S. N. Shevchenko

Landau–Zener–Stückelberg interferometry

Abstract A transition between energy levels at an avoided crossing is known as a Landau–Zener transition. When a two-level system (TLS) is subject to periodic driving with sufficiently large amplitude, a sequence of transitions occurs. The phase accumulated between transitions (commonly known as the Stuckelberg phase) may result in constructive or destructive interference. Accordingly, the physical observables of the system exhibit periodic dependence on the various system parameters. This phenomenon is often referred to as Landau–Zener–Stuckelberg (LZS) interferometry. Phenomena related to LZS interferometry occur in a variety of physical systems. In particular, recent experiments on LZS interferometry in superconducting TLSs (qubits) have demonstrated the potential for using this kind of interferometry as an effective tool for obtaining the parameters characterizing the TLS as well as its interaction with the control fields and with the environment. Furthermore, strong driving could allow for fast and reliable control of the quantum system. Here we review recent experimental results on LZS interferometry, and we present related theory.

Consistency of ground state and spectroscopic measurements on flux qubits

We compare the results of ground state and spectroscopic measurements carried out on superconducting flux qubits which are effective two-level quantum systems. For a single qubit and for two coupled qubits we show excellent agreement between the parameters of the pseudospin Hamiltonian found using both methods. We argue that by making use of the ground state measurements the Hamiltonian of N coupled flux qubits can be reconstructed as well at temperatures smaller than the energy level separation. Such a reconstruction of a many-qubit Hamiltonian can be useful for future quantum information processing devices.

Multiphoton transitions between energy levels in a phase-biased Cooper-pair box

We investigated both theoretically and experimentally dynamic features of a phase-biased charge qubit consisting of a single-Cooper-pair transistor closed by a superconducting loop. The effective inductance of the qubit was probed by a high-quality tank circuit. In the presence of a microwave power, with a frequency of the order of the qubit energy level separation, an alteration of the qubit inductance was observed. We demonstrate that this effect is caused by the redistribution of the qubit level population. The excitation of the qubit by one-, two-, and three-photon processes was detected. Quantitative agreement between theory and experimental data was found.

Quantum behavior of a flux qubit coupled to a resonator

A detailed theory for a system of a superconducting qubit coupled to a transmission line resonator is presented. We describe the system by solving analytically and numerically the master equation for the density matrix, which includes a dissipative Lindblad term. We calculate the transmission coefficient, which provides a way to probe the dressed states of a qubit. The theoretical results are related to experiments with intermediate coupling between a qubit and a resonator when the coupling energy is of the same order as the qubit relaxation rate.

Resonant excitations of single and two-qubit systems coupled to a tank circuit

The interaction of flux qubits with a low frequency tank circuit is studied. It is shown that changes in the state of the interacting qubits influence the effective inductance and resistance of the circuit, which is the essence of the so-called impedance measurement technique. The multiphoton resonant excitations in both single flux qubits and pairs of coupled flux qubits are investigated. In particular, we compare our theoretical results with recent spectroscopy measurements, Landau-Zener interferometry, and the multiphoton fringes.

Dynamic behavior of Josephson-junction qubits: crossover between Rabi oscillations and Landau–Zener transitions

We study the dynamic behavior of a quantum two-level system with periodically varying parameters by solving numerically the master equation for the density matrix. Two limiting cases are considered: multiphoton Rabi oscillations and Landau–Zener transitions. The approach is applied to the description of the dynamics of superconducting qubits. In particular, the case of the interferometer-type charge qubit with periodically varying parameters (gate voltage or magnetic flux) is investigated. The time-averaged energy level populations are calculated as functions of the qubit’s control parameters.

Gate-Sensing Coherent Charge Oscillations in a Silicon Field-Effect Transistor

Quantum mechanical effects induced by the miniaturization of complementary metal-oxide-semiconductor (CMOS) technology hamper the performance and scalability prospects of field-effect transistors. However, those quantum effects, such as tunneling and coherence, can be harnessed to use existing CMOS technology for quantum information processing. Here, we report the observation of coherent charge oscillations in a double quantum dot formed in a silicon nanowire transistor detected via its dispersive interaction with a radio frequency resonant circuit coupled via the gate. Differential capacitance changes at the interdot charge transitions allow us to monitor the state of the system in the strong-driving regime where we observe the emergence of Landau-Zener-Stückelberg-Majorana interference on the phase response of the resonator. A theoretical analysis of the dispersive signal demonstrates that quantum and tunneling capacitance changes must be included to describe the qubit-resonator interaction. Furthermore, a Fourier analysis of the interference pattern reveals a charge coherence time, T2 ≈ 100 ps. Our results demonstrate charge coherent control and readout in a simple silicon transistor and open up the possibility to implement charge and spin qubits in existing CMOS technology.

Amplification and attenuation of a probe signal by doubly dressed states

We analyze a system composed of a qubit coupled to the electromagnetic fields in two high quality quantum oscillators. A particular realization of such a system is the superconducting qubit coupled to a transmission-line resonator driven by two signals with frequencies close to the resonators harmonics. This doubly driven system can be described in terms of the doubly dressed qubit states. Our calculations demonstrate the possibility to change the number of photons in the resonator and the transmission of the fundamental-mode signal over a wide parameter range exploiting resonances with the dressed qubit. Experiments show that in the case of high quality resonators the dressed energy levels and corresponding resonance conditions can be probed, even for high driving amplitudes. The interaction of the qubit with photons of two harmonics can be used for the creation of quantum amplifiers or attenuators.

Inverse Landau-Zener-Stückelberg problem for qubit-resonator systems

We consider theoretically a superconducting qubit - nanomechanical resonator (NR) system, which was realized by LaHaye et al. [Nature 459, 960 (2009)]. First, we study the problem where the state of the strongly driven qubit is probed through the frequency shift of the low-frequency NR. In the case where the coupling is capacitive, the measured quantity can be related to the so-called quantum capacitance. Our theoretical results agree with the experimentally observed result that, under resonant driving, the frequency shift repeatedly changes sign. We then formulate and solve the inverse Landau-Zener-Stuckelberg problem, where we assume the driven qubits state to be known (i.e. measured by some other device) and aim to find the parameters of the qubits Hamiltonian. In particular, for our system the qubits bias is defined by the NRs displacement. This may provide a tool for monitoring of the NRs position.

Multiphoton transitions in Josephson-junction qubits (Review Article)

Two basic physical models, a two-level system and a harmonic oscillator, are realized on the mesoscopic scale as coupled qubit and resonator. The realistic system includes moreover the electronics for controlling the distance between the qubit energy levels and their populations and to read out the resonator’s state, as well as the unavoidable dissipative environment. Such rich system is interesting both for the study of fundamental quantum phenomena on the mesoscopic scale and as a promising system for future electronic devices. We present recent results for the driven superconducting qubit–resonator system, where the resonator can be realized as an LC circuit or a nanomechanical resonator. Most of the results can be described by the semiclassical theory, where a qubit is treated as a quantum two-level system coupled to the classical driving field and the classical resonator. Application of this theory allows to describe many phenomena for the single and two coupled superconducting qubits, among which ar...