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Dissipative and dispersive measurements of a Cooper pair box

Zaeill Kim

Published Jan 1, 2010• 2 citations
OpenAlex

Abstract

The quantum states of an Al/AlOx/Al Cooper pair box (CPB) qubit were measured at temperatures below 100 mK. Detailed spectroscopic measurements of the excited state of the CPB were made along with detailed measurements of the lifetime T1 of the first excited state. The CPB states were probed using radio-frequency (rf) techniques to read out using either an rf - single-electron transistor (rf-SET) or a low-loss superconducting resonator. Using an rf-SET, I measured the excited state spectrum of a CPB from 15 to 50 GHz. In this spectrum, a few anomalous avoided level crossings (ALC) were observed. These ALCs exhibited a strong gate voltage dependence and Josephson energy (Ej) dependence, consistent with a charge fluctuator coupled to the CPB island. A model Hamiltonian was used to fit the measured spectrum. Fitting parameters such as the charging energy Ec/h = 12.1 GHz and the Josephson energy Ej/h tuned between 2 GHz and 21 GHz for the CPB, and the well asymmetry, tunneling amplitude, and the minimum hopping distance for each fluctuator were extracted. The tunneling rates ranged from less than 3.5 to 13 GHz, i.e. values between 5 % and 150 % of the well asymmetry, and the dipole moments yield a minimum hopping distance of 0.3 to 0.8 Angstroms. I also made detailed measurements of the lifetime of the first excited state away from the CPB charge degeneracy point and found that the lifetime varied from less than 50 ns up to a few us as the Josephson energy Ej decreased, consistent with a charge noise (Sq~10-11 e2/Hz around 37 GHz to Sq~10-12 e2/Hz around 27 GHz) coupled to the qubit. I also found that at frequencies where an ALC was observed in the spectrum, a decrease in T1 occurred, suggesting that the discrete charge defects are a significant source of dissipation in the CPB. I also designed and fabricated a quasi-lumped element thin-film superconducting Al microwave resonator on sapphire to be used for a dispersive read-out of the CPB. The resonator consists of a meandering inductor and an interdigitated capacitor coupled to a transmission line. At T = 30 mK and on resonance at 5.578 GHz, the transmission through the transmission line decreased by 15 dB and the loaded quality factor was 60,000. I measured the temperature dependence of the resonator frequency and loss at temperatures as high as 500 mK and found reasonable agreement with the Mattis-Bardeen theory. Finally, I coupled a quasi-lumped element microwave resonator (f0~5.443 GHz), made of superconducting Al on sapphire, to an Al/AlOx/Al CPB qubit. Most of my measurements were made in the dispersive regime where Ej-hf0 is much larger than the coupling strength. In this case, the qubit causes a small state-dependent frequency shift in the resonator's resonant frequency. By sending down a second microwave tone (the pump), I was able to excite the CPB qubit. In zero magnetic field with the CPB far detuned from the resonator, I measured a 50 kHz decrease in f0 with the qubit in the ground state and biased near the degeneracy point of the CPB. The charging energy and Josephson energy of the CPB were determined from spectroscopy taken by saturating the CPB with a second microwave tone and measuring the transmission through the resonator. The first device had Ec/h = 12.5 GHz and maximum Ej/h = 9 GHz. The second device had Ec/h = 6.24 GHz and Ej/h tuned between 4 GHz and 8 GHz. By changing the external magnetic field, I could decrease the effective Ej of the CPB. From modeling, I extracted coupling strengths g/2&pi = 11 MHz and 5 MHz for the first and second device, respectively. Finally I did single and two-tone spectroscopy, and measured the relaxation and Rabi oscillations of the CPB. From the first device, I was able to obtain relaxation times T1 of 10.3 us at Ej/h = 7 GHz on the CPB degeneracy point and spectroscopic coherence times T2 *~ 100 ns. From the second device, I found relaxation times T1 of 200 us at Ej/h = 4 GHz to 4.5 GHz decreasing down to 4 us around 8 GHz. There was also a depression in T1 around the resonant frequency of the resonator. The Rabi decay times were found to be up to T'~ 330 ns.

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Decoherence And Defects In Cooper-Pair Boxes

V. Zaretskey

This dissertation describes my detailed investigation of decoherence and defects in two Al/AlOx/Al Cooper-pair box (CPB) charge qubits. Both devices were coupled to thin-film lumped-element superconducting aluminum LC resonators at a temperature of 25 mK. Device 1 was previously found to have an exceptionally long energy relaxation time of T1=205 μs and a strong correlation between the lifetime T1 and the decoupling from the microwave drive line dVg,rms/dΩR,0. I determined the dephasing properties of this CPB though a series of experiments. I measured Ramsey fringes, extracted dephasing times Tφ in the range200-500 ns, and determined a corresponding bound of Sq(f=1 Hz)≤(3×10-3)2 e2/Hz on the amplitude of the 1/f charge noise affecting the qubit. I then carried out a spin echo experiment and found echo decay times Techo in the 2.4-3.3 μs range, implying a high frequency 1/f charge noise cutoff of ωc/2π≈0.2 MHz. I followed this up by fabricating and characterizing a nearly identical Device 2. This CPB had a reasonably long relaxation time T1≈4-30 μs and again the lifetime T1 and decoupling dVg,rms/dΩR,0 were correlated. Although the lifetime of Device 2 was shorter than that of Device 1, the results suggest that the exceptional relaxation time was somewhat reproducible and that this approach may lead to further improvements in qubit coherence. During my initial characterization of Device 2, I discovered that it displayed an anomalously twinned transition spectrum. I studied this feature in detail in parallel with my decoherence experiments. I found that above the resonator resonance at ω/2π=5.472 GHz the system spectrum was twinned but below it was quadrupled. This behavior was consistent with a pair of two-level systems (TLS) coupled non-resonantly to the CPB via both charge and critical current. I developed a model that matched this scenario and successfully fit the predicted spectrum to my data. Both the coherent non-resonant interaction and joint charge and critical current CPB-TLS coupling are novel observations. From the fits I extracted microscopic parameters of the fluctuators including the well asymmetry, tunneling rate, and a minimum hopping distance of 0.2-0.45 Å. I also found a large fractional change of the Josephson energy ΔEJ,k/EJ≈30-40%, consistent with a non-uniform tunnel barrier containing a few dominant conduction channels and a defect that modulates one of them.

•Jan 1, 2013•0 citations

Decoherence in a pair of long-lived Cooper-pair boxes

V. Zaretskey, S. Novikov, B. Suri +3 more

We have investigated the decoherence of quantum states in two Al/AlOx/Al Cooper-pair boxes coupled to lumped element superconducting inductor-capacitor resonators. At 25 mK, the first qubit had an energy relaxation time T1 that varied from 30 μs to 200 μs between 4 and 8 GHz and displayed an inverse correlation between T1 and the coupling to the microwave drive line. The Ramsey fringe decay times T2* were in the [200–500] ns range while the spin echo envelope decay times Techo varied from 2.4–3.3 μs, consistent with 1/f charge noise with a high frequency cutoff of 0.2 MHz. A second Cooper-pair box qubit with similar parameters showed T1=4−30 μs between 4 and 7.3 GHz, and that the T1 and the coupling were again inversely correlated. Although the lifetime of the second device was shorter than that of the first device, the dependence on coupling in both devices suggests that further reduction in coupling should lead to improved qubit performance.

arXiv:1308.1043•Journal of Applied Physics•Sep 4, 2013•6 citations

Anomalous avoided level crossings in a Cooper-pair box spectrum

Z. Kim, V. Zaretskey, Y. Yoon +5 more

We have observed a few distinct anomalous avoided level crossings and voltage-dependent transitions in the excited-state spectrum of an $\text{Al}/{\text{AlO}}_{x}/\text{Al}$ Cooper-pair box (CPB). The device was measured at 40 mK in the 15--50 GHz frequency range. We find that a given level crosses the CPB spectrum at two different gate voltages; the frequency and splitting size of the two crossings differ and the splitting size depends on the Josephson energy of the CPB. We show that this behavior is not only consistent with the CPB being coupled to discrete charged ``two-level'' quantum systems, which move atomic distances in the CPB junctions, but that the spectra provide information about the fluctuators, which is not available from phase qubit spectra of anomalous avoided levels. In particular by fitting a model Hamiltonian to our data, we extract microscopic parameters for each fluctuator, including well asymmetry, tunneling amplitude, and the minimum hopping distance for each fluctuator. The tunneling rates range from less than 3.5--13 GHz, which represent values between 5 and 150% of the well asymmetry, and the dipole moments give a minimum hopping distance of $0.3--0.8\text{ }\text{\AA{}}$. We have also found that these discrete two-level systems have a pronounced effect on the relaxation time $({T}_{1})$ of the quantum states of the CPB and hence can be a source of dissipation for superconducting quantum bits.

arXiv:0810.2455•Physical Review B•Oct 10, 2008•47 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

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

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 quantum states of an Al/AlOx/Al Cooper pair box (CPB) qubit were measured at temperatures below 100 mK. Detailed spectroscopic measurements of the excited state of the CPB were made along with detailed measurements of the lifetime T1 of the first excited state. The CPB states were probed using radio-frequency (rf) techniques to read out using either an rf - single-electron transistor (rf-SET) or a low-loss superconducting resonator. Using an rf-SET, I measured the excited state spectrum of a CPB from 15 to 50 GHz. In this spectrum, a few anomalous avoided level crossings (ALC) were observed. These ALCs exhibited a strong gate voltage dependence and Josephson energy (Ej) dependence, consistent with a charge fluctuator coupled to the CPB island. A model Hamiltonian was used to fit the measured spectrum. Fitting parameters such as the charging energy Ec/h = 12.1 GHz and the Josephson energy Ej/h tuned between 2 GHz and 21 GHz for the CPB, and the well asymmetry, tunneling amplitude, and the minimum hopping distance for each fluctuator were extracted. The tunneling rates ranged from less than 3.5 to 13 GHz, i.e. values between 5 % and 150 % of the well asymmetry, and the dipole moments yield a minimum hopping distance of 0.3 to 0.8 Angstroms. I also made detailed measurements of the lifetime of the first excited state away from the CPB charge degeneracy point and found that the lifetime varied from less than 50 ns up to a few us as the Josephson energy Ej decreased, consistent with a charge noise (Sq10-11 e2/Hz around 37 GHz to Sq10-12 e2/Hz around 27 GHz) coupled to the qubit. I also found that at frequencies where an ALC was observed in the spectrum, a decrease in T1 occurred, suggesting that the discrete charge defects are a significant source of dissipation in the CPB. I also designed and fabricated a quasi-lumped element thin-film superconducting Al microwave resonator on sapphire to be used for a dispersive read-out of the CPB. The resonator consists of a meandering inductor and an interdigitated capacitor coupled to a transmission line. At T = 30 mK and on resonance at 5.578 GHz, the transmission through the transmission line decreased by 15 dB and the loaded quality factor was 60,000. I measured the temperature dependence of the resonator frequency and loss at temperatures as high as 500 mK and found reasonable agreement with the Mattis-Bardeen theory. Finally, I coupled a quasi-lumped element microwave resonator (f05.443 GHz), made of superconducting Al on sapphire, to an Al/AlOx/Al CPB qubit. Most of my measurements were made in the dispersive regime where Ej-hf0 is much larger than the coupling strength. In this case, the qubit causes a small state-dependent frequency shift in the resonator's resonant frequency. By sending down a second microwave tone (the pump), I was able to excite the CPB qubit. In zero magnetic field with the CPB far detuned from the resonator, I measured a 50 kHz decrease in f0 with the qubit in the ground state and biased near the degeneracy point of the CPB. The charging energy and Josephson energy of the CPB were determined from spectroscopy taken by saturating the CPB with a second microwave tone and measuring the transmission through the resonator. The first device had Ec/h = 12.5 GHz and maximum Ej/h = 9 GHz. The second device had Ec/h = 6.24 GHz and Ej/h tuned between 4 GHz and 8 GHz. By changing the external magnetic field, I could decrease the effective Ej of the CPB. From modeling, I extracted coupling strengths g/2&pi = 11 MHz and 5 MHz for the first and second device, respectively. Finally I did single and two-tone spectroscopy, and measured the relaxation and Rabi oscillations of the CPB. From the first device, I was able to obtain relaxation times T1 of 10.3 us at Ej/h = 7 GHz on the CPB degeneracy point and spectroscopic coherence times T2 * 100 ns. From the second device, I found relaxation times T1 of 200 us at Ej/h = 4 GHz to 4.5 GHz decreasing down to 4 us around 8 GHz. There was also a depression in T1 around the resonant frequency of the resonator. The Rabi decay times were found to be up to T'~ 330 ns.
This dissertation describes my detailed investigation of decoherence and defects in two Al/AlOx/Al Cooper-pair box (CPB) charge qubits. Both devices were coupled to thin-film lumped-element superconducting aluminum LC resonators at a temperature of 25 mK. Device 1 was previously found to have an exceptionally long energy relaxation time of T1=205 μs and a strong correlation between the lifetime T1 and the decoupling from the microwave drive line dVg,rms/dΩR,0. I determined the dephasing properties of this CPB though a series of experiments. I measured Ramsey fringes, extracted dephasing times Tφ in the range200-500 ns, and determined a corresponding bound of Sq(f=1 Hz)≤(3×10-3)2 e2/Hz on the amplitude of the 1/f charge noise affecting the qubit. I then carried out a spin echo experiment and found echo decay times Techo in the 2.4-3.3 μs range, implying a high frequency 1/f charge noise cutoff of ωc/2π≈0.2 MHz. I followed this up by fabricating and characterizing a nearly identical Device 2. This CPB had a reasonably long relaxation time T1≈4-30 μs and again the lifetime T1 and decoupling dVg,rms/dΩR,0 were correlated. Although the lifetime of Device 2 was shorter than that of Device 1, the results suggest that the exceptional relaxation time was somewhat reproducible and that this approach may lead to further improvements in qubit coherence. During my initial characterization of Device 2, I discovered that it displayed an anomalously twinned transition spectrum. I studied this feature in detail in parallel with my decoherence experiments. I found that above the resonator resonance at ω/2π=5.472 GHz the system spectrum was twinned but below it was quadrupled. This behavior was consistent with a pair of two-level systems (TLS) coupled non-resonantly to the CPB via both charge and critical current. I developed a model that matched this scenario and successfully fit the predicted spectrum to my data. Both the coherent non-resonant interaction and joint charge and critical current CPB-TLS coupling are novel observations. From the fits I extracted microscopic parameters of the fluctuators including the well asymmetry, tunneling rate, and a minimum hopping distance of 0.2-0.45 Å. I also found a large fractional change of the Josephson energy ΔEJ,k/EJ≈30-40%, consistent with a non-uniform tunnel barrier containing a few dominant conduction channels and a defect that modulates one of them.
We have investigated the decoherence of quantum states in two Al/AlOx/Al Cooper-pair boxes coupled to lumped element superconducting inductor-capacitor resonators. At 25 mK, the first qubit had an energy relaxation time T1 that varied from 30 μs to 200 μs between 4 and 8 GHz and displayed an inverse correlation between T1 and the coupling to the microwave drive line. The Ramsey fringe decay times T2* were in the [200–500] ns range while the spin echo envelope decay times Techo varied from 2.4–3.3 μs, consistent with 1/f charge noise with a high frequency cutoff of 0.2 MHz. A second Cooper-pair box qubit with similar parameters showed T1=4−30 μs between 4 and 7.3 GHz, and that the T1 and the coupling were again inversely correlated. Although the lifetime of the second device was shorter than that of the first device, the dependence on coupling in both devices suggests that further reduction in coupling should lead to improved qubit performance.
We have observed a few distinct anomalous avoided level crossings and voltage-dependent transitions in the excited-state spectrum of an Al/AlOx/Al\text{Al}/{\text{AlO}}_{x}/\text{Al}Al/AlOx​/Al Cooper-pair box (CPB). The device was measured at 40 mK in the 15--50 GHz frequency range. We find that a given level crosses the CPB spectrum at two different gate voltages; the frequency and splitting size of the two crossings differ and the splitting size depends on the Josephson energy of the CPB. We show that this behavior is not only consistent with the CPB being coupled to discrete charged ``two-level'' quantum systems, which move atomic distances in the CPB junctions, but that the spectra provide information about the fluctuators, which is not available from phase qubit spectra of anomalous avoided levels. In particular by fitting a model Hamiltonian to our data, we extract microscopic parameters for each fluctuator, including well asymmetry, tunneling amplitude, and the minimum hopping distance for each fluctuator. The tunneling rates range from less than 3.5--13 GHz, which represent values between 5 and 150% of the well asymmetry, and the dipole moments give a minimum hopping distance of 0.3−−0.8 A˚0.3--0.8\text{ }\text{\AA{}}0.3−−0.8 A˚. We have also found that these discrete two-level systems have a pronounced effect on the relaxation time (T1)({T}_{1})(T1​) of the quantum states of the CPB and hence can be a source of dissipation for superconducting quantum bits.
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 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.
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.