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arXiv:1305.3962 · Physical Review B

Spectroscopy of a Cooper-pair box coupled to a two-level system via charge and critical current

V. Zaretskey, Baladitya Suri, S P Novikov, F. C. Wellstood, B. S. Palmer

Published May 28, 2013• 18 citations
arXivHTMLPDF

Abstract

We report on the quadrupling of the transition spectrum of an Al/AlO${}_{x}$/Al Cooper-pair box (CPB) charge qubit in the $4.0--7.3$ GHz frequency range. The qubit was coupled to a quasilumped-element Al superconducting resonator and measured at a temperature of 25 mK. We obtained good matches between the observed spectrum and the spectra calculated from a model Hamiltonian containing two distinct low-excitation energy two-level systems (TLS) coupled to the CPB. In our model, each TLS has a charge that tunnels between two sites in a local potential and induces a change in the CPB critical current. By fitting the model to the spectrum, we have extracted microscopic parameters of the fluctuators including the well asymmetry, tunneling rate, and a surprisingly large fractional change ($30%--40%$) in the critical current (12 nA). This large change is consistent with a Josephson junction with a nonuniform tunnel barrier containing a few dominant conduction channels and a TLS that modulates one of them.

DOI
10.1103/physrevb.87.174522

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We report on the quadrupling of the transition spectrum of an Al/AlOx{}_{x}x​/Al Cooper-pair box (CPB) charge qubit in the 4.0−−7.34.0--7.34.0−−7.3 GHz frequency range. The qubit was coupled to a quasilumped-element Al superconducting resonator and measured at a temperature of 25 mK. We obtained good matches between the observed spectrum and the spectra calculated from a model Hamiltonian containing two distinct low-excitation energy two-level systems (TLS) coupled to the CPB. In our model, each TLS has a charge that tunnels between two sites in a local potential and induces a change in the CPB critical current. By fitting the model to the spectrum, we have extracted microscopic parameters of the fluctuators including the well asymmetry, tunneling rate, and a surprisingly large fractional change (3030%--40%30) in the critical current (12 nA). This large change is consistent with a Josephson junction with a nonuniform tunnel barrier containing a few dominant conduction channels and a TLS that modulates one of them.
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.
We analyze the behavior of a Cooper Pair Box (CPB) that is coupled to charge fluctuators that reside in the dielectric barrier layer in the box's ultra-small tunnel junction. We derive the Hamiltonian of the combined system and find the coupling between the CPB and the fluctuators as well as a coupling between the fluctuators that is due to the CPB. We then find the energy levels and transition spectrum numerically for the case of a CPB coupled to a single charge fluctuator, where we treat the fluctuator as a two-level system that tunnels between two sites. The resulting spectra show the usual transition spectra of the CPB plus distinctive transitions due to excitation of the fluctuator; the fluctuator transitions are 2-e periodic and resemble saw-tooth patterns when plotted as a function of the gate voltage applied to the box. The combined CPB fluctuator spectra show small second-order avoided crossings with a size that depends on the gate voltage. Finally, we discuss how the microscopic parameters of the model, such as the charge times the hopping distance, the tunneling rate between the hopping sites, and the energy difference between the hopping sites, can be extracted from CPB spectra, and why this yields more information than can be found from similar spectra from phase qubits.
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.
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 report spectroscopic measurements of discrete two-level systems (TLSs) coupled to a dc superconducting quantum interference device phase qubit with a $16\text{ }\ensuremath{\mu}{\text{m}}^{2}$ area Al/AlOx/Al\text{Al}/{\text{AlO}}_{x}/\text{Al}Al/AlOx​/Al junction. Applying microwaves in the 10--11 GHz range, we found eight avoided level crossings with splitting sizes from 10 to 200 MHz and spectroscopic lifetimes from 4 to 160 ns. Assuming the transitions are from the ground state of the composite system to an excited state of the qubit or an excited state of one of the TLS states, we fit the location and spectral width to get the energy levels, splitting sizes, and spectroscopic coherence times of the phase qubit and TLSs. The distribution of splittings is consistent with noninteracting individual charged ions tunneling between random locations in the tunnel barrier and the distribution of lifetimes is consistent with the AlOx{\text{AlO}}_{x}AlOx​ in the junction barrier having a frequency-independent loss tangent. To check that the charge of each TLS couples independently to the voltage across the junction, we also measured the spectrum in the 20--22 GHz range and found tilted avoided level crossings due to the second excited state of the junction and states in which both the junction and a TLS were excited.
We observe the effect of the Aharonov-Casher (AC) interference on the spectrum of a superconducting system containing a symmetric Cooper pair box (CPB) and a large inductance. By varying the charge n_{g} induced on the CPB island, we observe oscillations of the device spectrum with the period Δn_{g}=2e. These oscillations are attributed to the charge-controlled AC interference between the fluxon tunneling processes in the CPB Josephson junctions. The measured phase and charge dependences of the frequencies of the |0⟩→|1⟩ and |0⟩→|2⟩ transitions are in good agreement with our numerical simulations. Almost complete suppression of the single fluxon tunneling due to destructive interference is observed for the charge n_{g}=e(2n+1). The CPB in this regime enables fluxon pairing, which can be used for the development of parity-protected superconducting qubits.