nonlinear magneto-optical rotation with frequency-modulated light

Nonlinear Magneto-Optical Nonlinear Magneto-Optical RotationRotation

with with Frequency-Modulated LightFrequency-Modulated Light

Derek KimballDmitry Budker

Simon RochesterValeriy Yashchuk

Max Zolotorevand many others...

Some of the many others:

D. EnglishK. KernerC.-H. LiT. MilletA.-T. NguyenJ. StalnakerA. Sushkov

E. B. AlexandrovM. V. BalabasW. GawlikYu. P. MalakyanA. B. MatskoI. Novikova A. I. OkunevichS. PustelnyA. WeisG. R. Welch

Budker Group:Non-Berkeley Folks:

Technical Support:M. SolarzA. VaynbergG. WeberJ. Davis Funding: ONR, NSF

Plan: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), 1153-1201.

Linear Magneto-Optical (Faraday) Linear Magneto-Optical (Faraday) RotationRotation

Medium

Linear Polarization

Circular Components

MagneticField

= (n+-n-)0l2c

= (n+-n-) l

1846-1855: Faraday discovers magneto-optical rotation

1898,1899: Macaluso and Corbino discover resonant character of Faraday rotation

Linear Magneto-Optical (Faraday) Linear Magneto-Optical (Faraday) RotationRotation

1898: Voigt connects Faraday rotation to the Zeeman effect

Linear Magneto-Optical (Faraday) Linear Magneto-Optical (Faraday) RotationRotation

Linear Magneto-Optical (Faraday) Linear Magneto-Optical (Faraday) RotationRotation

-5 -4 -3 -2 -1 0 1 2 3 4 5

Normalized magnetic field (b = 2gF0B / )

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3

0.4

0.5

0.6

Rot

atio

n an

gle

(rad

)

20

0

0 /21

/2

2

Bg

Bg

l

l

F

F

B ~ 400 G

Nonlinear Magneto-Optical RotationNonlinear 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:

-2 -1 0 1 2 3

0.2

0.4

0.6

0.8

1

B = 0

Spectral hole-burning:

-2 -1 0 1 2 3

-1

-0.5

0

0.5

Num

ber

of a

tom

s

Atomic velocity

Light detuning

Inde

x of

ref

ract

ion

Re[n+-n-]

B 0

Small field NMOR enhanced!

Nonlinear Magneto-optical Nonlinear Magneto-optical RotationRotation

due to atomic polarizationdue to atomic polarizationThree stage process:

Opticalpumping

Precessionin B-field

Probingvia opticalrotation

Circularly polarized light consists of photons

with angular momentum = 1 ħ along z.

M = 1

Optical pumpingOptical pumping

MF = -1 MF = 0 MF = 1

z

F = 1

F’ = 0

Fluorescence has randomdirection and polarization.

Circularly polarized light propagating in z directioncan create orientation along z.

MF = -1 MF = 0 MF = 1

z

F = 1

F’ = 0

Medium is now transparent to lightwith right circular polarization in z direction!

Circularly polarized light propagating in z directioncan create orientation along z.

Optical pumpingOptical pumping

MF = -1 MF = 0 MF = 1

z

F = 1

F’ = 0

Light linearly polarized along z can create alignment along z-axis.

Optical pumpingOptical pumping

MF = -1 MF = 0 MF = 1

z

F = 1

F’ = 0

Light linearly polarized along z can create alignment along z-axis.

Medium is now transparent to lightwith linear polarization along z!

Optical pumpingOptical pumping

MF = -1 MF = 0 MF = 1

z

F = 1

F’ = 0

Light linearly polarized along z can create alignment along z-axis.

Medium strongly absorbs lightpolarized in orthogonal direction!

Optical pumpingOptical pumping

.

Aligned“Peanut” with axis

along z preferred axis.

z

x

y

Oriented“Pumpkin” pointing

in z-direction preferred direction.

z

x

y

UnpolarizedSphere centered

at origin,equal probabilityin all directions.

z

x

y

Visualization of Atomic Visualization of Atomic PolarizationPolarization

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 whenlaser frequency (l) is tuned to

atomic resonance frequency (0).

Optical pumpingOptical pumping

Precession in Magnetic FieldPrecession in Magnetic Field

Interaction of the magnetic dipole momentwith a magnetic field causes the angular momentum

to precess – just like a gyroscope!

= dF

dt

= B = B

gF B F B

dFdt B

= =L = gF B B

B

, F

torque causes polarized atoms to precess: B Precession in Magnetic FieldPrecession in Magnetic Field

Relaxation and probing of atomic polarizationRelaxation 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 NMORCoherence Effects in NMOR

2

rel0

rel0

00

00 /21

/2

22sin

2rel

Bg

Bg

l

lBtgedt

l

l

F

FF

t

t

Magnetic-field dependence of NMOR due to atomic polarizationcan be described by the same formula we used for linear Faradayrotation, but rel :

How can we get slowest possible rel?

Paraffin-coated cellsParaffin-coated cells

Academician Alexandrov hasbrought us some beautiful“holiday ornaments”...

Paraffin-coated cellsParaffin-coated cells

Alkali atoms work best with paraffin coating...

Most of our work involves Rb:

D1

(794

.8 n

m)

D2

(780

.0 n

m)

5 S 1 /2

2

5 P 1 /2

2

5 P 3 /2

2

6 8 3 5 M H z

8 1 2 M H z

4 9 6 M H z

F = 1

F = 2

F = 1

F = 2

F = 0

F = 3

F = 2F = 1

~~

~~87Rb (I = 3/2)

Paraffin-coated cellsParaffin-coated cells

Polarized atoms can bounce off the walls of a paraffin-coatedcell ~10,000 times before depolarizing!

This can be seen using the method of “relaxation in the dark.”

B

4

Relaxation in the DarkRelaxation in the Dark

MF = -1 MF = 0 MF = 1F = 1

F’ = 0

Light can be used to probe ground state atomic polarization:

No absorption of right circularly polarized light.

z

Photodiode

MF = -1 MF = 0 MF = 1F = 1

F’ = 0

Light can be used to probe ground state atomic polarization:

Significant absorptionof left circularly polarized light.

z

Photodiode

Relaxation in the DarkRelaxation in the Dark

Paraffin-coated cellsParaffin-coated cells

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Time (s)

0.22

0.23

0.24

0.25

0.26

Pro

be t

rans

mis

sion

(ar

b. u

nits

)

Bx = 100 G

rel = 2 1.004(2) Hz

+ -

DCpolarimetercalibration

polarizer

magnetic shield

magnetic coil

Rb-cell

lock-in

reference

pre-amplifier

analyzer

polarization-modulator

polarization-rotator

PD1

PD2attenuator

spectrum analyzer

diode laser

P

uncoated Rb cell in magnetic field

/4BS

PD

PD

Dichroic Atomic Vapor Laser Lockdifferentialamplifier

PD

light-pipe

feedbacklaser frequency control

fluorescencecontrol

and dataacquisition

absorption

magnetic field current

first harmonic

Experimental SetupExperimental Setup

Magnetic ShieldingMagnetic Shielding

ø 24.5" ø 21"

12"

ø 1

8"

25"

20"

16"

Four-layer ferromagnetic magnetic shielding with nearly spherical geometry reduces fields in all directions

by a factor of 106!

Magnetic ShieldingMagnetic Shielding

3-D coils allow controland cancellation of fieldsand gradients inside shields.

NMOR Coherence Effect in Paraffin-coated CellNMOR Coherence Effect in Paraffin-coated Cell

-10 -8 -6 -4 -2 0 2 4 6 8 10

Magnetic Field (G)

-10

-8

-6

-4

-2

0

2

4

6

8

10R

otat

ion

Ang

le (

mra

d)

85Rb D2 Line, I = 50 W/cm2,F=3 F’=4 component

rel = 2 0.9 HzKanorsky, Weis, Skalla (1995). Appl. Phys. B 60, 165.Budker, Yashchuk, Zolotorev (1998). PRL 81, 5788.Budker, Kimball, Rochester, Yashchuk, Zolotorev (2000). PRA 62, 043403.

Sensitive measurement of magnetic fieldsSensitive measurement of magnetic fields

85Rb D2 line, F=3 F’ component,I = 4.5 mW/cm2

0

10

20

30

40

50

B

z) (

10-1

2 G/H

z1/2 )

-1.2 -0.8 -0.4 -0.0 0.4 0.8 1.2

Relative Frequency (GHz)

0.7

0.8

0.9

1.0

Tra

nsm

issi

on

HzG/ 103 12

The dynamic range of an NMOR-based magnetometer islimited by the width of the resonance:

-10 -8 -6 -4 -2 0 2 4 6 8 10

Magnetic Field (G)

-10

-8

-6

-4

-2

0

2

4

6

8

10R

otat

ion

Ang

le (

mra

d)

B ~ 2 G

How can we increase the dynamic range?

NMOR with Frequency-Modulated NMOR with Frequency-Modulated LightLight

• Magnetic field modulates optical properties of medium at 2L.

• There should be a resonance when the frequency of light is modulated at the same rate!

ExperimentalSetup:

Inspired by:Barkov, Zolotorev (1978). JETP Lett. 28, 503.Barkov, Zolotorev, Melik-Pashaev (1988). JETP Lett. 48, 134.

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

Firs

t H

arm

onic

Am

plitu

de (

mra

d)

-1600 -1200 -800 -400 0 400 800 1200 1600

Longitudinal Magnetic Field (G)

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

In-phase component

Out-of-phase (quadrature) component

m = 21 kHz

= 2220 MHz

P 15 W

87Rb D1 LineF = 2 1

Budker, Kimball,Yashchuk, Zolotorev (2002).PRA 65, 055403.

Nonlinear Magneto-optical Nonlinear Magneto-optical RotationRotation

Low-field resonance is due to equilibriumrotated atomic polarization – at constant

angle due to balance of pumping, precession, and relaxation.

Low field resonance:L rel

Nonlinear Magneto-optical Nonlinear Magneto-optical RotationRotation

On resonanceOn resonance::Light polarized alongLight polarized along

atomic polarization is transmitted,atomic polarization is transmitted,light of orthogonal polarizationlight of orthogonal polarization

is absorbed.is absorbed.

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

Firs

t H

arm

onic

Am

plitu

de (

mra

d)

-1600 -1200 -800 -400 0 400 800 1200 1600

Longitudinal Magnetic Field (G)

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

In-phase component

Out-of-phase (quadrature) component

m = 21 kHz

= 2220 MHz

P 15 W

87Rb D1 LineF = 2 1

Nonlinear Magneto-optical Nonlinear Magneto-optical RotationRotation

• 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 Nonlinear Magneto-optical RotationRotation

• Optical properties of the atomic medium are modulated at 2L.

• A resonance occurs when m = 2L.

Nonlinear Magneto-optical Nonlinear Magneto-optical RotationRotation

• Quadrature signals arise due to difference in phase between rotating medium and probe light.

• Second harmonic signals appear for m = L.

Nonlinear Magneto-optical Nonlinear Magneto-optical RotationRotation

NMOR with Frequency-Modulated NMOR with Frequency-Modulated LightLight

L ase r F req u en cy D e tu n in g (G H z)

0 .9

0

(a )

(b )

Fir

st H

arm

onic

Am

plit

ude

(mra

d)

(c )

(d )F = 2

R b

F = 1, ,

F = 2,

F = 1,

8 5

R b , F = 28 7 R b , F = 18 7

Rot

atio

n (m

rad)

Tra

nsm

issi

on

0 .7

0 .8

1 .0

4

8

1 2

0

1

2

-2

-1

0

1

2

-2

-1

Low field resonance

High field resonance

Note that spectrum ofFM NMOR First Harmonicis related to NMOR spectrum:

f

For 2nd harmonic (not shown):

2

2

s

0 20 40 60 80 100 120 140 160 180 200Time (s)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Firs

t Har

mon

ic (

mra

d)

1 G

Measurement of magnetic field with FM NMOR

Demonstrated sensitivity ~ 510-10 Hz/G

MagnetometryMagnetometry

MagnetometryMagnetometry

Magnetic resonance imaging (MRI) in Earth field?

Measurement of Xe nuclear spins.

MagnetometryMagnetometry

Magnetic resonance imaging (MRI) in Earth field?

Tim e (m in )

5

0

-5

Mag

neti

c F

ield

(nG

)

1 0

1 5

2 0

129Xe 26% natural abundance, pressure = 5 bar

A A mystery...mystery...

m = 4 L

See new resonances at

for high light power!

L o n g itu d in a l M ag n e tic F ied ( G )

Qua

drat

ure

Sig

nal

(a

rb. u

n.)

0

4 0 0

P = 2 1 0 W

P = 8 0 0 W

P = 8 0 0 W

-4 0 0

-8 0 0

8 0 0

0 2 0 0

-2 0 0

-4 0 0

4 0 0

0

2 0 0

-2 0 0

-4 0 0

4 0 0 I

n-ph

ase

Sig

nal

(arb

. un.

) I

n-ph

ase

Sig

nal

(arb

. un.

)

(c )

(b )

(a )

x 5

x 5

L a rm o r F req u en cy (H z)

m = 200 Hz

Hexadecapole Hexadecapole ResonanceResonance

Arises due to creation and probing of

hexadecapole moment ( = 4).Yashchuk, Budker, Gawlik, Kimball, Malakyan, Rochester (2003). PRL 90,253001.

Hexadecapole Hexadecapole ResonanceResonance

Highest moment possible: = 2F

No resonancefor F=1

Hexadecapole Hexadecapole ResonanceResonance

At low light powers:

Quadrupole signal I2

Hexadecapole signal I4