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The Single Cooper Pair Box - a Quantum-Mechanical Electronic Circuit

David Gunnarsson

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

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.

Related papers

Quantum Coherence in the Single Cooper Pair Box

Kevin Bladh

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.

•Jan 1, 2005•3 citations

The single Cooper-pair box as a charge qubit

Kevin Bladh, T. Duty, David Gunnarsson +1 more

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.

New Journal of Physics•Aug 26, 2005•37 citations

Towards Single Shot Measurements of a Cooper Pair Box Qubit Using an RF Single Electron Transistor

Carlos Alberto Sánchez

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.

•Dec 5, 2005•0 citations

Free evolution of superposition states in a single Cooper pair box

Alexandre Guillaume, Justin F. Schneiderman, Per Delsing +2 more

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 $2e$ periodic charge quantization. We performed spectroscopy measurements to extract the charging energy ${(E}_{C})$ and the Josephson coupling energy ${(E}_{J}).$ 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.

arXiv:cond-mat/0312544•Physical Review B•Apr 21, 2004•30 citations

Coherent dynamics of a Josephson charge qubit

T. Duty, David Gunnarsson, Kevin Bladh +1 more

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 $2e$ 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 ${E}_{J}{/E}_{C}$ ratio and finite rise time of the dc pulses. In addition, we are able to demonstrate nearly $100%$ initial charge state polarization. We also present a method to determine the relaxation time ${T}_{1}$ when it is shorter than the measurement time ${T}_{\mathrm{meas}}.$

arXiv:cond-mat/0305433•Physical Review B•Apr 5, 2004•178 citations

Fast dynamics and measurements of single-charge devices

Fredrik Persson

In this thesis we present measurements on several different devices, including the single-Cooper-pair box (SCB), the single-electron box, tunable superconducting resonators and so called transmon qubits.By exploiting the quantum capacitance of the SCB, its state was measured by embedding the box in a radio-frequency resonator. The state dependent change in the quantum capacitance then leads to a measurable shift in the resonator’s frequency. On-chip, lumped-element resonators, optimized for speed, were fabricated together with the SCB, which allowed for very fast measurements of the state. We have performed pulsed measurements, and were able to show that the state of the SCB can be measured in approximately 50 ns.The normal state of the box was also measured. We observed a gate dependent absorption in the single-electron box, centered around its degeneracy points, when it was driven by an radio-frequency signal. The internal dynamics of the system was analyzed by solving a two-level master equation. We model the dissipation in the box by defining an effective resistance, which we name the “Sisyphus resistance”. With the developed model, we were able tovery accurately reproduce the measured response over a wide range of RF amplitudes and temperatures.Measurements on a strongly driven SCB are presented. The response of the driven SCB is explained in terms of longitudinal dressed states, defining an effective two-level system. The coherence times of the dressed two-level system are extracted. Relaxation and dephasing in the dressed basis was analyzed.We analyze the possibility of using tunable superconducting resonators for coupling qubits. The resonance frequency is changed by modifying the inductance of a superconducting quantum interference device incorporated into the resonator. The tunability and tuning speed are investigated.We also present measurements on transmon qubits coupled to the tunable cavity. We show Rabi oscillations and observe coherent interaction between two coupled transmon qubits.

•Jan 1, 2010•0 citations
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.
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 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.
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.
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 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​.
In this thesis we present measurements on several different devices, including the single-Cooper-pair box (SCB), the single-electron box, tunable superconducting resonators and so called transmon qubits.By exploiting the quantum capacitance of the SCB, its state was measured by embedding the box in a radio-frequency resonator. The state dependent change in the quantum capacitance then leads to a measurable shift in the resonator’s frequency. On-chip, lumped-element resonators, optimized for speed, were fabricated together with the SCB, which allowed for very fast measurements of the state. We have performed pulsed measurements, and were able to show that the state of the SCB can be measured in approximately 50 ns.The normal state of the box was also measured. We observed a gate dependent absorption in the single-electron box, centered around its degeneracy points, when it was driven by an radio-frequency signal. The internal dynamics of the system was analyzed by solving a two-level master equation. We model the dissipation in the box by defining an effective resistance, which we name the “Sisyphus resistance”. With the developed model, we were able tovery accurately reproduce the measured response over a wide range of RF amplitudes and temperatures.Measurements on a strongly driven SCB are presented. The response of the driven SCB is explained in terms of longitudinal dressed states, defining an effective two-level system. The coherence times of the dressed two-level system are extracted. Relaxation and dephasing in the dressed basis was analyzed.We analyze the possibility of using tunable superconducting resonators for coupling qubits. The resonance frequency is changed by modifying the inductance of a superconducting quantum interference device incorporated into the resonator. The tunability and tuning speed are investigated.We also present measurements on transmon qubits coupled to the tunable cavity. We show Rabi oscillations and observe coherent interaction between two coupled transmon qubits.