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arXiv:2401.03627 · Ukrainian Journal of Physics

Interferometry and Dynamics of a Transmon-Type Qubit in Front of a Mirror

M. P. Liul

Published Jan 15, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract We theoretically describe the stationary regime and coherent dynamics of a capacitively shunted transmon-type qubit which is placed in front of a mirror. The considered qubit is irradiated by two signals: pump (dressing) and probe. By changing amplitudes and frequencies of these signals we study the system behaviour. The main tool of our theoretical analysis is solving of the Lindblad equation. We also consider Lindblad superoperators in charge and energy bases and compare the results. Theoretically obtained occupation probability is related to the experimentally measured value. This study helps to understand better the properties of qubit-mirror system and gives new insights about the underlying physical processes.

DOI
10.15407/ujpe70.1.16

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Abstract We theoretically describe the stationary regime and coherent dynamics of a capacitively shunted transmon-type qubit which is placed in front of a mirror. The considered qubit is irradiated by two signals: pump (dressing) and probe. By changing amplitudes and frequencies of these signals we study the system behaviour. The main tool of our theoretical analysis is solving of the Lindblad equation. We also consider Lindblad superoperators in charge and energy bases and compare the results. Theoretically obtained occupation probability is related to the experimentally measured value. This study helps to understand better the properties of qubit-mirror system and gives new insights about the underlying physical processes.
We consider the dynamics and stationary regime of a capacitively shunted transmon-type qubit in front of a mirror. The qubit is affected by probe and dressing signals. By varying the parameters of these signals and then analyzing the probe signal (reflected by the ``atom plus mirror'' system), it is possible to explore the system dynamics, which can be described by the Bloch equation. The obtained time-dependent occupation probabilities are related to the experimentally measured reflection coefficient. The study of this type of dynamics opens up new horizons for better understanding the ``qubit plus mirror'' circuit properties and the underlying physical processes, such as Landau-Zener-St\"uckelberg-Majorana transitions.
We investigate the Landau-Zener-St\"uckelberg-Majorana interferometry of a superconducting qubit in a semi-infinite transmission line terminated by a mirror. The transmon-type qubit is at the node of the resonant electromagnetic (EM) field, ``hiding'' from the EM field. We modulate the resonant frequency of the qubit by applying a sinusoidal flux pump. We perform spectroscopy by measuring the reflection coefficient of a weak probe in the system. Remarkable interference patterns emerge in the spectrum, which can be interpreted as multiphoton resonances in the dressed qubit. Our calculations agree well with the experiments.
We study the dynamics of a qubit-resonator system, when the resonator is driven by two signals. The interaction of the qubit with the high-amplitude driving we consider in terms of the qubit dressed states. Interaction of the dressed qubit with the second probing signal can essentially change the amplitude of this signal. We calculate the transmission amplitude of the probe signal through the resonator as a function of the qubit’s energy and the driving frequency detuning. The regions of increase and attenuation of the transmitted signal are calculated and demonstrated graphically. We present the influence of the signal parameters on the value of the amplification, and discuss the values of the qubit-resonator system parameters for an optimal amplification and attenuation of the weak probe signal.
We study the dynamics of a qubit-resonator system, when the resonator is driven by two signals. The interaction of the qubit with the high-amplitude driving we consider in terms of the qubit dressed states. Interaction of the dressed qubit with the second probing signal can essentially change the amplitude of this signal. We calculate the transmission amplitude of the probe signal through the resonator as a function of the qubit's energy and the driving frequency detuning. The regions of increase and attenuation of the transmitted signal are calculated and demonstrated graphically. We present the influence of the signal parameters on the value of the amplification, and discuss the values of the qubit-resonator system parameters for an optimal amplification and attenuation of the weak probe signal.
Waveguide circuit quantum electrodynamics (waveguide circuit QED) studies light-matter interaction with superconducting circuits in one dimension. In circuit QED, natural atoms are replaced by superconducting qubits consisting of a non-linear Josephson junction, resulting in an anharmonic energy spectrum just like real atoms. With superconducting qubits, it is possible to study quantum optical phenomena and reach new regimes hard to achieve with real atoms due to weak coupling to the electromagnetic field. The reduction to one dimension in waveguide QED increases the electromagnetic field's directionality, which results in reduced losses. In this thesis, we first introduce circuit quantisation, giving the basis for the next part, where we investigate a transmon, a charge-insensitive artificial atom, coupled to a semi-infinite transmission line. An atom coupled to a semi-infinite waveguide is referred to as an atom in front of a mirror and is the subject of all appended papers. We proceed by summarising Paper I and III's main results: in Paper I, we investigate the spontaneous emission of a transmon coupled to a semi-infinite transmission line, where we take time-delay effects into account. We find that the system dynamics strongly depend on the coupling strength to the transmission line and the atom's position with respect to the electromagnetic field, leading to the Purcell effect or the convergence to a dark state with finite excitation probability. In the high-impedance regime, which we investigate in Paper III, the properties of the transmon coupled to the high-impedance transmission line change drastically. It becomes highly reflective and creates its own cavity with the mirror, resulting in the emergence of cavity modes and vacuum Rabi oscillations in the spontaneous emission dynamics. In the next chapter of the thesis, we demonstrate how to quantise an electromagnetic field and derive a light-matter interaction Hamiltonian within dipole approximation. We then give an introduction to open quantum systems and derive the quantum-optical master equation in Lindblad form. Furthermore, we introduce the dressed state picture, where the interaction of light and matter is so strong that the individual energy levels of light and matter are no longer separable. Both the quantum optical master equation and the dressed state picture are relevant in Paper II and IV. In Paper II, an experimental collaboration, we perform several experiments to characterise and discriminate different decay rates of a superconducting qubit coupled to the end of a transmission line. One experiment measured the atomic fluorescence spectral density, which shows an asymmetry for off-resonant driving, resulting from pure dephasing: an effect that we explain in more detail in this thesis and Paper II. In Paper IV, we theoretically investigate amplification mechanisms realised by different set-ups of an atom coupled to a semi-infinite waveguide. In the considered systems, the amplification of a probe field happens either due to population inversion between the pure states or dressed states or multi-photon processes. We find that compared to an open waveguide, we can achieve a higher gain in the amplification with a semi-infinite waveguide.
We study theoretically dynamic properties of an interferometer-type Josephson charge qubit consisting of a single-Cooper-pair transistor (with capacitive gate) closed by a superconducting inductive loop. Analyzing the radio frequency response of the device to microwave fields within a two-band model, we present basic features of a method for observation of various Rabi oscillation modes. For this purpose, we apply the dressed-state approach in conjunction with the rotating-wave approximation to the strongly driven quantum bit. We demonstrate the existence of multiphoton excitations and Rabi frequency transitions between quantum levels connected with them. The resulting oscillations of the supercurrent in the interferometer modify the spectral loop susceptibility. This effect may be detected via the coil of a high-quality tank circuit inductively coupled to the qubit by measuring the tank-qubit impedance variations belonging to it.