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arXiv:2611.02798

Coulomb Blockade in a One-Dimensional Array of Strong Josephson Junctions

Bayan Karimi, Ze-Yan Chen, Jukka P. Pekola, T. Yamamoto, M. Houzet, L. I. Glazman

cond-mat.mes-hallquant-phSubmitted Nov 2, 2026• Announced Nov 3, 2026
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Abstract

We measure current across a voltage-biased, long array of Josephson junctions. Current-voltage characteristics do not reveal any critical current and instead display the presence of threshold voltage and finite activation energy, consistent with the Coulomb blockade. However, the respective energy scale is well below the value prescribed by a single-island geometric capacitances. We explain it by strong quantum fluctuations of charge associated with a large value of Josephson energy, $E_J\gg E_C$, where $E_C$ is the charging energy associated with the inter-island capacitance. Comparison between the data and theory points to an exponential (in parameter $E_J/E_C$) suppression of the threshold. We achieve a quantitative agreement with theory by accounting for charge-$e$ solitons associated with the quasiparticles entering the array from its leads. This scenario however underestimates the current below threshold in comparison with the experiment.

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We measure current across a voltage-biased, long array of Josephson junctions. Current-voltage characteristics do not reveal any critical current and instead display the presence of threshold voltage and finite activation energy, consistent with the Coulomb blockade. However, the respective energy scale is well below the value prescribed by a single-island geometric capacitances. We explain it by strong quantum fluctuations of charge associated with a large value of Josephson energy, EJ≫ECE_J\gg E_CEJ​≫EC​, where ECE_CEC​ is the charging energy associated with the inter-island capacitance. Comparison between the data and theory points to an exponential (in parameter EJ/ECE_J/E_CEJ​/EC​) suppression of the threshold. We achieve a quantitative agreement with theory by accounting for charge-eee solitons associated with the quasiparticles entering the array from its leads. This scenario however underestimates the current below threshold in comparison with the experiment.
We present measurements of the temperature-dependent conductance for series arrays of small-capacitance superconducting quantum interference devices (SQUIDs). At low-bias voltages, the arrays exhibit a strong Coulomb blockade, which we study in detail as a function of temperature and Josephson energy EJ{E}_{J}EJ​. We find that the zero-bias conductance is dominated by a thermally activated hopping of Cooper pairs between neighboring superconductive islands with the activation energy of the order of $\ensuremath{\Lambda}{E}_{C}$, where $\ensuremath{\Lambda}$ is the charge screening length in the array and EC{E}_{C}EC​ is the charging energy of a single SQUID.
This thesis presents the results of an experimental study on single-charge effects in nanoscale Josephson junctions and Cooper pair transistors (CPTs). In nanoscale Josephson junctions the charging energy EC becomes significant at sub-Kelvin temperatures and single-charge effects, such as the Coulomb blockade of Cooper pair tunneling, influence the transport properties. In order to observe charging effects in a single Josephson junction, the impedance of the electromagnetic environment surrounding the junction has to be larger than the quantum resistance (RQ=h/4e2≈6.45kΩ). In this work the high impedance environment is obtained by biasing the sample under test (single Josephson junction or CPT) with four one-dimensional Josephson junction arrays having SQUID geometry. The advantage of this configuration is the possibility of tuning in situ the effective impedance of the electromagnetic environment. By applying a magnetic field perpendicular to the SQUID loops, the Josephson energy EJ of the SQUIDs is suppressed, resulting in an increase of the measured zero bias resistance of the arrays of several orders of magnitude (104< R0 (Ω) <109). This bias method enables the measurement of the same sample in environments with different impedance. As the impedance of the environment is increased, the current-voltage characteristics (IVCs) of the single Josephson junction and of the CPT show a well defined Coulomb blockade feature with a region of negative differential resistance, signature of the coherent tunneling of single Cooper pairs. The measured IVCs of a single Josephson junction with SQUID geometry in the high impedance environment show a qualitative agreement with the Bloch band theory as the EJ/EC ratio of the junction is tuned with the magnetic field. We also studied a single nontunable Josephson junction with strong coupling (EJ/EC > 1), where the exact dual of the overdamped Josephson effect is realized, resulting in a dual shape of the IVC, where the roles of current and voltage are exchanged. Here, we make for the first time a detailed quantitative comparison with a theory which includes the effect of fluctuations due to the finite temperature of the environment. The measurements on CPTs in the high impedance environment showed that the Coulomb blockade voltage is modulated periodically by the gate-induced charge. The gate-voltage dependence of the CPT changes from e-periodic to 2e-periodic as the impedance of the environment is increased. The high impedance environment reduces quasiparticle tunneling rates, thereby restoring the even parity of the CPT island. This behavior suggests that high impedance leads can be used to effectively suppress quasiparticle poisoning.
We present observations of charging effects for Cooper pairs in short linear arrays of small-capacitance Josephson junctions. Current-voltage characteristics show a Coulomb gap for Cooper-pair tunneling when the charging energy exceeds the Josephson coupling energy. In a double junction a zero-voltage current is observed that is modulated by a gate voltage applied to the metal island between the junctions. For longer arrays a crossover from Coulomb blockade of Cooper-pair tunneling to a supercurrent is observed when the ratio of Josephson coupling to charging energy is increased.
We have measured the current-voltage $(I\ensuremath{-}V)$ characteristics of small-capacitance single Josephson junctions at low temperatures (T=0.02−−0.6K),(T=0.02--0.6\mathrm{K}),(T=0.02−−0.6K), where the strength of the coupling between the single junction and the electromagnetic environment was controlled with one-dimensional arrays of dc superconducting quantum interence devices (SQUIDs). The single-junction $I\ensuremath{-}V$ curve is sensitive to the impedance of the environment, which can be tuned in situ. We have observed Coulomb blockade of Cooper-pair tunneling and even a region of negative differential resistance, where the zero-bias resistance ${R}_{0}^{\ensuremath{'}}$ of the SQUID arrays is much higher than the quantum resistance ${R}_{K}\ensuremath{\equiv}{h/e}^{2}\ensuremath{\approx}26\mathrm{k}\ensuremath{\Omega}.$ The negative differential resistance is evidence of the coherent single-Cooper-pair tunneling within the theory of current-biased single Josephson junctions. Based on this theory, we have calculated the $I\ensuremath{-}V$ curves numerically in order to compare then with the experimental ones at ${R}_{0}^{\ensuremath{'}}\ensuremath{\gg}{R}_{K}.$ The numerical calculation agrees with the experiments qualitatively. We also discuss the ${R}_{0}^{\ensuremath{'}}$ dependence of the single-Josephson-junction $I\ensuremath{-}V$ curve in terms of the superconductor-insulator transition driven by changing the coupling to the environment.
We have measured the current-voltage characteristics of small-capacitance single Josephson junctions at low temperatures ( T< or =0.04 K), where the strength of the coupling between the single junction and the electromagnetic environment was controlled with one-dimensional arrays of dc SQUIDs. We have clearly observed Coulomb blockade of Cooper-pair tunneling and even a region of negative differential resistance, when the zero-bias resistance of the SQUID arrays is much higher than the quantum resistance h/e(2) approximately 26 kOmega. The negative differential resistance is evidence of coherent single-Cooper-pair tunneling in the single Josephson junction.
We have measured the current–voltage characteristics of small-capacitance single Josephson junctions at low temperatures T ≤ 0.04 K, where the strength of the coupling between the single junction and the electromagnetic environment was controlled with one-dimensional arrays of dc SQUIDs. When the zero-bias resistance of the SQUID arrays is much higher than the quantum resistance h/e 2 ≈ 26 kΩ, a transition to a clear Coulomb blockade of Cooper-pair tunnelling is induced in the single junction. The measured blockade voltage agrees with the theoretical prediction.