Zeeman Effect Induced by Intense Laser Light

E. Stambulchik and Y. MaronFaculty of Physics, Weizmann Institute of Science, Rehovot 7610001, Israel

(Received 14 May 2014; published 19 August 2014; corrected 20 August 2014)

We analyze spectral line shapes of hydrogenlike species subjected to fields of electromagnetic waves. Itis shown that the magnetic component of an electromagnetic wave may significantly influence the spectra.In particular, the Zeeman effect induced by a visible or infrared light can be experimentally observed usingpresent-day powerful lasers. In addition, the effect may be used for diagnostics of focused beam intensitiesachieved at existing and newly built laser facilities.

DOI: 10.1103/PhysRevLett.113.083002 PACS numbers: 32.60.+i, 32.70.-n, 52.70.La

Effects of oscillating electric fields on atomic spectrahave been studied since the work of Blokhintsev [1] at thedawn of modern quantum mechanics. It follows fromRef. [1] that a single-Stark-component model hydrogenline in an ~F0 cosΩt field is split into an infinite number ofsatellites separated from the line center bykΩ, where k isan integer. If, on the other hand, the field is circularlypolarized, each static Stark-shifted component is split intotwo satellites, separated approximately by Ω, as foundby Ishimura [2]. For nondegenerate atomic systems withthe quadratic Stark effect, the influence of oscillatingelectric fields was first considered by Baranger and Mozer[3], who showed, in the weak-field limit, that a dipole-forbidden transition is split in the presence of such afield into two Ω-separated satellites. This work wasextended by Hicks et al. [4] to arbitrarily strong electricfields. These effects have been employed in numerousstudies, with the oscillating electric fields being due toplasma waves or powerful microwave or laser radiation;for details and an extensive bibliography up to the mid-1990s, see monographs [5–7].Optical lasers generate electromagnetic (EM) waves in

the visible or near-IR bands, i.e., ℏΩ ∼ 1 eV. Evidently,observing satellites separated by a few eV is practicalonly for radiative transitions with energies in the range ofhundreds of eV to a few keV, i.e., x rays (e.g., Refs. [8,9]);otherwise, the satellites may become intermixed with otherspectral lines present in the same spectral region.Spontaneous x-ray radiation is characteristic of inner-shelltransitions in species with atomic number Z ≳ 10, since theelectron binding energies grow as ∼Z2. However, thecoupling of the atom to the electric field is −~d · ~F, where~d is the dipole-moment operator, and its matrix elementsdecrease as ∼Z−1.Contrary to the electric field, the coupling to the

magnetic field −~μ · ~B does not depend on Z (here, ~μ isthe magnetic dipole moment of the atom). Therefore, theimportance of the magnetic component of the EM field,relative to that of the electric component, should growlinearly with Z. Indeed, a simple estimation of the ratio

between the magnetic and electric couplings of a charge,

moving with velocity v, to an EM wave in vacuum isv=c, where c is the speed of light. With a typical electronvelocity αZc=n in the Coulomb field of charge Z,this gives αZ=n, where α ≈ 1=137 is the fine-structureconstant and n is the principal quantum number.However, to the best of our knowledge, all studies haveso far neglected the magnetic field, considering only theelectric component.For concreteness, let us consider the Lyman-α transition

(n ¼ 2 to n ¼ 1) in a hydrogenlike atom or ion. At first, we

assume static ~F and ~B crossed at an arbitrary angle θ.Without loss of generality, we choose the quantization axis

z along ~F, while ~B lies in the x-z plane. We further assume apurely linear Stark effect, neglecting the spin degree offreedom and interactions between levels with different n.With these assumptions, the ground 1s state remainsunperturbed (ΔE1 ¼ 0), while the n ¼ 2 manifold of the

total perturbation V ¼ −~d · ~F − ~μ · ~B reads as

V2 ¼

0BBBBB@

0 0 3Z F 0

0 − α cos θ2

B α sin θ2ffiffi2

p B 0

3Z F

α sin θ2ffiffi2

p B 0 α sin θ2ffiffi2

p B

0 0 α sin θ2ffiffi2

p B α cos θ2

B

1CCCCCA

ð1Þ

[from now on, we use the atomic units whereme ¼ e ¼ ℏ ¼ 1=ð4πϵ0Þ ¼ 1; in these units, the speed oflight c is 1=α, while the Bohr magneton μ0 ¼ α=2]. Weused the (n, l, m) representation; i.e., the four n ¼ 2 basisvectors are j2s0i, j2p−1i, j2p0i, and j2pþ1i.With only F or B present, the eigenvalue solutions of

det ðV2 − ΔE2Þ ¼ 0 are 0 (doubly degenerate) and 3F=Z

or αB=2, respectively. With ~F parallel to ~B, ~d · ~F and

~μ · ~B commute; therefore, the eigenvalues are 3F=Z andαB=2, i.e., a combination of the lateral Stark andZeeman patterns. The relative effect of the magneticfield is, thus,

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ξ≡ αZ6

BF: ð2Þ

However, if ~F and ~B are perpendicular, the solution isdifferent: There again appears the doubly degenerateunshifted component, while the two lateral componentsare shifted by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3F=ZÞ2 þ ðαB=2Þ2

p, i.e.,

ΔE2 ¼0; 3F

Z

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ξ2

p : ð3Þ

Therefore, the Stark-Zeeman pattern becomes that of the

pure Stark effect in an “effective” field F0 ¼ Fffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ξ2

p. In

an EM wave propagating in vacuum or in a media with therefraction index close to unity (such as an underdense

plasma), ~F and ~B are perpendicular and equal by absolutevalue, i.e., ξ ¼ αZ=6. Therefore, even for transuraniumelements, the correction due to the magnetic field is onlyabout 1%.Thus, at first sight, the neglect of the magnetic compo-

nent appears to be justified. However, so far, we haveomitted the spin degree of freedom. Including it makes acrucial difference. The fine structure due to the spin-orbit

interaction Vls ¼ Anl~L · ~S results in splitting of an nl

(l > 0) level into j ¼ lþ 12and j ¼ l − 1

2levels shifted by

Anll=2 and −Anlðlþ 1Þ=2, respectively, where

Anl ¼ α2Z4

n3lðlþ 1Þð2lþ 1Þ ð4Þ

(e.g., see Ref. [10]; the Lamb shift is neglected). In the(n, l, m, ms) representation, the n ¼ 2 subset of theperturbation is then

V2 ¼

0BBBBBBBBBBBBBBBBBB@

−a b 0 0 f 0 0 0

−a 0 0 0 f 0 0

a2

b bffiffi2

p 0 0 0

− a2

affiffi2

p bffiffi2

p 0 0

0 b bffiffi2

p 0

0 affiffi2

p bffiffi2

p

− a2

ba2

1CCCCCCCCCCCCCCCCCCA

; ð5Þ

where f ≡ 3F=Z, b≡ αB=2, and a≡ A2p (the lowertriangle of the symmetric matrix is not shown).Assuming smallness of b (i.e., omitting terms containingthe second and higher orders of b in the characteristicpolynomial), one obtains doubly degenerate ΔE2 ¼ a=2and six roots of cubic equations

2ΔE23 þ ΔE2

2ð3aþ 4bÞ − 2ΔE2ðf2 − 3abÞ¼ aðf2 − 2abþ a2Þ ð6Þ

and a similar one with b → −b. The explicit expressions arestraightforward but cumbersome.Evidently, if f, b ≫ a, the Stark-Zeeman pattern reduces

to that of Eq. (3). The opposite, weak-field limit (f, b ≪ a)is obviously not interesting. Let us instead consider the caseof b≲ a ≪ f. In this case,

ΔE2 ≈ a2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2 þ b2

q b −

a2

: ð7Þ

The n ¼ 1 perturbation Hamiltonian is

V1 ¼0 b

b 0

; ð8Þ

giving

ΔE1 ¼ b: ð9Þ

Importantly, the upper-state energies depend, in general,on f and b, while those of the lower states depend only onb. The b contribution to the energies of the upper lateraland lower states correspond to different spin orientationsalong the magnetic field; since spin-flipping transitionsare dipole forbidden, the lateral components of theLyman-α transition appear at

ΔEðlateralÞ ≈ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2 þ b2

q−a2

¼ 3FZ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ξ2

p−A2p

2; ð10Þ

i.e., up to a constant shift, the same as in the no-spinapproximation (3), with a negligible magnetic-fieldeffect. However, the central component that remainsunshifted in the no-spin approximation becomes split intofour components

ΔEðcentralÞ ≈ a2 b ¼ 1

2ðA2p αBÞ; ð11Þ

the positions of which depend linearly on B.Although derived assuming f ≫ a, these results also

remain qualitatively valid if f ≃ a. As an example, thecombined Stark-Zeeman effect of the Lyman-α transitionin Kr XXXVI (hydrogenlike Kr ion) under the static Fand B fields, approximately corresponding to those inthe laser light focused to an intensity of 1020 W=cm2, isshown in Fig. 1. For the numerical calculations, theatomic energies were taken from the NIST database[11], thus also accounting for the Lamb shift; the effectof the latter was, as expected, mostly negligible for the

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083002-2

field strengths assumed. Contrary to the calculationswith the fine-structure effect omitted, the magneticfield causes a significant, ≈αB, splitting of the centralcomponents.An important characteristic of the spectrum is its

polarization properties, unsurprisingly very similar to thoseof the pure Stark effect: The central and lateral componentsare mostly σ and π polarized, respectively. Therefore, witha proper direction of observation (parallel to ~F) or using apolarizer, the π components can be suppressed, thussimplifying observation of the Zeeman effect in the centralpart of the spectrum.Let us now consider the time-dependent effect in the

presence of a harmonic EM field. To this end, we employ acomputer simulation technique [12], modified to use,instead of the plasma microfields, macroscopic fields asa perturbation:

~FðtÞ ¼ ~F0 sin ðΩtÞ;~BðtÞ ¼ ~B0 sin ðΩtÞ; ð12Þ

where ~F0 and ~B0 are orthogonal and equal in absolutevalue; that is, they are components of a linearly polarizedEM wave. The results are shown in Fig. 2(a). Althoughthe spectrum is composed of an infinite number ofsatellites, separated by ℏΩ (the inset shows a small portionof the graph zoomed in), it retains the important featuresinferred from the static solution (Fig. 1): (i) the centralcomponents are split by the magnetic field, and (ii) theZeeman effect can be isolated and enhanced by choosingan appropriate direction of observation or polarization.These features become especially apparent after convo-lution of the spectrum with a smoothing function, asshown in Fig. 2(b).

Evidently, higher intensities and, hence, stronger mag-netic fields would result in a more pronounced Zeemaneffect. For such strong fields, however, the linear-Stark-effect approximation is no longer justified, andhigher-order Stark effects (arising mostly due to electric-field-induced mixing between the n ¼ 2 and n ¼ 3 states)need to be accounted for, resulting in “redshifts” of theLy-α components. Such calculations are shown in Fig. 3.One sees that by varying the laser intensity, the spectralpattern changes significantly.The effect is important and can be observed in many

species, provided a few criteria are satisfied. The firstone, already discussed, is that b≲ a≲ f. With the lasermean-squared field magnitudes hFi ¼ hBi ¼ ffiffiffiffiffiffiffiffiffiffi

4παIp

,where I is the laser beam intensity, this condition can berewritten as

α3=2Z5

288≲ ffiffiffiffiffi

πIp ≲ α1=2Z4

48: ð13Þ

The natural linewidth Γ, determined by the spontaneousradiative rate, should be smaller than the Zeeman splitting,i.e., Γ≲ b, or [10]

0.00

0.05

0.10

0.15

Inte

nsity

(ar

b. u

nits

)-150 -100 -50 0 50 100

∆E (eV)

0.00

0.01

0.02

0.03

0.04

Inte

nsity

(ar

b. u

nits

)

Observation FObservation B

(a)

(b)

FIG. 2 (color online). (a) Combined Stark-Zeeman effect ofKr XXXVI Ly-α in the EM field of a linearly polarized lightwith intensity 1020 W=cm2 and angular frequency Ω ¼2 × 1015 rad=s (λ ≈ 0.94 μm), averaged over all directions ofobservation. (b) Same after convolution with a 4-eV FWHMGaussian and shown separately for directions of observationparallel to ~F or ~B of the laser field.

-200 -100 0 100 200∆E (eV)

0.10

0.05

0.00

0.05

0.10

Inte

nsity

(ar

b. u

nits

)

No fieldsOnly FF ⊥ B

σ

π

FIG. 1 (color online). Combined static Stark-Zeeman effect ofKr XXXVI Ly-α in the perpendicular electric and magnetic fields.The σ and π polarization components are given separately. Forcomparison, zero-field and electric-field-only spectra are alsoshown. Fz ¼ 30 TV=m and Bx ¼ 100 kT are assumed. Zero onthe energy axis corresponds to the Kr XXXVI Ly-α unperturbedenergy of ≈13.5 keV [11].

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ffiffiffiffiffiπI

p ≳ 64α3=2Z4

2187: ð14Þ

In addition, the laser intensity should not be too high tocause field ionization of the upper (n ¼ 2) level. Using theclassical ionization field (e.g., Ref. [13]) as an estimate, thiscriterion gives

ffiffiffiffiffiπI

p ≲ Z3

512α1=2: ð15Þ

Inequalities (13)–(15) are represented graphically in Fig. 4;the area in the ðZ; IÞ plane for which all these fourconditions are simultaneously satisfied is filled with alight gray color. Two additional criteria are less general.Similarly to the f=Ω≳ 1 [5,6] requirement for the Starkeffect to be pronounced, in the case of the Zeeman effect,for the same reason, b=Ω≳ 1 should be satisfied, i.e.,

ffiffiffiffiffiπI

p ≳ Ωα3=2

: ð16Þ

Finally, for observing the effect, it must be resolved, i.e.,b≳ ΔE, where ΔE is the contribution to the linewidth dueto both the Doppler and instrumental broadenings. Thus,

ffiffiffiffiffiπI

p ≳ 3Z2

8α3=2R; ð17Þ

where R ¼ Etr=ΔE, and Etr ≈ 3Z2=8 is the Ly-α transitionenergy. With Ω ¼ 2 × 1015 rad=s and R ¼ 3000 (seebelow), these additional criteria result in a further narrowingdown of the region of observability, shown by the darkergray area in Fig. 4. One can see, for example, that the effectis important for the Ar Ly-α transition in the laser-intensityrange of approximately ð1–4Þ × 1019 W=cm2, whereas forKr Ly-α, the relevant range is ð1–20Þ × 1020 W=cm2.For a direct observation of the effect, high temporal and

spatial resolutions are required, since achieving irradiationintensities ∼1020 W=cm2 implies the use of subpicosecondlaser pulses focused to a few-micrometer-diameter spot.Evidently, acquiring radiation from a larger volume or duringa longer time would result in mixing of the laser-affectedspectrum with the unperturbed line, making the discrimina-tion of the interesting effect challenging. We performedcalculations assuming a measurement where only one-tenthof the recorded signal is useful, for example, a laser pulsewitha Gaussian envelope of 100-fs FWHM and a 1-ps timeresolution of the data acquisition system; alternatively, thesecalculations would correspond, e.g., to a 4 times better timeresolution [14] but an inferior spatial resolution, exceedingthe laser-focal-spot diameter twofold. The results are shownin Fig. 5. Even with these unfavorable assumptions, the

-75 -50 -25 0 25 50∆E (eV)

0

0.01

0.02

0.03

0.04

0.05

0.06In

tens

ity (

arb.

uni

ts)

1020

W/cm2

3×1020

W/cm2

1021

W/cm2

FIG. 3 (color online). Combined Stark-Zeeman effect of KrXXXVI Ly-α in the EM field of a linearly polarized light ofvarying intensity. The direction of observation is parallel to thepolarization vector. The spectra are convolved with a 4-eVFWHM Gaussian.

0 20 40 60 80Z

1x10+24

1x10+22

1x10+20

1x10+18

1x10+16

1x10+14

1x10+12

1x10+10

1x10+8

I (W

/cm

2 )

Ar Kr

f = ab = aField ionizationb = Γb = h- Ω (Ω = 2×10

15 rad/s)

b/Etr = R

-1 (R = 3000)

FIG. 4 (color online). Criteria and regions of observability ofthe Zeeman effect on Ly-α of species with the atomic number Z inthe EM field with intensity I (see the text).

-60 -40 -20 0 20 40∆E (eV)

0

0.05

0.1

0.15

Inte

nsity

(ar

b. u

nits

)

0 200 400 600 800 1000t (fs)

0

1021

Las

er e

nvel

ope

(W/c

m2 )

× 10

FIG. 5 (color online). Modeling of the Kr XXXVI Ly-α spectrumcollected over a 1-ps exposure and subjected to an EM field of alinearly polarized laser with a 100-fs Gaussian envelope (shownin the inset). The direction of observation is parallel to thepolarization vector. For clarity, a ×10 enlarged spectrum is shownby the dashed line.

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Zeeman splitting is clearly seen at the “blue” side of theunperturbed Ly-α1=2 (1s1=2 − 2p1=2) component. Theplasma electrons should have a sufficiently high temperature(kTe ∼ 3–6 keV) in order to cause a significant population ofthe H-like n ¼ 2 states. We note that the Doppler broadeningis likely to be negligible, since the thermal electron-ionrelaxation time for an underdense (ne ≲ 1021 cm−3) Krplasma is on the order of 10 ns (e.g., Ref. [13]), much longerthan the laser-pulse duration. Thus, the ΔE ¼ 4 eV broad-ening assumed in this and previous figures should be viewedlargely as an instrumental broadening, corresponding to aresolving powerR ¼ E=ΔE ≈ 3400. Finally, the effect of theplasma microfields for the conditions assumed is completelynegligible [13].To conclude, contrary to naive expectations, the

magnetic component of an EM wave may significantlyinfluence spectra of radiative transitions. Amazingly, theZeeman effect induced by a visible or infrared light can beexperimentally observed using present-day powerful lasers.Additionally, the effect may, in principle, serve for diag-nostics of focused beam intensities achieved at existing andnewly built laser facilities.

We thank E. Kroupp, G. G. Paulus, O. Renner, andI. Uschmann for fruitful discussions. Valuable comments ofthe referees are highly appreciated. This work was sup-ported in part by the Minerva Foundation (Germany) andthe Cornell Center of Excellence for the Study of Pulsed-Power-Driven High Energy Density Plasma Studies.

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490 (1972).[5] V. S. Lisitsa, Atoms in Plasmas, Springer Series on Atoms

and Plasmas Vol. 14 (Springer, Berlin, 1994).[6] E. Oks, Plasma Spectroscopy: The Influence of Microwave

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[7] H. R. Griem, Principles of Plasma Spectroscopy(Cambridge University Press, Cambridge, England, 1997).

[8] O. Renner, O. Peyrusse, P. Sondhauss, and E. Förster,J. Phys. B 33, L151 (2000).

[9] N. C. Woolsey, D. M. Chambers, C. Courtois, E. Förster,C. D. Gregory, I. M. Hall, J. Howe, O. Renner, andI. Uschmann, J. Quant. Spectrosc. Radiat. Transfer 99,680 (2006).

[10] H. A. Bethe and E. E. Salpeter, Quantum Mechanics ofOne- and Two-Electron Atoms (Plenum, New York,1977).

[11] A. Kramida, Yu. Ralchenko, J. Reader, and the NIST ASDTeam, NIST Atomic Spectra Database (version 5.1), http://physics.nist.gov/asd.

[12] E. Stambulchik and Y. Maron, J. Quant. Spectrosc. Radiat.Transfer 99, 730 (2006).

[13] E. Stambulchik and Y. Maron, JINST 6, P10009(2011).

[14] J. Feng, H. J. Shin, J. R. Nasiatka, W. Wan, A. T. Young,G. Huang, A. Comin, J. Byrd, and H. A. Padmore,Appl. Phys. Lett. 91, 134102 (2007).

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