Hanle effect

The first experimental evidence for the effect came from Robert W. Wood,{{cite journal | last=Wood | first=R.W. |author-link=Robert W. Wood| title=LXVII. Selective reflexion, scattering and absorption by resonating gas molecules | journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science | volume=23 | issue=137 | year=1912 | issn=1941-5982 | doi=10.1080/14786440508637267 | pages=689–714|url=https://babel.hathitrust.org/cgi/pt?id=uc1.c032867996&view=1up&seq=707}}{{cite journal | last1=Wood | first1=R. W. |author-link=Robert W. Wood| last2=Ellett | first2=A. | title=On the Influence of Magnetic Fields on the Polarisation of Resonance Radiation | journal=Proceedings of the Royal Society A | volume=103 | issue=722 | date=1923-06-01 | issn=1364-5021 | doi=10.1098/rspa.1923.0065 | pages=396–403|url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015006589025&view=1up&seq=440| bibcode=1923RSPSA.103..396W | doi-access=free }} and Lord Rayleigh.{{cite journal | last=Rayleigh | first=L. |author-link=John William Strutt, 3rd Baron Rayleigh| title=Polarisation of the Light Scattered by Mercury Vapour Near the Resonance Periodicity | journal=Proceedings of the Royal Society A | volume=102 | issue=715 | date=1922-11-01 | issn=1364-5021 | doi=10.1098/rspa.1922.0080 | pages=190–196|url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015011351098&view=1up&seq=218| bibcode=1922RSPSA.102..190R | doi-access=free }} The effect is named after Wilhelm Hanle, who was the first to explain the effect, in terms of classical physics, in Zeitschrift für Physik in 1924.{{Cite journal|last=Hanle|first=Wilhelm|author-link=Wilhelm Hanle|date=1924-12-01|title=Über magnetische Beeinflussung der Polarisation der Resonanzfluoreszenz|journal=Zeitschrift für Physik|language=de|volume=30|issue=1|pages=93–105|doi=10.1007/bf01331827|issn=0044-3328|bibcode=1924ZPhy...30...93H|s2cid=120528168}}{{cite book | last=Hanle | first=W. |author-link=Wilhelm Hanle| title=Ergebnisse der Exakten Naturwissenschaften | chapter=Die magnetische Beeinflussung der Resonanzfluoreszenz | publisher=Springer Berlin Heidelberg | location=Berlin, Heidelberg | year=1925 | isbn=978-3-642-93859-7 | doi=10.1007/978-3-642-94259-4_7 | pages=214–232|language=de}} Initially, the causes of the effect were controversial, and many theorists mistakenly thought it was a version of the Faraday effect. Attempts to understand the phenomenon were important in the subsequent development of quantum physics.{{cite journal |author1=J Alnis |author2=K Blushs |author3=M Auzinsh |author4=S Kennedy |author5=N Shafer-Ray |author6=E R I Abraham |year=2003 |title=The Hanle effect and level crossing spectroscopy in Rb vapour under strong laser excitation |journal=Journal of Physics B |volume=36 |issue=6 |pages=1161–1173 |doi=10.1088/0953-4075/36/6/307 |url=http://www.nhn.ou.edu/~abe/research/OUReprints/The%20Hanle%20effect%20in%20Rb%20vapour%20under%20strong%20laser%20excitation.pdf |bibcode=2003JPhB...36.1161A |s2cid=250734473 |access-date=2012-03-06 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303185402/http://www.nhn.ou.edu/~abe/research/OUReprints/The%20Hanle%20effect%20in%20Rb%20vapour%20under%20strong%20laser%20excitation.pdf |url-status=dead }}

An early theoretical treatment of level crossing effect was given by Gregory Breit.{{cite journal | last=Breit | first=G. |author-link=Gregory Breit| title=Quantum Theory of Dispersion (Continued). Parts VI and VII | journal=Reviews of Modern Physics | volume=5 | issue=2 | date=1933-04-01 | issn=0034-6861 | doi=10.1103/revmodphys.5.91 | pages=91–140| bibcode=1933RvMP....5...91B }}

The classical explanation for this effect involves the Lorentz oscillator model, which treats the electron bound to the nucleus as a classical oscillator. When light interacts with this oscillator, it sets the electron in motion in the direction of its polarization. Consequently, the radiation emitted by this moving electron is polarized in the same direction as the incident light, as explained by classical electrodynamics.

Consider light propagating along the y-axis, linearly polarized with the electric field along the x-axis, incident on a single atom in a vapor cell. The vapor cell is placed in a uniform magnetic field along the z-axis. A detector is placed to monitor the fluorescent light emitted along the z-axis from the vapor cell and measure its polarization in the x-y plane.

The oscillating electric field of the incident light induces oscillations in the electron along the x axis. The electric field generated by the induced dipole along the z-axis is given by

\mathbf E(z, t) = \frac{\omega_0^2}{4\pi\epsilon_0 c^2} \mathbf{p}'(t - z/c)\frac{\cos[kz - \omega_0(t - t_0)]}{z}

where $\backslash omega\_0$ is the angular frequency of the incident light and $t\_0$ is the time at which the electron is excited.

The component of the fluorescent light polarized at angle $\backslash alpha$ with respect to the x axis at the detector has intensity given by

I(\alpha, t) = \frac{\omega_0^4 p_0^2}{64\pi^2\epsilon c^3} e^{-(t-t_0)/\tau \{1 + cos[2(\omega_L(t - t_0) - 2\alpha]\}}

where $\backslash omega\_L\; =\; g\_J\; eB/(2m\_0)$ is the Larmor frequency and $1/\backslash tau$ is the damping rate of the fluorescence radiation.

Consider $N$ atoms in the vapor cell, excited at a constant rate $R$. The steady-state intensity at the detector can be obtained by integrating over the excitation time $t\_0$ from $-\backslash infty$ to $t$, which gives

I(\alpha) = NR \frac{p_0^2 \omega_0^4}{128 \pi^2 \epsilon_0 c^3} \left[\frac{1}{\gamma} + \frac{\gamma \cos(2\alpha)}{\gamma^2 + \omega_L^2} + \frac{\sin(2\alpha}{\gamma^2 + \omega_L^2}\right]

The polarization degree is

P = \frac{I_x - I_y}{I_x + I_y} = \frac{1}{1 + (g_Je\tau/m_0c)^2 B^2}

where $I\_x$ and $I\_y$ are the intensities of the x-polarized component ($\backslash alpha\; =\; 0$) and the y-polarized component ($\backslash alpha\; =\; \backslash pi/2$). This has a Lorentzian shape as a function of the magnetic field strength $B$.

Observation of the Hanle effect on the light emitted by the Sun is used to indirectly measure the magnetic fields within the Sun, see:

• Polarization in astronomy
• Imaging spectroscopy

The effect was initially considered in the context of gasses, followed by applications to solid state physics.{{cite book | last1=Pikus | first1=G. E. |author-link=Gregory Pikus| last2=Titkov | first2=A. N. | title=The Hanle Effect and Level-Crossing Spectroscopy | chapter=Applications of the Hanle Effect in Solid State Physics | publisher=Springer US | location=Boston, MA | year=1991 | isbn=978-1-4613-6707-9 | doi=10.1007/978-1-4615-3826-4_6 | pages=283–339}} It has been used to measure both the states of localized electrons{{cite journal | last1=Karlov | first1=N.V. | last2=Margerie | first2=J. | last3=Merle-D'Aubigné | first3=Y. | title=Pompage optique des centres F dans KBr | journal=Journal de Physique | volume=24 | issue=10 | year=1963 | issn=0368-3842 | doi=10.1051/jphys:019630024010071700 | pages=717–723| s2cid=95183756 |language=fr| url=https://hal.archives-ouvertes.fr/jpa-00205553/file/ajp-jphys_1963_24_10_717_0.pdf }} and free electrons.{{cite journal | last=Parsons | first=R. R. | title=Band-To-Band Optical Pumping in Solids and Polarized Photoluminescence | journal=Physical Review Letters | volume=23 | issue=20 | date=1969-11-17 | issn=0031-9007 | doi=10.1103/physrevlett.23.1152 | pages=1152–1154| bibcode=1969PhRvL..23.1152P }} For spin-polarized electrical currents, the Hanle effect provides a way to measure the effective spin lifetime in a particular device.{{cite journal | last1=van ’t Erve | first1=O. M. J. | last2=Friedman | first2=A. L. | last3=Li | first3=C. H. | last4=Robinson | first4=J. T. | last5=Connell | first5=J. | last6=Lauhon | first6=L. J. | last7=Jonker | first7=B. T. | title=Spin transport and Hanle effect in silicon nanowires using graphene tunnel barriers | journal=Nature Communications | volume=6 | issue=1 | date=2015-06-19 | issn=2041-1723 | doi=10.1038/ncomms8541 | page=7541| pmid=26089110 |doi-access=free| bibcode=2015NatCo...6.7541V }}

The zero-field Hanle level crossings involve magnetic fields, in which the states which are degenerate at zero magnetic field are split due to the Zeeman effect. There is also the closely analogous zero-field Stark level crossings with electric fields, in which the states which are degenerate at zero electric field are split due to the Stark effect. Tests of zero field Stark level crossings came after the Hanle-type measurements, and are generally less common, due to the increased complexity of the experiments.{{cite journal | last1=Bylicki | first1=F. | last2=Weber | first2=H.G. | title=Zero-field Stark level crossing and Stark—Zeeman recrossing experiments in the 593 nm band of NO₂ | journal=Chemical Physics | volume=70 | issue=3 | year=1982 | issn=0301-0104 | doi=10.1016/0301-0104(82)88099-3 | pages=299–305| bibcode=1982CP.....70..299B }}

• Larmor precession
• Resonance fluorescence
• Optical pumping

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