A9: Nonlinear Optics and Laser Spectroscopy

Nonlinear Optics and Laser SpectroscopyINTRODUCTION:

In this experiment several techniques from modern nonlinear optical physics andlaser spectroscopy are investigated. The experiment utilizes a coherent source of light,provided by an extended cavity diode laser, to probe the spectroscopic properties ofatomic Rubidium vapor, as well as novel coherent phenomena that arise from theinteraction of light and atoms. Proposed experiments include: a) observation and identifi-cation of Rb absorption lines; b) investigation of sub-Doppler resonances in a standing-wave laser field; c) investigation of the absorptive and dispersive properties of “dark-resonances” through measurement of non-linear polarization phenomena such as thenonlinear Faraday effect; and d) measurement of nonlinearly induced ultra-slow groupvelocities for light pulses.BACKGROUND

Atomic Absorption, Optical Pumping and Doppler Free Spectroscopy

Figure 1: The fine structure of Rb85 (left) and Rb87 (right).

The relevant atomic energy levels of the two stable isotopes, Rb85 (nuclear spin of3/2) and Rb87 (nuclear spin 1/2) are illustrated in Figure 1. In this experiment you willstudy the optical properties of the D1 transition between the 52S1/2 and 52P1/2 levels. In theabsence of an external magnetic field that induces Zeeman splitting of the hyperfinelevels there are four optical transitions for each isotope. A sketch of the D1 spectrum forRb that assumes each transition has a Doppler broadened full width at half maximum ~500MHz is shown in Figure 2. The intensities, which are adjusted to replicate the naturalabundances of the two isotopes, assume equal populations in the two hyperfine splitground states.

The basic non-linear optical phenomena involving these isotopes is known asoptical pumping. For a circularly polarized incident optical beam the optical selectionrules will only allow absorption between states in which the magnetic quantum numberchanges by Δm = ±1 , where the sign depends on the sense of the polarization (i.e. right orleft). For example, for one sense of circular polarization the only D1 absorptions from theF=1 ground state to the F=1 excited of Rb87 will be from m = −1→ 0( ) and 0→1( ) .There will not be any D1 absorption to the F=1 state from the m=+1 ground state. If theoptical intensity is strong enough that the rate at which atoms are excited is larger than

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the rate at which they decay back to the ground state and the number of atoms in them=+1 ground state will build up while the number in the m=-1 and 0 become depleted.The consequence is that theD1 absorption with this circular polarization will decrease.Along with the decease in absorption optical fluoresence of the D1 lines will alsodecrease. This is the basic optical pumping phenomenon. The ground state that has beenfilled by the pumping process is known as the dark state since it does not interact with theoptical field.

Figure 2: Calculated optical spectrum for the D1 lines of Rb85 and Rb87

assuming a Doppler broadened full width at half maximum of ~ 500 MHz.(Vanier, et al. 1989)

The rate at which atoms decay back to the ground state via spontaneous emission,which determines the intrinsic width of the atomic absorption, is considerably less thanthe Doppler broadened width due to the thermal motions of the atomic gas. This intrinsicwidth (≈6 MHz), which is narrower than the full Doppler width (≈500 MHz), can beobserved using the method know as Doppler Free absorption. In this technique, twocounterpropagating, overlapping laser beams of exactly the same frequency interact withan ensemble atoms. To understand the effect of the light, one can divide the atoms in theensemble into groups, each of which is characterized by a particular projection of theatomic velocity vz onto the direction of the laser beam path When the laser frequencydiffers by Δ from the resonant frequency of the atoms, _0, one beam interacts resonantlywith the group of atoms with velocity component vz = Δ/k (where k ≈ω c is the lightwavenumber), and the other beam interacts with a different group of atoms, those withvelocity component vz = -Δ/k.. Each of the beams induces absorption at a rate proportionalto EO

2 where E0 is the amplitude of the electric fields for the two beams. However,when the laser frequency is on resonance with the atoms (Δ= 0), the two beams interactwith the same group of atoms, those with velocity component parallel to the beams ≈ 0.Under these circumstances the pair of beams can act together and the transitionprobability for absorption is proportional to 4 E0

2 , which is four times the rate of theindividual beams. If the individual laser fields are just strong enough to induce partialoptical pumping the effect with the pair acting together will induce considerably greaterpumping. Under the proper conditions the line width of this effect can approach thenatural width determined by the spontaneous emission lifetime of the atomic transition.

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In the simplest case this implies that, for example, there should be two narrowlines superposed on each of the two Doppler broadened transition to the 52P1/2 state.Taking these two narrow lines to be a frequencies ν1 and ν2, where ν2 −ν1 = 362.2MHzlines should also be a third combination line at ν3 = ν2 + ν1( ) 2 . We will leave it to youto explain why this is so.

Dielectric ConstantsThe basis of the Faraday effect is the magnetic field induced difference between

the dielectric constants for Right/Left circular polarization. The effect can also beascribed to the linear magnetic field dependence of the off diagonal element of thedielectric tensor, ε xy ~ B0 where B0 is the DC magnetic field. To illustrate, takingν ≡ω c with

E ~ exp i kz −ωt[ ]( ) the optical wave equation can be expressed as

Eq. 1

k2E = ν 2

ε0 iε xy−iε xy ε0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

E

With

E = 1

±i⎡

⎣⎢

⎤

⎦⎥ the wavevector is determined by

Eq. 2 k2 1±i

⎡

⎣⎢

⎤

⎦⎥ = ν

2ε0 iε xy

−iε xy ε0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1±i

⎡

⎣⎢

⎤

⎦⎥ = ν

2ε±1±

⎡

⎣⎢

⎤

⎦⎥

where ε± = ε0 ε xy .Assuming a wave enters the material at z=0 with a polarization

E 0( ) = E0 x the

solution with k ≈ν ε0 and δk = ν ε xy 2 ε0

Eq. 3

E = xE0 cos δkz[ ]+ yE0 sin δkz[ ]{ }exp ikz( )

implies rotation of the polarization by an angle

Eq. 4 dφ dz = νε xy 2 ε0 = VB0

where V is the Verdet coefficient for the linear Faraday effect. The nonlinear Faradayoccurs when V depends on the laser intensity.

Linear Faraday EffectThe simplest classical model for the dielectric constant is to assume that in

response to an external electric field the electron displacement of an atom can be treatedas a simple harmonic oscillator

r = −eE0 exp +iωt( ) / m ω02 −ω 2 + iγω⎡⎣ ⎤⎦

ω ≈ω0~ −e 2mω0[ ]E0 exp +iωt( ) ω0 −ω + i γ 2( )⎡⎣ ⎤⎦

.

If the damping is neglected the real part of the dielectric constant for an assembly of Natoms per unit volume has the form

Eq. 5 ε ω( )−1≈ 4πN e2 2mω0( ) ω −ω0( )−1 .

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An equivalent quantum mechanical result can be obtained by elementary time dependentperturbation theory, as outlined in the appendix below. In the quantum treatment thelinear Faraday effect follows from the Zeeman splitting of a degenerate ground state asω0 →ω0 ±δω0 for whichε± ω( )−14π

~ (ω0 ±δω0 −ω )−1 .

Leading to the rotation expected for the linear Faraday effect

Eq. 6 dφ dz ~ ε+ − ε− = δω∂ε∂ω

~ 2δω01

ω0 −ω( )2.

In practice the linear Faraday effect is too small to measure in Rb vapor at normalpressures.

Non-Linear Faraday Effect.In the appendix below we will derive a quantum mechanical expression for the.

non-linear Faraday effect based on a simplified atomic model for which the ground stateconsists of a single doublet g1,2 → ±1 with magnetic quantum numbers of ±1 and asinglet excited state e with magnetic quantum number of zero. When this system ispumped with circularly polarized light the dark state in which the atoms accumulatecorresponds to that ground states with m=+1 or -1 that does not couple to the incidentlight. However, in the absence of Zeeman splitting this dark state is not unique, as itdepends on the incident light polarization. For example if the system is pumped with alinearly polarized light beam, say along x, than the depleted state isx = 2−1 2 +1 + −1⎡⎣ ⎤⎦ and the populated or dark state is y = −i2−1 2 +1 − −1⎡⎣ ⎤⎦ .

Since the incident field when measuring the Faraday effect is linearly polarized thiswould be the dark state in the absence of the Zeeman splitting. Similar bright/dark statesare formed for any type of elliptically polarized wave.

On the other hand, when a magnetic field is applied parallel to k the Zeeman

splitting of the ground state forces an admixture of x and y such that y is no longerdark. The effect of the admixture is to induce a non-vanishing amplitude for ε xy thatgives rise to the non-linear Faraday effect. The essence of the result is that

dφ dz ~hµ E

4h2 + µ2 E 2

where h is proportional to the Zeeman splitting of the ground state and E is theamplitude of the optical field. Note that if either the Zeeman splitting or the opticalintensity is too large the effect is quenched. Again, refer to the appendix for a moredetailed calculation.

Slow LightFor this experiment the slow light effect is observed in the absence of Zeeman

splitting with a time dependent elliptically polarized optical state such asEx = 1+ am cosωmt( )Eo exp iωt

Ey = i 1− am sinωmt( )E0 exp iωt ,

which is made up of an intense wave of one circular polarization and a weak wave of theother, i.e., with am << 1 . If ωm → 0 the unmodulated wave is elliptically polarized and,

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as just explained there will be an elliptically polarized dark state. This dark state willpersist even for ω m ≠ 0 so long as ωmTg < 1 where Tg is the slow relaxation time bywhich atoms can relax between the ground states g1 and g2 . Typically Tg ≈ 0.1-1 ms,depending on the experimental details. On the other hand, when ωmTg >> 1 the atomicsystem can not follow the modulation of the field. In this situation the system changes toone in which one sense of circular polarization strongly pumps the atomic system whilethe other field simply probes the atomic distributions produced by the pump field. Sincethe pumping field depletes the atomic system of atoms with which it interacts this fieldpropagates through the medium at a speed that is very close to the speed of light invacuum. On the other hand, the weak probe field sees a relatively large dielectricconstant, due to the large atomic populations of the dark state, with a very strongdispersion dε dω associated with the narrow range of frequencies for whichω −ωm ≤ 1 Tg . The rapid change in ε over this small range of ω can lead to a large value

of dε dω which, in turn, results in a very slow group velocity for the probe beam. Seethe discussion in Scully and Zubairy, 1997[PP1].

Eq. 7 vg ≈ c 1+ ω 2( ) dε dω( )⎡⎣ ⎤⎦−1

EXPERIMENT

λ/4

ND 0.6

Figure 3: Schematic illustration one possible configuration for the opticalelements in this experiment. The component labeling is as follows: AW is anattenuator wheel; PD1,2,3,4 are photo diodes; λ/4 and λ/2 are quarter and halfwave plates respectively; PBS1,2 are polarizing beam splitters. The two Rb cells arelabeled Rb.

Laser spectroscopy of Rb:

Use 1st the λ/4 plate to produce circular polarization such the PBS1 splits be beam50:50.as shown in Figure 3. You should observe the Rb absorption lines both intransmission measurements (PD2) and fluorescence (PD1). Identify each of the lines witha corresponding transition between a pair of hyperfine sublevels of the isotopes Rb85 andRb87. What determines the relative strength and shapes of the lines?

Doppler-free saturation spectroscopy: Using a mirror to retro-reflect the beam back into the absorption cell and thus to

create an optical standing-wave, observe and study narrow, sub-Doppler features influorescence. Zoom in on each of the transition groups. Determine the atomic transition

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corresponding to each resonance. Identify also the crossover resonances and explain theirorigin. Investigate what determines the width of the observed features.

Nonlinear polarization spectroscopy:For the Faraday measurement the Pockels cell is not necessary. If it is in place it

should be removed. The laser beam should be collimated to have a diameter of a few mmand should pass through the magnetically shielded Rb vapor cell. The cell should beheated to produce Rb atomic densities in the range of 1011 to 1012 cm_3. (What is thecorresponding temperature?) A longitudinal magnetic field is created by a solenoidplaced inside the magnetic shields. (What field strength should you use?)

Before starting this experiment you should understand how polarizationmeasurement works. To this end, the following calculation is important. Consider laserlight that passes through a linear polarizer and a near-resonant atomic medium (whichrotates the light polarization by an angle _ . It will also attenuate the light. The light thenpasses through another polarizer with an optical axis that is rotated by an angle _ withrespect to the first polarizer, and then finally through a photodetector. Calculate theexpected signal dependance on _ for different orientations of _ = 0, 45° and 90°. Forwhich setting of _ would you expect to see the largest signal in the case of small _ ?

Equipped with this knowledge, you are now in a position to observe andunderstand nonlinear Faraday resonances. For the most optimal setting of _, tune the lasermanually across the Rb absorption lines while monitoring the changes in lightpolarization (using, e.g., two crossed polarizers) as the magnetic field is variedperiodically. (What is a convenient frequency for this field variation?) Observe thenonlinear Faraday resonance and study the signal as a function of light intensity, atomicdensity and laser detuning (i.e., tune the laser to the different hyperfine components).

Calibrate your magnetic field scan and estimate the width of the resultingresonances. For the several specific transitions, study the signal dependence on _, as wellas on light power. In addition, observe the dark-resonance effect by monitoring the totaltransmitted intensity of linearly polarized light through the cell as a function of magneticfield and without a second polarizer (this corresponds to a measurement of absorptiveproperties).

The following investigations should be pursued: a) study the width of the dark-resonance as function of light power; b) determine the polarization rotation angle fromyour data sets, and determine the dependence on laser power; c) from thesemeasurements calculate the absolute change in the refractive index near the dark-resonance as well as the slope of the refractive index, and then estimate thecorresponding group velocity; d) determine the absolute change in the total transmittedintensity across the dark-resonance; e) explain why Faraday resonances differ so mucheven for nearby transitions, e.g., for F = 1 _ F = 1 and F = 1 _ F = 2 transitions in Rb.APPARATUS

LaserBefore turning on the laser, discuss operating procedures and safety precautions

with the faculty or staff. The laser is a Class IIIb device. Your vision can be damaged byeven a momentary exposure. Never place your eye at the level of the laser beam. Keepall beams confined within the footprint of the optical table. Wear laser safety glasseswhenever putting any component into or out of the beam. When using the IR card toview the beam, hold the card at such an angle that the specular reflection will travel

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downwards.

The light source used in this experiment is a Vortex 6017 tunable external-cavitydiode laser, manufactured by New Focus. In a tunable diode laser, the rear facet of thediode crystal together with a diffraction grating and mirror define an external cavity. (Seethe figure in the laser operating manual, page 49.) Small changes in the position of themirror make small changes in the length of the cavity, and thus in the wavelength of thelight. A piezoelectric transducer (PZT) is used to move the mirror.

The button labeled “Display” on the laser control unit toggles between lasercurrent, laser power, PZT voltage, and an auxiliary function, which is not used. WhenPZT voltage is selected and the “Set” button is illuminated, one can manually tune thePZT voltage and thus the laser wavelength. The center frequency of the laser is794.99nm (vacuum). A rear-panel voltage input makes it possible to sweep through arange of wavelengths repeatedly. An input of ±4.5 V sweeps across the entire range ofthe PZT (approximately 0.25 nm).

The laser control unit includes a temperature controller, which stabilizes the laserdiode at about 18˚C. When the key-switch is activated, the temperature controller turnson. Several minutes are needed to achieve stable temperature.

To turn on the laser itself, use the button marked “Power.” When the button ispressed, the laser current ramps up above the lasing threshold.

In order to prolong the life of the laser, minimize cycling the Power. In practicethis usually means turning on the beam at the beginning of a lab session and turning it offat the end. The temperature controller (key switch) can be left on for the duration of theexperiment. The current setpoint is generally left at the factory-recommended value of33.1mA. The laser is normally operated in constant current mode as opposed to constantpower mode.

Rubidium Vapor CellsFour rubidium vapor cells are available. Each cell is cylindrical - three inches

long and one inch in diameter. The ends are made from optical quality glass. The firstcell contains only rubidium vapor; the pressure in the cell is simply the vapor pressure ofrubidium at room temperature. Small droplets of solid rubidium are visible on the wall ofthe cell. This is the cell mounted in the saturated absorption measurement at 90˚ to theprimary beam axis on the optical table.

Three other cells contain a neon buffer gas in addition to rubidium. The cell mostlikely found already mounted in the magnetic shield contains Rb-87 only and 3 Torr ofneon. The other two cells contain rubidium in the naturally occurring ratio. The pressureof neon in these cells is 3 Torr and 10 Torr respectively. Any of these cells can be used toobserve dark resonance and slow light. The buffer gas minimizes collisions of rubidiumatoms with the walls.

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Wave Plates and Polarizing CubesManipulating the polarization of light is critical in this experiment because

selection rules for the atomic transitions are determined by how the polarization of thelight is chosen. If you are not completely comfortable with the use of wave plates tocreate circular and elliptical polarization, take some time to investigate their behaviorearly on in your work.

It can be useful to look at computer simulations such as http://home3.netcarrier.com/~chan/EM/PROGRAMS/POLARIZATION/which demonstrates how polarization components add after passing through wave plates.

Light traveling in a medium slows down by a factor of the index of refraction ofthe medium: vn = c/n. Wave plates are used to preferentially retard the light wave alonga certain axis. Wave plates are made of birefringent material i.e., a crystalline material inwhich the index of refraction which varies with crystal direction. The direction in whichthe light slows down least is called the “fast axis.”

Quarter Wave Plate:A quarter wave plate retards light polarized along its fast axis by 90˚ less than it

retards light polarized along the perpendicular direction. Consider a quarter wave plateoriented with its fast axis vertical (along the y-axis). Linearly polarized light is incidentupon the wave plate. Define θ as the angle between the polarization direction and thepositive y-axis.

If θ = 45˚, Ey = Ex. Ex, however is delayed by one-quarter period. Thus the netelectric field vector leaving the plate rotates. The rotation can be either clockwise (σ+) orcounterclockwise (σ-) as viewed upstream along the propagation direction depending onwhether plus or minus 45˚ is chosen.

If θ = 0˚ or 90˚, no net relative phase change occurs; the polarization isunaffected. If θ is between 0˚ and 45˚, elliptical polarization results. In the computersimulation, you can create elliptical polarization by choosing a 90˚ phase differencebetween Ey and Ex and making the magnitudes different.

Half Wave Plate:As for the quarter wave plate, if the fast axis is parallel or perpendicular to linear

input polarization, nothing happens.

A half wave plate can be thought of simply as two quarter wave plates in series.So consider linearly polarized light at θ = 45˚ incident upon the first of two quarter waveplates. The output of the first quarter wave plate is circularly polarized. Passing throughthe second quarter wave plate, the light becomes linearly polarized again, but at an angleof 90˚ relative to the original polarization direction. (Rays are reversible.) This is aspecial case, easy to visualize, which shows that the half wave plate rotates linearpolarized light through 2θ. In general, when the input polarization is at an angle θrelative to the fast axis of a half wave plate, the output polarization is rotated through 2θrelative to the input polarization.

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Linear Polarizer:A linear polarizer optimized for infrared wavelengths is available. It is mounted

in a rotary mount labeled “Melles Griot,” the manufacturer. The easy axis of thepolarizer is marked with two small white tick marks on the inner housing of the polarizer.Use it to verify the direction and type of polarization of the laser beam and the action ofthe waveplates. The polarizer will extinguish linearly polarized light (at 90˚) and appearisotropic to circularly polarized light.

Linearly polarized light can be thought of as a superposition of right- and left-circularly polarized light of equal intensities.

Jones Vectors and Matrices are a convenient way of expressing the actions ofwave plates on polarized light. See

http://scienceworld.wolfram.com/physics/JonesVector.html

Polarizing Cubes:Four cubes, three of which are sensitive to polarization, are used as beamsplitters.

The cubes are composed of two right-angle prisms cemented together. For the polarizingcubes, the interface between the prisms reflects vertically-polarized light at 90˚ andtransmits horizontally-polarized light.

The first cube splits the beam between the low-pressure rubidium cell (for thesaturated absorption spectroscopy) and the magnetic shield (for dark resonance and slowlight). A half-wave plate upstream of this cube can be used to rotate the polarization ofthe input light. Thus the intensity of the reflected and transmitted beams can be varied.

The second, smaller cube is not sensitive to polarization. It splits off a beam usedas a reference in the slow light measurement. A third cube selects the component of thereference beam which has been rotated by the Pockels cell (more below).

The fourth cube, installed downstream of the magnetic shield, is used as apolarization analyzer. A quarter wave plate upstream of it but downstream of themagnetic shield re-linearizes circularly polarized light. Thus in conjunction with thequarter wave plate and two photodetectors, this cube acts as an analyzer to resolve theright- and left-circular components of the beam.

Pockels CellThe Pockels Cell (Conoptics 350-80LA or Lasermetrics 3079-4), made of a

birefringent material, acts as a voltage-variable wave plate. The voltage required forhalf-wave retardation is several hundred volts. A dc amplifier (Lasermetrics AF3)provides a ±650V output, which can be a combination of dc and ac signals. A gaussianpulse from the Stanford Research Systems DS345 function generator can be used forinput to the AF3. The input signal should be 1V peak-to-peak maximum. Start with thegain control set to minimum and the dc offset switch set to its center position (off). Totest the functionality of the Pockels cell, monitor horizontal and vertical polarizationcomponents of the beam using the polarization analyzer. Begin by switching on thepositive dc offset and slowly increasing its value. You should see one polarization

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component increase and the other decrease. Next connect the input signal, and slowlyincrease the gain. The polarization analyzer should detect anticorrelated modulations ofthe polarization components, synchronized with the input signal.

PhotodetectorsTwo types of silicon photodetectors are in use: Thorlabs PDA55 and Thorlabs

DET110. The PDA55s require a dc input, and incorporate a variable-gaintransimpedance amplifier. The DET110s are battery operated; please switch them off atthe end of the lab period. Since the output is a current, they require a terminating resistor(10kΩ recommended).

Magnetic Shield, Oven and SolenoidA schematic end-view of the magnetic shield assembly is shown below.

Magnetic Shield:A three-layer cylindrical mu-

metal shield screens the rubidium cellfrom the earth’s magnetic field as wellas stray fields generated in thelaboratory. The cylindrical portions ofthe shield are held in place by sheets offoam, which also act as thermalinsulation. The shields can bedisassembled to change the rubidiumcell or for viewing. The easiest way todo this is by removing the mu-metalcaps from the downstream end of theapparatus. The rubber grommets thathold the copper supports in place are cutin half on the downstream end tofacilitate disassembly.

Oven:A heater is wound around the inner cylinder. The heater is designed in a non-

inductive manner to minimize magnetic disturbances. Even so, you may get strongersignals by turning off the heater once the oven is heated up to the desired temperature.The heater has a resistance of about 92Ω. Two resistors (6Ω total) are wired in serieswith the heater to protect the power supply in case of an accidental short circuit. AHarrison 6200A regulated dc power supply provides power to the heater. The timeconstant for heating is on the order of hours, so it’s best to turn the oven on in themorning before lab. At 28 V, (300 mA), the oven reaches a temperature of about 60˚Cwhich is sufficient to observe a robust dark resonance, and to detect a slow-light signal.

SolenoidThe solenoid is wound on a slotted aluminum frame. It consists of two layers of

#28 high temperature magnet wire. The bottom layer has 544 turns and the top layer 518turns. The solenoid length is 8 inches, and the inner diameter is 107 mm. The resistanceof the solenoid is 78Ω. It can be driven directly by one of the function generators orthrough a series resistor/monitor box. An approximate calibration is: field (mG) =

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65*current (mA) + 39.APPENDIX: QUANTUM TREATEMENT OF THE NON-LINEARFARADAY EFFECT.

Hamiltonian and Wave EquationFor heuristic purposes it will be best to discuss the physics of both the Faraday

effect and slow light in terms of the simplified energy level scheme for which the groundstate consists of a single doublet g1,2 → ±1 with magnetic quantum numbers of ±1 anda singlet excited state e with magnetic quantum number of zero. We will take the

optical matrix elements between the states to be e er g1,2 = µ and the energy difference

between the ground and excited states to be hω0 . If a DC magnetic field is appliedparallel to the optical wavevector it causes a Zeeman splitting, 2h, proportional to theexternal magnetic field. Furthermore we will simplify our notation by taking h = 1 .

With these assumptions the Hamiltonian can be expressed as:Eq. 8 H = H0 + H1

H0 =ω0 e e − µ E+e+ iω t e g1 + E−e

+ iω t e g2 + E+*e− iω t g1 e + E−

*e− iω t g2 e{ }H1 = h g2 g2 − g1 g1{ }The wave equation is

Eq. 9 idψdt

= Hψ

where ψ is a linear combination of g1,2 & eThe physics that we want to discuss can be most easily treated by transforming to

Eq. 10 e = e` e− iωt .

With the definitionEq. 11 Δ =ω0 −ω

the Hamiltonian becomes

Eq. 12H 0= Δ e` e` − µ E+ e` g1 + E− e` g2 + E+

* g1 e` + E−* g2 e`{ }

H `= H 0+H1

The wave function in the primed reference frame can be written as

Eq. 13 ψ ` = b` t( ) e ' + a1 t( ) g1 + a2 t( ) g2From which the Schrodinger wave equation can be written

Eq. 14

ib`a1a2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

Δ −µE+ −µE−

−µE+* −h 0

−µE−* 0 h

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

b`a1a2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

The steady state the solution takes the form

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Eq. 15

Δ − λ −µE+ −µE−

−µE+* −h − λ 0

−µE−* 0 h − λ

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

b 'a1a2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= 0

where λ are the energy eigen values.Linear Faraday EffectIn order to obtain expressions for the 2 × 2 dielectric constant tensor in the

following section we will need to evaluate the matrix elements of the polarizationoperator in the laboratory frame

Eq. 16

p =e g1 µ2

x − iy( ) + µ e g22

x + iy( )

+g1 e2

µ x + iy( ) + g2 e2

µ x − iy( );

however, before apply this formalism to the non-linear effect it would be useful to seehow it applies for the linear effect. When Ω ≈ 0 the wave equation becomes

Eq. 17 Δ − λ 0 00 −h − λ 00 0 h − λ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

ba1a2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

with eigen solutionsλ = ±h & a1,2 = 1

λ = Δ & b = 1These are the unperturbed states. For small E Eq. 15 can be solved using a perturbationapproximation. In the transformed frame the solution for the lower energy states has theform

Eq. 18

ψ '1,2 → g1,2 +µE±

Δ he` .

In the laboratory frame the solution is

Eq. 19

ψ 1,2 → g1,2 +µE±

Δ he e+ iω t

For these states the expectation of the polarization operator is thus

Eq. 20

ψ 1p ψ 1 =

x − iy( )2

µ2E+

Δ − he+ iω t + cc

ψ 2p ψ 2 =

x + iy( )2

µ2E−

Δ + heiω t + cc

The polarizability for left/right circular polarizations are thus

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Eq. 21

α± =ψ 1,2

p ψ 1,2

E±

~ µ2

Δ h

and to lowest order in h the difference α+ −α− ~ 2h / Δ2 has the same from as the

classical result that was given by Eq. 6.Non-Linear Effect.The linear Faraday effect corresponds to the perturbation solution to Eq. 15 when

E << h . For the non-linear Faraday effect the problem must be solved in the oppositelimit for which E >> h . To do this we first solve Eq. 15 with h=0.

Eq. 22

Δ − λ −µE+ −µE−

−µE+* −λ 0

−µE−* 0 −λ

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

b`a1a2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= 0

Δ − λ( )λ2 + λµ2 E 2 = 0 where E 2 = E+2 + E−

2

There are three solutions for the eigen values

Eq. 23 λ 0 = 0 and λ± =Δ2±

Δ2

⎛⎝⎜

⎞⎠⎟2

+ µ2 E 2 .

The state with λ0 is a dark state that does not couple to the excited state.

Eq. 24 . D =E− g1 − E+ g2

E

for which D p D = 0. The two other solutions with λ± correspond to linear

combinations of the higher energy state e and a bright state

Eq. 25 B =E+* g1 + E−

* g2E

The bright state is not itself a solution to Eq. 15. The two solutions corresponding to λ±

are what are known as the dressed states. They correspond to linear combinations of theexcited and bright states that are orthogonal to the dark state. λ± = c1

± e` + c2± B where

c1,2± can be determined from Eq. 15 in analogy with Eq. 22. (See below)

One view of the physics of these two states can be understood by assuming that atsome time t=0 an atom is initially in the bright state B that is formed from a linearcombination of the two dressed states λ± . The effect of the energy splitting energy (Eq.23) is that the relative phase of these two states changes with timeexp iΔφ[ ] ≈ exp i λ+ − λ−( )t⎡⎣ ⎤⎦ such that the resultant state oscillates with Rabioscillations between the initial bright B and e . Of course the effect of relaxationprocesses is to interrupt the oscillations and induce absorptive transitions to e thateventually decay in a way that depopulates B and moves a number of the atoms into

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the dark state D . The fraction that are pumped to D will naturally depend on therelative absorption and relaxation rates.

If the effects of damping are neglected the solutions for the wave function forthese states can be expressed in terms of the basis state = e , B , D⎡⎣ ⎤⎦ . The twoterms in the Hamiltonian matrix for H `= H 0+H1 have the form

Eq. 26 H 0 =Δ −µ E 0

−µ E 0 00 0 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

H1 =hE2

0 0 00 E−

2 − E+2 −2E−E+

0 −2E−*E+

* E+2 − E−

2

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Taking the wave function to be ψ ' = b ' e + β B + γ D the wave equation becomes

Eq. 27 H `−λI[ ] ψ =

Δ − λ −µ E 0−µ E η − λ −τ

0 −τ * −η − λ

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

b 'βγ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= 0

where η =hΩ 2 Ω−

2 − Ω+2⎡

⎣⎤⎦ & τ = −

2hΩ 2 Ω−Ω+

The eigen values are obtained from the solutions toΔ − λ( ) λ2 −η2 − τ 2( ) + µ2 Ω 2 η + λ( ) = 0

however since η2 + τ 2 = h2 this can be written

Eq. 28 Δ − λ( ) λ2 − h2( ) + µ2 Ω 2 η + λ( ) = 0In the absence of a Zeeman splitting (i.e. h=η=0) Eq. 27 reduces to Eq. 22 and

the eigen value solutions are given by Eq. 23 and the eigen vectors correspond to thedressed states alluded to above. On the other hand, for h≠0 the eigen values solutions toEq. 28 depend on h. The eigen vectors appropriate Eq. 27 must be orthogonal to any ofthe row vectors. These can be obtained by a simple cross product. For example if

Det M = Deta b cd e fg h l

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= 0 the vector V =

el − fhdl − fgdh − eg

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

which is formed from the

Det V = Det1 2 3d e fg h l

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

is obviously satisfies M •V = 0 . Thus one expression for the

eigen vector for the h≠0 problem will be proportional to

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Eq. 29b`βγ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥~

µ E τΔ − λ( )τ

Δ − λ( ) η − λ( ) − µ2 E 2

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Dielectric ConstantConsider the h≠0 problem for which the optical field is at the atomic resonance,

Δ=0, and the polarization is linear, η=0. From Eq. 29 the eigen vector for the dark state(i.e. λ=0) has the form

Eq. 30 ψ `D−h =b`βγ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥~

µ E τ0

−µ2 E 2

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⇒−τ µ E01

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

This can be written as ψ `D−h = D +2hµ E

e`⎡

⎣⎢

⎤

⎦⎥ or for the assumed linear polarization

Eq. 31 ψ `D−h ≈g1 − g2

2+2hµ E

e⎡

⎣⎢

⎤

⎦⎥ .

It is straightforward to demonstrate that

Eq. 32 ψ D−h py ψ D−h = −i4hE

= −i4hE2 Ex

or

Eq. 33 ε yx ~ −i 4h E 2( )This is the term responsible for the Non-Linear Faraday effect.

The more general solution to Eq. 30 has the form

ψ `D−h =1

4h2 + µ2 E 2

2h0

µ E

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

and

Eq. 34 α yx ~hµ E

4h2 + µ2 E 2

Dispersion Δ≠0For linear polarization (η=0), h=0 and a small offset from resonance

( Δ << µ E )the solutions to Eq. 28 are λ=0 and λ± ≈ µ E ;however, as Δ increases tothe point that Δ >> µ E the two non-zero eigen values approach

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Eq. 35 λ+ ≈ Δ + µ2 Ω 2 Δ

λ− ≈ −µ2 Ω 2 Δ

In this limit λ+ → e and the λ− solution Β approaches one of the low energy eigensolutions to the Hamiltonian. In this limit the wave solutions and the dielectric constantsare only slightly different from those of the linear Faraday effect.

For a real system, like the Rb-vapor, the full Doppler width of the atomicresonances is generally much larger than either µ E . As a consequence the only atomsthat experience the non-linear effects discussed above are those whose Doppler shifts arerelatively closely matched to the laser frequency. The net Faraday rotation is thus asuperposition of the Non-Linear effect arising from the atoms near resonance, Eq. 33,and the weaker linear effect.

Slow LightSlow light is produced by a combination of one strong circularly polarized

incident laser beam (the pump beam) that oscillates as E+ exp iνt( ) and a second weakerbeam (probe beam) that oscillates with the other sense of polarization at a slightlydifferent frequency, i.e. E− exp i ν + δν( )t[ ] . Assuming Δ=0, if the difference frequency,δν, is sufficiently slow the atoms respond to the instantaneous polarization and formdark/bright states just as was discussed above for the h=0 problem. Observe that for Δ=0and h=0 the dark state(λ=0) described byEq. 27 and Eq. 29 can have any polarization.On the other hand, if δν is large in comparison with the relaxation time by which atomsmove between the ±1 or g1,2 states the effect of two fields on the atoms becomesindependent.

The solution to the Hamiltonian (Eq. 15) for the strong pump beam alone atresonance corresponds to a dark state, g2 at λ=0, and two dressed states at

λ± = ±µ Ω+ of the form λ± =12

e ± g 1⎡⎣ ⎤⎦ . If optical pumping to g2 is complete

the time dependent wave equation or the probe beam can be reduced to

Eq. 36

ib`a 2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

Δ± −µE−e− iδνt

−µE−*e+ iδνt 0

⎡

⎣⎢⎢

⎤

⎦⎥⎥

b`a2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

where the term Δ± account for the AC Stark effect by which the exited state has beenshifted by the strong pump beam (i.e. λ± ). Neglecting the AC Stark effect the solution isto induce an atomic polarizability analogous to the linear result given by Eq. 21 but witha very narrow spectral width of the order of µ E , which is considerably narrower thaneither the full Doppler width or the intrinsic zero-velocity optical lifetime. (Note that fortypical experimental parameters, the power broadening rate is much greater than the slowground state relaxation rate ~1 kHz, but still very narrow.) See the discussion regardingEq. 35.

The slow light effect is evident in the difference between the group velocity of thepump beam, which is the speed of light in vacuum, and the group velocity of the probebeam, which is given by

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Eq. 37 vg ≈ c 1+ ω 2( ) dε dω( )⎡⎣ ⎤⎦−1

.The consequence of the very narrow width of the non-linearly induced polarizabilty is tocause a large value of dε dω and a correspondingly small group velocity.

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.

References

Vanier, J. and Audoin, C. 1989. The quantum physics of atomic frequency standards.Bristol ; Philadelphia: A. Hilger.

Scully, M.O. and Zubairy, M.S. 1997. Quantum Optics. Cambridge Unievrsity Press.