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:2303.04114 · Scientific Reports

Extremely large Lamb shift in a deep-strongly coupled circuit QED system with a multimode resonator

Ziqiao Ao, Sahel Ashhab, Fumiki Yoshihara, Tomoko Fuse, Kosuke Kakuyanagi, Shiro Saito, Takao Aoki, Kouichi Semba

Published Jul 13, 2023• 13 citations
arXivHTMLPDF

Abstract

We report experimental and theoretical results on the extremely large Lamb shift in a multimode circuit quantum electrodynamics (QED) system in the deep-strong coupling (DSC) regime, where the qubit-resonator coupling strength is comparable to or larger than the qubit and resonator frequencies. The system comprises a superconducting flux qubit (FQ) and a quarter-wavelength coplanar waveguide resonator ([Formula: see text] CPWR) that are coupled inductively through a shared edge that contains a Josephson junction to achieve the DSC regime. Spectroscopy is performed around the frequency of the fundamental mode of the CPWR, and the spectrum is fitted by the single-mode quantum Rabi Hamiltonian to obtain the system parameters. Since the qubit is also coupled to a large number of higher modes in the resonator, the single-mode fitting does not provide the bare qubit energy but a value that incorporates the renormalization from all the other modes. We derive theoretical formulas for the Lamb shift in the multimode resonator system. As shown in previous studies, there is a cut-off frequency [Formula: see text] for the coupling between the FQ and the modes in the CPWR, where the coupling grows as [Formula: see text] for [Formula: see text] and decreases as [Formula: see text] for [Formula: see text]. Here [Formula: see text] is the frequency of the nth mode. The cut-off effect occurs because the qubit acts as an obstacle for the current in the resonator, which suppresses the current of the modes above [Formula: see text] at the location of the qubit and results in a reduced coupling strength. Using our observed spectrum and theoretical formulas, we estimate that the Lamb shift from the fundamental mode is 82.3% and the total Lamb shift from all the modes is 96.5%. This result illustrates that the coupling to the large number of modes in a CPWR yields an extremely large Lamb shift but does not suppress the qubit energy to zero, which would happen in the absence of a high-frequency cut-off.

DOI
10.1038/s41598-023-36547-w

Related papers

Ultrastrong coupling in two-resonator circuit QED

A. Baust, Eric A. Hoffmann, M. Haeberlein +17 more

We report on ultrastrong coupling between a superconducting flux qubit and a resonant mode of a system comprised of two superconducting coplanar stripline resonators coupled galvanically to the qubit. With a coupling strength as high as $17.5%$ of the mode frequency, exceeding that of previous circuit quantum electrodynamics experiments, we observe a pronounced Bloch-Siegert shift. The spectroscopic response of our multimode system reveals a clear breakdown of the Jaynes-Cummings approximation. In contrast to earlier experiments, the high coupling strength is achieved without making use of an additional inductance provided by a Josephson junction.

arXiv:1412.7372•Physical review. B./Physical review. B•Jun 1, 2016•95 citations

High-frequency suppression of inductive coupling between flux qubit and transmission line resonator

Sahel Ashhab, Ziqiao Ao, Fumiki Yoshihara +2 more

Abstract We perform theoretical calculations to investigate the naturally occurring high-frequency cutoff in a circuit comprising a flux qubit coupled inductively to a transmission line resonator (TLR). Specifically, a decoupling occurs between the qubit and the high-frequency modes. The coupling strength between the qubit and resonator modes increases with mode frequency ω as <?CDATA $\sqrt{\omega }$?> ω at low frequencies and decreases as <?CDATA $1/\sqrt{\omega }$?> 1 / ω at high frequencies. This result is similar to those of past studies that considered somewhat similar circuit designs. By avoiding the approximation of ignoring the qubit-TLR coupling in certain steps in the analysis, we obtain effects not captured in previous studies. In particular, we obtain a resonance effect that shifts the TLR mode frequencies close to qubit oscillation frequencies. We derive expressions for the TLR mode frequencies, qubit-TLR coupling strengths and qubit Lamb shift. We identify features in the spectrum of the system that can be used in future experiments to test and validate the theoretical model.

arXiv:2308.07849•Physica Scripta•Feb 19, 2024•1 citation

Convergence of the multimode quantum Rabi model of circuit quantum electrodynamics

Mario F. Gely, A. Parra-Rodriguez, Daniel Bothner +4 more

Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines, and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number $n$ scales as $\sqrt{n}$ for $n$ up to $\ensuremath{\infty}$, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a nonzero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultrastrong coupling regime of a multimode circuit QED.

arXiv:1701.05095•Physical review. B./Physical review. B•Jun 14, 2017•72 citations

Inversion of Qubit Energy Levels in Qubit-Oscillator Circuits in the Deep-Strong-Coupling Regime

Fumiki Yoshihara, T. Fuse, Z. Ao +6 more

We report on experimentally measured light shifts of superconducting flux qubits deep-strongly coupled to LC oscillators, where the coupling constants are comparable to the qubit and oscillator resonance frequencies. By using two-tone spectroscopy, the energies of the six lowest levels of each circuit are determined. We find huge Lamb shifts that exceed 90% of the bare qubit frequencies and inversions of the qubits' ground and excited states when there are a finite number of photons in the oscillator. Our experimental results agree with theoretical predictions based on the quantum Rabi model.

arXiv:1712.05039•Physical Review Letters•May 3, 2018•109 citations

Ultrastrong coupling regime of cavity QED with phase-biased flux qubits

Jérôme Bourassa, Jay Gambetta, A. A. Abdumalikov +3 more

We theoretically study a circuit QED architecture based on a superconducting flux qubit directly coupled to the center conductor of a coplanar waveguide transmission-line resonator. As already shown experimentally [A. A. Abdumalikov, Jr. et al., Phys. Rev. B 78, 180502(R) (2008)], the strong coupling regime of cavity QED can readily be achieved by optimizing the local inductance of the resonator in the vicinity of the qubit. In addition to yielding stronger coupling with respect to other proposals for flux qubit based circuit QED, this approach leads to a qubit-resonator coupling strength $g$ which does not scale as the area of the qubit but is proportional to the total inductance shared between the resonator and the qubit. Strong coupling can thus be attained while still minimizing sensitivity to flux noise. Finally, we show that by taking advantage of the large kinetic inductance of a Josephson junction in the center conductor of the resonator can lead to coupling energies of several tens of percent of the resonator frequency, reaching the ultrastrong coupling regime of cavity QED where the rotating-wave approximation breaks down. This should allow an on-chip implementation of the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ Jahn-Teller model.

arXiv:0906.1383•Physical Review A•Sep 15, 2009•252 citations

All-microwave Lamb shift engineering for a fixed frequency multi-level superconducting qubit

Byoung-moo Ann, Gary Alexander Steele

It is known that the electromagnetic vacuum is responsible for the Lamb shift, which is a crucial phenomenon in quantum electrodynamics (QED). In circuit QED, the readout or bus resonators that are dispersively coupled can result in a significant Lamb shift of the qubit. However, previous approaches or proposals for controlling the Lamb shift in circuit QED demand overheads in circuit designs or non-perturbative renormalization of the system’s eigenbases, which can impose formidable limitations. In this work, we propose and demonstrate an all-microwave method for controlling the Lamb shift of fixed-frequency transmons. We employ the drive-induced longitudinal coupling between the transmon and resonator. By simply using an off-resonant monochromatic drive near the resonator frequency, we can control the net Lamb shift up to ±30 MHz and engineer it to zero with the drive-induced longitudinal coupling without facing the aforementioned challenges. Our work establishes an efficient way of engineering the fundamental effects of the electromagnetic vacuum and provides greater flexibility in non-parametric frequency controls of multilevel systems. Engineering the Lamb shifts of superconducting qubits opens new opportunities in resonant frequency tunings and other applications. Here, the authors devise and demonstrate an all-microwave approach that can be utilized with fixed-frequency superconducting qubits.

arXiv:2304.11782•Communications Physics•Oct 25, 2024•2 citations
We report experimental and theoretical results on the extremely large Lamb shift in a multimode circuit quantum electrodynamics (QED) system in the deep-strong coupling (DSC) regime, where the qubit-resonator coupling strength is comparable to or larger than the qubit and resonator frequencies. The system comprises a superconducting flux qubit (FQ) and a quarter-wavelength coplanar waveguide resonator ([Formula: see text] CPWR) that are coupled inductively through a shared edge that contains a Josephson junction to achieve the DSC regime. Spectroscopy is performed around the frequency of the fundamental mode of the CPWR, and the spectrum is fitted by the single-mode quantum Rabi Hamiltonian to obtain the system parameters. Since the qubit is also coupled to a large number of higher modes in the resonator, the single-mode fitting does not provide the bare qubit energy but a value that incorporates the renormalization from all the other modes. We derive theoretical formulas for the Lamb shift in the multimode resonator system. As shown in previous studies, there is a cut-off frequency [Formula: see text] for the coupling between the FQ and the modes in the CPWR, where the coupling grows as [Formula: see text] for [Formula: see text] and decreases as [Formula: see text] for [Formula: see text]. Here [Formula: see text] is the frequency of the nth mode. The cut-off effect occurs because the qubit acts as an obstacle for the current in the resonator, which suppresses the current of the modes above [Formula: see text] at the location of the qubit and results in a reduced coupling strength. Using our observed spectrum and theoretical formulas, we estimate that the Lamb shift from the fundamental mode is 82.3% and the total Lamb shift from all the modes is 96.5%. This result illustrates that the coupling to the large number of modes in a CPWR yields an extremely large Lamb shift but does not suppress the qubit energy to zero, which would happen in the absence of a high-frequency cut-off.
We report on ultrastrong coupling between a superconducting flux qubit and a resonant mode of a system comprised of two superconducting coplanar stripline resonators coupled galvanically to the qubit. With a coupling strength as high as 17.517.5%17.5 of the mode frequency, exceeding that of previous circuit quantum electrodynamics experiments, we observe a pronounced Bloch-Siegert shift. The spectroscopic response of our multimode system reveals a clear breakdown of the Jaynes-Cummings approximation. In contrast to earlier experiments, the high coupling strength is achieved without making use of an additional inductance provided by a Josephson junction.
Abstract We perform theoretical calculations to investigate the naturally occurring high-frequency cutoff in a circuit comprising a flux qubit coupled inductively to a transmission line resonator (TLR). Specifically, a decoupling occurs between the qubit and the high-frequency modes. The coupling strength between the qubit and resonator modes increases with mode frequency ω as <?CDATA ω\sqrt{\omega }ω​?> ω at low frequencies and decreases as <?CDATA 1/ω1/\sqrt{\omega }1/ω​?> 1 / ω at high frequencies. This result is similar to those of past studies that considered somewhat similar circuit designs. By avoiding the approximation of ignoring the qubit-TLR coupling in certain steps in the analysis, we obtain effects not captured in previous studies. In particular, we obtain a resonance effect that shifts the TLR mode frequencies close to qubit oscillation frequencies. We derive expressions for the TLR mode frequencies, qubit-TLR coupling strengths and qubit Lamb shift. We identify features in the spectrum of the system that can be used in future experiments to test and validate the theoretical model.
Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines, and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number nnn scales as n\sqrt{n}n​ for nnn up to $\ensuremath{\infty}$, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a nonzero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultrastrong coupling regime of a multimode circuit QED.
We report on experimentally measured light shifts of superconducting flux qubits deep-strongly coupled to LC oscillators, where the coupling constants are comparable to the qubit and oscillator resonance frequencies. By using two-tone spectroscopy, the energies of the six lowest levels of each circuit are determined. We find huge Lamb shifts that exceed 90% of the bare qubit frequencies and inversions of the qubits' ground and excited states when there are a finite number of photons in the oscillator. Our experimental results agree with theoretical predictions based on the quantum Rabi model.
We theoretically study a circuit QED architecture based on a superconducting flux qubit directly coupled to the center conductor of a coplanar waveguide transmission-line resonator. As already shown experimentally [A. A. Abdumalikov, Jr. et al., Phys. Rev. B 78, 180502(R) (2008)], the strong coupling regime of cavity QED can readily be achieved by optimizing the local inductance of the resonator in the vicinity of the qubit. In addition to yielding stronger coupling with respect to other proposals for flux qubit based circuit QED, this approach leads to a qubit-resonator coupling strength ggg which does not scale as the area of the qubit but is proportional to the total inductance shared between the resonator and the qubit. Strong coupling can thus be attained while still minimizing sensitivity to flux noise. Finally, we show that by taking advantage of the large kinetic inductance of a Josephson junction in the center conductor of the resonator can lead to coupling energies of several tens of percent of the resonator frequency, reaching the ultrastrong coupling regime of cavity QED where the rotating-wave approximation breaks down. This should allow an on-chip implementation of the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ Jahn-Teller model.
It is known that the electromagnetic vacuum is responsible for the Lamb shift, which is a crucial phenomenon in quantum electrodynamics (QED). In circuit QED, the readout or bus resonators that are dispersively coupled can result in a significant Lamb shift of the qubit. However, previous approaches or proposals for controlling the Lamb shift in circuit QED demand overheads in circuit designs or non-perturbative renormalization of the system’s eigenbases, which can impose formidable limitations. In this work, we propose and demonstrate an all-microwave method for controlling the Lamb shift of fixed-frequency transmons. We employ the drive-induced longitudinal coupling between the transmon and resonator. By simply using an off-resonant monochromatic drive near the resonator frequency, we can control the net Lamb shift up to ±30 MHz and engineer it to zero with the drive-induced longitudinal coupling without facing the aforementioned challenges. Our work establishes an efficient way of engineering the fundamental effects of the electromagnetic vacuum and provides greater flexibility in non-parametric frequency controls of multilevel systems. Engineering the Lamb shifts of superconducting qubits opens new opportunities in resonant frequency tunings and other applications. Here, the authors devise and demonstrate an all-microwave approach that can be utilized with fixed-frequency superconducting qubits.