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 destructive interference of multiple photon processes at 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 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 conditions of experiments with a nitrogen vacancy center in diamond.
A superconducting qubit was driven in an ultrastrong fashion by an oscillatory microwave field, which was created by coupling via the nonlinear Josephson energy. The observed Stark shifts of the `atomic' levels are so pronounced that corrections even beyond the lowest-order Bloch-Siegert shift are needed to properly explain the measurements. The quasienergies of the dressed two-level system were probed by resonant absorption via a cavity, and the results are in agreement with a calculation based on the Floquet approach.
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 strong drive (classical p…
A cavity quantum electrodynamical (QED) system beyond the strong-coupling regime is expected to exhibit intriguing quantum phenomena. Here we report a direct measurement of the photon-dressed qubit transition frequencies up to four photons by harnessing the same type of state transitions in an ultrastrongly coupled circuit-QED system realized by inductively coupling a superconducting flux qubit to a coplanar-waveguide resonator. This demonstrates a convincing observation of the photon-dressed Bloch-Siegert shift in the ultrastrongly coupled quantum system. Moreover, our results show that the p…
Coherent light-matter interaction can be used to manipulate the energy levels of atoms, molecules and solids. When light with frequency ω is detuned away from a resonance ωo, repulsion between the photon-dressed (Floquet) states can lead to a shift of energy resonance. The dominant effect is the optical Stark shift (1/(ω0-ω)), but there is an additional contribution from the so-called Bloch-Siegert shift (1/(ωo+ω)). Although it is common in atoms and molecules, the observation of Bloch-Siegert shift in solids has so far been limited only to artificial atoms since the shifts were small (<1 μeV)…
We report on the theoretical and experimental investigation of valley-selective optical Stark and Bloch-Siegert shifts of exciton resonances in monolayers WSe$_2$ and MoS$_2$ induced by strong circularly polarized nonresonant optical fields. We predict and observe transient shifts of both 1sA and 1sB exciton transitions in the linear interaction regime. The theoretical description is based on semiconductor Bloch equations. The solutions of the equations are obtained with a modified perturbation technique, which takes into account many-body Coulomb interaction effects. These solutions allow to …
Coherent interaction between matter and periodic light field induces both optical Stark effect (OSE) and Bloch-Siegert shift (BSS). Observing the BSS has been historically challenging, not only because it is weak but it is often accompanied by a much stronger OSE. Herein, by controlling the light helicity, we can largely restrict the OSE and BSS to different spin-transitions in CsPbI3 perovskite quantum dots, achieving room-temperature BSS as strong as 4 meV with near-infrared pulses. The ratio between the BSS and OSE magnitudes is however systematically higher than the prediction by the non-i…
Comprehending the dynamical behaviour of quantum systems driven by time-varying Hamiltonians is particularly difficult. Systems with as little as two energy levels are not yet fully understood as the usual methods including diagonalisation of the Hamiltonian do not work in this setting. In fact, since the inception of Magnus' expansion in 1954, no fundamentally novel mathematical approach capable of solving the quantum equations of motion with a time-varying Hamiltonian has been devised. We report here of an entirely different non-perturbative approach, termed path-sum, which is always guarant…
We theoretically study coherent subharmonic (multi-photon) transitions of a harmonically driven spin. We consider two cases: magnetic resonance (MR) with a misaligned, i.e., non-transversal driving field, and electrically driven spin resonance (EDSR) of an electron confined in a one-dimensional, parabolic quantum dot, subject to Rashba spin-orbit interaction. In the EDSR case, we focus on the limit where the orbital level spacing of the quantum dot is the greatest energy scale. Then, we apply time-dependent Schrieffer-Wolff erturbation theory to derive a time-dependent effective two-level Hami…
We investigate the dynamics of a driven two-level system (classical Rabi model) using the counter-rotating-hybridized rotating wave method (CHRW), which is a simple method based on a unitary transformation with a parameter $ξ$. This approach is beyond the traditional rotating-wave approximation (Rabi-RWA) and more importantly, remains the RWA form with a renormalized tunneling strength and a modified driving strength. The reformulated rotating wave method not only possesses the same mathematical simplicity as the Rabi-RWA but also allows us to explore the effects of counter-rotating (CR) compo…
In the framework of the non-secular 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 have been 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…
We study the coherent dynamics of relaxing qubits driven by a bichromatic radiation in the dispersive regime, when detuning of the frequency $ω_{rf}$ of a longitudinal radiofrequency field from the Rabi frequency $ω_{1}$ in a transverse microwave field is comparable in magnitude to $ω_{rf}$ and $ω_{1}$. We analytically describe this regime beyond the rotating wave approximation and find that the dominant feature of dynamics of qubits is the shift of the Rabi frequency caused by the dynamical Zeeman and Bloch-Siegert-like effects. These fundamental effects can be experimentally separated becaus…
Rabi flopping between Bloch bands induced by a weak ac resonant field is a coherent effect involving interband transitions. Here we consider the fundamental processes of emission/absorption of quanta and Rabi oscillations in non-Hermitian two-band lattices exhibiting unbalanced non-Hermitian skin effect, and unveil an unprecedented scenario of Rabi flopping. The effective dipole moment of the transition - usually considered a bulk property - is however strongly dependent on boundary conditions, being greatly enhanced with increased Rabi frequency only when open boundaries are present. As the f…
Carrier dynamics at energy level anticrossings in biased semiconductor superlattices, was studied in the time domain by solving the time-dependent Schroedinger equation. The resonant nature of interminiband Rabi oscillations has been explicitly demonstrated to arise from interference of intrawell and Bloch oscillations. We also report a simulation of direct Rabi oscillations across three minibands, in the high field regime, due to interaction between three strongly coupled minibands.
Electrically driven spin resonance (EDSR) is an established tool for controlling semiconductor spin qubits. Here, we theoretically study a frequency-mixing variant of EDSR, where two driving tones with different drive frequencies are applied, and the resonance condition connects the spin Larmor frequency with the sum of the two drive frequencies. Focusing on flopping-mode operation, we calculate the parameter dependence of the Rabi frequency and the Bloch-Siegert shift. A shared-control spin qubit architecture could benefit from this bichromatic EDSR scheme, as it enables simultaneous single-q…
We study a realistic model for driven qubits using the numerical solution of the Bloch-Redfield equation as well as analytical approximations using a high-frequency scheme. Unlike in idealized rotating-wave models suitable for NMR or quantum optics, we study a driving term which neither is orthogonal to the static term nor leaves the adiabatic energy value constant. We investigate the underlying dynamics and analyze the spectroscopy peaks obtained in recent experiments. We show, that unlike in the rotating-wave case, this system exhibits nonlinear driving effects.We study the width of spectros…
Limitations of the $g$-tensor formalism of semiconductor spin qubits
ELTE Eötvös Loránd University（厄特沃什·罗兰大学）；Institute of Physics, Budapest University of Technology and Economics（布达佩斯技术与经济大学物理研究所）；HUN-REN-BME-BCE Quantum Technology Research Group（匈牙利研究网络-布达佩斯技术与经济大学-布达佩斯经济与商业学院量子技术研究组）
We investigate the near infrared interband absorption of semiconductor quantum wells driven by intense terahertz radiation in the regime of ultrastrong coupling, where the Rabi frequency is a significant fraction of the frequency of the strongly driven transition. With the driving frequency tuned just below the lowest frequency transition between valence subbands, a particularly interesting phenomenon is observed. As the THz power increases, a new peak emerges above the frequency of the undriven exciton peak which grows and eventually becomes the larger of the two. This reversal of relative pe…
Rabi oscillations are coherent transitions in a quantum two-level system under the influence of a resonant perturbation, 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 [N. Gronbech-Jensen and M. Cirillo, Phys. Rev. Lett. 95, 067001 (2005)] 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…
Motivated by recent efforts to achieve cold fermions pairing near a Feshbach resonance, we consider the dynamics of formation of the Bardeen-Cooper-Schrieffer (BCS) state. At times shorter than the quasiparticle energy relaxation time, after the interaction is turned on, the dynamics of the system is nondissipative. We show that this collective nonlinear evolution of the BCS-Bogoliubov amplitudes (u,v) along with the pairing function, is an integrable dynamical problem, and obtain a family of exact solutions in the form of single solitons and soliton trains. We interpret the collective oscilla…
We studied the semiconductor response with respect to high intensity resonant excitation on short time scale when the contribution of the Fermi statistics of the electrons and holes prevails. We studied both the single and double pulse excitations. For the latter case we considered the time evolution of the multi-wave mixing exciton polarization. The main difference between the excitation by a single pulse or by two non-collinear pulses is that the Rabi oscillations of the multi-wave mixing response are characterized by two harmonics. Analyzing the operator dynamics governed by the external ex…
We achieve the strong coupling regime between an ensemble of phosphorus donor spins in a highly enriched $^{28}$Si crystal and a 3D dielectric resonator. Spins were polarized beyond Boltzmann equilibrium using spin selective optical excitation of the no-phonon bound exciton transition resulting in $N$ = $3.6\cdot10^{13}$ unpaired spins in the ensemble. We observed a normal mode splitting of the spin ensemble-cavity polariton resonances of 2$g\sqrt{N}$ = 580 kHz (where each spin is coupled with strength $g$) in a cavity with a quality factor of 75,000 ($γ\ll κ\approx$ 60 kHz where $γ$ and $κ$ a…
Semiconductor quantum dots in photonic cavities are strongly coupled light-matter systems with prospective applications in optoelectronic devices and quantum information processing. Here we present a theoretical study of the coupled exciton--light field dynamics of a planar quantum dot ensemble, treated as two-level systems, embedded in a photonic cavity modeled by Maxwell's equations. When excited by coupling an external short laser pulse into the cavity, we find an exciton-polariton-like behavior for weak excitation and Rabi oscillations for strong excitation with a sharp transition between …
We study Rabi oscillations detected in the coherent optical response from various exciton complexes in a 20~nm-thick CdTe/(Cd,Mg)Te quantum well using time-resolved photon echoes. In order to evaluate the role of exciton localization and inhomogeneous broadening we use selective excitation with spectrally narrow ps-pulses. We demonstrate that the transient profile of the photon echo from the localized trion (X$^-$) and the donor-bound exciton (D$^0$X) transitions strongly depends on the strength of the first pulse. It acquires a non-Gaussian shape and experiences significant advancement for pu…
This thesis is devoted to the study of quantum mechanical effects that arise in systems of reduced dimensionality. Specifically, we investigate coherence and correlation effects in quantum transport models. In the first part, we present a theory of Markovian and non- Markovian current correlations in nanoscopic conductors. The theory is applied to obtain the spectrum of quantum noise and high-order current correlations at finite frequencies in quantum-dot systems. One of the main conclusions is that only the non-Markovian approach contains the physics of vacuum fluctuations. In the second part…
Continuous spectroscopic measurements of a strongly driven superconducting qubit by means of a high-quality tank circuit (a linear detector) are under study. Output functions of the detector, namely, a spectrum of voltage fluctuations and an impedance, are expressed in terms of the qubit spectrum and magnetic susceptibility. The nonequilibrium spectrum of the current fluctuations in the qubit loop and the linear response function of the driven qubit coupled to a heat bath are calculated with Bloch-Redfield and rotating wave approximations. Backaction effects of the qubit on the tank and the ta…
The Bloch sphere is a generic picture describing a coupled two-level system and the coherent dynamics of its superposition states under control of electromagnetic fields. It is commonly employed to visualise a broad variety of phenomena ranging from spin ensembles and atoms to quantum dots and superconducting circuits. The underlying Bloch equations describe the state evolution of the two-level system and allow characterising both energy and phase relaxation processes in a simple yet powerful manner. Here we demonstrate the realisation of a nanomechanical two-level system which is driven by ra…
We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combing Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equa…
We theoretically investigate basic properties of nonequilibrium steady states of periodically-driven open quantum systems based on the full solution of the Maxwell-Bloch equation. In a resonantly driving condition, we find that the transverse relaxation, also known as decoherence, significantly destructs the formation of Floquet states while the longitudinal relaxation does not directly affect it. Furthermore, by evaluating the quasienergy spectrum of the nonequilibrium steady states, we demonstrate that the Rabi splitting can be observed as long as the decoherence time is as short as one thir…
The quantum state of a flux qubit was successfully pulse-controlled by using a resonant microwave. We observed Ramsey fringes by applying a pair of phase-shifted pi/2 microwave pulses without introducing detuning. With this method, the qubit state can be rotated on an arbitrary axis in the x-y plane of the Bloch sphere in a rotating frame. We obtained a qubit signal from a coherent oscillation with an angular velocity of up to 2pi*11.4 Grad/s. In combination with Rabi pulses, this method enables us to achieve full control of single qubit operation. It also offers the possibility of orders of m…
We characterize numerically and analytically the signatures of the spin-orbit interaction in a two-electron GaAs double quantum dot in the presence of an external magnetic field. In particular, we obtain the return probability of the singlet state by simulating Landau-Zener voltage detuning sweeps which traverse the singlet-triplet ($S-T_+$) resonance. Our results indicate that non-spin-conserving interdot tunneling processes arising from the spin-orbit interaction have well defined signatures. These allow direct access to the spin-orbit interaction scales and are characterized by a frequency …
