aa r X i v : . [ qu a n t - ph ] D ec Effect of a Strong Laser on Spin Precession
Arsen Khvedelidze
A.Razmadze Mathematical Institute, Tbilisi, GE-0193, GeorgiaJoint Institute for Nuclear Research, Dubna, 141980, Russia
Abstract
The semiclassical dynamics of a charged spin-1/2 particle in an intense electro-magnetic plane wave is analyzed beyond the electric dipole approximation and takinginto account the leading relativistic corrections to the Pauli equation. It is arguedthat the adiabatic spin evolution driven by a low intensity radiation changes its char-acter drastically as an intensity of a laser is increasing. Particularly, it is shown thata charged particle exposes a spin flip resonance at a certain pick value of a laser fieldstrength, which is determined by a particle’s gyromagnetic ratio.
The problem and result.
A good deal of a considerable knowledge on a chargedspinning particle interaction with a low intensity laser has been gleaned from the extensiveuse of the electric dipole approximation [1]. This approximation works perfectly to describethe particle’s classical trajectory as well as to understand the adiabatic evolution of spin,represented by the intrinsic angular momentum [2]. With the growing intensity of a radi-ation different relativistic corrections to the charge motion become relevant [3], [4]. Thisdemands to refuse the electric dipole approximation and to take into account the influenceof the magnetic part of the Heaviside-Lorentz force. Entering to this non-dipole regiona new physics become tangible. In this context, the present talk aims to report on themanifestation of a such non-dipole physics: a charged particle’s spin flip resonance inducedby a strong laser field .It is arguably the best to describe the spin-flip resonance in the so-called average restframe, frame where the mean particle’s velocity vanishes. In this frame, as our calculationsshow, the probability to flip for a spin, that is initially polarised along the direction ofpropagation of the circularly polarised monochromatic plane wave, is given by an analog ofthe well-known formula from the Rabi magnetic resonance problem [5]: P ↓↑ = A ↓↑ ( η ) sin ( ω S t ) , ω S := ω L | − g | q κ η + ( η − η ∗ ) , (1)The frequency ω S differs from a laser circular frequency ω L and depends nonlinearly on aparticle’s gyromagnetic ratio g , as well as on a laser field strength parameter [6]: η = − e m c h A i , h· · ·i − time average , A − a laser gauge potential . (2)The flipping amplitude A ↑↓ ( η ) in (1) has the following resonance form A ↓↑ ( η ) = κ η κ η + ( η − η ∗ ) , κ := 2 g (1 − g ) , η ∗ := 4 g − . (3)1 ketch of the calculations. To get the above results the recently elaborated method[7] is extended from the classical to the quantum case. The conventional semiclassical ap-proach attitude, when a charged particle motion in a given electromagnetic background isstudied classically within the non-relativistic Hamilton-Jacobi theory, is adopted. At thesame time, the spin evolution is treated quantum mechanically, as required by the spin na-ture, using the Pauli equations with the leading relativistic corrections. The spin-radiationinteraction is encoded in the effective spatially homogeneous magnetic field configuration,which is determined by the geometry of a particle’s classical trajectory. The describedapproximation is formulated mathematically as follows. The laser radiation is modelled bythe elliptically polarized monochromatic plane wave propagated along the z -axis A µ := a (cid:18) , ε cos( ω L ξ ) , √ − ε sin( ω L ξ ) , (cid:19) , ξ = t − zc , (4)where 0 ≤ ε ≤ a is related to thelaser field strength (2) , η = e a /m c . A charged spin-1/2 particle is in a pure quantumstate Ψ admitting the semiclassical charge & spin decomposition, | Ψ i = X i =0 , X α = ± c α,i | ψ α i ⊗ | χ i i . (5)Two states, | ψ ± i , are the linearly independent WKB solutions to the Schr¨odinger equationfor a charged spinless particle moving in the background (4). According to the semiclassicalcalculations the spin state vectors | χ i in the decomposition (5) satisfy the spin evolutionequation: i dd t | χ i = − ge mc B ′ ( t ) · σ | χ i , (6)where the spatially homogeneous magnetic field B ′ ( t ) is accounted for a laser filed couplingto a spin moving in the laboratory frame with the velocity v and acceleration a , B ′ ( t ) := (cid:18) B − c v × E (cid:19) + megc v × a . (7)Here the term in parenthesis is magnetic field seen in the particle’s instantaneous rest frameand evaluated along the particle’s classical orbit. The last contribution in (7) correspondsto the leading part of the so-called Thomas precession correction due to the non-vanishingcurvature of a particle’s trajectory [2].Analysis of the equations begins with the derivation of the exact solution to the classicalnon-relativistic Hamilton-Jacobi equation for spinless particle moving in the electromag-netic background (4). Solving this Hamilton-Jacobi problem [7] one determine both, theWKB solution to the Schr¨odinger equation and the effective magnetic field (7). We assumebelow that the particle’s classical trajectory x ( t ) fulfills the initial condition x (0) = 0 andfix also the frame, where the average of a particle’s velocity component, orthogonal to thewave propagation direction vanishes, < υ ⊥ > = 0 . To find the effective magnetic field weowe from [7] the expression for a particle’s velocity υ = h − cηε cn ( ω ′ L t, µ ) , − cη √ − ε sn ( ω ′ L t, µ ) , c − c (1 − β z ) dn ( ω ′ L t, µ ) i , (8) To simplify expressions, the initial state is assumed to have only one nonzero coefficient, c + , . Notethat, the unit normalization condition on the WKB wave function fixes this coefficient πc , = 2 m ω P , ω p – a particle’s fundamental frequency.
2s well as the expression for its z -coordinate: z ( t ) = ct − cω L am( ω ′ L t, µ ) . The argumentof the elliptic Jacobian functions and the Jacobian amplitude function, am( u, µ ) , is thelaboratory frame time t , scaled by the non-relativistically Doppler shifted laser frequency ω ′ L = γ z ω L , γ z = 1 − υ z (0) /c and the modulus µ is γ z µ = η (1 − ε ) . Using this solutionthe exact expression for the effective magnetic field B ′ ( t ) can be found: B ′ = aω ′ L gc (cid:16) √ − ε [( g + 1)dn − γ z ] cn , ε h ( g + 1)dn − γ z (1 − µ ) i sn , ε √ − ε h dn − gγ − z i(cid:17) . The resonant oscillations.–
In the average rest frame, < υ > = 0 , when laser beam iscircularly polarised, ε = 1 / , the expression for the effective magnetic field B ′ simplifiesto the constant magnitude field: B ′ ( t ) = | B ′ | n ( t ) , | B ′ | := aω L | − g | gc q κ + η , (9)aligned the unit time dependent vector n ( t ) := (sin θ cos ω L t, sin θ sin ω L t, cos θ ) . Theeffective magnetic field (9) rotates with the frequency ω L about the axis inclined withrespect to the field. The inclination angle θ is determined from the relation: η tan θ = κ . Therefore, the effective laser-spin interaction for the circular polarised radiation is pre-cisely the famous rotated magnetic field describing the nuclear magnetic resonance phe-nomenon !
Having in mind this observation one can use the well-known exact Rabi solution[5] to find the semiclassical evolution of spin-1/2 particle. Particularly, a straightforwardcalculations lead to the expressions (1) and (3) for the spin flipping probability announcedat the beginning of this report.
Conclusion.
In the present talk the non-dipole effect of a strong laser on the spin of acharged particle was described quantum mechanically, while the evolution of position andmomentum of a particle itself were treated according to the classical Newton equations withcomplete Heaviside-Lorentz force. The derived results indicate a very different spin physicsin a high intensity laser field versus to a low intensity adiabatic regime.
Acknowledgment.–
The research was supported in part by the Georgian National Sci-ence Foundation under Grant No. GNSF/ST06/4-050, the Russian Foundation for BasicResearch Grant No. 08-01-00660 and by the Ministry of Education and Science of theRussian Federation Grant No. 1027.2008.2.