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Physical Review B

Tunability of a 2e periodic single Cooper pair box

David Gunnarsson, T. Duty, Kevin Bladh, Per Delsing

Published Dec 30, 2004• 15 citations
Publisher

Abstract

We have measured the fully $2e$ periodic Coulomb staircase of a single Cooper pair box (SCB) in superconducting quantum interference design geometry, using a radio-frequency single-electron transistor. We have determined the energies of the SCB with microwave spectroscopy and compared the calculated shape of the Coulomb staircases to the measured staircases. We find excellent agreement as the Josephson coupling energy is tuned by an external magnetic field.

DOI
10.1103/physrevb.70.224523

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We have measured the fully 2e2e2e periodic Coulomb staircase of a single Cooper pair box (SCB) in superconducting quantum interference design geometry, using a radio-frequency single-electron transistor. We have determined the energies of the SCB with microwave spectroscopy and compared the calculated shape of the Coulomb staircases to the measured staircases. We find excellent agreement as the Josephson coupling energy is tuned by an external magnetic field.
We present a series of measurements on nine single Cooper-pair boxes (SCBs), where the charging energy, E C , and the Josephson coupling energy, E J , have been varied. We have investigated both the ground state properties of the SCBs and their quantum coherent properties. The state of the SCBs could be manipulated by an external gate voltage and the charge was measured by coupling it capacitatively to a radio-frequency single-electron-transistor (RF-SET). By ramping the gate voltage and simultaneously measuring the charge of the SCBs using the RF-SET, we could measure the Coulomb staircases of the SCBs. For sufficiently low E C the SCBs showed a fully 2 e periodic Coulomb staircase. For samples with higher E C the staircase showed a short step for odd number of charges indicating quasi-particle 'poisoning'. However, if E C was not too large, the short step could be removed by applying a parallel magnetic field. We attribute this effect to a stronger suppression of the superconducting energy gap in the reservoir than in the box. Using microwave spectroscopy we have determined E C and E J for the SCBs. These values agree well with the shape of the Coulomb staircases which we measure. For a limited range of gate voltage, the SCBs were found to behave as model two-level quantum-mechanical systems. A non-adiabatic change in the induced island charge was used to bring two charge states into resonance. The resulting time evolution showed clear charge oscillations between the ground and excited state, with an amplitude above 70% and a frequency given by the energy level separation divided by Planck's constant. These oscillations had a longest coherence time of T 2 = 9 ns, at a point where the pure charge states are degenerate. The coherence time at this point was found to be limited by the relaxation rate. Away from the charge degeneracy point, the coherence time was limited by the pure dephasing rate. The dependence of T 2 on gate charge suggested that low frequency fluctuators were the main source of dephasing away from the degeneracy point.
In this thesis the single Cooper pair box (SCB) is investigated with a radio frequency single electron transistor (RF-SET). The RF-SET is used as an electrometer to detect the charge of the SCB. The SCB can be regarded as a quantum two-level system and could therefore be a possible qubit. Both the SCB and the RF-SET have been fabricated with e-beam lithography followed by two-angle shadow evaporation of aluminium. The system has been measured in a dilution refrigerator at temperatures of ~30 mK. A charge sensitivity of 2.3 μe/square root of Hz was measured for an optimized RF-SET with a bandwidth of 7 MHz, operated in the superconducting state. In a phase sensitive measurement of the RF-SET reflected signal, the gate dependence of the quantum capacitance, due to avoided level crossing, was detected in a superconducting SET. Coulomb staircases for the SCB were obtained by changing the charge of the SCB by ramping a gate voltage and continuously measuring the charge with the RF-SET. To prevent quasiparticle poisoning of the SCB island, we have used a method of graded gaps. Here we exploit the greater reduction of the superconducting gap for the connecting leads with an external magnetic field applied, resulting in a higher gap for the island. This creates an effective barrier against quasiparticle tunnelling. The charging energy and the Josephson coupling energy of the SCB were measured with microwave spectroscopy. The SCBs were fabricated in a SQUID geometry, which made it possible to tune the Josephson coupling energy of the box. This tuning was verified through microwave spectroscopy. The effect of tuning could also be seen in the pure 2e Coulomb staircase, as a predicted smearing of the staircase, which was in excellent agreement with the independent spectroscopy data. Coherent quantum oscillation of the SCB charge state was measured by applying nonadiabatic DC-pulses to the SCB gate. The measured amplitude of the charge oscillations was higher than 70%. The deviation from 100% can be explained by the initial charge state purity and the finite rise time of the pulses. The longest decoherence time of the charge oscillations was T2 = 9 ns at the charge degeneracy point. It was found to be limited by relaxation. Away from the degeneracy point the decoherence time was limited by pure dephasing, due to low frequency charge fluctuators.
We have fabricated a single Cooper-pair box (SCB) in close proximity to a single electron transistor (SET) operated in the radio-frequency mode (RF-SET) with an inductor and capacitor lithographed directly on chip. The RF-SET was used to measure the charge state of the SCB revealing a 2e2e2e periodic charge quantization. We performed spectroscopy measurements to extract the charging energy (EC){(E}_{C})(EC​) and the Josephson coupling energy (EJ).{(E}_{J}).(EJ​). Control of the temporal evolution of the quantum charge state was achieved by applying fast dc pulses to the SCB gate. The dephasing and relaxation times were extracted from these measurements.
We investigate the backaction of superconducting single-electron transistor (SSET) continuously measuring a Cooper-pair box. Due to the minimized backaction of the SSET, we observe a 2e periodic Coulomb staircase according to the two-level system Hamiltonian of the Cooper-pair box. We demonstrate that we can control the quantum broadening of the ground state in-situ. We perform spectroscopy measurements and demonstrate that we have full control over the Cooper-pair box Hamiltonian. The ability to reduce the backaction is a necessary condition to use the SSET as a quantum state readout for the CPB as a qubit.
We have fabricated a Josephson charge qubit by capacitively coupling a single-Cooper-pair box (SCB) to an electrometer based upon a single-electron transistor (SET) configured for radio-frequency readout (rf-SET). Charge quantization of 2e2e2e is observed and microwave spectroscopy is used to extract the Josephson and charging energies of the box. We perform coherent manipulation of the SCB by using very fast dc pulses and observe quantum oscillations in time of the charge that persist to $\ensuremath{\simeq}10\mathrm{ns}.$ The observed contrast of the oscillations is high and agrees with that expected from the finite EJ/EC{E}_{J}{/E}_{C}EJ​/EC​ ratio and finite rise time of the dc pulses. In addition, we are able to demonstrate nearly 100100%100 initial charge state polarization. We also present a method to determine the relaxation time T1{T}_{1}T1​ when it is shorter than the measurement time Tmeas.{T}_{\mathrm{meas}}.Tmeas​.
I report an investigation into using an rf Single Electron Transistor (rf-SET) to implement single shot readout of the quantum state of a Cooper pair box. I fabricated Al/AlOx/Al SETs and Cooper pair boxes using e-beam lithography and double-angle evaporation. The devices had typical charging energies of about 1 K and were measured at 30 mK in a dilution refrigerator. I built rf-SETs with an LC resonant circuit integrated on the chip and operating frequencies of 650 MHz and 1.5 GHz. The rf-SETs had a typical bandwidth of 30 MHz and the best achieved charge sensitivity was 8X10-6 erms/(Hz)1/2 at 1.5 GHz. The rf-SETs were used to measure the Coulomb staircase on the Cooper pair box. Measurements of the staircase show features due to quasiparticle poisoning that are dependent on temperature and SET bias. The data was compared to a theory I developed to extend existing theories of quasiparticle poisoning. My model is based on the calculation of tunneling rates to explain observations of parity effects in the Cooper pair box over a wide temperature range. Quantum state control of the Cooper pair box was demonstrated by using fast voltage pulses applied to the gate. Temporal oscillations of charge states and spectroscopy measurements reveal a dephasing time T2=200 ps at 30 mK, for a drive frequency of 30 GHz. The energy relaxation time T1=140 ns was determined from time domain measurements of the decay of the Cooper pair box charge. The fidelity of single shot measurements and the feasibility of this quantum readout scheme based on the results are discussed.