Abstract
The interplay between superconductivity and Coulomb interactions has been studied for more than 20 years now^{1,2,3,4,5,6,7,8,9,10,11,12,13}. In lowdimensional systems, superconductivity degrades in the presence of Coulomb repulsion: interactions tend to suppress fluctuations of charge, thereby increasing fluctuations of phase. This can lead to the occurrence of a superconducting–insulator transition, as has been observed in thin superconducting films^{5,6}, wires^{7} and also in Josephson junction arrays^{4,9,11,12,13}. The last of these are very attractive systems, as they enable a relatively easy control of the relevant energies involved in the competition between superconductivity and Coulomb interactions. Josephson junction chains have been successfully used to create particular electromagnetic environments for the reduction of charge fluctuations^{14,15,16}. Recently, they have attracted interest as they could provide the basis for the realization of a new type of topologically protected qubit^{17,18} or for the implementation of a new current standard^{19}. Here we present quantitative measurements of quantum phase slips in the ground state of a Josephson junction chain. We tune in situ the strength of quantum phase fluctuations and obtain an excellent agreement with the tightbinding model initially proposed by Matveev and colleagues^{8}.
Main
In superconducting circuits, each electrical element such as an inductor, a capacitor or the Josephson element can add a degree of freedom. In the case of small circuits, by applying Devoret’s circuit theory^{20}, a complete analytical description that takes into account all degrees of freedom can be obtained. However, when the circuits contain an increasing number of elements, as for example Josephson junction chains, even numerical solutions of the problem become difficult to obtain when taking into account all degrees of freedom. Nevertheless, our measurements demonstrate that the ground state of a phasebiased Josephson junction chain (see Fig. 1a) can be described by a single degree of freedom. Although the chain is a multidimensional object, the effect of quantum phase slips can be described by a single variable, m, that counts the number of phase slips in the chain.
We start by giving a short introduction on the lowenergy properties of a Josephson junction chain analysed in terms of quantum phase slips^{8}. Let us consider the Josephson junction chain shown in Fig. 1a. The chain contains N junctions and is biased with a phase γ. We denote E_{J} the Josephson energy of a single junction and E_{C}=e^{2}/2C its charging energy. Here we consider E_{J}≫E_{C}. Let Q_{i} be the charge on each junction and θ_{i} the phase difference. In the nearestneighbourcapacitance limit, the Hamiltonian can be written as:
Ignoring the charging energy for the moment, in the classical ground state the phase bias γ is equally distributed on the N junctions: θ_{i}=γ/N, as illustrated by the solid line in Fig. 1b. The resulting Josephson energy hence reads E_{0}=E_{J}γ^{2}/2N and the chain is equivalent to a large inductance. If a phase slip occurs on one of the junctions, say the jth junction, then θ_{j}→θ_{j}+2π. The constraint would be violated after such a phaseslip event if the phases across all other junctions do not adjust. Therefore, the phase difference θ_{i} over all other junctions changes a little from γ/N to (γ−2π)/N to accommodate the bias constraint (see the dashed line in Fig. 1b). A phase slip on a single junction leads to a collective response of all junctions. Consequently, the Josephson energy of the entire chain changes from E_{0}=E_{J}γ^{2}/2N to E_{m}=E_{J}(γ−2πm)^{2}/2N after m phase slips. The classical ground state energy of the chain consists of shifted parabolae that correspond respectively to a fixed number m of phase slips (see Fig. 1c). For the special values γ=π(2m+1), the energies E_{m} and E_{m+1} are degenerate.
Taking now into account the finite charging energy E_{C}, quantum phase slips can lift the degeneracy at the points γ=π(2m+1). In the limit of rare phase slips, that is, E_{J}≫E_{C}, the hopping element for the quantum phase slip can be approximated by^{21,22}: . As a phase slip can take place on any of the N junctions, the hopping term between the two states m〉 and m+1〉 is given by N v. Therefore, using a tightbinding approximation, the total Hamiltonian for the chain is given by:
Figure 1c shows the numerical calculation of the two lowest eigenenergies of the Hamiltonian (1) for three different ratios E_{J}/E_{C}=20, 3 and 1.3 in the case of a sixjunction chain. Figure 1d shows the corresponding current–phase relation of the chain in the ground state. The chain’s supercurrent is obtained by the calculation of the derivative of the groundstate energy E_{g}: i_{S}=(2e/ℏ)(∂ E_{g}/∂ γ). For large values of E_{J}/E_{C}, quantum phase fluctuations are very small (v∼0) and the current–phase relation has a sawtoothlike dependence with a critical current that is approximatively N/π times smaller than that of a single junction of the chain. We call this regime the ‘classical’ phaseslip regime. When quantum phase fluctuations increase, that is, E_{J}/E_{C} decreases, the current–phase relation becomes sinusoidal and the critical current becomes exponentially suppressed with N and E_{J}/E_{C} (ref. 8).
To measure the effect of quantum phase slips on the ground state of a Josephson junction chain, we have studied a chain of six junctions. Our measurement setup and the junction parameters are presented in Fig. 2 and Table 1. Each junction in the chain is realized by a superconducting quantum interference device (SQUID) to enable tunable Josephson coupling E_{J}. In this way we can tune in situ the E_{J}/E_{C} ratio by applying a uniform magnetic flux Φ_{S} through all SQUIDs, and consequently we can control the strength of quantum phase fluctuations. For our measurements we placed this chain in a closed superconducting loop, threaded by the flux Φ_{C}, containing an extra shunt Josephson junction that is used for the readout of the chain state. The flux Φ_{C} enables the control of the bias phase γ=Φ_{C}−δ over the chain.
We have measured the switching current of the entire Josephson junction circuit containing both the chain and the readout junction. The switching current was determined from the switching probability at 50%. The switching probability as a function of bias current I_{bias} has a width of ≈20 nA. We apply typically 10,000 biascurrent pulses of amplitude I_{bias} and measure the switching probability as the ratio between the number of switching events and the total number of pulses. The current pulses have a rise time of 8 μs and a total duration of 20 μs. The results of the switchingcurrent measurements as a function of flux Φ_{C} are shown in Fig. 3. From these switchingcurrent measurements we deduce the effect of quantum phase slips on the ground state of the chain.
The measured switching current corresponds to the escape process out of the total potential energy U_{tot} containing the contributions of the readout junction and the chain:
Here E_{g} is the ground state of the sixSQUID chain calculated by solving the Hamiltonian (1). As E_{J}^{RO}≫E_{g} the main component in U_{tot} is the potential of the currentbiased readout junction E_{J}^{RO}cos(δ)−(ℏ/2e)I_{bias}δ. Figure 1f shows the escape potential at constant bias current for three different flux values φ_{C} corresponding to three different biasing phases γ over the chain. Let us point out that the position of the minimum of the potential U_{tot} is in good approximation independent of the value of the flux φ_{C}. Therefore, the bias phase difference γ over the chain depends only on the flux φ_{C}. As a consequence, the φ_{C} dependence of the measured switching current results from the γ dependence of the chain’s ground state.
The escape from the potential U_{tot} occurs by means of macroscopic quantum tunnelling (MQT). The MQT rate for an arbitrary potential can be calculated in the limit of weak tunnelling using the dilute instantongas approximation^{23}. Within this model, the escape rate Γ out of the washboard potential U_{tot}(γ) reads^{24}:
where A and B are given by:
We have denoted by σ the width of the barrier and by x the phase coordinate measured from the minimum of the washboard potential. The plasma frequency is , where E_{C}^{RO} is the charging energy of the readout junction.
Knowing the escape rate Γ, we can calculate the switching probability:
The results of numerical calculations and the experimental data are shown in Fig. 3. The theory fits very well both in amplitude and shape the oscillations of the measured switching current. Let us point out that we have used the nominal values for E_{J} and E_{C} calculated from the characteristics of the sample indicated in Table 1. The normalstate resistance for a single chain junction has been deduced from the measured normalstate resistance of the readout junction by considering the size ratio between the two. We evaluate the precision of the determination of E_{J} and E_{C} to be in the range of ±10%. This error bar on E_{J} and E_{C} yields an uncertainty of ±15% for the phaseslip amplitude N v. The eventual presence of junction inhomogeneity or an important effect of background charges would imply a significantly larger decrease of the phaseslip amplitude^{8}. The good agreement between theory and experiment confirms the homogeneity of our junctions. It excludes a significant contribution of background charges in the overall shape of the switching curve and demonstrates the collective nature of the phaseslip events.
From the measurements in Fig. 3, we define the switchingcurrent amplitude ΔI_{SW} as half of the peaktopeak variation of the switching current with the flux Φ_{C}. Figure 4 shows the measured ΔI_{SW} and the corresponding theoretical calculations as a function of E_{J}/E_{C}. For each measurement, E_{J} has been calculated using the flux dependence of the SQUID’s Josephson coupling: E_{J}(Φ_{S})=(ℏ/2e)i_{0}(Φ_{S}) with i_{0}(Φ_{S})=I_{C}^{SQ}cos(πΦ_{S}/Φ_{0}). To distinguish between the suppression of the switching current that is due to quantum phase fluctuations and the one that is simply due to the wellknown cancellation of the SQUID’s critical current as a function of flux, we plot the switchingcurrent amplitude divided by the critical current of a single SQUID i_{0}. We see that the measured switchingcurrent amplitude follows very well the predicted theoretical suppression of the switchingcurrent oscillations in the presence of quantum phase fluctuations. From our measurements we can also deduce the strength of the quantum phaseslip amplitude. With decreasing E_{J}/E_{C} ratio from 3 to 1 the quantum phaseslip amplitude increases from 0.8 to 2.7 GHz. In addition, in Fig. 4 we have plotted for comparison the calculation for the switchingcurrent amplitude in the case when quantum phase fluctuations would be negligibly small: v∼0. As expected, we get a practically flat dependence on E_{J}/E_{C}.
Further on, the upper x axis of Fig. 4 shows the ratio E_{J}/k_{B}T of the Josephson energy with respect to the thermal energy at T=50 mK. As E_{J}≫k_{B}T, thermal fluctuations are excluded to explain the suppression of the switching current with decreasing E_{J}/E_{C}. Further measurements (not shown here) reveal a constant switchingcurrent amplitude and width of the switching distribution up to a temperature of T=100 mK.
We present a detailed experimental characterization of the effect of quantum phase slips on the ground state of a Josephson junction chain. These phase slips are the result of fluctuations induced by the finite charging energy of each Josephson junction in the chain. The experimental results can be fitted in very good agreement by considering a simple tightbinding model for the phase slips^{8}. Our measurements also show that a Josephson junction chain under phasebias constraint can behave in a collective way very similar to a single macroscopic quantum object.
These results open the way for the use of quantum phase slips in Josephson junction networks for the implementation of a new current standard, the observation of Bloch oscillations^{19}, the fabrication of topologically protected qubits^{25} and the design of new superconducting circuit elements.
Methods
The circuit was fabricated on a Si/SiO_{2} substrate and the Al/AlO_{x}/Al junctions were obtained using standard shadow evaporation techniques. The aluminium oxide was obtained by natural oxidation in a controlled O_{2} atmosphere. Roomtemperature measurements on an ensemble of ∼100 junctions revealed a variance of 4% of the normalstate resistance of the junctions.
The sample was mounted in a closed copper block that was thermally connected to the cold plate of a dilution refrigerator at 50 mK. All lines were strongly filtered by lowpass filters at the cryostat entrance and by thermocoaxial cables and π filters at low temperatures.
The switching current I_{SW} of the circuit is obtained by carrying out the following sequence. We use a series of M current steps of equal amplitude I_{bias} to bias the junction. We count the number of transitions to the voltage state M_{SW} and thus obtain the value of the switching probability P_{SW}=M_{SW}/M corresponding to the applied I_{bias}. By sweeping the I_{bias} amplitude and repeating the above sequence, we measure a complete switching histogram, P_{SW} versus I_{bias}. The P_{SW}=50% bias current is called the switching current of the circuit, I_{SW}.
The principle of the readout scheme was first implemented by Vion et al. ^{26} and has also been used for the measurement of the ground state of superconducting atomic contacts^{27}. The choice of the readout junction critical current I_{C}^{RO} for an optimal measurement of i_{S} is not straightforward. On the one hand one would like to have I_{C}^{RO}≫i_{S}, but on the other hand the width of the switching histograms w increases with I_{C}^{RO} and so do the statistical fluctuations resulting from finite ensemble size . For reasonable measuring timescales, the number of current steps M is limited to values of about 10^{4}. If we want to measure supercurrents for the SQUID chain in the range of 1 nA, I_{C}^{RO} needs to be in the range of 100 nA. We have used a readout junction with a critical current I_{C}^{RO}=330 nA that offers a good tradeoff.
In our MQT analysis we neglect the effect of dissipation on the escape rate. Small dissipation can add a prefactor in front of the exponential in the switching probability formula (3). However, this factor is independent of Φ_{C}, so it will change only the offset value of I_{SW} in Fig. 3, but not the shape nor the amplitude of the I_{SW} oscillations.
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Acknowledgements
We thank B. Douçot, D. Estève, F. Hekking, L. Ioffe and G. Rastelli for fruitful discussions. We are grateful to the team of the Nanofab facility in Grenoble for their technical support in the sample fabrication. Our research is supported by the European STREP MIDAS and the French ANR ‘QUANTJO’.
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I.M.P. fabricated the sample and carried out the experiments. I.M.P., B.P., O.B. and W.G. designed the experiment, analysed the data and wrote the paper. I.M.P. and I.P. carried out the numerical calculations. Z.P. and F.L. contributed to the carrying out of experiments and sample fabrication. W.G. supervised the project. All authors discussed the results and commented on the manuscript at all stages.
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Pop, I., Protopopov, I., Lecocq, F. et al. Measurement of the effect of quantum phase slips in a Josephson junction chain. Nature Phys 6, 589–592 (2010). https://doi.org/10.1038/nphys1697
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DOI: https://doi.org/10.1038/nphys1697
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