Download Moed A

Moed A
A. Waxman
(Dated: 2009)
I.
PART A
1. fine splitting - 6 points The fine splitting of the energy levels is a result of the spin-orbit coupling: namely, the
splitting exhibits the coupling between the electron’s orbital momentum L and its spin S. The interaction Hamiltonian
is:
~ ·S
~
Hf = αf L
(1)
~ + S),
~ where the possible
The energy levels will be split according to the total angular momentum J values (J~ = L
87
values are |L − S| ≤ J ≤ |L + S| with intervals of 1. In Rb atoms (as in all Alkali atoms) there is one electron in
the outer shell (S=1/2). The ground state 5S (L=0) is thus not split. The first excited state in 5P (L=1) is split to
52 P1/2 and 52 P3/2 where 1/2 and 3/2 are J values. An intuitive picture is that in the frame of an orbiting electron (i.e.
not in the S shell), the proton orbits around the electron creating a magnetic field which interacts with the electron
spin.
hyperfine splitting - 6 points The hyperfine splitting exhibits the coupling between the angular momentum of
the electron J and the angular momentum of the nucleon I. The interaction Hamiltonian is:
Hhf = αhf I~ · J~
(2)
~ where the possible values are |I − J| ≤ F ≤ |I + J|
The energy levels will be split according to F values (F~ = J~ + I),
with intervals of 1. The splitting of the states 5S1/2 , 52 P1/2 and 52 P3/2 in 87 Rb (I=3/2) is shown in the diagram
below (Fig. 1).
Zeeman splitting - 6 points The Zeeman splitting is a result of external magnetic fields applied onto the atom.
The interaction Hamiltonian is
~
HZeeman = −~
µ·B
(3)
Since the energy of the interaction (in a weak field approximation) is small compared to the hyperfine splitting energy,
we can assume that F is still a good quantum number, and the splitting is according to mF values (the projection of
the angular momentum along the z-axis). This is demonstrated for 87 Rb in the diagram bellow. The Zeeman energy
shift to first order is given by:
∆E = gF µB mF B
(4)
where µB is the Bohr magneton and gF is the Lande factor. From its definition above, the Zeeman splitting is smaller
in energy than the fine and hyperfine splitting (0.7 MHz/G in 87 Rb ground state).
2
Fine
structure
Hyperfine
structure
F=3
5P3/2
F=2
5P
F=1
F=0
F=2
5P1/2
816 MHz
F=1
F=2
Zeeman
splitting
MF=3
2
1
0
-1
-2
-3
2
1
0
-1
-2
1
0
-1
0
2
1
0
-1
-2
-1
0
1
2
1
0
-1
-2
5S
5S1/2
6.8 GHz
F=1
FIG. 1: A level diagram for
87
-1
0
1
Rb ground state and first excited state. Fine, hyperfine and Zeeman splitting are shown.
Experimental observation - 7 points The hyperfine splitting is smaller in energy relative to the fine splitting
since the magnetic moment of the nucleon is much smaller than the magnetic moment of the electron (or equivalently
αf À αhf ). The nucleon’s Lande-factor (proportional to the magnetic moment) in 87 Rb, for example, is gI ≈ 0.0007
while gL ≈ 1 and gS ≈ 2. Hence the hyperfine splitting (in frequency) is from the order of hundreds of MHz (up to
several GHz) while the fine splitting (between 52 P1/2 and 52 P3/2 ) is on the order of several THz. Consequently, in
the lab a simple spectroscopy experiment will reveal two separate doppler dips (several nano meters apart) for the
two fine structure transitions from the ground state. To observe the hyperfine structure, one has to use doppler-free
spectroscopy, in order to resolve the lines which are typically 100MHz apart. The Zeeman sublevels are typically a few
MHz apart (for an external field of a few Gauss) and this can be observed in the lab by inducing transition between
the levels with an RF radiation, or by scanning the difference between two raman beams in a CPT experiment.
2. how a clock works incl. schematic - 15 points An atomic clock is a type of clock that uses an atomic
resonance frequency standard as its timekeeping element. The simplest clock has two ground state levels (e.g. F=1
and F=2 for 87 Rb). The frequency difference between these states serves as a feedback loop to correct the accuracy
of a frequency reference (usually a 10 MHz electronic device, see Fig. 2). In the case of the 87 Rb atom, when the 6.8
GHz radiation based on the frequency reference is indeed accurate, atoms are excited from the F=1 level to the F=2.
A laser probe tuned to the |F = 2 >←→ |F 0 = 2 > then shows more absorption (or less intensity, see Fig. 3).
what affects the clock uncertainty - 10 points The accuracy of the clock is affected by:
(a) Collisions between atoms or between atoms and the cell walls. Such collisions may change the atomic state or
simply shift the energy of the line, leading to a broadening of the transition line.
(b) Small Transient magnetic fields. These fields, originating from the noisy environment, change the transition
frequency due to Zeeman effect, leading to a broadening and shift of the transition line.
(c) Non uniformity of the driving microwave field along the atomic cell. This will lead to a ”location dependent”
3
interaction between atom and field.
(d) smaller affects are power broadening and doppler.
Laser
Laser
beam
PD
87
Rb atoms
Electric
signal
Lock in
amplifier
Microwave signal
6.8 GHz
F'=2
Error
signal
6800
Laser light
10 MHz
ref.
PID
correction
signal to the
frequency
F=2
Microwave signal
F=1
FIG. 2: A block diagram describing a simple atomic clock. In the inset, the relevant level scheme and radiation fields are
shown.
210
200
F'=2
Laser light
F=2
F=1
Microwave signal
Light intensity [arb. units]
190
180
170
160
150
140
130
120
110
100
6.60
6.65
6.70
6.75
6.80
6.85
6.90
6.95
7.00
microwave frequency [GHz]
FIG. 3: The intensity of the laser beam tuned to the |F = 2 >←→ |F 0 = 2 > transition, as a function of the scanned microwave
frequency.
4
3.
Atoms will be trapped by either an optical or magnetic force. As the electric force can only attract atoms, the
electric force may also be used only in combination with another force. The force acting on a particle is of course the
derivative of the potential with respect to space.
Optical dipole trap - 15 points - The interaction between laser light and the atoms results in a shift of the atomic
energy levels (Stark shift). The interaction strength depends on the laser’s intensity I(r) and detuning ∆:
I(r)
(5)
∆
Tightly focusing the light, one can create a location dependent potential (differential stark shift) which is either
attractive (the light is red detuned- ∆ < 0) or repulsive (the light is blue detuned- ∆ > 0). A ”red detuned” trap is
hence constructed by a simple focused gaussian beam, while a blue detuned trap demands the production of a dark
area by using e.g. a hollow beam. In the latter case the atoms will be trapped ”in the dark” between the two potential
peaks (see Fig. 4). The trap light has to be far detuned from the optical transition so as to minimize the scattering
rate (proportional to I(r)/∆2 )while keeping a tight trapping potential (proportional to I(r)/∆).
U∝
FIG. 4: The potential curve of a ”red” (left) and a ”blue” (right) dipole trap
Magnetic trap - 10 points - The simplest magnetic trap is formed by the magnetic quadrupole field that is
created in the center of two coils which carry currents in opposite directions (see Fig. 5a) The modulus (absolute)
magnetic field grows linearly with the distance from the center, where the magnetic field is zero. The magnetic
potential of atoms in a magnetic field is:
U =µ·B
(6)
(where at times, the equation is given a minus sign depending on the convention of the sign of gF ). For small fields
we can write µ = gF mF µB , where µB is the Bohr magneton, gF is Lande factor and mF is the projection of F along
the quantization axis. Only atoms in a low field seeking state (the magnetic moment of the trapped atom is oriented
such that it experiences a force towards the minimum of the magnetic field) can be trapped in a potential with a local
minimum of |B|. The low field seeking states are states where µ > 0. In87 Rb the possible states are |F = 2; mF = 2 >,
|F = 2; mF = 1 >, |F = 1; mF = −1 >. Due to Larmor precession and adiabatic evolution, the magnetic moment of
the atom maintains its orientation with respect to the local field leading to a trapping potential which is a function
of |B| (Fig.5b). The major shortcoming of the quadrupole trap is that cold atoms which are close to the trap center
are removed from the trap due to non-adiabatic spin flips (as the Larmor precession frequency drops to zero in areas
with no magnetic field). Geometrically more complicated traps with a non-zero minimum are hence applied in most
experiments.
4. main pumping beam - 15 points Let us first compute the linewidth of the beams tuned to the |F = 4 >←→
|F 0 = 4 > transition. In order to pump the atoms to |F = 4, mF = 4 > this beam has to be σ + polarized meaning
mF 0 = mF + 1. Since the atoms ”fall back” by spontaneous emission with an average of ∆mF = 0 they will be
”pushed” after a short time to the highest mF state which is a dark state. The frequency of a beam resonant with a
|F = 4, mF >←→ |F 0 = 4, mF + 1 > transition depends on the zeeman shift of the mF state and is given by:
¸
·
2 mF
+
/h
(7)
νres (mF ) = ν0−0 + µB B [gF 0 (mF + 1) − gF mF ] /h = ν0−0 + µB B
3
6
where ν0−0 is the frequency of the transition |F = 4, mF = 0 >←→ |F 0 = 4, mF 0 = 0 > and the lande factors are
1/2 and 2/3 for F = 4 and F 0 = 4 respectively. The linewidth of the laser has to ensure all the transition between
different mF states will be in resonance. Since νres (mF ) is linear and ascending with mF the maximum and minimum
values of it are given for mF = 3 and mF = −4 respectively. The laser linewidth will hence be:
∆ν = νres (mF = 3) − νres (mF = −4) =
7 µB B
6 h
(8)
5
(a)
(b)
FIG. 5: The magnetic field produced (a) and the potential across the x axes (b) for the quadropole trap configuration
repumping beam - 10 points The purpose of the second beam tuned to the |F = 3 >←→ |F 0 = 4 > transition
is to return atoms that fell to |F = 3 > by spontaneous emission to the cycle of the pumping. For this beam we
perform a similar calculation. We just change to gF = −1/2 (according to the value given in the question) in Eq. 5.
to get:
·
¸
2 7mF
νres (mF ) = ν0−0 + µB B [gF 0 (mF + 1) − gF mF ] /h = ν0−0 + µB B
+
/h
(9)
3
6
Here we get maximum and minimum values for mF = 3 and mF = −3 respectively, leading to
∆ν = νres (mF = 3) − νres (mF = −3) = 7
µB B
h
(10)
(A remark: Considering a linearly polarized beam for the |F = 3 >←→ |F 0 = 4 > transition (mF 0 = mF ) we would
have achieved the same ”repumping” effect and the same linewidth)
II.
PART B
5. 6 beam MOT in two reference frames - 10 points
A MOT works by optically cooling atoms (via recoil of photon absorption) in a linearly inhomogeneous magnetic field
B = B(z) ≈ Az, such as that formed by a magnetic quadrupole field. While the light, which is red detuned to interact
with atoms moving against the incoming photon momemntum, is responsible for the atom cooling, the quadrupole
magnetic field enables the trapping of the atoms, though not through the µ · B potential, but rather via the flipping
of the Zeeman axis which ensures that light coming from the left interacts only with the atoms on the left side of
the trap and light coming from the right interacts with atoms on the right side of the trap, thus pushing the atoms
from all directions towards the center. This position dependent interaction works as follows (in a simplified scheme):
atomic transitions with Jground = 0 → Jexcited = 1 have three excited level Zeeman components in a magnetic field,
excited by each of three light polarizations. Defining the spin direction relative to a spatial axis in the lab frame,
on one side of the trap the Zeeman interaction shifts the excited state me = +1 upwards, whereas the state with
me = −1 is shifted down. Because the magnetic field flips its direction at the center of the trap, the other side of the
trap has opposite signs for the above description of me . At one side of the trap, the magnetic field therefore tunes
the ∆m = −1 transition closer to resonance and the ∆m = +1 transition further out of resonance (as the light is
red detuned). If the polarization of the laser beam incident from this side is chosen to be σ − , and correspondingly
σ + for the other beam, then more light is scattered from the σ − beam than from the σ + beam. Thus the atoms are
driven toward the center of the trap. On the other side of the center of the trap, the roles of the me = ±1 states are
reversed and now more light is scattered from the σ + beam, again driving the atoms toward the center. Hence in this
description, the two counter propagating beams have opposite circular polarization. In 3D one would find 3 pairs of
counter propagating beams, each pair having the two types of circular polarizations.
A second view, will be one where the quantum axis is now not the spatial axis in the lab frame as before but an
axis parallel to the local magnetic field. In this reference frame, on both sides of the trap center the me = −1 level
is shifted down and as the light is red detuned, photons coming from both sides will have to have a σ − polarization,
again relative to the direction of the local magnetic field. In this view, the MOT counter propagation beams have
the same circular polarization, as now the photon polarization is determined by its helicity i.e. the direction of its
angular momentum relative to the direction of its momentum. In 3D one would find 3 pairs of counter propagating
beams: e.g. the two pairs going through the coil axis will have σ − while the other 4 beams will have σ + . This is a
result of the cylindrical symmetry of the magnetic field. The exact sign of the polarization is determined by the sign
of gF and by the direction of the current in the coils.
6
simple experiment to check the polarizability - 3 points One can check the circularly polarized light, with
a polarizing beam splitter cube (PBC) and a λ/4 wave plate rotated by 45 degrees relative to the axes of the cube.
If you position this ’device’ in front of the beam, according to the output direction of the linear light at the PBC one
can determine the original circular polarization of the light (according to the helicity i.e. in the local frame relative
to the monetum direction of the photon). In accordance with the second description offered above, the device will
give the same result for any two counter propagating beams at the MOT.
blue detuned light - 2 points A blue detuned light will change the interaction in such a way that more light
is scattered from the σ − beam than from the σ + beam on one side and vice versa on the other, and therefore will
push the atoms away from the center of the trap. In addition, blue detuned light will not interact with atoms moving
towards the light and will therefore not slow (cool) the atoms.
photon recoil - 5 points The light scattering (absorption) force the atom feels in a light field is proportional to
the scattering rate γs times the momentum transfer per absorption h̄k. This should be higher than the gravitational
force, namely:
mg = h̄kγs → γs =
mg
h̄k
(11)
reflection MOT - 5 points In a reflection MOT (mirror MOT), one uses 4 beams instead of 6 (see Fig. 6), but
two of the beams are reflected from a mirror, thus creating the required 6 beams. Since the polarization of the light
changes when it hits a mirror, it is possible to do it. In this scheme the quadrupole field axis needs to be angled by
45◦ to the mirror axis.
To get the 5 points one must show the right polarizations and field axes in the drawing.
+
-
+
-
FIG. 6: A diagram of a Mirror MOT
7
6. what is EIT, why does it happen and how do you observe it experimentally - 10 points EIT
(Electro-Magnetic Transparency) happens in an atomic Λ-system consisting of three levels (2 ground and 1 excited)
and in the presence of two light fields (see Fig. 7a). When the frequency difference between the two light fields
corresponds to the Hyperfine splitting (the frequency difference between the two lower levels), a sharp reduction in
the light absorption and scattering occurs (see Fig. 7b). In order to see the essential physics which is happening in
the three-level Λ-system under two light fields one can choose a more suitable basis:|3 >, |+ >, |− >, consisting of the
excited state and of what is essentially the symmetric and the antisymmetric superposition of the two ground states,
respectively. In particular, state |− > looks like this in the standard energy eigenstate basis:
|− >= (Ω1 |1 > −Ω2 |2 >)/Ωef f
(12)
where Ω1 and Ω2 are the Rabi frequencies of the transitions |1 >→ |3 > and |2 >→ |3 >, respectively. Ωef f is simply:
Ωef f = (Ω21 + Ω22 )1/2 .
(13)
It can be easily shown that the transition amplitude between this state and the excited state is zero, given that the
frequency difference between the two light fields is equal to the hyperfine split. Atoms that spontaneously decayed to
state |− > will thus stay there. In analogy to optical pumping one can say that |− > is a dark state.
The two ways to observe EIT is first to have a photodiode in the line of the incoming laser beam which then shows
more incident light when EIT happens, or a photodiode on the side of the beam facing the axis perpendicular to the
beam axis, which shows less light when EIT happens.
why on resonance with the excited state - 5 points Since the ”pumping” of atoms to the dark state is
through spontaneous emission for the upper excited state, the maximal pumping rate (or the maximal EIT signal) is
achieved when the two laser beams are in resonance with the optical transitions to the excited state, causing maximal
spontaneous emission.
light speed in EIT - 10 points The group velocity of the light in the atomic medium is given by:
vg =
c
dω
=
dn
dk
n + ω dω
(14)
where k is the wave number, ω is the light frequency and n is the refractive index. Since the refractive index as a
function of frequency looks like a derivative of the absorption signal, and since the EIT dip is narrow in frequency
dn
space, the value of dω
at the EIT peak will be very high, leading to a decrease in the group velocity around the EIT
resonance.
III.
PART C.
7. side band cooling - 5 points Side band cooling is a technique enabling further cooling of atoms by transferring
them to the ground vibrational state of the trapping potential. The cooling transition is for example |F = 3, mF =
3, ν = n >7→ |F = 3, mF = 2, ν = n − 1 >7→ |F = 3, mF = 1, ν = n − 2 > where ν is the vibrational quantum
number (ν = 1 is the ground state). The total energy states are degenerated in energy by applying a magnetic field
B such that the Zeeman splitting is exactly equal to the vibrational level splitting (see Fig. 8). First, we optically
pump the atoms to the state |F = 3, mF = 3 >. A ”Raman” laser beam detuned by ∼1 GHz from the excited state
(F 0 = 1) transfers the atoms in two steps to the lower vibrational state (|F = 3, mF = 1, ν = n − 2 >) followed by a
σ(+) optical pumping to |F = 3, mF = 3, ν = n − 2 >, at which point the cycle starts again. Note that the optical
pumping only changes the electronic state of the atoms and does not affect the vibrational state. This process may
be repeated until the atoms are in the lowest vibrational level. If the atom gets ”stuck” in |F = 3, mF = 2, ν = 2 >,
a weak linear light will pump it to |F = 3, mF = 3, ν = 1 > which is a dark state for the Raman beams.
8
(a)
(b)
FIG. 7: (a)The level diagram of an atomic Λ-system, and (b) the EIT signal as a function of the frequency offset between the
two lasers
8. cross over peaks - 5 points In doppler free spectroscopy we use a retro-reflected beam to resolve the absorption
peaks from the doppler broadened signal. Aside from the absorption peaks we get crossover peaks located exactly in
the middle between two absorption peaks. The crossover peaks can be either positive or negative:
(a)Positive crossover(87 Rb)
The hyperfine splitting of the ground state is larger than the splitting of the excited state and the width of the doppler
dip and hence only one ground state takes part in the cross overs of each doppler dip. When the laser frequency
is an average of two different transitions to levels in the excited state (ω = (ω1 + ω2 )/2, see Fig. 9a), the counter
propagating pump and the probe will be in resonance with the same atoms; if the atoms move towards the probe
beam, the pump beam will excite the ω2 transition relative to which it is blue detuned, and vice versa. Since the
atoms absorb the pump light, some of them will spontaneously decay to the other level of the ground state which is
a dark state for them. Hence, the absorption of the counter propagating probe will be decreased which means the
intensity of light will be increased leading to a peak in the intensity profile (see Fig. 9b)
9
FIG. 8: A scheme of sideband cooling. The straight lines denote the optical pumping beams while the bent lines denote the
cooling Raman transition
(b)Negative crossover(Lithium)
Here, the hyperfine splitting of the ground state is smaller than the Doppler dip width and hence both ground states
take part in the interaction. As the splitting between the excited states is very small, effectively only one excited state
takes part (see Fig. 10a). When the laser frequency is equal to the average of the transition frequencies between the
two levels in the ground state and the excited state, the pump will be in resonance with atoms in level 1 moving in
the direction of the beam. The atoms will absorb the pump light and immediately decay spontaneously to both levels
in the ground state. This process can be looked at as a pumping of the atoms to level 2. The same atoms will move
towards the probe and will thus be absorbed only if they are in level 2. Since the pump increases the atom number
in level 2 the absorption of the probe will be enhanced leading to a decrease in the intensity (see Fig. 10b).
10
(a)
2
1
1
2
2
1
2
0
(b)
F’=1
Intensity [arb. units]
0.055
1-2
1-3
2-3
0.000
F’=3
F’=2
-0.055
0.000
0.005
0.010
0.015
0.020
0.025
Laser detuning [arb. units]
FIG. 9: The level structure leading to a positive crossover peak (a) and experimental results demonstrating the phenomenon
in 87 Rb (b).
(a)
(b)
1
2
1
2
2
F=
FIG. 10: The level structure leading to a negative crossover peak (a) and experimental results demonstrating the phenomenon
in 7 Li (b).