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:1210.5326 · Physical Review A

Generalized rotating-wave approximation to biased qubit-oscillator systems

Yuyu Zhang, Qing‐Hu Chen, Yang Zhao

Published Mar 25, 2013• 31 citations
arXivHTMLPDF

Abstract

The generalized rotating-wave approximation with counter-rotating interactions has been applied to a biased qubit-oscillator system. Analytical expressions are explicitly given for all eigenvalues and eigenstates. For a flux qubit coupled to superconducting oscillators, spectra calculated by our approach are in excellent agreement with experiment. Calculated energy levels for a variety of biases also agree well with those obtained via exact diagonalization for a wide range of coupling strengths. Dynamics of the qubit has also been examined, and results lend further support to the validity of the analytical approximation employed here. Our approach can be readily implemented and applied to superconducting qubit-oscillator experiments conducted currently and in the near future with a biased qubit and for all accessible coupling strengths.

DOI
10.1103/physreva.87.033827

Related papers

Unified analytical treatments of qubit—oscillator systems

Shu He, Yuyu Zhang, Qing‐Hu Chen +3 more

An effective scheme within two displaced bosonic operators with equal positive and negative displacements is extended to study qubit—oscillator systems analytically in a unified way. Many previous analytical treatments, such as generalized rotating-wave approximation (GRWA) [ Phys. Rev. Lett. 99 , 173601 (2007) ] and an expansion in the qubit tunneling matrix element in the deep strong coupling regime [ Phys. Rev. Lett. 105 , 263603 (2010) ] can be recovered straightforwardly within the present scheme. Moreover, further improving GRWA and the extension to the finite-bias case are implemented easily. The algebraic formulae for the eigensolutions are then derived explicitly and uniquely, which work well in a wide range of the coupling strengths, detunings, and static bias including the recent experimentally accessible parameters. The dynamics of the qubit for an oscillator in the ground state is also studied. At the experimentally accessible coupling regime, GRWA can always work well. When the coupling is enhanced to the intermediate regime, only the improving GRWA can give the correct description, while the result of GRWA shows strong deviations. The previous Van Vleck perturbation theory is not valid to describe the dynamics in the present-day experimentally accessible regime, except for the strongly biased cases.

arXiv:1204.0953•Chinese Physics B•Jun 1, 2013•10 citations

Generalized rotating-wave approximation for the two-qubit quantum Rabi model

Yuyu Zhang, Qing‐Hu Chen

The generalized rotating-wave approximation (GRWA) is presented for the two-qubit quantum Rabi model. The analytical expressions in the zeroth-order approximation recover the previous adiabatic ones. The counter-rotating-wave terms can be eliminated by performing the first-order corrections. An effective solvable Hamiltonian with the same form as the ordinary RWA one is then obtained, giving very accurate eigenvalues and eigenstates. Energy levels in the present GRWA are in accordance with the numerical exact diagonalization ones in a wide range of coupling strengths. The population dynamics in the GRWA are in quantitative agreement with the numerical results and exhibit the absence of collapses clearly, revealing the effects of the counter-rotating wave.

arXiv:1410.1965•Physical Review A•Jan 8, 2015•37 citations

Qubit-oscillator dynamics in the dispersive regime: Analytical theory beyond the rotating-wave approximation

David Zueco, Georg M. Reuther, Sigmund Kohler +1 more

We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently employed rotating-wave approximation (RWA) in the coupling between the two-level system and the resonator. For a detuning sufficiently larger than the qubit-oscillator coupling, we diagonalize the non-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extend the regime in which dispersive qubit readout is possible. If several qubits are coupled to one resonator, an effective qubit-qubit interaction of Ising type emerges, whereas RWA leads to isotropic $XY$ interaction. This impacts on the entanglement characteristics of the qubits.

arXiv:0907.3516•Physical Review A•Sep 30, 2009•201 citations

Dissipative dynamics of a biased qubit coupled to a harmonic oscillator: Analytical results beyond the rotating wave approximation

Johannes Hausinger, Milena Grifoni

We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using Van-Vleck perturbation theory and going to second order in the coupling between TLS and HO, we show how the Hamiltonian of the TLS-HO system can be diagonalized analytically. Our model represents an improvement to the usually used Jaynes-Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are\nfound: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in closed analytic form.

•Jan 1, 2008•43 citations

Dissipative dynamics of a biased qubit coupled to a harmonic oscillator: analytical results beyond the rotating wave approximation

Johannes Hausinger, Milena Grifoni

We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using the Van Vleck perturbation theory and going to second order in the coupling between the TLS and the HO, we show how the Hamiltonian of the TLS–HO system can be diagonalized analytically. Our model represents an improvement on the usually used Jaynes–Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are found: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in a closed analytic form.

arXiv:0806.3387•New Journal of Physics•Nov 20, 2008•46 citations

Superconducting phase qubit coupled to a nanomechanical resonator: Beyond the rotating-wave approximation

Andrew T. Sornborger, Andrew N. Cleland, Michael R. Geller

We consider a simple model of a Josephson junction phase qubit coupled to a solid-state nanoelectromechanical resonator. This and many related qubit-resonator models are analogous to an atom in an electromagnetic cavity. When the systems are weakly coupled and nearly resonant, the dynamics is accurately described by the rotating-wave approximation (RWA) or the Jaynes-Cummings model of quantum optics. However, the desire to develop faster quantum-information-processing protocols necessitates approximate, yet analytic descriptions that are valid for more strongly coupled qubit-resonator systems. Here we present a simple theoretical technique, using a basis of dressed states, to perturbatively account for the leading-order corrections to the RWA. By comparison with exact numerical results, we demonstrate that the method is accurate for moderately strong coupling and provides a useful theoretical tool for describing fast quantum information processing. The method applies to any quantum two-level system linearly coupled to a harmonic oscillator or single-mode boson field.

arXiv:quant-ph/0407106•Physical Review A•Nov 17, 2004•67 citations
The generalized rotating-wave approximation with counter-rotating interactions has been applied to a biased qubit-oscillator system. Analytical expressions are explicitly given for all eigenvalues and eigenstates. For a flux qubit coupled to superconducting oscillators, spectra calculated by our approach are in excellent agreement with experiment. Calculated energy levels for a variety of biases also agree well with those obtained via exact diagonalization for a wide range of coupling strengths. Dynamics of the qubit has also been examined, and results lend further support to the validity of the analytical approximation employed here. Our approach can be readily implemented and applied to superconducting qubit-oscillator experiments conducted currently and in the near future with a biased qubit and for all accessible coupling strengths.
An effective scheme within two displaced bosonic operators with equal positive and negative displacements is extended to study qubit—oscillator systems analytically in a unified way. Many previous analytical treatments, such as generalized rotating-wave approximation (GRWA) [ Phys. Rev. Lett. 99 , 173601 (2007) ] and an expansion in the qubit tunneling matrix element in the deep strong coupling regime [ Phys. Rev. Lett. 105 , 263603 (2010) ] can be recovered straightforwardly within the present scheme. Moreover, further improving GRWA and the extension to the finite-bias case are implemented easily. The algebraic formulae for the eigensolutions are then derived explicitly and uniquely, which work well in a wide range of the coupling strengths, detunings, and static bias including the recent experimentally accessible parameters. The dynamics of the qubit for an oscillator in the ground state is also studied. At the experimentally accessible coupling regime, GRWA can always work well. When the coupling is enhanced to the intermediate regime, only the improving GRWA can give the correct description, while the result of GRWA shows strong deviations. The previous Van Vleck perturbation theory is not valid to describe the dynamics in the present-day experimentally accessible regime, except for the strongly biased cases.
The generalized rotating-wave approximation (GRWA) is presented for the two-qubit quantum Rabi model. The analytical expressions in the zeroth-order approximation recover the previous adiabatic ones. The counter-rotating-wave terms can be eliminated by performing the first-order corrections. An effective solvable Hamiltonian with the same form as the ordinary RWA one is then obtained, giving very accurate eigenvalues and eigenstates. Energy levels in the present GRWA are in accordance with the numerical exact diagonalization ones in a wide range of coupling strengths. The population dynamics in the GRWA are in quantitative agreement with the numerical results and exhibit the absence of collapses clearly, revealing the effects of the counter-rotating wave.
We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently employed rotating-wave approximation (RWA) in the coupling between the two-level system and the resonator. For a detuning sufficiently larger than the qubit-oscillator coupling, we diagonalize the non-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extend the regime in which dispersive qubit readout is possible. If several qubits are coupled to one resonator, an effective qubit-qubit interaction of Ising type emerges, whereas RWA leads to isotropic XYXYXY interaction. This impacts on the entanglement characteristics of the qubits.
We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using Van-Vleck perturbation theory and going to second order in the coupling between TLS and HO, we show how the Hamiltonian of the TLS-HO system can be diagonalized analytically. Our model represents an improvement to the usually used Jaynes-Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are\nfound: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in closed analytic form.
We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using the Van Vleck perturbation theory and going to second order in the coupling between the TLS and the HO, we show how the Hamiltonian of the TLS–HO system can be diagonalized analytically. Our model represents an improvement on the usually used Jaynes–Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are found: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in a closed analytic form.
We consider a simple model of a Josephson junction phase qubit coupled to a solid-state nanoelectromechanical resonator. This and many related qubit-resonator models are analogous to an atom in an electromagnetic cavity. When the systems are weakly coupled and nearly resonant, the dynamics is accurately described by the rotating-wave approximation (RWA) or the Jaynes-Cummings model of quantum optics. However, the desire to develop faster quantum-information-processing protocols necessitates approximate, yet analytic descriptions that are valid for more strongly coupled qubit-resonator systems. Here we present a simple theoretical technique, using a basis of dressed states, to perturbatively account for the leading-order corrections to the RWA. By comparison with exact numerical results, we demonstrate that the method is accurate for moderately strong coupling and provides a useful theoretical tool for describing fast quantum information processing. The method applies to any quantum two-level system linearly coupled to a harmonic oscillator or single-mode boson field.