Coupled superconducting qudit-resonator system: Energy spectrum, state population, and state transition under microwave drive

W. Y. Liu, H. K. Xu, F. F. Su, Z. Y. Li, Ye Tian, Siyuan Han, and S. P. Zhao

Phys. Rev. B 97, 094513 – Published 26 March 2018

Abstract

Superconducting quantum multilevel systems coupled to resonators have recently been considered in some applications such as microwave lasing and high-fidelity quantum logical gates. In this work, using an rf-SQUID type phase qudit coupled to a microwave coplanar waveguide resonator, we study both theoretically and experimentally the energy spectrum of the system when the qudit level spacings are varied around the resonator frequency by changing the magnetic flux applied to the qudit loop. We show that the experimental result can be well described by a theoretical model that extends from the usual two-level Jaynes-Cummings system to the present four-level system. It is also shown that due to the small anharmonicity of the phase device a simplified model capturing the leading state interactions fits the experimental spectra very well. Furthermore we use the Lindblad master equation containing various relaxation and dephasing processes to calculate the level populations in the simpler qutrit-resonator system, which allows a clear understanding of the dynamics of the system under the microwave drive. Our results help to better understand and perform the experiments of coupled multilevel and resonator systems and can be applied in the case of transmon or Xmon qudits having similar anharmonicity to the present phase device.

Received 17 July 2017
Revised 10 January 2018

DOI:https://doi.org/10.1103/PhysRevB.97.094513

Physics Subject Headings (PhySH)

Superconducting qubits

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

W. Y. Liu^1,2, H. K. Xu^1,2, F. F. Su^1,2, Z. Y. Li^1,3, Ye Tian¹, Siyuan Han^4,1, and S. P. Zhao^1,2,5

¹Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
²School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
³School of Physics, Nanjing University, National Laboratory of Solid State Microstructures, Nanjing 210093, China
⁴Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA
⁵CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China

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Vol. 97, Iss. 9 — 1 March 2018

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Images

Figure 1
(a) Schematic of the coupled rf-SQUID type phase qudit (red) and resonator (blue) system. The qudit has a Josephson critical current $I_{c}$ , shunt capacitance $C_{q}$ , and loop inductance $L_{q}$ while the resonator is characterized by its effective inductance $L_{r}$ and capacitance $C_{r}$ . They are coupled by a capacitance $C_{c}$ . The flux bias provides both dc and rf components of $Φ_{ext} = Φ_{dc} + Φ_{rf} (t)$ . (b) Energy potential and level diagram of a superconducting phase qudit. (c) Energy-level diagram of a resonator.
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Figure 2
Energy-level diagram of the coupled qudit and resonator system. The levels are shown for three different resonant cases of $ω_{10} = ω_{r}$ (one-photon), $ω_{20} = 2 ω_{r}$ (two-photon), and $ω_{30} = 3 ω_{r}$ (three-photon), respectively.
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Figure 3
Left panels: Experimental spectroscopic scans as a function of flux bias for three microwave driving powers. (a) $- 100$ dBm; (b) $- 95$ dBm; (c) $- 70$ dBm. Right panels: Dressed state energy levels. Experimental (symbols) and theoretical (solid lines) spectral lines involving the first (d), the second (e), and the third (f) excited states of the qudit and resonator system shown in Fig. 2. Experimental data are taken from the spectra in (a)–(c) while the theoretical ones in (e) and (f) are the calculated results divided by a factor of 2 and 3, respectively. Note that the horizontal scales are different in (d), (e), and (f). The straight broken lines show the uncoupled qudit and resonator levels as indicated. The circle, squares, upward triangles, and downward triangle indicate the nearby bare state interaction strengths of $g / π, \sqrt{2} g / π, \sqrt{3} g / π$ , and $2 g / π$ , respectively (see Fig. 2). Single and double arrows indicate weaker interactions between two states via one and two intermediate states, respectively.
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Figure 4
Calculated spectra a function of flux bias at the microwave driving powers of (a) $Ω / 2 π = 1.1$ MHz and (b) $Ω / 2 π = 3.5$ MHz. The arrow in (a) indicates the resonant point. Three vertical dashed lines in (b) indicate the cuts at flux biases of $Φ_{dc} = 0.6766$ , 0.6786, and 0. $6806 Φ_{0}$ , respectively.
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Figure 5
Calculated spectra at the microwave driving powers of $Ω / 2 π = 3.5$ MHz (lines). (a) $Φ_{dc} = 0.6766 Φ_{0}$ , (b) $Φ_{dc} = 0.6786 Φ_{0}$ , and (c) $Φ_{dc} = 0.6806 Φ_{0}$ , corresponding to the three cuts in Fig. 4. Symbols are the experimental data measured at microwave power of $- 95$ dBm.
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Figure 6
Calculated average photon number $\bar{N}$ in the resonator versus qudit flux bias and microwave frequency at the microwave powers of (a) $Ω / 2 π = 1.1$ MHz and (b) $Ω / 2 π = 3.5$ MHz, corresponding to those in Fig. 4. Inset shows the probabilities ${\tilde{P}}_{1}$ and ${\tilde{P}}_{2}$ with one and two photons in the resonator along the dashed line in (a). Note that $\bar{N} = {\tilde{P}}_{1} + 2 {\tilde{P}}_{2}$ (see Appendix pp2).
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