We develop a theory for the quantum circuit consisting of a superconducting loop interrupted by four Josephson junctions and pierced by a magnetic flux (either static or time-dependent). In addition to the similarity with the typical three-junction flux qubit, we demonstrate the difference of the four-junction circuit from its three-junction analogue, especially its distinct advantages over the latter. Moreover, the four-junction circuit in the phase regime is also investigated. Our theory provides a tool to explore the physical properties of this four-junction superconducting circuit.We develop a theory for the quantum circuit consisting of a superconducting loop interrupted by four Josephson junctions and pierced by a magnetic flux (either static or time-dependent). In addition to the similarity with the typical three-junction flux qubit, we demonstrate the difference of the four-junction circuit from its three-junction analogue, especially its distinct advantages over the latter. Moreover, the four-junction circuit in the phase regime is also investigated. Our theory provides a tool to explore the physical properties of this four-junction superconducting circuit.

Jan 30, 20160 citations

Four-Qubit Device with Mixed Couplings

M. Grajcar, A. Izmalkov, S. H. W. van der Ploeg +12 more

We present the first experimental results on a device with more than two superconducting qubits. The circuit consists of four three-junction flux qubits, with simultaneous ferro- and antiferromagnetic coupling implemented using shared Josephson junctions. Its response, which is dominated by the ground state, is characterized using low-frequency impedance measurement with a superconducting tank circuit coupled to the qubits. The results are found to be in excellent agreement with the quantum-mechanical predictions.We present the first experimental results on a device with more than two superconducting qubits. The circuit consists of four three-junction flux qubits, with simultaneous ferro- and antiferromagnetic coupling implemented using shared Josephson junctions. Its response, which is dominated by the ground state, is characterized using low-frequency impedance measurement with a superconducting tank circuit coupled to the qubits. The results are found to be in excellent agreement with the quantum-mechanical predictions.

arXiv:cond-mat/0509557Physical Review LettersFeb 2, 200675 citations

A general flux-Based Circuit Theory for Superconducting Josephson Junction Circuits

Yongliang Wang

Superconducting quantum interference devices (SQUIDs), single flux-quantum (SFQ) logic circuits, and quantum Josephson junction circuits have been developed into a family of superconductor integrated circuit, and are widely applied for subtle magnetic-field measurements, energy-efficient computing, and quantum computing, respectively. They are Josephson junction networks composed of Josephson junctions and normal resistor-inductor-capacitor (RLC) components, working with the fluxoid-quantization principle and Josephson effects to achieve unique flux-modulated dynamics and characteristics; they react to the vector potential of magnetic fields rather than the electric potential. However, the conventional circuit diagrams and nodal analysis methods focus on the electric charges flowing though branches and nodes, ignoring dynamics of the magnetic fluxes flowing from loop to loop. This article introduces a general flux-based circuit theory to unify the analyses of Josephson junction circuits and normal RLC circuits. This theory presents a magnetic-flux-generator (MFG) concept to unify Josephson junctions and normal circuit elements, and abstract both Josephson junction circuits and normal RLC circuits as MFG network; it derives a general network equation to describe dynamics of Josephson junction circuits, and invents a kind of magnetic-flux flow (MFF) diagram to depict the working principles of magnetic-flux flows inside Josephson junction circuits. The flux-based theory is complementary to the conventional circuit theories in the design and analysis of superconductor integrated circuits.Superconducting quantum interference devices (SQUIDs), single flux-quantum (SFQ) logic circuits, and quantum Josephson junction circuits have been developed into a family of superconductor integrated circuit, and are widely applied for subtle magnetic-field measurements, energy-efficient computing, and quantum computing, respectively. They are Josephson junction networks composed of Josephson junctions and normal resistor-inductor-capacitor (RLC) components, working with the fluxoid-quantization principle and Josephson effects to achieve unique flux-modulated dynamics and characteristics; they react to the vector potential of magnetic fields rather than the electric potential. However, the conventional circuit diagrams and nodal analysis methods focus on the electric charges flowing though branches and nodes, ignoring dynamics of the magnetic fluxes flowing from loop to loop. This article introduces a general flux-based circuit theory to unify the analyses of Josephson junction circuits and normal RLC circuits. This theory presents a magnetic-flux-generator (MFG) concept to unify Josephson junctions and normal circuit elements, and abstract both Josephson junction circuits and normal RLC circuits as MFG network; it derives a general network equation to describe dynamics of Josephson junction circuits, and invents a kind of magnetic-flux flow (MFF) diagram to depict the working principles of magnetic-flux flows inside Josephson junction circuits. The flux-based theory is complementary to the conventional circuit theories in the design and analysis of superconductor integrated circuits.

arXiv:2308.01693Aug 3, 20230 citations

Quantum behavior of a superconducting loop interrupted by a double-barrier Josephson junction

Roberto De Luca

The Hamiltonian of a superconducting thin loop interrupted by a double-barrier Josephson junction is derived. The parameter ε, which takes account of the difference in the Josephson coupling of the two single junctions in the SISIS structure, is seen to allow variation of the energy gap ΔE01 between the ground state and the first excited flux number state of the system. An additional parameter γ, describing the coupling between the outer electrodes of the double-barrier Josephson junction, is introduced. Persistent currents in the system are studied in terms of the usual parameter α, representing the ratio between the Josephson energy and the magnetic energy, and of the externally applied flux.The Hamiltonian of a superconducting thin loop interrupted by a double-barrier Josephson junction is derived. The parameter ε, which takes account of the difference in the Josephson coupling of the two single junctions in the SISIS structure, is seen to allow variation of the energy gap ΔE01 between the ground state and the first excited flux number state of the system. An additional parameter γ, describing the coupling between the outer electrodes of the double-barrier Josephson junction, is introduced. Persistent currents in the system are studied in terms of the usual parameter α, representing the ratio between the Josephson energy and the magnetic energy, and of the externally applied flux.

Journal of Applied PhysicsDec 1, 20103 citations

Variable superconducting quantum-interference device: Theory

H. J. Fink, V. Grünfeld, S. M. Roberts

A dc superconducting quantum interference device (SQUID) consisting of a symmetric ring circuit with four Josephson junctions and two independent external currents ${I}_{1}$ and ${I}_{2}$ , which enter the circuit at different points, has a phase boundary between the fully superconducting and the normal states that is different from that of a conventional SQUID. The control current ${I}_{2}$ modifies the magnetic flux (\ensuremath{\varphi}) periodicity and the value of the maximum measuring current ${I}_{1m}$ (\ensuremath{\varphi}). For values of ${I}_{2}$ > ${I}_{c}$ (critical current of Josephson junction) the fully resistive state exists over flux intervals centered around (n+(1/2)${\ensuremath{\varphi}}_{0}$, where n is an integer and ${\ensuremath{\varphi}}_{0}$ is the fluxoid quantum. A superconducting circuit without Josephson junctions is suggested which should have a phase boundary similar to that found here. Minimization procedures which are usually employed in analyzing SQUID's are discussed.A dc superconducting quantum interference device (SQUID) consisting of a symmetric ring circuit with four Josephson junctions and two independent external currents ${I}_{1}$ and ${I}_{2}$, which enter the circuit at different points, has a phase boundary between the fully superconducting and the normal states that is different from that of a conventional SQUID. The control current ${I}_{2}$ modifies the magnetic flux (\ensuremath{\varphi}) periodicity and the value of the maximum measuring current ${I}_{1m}$(\ensuremath{\varphi}). For values of ${I}_{2}$>${I}_{c}$ (critical current of Josephson junction) the fully resistive state exists over flux intervals centered around (n+(1/2)${\ensuremath{\varphi}}_{0}$, where n is an integer and ${\ensuremath{\varphi}}_{0}$ is the fluxoid quantum. A superconducting circuit without Josephson junctions is suggested which should have a phase boundary similar to that found here. Minimization procedures which are usually employed in analyzing SQUID's are discussed.

Physical review. B, Condensed matterJul 1, 19875 citations

Decoherence rates in complex Josephson qubit circuits

David Peter Divincenzo, Frederico Brito, R. H. Koch

A complete analysis of the decoherence properties of a Josephson junction qubit is presented. The qubit is of the flux type and consists of two large loops forming a gradiometer, one small loop, and three Josephson junctions. We develop a complete, quantitative description of the inductances and capacitances of the circuit. Including two stray capacitances makes the quantum mechanical modeling of the system five dimensional. To make the required calculation tractable, we devise a general Born-Oppenheimer approximation which reduces the effective dimensionality in the calculation to one. Contributions to relaxation $({T}_{1})$ and dephasing $({T}_{\ensuremath{\phi}})$ arising from two different control circuits, one coupled to the small loop and one coupled to a large loop, is computed. We explore ${T}_{1}$ and ${T}_{\ensuremath{\phi}}$ along an optimal line in the space of applied fluxes; along this ``S line'' we see significant and rapidly varying contributions to the decoherence parameters, primarily from the circuit coupling to the large loop.A complete analysis of the decoherence properties of a Josephson junction qubit is presented. The qubit is of the flux type and consists of two large loops forming a gradiometer, one small loop, and three Josephson junctions. We develop a complete, quantitative description of the inductances and capacitances of the circuit. Including two stray capacitances makes the quantum mechanical modeling of the system five dimensional. To make the required calculation tractable, we devise a general Born-Oppenheimer approximation which reduces the effective dimensionality in the calculation to one. Contributions to relaxation $({T}_{1})$ and dephasing $({T}_{\ensuremath{\phi}})$ arising from two different control circuits, one coupled to the small loop and one coupled to a large loop, is computed. We explore ${T}_{1}$ and ${T}_{\ensuremath{\phi}}$ along an optimal line in the space of applied fluxes; along this ``S line'' we see significant and rapidly varying contributions to the decoherence parameters, primarily from the circuit coupling to the large loop.

arXiv:cond-mat/0510843Physical Review BJul 25, 200628 citations

A dc superconducting quantum interference device (SQUID) consisting of a symmetric ring circuit with four Josephson junctions and two independent external currents

{I}_{1}

and

{I}_{2}

, which enter the circuit at different points, has a phase boundary between the fully superconducting and the normal states that is different from that of a conventional SQUID. The control current

{I}_{2}

modifies the magnetic flux (\ensuremath{\varphi}) periodicity and the value of the maximum measuring current

{I}_{1m}

(\ensuremath{\varphi}). For values of

{I}_{2}

{I}_{c}

(critical current of Josephson junction) the fully resistive state exists over flux intervals centered around (n+(1/2)${\ensuremath{\varphi}}_{0}$, where n is an integer and ${\ensuremath{\varphi}}_{0}$ is the fluxoid quantum. A superconducting circuit without Josephson junctions is suggested which should have a phase boundary similar to that found here. Minimization procedures which are usually employed in analyzing SQUID's are discussed.

({T}_{1})

and dephasing $({T}_{\ensuremath{\phi}})$ arising from two different control circuits, one coupled to the small loop and one coupled to a large loop, is computed. We explore

{T}_{1}

and ${T}_{\ensuremath{\phi}}$ along an optimal line in the space of applied fluxes; along this ``S line'' we see significant and rapidly varying contributions to the decoherence parameters, primarily from the circuit coupling to the large loop.