Tunneling of Cooper pairs across voltage-biased asymmetric single-Cooper-pair transistors

We analyze tunneling of Cooper pairs across voltage biased asymmetric single-Cooper-pair transistors. Also tunneling of Cooper pairs across two capacitively coupled Cooper-pair boxes is considered, when the capacitive coupling and Cooper pair tunneling are provided by a small Josephson junction between the islands. The theoretical analysis is done at subgap voltages, where the current-voltage characteristics depend strongly on the macroscopic eigenstates of the island(s) and their coupling to the dissipative environment. As the environment we use an impedance which satisfies $\mathrm{Re}\left[Z\left(\omega \right)\right]⪡R_Q$ and a few $LC$ oscillators in series with $Z\left(\omega \right)$. The numerically calculated $I\text{−}V$ curves are compared with experiments where the quantum states of mesoscopic SQUIDs are probed with inelastic Cooper pair tunneling. The main features of the observed $I\text{−}V$ data are reproduced. Especially, we find traces of band structure in the higher excited states of the Cooper-pair boxes as well as traces of multiphoton processes between two Cooper-pair boxes in the regime of large Josephson coupling $E_J⪢E_C$.

Figure 1

A voltage biased SCPT. We study the case $E_{J1}⪢E_{J2}$ and the smaller JJ is called the probe. The superconducting leads (and the spurious environment of the island) are modeled through an impedance $Z\left(\omega \right)$.

Figure 2

Three Josephson junctions in series with the voltage source. One of the large JJs could as well as be an $LC$ oscillator, modeling the effect of a spurious resonance.

Figure 3

(Color online) A 2D-surface plot of the measured probe junction current as a function of voltage and flux through the SQUID loop. The resonances originating from the SQUIDs show ${\Phi }_0$-periodicity. The figure also shows a resonance of Fraunhofer/Airy type with weaker dependence on $\Phi$ (pointed by the left arrow), and multiphoton transitions between the SQUID and the fluctuator behind this resonance (the right arrow). The $I\text{−}V$ characteristics of this 4-SQUID sample are analyzed more quantitatively in Sec. 5B.

Figure 4

The positions of the resonances in the 1-SQUID experiment (data points) compared with those resulting from the model of a SQUID and an $LC$ oscillator in series with the probe junction (lines). The resonances have been labeled by the corresponding transitions behind them. For example, ∣1,0⟩ means that the resonance occurs due the transition from the ground state to the product state, in which the CPB is in its first excited state and the $LC$ oscillator stays in its ground state. The energy levels of the CPBs are actually bands (shaded), meaning that the levels are quasicharge $Q_0=C_{0}U-(C_2+C_0∕2)V$ dependent. The resonances due to the band edges $Q_0=0$, $Q_0=e$ have been plotted explicitly. The parameters in the numerical modeling are summarized in Table .

Figure 5

The maximum current of the resonance ∣1,0⟩ in the 1-SQUID experiment (triangles) compared with that resulting from the model of a SQUID and an $LC$ oscillator in series with the probe junction. The $I\text{−}V$ curves of the model were averaged over all values of the quasicharge. Different from Fig. 4, the $LC$ oscillator (“$LC2$” in Table ) describes a small $I\text{−}V$ peak seen at a voltage $\approx 90\mu \mathrm{V}$ (not shown in Fig. 4). In the region $\Phi >0.3{\Phi }_0$ the state ∣1,0⟩ suffers from a slow relaxation and the current is determined by the photon emission to the Ohmic low frequency environment. A resonance occurs nearby $\approx 0.2{\Phi }_0$ when the states ∣1,0⟩ and ∣0,1⟩ are coupled, the latter having a faster relaxation to its ground state. The current between $\Phi =0$ and $\Phi =0.1{\Phi }_0$ seems also to be enhanced, probably due to further entanglement with the environment. In the calculation of the relaxation rates we have used a perturbation $C_2{Q}_{\text{SQUID}}{V}_f∕(C_1+C_2)+0.13Q_{LC}{V}_{f^{\prime }}$ (Sec. 4).

Figure 6

Left: Schematic drawing of the 4-SQUID experiment. The SQUIDs are drawn as JJs. In this situation, the two SQUIDs (at the same side) behave as a single JJ, but with double the coupling energy $E_J$ and capacitance $C$ compared to the individual SQUIDs. Right: The resulting theoretical model of the system.

Figure 7

The positions of the resonances in the 4-SQUID experiment (data points) compared with the edges of the energy bands (lines) and the band structure (shaded) calculated from the model of two SQUIDs and two $LC$ oscillators in series with the probe junction. The experimental data is based on $I\text{−}V$ curves of which two examples are shown in Fig. 8. Also shown is a schematic diagram of the model circuit and the code used for labelling the states. The parameters in the numerical model are summarized in Table .

Figure 8

The current across the 4-SQUID sample when $\Phi =0$ (main frame) and $\Phi =0.45{\Phi }_0$ (inset). The solid line is the experimental and the dashed line the theoretical $I\text{−}V$ curve. For the theoretical fit we have averaged over all values of $Q_0$ and $Q_0^{\prime }$ and the widths of the bands are indicated as arrows. In the inset one can see that the ∣1,0,0⟩ resonance has widened and centers at $\sim 32\mu \mathrm{V}$. The $\mid 2^{*},0,0⟩$ resonance lies almost at the same point as the upper edge of the ∣2,0,0⟩ resonance, making the overall structure asymmetric. The $\mid 3^{*},0,0⟩$ resonance lies between $\sim 88$ and $\sim 108\mu \mathrm{V}$, and the ∣3,0,0⟩ resonance lies between $\sim 78$ and $\sim 112\mu \mathrm{V}$. In the calculation of the relaxation rates we have used $Q_{\mathrm{int}}=0.2(Q_1+Q_3+Q_{LC1}+Q_{LC2})$ (Sec. 4).

Figure 9

Experimental linewidths (data points) compared with theoretical linewidths (lines) for three of the resonances as a function of the magnetic flux. The theoretical values are obtained by adding a constant term $10\mu \mathrm{eV}$ to the widths of the bands.