Effect of decoherence on resonant Cooper-pair tunneling in a voltage-biased single-Cooper-pair transistor

We analyze how decoherence appears in the $I\text{−}V$ characteristics of a voltage-biased single-Cooper-pair transistor. Especially the effect on resonant single or several Cooper-pair tunneling is studied. We consider both a symmetric and an asymmetric transistor. As a decoherence source we use a small resistive impedance $\left(\text{Re}\left[Z\left(\omega \right)\right]⪡R_Q=h/4e^2\right)$ in series with the transistor, which provides both thermal and quantum fluctuations of the voltage. Additional decoherence sources are quasiparticle tunneling across the Josephson junctions and quantum $f$ noise caused by spurious charge fluctuators nearby the island. The analysis is based on a real-time diagrammatic technique which includes Zeno-type effects in the charge transport, where the tunneling is slowed down due to strong decoherence. As compared to the Pauli-master-equation treatment of the problem, the present results are more consistent with experiments where many of the predicted sharp resonant structures are missing or weakened due to decoherence.

Figure 1

(Color online) A voltage-biased SCPT consisting of two JJs (crossed boxes) in series with the voltage source $V$. The island is capacitively coupled to a gate lead. Also drawn is the EE described by an impedance $Z\left(\omega \right)$ and spurious charge fluctuators nearby the island (Sec. 3).

Figure 2

(Color online) A two-dimensional map of the current $I$ as a function of $Q_0$ and $V$ according to the CTM. A contour line for the current 10 pA is drawn for clarity. The resonances originate in the first- or higher-order Cooper-pair tunneling across the probe and simultaneous excitation of the CPB, described by $H_{\text{SCPT}}$ [Eq. (3)]. The resonance positions oscillate as a function of $Q_0$ with the period of $2e$ due to the band structure of the CPB eigenstates. Since $Q_0^{\prime }=Q_0-C_{2}V$ (the quasicharge in the CPB Hamiltonian), the change in $V$ also leads to a change in $Q_0^{\prime }$ and therefore the band structure is skewed when plotted as a function of $Q_0$. The parameters are $E_{J1}=5E_C=12.5E_{J2}=\Delta /2.2=100\mu \text{eV}$, $C_2/C_1=0.21$, $T=50\text{mK}$, and $R=50\Omega$.

Figure 3

(Color online) A two-dimensional map of the current calculated using the DMA and the same parameters as in Fig. 2. A contour line for the current 10 pA is drawn. As compared to the results obtained by the CTM (Fig. 2), the characteristics are similar but differ for the weak higher-order resonances. Especially, the higher-order resonances between the first and the third bands are barely visible. The first-order resonances are approximately of equal strength in both models.

Figure 4

(Color online) The effect of the quantum noise according to the DMA. A first-order resonance between the ground and the first excited state (which is a level since the bandwidth is very small) is located at $V\sim 99\mu \text{V}$ and a second-order resonance between the ground and the second excited state at $V\sim 94.5\mu \text{V}$. With increasing $R$, the resonant transport first increases $\propto R$ (coherent regime) and finally starts to decrease $\propto 1/R$ (incoherent regime). The width of the resonance in the incoherent regime is determined by the lifetime of the CPB excited state (Ref. 27) and in the coherent regime by the splitting of the SCPT eigenstates. The second-order transition has much smaller splitting and therefore it is more easily washed out by the quantum noise of the EE. The parameters are $E_{J1}=48.4E_C=500\mu \text{eV}$, $E_{J2}=3\mu \text{eV}$, $C_2/C_1=0.1$, and $T=0$. No quasiparticles are included as their effect is small for $\Delta >200\mu \text{eV}$.

Figure 5

(Color online) The effect of thermal noise on the first- and second-order $I\left(V\right)$ resonances of Fig. 4 with a fixed $R=50\Omega$. The evolution in the first-order resonance is mainly coherent (linear regime in Fig. 4) and the effect of thermal noise is fairly small (the area of the peak increases). Qualitatively this is because the bottleneck of the current is the slow relaxation of the populated excited state. The result is similar as obtained using the CTM (not plotted). However, the tunneling in the second-order resonance is partially incoherent and the increase in temperature tends to broaden (with a constant area) and finally flush the resonance. Qualitatively this is because the bottleneck of the current is the excitation of the CPB which is strongly perturbed by thermal fluctuations of $V$. In the CTM the second-order resonance widens but does not substantially lose its height (the area increases) as the temperature is increased.

Figure 6

(Color online) The current nearby two higher-order resonances of Fig. 3 for $Q_0=0$ while changing $E_{J2}$ for three different environments. The solid lines correspond to $E_{J2}=5\mu \text{eV}$, the dashed lines to $E_{J2}=10\mu \text{eV}$, and the dotted lines to $E_{J2}=15\mu \text{eV}$. The calculation for the CF (and no EE) is made using an effective resistance $R_{\text{CF}}={\left(C_2/C_{\Sigma }\right)}^{2}50\Omega$ and the calculation for the EE (and no CF) with $R=50\Omega$, $T=50\text{mK}$, but also with $T=0$, i.e., when no thermal noise is present. The zero-temperature currents for the EE and CF are similar except that the latter has no nonresonant “background” current $\propto E_{J2}^{2}R/V$. The thermal noise of EE with $T=50\text{mK}$ broadens the resonances and flushes them in the case of small $E_{J2}$.

Figure 7

(Color online) A two-dimensional map of the current calculated using the DMA and the same parameters as in Fig. 2 except $\Delta =150\mu \text{eV}$. A contour line for the current 10 pA is drawn. The resonances below $V\sim 80\mu \text{V}$ maintain $2e$ periodicity as the quasiparticle tunneling is very small at these voltages. The resonances above start to show $e$ periodicity as the decay process releases enough energy for a Cooper pair to break and the quasiparticle to tunnel across the probe. The arrows point to the double-resonance points of the third band (see text). All quasiparticle-tunneling thresholds have been broadened by the factor ${\Gamma }_{2e}$ (Sec. 3C2).

Figure 8

(Color online) Some Cooper-pair tunneling resonances (solid lines), quasiparticle-tunneling thresholds (dashed lines), and the odd-parity-state decay thresholds (dotted lines) in a symmetric SCPT with $E_J<E_C$. The quasiparticle thresholds are calculated assuming $\Delta =2E_c$.

Figure 9

(Color online) A two-dimensional map of the current across the symmetric SCPT according to the DMA. Some of the resonances and thresholds are identified in Fig. 8. In the white regions the current exceeds the plotting range. All quasiparticle-tunneling thresholds have been broadened by the factor ${\Gamma }_{2e}$. The parameters are $E_C=3.3E_J=\Delta /2=100\mu \text{eV}$, $R=50\Omega$, $T=50\text{mK}$, and $R_{\text{CF}}=3\Omega$.

Figure 10

(Color online) The second-order irreducible diagrams describing interaction with the environment. The dashed lines represent the mirror diagrams. Their contribution is the complex conjugate of the original ones with switched $a\leftrightarrow b$ and $m\leftrightarrow n$ (a mirror rule). The arrows point to the directions of time in the two branches.