The effect of decoherence on the Josephson current between two superconductors when Cooper pair transport is achieved through the shuttle mechanism can be of great relevance. In this paper we analyze the DC Josephson current in several experimentally relevant situations. We show that decoherence effect, due to gate voltage fluctuations, can either suppress or enhance the critical current and also change its sign. The current noise spectrum displays a peak at Josephson energy and it is phase sensitive. We finally propose a device that has the shuttle properties but requires no mechanically moving part.
The supercurrent of a Josephson junction is reduced by phase diffusion. For ultrasmall capacitance junctions the current may be further decreased by Coulomb blockade effects. We calculate the Cooper pair current by means of time-dependent perturbation theory to all orders in the Josephson coupling energy and obtain the current-voltage characteristic in closed form in a range of parameters of experimental interest. The results comprehend phase diffusion of the coherent Josephson current in the classical regime as well as the supercurrent peak due to incoherent Cooper pair tunneling in the Coulomb blockade regime.
We discuss coherent transport of Cooper pairs through a Cooper pair shuttle. We analyze both the dc and ac Josephson effects in the two limiting cases where the charging energy
EC is either much larger or much smaller than the Josephson coupling
EJ. In the limit
EJ⪡EC we present the detailed behavior of the critical current as a function of the damping rates and the dynamical phases. The ac effect in this regime is very sensitive to all dynamical scales present in the problem. The effect of fluctuations of the external periodic driving is discussed as well. In the opposite regime the system can be mapped onto the quantum kicked rotator, a classically chaotic system. We investigate the transport properties also in this regime, showing that the underlying classical chaotic dynamics emerges as an incoherent transfer of Cooper pairs through the shuttle. For an appropriate choice of the parameters the Cooper pair shuttle can exhibit the phenomenon of dynamical localization. We discuss in detail the properties of the localized regime as a function of the phase difference between the superconducting electrodes and the decoherence due to gate voltage fluctuations. Finally we point out how dynamical localization is reflected in the noise properties of the shuttle.
We analyze how decoherence appears in the $I\text{\ensuremath{-}}V$ 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 $(\text{Re}[Z(\ensuremath{\omega})]⪡{R}_{Q}=h/4{e}^{2})$ 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.
We have studied decoherence in a system where two Josephson-junction flux qubits share a part of their superconducting loops and are inductively coupled. By tuning the flux-bias condition, we control the sensitivities of the energy levels to flux noises in each qubit. The dephasing rate of the first excited state is enhanced or suppressed depending on the amplitudes and the signs of the sensitivities. We have quantified the
1/f flux noises and their correlations and found that the dominant contribution is by local fluctuations.
We compute the decoherence caused by
1∕f fluctuations at low frequency
f in the critical current
I0 of Josephson junctions incorporated into flux, phase, charge, and hybrid flux-charge superconducting quantum bits (qubits). The dephasing time ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ scales as ${I}_{0}∕\ensuremath{\Omega}\ensuremath{\Lambda}{S}_{{I}_{0}}^{1∕2}(1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz})$, where $\ensuremath{\Omega}∕2\ensuremath{\pi}$ is the energy-level splitting frequency,
SI0(1Hz) is the spectral density of the critical-current noise at
1Hz, and $\ensuremath{\Lambda}\ensuremath{\equiv}\ensuremath{\mid}{I}_{0}d\ensuremath{\Omega}∕\ensuremath{\Omega}d{I}_{0}\ensuremath{\mid}$ is a parameter computed for given parameters for each type of qubit that specifies the sensitivity of the level splitting to critical-current fluctuations. Computer simulations show that the envelope of the coherent oscillations of any qubit after time
t scales as $\mathrm{exp}(\ensuremath{-}{t}^{2}∕2{\ensuremath{\tau}}_{\ensuremath{\phi}}^{2})$ when the dephasing due to critical-current noise dominates the dephasing from all sources of dissipation. We compile published results for fluctuations in the critical current of Josephson tunnel junctions fabricated with different technologies and a wide range in
I0 and area
A, and show that their values of
SI0(1Hz) scale to within a factor of 3 of $[144{({I}_{0}∕\ensuremath{\mu}\mathrm{A})}^{2}∕(\mathcal{A}∕\ensuremath{\mu}{\mathrm{m}}^{2})]{(\mathrm{pA})}^{2}∕\mathrm{Hz}$ at
4.2K. We empirically extrapolate
SI01∕2(1Hz) to lower temperatures using a scaling
T(K)∕4.2. Using this result, we find that the predicted values of ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ at
100mK range from $0.8\phantom{\rule{0.5em}{0ex}}\text{to}\phantom{\rule{0.5em}{0ex}}12\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$, and are usually substantially longer than values measured experimentally at lower temperatures.