Interferometry and dynamics of a transmon-type qubit in front of a mirror

Considerable attention is being drawn to topics related to quantum computing [Nielsen and Chuang, 2010]. Among the promising building blocks for such devices, superconducting qubits stand out, operating at nanosecond scales with millisecond coherence times [Kockum and Nori, 2019]. These qubits, controlled by microwaves and featuring lithographic scalability [Oliver and Welander, 2013], hold great potential for the advancement of quantum computers.

An essential aspect is the exploration of superconducting qubits within a semi-infinite transmission line [Hoi et al., 2015], crucial for quantum electrodynamics, particularly waveguide quantum electrodynamics (WQED) [Kannan et al., 2020]. Notably, in Ref. [Wen et al., 2018], it was revealed that a transmon qubit at the transmission line’s end could amplify a probe signal with an amplitude gain of up to 7%. In comparison, single quantum dot [Xu et al., 2007] and natural atoms [Wu et al., 1977] exhibit much lower signal amplifications: 0.005% and 0.4%, respectively. Our investigation could address pertinent physics issues in WQED, including dynamics in atom-like mirrors [Mirhosseini et al., 2019], collective Lamb shift [Wen et al., 2019], the dynamical Casimir effect [Wilson et al., 2011], cross-Kerr effect [Hoi et al., 2013], generation of non-classical microwaves [Hoi et al., 2012], probabilistic motional averaging [Karpov et al., 2020], photon routing [Hoi et al., 2011], and more.

Driven quantum systems find description through Landau-Zener-Stückelberg-Majorana (LZSM) transitions [Ivakhnenko et al., 2023; Shevchenko, 2019; Liul et al., 2023a]. When driven periodically, these systems exhibit interference. LZSM interferometry, crucial for studying fundamental quantum phenomena and characterizing quantum systems, was explored in Refs. [Ivakhnenko et al., 2023]. Additionally, quantum logic gates can be implemented using LZSM dynamics [Campbell et al., 2020].

This research is an extension of Ref. [Liul et al., 2023b], where the developed theory was approved experimentally. In this article we build the dependencies which were not considered before. The interferograms and dynamic plots presented in this paper allow to understand better underlying physical processes and thus could be interesting and useful for setting up future experiments. Also we solve the Lindblad equation with Lindblad superoperators in different bases: a charge and an energy one and then conclude that the superoperators have to be in the charge basis, since the equation is also written in this basis. Such a comparison is very important from the methodological point of view.

The rest of the paper is organized as follows. Section II describes the theoretical aspects of the research: the system Hamiltonian and the Lindblad master equation, which was used for obtaining the results, are introduced. In Sec. III we study the relations between charge and energy bases. Section IV shows the obtained interferograms and Sec. V time dependencies. In Sec. VI our conclusions are presented.

In the study [Wen et al., 2020], the experimentally studied reflection coefficient $\left|r\right|$ is linked to the theoretically calculated probability of an upper charge level occupation $P_{1}$. An increase in $P_{1}$ corresponds to a decrease in $\left|r\right|$. In that article the computations were performed in the charge basis. In our analysis, we maintain this alignment between theoretical predictions and experimental results, conducting calculations in the charge basis. The transmon-type qubit which placed in front of a mirror can be characterized by the Hamiltonian:

where the diagonal part describes the energy-level modulation

the off-diagonal part corresponds to the coupling to the probe signal

To exclude the fast driving from the considered Hamiltonian the authors of Ref. [Wen et al., 2020] considered the unitary transformation $U=\exp\left(-i\omega_{\mathrm{p}}\sigma_{z}t/2\right)$ and the rotating-wave approximation [Silveri et al., 2015] and as a result one obtained the new Hamiltonian

where

Here $\delta$ is the energy-level modulation amplitude, $G$ describes the coupling to the probe signal (Rabi frequency of the probe signal). According to Ref. [Wen et al., 2020]

where $\omega_{\mathrm{node}}$ characterizes the qubit location in a semi-infinite transmission line [corresponding to the blue curve in Fig. 1], and $G_{0}$ is proportional to the probe signal amplitude. Moreover, we should mention that if $\omega_{\mathrm{p}}=\omega_{\mathrm{node}}$ the qubit is “hidden” or “decoupled” from the transmission line, with $G=0$.

To describe the qubit dynamics, we utilize the Lindblad equation, which in the charge basis with the Hamiltonian (4) has the form:

where $\rho\leavevmode\nobreak\ =\leavevmode\nobreak\ \begin{pmatrix}\rho_{00}&\rho_{01}\\ \rho^{*}_{01}&1-\rho_{00}\end{pmatrix}$ is the density matrix, such that $P_{1}=1-\rho_{00}$. The Lindblad superoperator $\breve{L}_{\alpha}$ describes the system relaxation induced by interactions with the environment,

where $\left\{a,b\right\}=ab+ba$ is the anticommutator. For a qubit there exist two relaxation channels: dephasing (described by $L_{\phi}$) and energy relaxation (described by $L_{\mathrm{relax}}$). In the energy basis the corresponding operators can be expressed in the following form:

with $\sigma^{+}=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$, $\sigma_{z}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$, $\mathrm{\Gamma}_{1}$ being the qubit relaxation, $\mathrm{\Gamma}_{2}=\mathrm{\Gamma}_{1}/2+\mathrm{\Gamma}_{\phi}$ is the decoherence rate, $\mathrm{\Gamma}_{\phi}$ is the pure dephasing rate.

In this section we will discuss the question related to the selection of an appropriate basis. As it was mentioned, the Hamiltonian (4) and the Lindnlad equation (9) were written in a charge basis, while the relaxations $\Gamma_{1}$ and $\Gamma_{\phi}$ are determined in the energy basis. So one should transfer Lindblad operators from the energy basis to the charge one. To study this question let’s rewrite the Hamiltonian (4) in the form:

where the constant component is equal to

and time-dependent part can be written as

In other words the Hamiltonian (4) can be considered as an effective Hamiltonian of a two-level system with tunneling amplitude $G$, and excitation $h(t)\leavevmode\nobreak\ =\leavevmode\nobreak\ -\leavevmode\nobreak\ \frac{\hbar}{2}\left(\Delta\omega+\delta\sin\omega_{\mathrm{pump}}t\right)$.

The transfer matrix which provides transfer from a charge basis to an energy one has a form:

where

The matrix $S$ diagonalizes time-independent part of the Hamiltonian $H_{0}$ (13):

where $E_{0}$, $E_{1}$ — are eigenvalues of the Hamiltonian $H_{0}$.

Therefore, one can transfer Lindblad superoperators $L_{\alpha}^{\mathrm{energy}}$ from energy basis to the charge one $L_{\alpha}$ by using the relation:

Substitution of Eqs. (11, 19) to the Lindblad equation (9) will give the necessary dependence $P_{1}\leavevmode\nobreak\ =\leavevmode\nobreak\ P_{1}(t)$, which is used for building theoretical plots.

By solving Eq. (9), we get $P_{1}$ as a function of time $t$, the pump frequency $\omega_{\rm pump}$, the pump amplitude $\delta$, the probe frequency $\omega_{\mathrm{p}}$, the signal amplitude $G$. The occupation probability of the upper charge level is a function of all these parameters, $P_{1}\leavevmode\nobreak\ =\leavevmode\nobreak\ P_{1}(t,\leavevmode\nobreak\ \omega_{\rm pump},\leavevmode\nobreak\ \omega_{\rm p},\leavevmode\nobreak\ \delta,\leavevmode\nobreak\ G)$. We can also calculate the dependencies for $P_{1}$ in the stationary regime by averaging the results over time.

To obtain the time-averaged values, we analyzed the curve $P_{1}\leavevmode\nobreak\ =\leavevmode\nobreak\ P_{1}(t)$ to select the minimum time $t_{\mathrm{min}}$ after which the amplitude of oscillations does not change. And then averaging was applied for the interval $[t_{\mathrm{min}},t_{\mathrm{final}}]$, where $t_{\mathrm{final}}$ corresponds to the pulse off time. We determined that for our case $t_{\mathrm{min}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 1.5\leavevmode\nobreak\ \mu{\rm s}$ and $t_{\mathrm{final}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 2.0\leavevmode\nobreak\ \mu{\rm s}$ (see Ref. [Liul et al., 2023b] for more details).

In this section, we theoretically investigate the dependence of the qubit upper charge level occupation probability $P_{1}$ on the pump frequency $\omega_{\rm pump}$ and the probe frequency $\omega_{\rm p}$. Note that the experimental study of such dependence was not conducted. But since one found the values of the fitting parameters in Ref. [Liul et al., 2023b] we can construct such a dependence. It is shown in Fig. 2, (a) corresponds to the pump amplitude $\delta\leavevmode\nobreak\ =\leavevmode\nobreak\ 10\,\mathrm{MHz}$, for (b) the pump amplitude is equal to $\delta\leavevmode\nobreak\ =\leavevmode\nobreak\ 20\,\mathrm{MHz}$. From the analysis of interferograms, it becomes clear that the larger the pumping amplitude $\delta$, the larger the amplitude along the $\omega_{\rm p}$ axis. Similar dependencies can be observed in Ref. [Wen et al., 2020], where only the stationary case was considered.

Such interferograms are useful not only for obtaining fitting parameters between theory and experiment, but they also play a key role in system characterization. In particular, they can help to:

• •

  estimate the system decoherence time;

• •

  provide a tool for calibrating signal strength during experimental studies;

• •

  open up new possibilities for multiphoton spectroscopy.

We considered the dynamics and stationary mode of a transmon-type capacitive shunted qubit in front of a mirror, which is affected by two signals: probing and pumping.

To analyze the steady state, the upper charge level occupation probability $P_{1}$ was calculated as a function of the pump frequency $\omega_{\rm pump}$ and the probe frequency $\omega_{\rm p}$ at the fixed pump amplitude $\delta$ and the probe amplitude $G$. The resulting dependencies can be used to obtain fitting parameters; system decoherence time estimation; power calibration; studying multiphoton spectroscopy.

To analyze the dynamics, we plotted the dependencies of the upper charge level occupation probability $P_{1}$ on (a) time $t$ and the probe frequency $\omega_{\rm p}$ at different pump frequencies $\omega_{\rm pump}$; (b) time $t$ and the probe frequency $\omega_{\rm p}$ at different values of the probe amplitude $G$ and the relaxation rate $\Gamma_{1}$ for the pump frequency $\omega_{\rm pump}/2\pi\leavevmode\nobreak\ =\leavevmode\nobreak\ 5\,\mathrm{MHz}$. The obtained results can be used in setting up future experiments.

The Lindblad equation with Lindblad superoperators in different bases: a charge and an energy one was solved. One concluded that the superoperators had to be in the charge basis, since the equation is also written in this basis. Such a comparison should be useful from the methodological point of view.