Basic superconducting circuits for qubit applications: current-biased Josephson junction ; dc SQUID ; rf SQUID ; single Cooper pair box ; the crossed box indicates a combination of a Josephson tunneling element and a junction capacitor connected in parallel.
Single Cooper pair transistor (SCT): SCB with loop-shape bulk electrode connected to the island via two JJs; charge fluctuation on the island produces current fluctuation in the loop.
Quantized energy levels in the potential of a current-biased JJ; the two lower levels form the JJ qubit, the dashed line indicates a leaky level with higher energy.
Double-well potential of the rf SQUID with degenerate quantum levels in the wells. Macroscopic quantum tunneling through the potential barrier introduces a level splitting , and the lowest level pair forms a qubit ; truncation of the junction Hamiltonian, dashed lines indicate potentials of the left and right wells with ground energy levels .
Energy spectrum of the flux qubit versus bias flux (solid lines): it results from hybridization of the flux states (dashed lines).
Persistent current flux qubit with 3 junctions (bold line) connected inductively (left), and galvanically (right) to a measurement dc SQUID.
SCB energy spectrum (bold) versus gate potential: it results from hybridization of the charge states (dashed) due to Josephson tunneling; level anticrossings occur at .
Energy spectrum of microscopic bound Andreev levels; the level splitting is determined by the contact reflectivity.
The Bloch sphere: the Bloch vector represents the states of the two-level system; the vector represents the two-level Hamiltonian; the Bloch vector of the energy eigenstate is parallel (antiparallel) to the vector ; free evolution of the Bloch vector (precession) ; rotation of the Bloch vector under a time dependent perturbation—Rabi oscillation .
Single electron transistor (SET) capacitively coupled to an SCB.
Probability distributions of counted electrons as functions of time after the turning on of the measurement beam of electrons. Courtesy of G. Johansson, Chalmers.
SCT qubit coupled to a readout oscillator. The qubit is operated by input pulses . The readout oscillator is controlled and driven by ac microwave pulses . The output signal will be ac voltage pulses , the amplitude or phase of which may discriminate between the qubit “0” and “1” states.
SCT qubit coupled to a JJ readout quantum oscillator. The JJ oscillator is controlled by dc/ac current pulses adding to the circulating currents in the loop due to the SCT qubit. The output will be dc/ac voltage pulses discriminating between the qubit “0” and “1” states.
Josephson potential energy of the measurement junction during the measurement (left): for the “0” qubit eigenstate there is a well (solid line) confining a level, while for the “1” qubit state there is no well (dashed line). Switching event on the current–voltage characteristic (right).
Control pulse sequences involved in quantum state manipulations and measurement. Top: microwave voltage pulses are applied to the control gate for state manipulation. Middle: a readout dc pulse (DCP) or ac pulse (ACP) is applied to the threshold detector/discriminator a time after the last microwave pulse. Bottom: output signal from the detector. The occurrence of a output pulse depends on the occupation probabilities of the energy eigenstates. A discriminator with threshold converts into a boolean 0/1 output for statistical analysis.
Qubit energy level scheme. The qubit working point and transition energy is marked by the dashed line. The arrow marks the detuned microwave excitation . Population of the upper level as a function of the detuning; the inverse of the half-width of the resonance line gives the total decoherence time .
Decay of the switching probability of the charge–qubit readout junction as a function of the delay time between the excitation and readout pulses. Courtesy of D. Esteve, CEA-Saclay.
Rabi oscillations of the switching probability measured just after a resonant microwave pulse of duration (left); measured Rabi frequency (dots) varies linearly with microwave amplitude (voltage) as expected (right). Courtesy of D. Esteve, CEA-Saclay.
Ramsey fringes of the switching probability after two phase-coherent microwave pulses separated by the time delay . The continuous line represents a fit by exponentially damped cosine function with time constant . The oscillation period coincides with the inverse of the detuning frequency (here ). Courtesy of D. Esteve, CEA-Saclay.
Fixed inductive (flux) coupling of elementary flux qubit. The loops can be separate, or have a common leg like in the figure.
Fixed capacitive coupling of charge qubits.
Capacitive coupling of single JJ qubits.
Two charge qubits coupled to a common oscillator.
Charge (charge–phase) qubits coupled via a common Josephson junction providing phase coupling of the two circuits.
Flux transformer with variable coupling controlled by a SQUID.
Coupled charge qubits with current-controlled phase coupling: the arrow indicates the direction of the controlling bias current.
Variable capacitance tuned by a voltage-controlled SCB.
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