Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2304.13351 · New Journal of Physics

A quantum fluctuation description of charge qubits

Fabio Benatti, Federico Carollo, Roberto Floreanini B, Heide Narnhofer, F. Valiera

Published Dec 29, 2023• 4 citations
arXivHTMLPDF

Abstract

Abstract We consider a specific instance of a superconducting circuit, the so-called charge-qubit, consisting of a capacitor and a Josephson junction that we describe by means of the BCS microscopic model in terms of two tunnelling superconducting systems in the strong-coupling quasi-spin formulation. Then, by means of collective observables we derive the Hamiltonian governing the quantum behaviour of the circuit in the limit of a large numberNof quasi-spins. Our approach relies on suitablequantum fluctuations, i.e. on collective quasi-spin operators, different from mean-field observables, that retain a quantum character in the large-Nlimit. These collective operators generate the Heisenberg algebra on the circle and we show that their dynamics reproduces the phenomenological one generated by the charge qubit Hamiltonian obtained by quantizing the macroscopic classical Hamiltonian of the circuit. The microscopic derivation of the emergent, large-Nbehaviour provides a rigorous setting to investigate more in detail both general quantum circuits and quantum macroscopic scenarios; in particular, in the specific case of charge-qubits, it allows to explicitly obtain the temperature dependence of the critical Josephson current in the strong coupling regime, a result not accessible using standard approximation techniques.

DOI
10.1088/1367-2630/ad19ac

Related papers

Superconducting charge qubits from a microscopic many-body perspective

D A Rodrigues, T P Spiller, James F. Annett +1 more

The quantized Josephson junction equation that underpins the behaviour of charge qubits and other tunnel devices is usually derived through canonical quantization of the classical macroscopic Josephson relations. However, this approach may neglect effects due to the fact that the charge qubit consists of a superconducting island of finite size connected to a large superconductor. We show that the well-known quantized Josephson equation can be derived directly and simply from a microscopic many-body Hamiltonian. By choosing the appropriate strong-coupling limit we produce a highly simplified Hamiltonian that nevertheless allows us to go beyond the mean-field limit and predict further finite-size terms in addition to the basic equation.

arXiv:0709.3460•Journal of Physics Condensed Matter•Sep 28, 2007•5 citations

BCS models of Josephson qubits I. Energy spectra

Robert Alicki, W. Miklaszewski

There exists a large number of experimental and theoretical results supporting the picture of macroscopic qubits implemented by nanoscopic Josephson junctions of three different types -- charge qubit, flux qubit and phase qubit. The standard unified description of such systems is based on the formal quantization of the phenomenological Kirchhoff equations for the corresponding circuits. In this paper a simplified version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions. For these models the state-dependent individual tunneling of Cooper pairs couples ground pair states with excited pair states what leads to a more complicated structure of the lowest lying energy levels. In particular, the highly degenerate levels emerge, which act as probability sinks for the qubit. These models allow also for the coupling to phonons as an efficient mechanism of relaxation for all types of junctions. The alternative formulas concerning basic spectral parameters of superconducting qubits are presented and compared with the experimental data. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.

arXiv:1012.0140•Dec 1, 2010•0 citations

Strong-coupling BCS models of Josephson qubits

Robert Alicki, W. Miklaszewski

The strong-coupling version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions--charge qubit, flux qubit and phase qubit. Applied to Josephson qubits it yields a more complicated structure of the lowest-lying energy levels than that obtained from phenomenological models based on quantization of the Kirchhoff equations. In particular, highly degenerate levels emerge, which act as probability sinks for the qubit. The alternative formulae concerning spectra of superconducting qubits are presented and compared with the experimental data. In contrast to the existing theories those formulae contain microscopic parameters of the model. In particular, for the first time, the density of Cooper pairs at zero temperature is estimated for an Al-based flux qubit. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.

Journal of Physics Condensed Matter•Dec 10, 2012•1 citation

BCS model of Cooper Pair Box

Robert Alicki

The standard phenomenological Hamiltonian of a small superconducting Josephson junction in the charge regime (Cooper Pair Box) produces a model of the effective charge qubit with possible applications to quantum information processing. In this note a new model based on the BCS Hamiltonian with individual tunneling yields an effective multi-level picture with a highly degenerated level placed between the ground state and the excited state. Unlike in the standard approach, the excited Cooper pairs play here an important role. For such a system coupled to a zero temperature bath the additional levels act as a probability sink. In contrast to the standard large-spin model the coupling to phonons can be an effective source of dissipation. This model provides also alternative explanations of various effects observed in experiments and sheds new light on the issue of Josephson junctions as macroscopic quantum systems.

arXiv:1002.3924•Feb 20, 2010•1 citation

The superconducting quasicharge qubit

Ivan Pechenezhskiy, Raymond Mencia, Long B. Nguyen +2 more

Abstract The non-dissipative nonlinearity of Josephson junctions1 converts macroscopic superconducting circuits into artificial atoms2, enabling some of the best-controlled qubits today3, 4. Three fundamental types of superconducting qubit are known5, each reflecting a distinct behaviour of quantum fluctuations in a Cooper pair condensate: single-charge tunnelling (charge qubit6, 7), single-flux tunnelling (flux qubit8) and phase oscillations (phase qubit9 or transmon10). Yet, the dual nature of charge and flux suggests that circuit atoms must come in pairs. Here we introduce the missing superconducting qubit, ‘blochnium’, which exploits a coherent insulating response of a single Josephson junction that emerges from the extension of phase fluctuations beyond 2π (refs. 11–14). Evidence for such an effect has been found in out-of-equilibrium direct-current transport through junctions connected to high-impedance leads15–19, although a full consensus on the existence of extended phase fluctuations is so far absent20–22. We shunt a weak junction with an extremely high inductance—the key technological innovation in our experiment—and measure the radiofrequency excitation spectrum as a function of external magnetic flux through the resulting loop. The insulating character of the junction is manifested by the vanishing flux sensitivity of the qubit transition between the ground state and the first excited state, which recovers rapidly for transitions to higher-energy states. The spectrum agrees with a duality mapping of blochnium onto a transmon, which replaces the external flux by the offset charge and introduces a new collective quasicharge variable instead of the superconducting phase23, 24. Our findings may motivate the exploration of macroscopic quantum dynamics in ultrahigh-impedance circuits, with potential applications in quantum computing and metrology.

•Jan 1, 2020•17 citations

Third Quantization for Order Parameters (II): Local Field Quantization in Superconducting Quantum Circuits

Miao-Miao Yi, Guo-Jian Qiao, Xin Yue +1 more

The quantization of superconducting transmission-line resonators is usually introduced phenomenologically by modeling the resonator as an effective LC circuit and imposing canonical commutation relations on macroscopic variables such as charge and flux. Although this approach is highly successful, it leaves open why these macroscopic variables should obey quantum commutation relations and how this behavior emerges from the superconducting state. In this work, starting from the microscopic pairing Hamiltonian underlying BCS superconductivity, we derive the low-energy effective Hamiltonian of a circuit-QED architecture containing a superconducting transmission line with distributed capacitive and inductive elements. We establish quantitative relations between macroscopic observables, including current and voltage, and the spatially local superconducting phase, as well as the microscopic parameters of the electron-phonon system. We then extend the third quantization of the superconducting order parameter, introduced in Paper (I) for the global phase, to the spatially local case. This gives a macroscopic field quantization of the superconducting phase. We show that, after restriction to the low-energy excitation subspace, the local superconducting phase becomes a genuine quantum dynamical variable. Thus, the quantum behavior of transmission-line resonators need not be postulated at the macroscopic level, but follows from the third quantization of the superconducting order parameter. These results suggest that capacitive and inductive superconducting circuit elements share the same microscopic origin, providing a unified framework for superconducting circuit quantization.

quant-pharXiv:2604.24092•Apr 28, 2026•0 citations
Abstract We consider a specific instance of a superconducting circuit, the so-called charge-qubit, consisting of a capacitor and a Josephson junction that we describe by means of the BCS microscopic model in terms of two tunnelling superconducting systems in the strong-coupling quasi-spin formulation. Then, by means of collective observables we derive the Hamiltonian governing the quantum behaviour of the circuit in the limit of a large numberNof quasi-spins. Our approach relies on suitablequantum fluctuations, i.e. on collective quasi-spin operators, different from mean-field observables, that retain a quantum character in the large-Nlimit. These collective operators generate the Heisenberg algebra on the circle and we show that their dynamics reproduces the phenomenological one generated by the charge qubit Hamiltonian obtained by quantizing the macroscopic classical Hamiltonian of the circuit. The microscopic derivation of the emergent, large-Nbehaviour provides a rigorous setting to investigate more in detail both general quantum circuits and quantum macroscopic scenarios; in particular, in the specific case of charge-qubits, it allows to explicitly obtain the temperature dependence of the critical Josephson current in the strong coupling regime, a result not accessible using standard approximation techniques.
The quantized Josephson junction equation that underpins the behaviour of charge qubits and other tunnel devices is usually derived through canonical quantization of the classical macroscopic Josephson relations. However, this approach may neglect effects due to the fact that the charge qubit consists of a superconducting island of finite size connected to a large superconductor. We show that the well-known quantized Josephson equation can be derived directly and simply from a microscopic many-body Hamiltonian. By choosing the appropriate strong-coupling limit we produce a highly simplified Hamiltonian that nevertheless allows us to go beyond the mean-field limit and predict further finite-size terms in addition to the basic equation.
There exists a large number of experimental and theoretical results supporting the picture of macroscopic qubits implemented by nanoscopic Josephson junctions of three different types -- charge qubit, flux qubit and phase qubit. The standard unified description of such systems is based on the formal quantization of the phenomenological Kirchhoff equations for the corresponding circuits. In this paper a simplified version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions. For these models the state-dependent individual tunneling of Cooper pairs couples ground pair states with excited pair states what leads to a more complicated structure of the lowest lying energy levels. In particular, the highly degenerate levels emerge, which act as probability sinks for the qubit. These models allow also for the coupling to phonons as an efficient mechanism of relaxation for all types of junctions. The alternative formulas concerning basic spectral parameters of superconducting qubits are presented and compared with the experimental data. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
The strong-coupling version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions--charge qubit, flux qubit and phase qubit. Applied to Josephson qubits it yields a more complicated structure of the lowest-lying energy levels than that obtained from phenomenological models based on quantization of the Kirchhoff equations. In particular, highly degenerate levels emerge, which act as probability sinks for the qubit. The alternative formulae concerning spectra of superconducting qubits are presented and compared with the experimental data. In contrast to the existing theories those formulae contain microscopic parameters of the model. In particular, for the first time, the density of Cooper pairs at zero temperature is estimated for an Al-based flux qubit. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
The standard phenomenological Hamiltonian of a small superconducting Josephson junction in the charge regime (Cooper Pair Box) produces a model of the effective charge qubit with possible applications to quantum information processing. In this note a new model based on the BCS Hamiltonian with individual tunneling yields an effective multi-level picture with a highly degenerated level placed between the ground state and the excited state. Unlike in the standard approach, the excited Cooper pairs play here an important role. For such a system coupled to a zero temperature bath the additional levels act as a probability sink. In contrast to the standard large-spin model the coupling to phonons can be an effective source of dissipation. This model provides also alternative explanations of various effects observed in experiments and sheds new light on the issue of Josephson junctions as macroscopic quantum systems.
Abstract The non-dissipative nonlinearity of Josephson junctions1 converts macroscopic superconducting circuits into artificial atoms2, enabling some of the best-controlled qubits today3, 4. Three fundamental types of superconducting qubit are known5, each reflecting a distinct behaviour of quantum fluctuations in a Cooper pair condensate: single-charge tunnelling (charge qubit6, 7), single-flux tunnelling (flux qubit8) and phase oscillations (phase qubit9 or transmon10). Yet, the dual nature of charge and flux suggests that circuit atoms must come in pairs. Here we introduce the missing superconducting qubit, ‘blochnium’, which exploits a coherent insulating response of a single Josephson junction that emerges from the extension of phase fluctuations beyond 2π (refs. 11–14). Evidence for such an effect has been found in out-of-equilibrium direct-current transport through junctions connected to high-impedance leads15–19, although a full consensus on the existence of extended phase fluctuations is so far absent20–22. We shunt a weak junction with an extremely high inductance—the key technological innovation in our experiment—and measure the radiofrequency excitation spectrum as a function of external magnetic flux through the resulting loop. The insulating character of the junction is manifested by the vanishing flux sensitivity of the qubit transition between the ground state and the first excited state, which recovers rapidly for transitions to higher-energy states. The spectrum agrees with a duality mapping of blochnium onto a transmon, which replaces the external flux by the offset charge and introduces a new collective quasicharge variable instead of the superconducting phase23, 24. Our findings may motivate the exploration of macroscopic quantum dynamics in ultrahigh-impedance circuits, with potential applications in quantum computing and metrology.
The quantization of superconducting transmission-line resonators is usually introduced phenomenologically by modeling the resonator as an effective LC circuit and imposing canonical commutation relations on macroscopic variables such as charge and flux. Although this approach is highly successful, it leaves open why these macroscopic variables should obey quantum commutation relations and how this behavior emerges from the superconducting state. In this work, starting from the microscopic pairing Hamiltonian underlying BCS superconductivity, we derive the low-energy effective Hamiltonian of a circuit-QED architecture containing a superconducting transmission line with distributed capacitive and inductive elements. We establish quantitative relations between macroscopic observables, including current and voltage, and the spatially local superconducting phase, as well as the microscopic parameters of the electron-phonon system. We then extend the third quantization of the superconducting order parameter, introduced in Paper (I) for the global phase, to the spatially local case. This gives a macroscopic field quantization of the superconducting phase. We show that, after restriction to the low-energy excitation subspace, the local superconducting phase becomes a genuine quantum dynamical variable. Thus, the quantum behavior of transmission-line resonators need not be postulated at the macroscopic level, but follows from the third quantization of the superconducting order parameter. These results suggest that capacitive and inductive superconducting circuit elements share the same microscopic origin, providing a unified framework for superconducting circuit quantization.