Two-photon Terms Research Articles

The mixed quantum Rabi model

The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4 × 4 determinants responsible for the exact solutions are derived. These so-called G-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of G-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point. All energy levels only collapse to the lower one, which diverges negatively. The level crossings in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM due to absence of parity symmetry. In the weak two-photon coupling regime, the mixed QRM is equivalent to an one-photon QRM with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling, which may pave a highly efficient and economic way to access the deep-strong one-photon coupling regime.

Generation of non-classical light in a photon-number superposition

The ability to generate light in a pure quantum state is essential for advances in optical quantum technologies. However, obtaining quantum states with control in the photon-number has remained elusive. Optical light fields with zero and one photon can be produced by single atoms, but so far it has been limited to generating incoherent mixtures, or coherent superpositions with a very small one-photon term. Here, we report on the on-demand generation of quantum superpositions of zero, one, and even two photons, via pulsed coherent control of a single artificial atom. Driving the system up to full atomic inversion leads to the generation of quantum superpositions of vacuum and one photon, with their relative populations controlled by the driving laser intensity. A stronger driving of the system, with $2\pi$-pulses, results in a coherent superposition of vacuum, one and two photons, with the two-photon term exceeding the one-photon component, a state allowing phase super-resolving interferometry. Our results open new paths for optical quantum technologies with access to the photon-number degree-of-freedom.

Generalized quantum Rabi model with both one- and two-photon terms: A concise analytical study

A generalized quantum Rabi Hamiltonian with both one- and two-photon terms has emerged in the circuit quantum electrodynamics system for a decade. The usual parity symmetry is broken naturally in the simultaneous presence of both couplings, which complicates analytical treatments, even in the rotating wave approximations. In this paper, we propose an adiabatic approximation to this generic model by using Bogoliubov operators, and obtain a very concise analytical solution for both eigenvalues and eigenstates. Although the adiabatic approximation is only exact in the vanishing limit of the qubit frequency, the results for some physical observables nevertheless agree well with the numerical ones in a wide parameter regime. In the rotating-wave approximations, we also derive an analytical eigensolution. Two dominant Rabi frequencies are found in the Rabi oscillations of this generalized model. We also apply the present analytical theory to the vacuum Rabi splitting. It is found that some new phenomena emerge just because of the presence of the additional two-photon coupling term.

Dirac Equation in (1+1)-Dimensional Curved Spacetime and the Multiphoton Quantum Rabi Model.

We introduce an exact mapping between the Dirac equation in (1+1)-dimensional curved spacetime (DCS) and a multiphoton quantum Rabi model (QRM). A background of a (1+1)-dimensional black hole requires a QRM with one- and two-photon terms that can be implemented in a trapped ion for the quantum simulation of Dirac particles in curved spacetime. We illustrate our proposal with a numerical analysis of the free fall of a Dirac particle into a (1+1)-dimensional black hole, and find that the Zitterbewegung effect, measurable via the oscillatory trajectory of the Dirac particle, persists in the presence of gravity. From the duality between the squeezing term in the multiphoton QRM and the metric coupling in the DCS, we show that gravity generates squeezing of the Dirac particle wave function.

Off-resonance and non-resonant dispersion of Kerr nonlinearity for symmetric molecules [Invited

The exact formula is derived from the "sum over states" (SOS) quantum mechanical model for the frequency dispersion of the nonlinear refractive index coefficient n₂ for centrosymmetric molecules in the off-resonance and non-resonant regimes. This expression is characterized by interference between terms from two-photon transitions from the ground state to the even-symmetry excited states and one-photon transitions between the ground state and odd-symmetry excited states. When contributions from the two-photon terms exceed those from the one-photon terms, the non-resonant intensity-dependent refractive index n₂>0, and vice versa. Examples of the frequency dispersion for the three-level SOS model are given. Comparison is made with other existing theories.

Adiabatic elimination, the rotating-wave approximation and two-photon transitions

The rotating-wave approximation (RWA) is a formalism of great utility in the description of the coherent excitation of atoms and molecules by laser light. Not only does it give results in agreement with experiment, it also provides a simple framework allowing the Hamiltonian of a system to be written down from inspection of the state-linkage diagram. Recent interest in systems with a two-photon coupling prompted an investigation of the structure of two-photon terms in RWA Hamiltonians. In carrying through the derivation, an interaction with adiabatic elimination was discovered. It is shown that adiabatic elimination must be performed before application of the RWA, else terms are dropped that ought to be retained. RWA Hamiltonians for three-state systems with one and two two-photon linkages are displayed.

On the f sum rule and its extensions

The f sum rule is derived in a nonrelativistic frame and connected, via Ward identities, to the two-photon term of the Compton scattering. A generalisation to isospin symmetry in the nuclear case is discussed and linked to the meson exchange currents. The extension to a fully relativistic theory is then discussed and it is shown that the energy-weighted sum rule becomes a relation between the particle–hole and particle–antiparticle emission. Moreover the generalisation to isospin symmetry is derived and provides nonperturbative results.

A Three-Level Model Useful for Exploring Structure/Property Relationships for Molecular Third Order Optical Polarizabilit1Es

ABSTRACTApproximations are applied to the three-level perturbation theory expressions for the third order optical polarizability. We conclude there will be three competing dominant terms, γ ≈ γc + γn + γtp each of which can be optimized at different molecular electronic structures. The two-photon term, γtp, has been commonly ascribed as the most dominant one for γ. However, the other two terms, which optimize at different structures, potentially offer equivalent or larger overall γ. The term, γn, will be optimized at structures with large second order polarizability β, while the term, γC, optimizes at certain centrosymmetric free electron structures with large first order polarization, α Optimization by γc is the preferred route, since this leads to elimination of the canceling terms which limit optimization by γtp or γn. Recent EFISH results have confirmed our earlier speculations that the free electron squarylium dye structures are particularly good for optimization by way of γc. Strategies are presented to selectively optimize to either of the three structure types and to obtain molecules with even larger γ.

Electron pair production from pulsed electromagnetic fields in relativistic heavy-ion collisions.

We present calculations of the electron pair production cross sections in relativistic heavy-ion collisions. The electron pairs arise from the decay of vacuum excitations induced by the very strong and sharply pulsed electromagnetic fields near nuclei which collide at relativistic velocities. We present an exact Monte Carlo evaluation of the two-photon terms describing this process, and we discuss at length the inadequacies of the approximation schemes that are inherent to the usual virtual-photon approaches. Typical results for collisions corresponding to experiments at the BNL Alternating Gradient Synchrotron, CERN, and the Relativistic Heavy Ion Collider are discussed.

Saturation effects in coherent anti-Stokes Raman scattering

Saturation effects in coherent anti-Stokes Raman scattering (CARS) spectroscopy are discussed. The discussion is limited to Raman-resonant CARS (${\ensuremath{\omega}}_{l}\ensuremath{-}{\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{21}$, where ${\ensuremath{\omega}}_{l,s}$ are the frequencies of the pump fields with powers ${P}_{l,s}$, and ${\ensuremath{\omega}}_{21}$ is the frequency of the Raman transition $|1〉\ensuremath{\rightarrow}|2〉$) with the possible addition of a one-photon resonance (${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ is the frequency of the electronic transition $|1〉\ensuremath{\rightarrow}|3〉$). For these cases we show that the CARS polarization ${\mathcal{p}}_{\mathrm{CARS}}$ is proportional to the off-diagonal density-matrix element ${\ensuremath{\rho}}_{21}$. In order to determine ${\ensuremath{\rho}}_{21}$, we use Laplace transforms to solve the Bloch equations for the effective two-level system $|1〉$ and $|2〉$ when ${\ensuremath{\omega}}_{l}$ is far from resonance, or for the three-level system $|1〉,|2〉$, and $|3〉$, when ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$. The steady-state expression for ${\mathcal{p}}_{\mathrm{CARS}}$ in the former case gives ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}{P}_{s}^{\frac{1}{2}}$ at low powers and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{\ensuremath{-}1}{2}}$ at high powers. In the three-level system, we show that when the pressure is low and at least one field is weak, the slow time dependence of ${\ensuremath{\rho}}_{21}$ must be considered. When one field is strong and the other weak, the CARS spectrum is Stark split. When ${P}_{l}$ is high, for example, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{1}{2}}$ for ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$ and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{({P}_{l}{P}_{s})}^{\frac{1}{2}}$ when ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}\ifmmode\pm\else\textpm\fi{}{V}_{13}$ where ${V}_{13}$ is the one-photon Rabi frequency for the $|1〉\ensuremath{\rightarrow}|3〉$ transition. The Wilcox-Lamb approximation is used to reduce the three-level Bloch equations to rate equations containing one- and two-photon terms. When the fields are so weak that both one- and two-photon terms are small compared to the decay terms, the usual expression for ${\mathcal{p}}_{\mathrm{CARS}}$ is reproduced. If only the direct two-photon processes are important, the effective-two-system results are reproduced. When both fields are intense and nearly resonant, the steady state is rapidly achieved. The results for the case where one field is much stronger than the other are essentially the same as those for one strong and one weak field. When the fields are of comparable strength, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{\frac{1}{2}}{P}_{s}^{0}$ for ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ and ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$, and the CARS spectrum is split into five components.

Dynamics of the nonlinear four-wave mixing interaction

We present an analysis of the impulse response of four-wave mixing, in the case where the “two-photon term” of the nonlinear susceptibility is dominant. The nonlinear interaction itself will be considered to be steady state. The conditions of negligible pump depletion are considered first. It is shown that the reflected beam has only the character of perfect phase conjugation if the pulse length is longer than the medium length, and the latter is short compared to the characteristic distance of the problem. The “weak-pulse” approximation involving only two coupled equations is compared to the exact four-wave solution, when the medium thickness approaches the critical oscillation threshold. Finally, the stability and existence of steady-state solutions is analyzed for very long media (compared with the oscillation length).

Enhancement of asymmetry in lepton annihilation due to resonances

We show that the asymmetry in $e+\overline{e}\ensuremath{\rightarrow}2\ensuremath{\pi},3\ensuremath{\pi}$, due to interference between one-photon and two-photon terms is quite large near the $f$, ${A}_{1}$, and ${A}_{2}$ resonance positions, but goes through a zero. The analysis of this asymmetry can yield important information about the resonances and theories describing them.