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arXiv:cond-mat/0305190 · Physical Review Letters

Effect of Measurement on the Periodicity of the Coulomb Staircase of a Superconducting Box

Jaan Männik, James E. Lukens

Published Feb 5, 2004• 44 citations
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Abstract

We report on the effect of the backaction of a single Cooper pair transistor electrometer ($\mathbf{E}$) on the charge state of a superconducting box ($\mathbf{B}$). The charge is $e$ periodic in the gate bias of $\mathbf{B}$ when $\mathbf{E}$ is operated near voltages $2\ensuremath{\Delta}/e$ or $4\ensuremath{\Delta}/e$. We show that this is due to quasiparticle poisoning of $\mathbf{B}$ at a rate proportional to the number of quasiparticle tunneling events in $\mathbf{E}$ per second. We are able to eliminate this backaction and recover $2e$-charge periodicity using a new measurement method based on switching-current modulation of $\mathbf{E}$.

DOI
10.1103/physrevlett.92.057004

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•Jan 1, 2005•0 citations
We report on the effect of the backaction of a single Cooper pair transistor electrometer (E\mathbf{E}E) on the charge state of a superconducting box (B\mathbf{B}B). The charge is eee periodic in the gate bias of B\mathbf{B}B when E\mathbf{E}E is operated near voltages 2\ensuremathΔ/e2\ensuremath{\Delta}/e2\ensuremathΔ/e or 4\ensuremathΔ/e4\ensuremath{\Delta}/e4\ensuremathΔ/e. We show that this is due to quasiparticle poisoning of B\mathbf{B}B at a rate proportional to the number of quasiparticle tunneling events in E\mathbf{E}E per second. We are able to eliminate this backaction and recover 2e2e2e-charge periodicity using a new measurement method based on switching-current modulation of E\mathbf{E}E.
The Cooper-pair tunneling in superconducting charge boxes in tunable electromagnetic environments is experimentally studied using a single-electron-transistor electrometer and via current-voltage characteristics measurement with small excitations. In the low-impedance regime, the environmental effect is quantified with an effective Josephson coupling energy, which is found to decrease as the environment impedance increases. In the high-impedance regime, the Cooper-pair tunneling is blockaded. For impedance above $250\phantom{\rule{0.3em}{0ex}}\mathrm{M}\ensuremath{\Omega}$, Coulomb oscillation of the zero-bias conductance suggests that Cooper-pair tunneling is completely suppressed and subgap quasiparticle tunneling becomes dominant.
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.
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 report an experimental observation of the backaction of a single-electron transistor (SET) measuring the Coulomb staircase of a single-electron box. As current flows through the SET, the charge state of the SET island fluctuates. These fluctuations capacitively couple to the box and cause changes in the position, width, and asymmetry of the Coulomb staircase. A sequential tunneling model accurately recreates these effects, confirming this mechanism of the backaction of a SET. This is a first step toward understanding the effects of quantum measurement on solid-state qubits.
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.