Method for identifying electromagnetically induced transparency in a tunable circuit quantum electrodynamics system

Qi-Chun Liu, Tie-Fu Li, Xiao-Qing Luo, Hu Zhao, Wei Xiong, Ying-Shan Zhang, Zhen Chen, J. S. Liu, Wei Chen, Franco Nori, J. S. Tsai, and J. Q. You

Phys. Rev. A 93, 053838 – Published 31 May 2016

Abstract

Electromagnetically induced transparency (EIT) has been realized in atomic systems, but fulfilling the EIT conditions for artificial atoms made from superconducting circuits is a more difficult task. Here we report an experimental observation of the EIT in a tunable three-dimensional transmon by probing the cavity transmission. To fulfill the EIT conditions, we tune the transmon to adjust its damping rates by utilizing the effect of the cavity on the transmon states. From the experimental observations, we clearly identify the EIT and Autler-Townes splitting (ATS) regimes as well as the transition regime in between. Also, the experimental data demonstrate that the threshold $Ω_{AIC}$ determined by the Akaike information criterion can describe the EIT-ATS transition better than the threshold $Ω_{EIT}$ given by the EIT theory.

Received 20 April 2015

DOI:https://doi.org/10.1103/PhysRevA.93.053838

Physics Subject Headings (PhySH)

Electromagnetically induced transparency Superconducting qubits

Atomic, Molecular & Optical

Authors & Affiliations

Qi-Chun Liu¹, Tie-Fu Li^1,2,3, Xiao-Qing Luo², Hu Zhao¹, Wei Xiong^2,4, Ying-Shan Zhang¹, Zhen Chen², J. S. Liu¹, Wei Chen^1,*, Franco Nori^3,5, J. S. Tsai^3,6, and J. Q. You^2,†

¹Institute of Microelectronics, Department of Microelectronics and Nanoelectronics and Tsinghua National Laboratory of Information Science and Technology, Tsinghua University, Beijing 100084, China
²Quantum Physics and Quantum Information Division, Beijing Computational Science Research Center, Beijing 100193, China
³RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
⁴Department of Physics, Fudan University, Shanghai 200433, China
⁵Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
⁶Department of Physics, Tokyo University of Science, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

^*weichen@tsinghua.edu.cn
^†jqyou@csrc.ac.cn

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Vol. 93, Iss. 5 — May 2016

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Figure 1
(a) Schematic diagram of the experimental setup. A tunable 3D transmon consisting of a symmetric SQUID and a copper 3D cavity is thermally anchored to the mixing chamber, biased with a static magnetic field. A network analyzer works at 8.219 50 GHz (the resonant frequency of the 3D cavity when the transmon is in the ground state $| 0 〉$ ) with a fixed power of $- 15$ dBm at its output port, corresponding to the average photon number in the cavity to be $\sim 0.7$ . At this frequency, the transmission coefficient $T$ of the cavity is measured. A microwave source provides the control tone at $ω_{c} = ω_{21} = 2 π \times 3.979 50$ GHz with various powers. Another microwave source provides the probe tone at $ω_{p} = ω_{20} - δ$ with a fixed power of $- 50$ dBm at the source output. The microwave signals are combined at room temperature by two splitters and then strongly attenuated and filtered before reaching the sample. The transmitted signal through the 3D cavity is amplified at 4 K and room temperature before received by the network analyzer. (b) The lowest three energy levels of the transmon driven by a control field (red) and a probe field (blue). The control field with frequency $ω_{c}$ is in resonance with the transition between $| 1 〉$ and $| 2 〉$ , and the probe field with frequency $ω_{p}$ has a detuning $δ$ with the transition between $| 0 〉$ and $| 2 〉$ . The corresponding driving strengths are $Ω_{c}$ and $Ω_{p}$ , respectively.
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Figure 2
(a) The normalized transmission coefficient $T^{'}$ of the cavity, where the control field sweeping from $- 32$ to $- 19$ dBm at source output is applied in resonance with the transition frequency $ω_{21} / 2 π = 3.979 50$ GHz and the probe field with power $- 50$ dBm at source output has a detuning $δ / 2 π$ from the transition frequency $ω_{20} / 2 π = 8.370 75$ GHz. The dashed (black) guide lines correspond to the two peaks in $T^{'}$ . In order to clearly show the two peaks in the weak control-field range, we present $T^{'}$ in (a) only up to $- 19$ dBm for the control-field power. (b)–(e) The measured transmission coefficient $T^{'}$ versus the frequency detuning $δ / 2 π$ (denoted by red open circles) at different powers of the control field: (b) $- 31$ dBm, (c) $- 26$ dBm, (d) $- 20$ dBm, and (e) $- 8$ dBm, where the results at $- 31, - 26$ , and $- 20$ dBm correspond to the three vertical (white) lines in (a) and the result at $- 8$ dBm is not shown there. The solid curves are the fitting results obtained using Eq. (11) when $Ω_{c} / 2 π = 2.06, 2.88, 5.29$ , and $19.7$ MHz, respectively. (f) The driving strength $Ω_{c}$ at each control-field power (denoted by red open circles), which is obtained by fitting the measured transmission coefficient $T^{'}$ with Eq. (11). The upper limit of the driving strength $Ω_{EIT}$ for realizing EIT is indicated by the dashed line.
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Figure 3
(a)–(d) Comparison of the measured transmission coefficient $T^{'}$ (denoted by red open circles) with the results obtained using Eq. (12), i.e., the EIT model (solid curves), and Eq. (13), i.e., the ATS model (dashed curves). The powers of the control field are (a) $- 31$ dBm, (b) $- 26$ dBm, (c) $- 20$ dBm, and (d) $- 8$ dBm, which correspond to $Ω_{c} / 2 π = 2.06, 2.88, 5.29$ , and $19.7$ MHz, respectively. (e) The AIC per-point weights for both EIT and ATS models at different values of the control-field power (i.e., different driving strength $Ω_{c}$ ). The red open circles (blue solid triangles) correspond to the per-point weights of the EIT (ATS) model obtained by fitting with the experimental data. The red (blue) solid curve corresponds to the theoretical AIC per-point weights obtained with $γ_{20} / 2 π = 6.90$ MHz and $γ_{10} / 2 π = 1.76$ MHz, as well as an additional $3 %$ experimental noise, while the red (blue) dash-dotted curve corresponds to the theoretical AIC per-point weights obtained without the additional experimental noise.
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Figure 4
The transition spectroscopy between states $| 0 〉$ and $| 2 〉$ of the tunable 3D transmon at a flux bias giving $E_{J} / E_{C} = 27.5$ . (a) One-photon transition process between states $| 0 〉$ and $| 1 〉$ and two-photon transition process between states $| 0 〉$ and $| 2 〉$ . (b) One-photon transition process between states $| 0 〉$ and $| 2 〉$ .
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Figure 5
The measured Rabi oscillations between states $| 0 〉$ and $| 2 〉$ of the 3D transmon by using a $10.028 - GHz$ driving field. The black circles are experimental data and the red curve is an exponentially damped sinusoidal fit. The Rabi oscillation period is $56.8$ ns and the oscillation decay time is $130.6$ ns from the fit.
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Figure 6
Electric- and magnetic-dipole transition matrix elements of the 3D transmon as a function of the ratio $E_{J} / E_{C}$ . Panels (a) and (b) correspond to the electric-dipole one-photon transitions for $n_{g} = 0.5$ and $0.25$ , respectively. Panels (c) and (d) correspond to the magnetic-dipole one-photon transitions for $n_{g} = 0.5$ and $0.25$ , respectively.
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