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.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.

arXiv:1511.00174Physical Review ADec 2, 201517 citations

Coupling a Transmon Qubit to a Superconducting Metamaterial Resonator

H. Wang, Matthew Hutchings, Sager Indrajeet +5 more

Bulletin of the American Physical SocietyMar 16, 20160 citations

Transmon qubits coupled to superconducting lumped element resonators

Baladitya Suri

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.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.

Jan 1, 20153 citations

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.

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.