Download Moed B Solution

Moed B Solution
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). As there is no orbital angular momentum in the 5S ground state (L=0), it is thus not split.
The first excited state in 5P (L=1) and it 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 magnetic moment originating from the electron spin.
hyperfine splitting - 6 points The hyperfine splitting is a result of 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)
The intuitive explanation is that the magnetic moment of the electron interact with the magnetic field produced by
~ where the possible values are
the nucleon spin. The energy levels will be split according to F values (F~ = J~ + I),
2
|I − J| ≤ F ≤ |I + J| with intervals of 1. The splitting of the states 5S1/2 , 5 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). The allowed mF values are described for 87 Rb in the diagram bellow. The
Zeeman energy shift to first order in B is given by:
∆E = gF µB mF B
(4)
where µB is the Bohr magneton and gF is the Lande factor. As noted, the Zeeman splitting is smaller in energy than
the fine and hyperfine splitting (0.7 MHz/G in 87 Rb ground state).
2
267MHz
157MHz
72MHz
384THz
377THz
FIG. 1: A level diagram for
87
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, 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
A CPT atomic clock is a type of clock that uses a CPT atomic resonance frequency standard as its timekeeping
element. CPT (Coherent Population Trapping) 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. 2a). When the frequency difference between the two
phase-locked light fields corresponds to the Hyperfine splitting of the ground state (the frequency difference between
the two lower levels), a sharp reduction in the light absorption and scattering occurs (see Fig. 2b). The resonance is
a result of the decay of the atomic system into a dark state. From this state the excitation probability to the upper
level of the Λ-system is zero.
The CPT resonance serves as a feedback signal to correct the accuracy of a signal generator which is used to
modulate the laser beam (see Fig. 3). The modulation generates the two coherent Laser fields with a frequency
3
difference which matches the Hyperfine splitting (for a frequency spectrum of a modulated beam look at the inset
of Fig. 3). In the case of the 87 Rb atom, the two laser beams will be tuned to the transitions F = 1 7→ F 0 = 2
and F = 2 7→ F 0 = 2. As the process is based on spontaneous emission both beams have to be tuned to the optical
resonance. A CPT resonance will occur when the frequency difference between the beams equals the Hyperfine
splitting (6.8 GHz).
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 CPT transition line.
(c) Smaller affects are power broadening and doppler.
(a)
(b)
FIG. 2: (a)The level diagram of an atomic Λ-system, and (b) the CPT signal as a function of the frequency offset between the
two lasers.
To laser
lock
Modulated laser
beam
Laser
87
Rb atoms
Modulation signal
3.4 GHz
I
∆υ≈ Hyperfine split
-1
0
+1 υlaser
Electric
signal
Lock in
amplifier
Error
signal
× 340
10 MHz
ref.
PD
correction
signal to the
frequency
PID
FIG. 3: A block diagram describing a simple CPT atomic clock. In the inset, the frequency spectrum of a modulated beam is
shown the difference between the two first optical side bands has to match the hyperfine splitting.
4
3. spin-orbit - 7 points In the absence of an external magnetic field, the only magnetic field is induced by the
movement of the nucleon in the reference frame of the electron (spin-orbit interaction). As the proton is charged, it
will create a magnetic field when it moves in a circle. The magentic moment of the electron will interact with that
magnetic field, and the direction of the spin will determine the energy shift. The interaction Hamiltonian is:
~ ·S
~
Hf = αf L
(5)
~ + S).
~ In 87 Rb the level 5P
The energy levels will be split according to the total angular momentum J values (J~ = L
is split into two levels (one where the electron spin direction is up and one where it is down). Consequently, in a
spectroscopic experiment, we will observe two transition lines (D1 and D2), 15 nm away from each other.
zeeman split - 7 points In the presence of a constant (DC) external magnetic field the atomic energy levels will
be split according to the Hamiltonian
~
HB = −~
µ · B.
(6)
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). The Zeeman energy shift to first order in B is given by:
∆E = gF µB mF B
(7)
where µB is the Bohr magneton and gF is the Lande factor. As µB is 1.4MHz/G, 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.
Stern-Gerlach - 11 points In a non-homogenous external field the atom will experience a force which is a
derivative of the Zeeman energy, namely
F = dE/dz = gF µB mF
∂B
∂z
(8)
where we assumed a magnetic gradient along the z-axis. The atomic cloud will hence be split to several clouds (2F+1)
according to the possible mF values.
If I=0 (i.e. no nuclear angular momentum) or we simply neglect the nuclear spin, the number of clouds will be equal
to how many different values of µz exist, and this can be calculated according to the formula: µz = µB [ml +2ms ], where
ml , ms are the projections along the z-axis of the orbital and spin angular momentum of the electron, respectively.
Using several detectors (or other spatial detection methods) we will be able to observe the different ensembles.
4. explanation-15 points The magnetic potential of atoms in a magnetic field is:
U = −µ · B
(9)
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. In our case these
are |F = 3; mF = −3 >, |F = 3; mF = −2 >, |F = 3; mF = −1 >, |F = 4; mF = 4 >, |F = 4; mF = 3 >,
|F = 4; mF = 2 > and |F = 4; mF = 1 >. In the first order in B, the ”Zeeman splitting” (the energy splitting
between two states with the same F and consecutive mF , for example |F = 3; mF = −3 > and |F = 3; mF = −2 >)
equals gF µB B. Electromagnetic radiation tuned to a frequency corresponding with this energy will induce spin-flips
(transition to high field seeking states) followed by trap losses.
calculation-10 points In the presence of 5 MHz radiation, a maximal spin-flip rate is induced when:
∆E = gF µB B = h · 5 · 106
(10)
and hence the external magnetic field is:
B=
where we used µB = h·1.4 [MHz/G]
h · 5 · 106
= 7.14G,
gF µB
(11)
5
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 momentum, 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.
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
(12)
reflection MOT - 5 points In a reflection MOT (mirror MOT), one uses 4 beams instead of 6 (see Fig. 4), 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 as shown in the following drawing.
6. side band cooling - 25 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. 5).
6
σ+
σσ+
σ-
FIG. 4: A diagram of a Mirror MOT
First, we optically pump the atoms to the state |F = 3, mF = 3 >. A ”Raman” laser beam (both beam have
the same frequency, so this light is a simple one frequency laser light) 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 > (via |F 0 = 2, mF = 2, ν = n − 2 > and spontaneous
emission), at which point the cycle starts again. Note that on average the optical pumping 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 > (again via
|F 0 = 2, mF = 2, ν = 2 > and spontaneous emission) which is a dark state for the Raman beams.
FIG. 5: A scheme of sideband cooling. The straight lines denote the optical pumping beams while the bent lines denote the
cooling Raman transition. The zig-zag line denotes the spontaneous decay and emission.
7
III.
PART C
7.verbal explanation-2 points The realization of |∆mF | = 2 transition in neutral 87 Rb atoms is of great
importance for quantum information experiments: Being trapped by the same magnetic force at a ”magic” magnetic
field of 3.23 G, the states |F = 1, mF = −1 > and |F = 2, mF = 1 > are perfect candidates to serve as the two
computational qubit states. The schematics of the Raman transition between these two states using two coherent
laser beams is shown in Fig. 6.
However, this |∆mF | = 2 Raman transition (i.e. with optical fields) has not been realized (the same transition
was realized by a combination of µ-wave and RF fields). Since two quanta of momentum has to be transferred, this
process involves the flipping of both the electron and the nucleus, while laser light operates only on the electron
which can only absorb one quanta when flipping from spin down to up or vice versa. The flipping of the nuclear
spin can be done only through the hyperfine interaction between the electron and the nucleus while the atom is in
the excited state. This interaction needs a time long enough to be done, of the order of the inverse of the hyperfine
splitting energy of the excited state (∼100 MHz). On the other hand, we would like to avoid the occupation of the
excited state during the Raman transition as it leads to spontaneous emission. For that reason the Raman beams,
have to be largely detuned (∼ 10GHz) from the excited state. That will reduce the probability of the nuclear spin
to flip, taking the transition amplitude to zero.
Mathematical description-3 points The 2 photon transition from |Ai = |F = 1, mF = −1i to |Bi = |F =
2, mF = 1i, is possible through |Ci = |F 0 = 1, mF = 0i and |Di = |F 0 = 2, mF = 0i (see Fig. 6), due to selection
rules. A σ + laser beam with the frequency ω1 is tuned between |Ai and the excited state, having the detunings ∆1C
and ∆1D from levels |Ci and |Di respectively, while a σ − laser beam with the frequency ω2 is tuned between |Bi and
the excited state, having the detunings ∆2C and ∆2D . The effective Rabi frequency (denotes the amplitude of the
transition) of the process is thus a sum over the two Raman transitions namely,
ΩAB =
ΩAC ΩBC
ΩAD ΩBD
hB|d · E|CihC|d · E|Ai hB|d · E|DihD|d · E|Ai
+
=
+
2∆C
2∆D
2∆C
2∆D
(13)
where we set ∆2C ≈ ∆1C ≡ ∆C and ∆2D ≈ ∆1D ≡ ∆D as otherwise the difference between the beams would not
be 6.8GHz and the transition would not be able to function. The electric diploe matrix elements relevant to that
transition satisfy (because of the Clebsch-Gordan coefficients)
r r
r r
1
3
5
1
1 1
0
hB|d · E|CihC|d · E|Ai + hB|d · E|DihD|d · E|Ai = hJ = |d · E|J = i ·
−
= 0,
(14)
2
2
24 40
24 8
followed by Ω∗ BC ΩAC + Ω∗ BD ΩAD = 0. Introducing that relation, Eq. 13 reduces to
ΩAB =
1
1
1
ΩAC ΩBC (
−
),
2
∆C
∆D
(15)
and since ∆D = ∆C + (ED − EC )/h̄, we can write
ΩAB =
1
(ED − EC )/h̄
ΩAC ΩBC
,
2
∆2
(16)
where ∆ is the average value of ∆C and ∆D . For a detunning much larger than the hyperfine split frequency, the
effective Rabi frequency will go to zero.
8
F=3
F=2
F=1
F=0
|D>
5P3/2
0
|C>
∆ AD
∆ AC
σ+
σ−
780nm
-2
-1
0
1
|B>
5S1/2
-1
|A>
1
0
2
F=2
F=1
FIG. 6: A level scheme of the two photon Raman transition
8. Einstein’s suggestion-2 points Einstein suggested to hang the first slit (S1) on a spring (see Fig. 7). After
the particle is ”fired” from the first slit (S1), it can be deflected up or down. As a result of momentum conservation
the screen itself (S1) will be deflected to the opposite direction. By measuring the momentum kick of the slit we will
be able to determine the direction in which the particle was deflected and as a consequence the slit (in S2) through
which it passed.
Bohr’s answer-3 points This experiment demands the momentum of the first slit will be measured accurately. Bohr
claimed that according to the uncertainty principle the latitude in position of the first screen (S1) will then grow. To
be more precise, if α is the small angle between the two paths going through the two different slits (a-b and a-c), the
difference of momentum transfer, in these two cases, will be:
∆p =
h
α
λ
(17)
where λ is De Broglie wavelength. According to the uncertainty principle, the latitude of the slit position will then
be
∆x ∼
λ
.
α
(18)
If the diaphragm with the two slits is placed in the middle between the first slit and the screen, the distance between
two identical interference fringes will also be αλ . Since an uncertainty in the position of the first slit will cause the
same uncertainty in the fringes location, the interference pattern will be washed out.
9
S1
F
S2
b
d
a
c
FIG. 7: A scheme of the experiment suggested by Einstein