Abstract

It is shown with the use of the solution of the Boltzmann equation for laser pumping in a cell with antirelaxation coating that the velocity distribution of atomic polarization moments (PMs) is essentially dependent on the value of the magnetic field H. The z-velocity distribution of PMs in a low field, H10-4 A/m, is a Maxwellian one with a small admixture of an almost monokinetic one. At larger field the same distribution remains for longitudinal alignment, but for transverse alignment the Maxwellian part of the distribution disappears (at H1 A/m). It appears that the radial velocity distribution is also dependent on the field H. A calculation accounting for wall Maxwellization in low field gives for the pumping power needed to saturate the magneto-optic rotation a value similar to the experimentally determined value. It is shown that the known semiphenomenological theory neglecting the Maxwellization gives an acceptable description of magneto-optic rotation only for high (≈1 A/m) magnetic field.

© 2005 Optical Society of America

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References

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  1. C. G. Aminoff, "Velocity-selective optical pumping and collision effects," Ann. Phys. (Paris) 10, 995-1006 (1985).
  2. E. B. Alexandrov, M. V. Balabas, A. S. Pazgalev, A. K. Vershovskii, and N. N. Yakobson, "Double-resonance atomic magnetometers: from gas discharge to laser pumping," Laser Phys. 6, 244-251 (1996).
  3. D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotarev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403/1-7 (2000).
    [CrossRef]
  4. D. Budker, W. Gawlik, D. F. Kimball, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
    [CrossRef]
  5. S. Nakayama, "Optical pumping theory in polarization spectroscopy of Na," J. Phys. Soc. Jpn. 50, 609-614 (1981).
    [CrossRef]
  6. S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
    [CrossRef] [PubMed]
  7. A. I. Okunevich, "Laser pumping and magneto-optical rotation of polarization plane of the light in the cell with antirelaxation coating of the walls. I. Raising and solving the problem. II. Calculation for the atoms with Lambda scheme of the levels," Opt. Spectrosc. 97, 890-904 (2004).
  8. As in the preceding work,7 we will consider the density matrix defined in the representation of polarization moments [M. I. Dyakonov, "Theory of resonance scattering of light in gas in magnetic field," Zh. Eksp. Teor. Fiz. 47, 2213-2221 (1964) [Sov. Phys. JETP 20 , 1484-1492 (1964)]. In this representation the density matrix is given by the set of components phiQK with 0< or =K< or =2j,−K< or =Q< or =K. We will call "truncated" the density matrix defined as a set of components phiQK with nonzero ranks K . PM with zero rank will be defined by the relation phi00 ≡Spphi=nF(v ), where n is the concentration of atoms.
  9. Ref. 7 consists of two parts; the references to the formulas in these parts will be given with the addition of "I-" and "II-", respectively, before the formula number.
  10. The orientation is absent because its arising at the transverse pumping (E ,H ) considered is the effect of the second order in light intensity.11
  11. D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088-2091 (2000).
    [CrossRef] [PubMed]
  12. In the case of the uncoated cell at H=1A/m we obtained for the quantities v~rho f01 ,v~rho Re f22 , and v~rho Im f22 exactly the same curves as in Fig. 4 b. For the quantity v~rho f02 we obtained the curve similar to the curve for the quantity v~rho f01 in Fig. 4 a.
  13. D. Budker, V. Yashchuk, and M. Zolotarev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
    [CrossRef]
  14. V. V. Yashchuk, E. Mikhailov, I. Novikova, and D. Budker, "Nonlinear magneto-optical rotation with separated light fields in 85Rb vapor contained in an anti-relaxation-coated cell," Technical Report LBNL-44762 (Lawrence Berkeley National Laboratory, 1999).

2004

A. I. Okunevich, "Laser pumping and magneto-optical rotation of polarization plane of the light in the cell with antirelaxation coating of the walls. I. Raising and solving the problem. II. Calculation for the atoms with Lambda scheme of the levels," Opt. Spectrosc. 97, 890-904 (2004).

2002

D. Budker, W. Gawlik, D. F. Kimball, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

2000

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotarev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403/1-7 (2000).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088-2091 (2000).
[CrossRef] [PubMed]

1998

D. Budker, V. Yashchuk, and M. Zolotarev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

1996

E. B. Alexandrov, M. V. Balabas, A. S. Pazgalev, A. K. Vershovskii, and N. N. Yakobson, "Double-resonance atomic magnetometers: from gas discharge to laser pumping," Laser Phys. 6, 244-251 (1996).

1993

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

1985

C. G. Aminoff, "Velocity-selective optical pumping and collision effects," Ann. Phys. (Paris) 10, 995-1006 (1985).

1981

S. Nakayama, "Optical pumping theory in polarization spectroscopy of Na," J. Phys. Soc. Jpn. 50, 609-614 (1981).
[CrossRef]

1964

As in the preceding work,7 we will consider the density matrix defined in the representation of polarization moments [M. I. Dyakonov, "Theory of resonance scattering of light in gas in magnetic field," Zh. Eksp. Teor. Fiz. 47, 2213-2221 (1964) [Sov. Phys. JETP 20 , 1484-1492 (1964)]. In this representation the density matrix is given by the set of components phiQK with 0< or =K< or =2j,−K< or =Q< or =K. We will call "truncated" the density matrix defined as a set of components phiQK with nonzero ranks K . PM with zero rank will be defined by the relation phi00 ≡Spphi=nF(v ), where n is the concentration of atoms.

Alexandrov, E. B.

E. B. Alexandrov, M. V. Balabas, A. S. Pazgalev, A. K. Vershovskii, and N. N. Yakobson, "Double-resonance atomic magnetometers: from gas discharge to laser pumping," Laser Phys. 6, 244-251 (1996).

Aminoff, C. G.

C. G. Aminoff, "Velocity-selective optical pumping and collision effects," Ann. Phys. (Paris) 10, 995-1006 (1985).

Balabas, M. V.

E. B. Alexandrov, M. V. Balabas, A. S. Pazgalev, A. K. Vershovskii, and N. N. Yakobson, "Double-resonance atomic magnetometers: from gas discharge to laser pumping," Laser Phys. 6, 244-251 (1996).

Budker, D.

D. Budker, W. Gawlik, D. F. Kimball, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotarev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403/1-7 (2000).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088-2091 (2000).
[CrossRef] [PubMed]

D. Budker, V. Yashchuk, and M. Zolotarev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

Dyakonov, M. I.

As in the preceding work,7 we will consider the density matrix defined in the representation of polarization moments [M. I. Dyakonov, "Theory of resonance scattering of light in gas in magnetic field," Zh. Eksp. Teor. Fiz. 47, 2213-2221 (1964) [Sov. Phys. JETP 20 , 1484-1492 (1964)]. In this representation the density matrix is given by the set of components phiQK with 0< or =K< or =2j,−K< or =Q< or =K. We will call "truncated" the density matrix defined as a set of components phiQK with nonzero ranks K . PM with zero rank will be defined by the relation phi00 ≡Spphi=nF(v ), where n is the concentration of atoms.

Gawlik, W.

D. Budker, W. Gawlik, D. F. Kimball, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Hansch, T. W.

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Kanorsky, S. I.

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Kimball, D. F.

D. Budker, W. Gawlik, D. F. Kimball, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotarev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403/1-7 (2000).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088-2091 (2000).
[CrossRef] [PubMed]

Nakayama, S.

S. Nakayama, "Optical pumping theory in polarization spectroscopy of Na," J. Phys. Soc. Jpn. 50, 609-614 (1981).
[CrossRef]

Okunevich, A. I.

A. I. Okunevich, "Laser pumping and magneto-optical rotation of polarization plane of the light in the cell with antirelaxation coating of the walls. I. Raising and solving the problem. II. Calculation for the atoms with Lambda scheme of the levels," Opt. Spectrosc. 97, 890-904 (2004).

Pazgalev, A. S.

E. B. Alexandrov, M. V. Balabas, A. S. Pazgalev, A. K. Vershovskii, and N. N. Yakobson, "Double-resonance atomic magnetometers: from gas discharge to laser pumping," Laser Phys. 6, 244-251 (1996).

Rochester, S. M.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotarev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403/1-7 (2000).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088-2091 (2000).
[CrossRef] [PubMed]

Vershovskii, A. K.

E. B. Alexandrov, M. V. Balabas, A. S. Pazgalev, A. K. Vershovskii, and N. N. Yakobson, "Double-resonance atomic magnetometers: from gas discharge to laser pumping," Laser Phys. 6, 244-251 (1996).

Weis, A.

D. Budker, W. Gawlik, D. F. Kimball, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Wurster, J.

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Yakobson, N. N.

E. B. Alexandrov, M. V. Balabas, A. S. Pazgalev, A. K. Vershovskii, and N. N. Yakobson, "Double-resonance atomic magnetometers: from gas discharge to laser pumping," Laser Phys. 6, 244-251 (1996).

Yashchuk, V.

D. Budker, V. Yashchuk, and M. Zolotarev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

Yashchuk, V. V.

D. Budker, W. Gawlik, D. F. Kimball, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotarev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403/1-7 (2000).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088-2091 (2000).
[CrossRef] [PubMed]

Zolotarev, M.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotarev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403/1-7 (2000).
[CrossRef]

D. Budker, V. Yashchuk, and M. Zolotarev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

Ann. Phys. (Paris)

C. G. Aminoff, "Velocity-selective optical pumping and collision effects," Ann. Phys. (Paris) 10, 995-1006 (1985).

J. Phys. Soc. Jpn.

S. Nakayama, "Optical pumping theory in polarization spectroscopy of Na," J. Phys. Soc. Jpn. 50, 609-614 (1981).
[CrossRef]

Laser Phys.

E. B. Alexandrov, M. V. Balabas, A. S. Pazgalev, A. K. Vershovskii, and N. N. Yakobson, "Double-resonance atomic magnetometers: from gas discharge to laser pumping," Laser Phys. 6, 244-251 (1996).

Opt. Spectrosc.

A. I. Okunevich, "Laser pumping and magneto-optical rotation of polarization plane of the light in the cell with antirelaxation coating of the walls. I. Raising and solving the problem. II. Calculation for the atoms with Lambda scheme of the levels," Opt. Spectrosc. 97, 890-904 (2004).

Phys. Rev. A

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotarev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403/1-7 (2000).
[CrossRef]

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Nonlinear magneto-optical rotation via alignment-to-orientation conversion," Phys. Rev. Lett. 85, 2088-2091 (2000).
[CrossRef] [PubMed]

D. Budker, V. Yashchuk, and M. Zolotarev, "Nonlinear magneto-optic effects with ultranarrow widths," Phys. Rev. Lett. 81, 5788-5791 (1998).
[CrossRef]

Rev. Mod. Phys.

D. Budker, W. Gawlik, D. F. Kimball, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Zh. Eksp. Teor. Fiz.

As in the preceding work,7 we will consider the density matrix defined in the representation of polarization moments [M. I. Dyakonov, "Theory of resonance scattering of light in gas in magnetic field," Zh. Eksp. Teor. Fiz. 47, 2213-2221 (1964) [Sov. Phys. JETP 20 , 1484-1492 (1964)]. In this representation the density matrix is given by the set of components phiQK with 0< or =K< or =2j,−K< or =Q< or =K. We will call "truncated" the density matrix defined as a set of components phiQK with nonzero ranks K . PM with zero rank will be defined by the relation phi00 ≡Spphi=nF(v ), where n is the concentration of atoms.

Other

Ref. 7 consists of two parts; the references to the formulas in these parts will be given with the addition of "I-" and "II-", respectively, before the formula number.

The orientation is absent because its arising at the transverse pumping (E ,H ) considered is the effect of the second order in light intensity.11

V. V. Yashchuk, E. Mikhailov, I. Novikova, and D. Budker, "Nonlinear magneto-optical rotation with separated light fields in 85Rb vapor contained in an anti-relaxation-coated cell," Technical Report LBNL-44762 (Lawrence Berkeley National Laboratory, 1999).

In the case of the uncoated cell at H=1A/m we obtained for the quantities v~rho f01 ,v~rho Re f22 , and v~rho Im f22 exactly the same curves as in Fig. 4 b. For the quantity v~rho f02 we obtained the curve similar to the curve for the quantity v~rho f01 in Fig. 4 a.

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Figures (7)

Fig. 1
Fig. 1

Z-velocity dependence of PMs at Wph=0.1 µW from (a and c) BECB theory and from (b and d) SPh theory. Solid curves: f01×2000 (a), f01 (b), and f01×4000 (c and d). Dotted curves: f02 (a, b, and d), f02×0.25 (c). Dashed curves: Re f22. Dashed-dotted curves: Im f22. Horizontal scale for dotted curve in c (f02) is the same as in a.

Fig. 2
Fig. 2

Z-velocity dependence of PMs at Wph=0.1 µW in uncoated cell. Solid curve, f01×4000; open circles, f02; dashed curve, Re f22; dashed-dotted curve, Im f22.

Fig. 3
Fig. 3

Z-velocity dependence of orientation f01 at Δ=-1, -0.5, 0, 0.5 and 1 (curves 1, 2, 3, 4, and 5, respectively). Wph=0.1 µW, Qc=1000, H=10-4 A/m.

Fig. 4
Fig. 4

Dependence of function v˜ρfˆ on radial velocity v˜ρ at Wph=0.1 µW, Qc=1000, Δ=0, v˜z=0.01. Solid curves: v˜ρf01×5000 (a), v˜ρf01×3000 (b). Dotted curves: v˜ρf02 (a), v˜ρf02×0.2 (b). Dashed curves: v˜ρ Re f22. Dashed-dotted curves: v˜ρ Im f22. Open circles: function 0.0006v˜ρ exp(-v˜ρ2) (a), 0.0002v˜ρ exp(-v˜ρ2) (b).

Fig. 5
Fig. 5

Dependence of the rotation angle φL and the transmission Ft on the magnetic field H. BECB theory at Qc=1000: solid curve, φL; dashed curve, Ft. SPh theory at γW=1.3×105 s-1: dotted curve, φL; dashed-dotted curve, Ft. Wph=20 µW.

Fig. 6
Fig. 6

Dependence of Hmin and φL min on pumping power Wph. BECB theory at Qc=1000: solid curve, Hmin 1; dashed curve, φL min 1; dotted curve, Hmin 2×10-4; dashed-dotted curve, φL min 2. SPh theory at γW=1.3×105 s-1: open circles, Hmin 2×104; open diamonds, φL min 2.

Fig. 7
Fig. 7

Dependence of Hmin 1 (solid curve) and φL min 1 (dotted curve) from SPh theory at γW=10 s-1.

Equations (18)

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vxxϕˆ+vyyϕˆ=Γp(Re σ˜)Lˆˆϕˆ+Γp(Re σ˜)nF(v)JˆL
ϕˆ<(A)=KreflF(v)JˆW,JˆW=2ϕˆ>(A)π.
Re σ˜=0.5(ΓJ,j)2(ωL-ωJ,j-kzvz)2+0.25(ΓJ,j)2.
Γpγtp2ω0γtp.
ϕˆL>(x˜)=ϕˆL<(-x˜)=KreflJˆWF(v)+Γp(x˜+x˜B)γtpv˜ρ(Re σ˜)nF(v)JˆL.
0=-γWϕˆ+Γp(Re σ˜)Lˆˆϕˆ+Γp(Re σ˜)nF(v)JˆL.
ϕˆ=[γW1ˆˆ-Γp(Re σ˜)Lˆˆ]-1Γp(Re σ˜)nF(v)JˆL.
ϕ01=0,ϕ02=-295Γp Re σ˜γWnF(v),
ϕ±22=-1315Γp Re σ˜(γW±i2ω0)nF(v).
(ω0)extr=±0.5γW.
fˆ(v˜z, v˜ρ)=up3n01ρ˜dρ˜02πdφφφ+2πϕˆLdφv,
ϕˆL=ϕˆL>atφv[φ, φ+π)ϕˆL<atφv(φ+π, φ+2π].
Ft=1-2 ln σ019R0+592R02+533 Re R22,
φL=-12 arc tan2 ln σ0Ft533Im R22+132J01.
R0=1d˜π-exp[-(Δ-δ˜/d˜)]Re σ˜dδ˜.
Rˆ=1πd˜-Re σ˜dδ˜0v˜ρdv˜ρfˆ,
Jˆ=1πd˜- Im σ˜dδ˜0v˜ρdv˜ρfˆ.
Im ϕ222ω0/(γW2+4ω02).

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