Quantized charge pumping in superconducting double barrier structure : Non-trivial correlations due to proximity effect

We consider quantum charge pumping of electrons across a superconducting double barrier structure in the adiabatic limit. The superconducting barriers are assumed to be reflectionless so that an incident electron on the barrier can either tunnel through it or Andreev reflect from it. In this structure, quantum charge pumping can be achieved (a) by modulating the amplitudes, ∆1 and ∆2 of the two superconducting gaps or alternatively, (b) by a periodic modulation of the order parameter phases, φ1 and φ2 of the superconducting barriers. In the former case, we show that the superconducting gap gives rise to a very sharp resonance in the transmission resulting in quantization of pumped charge, when the pumping contour encloses the resonance. On the other hand, we find that quantization is hard to achieve in the latter case. We show that inclusion of weak electron-electron interaction in the quantum wire leads to renormalisation group evolution of the transmission amplitude towards the perfectly transmitting limit due to proximity effects. Hence as we approach the zero temperature limit, we get destruction of quantized pumped charge. This is in sharp contrast to the case of charge pumping in a double barrier in a Luttinger liquid where quantized charge pumping is actually achieved in the zero temperature limit. We also propose an experimental set-up to study these effects. Introduction. – The phenomena of quantum charge pumping corresponds to a net flow of DC current between different electron reservoirs (at zero bias) connected via a quantum system whose (at least two) system parameters are periodically modulated in time [1]. The zero bias current is obtained in response to the time variation of the parameters of the quantum system which explicitly break the time reversal symmetry. When the frequency of modulation is smaller than any characteristic scale associated with the electron, adiabatic limit is reached. In this limit, the pumped charge in a unit cycle becomes independent of the pumping frequency. This is referred to as “adiabatic charge pumping”. It is worth mentioning that breaking of the time-reversal symmetry is necessary but not a sufficient condition in order to get a net pumped charge in a unit cycle. For obtaining a net pumped charge, parity must also be broken. In this letter, we explore possibilities of obtaining quantized pumped charge in unit of e for the double superconducting barrier system in onedimensional (1–D) quantum wire (QW). We show that electron−electron (e − e) interactions in the wire lead to destruction of the quantized pumped charge as one goes to low temperatures. This is in sharp contrast to the case of normal double barrier system in a Luttinger liquid [2–4]. Pumping of free electrons across 1–D quantum well and multi-wire junction was studied earlier in Ref. [5–7], where using Brouwer’s formula [8], it was shown that the pumped charge can be expressed as a sum of two contributions, viz., a dissipative part and a quantized topological part, the latter being independent of the details of the pumping contour [9]. The dissipative part was found to be proportional to the conductance through the system on the pumping contour while the topological part was non-zero only if the pumping contour enclosed a resonance. Hence in order to obtain quantized pumped charge, one needs to reduce the dissipative part as much as possible. This is very naturally achieved if one considers pumping through a quantum well in a 1–D interacting electron gas [3] (Lut-