Nonlinear Magneto-Optical Rotation with Frequency-Modulated Light Derek Kimball Dmitry Budker Simon Rochester Valeriy Yashchuk Max Zolotorev and many others...

Some of the many others: D. English K. Kerner C.-H. Li T. Millet A.-T. Nguyen J. Stalnaker A. Sushkov E. B. Alexandrov M. V. Balabas W. Gawlik Yu. P. Malakyan A. B. Matsko I. Novikova A. I. Okunevich S. Pustelny A. Weis G. R. Welch Budker Group: Non-Berkeley Folks: Technical Support: M. Solarz A. Vaynberg G. Weber J. Davis Funding: ONR, NSF

Plan: Linear Magneto-Optical (Faraday) Rotation Nonlinear Magneto-Optical Rotation (NMOR) –Polarized atoms –Paraffin-coated cells –Experiments NMOR with Frequency-Modulated light (FM NMOR) –Motivation –Experimental setup –Data: B-field dependence, spectrum, etc. A little mystery... Magnetometry Review: Budker, Gawlik, Kimball, Rochester, Yashchuk, Weis (2002). Rev. Mod. Phys. 74(4),

Linear Magneto-Optical (Faraday) Rotation  Medium Linear Polarization Circular Components Magnetic Field  = (n + -n - ) 0l0l 2c2c = (n + -n - ) ll : Faraday discovers magneto-optical rotation 1898,1899: Macaluso and Corbino discover resonant character of Faraday rotation

Linear Magneto-Optical (Faraday) Rotation 1898: Voigt connects Faraday rotation to the Zeeman effect

Linear Magneto-Optical (Faraday) Rotation

 B ~ 400 G

Nonlinear Magneto-Optical Rotation Faraday rotation is a linear effect because rotation is independent of light intensity. Nonlinear magneto-optical rotation possible when light modifies the properties of the medium: B = 0  Spectral hole-burning: Number of atoms Atomic velocity Light detuning Index of refraction Re[n + -n - ] B  0 Small field NMOR enhanced!

Nonlinear Magneto-optical Rotation due to atomic polarization Three stage process: Optical pumping Precession in B-field Probing via optical rotation

Circularly polarized light consists of photons with angular momentum = 1 ħ along z.  M = 1 Optical pumping M F = -1M F = 0M F = 1 z F = 1 F ’ = 0 Fluorescence has random direction and polarization. Circularly polarized light propagating in z direction can create orientation along z.

M F = -1M F = 0M F = 1 z F = 1 F ’ = 0 Medium is now transparent to light with right circular polarization in z direction! Circularly polarized light propagating in z direction can create orientation along z. Optical pumping

M F = -1M F = 0M F = 1 z F = 1 F ’ = 0 Light linearly polarized along z can create alignment along z-axis. Optical pumping

M F = -1M F = 0M F = 1 z F = 1 F ’ = 0 Light linearly polarized along z can create alignment along z-axis. Medium is now transparent to light with linear polarization along z! Optical pumping

M F = -1M F = 0M F = 1 z F = 1 F ’ = 0 Light linearly polarized along z can create alignment along z-axis. Medium strongly absorbs light polarized in orthogonal direction! Optical pumping.

Aligned “Peanut” with axis along z  preferred axis. z x y Oriented “Pumpkin” pointing in z-direction  preferred direction. z x y Unpolarized Sphere centered at origin, equal probability in all directions. z x y Visualization of Atomic Polarization Draw 3D surface where distance from origin equals the probability to be found in a stretched state ( M=F ) along this direction.

Optical pumping process polarizes atoms. Optical pumping is most efficient when laser frequency (  l ) is tuned to atomic resonance frequency (  0 ). Optical pumping

Precession in Magnetic Field Interaction of the magnetic dipole moment with a magnetic field causes the angular momentum to precess – just like a gyroscope!   =  dF dt  =   B     g F  B F  B   dF dt   B   ==  L = g F  B B B  , F    

torque causes polarized atoms to precess:   B   Precession in Magnetic Field

Relaxation and probing of atomic polarization Relaxation of atomic polarization: Plane of light polarization is rotated, just as if light had propagated through a set of “polaroid” films. Equilibrium conditions result in net atomic polarization at an angle to initial light polarization. (polarized atoms only)

Coherence Effects in NMOR Magnetic-field dependence of NMOR due to atomic polarization can be described by the same formula we used for linear Faraday rotation, but    rel : How can we get slowest possible  rel ?

Paraffin-coated cells Academician Alexandrov has brought us some beautiful “holiday ornaments”...

Paraffin-coated cells Alkali atoms work best with paraffin coating... Most of our work involves Rb: 87 Rb (I = 3/2)

Paraffin-coated cells Polarized atoms can bounce off the walls of a paraffin-coated cell ~10,000 times before depolarizing! This can be seen using the method of “relaxation in the dark.”

Relaxation in the Dark M F = -1M F = 0M F = 1 F = 1 F ’ = 0 Light can be used to probe ground state atomic polarization: No absorption of right circularly polarized light. z Photodiode

M F = -1M F = 0M F = 1 F = 1 F ’ = 0 Light can be used to probe ground state atomic polarization: Significant absorption of left circularly polarized light. z Photodiode Relaxation in the Dark

Paraffin-coated cells

DC polarimeter calibration polarizer magnetic shield magnetic coil Rb-cell lock-in reference pre-amplifier analyzer polarization- modulator polarization- rotator PD1 PD2 attenuator spectrum analyzer diode laser P uncoated Rb cell in magnetic field /4 BS PD Dichroic Atomic Vapor Laser Lock differential amplifier PD light-pipe feedback laser frequency control fluorescence control and data acquisition absorption magnetic field current first harmonic Experimental Setup

Magnetic Shielding Four-layer ferromagnetic magnetic shielding with nearly spherical geometry reduces fields in all directions by a factor of 10 6 !

Magnetic Shielding

3-D coils allow control and cancellation of fields and gradients inside shields.

NMOR Coherence Effect in Paraffin-coated Cell 85 Rb D2 Line, I = 50  W/cm 2, F=3  F’=4 component  rel = 2   0.9 Hz Kanorsky, Weis, Skalla (1995). Appl. Phys. B 60, 165. Budker, Yashchuk, Zolotorev (1998). PRL 81, Budker, Kimball, Rochester, Yashchuk, Zolotorev (2000). PRA 62,

Sensitive measurement of magnetic fields 85 Rb D2 line, F=3  F’ component, I = 4.5 mW/cm 2

The dynamic range of an NMOR-based magnetometer is limited by the width of the resonance:  B ~ 2  G How can we increase the dynamic range?

NMOR with Frequency-Modulated Light Magnetic field modulates optical properties of medium at 2  L. There should be a resonance when the frequency of light is modulated at the same rate! Experimental Setup: Inspired by: Barkov, Zolotorev (1978). JETP Lett. 28, 503. Barkov, Zolotorev, Melik-Pashaev (1988). JETP Lett. 48, 134.

In-phase component Out-of-phase (quadrature) component  m = 2  1 kHz  = 2  220 MHz P  15  W 87 Rb D1 Line F = 2  1 Budker, Kimball, Yashchuk, Zolotorev (2002). PRA 65, Nonlinear Magneto-optical Rotation

Low-field resonance is due to equilibrium rotated atomic polarization – at constant angle due to balance of pumping, precession, and relaxation. Low field resonance:  L   rel Nonlinear Magneto-optical Rotation On resonance: Light polarized along atomic polarization is transmitted, light of orthogonal polarization is absorbed.

In-phase component Out-of-phase (quadrature) component  m = 2  1 kHz  = 2  220 MHz P  15  W 87 Rb D1 Line F = 2  1 Nonlinear Magneto-optical Rotation

Laser frequency modulation  modulation of optical pumping. If periodicity of pumping is synchronized with Larmor precession, atoms are pumped into aligned states rotating at  L. High field resonances:  L >>  rel Nonlinear Magneto-optical Rotation

Optical properties of the atomic medium are modulated at 2  L. A resonance occurs when  m = 2  L. Nonlinear Magneto-optical Rotation

Quadrature signals arise due to difference in phase between rotating medium and probe light. Second harmonic signals appear for  m =  L. Nonlinear Magneto-optical Rotation

NMOR with Frequency-Modulated Light Low field resonance High field resonance Note that spectrum of FM NMOR First Harmonic is related to NMOR spectrum: For 2nd harmonic (not shown):

Demonstrated sensitivity ~ 5  Magnetometry

Magnetometry Magnetic resonance imaging (MRI) in Earth field? Measurement of Xe nuclear spins.

Magnetometry Magnetic resonance imaging (MRI) in Earth field? 129 Xe 26% natural abundance, pressure = 5 bar

A mystery...  m = 4  L See new resonances at for high light power!  m = 200 Hz

Hexadecapole Resonance Arises due to creation and probing of hexadecapole moment (  = 4). Yashchuk, Budker, Gawlik, Kimball, Malakyan, Rochester (2003). PRL 90,

Hexadecapole Resonance Highest moment possible:  = 2F No resonance for F=1

Hexadecapole Resonance At low light powers: Quadrupole signal  I 2 Hexadecapole signal  I 4