© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:0712.1910 · New Journal of Physics

Distinguishing quantum from classical oscillations in a driven phase qubit

Sergey N Shevchenko, A. N. Omelyanchouk, Alexandre M. Zagoskin, Sergey E. Savel'ev, Franco Nori

Published Jul 14, 2008• 23 citations

Abstract

Rabi oscillations are coherent transitions in a quantum two-level system under the influence of a resonant drive, with a much lower frequency dependent on the perturbation amplitude. These serve as one of the signatures of quantum coherent evolution in mesoscopic systems. It was shown recently (Grønbech-Jensen N and Cirillo M 2005 Phys. Rev. Lett. 95 067001) that in phase qubits (current-biased Josephson junctions) this effect can be mimicked by classical oscillations arising due to the anharmonicity of the effective potential. Nevertheless, we find qualitative differences between the classical and quantum effects. Firstly, while the quantum Rabi oscillations can be produced by the subharmonics of the resonant frequency ω 10 (multiphoton processes), the classical effect also exists when the system is excited at the overtones, n ω 10 . Secondly, the shape of the resonance is, in the classical case, characteristically asymmetric, whereas quantum resonances are described by symmetric Lorentzians. Thirdly, the anharmonicity of the potential results in the negative shift of the resonant frequency in the classical case, in contrast to the positive Bloch–Siegert shift in the quantum case. We show that in the relevant range of parameters these features allow us to distinguish confidently the bona fide Rabi oscillations from their classical Doppelgänger.

DOI: 10.1088/1367-2630/10/7/073026

Related papers

Pseudo-Rabi oscillations in superconducting flux qubits in the classical regime

A. N. Omelyanchouk, Sergey N Shevchenko, Alexandre M. Zagoskin +2 more

Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux qubit in the classical regime can produce low-frequency oscillations in the presence of an external field in resonance with the (high-frequency) harmonic mode $\ensuremath{\omega}$ of the system. Like in the case of quantum Rabi oscillations, the frequency of these pseudo-Rabi oscillations is much smaller than $\ensuremath{\omega}$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external drive not only at the resonance frequency $\ensuremath{\omega}$ and its subharmonics $(\ensuremath{\omega}/n)$, but also at its overtones, $n\ensuremath{\omega}$.

arXiv:0705.1768•Physical Review B•Aug 15, 2008•12 citations

Rabi-like oscillations of an anharmonic oscillator: Classical versus quantum interpretation

Julien Claudon, Alex Zazunov, F. W. J. Hekking +1 more

We have observed Rabi-like oscillations in a current-biased dc superconducting quantum interference device presenting enhanced coherence times compared to our previous realization [J. Claudon et al., Phys. Rev. Lett. 93, 187003 (2004)]. This Josephson device behaves as an anharmonic oscillator which can be driven into a coherent superposition of quantum states by resonant microwave flux pulses. Increasing the microwave amplitude, we study the evolution of the Rabi frequency from the two-level regime to the regime of multilevel dynamics. When up to three levels are involved, the Rabi frequency is a clear signature of a quantum behavior. At higher excitation amplitude, classical and quantum predictions for the Rabi frequency converge. This result is discussed in the light of a calculation of the Wigner function. In particular, our analysis shows that pronounced quantum interferences always appear in the course of the Rabi-like oscillations.

arXiv:0709.3787•Physical Review B•Nov 3, 2008•32 citations

Bloch–Siegert oscillations in the Rabi model with an amplitude-modulated driving field

A. P. Saǐko, S A Markevich, Ryhor Fedaruk

Abstract We study the coherent dynamics of a qubit excited by an amplitude-modulated electromagnetic field under the Rabi resonance when the frequency of the low-frequency modulation field matches the Rabi frequency in the high-frequency field. Due to the destructive interference of multiple photon processes within the ultrastrong coupling between the qubit and the low-frequency driving field, Rabi oscillations result exclusively from the Bloch–Siegert effect. It is directly observed in the time-resolved coherent dynamics as the Bloch–Siegert oscillation. In this case, triplets in the Fourier spectra of the coherent response are transformed into doublets with the splitting between the lines equal to twice the Bloch–Siegert shift. These unusual properties are demonstrated in the conditions of experiments with a nitrogen vacancy center in the diamond.

arXiv:2002.07165•Laser Physics•Oct 23, 2019•7 citations

Effects of the Bloch-Siegert Oscillation on the Precision of Qubit Rotations: Direct Two-Level vs. Off-Resonant Raman Excitation

Prabhakar Pradhan, George C. Cardoso, J. Morzinski +1 more

In a direct two-level qubit system, when the Rabi frequency is comparable to the resonance frequency, the rotating wave approximation is not appropriate. In this case, the Rabi oscillation is accompanied by another oscillation at twice the frequency of the driving field, the so called Bloch-Siegert oscillation (BSO), which depends on the initial phase of the driving field. This oscillation may restrict the precise rotation of a qubit made of direct two-level system. Here, we show that in case of an effectively two-level lambda system, the BSO is inherently negligible, implying a greater precision for rotation of a qubit made of such a lambda system when compared to a direct two-level qubit in a strong driving field.

arXiv:quant-ph/0402122•Feb 18, 2004•0 citations

Multiphoton Raman transitions and Rabi oscillations in driven spin systems

A. P. Saǐko, Siarhei A. Markevich, Ryhor Fedaruk

In the framework of the nonsecular perturbation theory based on the Bogoliubov averaging method, the coherent dynamics of multiphoton Raman transitions in a two-level spin system driven by an amplitude-modulated microwave field is studied. Closed-form expressions for the Rabi frequencies of these transitions are obtained beyond the rotating-wave approximation for the low-frequency driving component. It is shown that spin states dressed by the high-frequency component of the driving field are shifted due to the Bloch-Siegert-like effect caused by antiresonant interactions with the strong low-frequency driving. We predict that with increasing the order of the Raman transition, the Rabi frequency decreases and the contribution of the Bloch-Siegert shift to this frequency becomes dominant. It is found that the amplitude and phase of the Rabi oscillations strongly depend on the initial phase of the low-frequency field as well as on detuning from multiphoton resonance. The recent experimental data for the second- and third-order Raman transitions observed for the nitrogen-vacancy center in diamond (Z. Shu et al., arXiv:1804.10492) are well described in the frame of our approach. Our results may provide new possibilities for coherent control of quantum systems.

arXiv:1807.05086•Physical Review A•Oct 5, 2018•24 citations

Observation of the Bloch-Siegert shift in a driven quantum-to-classical transition

Iivari Pietikäinen, Sergey Danilin, K. S. Kumar +4 more

We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for a strong drive (classical phase) it presents an oscillatory behavior as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum.

arXiv:1610.09153•Physical review. B./Physical review. B•Jul 7, 2017•51 citations

Rabi oscillations are coherent transitions in a quantum two-level system under the influence of a resonant drive, with a much lower frequency dependent on the perturbation amplitude. These serve as one of the signatures of quantum coherent evolution in mesoscopic systems. It was shown recently (Grønbech-Jensen N and Cirillo M 2005 Phys. Rev. Lett. 95 067001) that in phase qubits (current-biased Josephson junctions) this effect can be mimicked by classical oscillations arising due to the anharmonicity of the effective potential. Nevertheless, we find qualitative differences between the classical and quantum effects. Firstly, while the quantum Rabi oscillations can be produced by the subharmonics of the resonant frequency ω 10 (multiphoton processes), the classical effect also exists when the system is excited at the overtones, n ω 10 . Secondly, the shape of the resonance is, in the classical case, characteristically asymmetric, whereas quantum resonances are described by symmetric Lorentzians. Thirdly, the anharmonicity of the potential results in the negative shift of the resonant frequency in the classical case, in contrast to the positive Bloch–Siegert shift in the quantum case. We show that in the relevant range of parameters these features allow us to distinguish confidently the bona fide Rabi oscillations from their classical Doppelgänger.

Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux qubit in the classical regime can produce low-frequency oscillations in the presence of an external field in resonance with the (high-frequency) harmonic mode $\ensuremath{\omega}$ of the system. Like in the case of quantum Rabi oscillations, the frequency of these pseudo-Rabi oscillations is much smaller than $\ensuremath{\omega}$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external drive not only at the resonance frequency $\ensuremath{\omega}$ and its subharmonics $(\ensuremath{\omega}/n)$, but also at its overtones, $n\ensuremath{\omega}$.

We have observed Rabi-like oscillations in a current-biased dc superconducting quantum interference device presenting enhanced coherence times compared to our previous realization [J. Claudon et al., Phys. Rev. Lett. 93, 187003 (2004)]. This Josephson device behaves as an anharmonic oscillator which can be driven into a coherent superposition of quantum states by resonant microwave flux pulses. Increasing the microwave amplitude, we study the evolution of the Rabi frequency from the two-level regime to the regime of multilevel dynamics. When up to three levels are involved, the Rabi frequency is a clear signature of a quantum behavior. At higher excitation amplitude, classical and quantum predictions for the Rabi frequency converge. This result is discussed in the light of a calculation of the Wigner function. In particular, our analysis shows that pronounced quantum interferences always appear in the course of the Rabi-like oscillations.

Abstract We study the coherent dynamics of a qubit excited by an amplitude-modulated electromagnetic field under the Rabi resonance when the frequency of the low-frequency modulation field matches the Rabi frequency in the high-frequency field. Due to the destructive interference of multiple photon processes within the ultrastrong coupling between the qubit and the low-frequency driving field, Rabi oscillations result exclusively from the Bloch–Siegert effect. It is directly observed in the time-resolved coherent dynamics as the Bloch–Siegert oscillation. In this case, triplets in the Fourier spectra of the coherent response are transformed into doublets with the splitting between the lines equal to twice the Bloch–Siegert shift. These unusual properties are demonstrated in the conditions of experiments with a nitrogen vacancy center in the diamond.

In a direct two-level qubit system, when the Rabi frequency is comparable to the resonance frequency, the rotating wave approximation is not appropriate. In this case, the Rabi oscillation is accompanied by another oscillation at twice the frequency of the driving field, the so called Bloch-Siegert oscillation (BSO), which depends on the initial phase of the driving field. This oscillation may restrict the precise rotation of a qubit made of direct two-level system. Here, we show that in case of an effectively two-level lambda system, the BSO is inherently negligible, implying a greater precision for rotation of a qubit made of such a lambda system when compared to a direct two-level qubit in a strong driving field.

In the framework of the nonsecular perturbation theory based on the Bogoliubov averaging method, the coherent dynamics of multiphoton Raman transitions in a two-level spin system driven by an amplitude-modulated microwave field is studied. Closed-form expressions for the Rabi frequencies of these transitions are obtained beyond the rotating-wave approximation for the low-frequency driving component. It is shown that spin states dressed by the high-frequency component of the driving field are shifted due to the Bloch-Siegert-like effect caused by antiresonant interactions with the strong low-frequency driving. We predict that with increasing the order of the Raman transition, the Rabi frequency decreases and the contribution of the Bloch-Siegert shift to this frequency becomes dominant. It is found that the amplitude and phase of the Rabi oscillations strongly depend on the initial phase of the low-frequency field as well as on detuning from multiphoton resonance. The recent experimental data for the second- and third-order Raman transitions observed for the nitrogen-vacancy center in diamond (Z. Shu et al., arXiv:1804.10492) are well described in the frame of our approach. Our results may provide new possibilities for coherent control of quantum systems.

We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for a strong drive (classical phase) it presents an oscillatory behavior as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum.