In this paper, we consider a system consisting of two capacitively coupled superconducting islands via Josephson junctions. We show that it can be reduced to two coupling harmonic oscillators under certain conditions, and can be solved exactly in terms of a displacing transformation, a beam-splitter-like transformation, and a squeezing transformation. It is found that the system evolves by a rotated-squeezed-coherent state when the system is initially in a coherent state. Quantum dynamics of the Cooper pairs in the two superconducting islands are investigated. It is shown that the number of the Cooper pairs in the two islands evolves periodically.In this paper, we consider a system consisting of two capacitively coupled superconducting islands via Josephson junctions. We show that it can be reduced to two coupling harmonic oscillators under certain conditions, and can be solved exactly in terms of a displacing transformation, a beam-splitter-like transformation, and a squeezing transformation. It is found that the system evolves by a rotated-squeezed-coherent state when the system is initially in a coherent state. Quantum dynamics of the Cooper pairs in the two superconducting islands are investigated. It is shown that the number of the Cooper pairs in the two islands evolves periodically.

Communications in Theoretical PhysicsJun 15, 20031 citation

Superposition of two mesoscopically distinct quantum states: Coupling a Cooper-pair box to a large superconducting island

Florian Marquardt, Christoph Bruder

We consider a system of two superconducting islands, each of which is coupled to a bulk superconductor via Josephson tunneling. One of the islands represents a ``Cooper-pair box,'' i.e., it is an effective two-level system. The other island has a smaller charging energy and approximates a harmonic oscillator. A capacitive interaction between the islands results in a dependence of the oscillator frequency on the quantum state of the box. Placing the latter in a coherent superposition of its eigenstates and exciting coherent oscillations in the large island will lead to a phase shift of these oscillations depending on the box quantum state, thereby producing a coherent superposition of two ``mesoscopically distinct'' quantum states in the large island.We consider a system of two superconducting islands, each of which is coupled to a bulk superconductor via Josephson tunneling. One of the islands represents a ``Cooper-pair box,'' i.e., it is an effective two-level system. The other island has a smaller charging energy and approximates a harmonic oscillator. A capacitive interaction between the islands results in a dependence of the oscillator frequency on the quantum state of the box. Placing the latter in a coherent superposition of its eigenstates and exciting coherent oscillations in the large island will lead to a phase shift of these oscillations depending on the box quantum state, thereby producing a coherent superposition of two ``mesoscopically distinct'' quantum states in the large island.

Physical review. B, Condensed matterJan 11, 200178 citations

Superposition of two mesoscopically distinct quantum states

Florian Marquardt, Christoph Bruder

We consider a system of two superconducting islands, each of which is coupled to a bulk superconductor via Josephson tunneling. One of the islands represents a "Cooper-pair box", i.e. it is an effective two-level system. The other island has a smaller charging energy and approximates a harmonic oscillator. A capacitive interaction between the islands results in a dependence of the oscillator frequency on the quantum state of the box. Placing the latter in a coherent superposition of its eigenstates and exciting coherent oscillations in the large island will lead to a phase-shift of these oscillations depending on the box quantum state, thereby producing a coherent superposition of two "mesoscopically distinct" quantum states in the large island.We consider a system of two superconducting islands, each of which is coupled to a bulk superconductor via Josephson tunneling. One of the islands represents a "Cooper-pair box", i.e. it is an effective two-level system. The other island has a smaller charging energy and approximates a harmonic oscillator. A capacitive interaction between the islands results in a dependence of the oscillator frequency on the quantum state of the box. Placing the latter in a coherent superposition of its eigenstates and exciting coherent oscillations in the large island will lead to a phase-shift of these oscillations depending on the box quantum state, thereby producing a coherent superposition of two "mesoscopically distinct" quantum states in the large island.

Jan 1, 20010 citations

Cooper-pair tunneling in small junctions with tunable Josephson coupling

Y. Harada, Hideaki Takayanagi, A. A. Odintsov

We investigate Cooper-pair tunneling in a circuit consisting of two dc-superconducting quantum interference devices in series, with a gate capacitively coupled to the central island. Measurements cover a wide range of values of the ratio between Josephson coupling energy ${E}_{J}$ and charging energy ${E}_{C}$ . The $\frac{{E}_{J}}{{E}_{C}}$ ratio dependence of the supercurrent is well described by the orthodox theory provided that strong fluctuations of the Josephson phase due to the electromagnetic environment are taken into account. Our data can be interpreted in terms of squeezing of the charge fluctuations with decreasing $\frac{{E}_{J}}{{E}_{C}}$ ratio.We investigate Cooper-pair tunneling in a circuit consisting of two dc-superconducting quantum interference devices in series, with a gate capacitively coupled to the central island. Measurements cover a wide range of values of the ratio between Josephson coupling energy ${E}_{J}$ and charging energy ${E}_{C}$. The $\frac{{E}_{J}}{{E}_{C}}$ ratio dependence of the supercurrent is well described by the orthodox theory provided that strong fluctuations of the Josephson phase due to the electromagnetic environment are taken into account. Our data can be interpreted in terms of squeezing of the charge fluctuations with decreasing $\frac{{E}_{J}}{{E}_{C}}$ ratio.

Physical review. B, Condensed matterSep 1, 199616 citations

Cooper Pair Box Coupled to a Current-Biased Josephson Junction

F. W. J. Hekking, O. Buisson, Franck Balestro +1 more

We study the dynamics of a quantum superconducting circuit which consists of a Josephson charge qubit, coupled capacitively to a current biased Josephson junction. Under certain conditions, the eigenstates of the qubit and the junction become entangled. We obtain the time evolution of these states in the limit of weak coupling. Rabi oscillations occur, as a result of the spontaneous emission and re-absorption of a single oscillation quantum in the junction. We discuss a possible way to experimentally determine the quantum state of the junction and hence observe the Rabi oscillations.We study the dynamics of a quantum superconducting circuit which consists of a Josephson charge qubit, coupled capacitively to a current biased Josephson junction. Under certain conditions, the eigenstates of the qubit and the junction become entangled. We obtain the time evolution of these states in the limit of weak coupling. Rabi oscillations occur, as a result of the spontaneous emission and re-absorption of a single oscillation quantum in the junction. We discuss a possible way to experimentally determine the quantum state of the junction and hence observe the Rabi oscillations.

arXiv:cond-mat/0201284Jan 16, 20027 citations

Quantum coherent dynamics of two coupled superconducting charge qubits

Yu. A. Pashkin, O. V. Astafiev, Tsuyoshi Yamamoto +5 more

We have analyzed and measured the quantum coherent dynamics of a circuit containing two coupled superconducting charge qubits. Each qubit is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two qubits are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation ofthe qubit circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each qubit by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup the measured pulse-induced probe currents are proportional to the probability for each qubit to have an extra Cooper pai1r after the manipulation. As expected from theory and observed experimentally the measured pulse-induced current in each probe has two frequency components whose position on the frequency axis can be externally controlled. This is a result ofthe inter-qubit coupling which is also responsible for the avoided level crossing that we observed in the qubits' spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape the system remains entangled most ofthe time reaching maximally entangled states at certain instances.We have analyzed and measured the quantum coherent dynamics of a circuit containing two coupled superconducting charge qubits. Each qubit is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two qubits are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation ofthe qubit circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each qubit by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup the measured pulse-induced probe currents are proportional to the probability for each qubit to have an extra Cooper pai1r after the manipulation. As expected from theory and observed experimentally the measured pulse-induced current in each probe has two frequency components whose position on the frequency axis can be externally controlled. This is a result ofthe inter-qubit coupling which is also responsible for the avoided level crossing that we observed in the qubits' spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape the system remains entangled most ofthe time reaching maximally entangled states at certain instances.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIEJun 1, 20050 citations

{E}_{J}

and charging energy

{E}_{C}

. The

\frac{{E}_{J}}{{E}_{C}}

ratio dependence of the supercurrent is well described by the orthodox theory provided that strong fluctuations of the Josephson phase due to the electromagnetic environment are taken into account. Our data can be interpreted in terms of squeezing of the charge fluctuations with decreasing

\frac{{E}_{J}}{{E}_{C}}

ratio.