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Transmon qubits coupled to superconducting lumped element resonators

Baladitya Suri

Published Jan 1, 2015• 3 citations
Publisher

Abstract

I discuss the design, fabrication and measurement at millikelvin-temperatures of Al/AlO$_x$/Al Josephson junction-based transmon qubits coupled to superconducting thin-film lumped element microwave resonators made of aluminum on sapphire. The resonators had a center frequency of around $6\,$GHz, and a total quality factor ranging from 15,000 to 70,000 for the various devices. The area of the transmon junctions was about $150\, \mathrm{nm} \times 150\, \mathrm{nm}$ and with Josephson energy $E_J$ such that $10\,\text{GHz} \leq E_J / h \leq 30\,$GHz. The charging energy of the transmons arising mostly from the large interdigital shunt capacitance, was $E_c / h \approx 300\,$MHz. I present microwave spectroscopy of the devices in the strongly dispersive regime of circuit quantum electrodynamics. In this limit the ac Stark shift due to a single photon in the resonator is greater than the linewidth of the qubit transition. When the resonator is driven coherently using a coupler tone, the transmon spectrum reveals individual ``photon number'' peaks, each corresponding to a single additional photon in the resonator. Using a weighted average of the peak heights in the qubit spectrum, I calculated the average number of photons $\bar{n}$ in the resonator. I also observed a nonlinear variation of $\bar{n}$ with the applied power of the coupler tone $P_{rf}$. I studied this nonlinearity using numerical simulations and found good qualitative agreement with data. In the absence of a coherent drive on the resonator, a thermal population of $5.474\,$GHz photons in the resonator, at an effective temperature of $120\,$mK resulted in a weak $n=1$ thermal photon peak in the qubit spectrum. In the presence of independent coupler and probe tones, the $n=1$ thermal photon peak revealed an Autler-Townes splitting. The observed effect was explained accurately using the four lowest levels of the dispersively dressed Jaynes-Cummings transmon-resonator system, and numerical simulations of the steady-state master equation for the coupled system. I also present time-domain measurements on transmons coupled to lumped-element resonators. From $T_1$ and Rabi oscillation measurements, I found that my early transmon devices (called design LEv5) had lifetimes ($T_1 \sim 1\,\mu$s) limited by strong coupling to the $50\,\Omega$ transmission line. This coupling was characterized by the the rate of change of the Rabi oscillation frequency with the change in the drive voltage ($\mathrm{d}f_{Rabi}\, / \mathrm{d}V$) -- also termed the Rabi coupling to the drive. I studied the design of the transmon-resonator system using circuit analysis and microwave simulations with the aim being to reduce the Rabi coupling to the drive. By increasing the resonance frequency of the resonator $\omega_r/2\pi$ from 5.4$\,$GHz to 7.2$\,$GHz, lowering the coupling of the resonator to the transmission line and thereby increasing the external quality factor $Q_e$ from 20,000 to 70,000, and reducing the transmon-resonator coupling $g/2\pi$ from 70$\,$MHz to 40$\,$MHz, I reduced the Rabi coupling to the drive by an order of magnitude ($\sim$ factor of 20). The $T_1 \sim 4\,\mu$s of devices in the new design (LEv6) was longer than that of the early devices, but still much shorter than the lifetimes predicted from Rabi coupling, suggesting the presence of alternative sources of noise causing qubit relaxation. Microwave simulations and circuit analysis in the presence of a dielectric loss tangent $\tan \delta \simeq 5\times10^{-6}$ agree reasonably well with the measured $T_1$ values, suggesting that surface dielectric loss may be causing relaxation of transmons in the new designs.

DOI
10.13016/m2633p

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We measure photon occupancy in a thin-film superconducting lumped element resonator coupled to a transmon qubit at $20\phantom{\rule{0.16em}{0ex}}\mathrm{mK}$ and find a nonlinear dependence on the applied microwave power. The transmon-resonator system was operated in the strong dispersive regime, where the ac Stark shift $(2\ensuremath{\chi})$ due to a single microwave photon present in the resonator was larger than the linewidth $(\mathrm{\ensuremath{\Gamma}})$ of the qubit transition. When the resonator was coherently driven at 5.474 325 GHz, the transition spectrum of the transmon at 4.982 GHz revealed well-resolved peaks, each corresponding to an individual photon number-state of the resonator. From the relative peak heights we obtain the occupancy of the photon states and the average photon occupancy $\overline{n}$ of the resonator. We observe a nonlinear variation of $\overline{n}$ with the applied drive power ${P}_{\mathrm{rf}}$ for $\overline{n}<5$ and compare our results to numerical simulations of the system-bath master equation in the steady state, as well as to a semiclassical model for the resonator that includes the Jaynes-Cummings interaction between the transmon and the resonator. We find good quantitative agreement using both models and analysis reveals that the nonlinear behavior is principally due to shifts in the resonant frequency caused by a qubit-induced Jaynes-Cummings nonlinearity.

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Dissipative and dispersive measurements of a Cooper pair box

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•Jan 1, 2010•2 citations
I discuss the design, fabrication and measurement at millikelvin-temperatures of Al/AlOx_xx​/Al Josephson junction-based transmon qubits coupled to superconducting thin-film lumped element microwave resonators made of aluminum on sapphire. The resonators had a center frequency of around 6 6\,6GHz, and a total quality factor ranging from 15,000 to 70,000 for the various devices. The area of the transmon junctions was about 150 nm×150 nm150\, \mathrm{nm} \times 150\, \mathrm{nm}150nm×150nm and with Josephson energy EJE_JEJ​ such that 10 GHz≤EJ/h≤30 10\,\text{GHz} \leq E_J / h \leq 30\,10GHz≤EJ​/h≤30GHz. The charging energy of the transmons arising mostly from the large interdigital shunt capacitance, was Ec/h≈300 E_c / h \approx 300\,Ec​/h≈300MHz. I present microwave spectroscopy of the devices in the strongly dispersive regime of circuit quantum electrodynamics. In this limit the ac Stark shift due to a single photon in the resonator is greater than the linewidth of the qubit transition. When the resonator is driven coherently using a coupler tone, the transmon spectrum reveals individual ``photon number'' peaks, each corresponding to a single additional photon in the resonator. Using a weighted average of the peak heights in the qubit spectrum, I calculated the average number of photons nˉ\bar{n}nˉ in the resonator. I also observed a nonlinear variation of nˉ\bar{n}nˉ with the applied power of the coupler tone PrfP_{rf}Prf​. I studied this nonlinearity using numerical simulations and found good qualitative agreement with data. In the absence of a coherent drive on the resonator, a thermal population of 5.474 5.474\,5.474GHz photons in the resonator, at an effective temperature of 120 120\,120mK resulted in a weak n=1n=1n=1 thermal photon peak in the qubit spectrum. In the presence of independent coupler and probe tones, the n=1n=1n=1 thermal photon peak revealed an Autler-Townes splitting. The observed effect was explained accurately using the four lowest levels of the dispersively dressed Jaynes-Cummings transmon-resonator system, and numerical simulations of the steady-state master equation for the coupled system. I also present time-domain measurements on transmons coupled to lumped-element resonators. From T1T_1T1​ and Rabi oscillation measurements, I found that my early transmon devices (called design LEv5) had lifetimes (T1∼1 μT_1 \sim 1\,\muT1​∼1μs) limited by strong coupling to the 50 Ω50\,\Omega50Ω transmission line. This coupling was characterized by the the rate of change of the Rabi oscillation frequency with the change in the drive voltage (dfRabi /dV\mathrm{d}f_{Rabi}\, / \mathrm{d}VdfRabi​/dV) -- also termed the Rabi coupling to the drive. I studied the design of the transmon-resonator system using circuit analysis and microwave simulations with the aim being to reduce the Rabi coupling to the drive. By increasing the resonance frequency of the resonator ωr/2π\omega_r/2\piωr​/2π from 5.4 \,GHz to 7.2 \,GHz, lowering the coupling of the resonator to the transmission line and thereby increasing the external quality factor QeQ_eQe​ from 20,000 to 70,000, and reducing the transmon-resonator coupling g/2πg/2\pig/2π from 70 \,MHz to 40 \,MHz, I reduced the Rabi coupling to the drive by an order of magnitude (∼\sim∼ factor of 20). The T1∼4 μT_1 \sim 4\,\muT1​∼4μs of devices in the new design (LEv6) was longer than that of the early devices, but still much shorter than the lifetimes predicted from Rabi coupling, suggesting the presence of alternative sources of noise causing qubit relaxation. Microwave simulations and circuit analysis in the presence of a dielectric loss tangent tan⁡δ≃5×10−6\tan \delta \simeq 5\times10^{-6}tanδ≃5×10−6 agree reasonably well with the measured T1T_1T1​ values, suggesting that surface dielectric loss may be causing relaxation of transmons in the new designs.
We measure photon occupancy in a thin-film superconducting lumped element resonator coupled to a transmon qubit at 20mK20\phantom{\rule{0.16em}{0ex}}\mathrm{mK}20mK and find a nonlinear dependence on the applied microwave power. The transmon-resonator system was operated in the strong dispersive regime, where the ac Stark shift $(2\ensuremath{\chi})$ due to a single microwave photon present in the resonator was larger than the linewidth $(\mathrm{\ensuremath{\Gamma}})$ of the qubit transition. When the resonator was coherently driven at 5.474 325 GHz, the transition spectrum of the transmon at 4.982 GHz revealed well-resolved peaks, each corresponding to an individual photon number-state of the resonator. From the relative peak heights we obtain the occupancy of the photon states and the average photon occupancy n‾\overline{n}n of the resonator. We observe a nonlinear variation of n‾\overline{n}n with the applied drive power Prf{P}_{\mathrm{rf}}Prf​ for n‾<5\overline{n}<5n<5 and compare our results to numerical simulations of the system-bath master equation in the steady state, as well as to a semiclassical model for the resonator that includes the Jaynes-Cummings interaction between the transmon and the resonator. We find good quantitative agreement using both models and analysis reveals that the nonlinear behavior is principally due to shifts in the resonant frequency caused by a qubit-induced Jaynes-Cummings nonlinearity.
This dissertation examines the design, fabrication, and characterization of a superconducting lumped-element tunable LC resonator that is used to vary the coupling between two superconducting qubits. Some level of qubit-qubit coupling is needed to perform gating operations. However, with fixed coupling, single qubit operations become considerably more difficult due to dispersive shifts in their energy levels transitions that depend on the state of the other qubit. Ideally, one wants a system in which the qubit-qubit coupling can be turned off to allow for single qubit operations, and then turned back on to allow for multi-qubit gate operations. I present results on a device that has two fixed-frequency transmon qubits capacitively coupled to a tunable thin-film LC resonator. The resonator can be tuned in situ over a range of 4.14 GHz to 4.94 GHz by applying an external magnetic flux to two single-Josephson junction loops, which are incorporated into the resonator’s inductance. The qubits have 0-to-1 transition frequencies of 5.10 GHz and 4.74 GHz. To isolate the system and provide a means for reading out the state of the qubit readout, the device was mounted in a 3D Al microwave cavity with a TE101 mode resonance frequency of about 6.1 GHz. The flux-dependent transition frequencies of the system were measured and fit to results from a coupled Hamiltonian model. With the LC resonator tuned to its minimum resonance frequency, I observed a qubit-qubit dispersive shift of 2χ_qq≈ 0.1 MHz, which was less than the linewidth of the qubit transitions. This dispersive shift was sufficiently small to consider the coupling “off”, allowing single qubit operations. The qubit-qubit dispersive shift varied with the applied flux up to a maximum dispersive shift of 2χ_qq≈ 6 MHz. As a proof-of-principle, I present preliminary results on performing a CNOT gate operation on the qubits when the coupling was “on” with 2χ_qq≈ 4 MHz. This dissertation also includes observations of the temperature dependence of the relaxation time T1 of three Al/AlOx/Al transmons. We found that, in some cases, T1 increased by almost a factor of two as the temperature increased from 30 mK to 100 mK. We found that this anomalous behavior was consistent with loss due to non-equilibrium quasiparticles in a transmon where one electrode in the tunnel junction had a smaller volume and slightly smaller superconducting energy gap than the other electrode. At sufficiently low temperatures, non-equilibrium quasiparticles accumulate in the electrode with a smaller gap, leading to an increased density of quasiparticles at the junction and a corresponding decrease in the relaxation time. I present a model of this effect, use the model to extract the density of non-equilibrium quasiparticles in the device, and find the values of the two superconducting energy gaps.
Metamaterial resonant structures made from arrays of superconducting lumped-circuit elements can exhibit microwave-mode spectra with left-handed dispersion for the standing-wave resonances, resulting in a high density of modes in the same frequency range where superconducting qubits are typically operated, as well as a bandgap at lower frequencies that extends down to dc. Using this regime for multimode circuit quantum electrodynamics, we perform a series of measurements of such a superconducting metamaterial resonator coupled to a flux-tunable transmon qubit. Through microwave measurements of the metamaterial, we observe the coupling of the qubit to each of the modes that it passes through. Using a separate readout resonator, we probe the qubit dispersively and characterize the qubit energy relaxation as a function of frequency, which is strongly affected by the Purcell effect in the presence of the dense mode spectrum. Additionally, we investigate the ac Stark shift of the qubit as the photon number in the various metamaterial modes is varied. Through numerical simulations, we explore designs based on this scheme with enhanced coupling so that the coupling energy between the qubit and the metamaterial modes can exceed the intermode spacing in future devices with achievable circuit parameters. The ability to tailor the dense mode spectrum through the choice of circuit parameters and manipulate the photonic state of the metamaterial through interactions with qubits makes this a promising platform for analog quantum simulation with microwave photons and quantum memories.
We report our development on design, fabrication and characterization of various superconducting resonators for quantum device applications. The resonators are developed mainly for superconducting qubit implementation in circuit QED structure, but future application will include photon detectors such as microwave kinetic inductance detectors. First we designed and fabricated half-wavelength coplanar waveguide resonators with niobium thin film on a high resistive silicon wafer, with the fundamental frequencies in the range of 1 to 8 GHz. The loaded quality factors are controlled by the coupling capacitors, and the lower bound of the intrinsic quality factor was measured down to 7 mK in the dilution refrigerator. We have also designed quarter-wavelength resonators and lumped LC resonators coupled to a transmission line, and characterized multiple resonators in a single transmission line. For the 3D transmon qubit application, we designed and fabricated a rectangular box resonator with superconducting aluminum. The intrinsic quality factor of the box resonator exceeds three million at 7 mK. We will also present basic results on the integration of a transmon qubit in the superconducting resonator.
Abstract Superconducting quantum circuits are a leading platform for quantum-information processing, but their development requires expertise spanning device design, microfabrication, cryogenic operation, microwave control, and quantum characterization. In this work, we report the realization of a flux-tunable superconducting transmon platform and present an integrated workflow encompassing design, fabrication, measurement, and numerical modeling. The device consists of an Al/AlOx/AlAl/AlO_x/AlAl/AlOx​/Al transmon coupled to a three-dimensional microwave cavity operating in the circuit quantum electrodynamics (cQED) dispersive regime. Two-tone spectroscopy reveals a fundamental transition frequency of ω01=5.0122 GHz\omega _{01}=5.0122~\textrm{GHz}ω01​=5.0122 GHz and an anharmonicity of α/2π=−202.4 MHz\alpha /2\pi =-202.4~\textrm{MHz}α/2π=−202.4 MHz , corresponding to EJ/EC≈82.9E_J/E_C \approx 82.9EJ​/EC​≈82.9 . Flux spectroscopy yields a cavity-qubit coupling strength of g/2π=260 MHzg/2\pi = 260~\textrm{MHz}g/2π=260 MHz , while the cavity linewidth gives a loaded quality factor of QL=6127Q_L=6127QL​=6127 and a Purcell limit of T1Purcell≈82 μsT_1^{\textrm{Purcell}}\approx 82~\mu \textrm{s}T1Purcell​≈82 μs . Time-domain measurements yield coherence times of T1=17.4 μsT_1 = 17.4~\mu \textrm{s}T1​=17.4 μs and T2∗=5.8 μsT_2^* = 5.8~\mu \textrm{s}T2∗​=5.8 μs . To investigate coherence and dissipation in the device, the experimental observations are analyzed within a Lindblad master-equation framework describing cavity photon loss and thermal excitation, qubit energy relaxation, and pure dephasing processes. Using a set of experimentally extracted parameters, the model reproduces the measured spectroscopic and time-domain observables, providing a quantitative description of the open-system dynamics of the transmon-cavity platform. The analysis shows that the measured coherence is primarily limited by pure dephasing and energy-relaxation processes, whereas thermal excitation plays a minor role under the experimental conditions. Inspired by the broader framework of quantum dissipation in macroscopic quantum systems pioneered by Caldeira and collaborators, this approach enables a quantitative analysis of relaxation and dephasing processes in superconducting quantum circuits. Beyond the characterization of a single device, the present work demonstrates the establishment of the expertise, infrastructure, and experimental workflow required for the design, fabrication, measurement, and modeling of superconducting quantum circuits, contributing to the development of quantum technologies in Brazil and related applications.
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.