slips (CQPS) Manucharyan2012 (); Pop2012 (), which constitute additional degrees of freedom, difficult to control experimentally. We show that we can completely suppress the CQPS by using large JJs with Josephson energy $E_J$$\backsimeq 100E_C$, where $E_C=e^2/\left(2C_J\right)$ is the charging energy of one junction.

Superconducting nanowires are also likely candidates for implementing superinductances Bezryadin2000 (); Mooij2006 (). Unfortunately, they show significant internal dissipation which is not yet well understood, and are more challenging to fabricate. The JJ arrays may also suffer from dissipation, either due to coupling to internal degrees of freedom Fazio2001 (); Rastelli2012 () or to a dissipative external bath Lobos2011 (). Indeed, transport measurements on large arrays of JJ show the appearance of a superconducting to insulating transition (SIT) with decreasing Josephson energy $E_J$ Chow1998 (); Kuo2001 (); Takahide2006 (). Previous implementations of resonators where the inductive energy is given by JJ arrays have yielded internal quality factors in the range of a few thousands Castellanos-Beltrana2007 (); Palacios-Laloy2008 (). We show here that the problem of dissipation can be alleviated by using large JJs to realize superinductances far from the SIT region with internal quality factors an order of magnitude above the previously reported values.

The superinductances are formed by an array of closely spaced Josephson junctions on a C-plane sapphire substrate, as shown in Fig. 1. We fabricated the junctions by e-beam lithography and double angle evaporation of aluminum using the bridge-free technique of BFT (); rigetti (). Prior to aluminum deposition, the substrate is cleaned of resist residues using an oxygen plasma pop (). We minimize the width of the connecting wires between junctions in order to reduce parasitic capacitances to ground, which ultimately lower the self-resonant frequency of the superinductance hutter2011 ().

We characterize our superinductances at low temperatures and microwave frequencies by incorporating them in lumped element LC resonators capacitively coupled to a co-planar waveguide in the hanger geometry. The resonator response is measured in transmission. An optical image of a typical device and its low-frequency circuit model are shown in Figs. 1(c,d).

The samples were mounted on the mixing chamber stage (15 mK) of a dilution refrigerator inside a copper box, enclosed in an aluminum-Cryoperm-aluminum shield with a Cryoperm cap. We used two 4-12 GHz Pamtech isolators and a 12 GHz K&L multi-section low-pass filter before the HEMT amplifier, and installed copper powder filters on the input and output lines.

The internal quality factor of a resonator is extracted by fitting the transmission data about the resonance with the response function geerlings ()

where $Q_{tot}$ and $Q_{ext}$ are the total and external quality factors, $f_R$ is the resonant frequency, and $\delta f$ characterizes asymmetry in the transmission response profile. The internal quality factor is given by $Q_{int}=\frac{Q_{ext}{Q}_{tot}}{Q_{ext}-Q_{tot}}$. We show a typical measured response for a resonator with an 80-junction superinductance in Fig. 2(a). This yields an internal quality factor of 37,000 for the resonator at 15 mK (stage temperature), corresponding to a loss in the superinductance of better than 27 ppm. We note that an unknown portion of the internal loss comes from the capacitors. The inset of Fig. 2(a) shows the dependence of the internal quality factor on the stage temperature. A similar resonator with two 80-junction arrays in parallel was measured to have a quality factor of 56,000 (Fig. 2(b)), corresponding to a superinductance loss of better than 18 ppm. The external quality factors, $Q_{ext}$, of both resonators were 5,000.

The data presented in Fig. 2 corresponds to measurements of the $k=1$ mode. The frequencies of higher modes, $k\ge 2$, of the array lie outside the band of our measurement setup. In order to observe these modes we exploit their cross-Kerr interaction, which is induced by the junction nonlinearity. This interaction leads to a frequency shift of the lowest mode ($k=1$) when higher modes are excited. In Fig. 3(a) we show the results of a two-tone measurement of an 80-junction resonator (same device as presented in Fig. 2(a)), where we continuously monitor the $k=1$ mode while sweeping the frequency of a probe tone. When the probe tone is resonant with a higher mode of the array, we observe a drop in the $k=1$ frequency.

Fig. 3(b) shows the comparison between the measured frequencies of the array modes and the theoretically predicted values. The black crosses show the renormalized mode frequencies calculated after incorporation of the corrections due to coupling capacitance pads (see supplementary information), which are in good agreement with the measured frequencies represented by colored markers. The parameters extracted from the fit were $C_0=0.04$ fF, $C_J=40$ fF and $L_J=1.9$ nH, with a confidence range of 20%. The simulated value of the capacitance to ground $C_0=0.09$ fF agrees within a factor of 2 with the inferred value from the fit. Room temperature resistance measurements of the junction arrays yield a value of $L_J=2.1$ nH, which agrees with the fit within 10%. Using the fit parameters we calculate the dispersion relation for the bare superinductance, shown as a gray curve in Fig. 3(b). We note that the lowest resonant frequency of the bare superinductance is 14.2 GHz, which meets the design criterion of having self resonances well above the frequency range of interest (1 - 10 GHz).

In order to characterize the phase slip rate of the superinductances, we monitor the frequency of the $k=1$ mode of a resonator formed by two superinductances in parallel, while sweeping an external magnetic field. As flux bias is increased, the persistent current induced in the loop increases, and results in a drop in frequency of the mode (Fig. 4(a)). This behavior can be described to lowest order in junction nonlinearity by the quasi-classical expression

where ${\Phi }_{ext}$ is the applied flux bias, and $m$ is the integer number of flux quanta inside the loop. A phase slip event is associated with an integer change in $m$, which we observe as a jump in resonator frequency. As we increase the flux bias, the phase slip rate is enhanced and frequency jumps become more probable mlg (). We swept the flux bias applied to the loop over several flux quanta before a phase slip event occurred. The typical duration between phase slips recorded in this experiment was over an hour, Fig. 4(a). This is a remarkably low phase slip rate of well under 1 mHz for a loop of 160 junctions, significantly lower than previously reported values on shorter arrays Manucharyan2012 (); Pop2010 ().

Due to the extremely low phase slip rate, in order to measure the resonator in the lowest flux state, we employ an active resetting scheme. We apply a high power pulse at one of the $k\ge 2$ modes before measuring the location of the lowest resonant frequency. This high power pulse activates phase slips, allowing the loop to settle into the lowest flux state. Using this protocol, we tracked the resonant frequency in the lowest flux state, as shown in Fig. 4(b). Discrete inverted parabolas are observed, which allow us to unambiguously calibrate the number of flux quanta in the loop. The fitted value for the total number of junctions differs by 6% from the actual number, which can be explained by the classical nature of the theory which does not take quantum fluctuations into account.

In conclusion, we have demonstrated the successful implementation of superinductances using arrays of Josephson junctions, and performed the first detailed characterization of a superinductance. We measured superinductances in the range of 100–300 nH, self resonant frequencies above 10 GHz, internal losses less than 20 ppm, and phase slip rates below 1 mHz. The long lifetimes of persistent current states in the array loop also demonstrate the low DC resistance of the arrays. With these parameters, the Josephson junction array superinductance enriches significantly the quantum electronics toolbox. Applications would include further suppression of offset charges in superconducting qubits Manucharyan2009 (), high Q and tunable lumped-element resonators, on-chip bias tees, kinetic inductance particle detectors Day2003 (), and precise measurements of Bloch oscillations bloch ().

We would like to acknowledge fruitful discussions with Luigi Frunzio, Kurtis Geerlings, Leonid Glazman, Wiebke Guichard, Zaki Leghtas, Mazyar Mirrahimi, Michael Rooks and Rob Schoelkopf. Facilities use was supported by YINQE and NSF MRSEC DMR 1119826. This research was supported by IARPA under Grant No. W911NF-09-1-0369, ARO under Grant No. W911NF-09-1-0514 and NSF under Grant No. DMR-1006060. As this manuscript was completed, we learned of a similar implementation of a superinductance using a different array topology Bell2012 ().