⇦Chinese Physics Letters, 2019, Vol. 36, No. 12, Article code 124204 Landau–Zener–Stückelberg Interference in Nonlinear Regime * Tong Wu (吴桐)1,2,3†, Yuxuan Zhou (周宇轩)2,3†, Yuan Xu (徐源)2,3, Song Liu (刘松)2,3,4, Jian Li (李剑)2,3,4** Affiliations 1Department of Physics, Harbin Institute of Technology, Harbin 150001 2Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055 3Shenzhen Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055 4Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518055 Received 27 September 2019, online 25 November 2019 *Supported by the National Natural Science Foundation of China under Grant No 11874065, the Key R&D Program of Guangdong Province under Grant No 2018B030326001, the Guangdong Innovative and Entrepreneurial Research Team Program under Grant No 2016ZT06D348, the Natural Science Foundation of Guangdong Province under Grant No 2017B030308003, the Natural Science Foundation of Hunan Province under Grant No 2018JJ1031, and the Science, Technology and Innovation Commission of Shenzhen Municipality under Grant Nos ZDSYS20170303165926217, JCYJ20170412152620376 and KYTDPT20181011104202253.
†Tong Wu and Yuxuan Zhou contributed equally to this work.
**Corresponding author. Email: lij33@sustech.edu.cn Citation Text: Wu T, Zhou Y X, Xu Y, Liu S and Li J et al 2019 Chin. Phys. Lett. 36 124204    Abstract Landau–Zener–Stückelberg (LZS) interference has drawn renewed attention to quantum information processing research because it is not only an effective tool for characterizing two-level quantum systems but also a powerful approach to manipulate quantum states. Superconducting quantum circuits, due to their versatile tunability and degrees of control, are ideal platforms for studying LZS interference phenomena. We use a superconducting Xmon qubit to study LZS interference by parametrically modulating the qubit transition frequency nonlinearly. For dc flux biasing of the qubit slightly far away from the optimal flux point, the qubit excited state population shows an interference pattern that is very similar to the standard LZS interference in linear regime, except that all bands shift towards lower frequencies when increasing the rf modulation amplitude. For dc flux biasing close to the optimal flux point, the negative sidebands and the positive sidebands behave differently, resulting in an asymmetric interference pattern. The experimental results are also in good agreement with our analytical and numerical simulations. DOI:10.1088/0256-307X/36/12/124204 PACS:42.50.Ct, 03.67.Lx, 74.50.+r, 85.25.Cp © 2019 Chinese Physics Society Article Text In recent years, superconducting circuit based quantum information processing has made enormous progress.[1] Superconducting qubits with long coherence time and weak nonlinearity, such as transmon[2] and Xmon,[3] pave the way for practical scalable quantum information processors. A split transmon or Xmon, in which the two Josephson junctions form a dc SQUID configuration, can have charge and flux degrees of controls. In most cases, single-qubit rotations around transverse ($X$, $Y$) axes on a Bloch sphere are realized by charge degree of controls induced by pulsed radio frequency (rf) gate voltages; flux degrees of controls are only used for tuning qubit transition frequency and realizing $Z$ (phase) gates with dc voltages.[4–6] In addition, an rf voltage pulse can also be used for the flux control to realize the parametric (longitudinal) modulation of qubit transition frequency. Flux induced parametric modulation is useful for studying phenomena such as motional averaging,[7] photon-assisted Landau–Zener–Stückelberg (LZS) interference,[7,8] and first-order sideband transitions between transmon and resonator.[9,10] Recently, two-qubit entangling gates activated by rf flux modulation have been realized between statically coupled transmons.[11,12] Switchable couplings between two qubits as well as between a qubit and a resonator have been experimentally realized,[13] based on a theoretical scheme[14,15] in which the qubit transition frequency is parametrically modulated. Very recently, antisymmetric spin exchange interaction and chiral dynamics have been demonstrated in all-to-all connected 5-qubit superconducting circuits by parametric modulations.[16] In most of these works, superconducting qubits are flux biased in near-linear regime, in which the change of qubit transition frequency is almost linearly dependent on the flux modulation. For example, if a sinusoidal flux modulation $\phi_{\rm rf}\cos({\it\Omega} t)$ on top of a dc flux $\phi_{\rm dc}$ is applied to the qubit, the qubit transition frequency $\omega_{10}$ will have a sinusoidal modulation, $$ \omega_{10} \approx \omega_{\rm dc} + A\cos({\it\Omega} t),~~ \tag {1} $$ where $\omega_{\rm dc}$ is the qubit transition frequency at $\phi_{\rm dc}$, ${\it\Omega}$ is the modulation frequency, and $A$ is the modulation amplitude (proportional to $\phi_{\rm rf}$). Since the flux dependent $|0\rangle\leftrightarrow|1\rangle$ transition frequency of a split transmon is approximately[2] $$ \hbar\omega_{10}(\phi) \approx \sqrt{8 E_{\rm C} E_{\rm J} |\cos(\phi)|} - E_{\rm C},~~ \tag {2} $$ with $E_{\rm C}$ the single electron charging energy and $E_{\rm J}$ the total Josephson energy (see Fig. 1), the near-linear regime can only be realized by dc flux biasing it far away from its optimal flux point ($\phi = 0$). The behavior of a superconducting qubit with parametric modulation in near-linear regime has been well studied and understood, whereas that in nonlinear regime is seldom studied experimentally. In this Letter, we present the spectroscopy of a superconducting Xmon qubit under rf flux in nonlinear regime. The experimental data shows unconventional LZS interference patterns, which are consistent with our analytical and numerical simulation results.
cpl-36-12-124204-fig1.png
Fig. 1. Readout resonator transmission amplitude $|S_{21}|$ (in dB) as a function of flux bias voltage and drive frequency $\omega$ for an Xmon qubit, without rf flux modulation. The (red) dotted curve is a fit of $\omega_{10}/2\pi$ by using Eq. (2), with $E_{\rm C}/h = 0.24$ GHz and $E_{\rm J}/h = 17.75$ GHz.
Our experimental setup is illustrated in Fig. 2. The circuit QED sample, consisting of a split Xmon qubit with a dispersive readout resonator, as well as an XY and a Z control lines, is mounted to the mixing chamber of a dilution refrigerator. To produce the dc + rf flux bias, we use a bias tee to combine the dc voltage from a voltage source and the sinusoidal rf voltage from a microwave signal generator at room temperature, and send the combined signal to the Z control line of the qubit. The qubit drive tone is generated by another microwave signal generator and sent through the XY control line. To measure the population of the qubit, a vector network analyzer (VNA) is used to probe the readout resonator. To calibrate the qubit, we obtain the two-tone spectroscopy to find the qubit transition frequencies corresponding to dc flux bias voltages, by sweeping the qubit drive tone frequency $\omega$ and measuring VNA tone transmission $S_{21}$ at each flux bias point. The spectroscopy data are shown in Fig. 1. Due to trapped flux during sample cooling down, $0$ bias voltage is not exactly corresponding to $0$ flux point. By fitting the flux dependent transition frequency with Eq. (2), we map the flux bias voltage $V$ to flux $\phi$, and find that they are of linear dependence as they should be. With dc + rf flux bias voltage applied to the $Z$ control line, and qubit drive tone applied to the $XY$ control line, the Hamiltonian of the qubit can be written as (take $\hbar = 1$) $$ H(t) = - \frac{1}{2}\omega_{10}(\phi)\sigma_z + R\cos(\omega t)\sigma_x,~~ \tag {3} $$ where $\sigma_{z(x)}$ is the Pauli-$Z(X)$ matrix, $R$ denotes the Rabi frequency, $\omega$ as mentioned above is the qubit drive frequency, $\omega_{10}(\phi)$ is given in Eq. (2), and $\phi = \phi_{\rm dc} + \phi_{\rm rf}\cos({\it\Omega} t)$. Including the energy relaxation and dephasing of the qubit, the dynamics of this system can be described by the Markov master equation $$\begin{align} \dot{\rho} = & -i\left[H(t), \rho\right] + \frac{{\it\Gamma}_2^\ast}{2} \left( \sigma_z\rho\sigma_z - \rho \right) \\ & + \frac{{\it\Gamma}_1}{2} \left( 2\sigma_-\rho\sigma_+ -\sigma_+\sigma_-\rho - \rho\sigma_+\sigma_- \right),~~ \tag {4} \end{align} $$ with $\rho$ being the density matrix, $\sigma_{+(-)}$ the qubit raising (lowering) operator, ${\it\Gamma}_1$ the energy relaxation rate, and ${\it\Gamma}_2^\ast$ the pure dephasing rate (note: the total dephasing rate ${\it\Gamma}_2 = {\it\Gamma}_2^\ast + {\it\Gamma}_1/2$).
cpl-36-12-124204-fig2.png
Fig. 2. Simplified schematic circuit diagram of the experimental setup. The circuitry inside the dashed box is at the mixing chamber of a dilution refrigerator, with base temperature of about 12 mK.
In this work, we consider the so-called second rotating-wave approximation (RWA) limit[17] in which the Rabi frequency is much smaller than the modulation frequency, $R\ll {\it\Omega}$. In this limit, if the near-linear regime Eq. (1) is valid, the master equation Eq. (4) will have a steady-state solution for the qubit excited state population,[8] $$ P_{\rm e} \approx \sum_{k = -\infty}^\infty \frac{\frac{{\it\Gamma}_2}{2{\it\Gamma}_1}[RJ_k(A/{\it\Omega})]^2}{{\it\Gamma}_2^2 \!+\! (\omega_{\rm dc}\! -\! \omega\! + \!k{\it\Omega})^2 \!+\! \frac{{\it\Gamma}_2}{{\it\Gamma}_1}[RJ_k(A/{\it\Omega})]^2},~~ \tag {5} $$ with $J_k(A/{\it\Omega})$ the $k$th Bessel function of the first kind, which determines the population of the $k$th sideband.
cpl-36-12-124204-fig3.png
Fig. 3. Upper: Measured $P_{\rm e}$ as a function of drive frequency $\omega$ and rf flux bias peak voltage $V_{\rm p}$, at dc flux bias voltage $-0.15$ V. Lower: $P_{\rm e}$ plotted by using Eq. (5) and replacing $\omega_{\rm dc}$ with $\widetilde\omega_{\rm dc}$ as shown in Eq. (6); ${\it\Gamma}_1 = {\it\Gamma}_2 = 1$ MHz is taken.
We first set the dc flux bias voltage to $-0.15$ V, corresponding to $\phi_{\rm dc} \approx -0.197\pi$, as indicated by the green solid arrow in Fig. 1. This dc flux bias point is slightly far away from the optimal flux point. A very weak qubit drive tone is applied to prevent excitation of higher levels, thus the Rabi frequency $R/2\pi$ is only about $5$ MHz. The modulation frequency ${\it\Omega}/2\pi$ is set to $55$ MHz to fulfill the second RWA limit. We measure the qubit excited state population $P_{\rm e}$ for various rf flux modulation amplitudes (in terms of peak voltage $V_{\rm p}$ of sinusoidal signals from the microwave signal generator) and qubit drive frequencies $\omega$. The experimental data are plotted in the upper panel of Fig. 3. It shows a certain interference pattern very similar to the LZS one (e.g., see Fig. 7 of Ref. [18]); however, the difference is obvious. In our case, the center band and all sidebands shift towards lower frequencies when increasing the rf flux modulation amplitude $V_{\rm p}$.
cpl-36-12-124204-fig4.png
Fig. 4. (a) and (b) Schematics of rf flux modulations (blue dashed curves) on the qubit been dc biased at $-0.15$ V [red circle in (a)] and 0.1 V [red circle in (b)], respectively. The (black) solid curves are the flux dependent qubit transition frequency, the same as the (red) dotted curve in Fig. 1. (c) and (d) Time evolutions of qubit transition frequency corresponding to (a) and (b), respectively.
To explain the shift towards lower frequencies, we consider the time evolution of $\omega_{10}(\phi)$ under rf flux modulation. Figure 4(a) shows a schematic energy diagram of the qubit dc biased at $\phi_{\rm dc} = -0.197\pi$ (indicated by the red circle) and rf modulated (modulation indicated by the blue-dashed curve). Due to the nonlinear relation between the transition frequency $\omega_{10}$ and the flux $\phi = \phi_{\rm dc} + \phi_{\rm rf}\cos({\it\Omega} t)$, the time evolution of $\omega_{10}(\phi)$ is not exactly sinusoidal, as indicated by the blue dash-dotted curve in Fig. 4(c). The mean value of $\omega_{10}(\phi)$ averaged over one period of $\cos({\it\Omega} t)$, $$ \widetilde\omega_{\rm dc} \equiv \frac{{\it\Omega}}{2\pi } \int_0^{2\pi / {\it\Omega}} \omega_{10}(\phi) dt,~~ \tag {6} $$ sets the center band frequency, as indicated by the green dashed line in Fig. 4(c). It is lower than the transition frequency set by the dc flux bias $\omega_{\rm dc}$ (the red solid line in the same plot). We approximate the modulation amplitude $A$ as $\left\{ \max[\omega_{10}(\phi)] - \min[\omega_{10}(\phi)] \right\} / 2$. By substituting $\widetilde\omega_{\rm dc}$ (replacing $\omega_{\rm dc}$) and $A$ into Eq. (5), as well as taking the measured ${\it\Gamma}_1\approx 1$ MHz and ${\it\Gamma}_2\approx 1$ MHz at dc flux bias $-0.15$ V, we plot the qubit excited state population versus drive frequency and rf flux amplitude in the lower panel of Fig. 3, and find a good agreement between it and the experimental data. The dotted curve in the lower panel of Fig. 3 indicates $\widetilde\omega_{\rm dc}$ as a function of $\phi_{\rm rf}$. We then set the dc flux bias voltage to $+0.1$ V, corresponding to $\phi_{\rm dc} \approx -0.041\pi$, which is close to the flux optimal point, as indicated by the blue dashed arrow in Fig. 1. We again measure the qubit excited state population $P_{\rm e}$ for different rf flux modulation peak voltage $V_{\rm p}$ and qubit drive frequencies $\omega$, with the same Rabi frequency $R/2\pi \approx 5$ MHz and modulation frequency ${\it\Omega} / 2\pi = 55$ MHz. The experimental data is shown in the upper panel of Fig. 5. The same as biased at $-0.15$ V, the center band and all sidebands shift towards lower frequencies when increasing the rf flux modulation amplitude. However, the sidebands have very different behaviors from the previous case.
cpl-36-12-124204-fig5.png
Fig. 5. Upper: Measured $P_{\rm e}$ as a function of drive frequency $\omega$ and rf flux bias peak voltage $V_{\rm p}$, at dc flux bias voltage $+0.1$ V. Lower: $P_{\rm e}$ plotted by numerically solving the master equation Eq. (4).
The interference pattern in $P_{\rm e}$ is basically due to the Bessel function $J_k(A/{\it\Omega})$. $P_{\rm e}$ vanishes when $J_k(A/{\it\Omega})$ goes to zero. When $\phi_{\rm dc}$ is slightly far away from the optimal flux point ($\phi = 0$), increasing $\phi_{\rm rf}$ can cause $J_k(A/{\it\Omega})$ reaches zero several times before $\phi_{\rm dc} + \phi_{\rm rf} = 0$. As long as $\phi_{\rm dc} + \phi_{\rm rf} \leq 0$, though $\omega_{10}(\phi)$ is not exactly sinusoidal, its oscillating frequency is still ${\it\Omega}$, as shown in Fig. 4(c). This means $P_{\rm e}$ vanishes at the same $A$ (so as $V_{\rm p}$) value for $-n$ and $+n$ sidebands. In the upper panel of Fig. 3, we highlight the second zeros (the first zeros are at $A = 0$ for all sidebands) of $-1$ sideband and $+1$ sideband by two dashed circles. For these two sidebands, $P_{\rm e}$ vanishes at the same $V_{\rm p}$ value. In the upper panel of Fig. 5, we also highlight the second zeros of $-1$ sideband and $+1$ sideband by two dashed circles. As one can see, in this $\phi_{\rm dc}$ close to $0$ case, $P_{\rm e}$ vanishes at very different $V_{\rm p}$ values for $-1$ sideband and $+1$ sideband. Figure 4(b) shows a schematic energy diagram of the qubit dc biased at $\phi_{\rm dc} = -0.041\pi$ and rf modulated. Since $\phi_{\rm dc}$ is so close to $0$, $\phi_{\rm dc} + \phi_{\rm rf}\cos({\it\Omega} t)$ can easily cover both positive and negative values, even before $J_k(A/{\it\Omega})$ reaches the second zero. The blue dash-dotted curve in Fig. 4(d) is the time evolution of $\omega_{10}(\phi)$ corresponding to such a flux modulation, which indicates that $\omega_{10}(\phi)$ does not oscillate at a single frequency. For $\omega_{10}(\phi) > \widetilde\omega_{\rm dc}$, the oscillating frequency is about $2{\it\Omega}$; and for $\omega_{10}(\phi) < \widetilde\omega_{\rm dc}$, the oscillating frequency is only about ${\it\Omega}$. Then $P_{\rm e}$ should vanish at $A/{\it\Omega} \approx 3.83$ for $-1$ sideband (below $\widetilde\omega_{\rm dc}$), and vanish at $A/(2{\it\Omega}) \approx 3.83$ for $+1$ sideband (above $\widetilde\omega_{\rm dc}$). This explains why in the upper panel of Fig. 5 two dashed circles are not at the same $V_{\rm p}$ value. We also calculate $P_{\rm e}$ by numerically solving the master equation (4). The numerical results plotted in the lower panel of Fig. 5 confirm the different behaviors of $-n$ sidebands and $+n$ sidebands. In summary, we have thoroughly studied the LZS interference in the nonlinear regime. With a superconducting Xmon qubit, we have experimentally implemented rf flux modulations of the qubit frequency and observed the interference patterns with the dc flux biasing both far away and close to the optimal flux point. The obtained experimental results are also in good agreement with our analytical and numerical simulations. References Quantum information processing with superconducting circuits: a reviewCharge-insensitive qubit design derived from the Cooper pair boxCoherent Josephson Qubit Suitable for Scalable Quantum Integrated CircuitsSuperconducting quantum circuits at the surface code threshold for fault toleranceFast adiabatic qubit gates using only σ z controlEfficient Z gates for quantum computingMotional averaging in a superconducting qubitStückelberg interference in a superconducting qubit under periodic latching modulationFirst-order sidebands in circuit QED using qubit frequency modulationFirst-order sideband transitions with flux-driven asymmetric transmon qubitsSuperconducting qubit in a waveguide cavity with a coherence time approaching 0.1 msParametrically Activated Entangling Gates Using Transmon QubitsAn efficient and compact switch for quantum circuitsQuantum Zeno switch for single-photon coherent transportCoexistence of single- and multi-photon processes due to longitudinal couplings between superconducting flux qubits and external fieldsSynthesis of antisymmetric spin exchange interaction and chiral spin clusters in superconducting circuitsPopulation dynamics and phase effects in periodic level crossingsLandau–Zener–Stückelberg interferometry
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