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arXiv:2502.05247 · Annalen der Physik

Excited States of Coherent Harmonic Qubits With Long‐Range Photon Coupling and Dissipation

Luca Gamberale, Giovanni Modanese

Published Sep 11, 2025• 0 citations
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Abstract

Abstract It is known that ensembles of interacting oscillators or qubits can exhibit the phenomenon of quantum synchronization. In this work, a set of identical two‐state systems that we call “harmonic qubits” is considered, because the kinetic part of their Hamiltonian is of the form , coupled through a multi‐state “photon” mode subject to dissipation. It has been proven numerically that when the coupling between the qubits and the photon is sufficiently strong, the ensemble condenses into a ground state with negative energy, the energy gap is proportional to and there are clear cross correlations . Here, the energy spectrum of the excited states of this system is the point of interest. In order to obtain information on the coherent transitions, a weak coupling of each qubit is introduced with an external oscillator of variable frequency and it is checked via Monte Carlo time evolution for which values of variations in the occupation of the external oscillator occur. After adding a second external oscillator coupled to the first only through the qubits, the energy transfer is also considered between the two external oscillators in dependence on their frequency, a transfer which is possible only through the excited states of the qubits. Above threshold (when ), resonant transfer is found at frequencies that are definitely higher, and growing with . This signals the presence of collective excited states, separated by large energy gaps, which are absent below threshold.

DOI
10.1002/andp.202500074

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Abstract It is known that ensembles of interacting oscillators or qubits can exhibit the phenomenon of quantum synchronization. In this work, a set of identical two‐state systems that we call “harmonic qubits” is considered, because the kinetic part of their Hamiltonian is of the form , coupled through a multi‐state “photon” mode subject to dissipation. It has been proven numerically that when the coupling between the qubits and the photon is sufficiently strong, the ensemble condenses into a ground state with negative energy, the energy gap is proportional to and there are clear cross correlations . Here, the energy spectrum of the excited states of this system is the point of interest. In order to obtain information on the coherent transitions, a weak coupling of each qubit is introduced with an external oscillator of variable frequency and it is checked via Monte Carlo time evolution for which values of variations in the occupation of the external oscillator occur. After adding a second external oscillator coupled to the first only through the qubits, the energy transfer is also considered between the two external oscillators in dependence on their frequency, a transfer which is possible only through the excited states of the qubits. Above threshold (when ), resonant transfer is found at frequencies that are definitely higher, and growing with . This signals the presence of collective excited states, separated by large energy gaps, which are absent below threshold.
We consider the quantum Rabi model as an open system that is subject to dissipation, dephasing, and sinusoidal qubit driving. One can change to an interaction picture where the qubit-driving term disappears at the expense of changing the free energy of the qubit which becomes time dependent. If the driving frequency is large with respect to the rest of the parameters with the exception of the driving strength, then one can obtain an effective Hamiltonian that accurately describes the dynamics of the system. The driving has two effects: the qubit-transition frequency is changed and the qubit has reduced dephasing. The driving strength can be chosen so that the qubit-transition frequency is reduced, made equal to zero, or even made negative so that the excited and ground states of the qubit are interchanged. Therefore, sinusoidal qubit driving offers another method to control the qubit-transition frequency and to reduce qubit dephasing. Adjusting the driving strength allows one to consider a qubit with degenerate energy levels. Not taking dissipation into account, the evolution operator of the qubit-harmonic oscillator system is given by a linear combination of the orthogonal projectors onto the eigenstates of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}$ followed by the evolution operator of a forced harmonic oscillator, the harmonic oscillator can be prepared in such a way that it is always found in a Schr\"odinger cat state, and the transition probability of the qubit can exhibit a collapse-revival behavior. In addition, the Born-Markov-secular master equation is deduced and the effects of dissipation are presented. In particular, smaller ultrastrong-coupling values are preferable over larger ultrastrong-coupling values and deep strong-coupling values in order to have long-lived, easily distinguishable Schr\"odinger cat states because the decoherence rate is inversely proportional to the square of the coupling. Finally, the qubit-harmonic oscillator system can be prepared in highly entangled states that are stable under dissipation.
We report a theoretical study of coherent collective quantum dynamic effects in an array of NNN qubits (two-level systems) incorporated into a low-dissipation resonant cavity. Individual qubits are characterized by energy level differences ${\ensuremath{\Delta}}_{i}$ and a spread of ${\ensuremath{\Delta}}_{i}$ is taken into account. Noninteracting qubits display coherent quantum beatings with NNN different frequencies, i.e., ${\ensuremath{\omega}}_{i}={\ensuremath{\Delta}}_{i}/\ensuremath{\hbar}$. Virtual emission and absorption of cavity photons provides a long-range interaction between qubits. In the presence of such interaction we analyze quantum correlation functions of individual qubits Ci(t){C}_{i}(t)Ci​(t) to obtain two collective quantum-mechanical coherent oscillations, characterized by frequencies ${\ensuremath{\omega}}_{1}=\overline{\ensuremath{\Delta}}/\ensuremath{\hbar}$ and ${\ensuremath{\omega}}_{2}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{R}$, where ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{R}$ is the resonant frequency of the cavity renormalized by interaction. The amplitude of these oscillations can be strongly enhanced in the resonant case when ${\ensuremath{\omega}}_{1}\ensuremath{\simeq}{\ensuremath{\omega}}_{2}$. These collective quantum oscillations can be directly observed, e.g., by measurements of frequency dependent transmission coefficient $D(\ensuremath{\omega})$ of electromagnetic field propagating in a transmission line coupled to the system.
The model of a two-level system (TLS) coupled to a harmonic oscillator has found a widespread application in physics and chemistry. In this thesis we focus on the field of circuit quantum electrodynamics where the TLS stands for a quantum bit (qubit), which is the basic building unit of a potential quantum computer, and the oscillator represents a transmission-line resonator, which can be used, e.g., to store information contained on the qubit or for communication between several qubits. Furthermore, the oscillator can model a superconducting quantum interference device (SQUID) determining the state of the qubit. In first experimental realizations the coupling between the qubit and the oscillator was small compared to characteristic frequencies of the two devices. In such a situation the Jaynes-Cummings model provides a realistic and completely analytical description of the energy spectrum and the dynamics of the setup. However, it relies on a rotating-wave approximation (RWA) which neglects counter-rotating terms in the full qubit-oscillator Hamiltonian. These counter-rotating terms become important under increase of the coupling strength, as it has been achieved in recent experiments. We present two analytical approaches to take these additional contributions into account: The first one treats the qubit-oscillator Hamiltonian perturbatively in the coupling strength and predicts a frequency shift in its energy spectrum, the Bloch-Siegert shift. The second one considers the qubit's tunneling splitting as small parameter and thus treats the problem to all orders in the coupling allowing to enter the ultrastrong coupling regime, where the coupling strength becomes comparable to the qubit's and/or oscillator's frequency. For both cases we give a thorough analysis of the qubit's dynamics. In order to perform operations the qubit is usually probed by an external driving force. We model this situation by coupling the TLS to a classical oscillator and examine the resulting time dependent Hamiltonian using a combination of Floquet theory and Van Vleck perturbation theory. Thus, we provide an analysis of the qubit's energy spectrum and dynamics which is nonperturbative in the driving amplitude. We investigate effects like coherent destruction of tunneling and driving-inducing tunneling oscillations. By coupling the driven TLS to a quantized oscillator we give for the first time an analytical treatment of the qubit-oscillator system being simultaneously exposed to both ultrastrong coupling and extreme driving. We observe interesting phenomena in the dynamics like the suppression of specific frequencies under a variation of the coupling strength. To mimic environmental influences on the qubit we apply the Caldeira-Leggett master equation approach to the driven TLS and to the qubit-oscillator system and provide for both cases analytical expressions for the relaxation and dephasing rates.
We study theoretically the collective quantum dynamics occurring in various interacting superconducting qubit arrays (SQAs) in the presence of a spread of individual qubit frequencies. The interaction is provided by mutual inductive coupling between adjacent qubits (short-range Ising interaction) or inductive coupling to a low-dissipative resonator (long-range exchange interaction). In the absence of interaction, the Fourier transform of the temporal correlation function of the total polarization (zzz projection of the total spin), i.e., the dynamic susceptibility $C(\ensuremath{\omega})$, demonstrates a set of sharp small magnitude resonances corresponding to the transitions of individual superconducting qubits. We show that even a weak interaction between qubits can overcome the disorder with a simultaneous formation of the collective excited states. This collective behavior manifests itself by a single large resonance in $C(\ensuremath{\omega})$. In the presence of a weak nonresonant microwave photon field in the low-dissipative resonator, the positions of dominant resonances depend on the number of photons, i.e., the collective ac Stark effect. Coupling of an SQA to the transmission line allows a straightforward experimental access of the collective states in microwave transmission experiments and, at the same time, to employ SQAs as sensitive single-photon detectors.
The device for the Josephson flux qubit can be considered as a solid-state artificial atom with multiple energy levels. When a large-amplitude harmonic excitation is applied to the system, transitions at the energy levels avoided crossings produce visible changes in the qubit population over many driven periods that are accompanied by a rich pattern of interference phenomena. We present a Floquet treatment of the periodically time-dependent Schr\"odinger equation of the strongly driven qubit beyond the standard two-level approach. For low amplitudes, the average probability of a given sign of the persistent current qubit exhibits, as a function of the static flux detuning and the driving amplitude, Landau-Zener-St\"uckelberg (LZS) interference patterns that evolve into complex diamondlike patterns for large amplitudes. In the case of highly coherent flux qubits we show that the higher-order diamonds can not be simply described relying on LZS transitions in each avoided crossing considered separately. In addition we propose a spectroscopic method based on starting the system in the first excited state instead of in the ground state, which can give further information on the energy-level spectrum and dynamics in the case of highly coherent flux qubits. We compare our numerical results with recent experiments that perform amplitude spectroscopy to probe the energy spectrum of the artificial atom.
We study numerically the behavior of a qubit coupled to a quantum dissipative driven oscillator (resonator). Above a critical coupling strength the qubit rotations become synchronized with the oscillator phase. In the synchronized regime, at certain parameters, the qubit exhibits tunneling between two orientations with a macroscopic change of the number of photons in the resonator. The lifetimes in these metastable states can be enormously large. The synchronization leads to a drastic change of qubit radiation spectrum with the appearance of narrow lines corresponding to recently observed single artificial-atom lasing [O. Astafiev, Nature (London) 449, 588 (2007)].