Qubit (quantum two-level system) coupled to a classical resonator. (a) Schematic diagram of the model qubit–resonator system. The qubit is represented by the two-level system with the two states, − and +, and with the energy difference ΔE. The resonator is demonstrated as the spring oscillator with the elasticity coefficient k 0. As described in the main text, influence of the qubit on the resonator can be described by introducing the effective elasticity coefficient k eff, which includes the qubit’s-state-dependent (or, parametric, for brevity) elasticity coefficient kq . (b) The flux qubit coupled via the mutual inductance M to the LC resonator. This can be described by introducing effective qubit’s-state-dependent inductance L eff, which includes the parametric inductance Lq in parallel to the tank’s inductance L 0. (c) The impact of the charge qubit on the nanomechanical resonator’s state can be described by introducing the effective qubit’s-state-dependent capacitance C eff, which includes the parametric capacitance Cq in parallel to the resonator’s capacitance C 0.
Flux qubit coupled inductively to an LCR (tank) circuit. The flux qubit is pierced by the magnetic flux Φ x induced by the current in the controlling coil and by the current in the tank’s inductor. The qubit is coupled via the mutual inductance M to the tank circuit. The resonant tank circuit consists of the inductor L 0, capacitor C 0, and resistor R 0; the circuit is biased with an RF current I bias. The tank voltage V is the measurable value.
Charge qubit probed by a nanomechanical resonator. The charge qubit is the Cooper-pair box, controlled by the magnetic flux Φ and the gate voltage VCPB + VMW . The resonator probing the qubit’s state here is the NR, which is characterized by the displacement at the midpoint x. The voltage-biased NR is measured through its resonance frequency shift Δω NR (Ref.88).
The equilibrium-state measurement. The dependence of the tank phase shift on the flux detuning f dc = Φdc/Φ0 − 1/2, when the qubit is thermally excited. The curves are plotted for kBT/h = 0.2, 0.5, 0.7, 1, 2, 4, and 8 GHz. Left inset: corresponding experimental results (Ref. 134). Right inset: temperature dependence of the width Δf dc of the dip at half-depth in the phase shift, shown in the main panel (Ref. 87).
Resonant excitation of the charge qubit probed by the tank circuit. The phase shift δ of the tank circuit coupled to the charge qubit, calculated theoretically (left) and measured (right). Panels a and b show the dependence on the gate voltage, while in c and d the dependence on the flux is demonstrated. Black and gray arrows in c demonstrate the positions of 1 - and 2-photon resonant transitions, and the arrows in d mark 1 -, 2 -, and 3-photon excitations (Ref. 30).
Low-amplitude one-photon resonant excitation of a flux qubit. (a) Energy levels E ±(f dc) matched by the driving at frequencies shown by the numbers and the arrows of the respective length. (b) and (c) Theoretically calculated and experimentally measured amplitude of the tank voltage v versus flux detuning f dc for different driving frequencies. (The upper curves are shifted vertically.) The one-photon excitations at ω/2π = 18, 5, and 3.5 GHz, demonstrated in (b) and (c), are explained by the arrows to the left in the energy diagram (a), while the arrows to the right of the length ω/2π = 4.15 GHz explain the multiphoton resonances in Fig. 7 (Refs. 41 and 87).
Multiphoton excitations of a flux qubit. Theoretically calculated dependence of the phase shift δ (a) and the amplitude v (b) on the bias current frequency ω p and the flux detuning f dc. (c, d) Experimentally measured phase shift δ and the amplitude v (Ref. 87).
LZS interferometry for the flux qubit probed by the tank circuit. The calculated (a) and measured (b) dependence of the tank phase shift on the flux detuning f dc and on the driving flux amplitude f ac (Refs. 41 and 87).
Scheme of two coupled qubits. The two flux qubits are coupled to each other, to the dc and μw lines, as well as to an unavoidable dissipative environment. The convenient model for description of the environment is the bath of harmonic oscillators. The system of two coupled qubits is also assumed to be coupled to the measuring resonant circuit (which is not shown here), as in Fig. 2 (Ref. 54).
Spectroscopy of the two-qubit system. The measured dependence of the phase shift δ on the flux biases fa and fb : ground-state measurement (without microwave excitation) (a); with weak microwave excitation at the driving frequencies ω/2π = 14.1 (b), 17.6 (c), and 20.7 (d) GHz (Ref. 41).
Characterizing strongly-driven two-qubit system. Calculated and plotted as functions of the bias f dc are four energy levels (a), total probability of the currents in two qubits to flow clockwise Z (b), the tank circuit voltage phase shift δ (c), the entanglement measure ɛ (d) (Ref. 87).
Imaging the multiphoton transitions in the two-qubit system. The resonant excitation of the qubits system is visualized by the tank voltage amplitude (a, c, e). The position of the resonant transitions can be understood by comparing with the respective energy contour lines (b, d, f) (Ref. 53).
Ladder-type transitions in the two-qubit system. Calculated as functions of the flux fb (at fa = 0.015): the energy levels (a), transition matrix elements Tnm (b), the occupation probabilities Pi (c) (Ref. 53).
Energy level structure with J = 0. (a) One-qubit and two-qubit energy levels as functions of the magnetic flux fb at fixed flux fa . The arrows show the fastest relaxation, which is assumed to relate to the qubit a. (b) and (c) Schemes for three- and four-level lasing at fb = fbL and fb = fbR . The driving magnetic flux pumps (P) the upper level; fast relaxation (R) creates the population inversion; the two operating levels can be used for lasing (L) (Ref. 54).
Two-qubit lasing and stimulated transition. The time-dependent occupation probabilities are plotted for one- (a) and two-photon (b) driving. The driving and fast relaxation create the inverse population between the levels |2〉 and |1〉; then the stimulating signal fL cos ω Lt is turned on Ref. 54.
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