Calibration of a 3D Optical Lattice By Aslak Sindbjerg Poulsen Department of Physics and Astronomy, Rice Univeristy Advisor Dr. Thomas C. Killian April 22, 2014 Abstract This paper presents the basic theory needed to understand optical lattices. The theory covers Bose-Einstein condenstation, making of ultracold samples, optical dipole traps and optical lattices and a describtion of atoms in such a lattice. The theory is then applied to calibrate the depth of an optical lattice consisting of three counter propagating laser beams as a function of the laser power. 1
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Calibration of a 3D Optical Lattice
By
Aslak Sindbjerg Poulsen
Department of Physics and Astronomy, Rice Univeristy
Advisor
Dr. Thomas C. Killian
April 22, 2014
Abstract
This paper presents the basic theory needed to understand optical lattices. The
theory covers Bose-Einstein condenstation, making of ultracold samples, optical dipole
traps and optical lattices and a describtion of atoms in such a lattice. The theory
is then applied to calibrate the depth of an optical lattice consisting of three counter
propagating laser beams as a function of the laser power.
1
Introduction
As a physicist, studying even the smallest phenomena and the world of quantum physics is of
great interest. The study of some of these fundamental quantum mechanical phenomenons
cannot be done a room temperature, so cooling them down to near absolute zero is necessary.
Being able to trap the atoms gives an extra parameter that can be used to manipulate the
atoms to give a greater insight. On top of that being able create a potential trap in the
shape of a lattice provides a valuable increase in controlling the atoms on an individual level,
enabling simulation of a number of systems. Creating optical lattices for trapping atoms
have proven to be extremely useful in many areas of research. Fields such as this include
making highly accurate optical clocks, simulating atoms and electrons as they appear in
solid state physics, and in the case of this group, controlling interactions between Strontium
atoms.
In this paper, I will review the basic theory of atoms in optical lattices. This includes the
theory concerning a Bose-Einstein Condensate (BEC), cooling of the atoms to reach obtain
BEC, describing the interaction between atoms and the electrical field of the laser, and
finally the band structure that arises from this lattice. This theory will then be used to
calibrate the depth of a lattice consisting of three arms as a function of the input voltage,
and laser power. This means its ability to trap atoms as a function of the laser strength.
Bose-Einstein Condensate
The concept of a state of matter in which a macroscopic number og particles occupy the
lowest energy level, was introduced in 1924-1925 by Einstein and Bose. The idea is that
below a critical temperature a macroscopic number of the particles will occupy the ground
2
state. In the grand canonical ensemble it can be shown that the total number of particles
Bosons is given by summing over all states in the ensemble:
N =∑i
nB(εi) where nB(εi) =1
eβ(εi−µ) − 1(1)
Here nB(εi) is the Bose-Einstein distribution, where β = 1/kT , µ is the chemical potential
and εi is the energy of the i’th state.
We consider all the atoms confined in a box of volume V. The total number of particles
given in eq. 1 can be approximated by an integral:
N =
∫g(ε)nB(ε)dε where g(ε) =
V
4π2
(2m
~2
)3/2√ε (2)
Here g(ε) is the density of states.
We wish to find the lowest temperature Tc, at which the total number of particles can
be accommodated in the exited states [4]. At T = Tc eq. 2 becomes (here we make the
substitution x = βε):
N =V
4π2
(2mkTc~2
)3/2 ∫ ∞0
√x
ex − 1dx (3)
Evaluating this expression at µ = 0, which is where N achieves its greatest value [4], we get:
N =1
2
√πζ
(3
2
)V
4π2
(2mkTc~2
)3/2
(4)
Where ζ(3/2) is the Riemann Zeta function. Solving for Tc we get:
Tc =2π
ζ(3/2)
(N
V
)3/2 ~2
mk≈ 3.3127
(N
V
)3/2 ~2
mk(5)
Below this temperature, a macroscopic number of bosons fall into the ground state, forming a
Bose-Einstein Condensate. Notice that the critical temperature is dependent on the density
(n = N/V ) of the particles, meaning a higher density gives a higher TC . In our case we have
3
a low density gas of Strontium particles, which usually have a critical temperature around
10−7K.
At temperatures T < Tc the number of particles in the exited state are approximately given
by eq. 1. The reason for this being the number of particles in the exited state, is because
at ε = 0 the transition from the sum to the integral is not valid. So in fact we ’cut of’ the
integral close to ε = 0 and thus exclude the ground state giving the very similar result for
T < Tc [9]:
Nex =1
2
√πζ
(3
2
)V
4π2
(2mkT
~2
)3/2
(6)
Since we have an expression for N at Tc we can simplify eq. 6 to:
Nex =
(T
Tc
)3/2
N (7)
From this we get the number of particles in the ground state:
N0 =
(1−
(T
Tc
)3/2)N (8)
At such low temperatures classical mechanics no longer applies, and it has to be considered
a quantum mechanical system, in which the gas is described by a wave function. At the
point where the BEC is created, the wavefunctions describing the individual atoms start to
overlap significantly for the system to be described by as a single particle [10]. Assuming
that number fluctuations are negligible, we can use a mean field approach to the wave
equation. Also we assume that all the particels are in the ground state (T = 0) [6]. In this
case the wave function that is used when describing this system of interacting particles is
the Gross-Pitaevskii equation [4],[6]:
− ~2m
∇2ψ(r) + V (r)ψ(r) + U0 |ψ(r)|2 ψ(r) = µψ(r) (9)
4
Where U0 = 4π~2aS/m is the effective interaction between particles at low energies and aS
is the s-wave scattering length. As is visible, this is a non-linear Schrodinger equation due
to the term describing the interaction between particles. If the interaction between particles
is minimal and can be neglected equation 9 reduces to the regular Schrodinger equation for
non-interacting particles:
− ~2m
∇2ψ(r) + V (r)ψ(r) = µψ(r) (10)
In our case, when we later want to solve the Schroeringer equation, we use the assumption
of non-interaction.
Cooling of the atoms
We wish to perform experiments on gasses that are in the nano-Kelvin scale such that
they can be characterised as a Bose-Einstein condensate and as such be described by the
Schrodinger equation, eqn. 10. To get to these temperatures three steps of cooling are used.
Before the atoms are cooled they are placed in an oven and heated to about 500m/s to get
a vapor. From this they have to be cooled down. The first step is to cool them to about
30m/S which is done by a Zeeman slower.
The Zeeman slower is based on the principle of doppler cooling in which a light beam exites
an atom and then due to momentum conservation slows it down. The beam photon causes an
average momentum change in only one direction because the direction of the spontaneously
emitted new photon is random. However, this does present a problem, since slowing an
atom will cause it to experience the wavelength of a new photon differently than before due
to Doppler shift and so, it no longer absorbs new photon. To account for this a magnetic
field is applied along the path of the atoms that through the Zeeman effect changes the
5
energy levels of the atoms and hence the resonance frequency enabling new absorptions that
will slow the atoms. This all together accounts for the Zeeman Slower (see ref [8]).
The next step is the magneto-optical-trap (MOT) which can cool down to ∼ 1mK. Here
the atoms are placed in a magnetic anti-Helmholtz configuration creating a quadropole-field.
The magnetic field gradient causes a Zeeman split in the atoms as a function of position
in the field. Also, a set of three perpendicular counter propagating beams help confine the
atoms. Circularly polarized counter propagating beams then illuminate the atoms which,
through selection rules, cause an imbalance in the force on the atoms from the beams. This
force imbalance pushes the atoms to the center of the configuration, i.e. the center of the
magnetic field and traps them [8]. The cooling works on the principles of Doppler cooling.
For Sr, two transitions are used to cool the atoms, the 1S0 →1 P1 transition which is a blue
laser and the 1S0 →3 P1 transition which is a red laser, see fig. 1. The use of laser cooling
that works at atomic transitions gives a minimum temperature that can be achieved, due
to constant absorption of photons, even if evaporative cooling is done. In this case, if the
atoms reached a lower temperature, the atoms would absorb a photon and be reheated to
the Doppler cooling limit.
The final step is to use evaporative cooling to get from the millikelvins to nanokelvins
necessary to get a BEC. The atoms are placed in a dipole trap where the light is detuned
from the transition frequencies of Sr, which means no reheating due to absorbed photons.
The trap depth can then be lowered such that the most energetic atoms can escape leaving
only atoms with the desired temperature. The temperature that is reached through this
process is on the nano-kelvin scale. At this point a macroscopic number of particles in the
gas have the necessary temperature and density to be characterised as a BEC (see section
on BEC) [8].
6
Figure 1: Selected trasitions and energy levels for strontium. Taken from ref. [5]
The Optical Lattice Potential
We would like to contain the BEC that has been created in an optical dipole trap that is
made up of three counterpropagating 532 nm lasers. Below we will derive the nature of the
potential which traps the atoms. The potential arises from the interaction of an induced
dipole moment in the atoms with an oscillating electric field.
The potential is determined by the beam paramters and the interaction of the light with
the atoms through the ac-Stark effect.
As it is known, an electric field E, can interact with both electrons and protons, but in
opposite direction. Hence in a neutral atom, (Ne = Np), the field can interact with these
respectively. Since the interaction works in opposite direction, the field will induce a dipole
moment, with the electrons in one end and the protons in the other. This induced dipole
can again interact with the E-field of the laser trapping them. This causes the atoms to
be placed in a periodic lattice with the lattice maxima corresponding to the minima and
maxima of the light field [6]. That is it both the minima and maxima is due to the squaring
of cos. Hence we have a lattice with spacing a = λ/2 where λ is the wavelength of the laser.
7
This effect can be treated as second order time-independent perturbation theory leading to
an energy shift, the ac Stark effect, given by (we are basically following [3]):
∆Ei =∑i6=j
|〈i|H |j〉|2
Ei − Ej(11)
The Hamiltonian describing the interaction between atom and light is given by H = −µE
where µ is the electric dipole operator. Ei is the unperturbed energy of the i’th state. The
main contribution of the energy shift usually comes from a few exited states. With this in
mind we consider a two level atom in which we have the ground state and one exited state.
This gives us the shift:
∆E =|〈e|H |g〉|Ee − Eg
=3πc2
ω30
Γ
∆I(r) ≈ −3πc2
ω30
Γ
ω0I(r) (12)
Here ∆ = ω − ω0 is the detuning, where is ω the frequency of the laser and ω0 is the
transition frequency between the ground state and the exited state. Finally we have the
spontaneous decay rate of the exited state Γ = ω3πε0~c3 |〈e|µ |g〉|
2and the intensity of the
laser I = ε0c|E|22 . The approximation made in the last expression is used when the trapping
frequency ω of the light is significantly smaller than the resonance frequency ω0. That is,
ω � ω0 which gives the approximation ∆ = ω − ω0 ≈= −ω0. Figure 2 shows schematically
what happens when the an atom interacts with the electric field. The expression above is
also known as the trapping potential or lattice depth:
Udip = −3πc2
ω30
Γ
ω0I(r) = −αI(r)
2ε0c(13)
Where α is the static polarization (static because ω � ω0). Note that adding more states
will increase the shift, and so the two level atom is an underestimate of the actual shift.
8
Figure 2: Stark shift. The left hand side shows that the shift is negative in the ground
state, while the shift is positive in the exited state when using a red-detuned laser. Right
hand side, the shift is proportional to the intensity of the field, which creates a ground state
potential well which can trap atoms in the center of the beam. Modified from [3] and [5]
The next step is to get an expression of the intensity in terms of a quantity we can
measure, which in this case is the power of the laser. The intensity of an electric field is
proportional to the square of the electric field [7]:
I(r) =ε0c|E(r)|2
2(14)
So by knowing how the electric field behaves, we know how the intensity behaves. In our
case, the electric field of choice is that which is produced by the light emanating from the
laser. We reflect the laser beam onto itself, i.e. a counter propagating wave, to make a
standing wave, leading to an E field given by:
E(x, t) = E1ei(kx−ωt) + E2e
i(−kx−ωt) (15)
Where E1 and E2 correspond to the amplitude of the two counter propagating beams
9
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
Position in units of a
Pot
entia
l in
units
of l
attic
e de
pth
[V0]
Figure 3: The potential associated with the lattice showing the cos2(kx) dependency
respectively. Squaring the E-field we get:
|E(x, t)|2 = |E(x, t)| · |E(x, t)|∗ = E21 + E2
2 + 2E1E2 cos(2kx) (16)
Using a the identity 1 + cos(2u) = 2 cos2(u) we get: