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.
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.
In this thesis, measurements on superconducting electrical circuits that exhibit quantum mechanical effects are presented. Nonlinear superconducting circuits can be regarded as artificial atoms with energy spectra that can be engineered. The artificial atoms can be used as quantum bits for quantum computation. By coupling the artificial atoms to the electromagnetic field inside a transmission line resonator, quantum optical effects can be studied in these electrical circuits.The possibility of using a frequency tunable transmission line resonators for coupling of superconducting quantum bits is investigated. Tunable superconducting resonators consisting of coplanar waveguide structures, terminated to ground in one end via a superconducting quantum interference device (SQUID), are characterized. By applying a magnetic field to the SQUID loop, the boundary condition of the resonator changes leading to a change in resonance frequency.A change of the resonance frequency by more than 250 linewidths is demonstrated. It is also demonstrated that the resonator can be tune by several hundred MHz on a time scale that is much shorter than the photon lifetime of the resonator.Circuits containing both one and two transmon qubits coupled to the resonator are characterized. Quantum mechanical interactions between the resonator and the qubits, in form of vacuum Rabi splitting, are observed. Coherent control of a single qubit coupled to the resonator is demonstrated by performing Rabi oscillations of the qubit. In the two qubit case, coherent interaction between the qubits is observed in terms of a nonlinear scaling of the coupling strength between the resonator and the collective two qubit system.Measurements on a strongly driven single Cooper-pair box, using a quantum capacitance readout, are also presented. A rich structure in the response is well explained in terms of longitudinal dressed states of the single-Cooper pair box.When measuring the response of a single-electron box, a gate dependent excess dissipation is observed. The dissipation is understood by considering the internal dynamics of the single electron box. The excess dissipation is modeled as an effective resistance named the Sisyphus resistance.
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.
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.
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 EJ/EC 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 T1 when it is shorter than the measurement time Tmeas.