Current-voltage characteristics of Josephson transistors interacting with dissipative environment

The quantum-mechanical effects originating in coherent tunneling of Cooper pairs (charge carriers in the superconducting state) in Josephson junctions (two superconductors separated by a thin insulator) have been investigated actively in recent years due to their great potential to be used in nanotechnological applications in the future [1]. A central obstacle has been that the effects are easily decohered by uncontrolled coupling between the studied system and its nearby environment. However, due to their high sensitivity to environmental fluctuations, the small Josephson junctions (JJs) can as well be used as probes of physics in mesoscopic low temperature devices.

Motivated by this, fundamental questions in dissipative quantum mechanics, and by recent experiments we study characteristics of superconducting devices whose operation is based on controlling single Cooper-pair tunneling through mesoscopic JJs. The devices interact with their nearby environment, mainly consisting of an electromagnetic environment formed my the leads attached to the device, of spurious charge fluctuators in nearby insulating materials, and of quasiparticles (unpaired electrons). They all can have drastic effect on the characteristics of the device, or the device can be based on their contribution. We study the interplay between coherent Cooper-pair tunneling and incoherent environmental processes. The analysis is based on theoretical methods typical for dissipative quantum mechanics.





Asymmetric SCPT. The Josephson junctions are drawn as crossed boxes. The electromagnetic environment is characterized by the impedance Z. Also is visuasized the possible presence of spurious charge fluctuators in the insulators surrounding the superconducting island.

A studied system is a Cooper-pair box (CPB) probed by a small Josephson junction (JJ) [6]. The circuit consists of two mesoscopic JJs (see the figure above) in series with a voltage source, and a gate lead connected capacitively to the island between the JJs. If the dimensions of the system are small enough (mesoscopic), single-tunneling events of Cooper pairs can be controlled, even the island possesses macroscopic number of particles. This is since the electrostatic energy of one extra particle in the island is noticeable. The smaller JJ is probing the excited states of the CPB (formed by the larger JJ and the island) since the measured I-V characteristics contain information of its energy-level/band structure. Experimentally these can be tuned in situ by applying magnetic field to the system (see the figure below).





Current through a small JJ attached to a larger JJ measured in LTL [6]. The energy-levels of the CPB are seen as current peaks. In this experiment two large parallel JJs work effectively as the single tunable (larger) JJ. Therefore the magnetic field (more precicely the magnetic flux through the loop) can be used to tune the energy levels of the CPB.

For the theoretical calculation of the Cooper-pair current through the smaller JJ, the standard theory of incoherent Cooper-pair tunneling [2] cannot be applied. This is since the Hamiltonian is anharmonic and can possess a band structure (the presence of the CPB), similar to that electrons have in periodic potentials. In Ref. [6] we show that the anharmonicity and the band structure can be taken into account by calculating the current-voltage characteristics by a more direct method, whereas the characteristics due to the electromagnetic environment of small impedance, except perhaps nearby few resonant frequencies, are included by the standard theory and by additional LC oscillators. We compare our numerically calculated I-V curves with experiments and show that the main features of the observed I-V data are reproduced. Especially, we find traces of band structure in the higher excited-states of the CPB as well as traces of multiphoton processes between the CPB and resonances in Z. Also simultaneous excitation of two CPBs are observed when the small JJ is surrounded by two CPBs (see the figure below).





Theoretical (solid lines) and experimental (dots) positions of the current peaks due to energy-level structure of the coupled CPBs. As the flux is increased, the energy levels broaden to energy bands. Two resonant modes in the transmission line are also identified.

The preceding circuit can also be seen as an asymmetric single-Cooper-pair transistor (SCPT). The theory of resonant Cooper-pair tunneling in SCPT [3] can then be applied. We have done this in Ref. [7] where we show that there exists possibilities for new (so called higher-order) resonances in the circuit. These were not included into the theory in Ref. [6]. However, the results are not consistent with experiments from which these resonances seem to be missing. The same discrepancy is present in the case of symmetric SCPT; this theory results in very cumbersome subgap characteristics not consistent with the experiments.

Motivated by this, we have constructed a model which describes the effect of the dissipative environment to the resonant Cooper-pair-tunneling more accurately [8]. The model is based on treating the environmental fluctuations as independent on the evolution of the system under study leading to a reduced master equation for the state of the transistor, describing its irreversible quantum dynamics. This approach produces both of the previous studies as the limiting cases of the dissipation strengths. Also all the three mentioned environments are considered, inspired by recent experimental observations [4]. By numerical simulations we demonstrate that most of the higher-order effects (in Cooper-pair tunneling) tend to be washed out by strong decoherence supplied by the environment. In order to see rich subgap characteristics, the relevant noise sources causing the wash out should be filtered more carefully. This explains why only few of them have been detected in the experiments. We give theoretical relations for preventing the wash out.





Numerical results for the current through symmetric SCPT interacting with typical environment. Q0 is proportional to the gate voltage. Only single higher-order resonant Cooper-pair tunneling process is seen (the resonant lines). The other structure is determined by simultaneous quasiparticle and Cooper-pair tunneling processes.

Partly ongoing study considers the dynamics of the Bloch oscillating transistor (BOT) [5]. The transistor is designed to operate via the interplay of coherent Cooper-pair tunneling across a JJ (emitter), continuous feed current through a large resistor (collector) and quasiparticle tunneling across another JJ or normal metal tunnel junction (base). We have constructed a model for this system in which the Cooper-pair tunneling is not treated treated perturbatively [9]. The model takes into account the environmental contribution similarly as in Ref. [8], leading to a master equation for the state of the BOT. In Ref. [9] we demonstrate, in the spirit of Ref. [6], that the base current can as well be used as the detector of the BOT dynamics. Our next step is to study how does the experimental data, measured in LTL, fit into the picture given by this model.





Simulated current through a base junction as functions of base voltage and collector voltage (proportional to I). Different regimes of the system can be identified from the current distribution.

References

  1. Yu. Makhlin, G. Schön, and A. Shnirman, Quantum-state engineering with Josephson-junction devices, Rev. Mod. Phys. 73, 357 (2001).

  2. G.-L. Ingold and Yu. V. Nazarov, Charge Tunneling Rates in Ultrasmall Junctions, cond-mat/0508728.

  3. A. Maassen van den Brink, A. A. Odintsov, P. A. Bobbert, and G. Schön, Coherent Cooper pair tunneling in systems of Josephson junctions: effects of quasiparticle tunneling and of the electromagnetic environment, Z. Phys. B 85, 459 (1991).

  4. O. Astafiev, Yu. A. Pashkin, Y. Nakamura, T. Yamamoto, and J. S. Tsai, Quantum Noise in the Josephson Charge Qubit, Phys. Rev. Lett. 93, 267007 (2004).

  5. J. Delahaye, J. Hassel, R. Lindell, M. Sillanpää, M. Paalanen, H. Seppä, and P. Hakonen, Low-Noise Current Amplifier Based on Mesoscopic Josephson Junction, Science 299 1045 (2003).

Publications

  1. J. Leppäkangas, E. Thuneberg, R. Lindell, and P. Hakonen, Tunneling of Cooper pairs across asymmetric single-Cooper-pair transistors, Phys. Rev. B 74, 054504 (2006).

  2. J. Leppäkangas and E. Thuneberg, Coherent tunneling of Cooper pairs in asymmetric single-Cooper-pair transistors, AIP Conference Proceedings 850, 947-948 (2006). Preprint.

  3. J. Leppäkangas and E. Thuneberg, Effect of decoherence on resonant Cooper-pair tunneling in a voltage-biased single-Cooper-pair transistor, Phys. Rev. B 78 144518 (2008).

  4. J. Leppäkangas and E. Thuneberg, Energy-band dynamics in a current-biased Josephson junction probed by incoherent Cooper-pair tunneling, to be published in the proceedings of LT-25, preprint.

  5. J. Leppäkangas, Spectroscopy of a many Josephson junction system using inelastic Cooper-pair tunneling, M. Sc. Thesis.

  6. J. Leppäkangas, Josephson transistors interacting with dissipative environment, PhD Thesis.


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4.11.2008, Juha Leppäkangas