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arXiv:1111.2332 · Physical Review B

Microwave spectroscopy of a Cooper-pair transistor coupled to a lumped-element resonator

M. T. Bell, L. B. Ioffe, M. E. Gershenson

Published Oct 11, 2012• 15 citations
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Abstract

We have studied the microwave response of a single Cooper-pair transistor (CPT) coupled to a lumped-element microwave resonator. The resonance frequency of this circuit, ${f}_{r}$, was measured as a function of the charge ${n}_{g}$ induced on the CPT island by the gate electrode and the phase difference across the CPT, ${\ensuremath{\varphi}}_{B}$, which was controlled by the magnetic flux in the superconducting loop containing the CPT. The observed ${f}_{r}({n}_{g},{\ensuremath{\varphi}}_{B})$ dependencies reflect the variations of the CPT Josephson inductance with ${n}_{g}$ and ${\ensuremath{\varphi}}_{B}$ as well as the CPT excitation when the microwaves induce transitions between different quantum states of the CPT. The results are in excellent agreement with our simulations based on the numerical diagonalization of the circuit Hamiltonian. This agreement over the whole range of ${n}_{g}$ and ${\ensuremath{\varphi}}_{B}$ is unexpected, because the relevant energies vary widely, from 0.1 to 3 K. The sensitivity of the CPT as an electrometer is peaked when the CPT excitation level approaches that of the microwave resonator.

DOI
10.1103/physrevb.86.144512

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We have studied the microwave response of a single Cooper-pair transistor (CPT) coupled to a lumped-element microwave resonator. The resonance frequency of this circuit, fr{f}_{r}fr​, was measured as a function of the charge ng{n}_{g}ng​ induced on the CPT island by the gate electrode and the phase difference across the CPT, \ensuremathφB{\ensuremath{\varphi}}_{B}\ensuremathφB​, which was controlled by the magnetic flux in the superconducting loop containing the CPT. The observed fr(ng,\ensuremathφB){f}_{r}({n}_{g},{\ensuremath{\varphi}}_{B})fr​(ng​,\ensuremathφB​) dependencies reflect the variations of the CPT Josephson inductance with ng{n}_{g}ng​ and \ensuremathφB{\ensuremath{\varphi}}_{B}\ensuremathφB​ as well as the CPT excitation when the microwaves induce transitions between different quantum states of the CPT. The results are in excellent agreement with our simulations based on the numerical diagonalization of the circuit Hamiltonian. This agreement over the whole range of ng{n}_{g}ng​ and \ensuremathφB{\ensuremath{\varphi}}_{B}\ensuremathφB​ is unexpected, because the relevant energies vary widely, from 0.1 to 3 K. The sensitivity of the CPT as an electrometer is peaked when the CPT excitation level approaches that of the microwave resonator.
The Cooper-pair transistor (CPT), a small superconducting island enclosed between two Josephson weak links, is the atomic building block of various superconducting quantum circuits. Utilizing gate-tunable semiconductor channels as weak links, the energy scale associated with the Josephson tunneling can be changed with respect to the charging energy of the island, tuning the extent of its charge fluctuations. Here, we directly demonstrate this control by mapping the energy level structure of a CPT made of an indium arsenide nanowire with a superconducting aluminum shell. We extract the device parameters based on the exhaustive modeling of the quantum dynamics of the phase-biased nanowire CPT and directly measure the even-odd parity occupation ratio as a function of the device temperature, relevant for superconducting and prospective topological qubits.
We have fabricated a Cooper-pair transistor (CPT) with parameters such that for appropriate voltage biases, it behaves essentially like a single Cooper-pair box (SCB). The effective capacitance of a SCB can be defined as the derivative of the induced charge with respect to gate voltage and has two parts, the geometric capacitance, C(geom), and the quantum capacitance C(Q). The latter is due to the level anticrossing caused by the Josephson coupling and is dual to the Josephson inductance. It depends parametrically on the gate voltage and its magnitude may be substantially larger than C(geom). We have detected C(Q) in our CPT, by measuring the in phase and quadrature rf signal reflected from a resonant circuit in which the CPT is embedded. C(Q) can be used as the basis of a charge qubit readout by placing a Cooper-pair box in such a resonant circuit.
We have investigated the Josephson inductance LJ{L}_{J}LJ​ of a superconducting Cooper pair transistor (SCPT). We traced the inductance using microwave reflection measurements on a tuned resonance circuit in which a SCPT was mounted in parallel to a $\ensuremath{\sim}200\text{ }\text{pH}$ strip line inductance. When the inverse of the Josephson inductance, determined on the charge-phase bias plane for a SCPT with a Josephson to Coulomb energy ratio of EJ/EC=1.75{E}_{J}/{E}_{C}=1.75EJ​/EC​=1.75, is integrated over the phase, we obtain a current-phase relation, which is strongly nonsinusoidal near the charge degeneracy point.
A single-Cooper-pair transistor (SCPT) is coupled capacitively to a voltage biased Josephson junction, used as a high-frequency generator. Thanks to the high energy of photons generated by the Josephson junction, transitions between energy levels, not limited to the first two levels, were induced and the effect of this irradiation on the dc Josephson current of the SCPT was measured. The phase and gate bias dependence of energy levels of the SCPT at high energy is probed. Because the energies of photons can be higher than the superconducting gap we can induce not only transfer of Cooper pairs but also transfer of quasiparticles through the island of the SCPT, thus controlling the poisoning of the SCPT. This can both decrease and increase the average Josephson energy of the SCPT: its supercurrent is then controlled by high-frequency irradiation.
This thesis presents measurements of quantum coherence in a single Cooper pair box, where quantum oscillations of charge between a reservoir and a small superconducting island have been observed. The single Cooper pair box (SCB) consists of a small metallic island which is connected to a reservoir via a tunnel junction. In the superconducting state, Cooper pairs are free to tunnel to and from the island, whose potential can be controlled by a gate voltage. A graded gap method was used to prevent quasiparticles from tunneling onto the island. For this purpose, a magnetic field suppressed the superconducting gap in the SCB leads more than in the island. This created an additional energy barrier for quasiparticles to tunnel to the island, making the gate voltage modulation periodic in Cooper pair charge. For a limited range of gate voltage, the SCB was found to behave as a model two-level quantum mechanical system. 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 a frequency given by the energy level separation divided by Planck's constant. These oscillations had a longest coherence time of T2=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, ?1. Away from this point, the coherence time was limited by the dephasing rate, ??. Away from the charge degeneracy, the dependence of T2 on gate charge suggested that low frequency fluctuators were the main source of dephasing. The charge of the Cooper pair box was measured by coupling the SCB capacitively to a radio frequency single electron transistor (RF-SET). The RF-SET was used as an electrometer by measuring the amount of reflection from a resonant microwave circuit in which a single electron transistor (SET) was embedded. A best charge sensitivity of 2.3·10-6 e/sqrt(Hz) and a 7 MHz bandwidth was reached in an improved setup. Another part of this thesis concerns an investigation of electrical filters for use in single electronics experiments. The characteristics of seven different types of cryogenic filters were evaluated.
We have studied dynamic properties of a special Josephson charge qubit. The device consists of a single-Cooper-pair transistor (with capacitive gate) closed by a superconducting inductive loop. The phase bias is implemented by an external flux that threads the loop. Restricting ourselves to a two-band model, we have derived analytical expressions for the oppositely circulating ring currents in terms of gate charge and phase bias. They permit the determination of important device characteristics such as critical current, loop parameters, and Josephson inductance. For the purpose of measurements in connection with macroscopic quantum coherence, we investigate the qubit's microwave response within the frame of Bloch equations with phenomenological relaxation rates. We find the (nonlinear) qubit impedance in terms of quality, detuning, and the quantum resistance unit, which reflects the competition between irradiation and dephasing.