Coulomb blockade due to quantum phase slips illustrated with devices

To illustrate the emergence of Coulomb blockade from coherent quantum phase-slip processes in thin superconducting wires, we propose and theoretically investigate two elementary setups, or devices. The setups are derived from the Cooper-pair box and Cooper-pair transistor, so we refer to them as the QPS box and QPS transistor, respectively. We demonstrate that the devices exhibit sensitivity to a charge induced by a gate electrode, this being the main signature of Coulomb blockade. Experimental realization of these devices will unambiguously prove the Coulomb blockade as an effect of coherence of phase-slip processes. We analyze the emergence of discrete charging in the limit of strong phase slips. We have found and investigated six distinct regimes that are realized depending on the relation between three characteristic energy scales: inductive energy, charging energy, and phase-slip amplitude. For completeness, we include a brief discussion of dual Josephson-junction devices.

Figure 1

(a) Cooper-pair box(CPB): generic Coulomb-blockade superconducting device. (b) In quantum phase-slip box (QPS box), the tunnel junction is replaced with a superconducting wire.

Figure 2

Energy spectrum of CPB vs induced charge $q$ in the limit of large charging energy. The discrete charging states give rise to a standard $2e$-periodic pattern of crossing parabolas. The spectrum of phase-slip devices considered in this paper follows the same pattern in the limit of large phase-slip amplitudes.

Figure 3

(a) Cooper-pair transistor (CPT). The dependence of its energy on external phase $\Phi$ (flux sensitivity) gives rise to the supercurrent that can be modulated by changing $q=C_g{V}_g$. (b) Quantum phase-slip transistor.

Figure 4

The magnitude of the charge-sensitive first-order correction [Eq. (13)] to the first four energy levels $n=0,\dots ,4$ vs ${\gamma }^2$.

Figure 5

QPS box: Small impedance regime. Bloch states: The energies vs quasiphase $\chi$. (a) Strong PS: $E_S=6.25E_L$. (b) Moderate PS: $E_S=2.5E_L$. (c) Weak PS: $E_S=0.25E_L$.

Figure 6

The renormalized inductive energy $E_L^{*}$ vs the phase-slip amplitude $E_S$. It becomes exponentially suppressed at $E_S\gg E_L$.

Figure 7

Small impedance regime ($\gamma =1.91$): The excitation energies vs $E_S/E_L$ for two values of the induced charge of (a) $q/e=0$ and (b) $q/e=1$. The sticking together of the excitation energies at sufficiently large $E_S$ indicates emergence of the charging states.

Figure 8

The charge sensitivity in the small impedance regime ($\gamma =1.91$) for (a) $E_S/E_L=5.48$, (b) $E_S/E_L=8.22$, and (c) $E_S/E_L=10.96$.

Figure 9

Excitation energies in the large impedance regime ($\gamma =0.32$) for (a) $q/e=0$ and (b) $q/e=1$.

Figure 10

The charge sensitivity in the large impedance regime ($\gamma =0.32$) for (a) $E_S/E_C=0.02$, (b) $E_S/E_C=0.1$, and (c) $E_S/E_C=0.3$.

Figure 11

The evolution of $E(Q)$ at $q/e=1$ upon increasing the phase-slip amplitude for (a) $E_S/E_C=0$, (b) $E_S/E_C=0.5$, and (c) $E_S/E_C=10$. The multiple potential minima formed correspond to discrete charging states.

Figure 12

Excitation energies in the intermediate impedance regime ($\gamma =1$) for (a) $q/e=0$ and (b) $q/e=1$.

Figure 13

The charge sensitivity at the intermediate impedance ($\gamma =1$) for (a) $E_S/\hslash {\omega }_0=1.0$, (b) $E_S/\hslash {\omega }_0=2.0$, and (c) $E_S/\hslash {\omega }_0=4.0$.

Figure 14

Distinct regimes in the QPS box. The lines in this schematic log-log plot present parameter regions in the space of $E_S$ and effective impedance $1/{\gamma }^2$. Region I: Perturbative regime; II: oscillator with renormalized capacitance; III: oscillator with renormalized inductance; IV: discrete charging states accompanied by oscillator excitations; V: direct mapping to CPB; and VI: pure charging states.

Figure 15

QPS transistor: The second-order correction to the ground-state energy {in units of ${\mathrm{Ẽ}}_1=-E_{S,1}^2/[4(E_{L,1}+E_{L,2})]$ and ${\mathrm{Ẽ}}_2=-E_{S,1}{E}_{S,2}/[4(E_{L,1}+E_{L,2})]$}. (a) Classical part of the correction vs external phase difference $\Phi$. From the uppermost curve downward: $\gamma \ll 1$ (asymptote), $\gamma =0.5$, $\gamma =1$, $\gamma =2$, $\gamma =3$, $\gamma \gg 1$ (asymptote). (b) Charge-sensitive (interference) part of the correction versus $\Phi$. From the uppermost curve down: $\gamma \ll 1$ (asymptote), $\gamma =4$, $\gamma =5.65$, and $\gamma =8$. Note the log-scale of the plot.

Figure 16

QPS transistor in the limit of large $E_S$. The positions of the potential minima in extended charge space $(Q_1,Q_2)$. The dotted lines connect equivalent minima, representing the set of points that are the same in this space. Dark dots give unique minima. Tunneling amplitudes through potential barriers [Hamiltonian equation (27)] are indicated by Greek letters.

Figure 17

Ground-state energy of the QPS transistor vs external phase $\Phi$ ($\gamma =1$) for (a) $q/e=0$ and (b) $q/e=1$. In both panels, the phase-slip amplitudes take the values $E_S/\hslash {\omega }_0=$ 0, 0.5, 1, and 2 from the uppermost curve to the lowest one.

Figure 18

Supercurrent in the QPS transistor vs $\Phi$ for (a) $q/e=0$ and (b) $q/e=1$. In both panels, the phase-slip amplitude takes the values $E_S/\hslash {\omega }_0=$ 0, 0.5, 0.75, 1, and 2. $E_S=0$ corresponds to the straight line.

Figure 19

Flux dependence of the lowest three energy levels for a symmetric QPS transistor ($E_{L,1}=E_{L,2};E_{S,1}=E_{S,2}$) for $\gamma =1$, $E_S/\hslash {\omega }_0=0.5$ and (a) $q/e=0$ and (b) $q/e=1$. Note the degeneracies at half-integer flux for $q/e=1$.

Figure 20

Flux sensitivity vs phase-slip amplitude for two values of charge ($\gamma =1$). The crosses represent $q/e=0$ and x symbols represent $q/e=1$.

Figure 21

The Josephson circuit dual to the QPS box.

Figure 22

The Josephson circuit dual to the QPS transistor.