Effective field and the Bloch-Siegert shift at bichromatic excitation of multiphoton EPR

Abstract

In view of the possibilities for changing the energy states of the quantum state with external electromagnetic fields and its interaction with the environment, interest in studying the dynamics of multiphoton processes in nonlinear optics, NMR and EPR, has increased recently [1][2][3][4][5][6][7][8]. In two-level spin systems, there are no real intermediate levels, and "dressed" spin systems formed by external electromagnetic fields can serve as intermediate levels. In view of this circumstance, a multilevel dynamical quantum system with controllable properties can be created on the basis of the two-level system. Analysis of the dynamics of the dressed spin systems opens new perspectives for developing methods in order to narrow spectral lines, to separate overlapping spectra, and to increase the existence time of coherent spin states.In pulsed EPR experiments based on the simultaneous absorption or emission of several photons with noticeably different frequencies, e.g., microwave and RF radiation, two-and three-photon electron spin echo [1], two-photon nutation [7], as well as RF-fieldinduced transparency [3] and nutation on dressed states [4,5], were observed. Owing to the small amplitude of the effective field (effective Rabi frequency) of multiphoton transitions, as well as to the presence of inhomogeneous line broadening in real systems, difficulties remain both in direct observation of the dynamics of these transitions and in its quantitative interpretation on the basis of adequate theoretical approaches.In this work, the dynamics of multiphoton transitions in two-level spin systems with its excitation by bichromatic (transverse microwave and longitudinal RF) radiation is studied theoretically and experimentally. To this end, the nonstationary nutation phenomenon, which is present the dynamics of quantum transitions, is used. The nutation frequency is directly determined by the effective interaction field. The direct observation of the dynamics of the interaction of radiation with the spin system makes it possible to reveal regions of multiphoton transitions caused by the inhomogeneous broadening.In the semiclassical approximation, the Hamiltonian of the electron spin system (spin S = 1/2) in the linearly polarized microwave field directed along the x axis of the laboratory coordinate system, as well as in the linearly polarized RF and static magnetic fields directed along the z axis, can be written in the form (1) where ω 0 = γ B 0 is the resonance frequency of spin variables in the magnetic field B 0 , γ is the electron gyromagnetic ratio, ω 1 = γ B 1 and ω 2 = γ B 2 are the Rabi frequency, and B 1 , B 2 , ω mw , ω rf , ϕ , and ψ are the amplitudes, frequencies, and phases of the microwave and RF fields, respectively. The dynamics of multiphoton transitions in a two-level spin system excited by transverse microwave and longitudinal RF fields with the frequencies ω mw and ω rf , respectively, is analyzed. The effective time-independent Hamiltonian describing the "dressed" spin states of the "spin + bichro...

Combating fluctuations in relaxation times of fixed-frequency transmon qubits with microwave-dressed states

“…A6, which involve high-order Bessel functions, can have non-negligible effects on the system dynamics. When considering ∆ d ≈ ∆, these terms contribute as offresonance transitions during the single-gate operations, thus shifting the frequency of the dressed qubit [35,60,61]. This can explain the frequency mismatch discussed in Sec.…”

“…Considering the unitary transformation U 1 = exp(−iθσ y /2) with θ = arctan(Ω s /∆ s ) that intends to diagonalizing H 0 in the dressed qubit basis, (i.e., the dressed basis is defined as the eigenstates of H 0 ) the following dressed system HamiltonianH dress = U † 1 H R U 1 , (i.e., the Hamiltonian in the dressed basis)H dress =[∆ − Ω d sin θ cos(∆ d t + φ)] Z Ω d cos θ cos(∆ d t + φ) X Ω d sin(∆ d t + φ)Y denotes the microwave-dressed qubit detuning, and {X = cos θσ x − sin θσ z , Y = σ y , Z = cos θσ z + sin θσ x } represent the Pauli operators defined on the dressed basis. According to the following unitary transformation[35,59,60]U 2 = exp −i Z 2 ∆t − Ω d sin θ ∆ d sin(∆ d t + φ) , (A5)and using the Jacobi-Anger relations, one can obtain the effective HamiltonianH J = U † 2 H dress U 2 + i∂ t (U † 2 )U 2 , i.e., H J = Ω d cos θ 4 (e i∆ d t+iφ + e −i∆ d t−iφ ) × e i∆t S + ∞ n=−∞ J n ( Ω d sin θ ∆ d )e −in(∆ d t+φ) + h.c. − Ω d 4 (e i∆ d t+iφ − e −i∆ d t−iφ ) × e i∆t S + ∞ n=−∞ J n ( Ω d sin θ ∆ d )e −in(∆ d t+φ) − h.c. ,(A6)where h.c. denotes the Hermitian conjugate, J n is the nth order Bessel function of the first kind, and S ± = (X ± iY )/2.Applying the RWA and dropping high-order Bessel functions, one can obtain the following effective Hamiltonian describing the usual single-qubit driven terms,H eff = Ω J 0 ( Ω d sin θ ∆ d ) e −iφ S + e i(∆−∆ d ) + h.c. .…”

Band-anisotropy induced Bloch–Siegert shift in graphene

“…The power series expansion has good agreement with quantum electrodynamic calculations, but a truncated continued fraction expression is shown to represent the exact values much better. The Bloch-Siegert shifts for multiphoton resonances at the inhomogeneous broadening of spectral lines reduce only the nutation amplitude but do not change their frequencies [37]. Beijersbergen et al [38] observe multiphoton resonances and Bloch-Siegert shifts in a strongly driven classical two-level system which is an optical ring resonator where the levels are two orthogonal linear polarizations.…”

Sideband transitions in a two-mode Josephson circuit driven beyond the rotating wave approximation

“…This work also uses multi-photon assisted transition, which is widely adopted when a desired transition is dipole forbidden [15,16,[30][31][32][33][34][35][36][37][38][39][40][41][42]. In a perspective that the studies of multi-photon transition beyond the RWA are mainly limited to theoretical cases [43][44][45], our work would attract attention.…”