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:

I(x, t) ∝ E21 + E2

2 − 2E1E2 + 4E1E2 cos(kx) = (E1 − E2)2 + 4E1E2 cos2(kx) (17)

Where k = 2π/λ is the wavenumber. Using that the lattice spacing of the lattice constructed

is half the wavelength, λ, we get a = 12λ = π/k and reciprocal lattice vector G = 2π

a = 2k,

see fig. 3.

From eq. 16 and 17 we know how the E-field behaves and from that we can describe the

intensity which becomes:

I(r) =(√

I1 −√I2

)2

+ 4√I1I2 cos2(kx) (18)

10

In the following, we use that the spotsize of the sample is much smaller than than the beam

waist, which means we can assume the intensity over the spotsize to be constant (whereas

the intensity over the entire beam waist is Gaussian). That is, we neglect the transverse

variation of the Gaussian beam and assume the atoms are on axis and hence at the peak

power. Finally, the intensity of the beam can be rewritten in terms of the power I = 2Pπw2

0

giving us:

I(r) =

(√2P1

πwx1wy1−

√2P2

πwx2wy2

)2

+8

π

√P1P2

wx1wy1wx2wy2cos2(kx) (19)

Where w1 and w2 are the waist size of the beams at the atoms. P1 and P2 are the power on

the sample from the two counterpropating beams. Both are linearly dependent on the input

laser power. Assuming the constant term is negligible we drop it and plug the remaining

oscillating term into the expression for Udip (eqn. 13) and get:

Udip = −α 8

2εcπ

√P1P2

wx1wy1wx2wy2cos2(kx) (20)

Finally this gives us an expression for the lattice depth in terms of the polarizability, the

waist dimensions and the power:

V0 = −α 8

2εcπ

√P1P2

wx1wy1wx2wy2(21)

Notice that we have only considered 1-D potentials when deriving the lattice depth. When

the experiment is running there will however be three beams creating a 3-D lattice. The 1-D

potentials are applicable only when the three beams are mutually perpendicular and have a

frequency offset controlled by an acousto optic modulator. In this case the three beams can

be considered indenpendent. If this is not the case, the interaction between the beams has

to be taken into account. When doing this all sorts of lattices can be constructed besides

the simple cubic structure created by the three perpendicular beams.

11

The band structure

To find the band structure of the optical lattice, which will be used when calibrating the

lattice depth, we need to solve the one-dimensional time-independent Schrodinger equation,

in which case eqn. 10 becomes (following [1] and [2]):

− ~2

2m

d2

dx2ψ(x) + V (x)ψ(x) = Eψ(x) (22)

In our case we consider the periodic potential V (x) = V0 cos2(kx) as we saw in the section

on the potential. Plugging this into the Hamiltonian, we have to solve the following:

− ~2

2m

d2

dx2ψ(x) + V0 cos2(kx)ψ(x) = Eψ(x) (23)

Or in terms of multiples, s, of the recoil energy ER = ~2k2/2m where V = sER and

E = εER:

− 1

k2

d2

dx2ψ(x) + s cos2(kx)ψ(x) = εψ(X) (24)

Due to the size of the BEC of 5−15µm and that our lattice constant is significantly smaller

at a = λ/2 = 266nm, we assume the potential to be infinite [1]. Using this assumption, we

can treat the case as a simple 1-D infinite lattice, corresponding to atoms on a string, which

means we can use regular band theory. Since we have a lattice we have a discrete number

of bands, n, that each satisfy Schrodingers equation. Finally we will in the following find

the solution in terms of the quasimomentum q = ~k, which indicates the velocity of lattice.

Using this, the equation to be solved for each band is:

− 1

k2

d2

dx2ψn,q(x) + s cos2(kx)ψn,q(x) = εn,qψn,q(x) (25)

To solve this we use that we have a periodic potential which means we can apply Bloch’s

theorem. Bloch’s theorem states that a wave function can be described as a plane wave eikx

12

multiplied by a periodic function, u(x) = u(x+ a) where a is the lattice constant. That is:

ψn,q(x) = eikxun,q(x) (26)

Since both the function ψn,q(x) and cos2(kx) are periodic they can be written as Fourier

series which both have periodicity λ/2:

un,q =

∞∑m=−∞

an,q(m)ei2kmx (27)

And

s cos2(kx) = s

(1

2+

1

2cos(2kx)

)= s

(1

2+

1

4

(e2ikx + e−2ikx

))(28)

By substituting eq. 27 into eq. 26 and defining ei(q/~+2km)x ≡ |φq+2m~k〉 we get

ψn,q =

∞∑m=−∞

an,q(m)ei(q/~+2km)x =

∞∑m=−∞

an,q(m) |φq+2m~k〉 (29)

Plugging this into the Schrodinger equation (eq. 25), differentiating with respect to x and

dividing through by |φq+2m~k〉 we get:

∞∑m=−∞

Hm′,m · an,q(m) = En · an,q(m) where Hm′,m =

(2m+ q)2

+ s/2, m = m′

−s/4, |m−m′| = 1

0, otherwise

(30)

From this we can find the population in each band and momentum group 2m~k, where m is

the momentum order, by solving the matrix for the eigenvalues an,q(m). Figure 4 shows the

population in bands 0,1 and 2 at lattice depth s = 10 and quasimomentum q = 0. Notice

that momentum orders higher that |m| = 2~k have negligible population. The coefficients

an,q(m) are instrumental in finding the populations when the lattice is turned on non-

adiabatically as we will see below.

13

−4−3−2−1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rel

ativ

e po

pula

tion

Band 0

−4−3−2−1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Momentum [2hk]

Band 1

−4−3−2−1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Band 2

Figure 4: The relative population in bands n=0,1,2 as a function of the momentum with

quasi momentum q=0

Furthermore, solving the Schrodinger equation as a function of quasimomentum enables

us to find the bandstructure as a function of q. Figure 5 shows the bandstructure for the

first four bands at q = −1..1 for three depths at s=5,10,15. Clearly the band gap increases

as the lattice depth increases. This coupled with the population in momuntum orders will

be used to calibrate the lattice depth.

Finding the frequency as a function of the Band gap

We now consider the situation where we load the BEC into the lattice non-adiabatically. A

BEC can be described as a plane wave given by φq = eiqx/~. Following [1] we have that the

BEC that is suddenly loaded into a lattice can described as a superpostion of Bloch states

|n, q〉 = ψn,q(x) =∑∞m=−∞ an,q(m) |φq+2m~k〉 giving us the expression:

Ψ(x, t = 0) =

∞∑n=0

|n, q〉 〈n, q|φq〉 (31)

14

−1 −0.5 0 0.5 10

2

4

6

8

10

12

14

16

18

20

Quasimomentum q [2π/λ]

Blo

ch S

tate

Ene

rgy

[ER

]

−1 −0.5 0 0.5 12

4

6

8

10

12

14

16

18

20

22

Quasimomentum q [2π/λ]

Blo

ch S

tate

Ene

rgy

[ER

]

−1 −0.5 0 0.5 10

5

10

15

20

25

Quasimomentum q [2π/λ]

Blo

ch S

tate

Ene

rgy

[ER

]

Figure 5: Band structure of optical lattice for a lattice depth of 5, 10 and 15ER. Here the

energies of the bands are plottet as a function of the quasimomentum q=-1..1 (this is the

first Brillouin zone)

Evaluating the term in brackets with the help of the orthogonality relation 〈φq+2m~k|φq+2m′~k〉 =

1 ⇐⇒ m = m′ we get:

〈n, q|φq〉 =

∞∑m=−∞

a∗n,q(m) 〈φq+2m~k|φq〉 = a∗n,q(0) (32)

The n’th term in eq. 31 is then given by

Ψn(x, t = 0) =

∞∑m=−∞

a∗n,q(0)an,q(m) |φq+2m~k〉 (33)

Adding the time dependency in the form e−iεn,qt/~ we get:

Ψn(x, t) =

∞∑m=−∞

a∗n,q(0)an,q(m)e−iεn,qt/~ |φq+2m~k〉 (34)

Summing over n we get the wave function gives:

Ψ(x, t) =

∞∑m=−∞

bq(m) |φq+2m~k〉 (35)

15

Where:

bq(m) ≡∞∑n=0

a∗n,q(0)an,q(m)e−iεn,q~ ∆t (36)

Figure 6 shows a plot of the population in different bands and in different momentum orders

as a function of lattice depth. From these we can see that the population in band 3 and up

is minimal for at lattice depth less than ∼ 15 ER. Also, the momentum orders m = ±2 and

above are relatively unpopulated up to ∼15 ER.

We wish to derive an expression for the population with a given momentum in terms of the

lattice depth, so as to later determine the lattice depth from measurements. As we will see,

we can obtain an expression of the oscillation of the momentum distribution as a function of

the band gap between the second excited band and the zeroth band, i.e. n=0,2. From this

we can calculate the lattice depth, since the band gap is determined by the lattice depth

through solving Schrodingers equation. When deriving the expression we assume that the

population is only in the zeroth and second exited band due to negligible population in

other bands at the depths up to ∼ 15ER. This means, that the expression we arrive at, will

only be useful in finding the lattice depth for depths up to 10-15 or so (see figure 6 a).

We evaluate the sum in bq(m), eq. 36, for n=0,2, giving us:

bq(m) = a∗0,q(0)a0,q(m)exp(−i ε0,q

~∆t)

+ a∗2,q(0)a2,q(m)exp(−i ε2,q

~∆t)

(37)

In the following we denote A = A(m) = a∗0,q(0)a0,q(m) and B = B(m) = a∗2,q(0)a2,q(m).

The population is given by multiplying by the complex conjugate giving us:

|bq(m)|2 = bq(m) ∗ bq(m)∗ (38)

= A2 +B2 +ABexp

(iε2,q − ε0,q

~∆t

)+ABexp

(−i ε2,q − ε0,q

~∆t

)(39)

16

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Lattice depth s [ER

]

Rel

ativ

e po

pula

tion

in B

and

(a) Relative population of band 0,2 and 4 as a function

of lattice depth s [ER] at q=0. Red is the zeroth band

(i.e. sum over b0,0(m)), green the second band and

blue the fourth band.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Lattice depth s [ER

]

Rel

ativ

e po

pula

tion

with

giv

en m

omen

tum

(b) Relative population of different momentums |m| =

0, 1, 2 as a function of lattice depth s [ER] at q=0.

Red is the zeroth momentum (i.e. sum over bn,0(0)),

the green is |m| = 1 and blue is |m| = 2

Figure 6: The relative population in the bands and momentum orders as function of lattice

depth when the lattice in switched on non-adiabatcally.

17

Figure 7: Theoretical calculations of oscillations of the relative population over time at q = 0

and s = 10ER. Red is momentum m=0 i.e. 0~k, blue |m| = 1 and green |m| = 2

By using Euler’s formula and noticing that the i sin terms cancel out, we end up with:

|bq(m)|2 = bq(m) · bq(m)∗ = A2 +B2 + 2AB cos

(ε2,q − ε0,q

~∆t

)(40)

From the above expression we have that the angular frequency is given by :

ω =ε2,q − ε0,q

~⇒ ω~ = Egap (41)

Where ε2,q − ε0,q = Egap. Equation 40 and 41 shows that the population in a given mo-

mentum group oscillates as a function of time. Figure 7 shows a plot of the oscillations of

the population in the momentum orders |m| = 0, 1, 2. Again the amplitude of the |m| = 2

order is small enough to be neglected. Hence from a measurement of the ±2~k population

as a function of time we should be able to find the band gap. Note also that assuming we

only occupy two bands, we get∑∞m=−∞ |bq(m)|2 = Ntot where Ntot is the total number of

particles in the lattice. In our experiment we have q=0, i.e. the lattice is stationary. Finding

18

0 5 10 15 200

5

10

15

Lattice depth s [ER

]

Ban

dgap

[E

R]

Figure 8: Theoretical calculation of band gap between ground state and second exited state

as a function of the lattice depth.

the lattice depth can then be done by finding the eigenvalues, i.e. the band energies, of the

Hamiltonian as a function of the lattice depth s. From this an expression for the band gap

as function of the lattice depth can be obtained:

Egap(s) = ε2,0(s)− ε0,0(s) (42)

Where ε2,0(s) is the energy in the second exited band as a function of lattice depth and

ε0,0(s) the ground state as a function of lattice depth. Figure 8 shows the band gap between

ground state and seconds exited state as a function of the lattice depth s at q=0.

The lattice depth is then found from equation 41 and 42 by solving for s in the equation:

ω~ = ε2,0(s)− ε0,0(s) (43)

19

Figure 9: Plot of a picture of the peaks produced when taking measurements at lattice depth

8.38ER and input voltage 2.54V. The center peak corresponds to the population in 0~k and

the two other corresponds to ±2~k, at ∆t = 15µs

Lattice depth calibration

To calibrate the lattice arms, we measure the population in the ±2~k momentum group,

which is |m|=1, as a function of time at a certain input voltage and hence power. The time

resolution of measurements is 2µs. For each measurement, a momuntum distribution of the

population is taken. A picture of a momentum distribution for a measurement can be seen in

fig. 9. Here the peaks have been normalized to add up to one. The center peak corresponds

to the zeroth momentum order and the two surrounding peaks corresponds to the |m| = 1

order. By having the populations in each momentum order as a function of time, we can

find the frequency at which the population oscillates (see fig. 7 and eq. 40). Figure 10 shows

the time evolution of the momentum distribution shown on a color scale. From this we can

see that the population oscillates as expected. By finding the total population in each peak,

we can then translate it into a plot as in fig. 11, which shows the total number of particles

20

Figure 10: Sequence of images of the peaks. The depth in this case is 8.38ER and input

voltage 2.54V. The first measurement (left) is taken at ∆t = 0.1µs increasing in steps of

0.2µs to the right ending with ∆t = 51µs

in the |m| = 1 momentum order along with the ratio to the total amount of particles. Note

that in that figure, there are three measurements. These three measurements are averaged

out so that we only have to fit to one curve for each lattice depth. The averaging is done

by finding the population ratio of the three measurements and then averaging those. The

reason to do it in this order, the other being taking the sum of the number of atoms in

±2~k of atoms and finding the ratio to the sum of the total number, is to eliminate number

fluctuations.

Based of the result obtained in eq. 40, we fit the averaged curve to a function of the form

f(t) = A+B · eC·t · cos(ωt+D) (44)

Note that in this expression, compared to eq. 40, we have added a phase shift constant and

an exponential term which accounts for the decaying oscillation amplitude. The decay is due

to inhomogeneity in the lattice beams meaning the interaction of the counter propagating

beams with each other is not completely constant over time. Also there is shot-to-shot

variation in the lattice, meaning that the lattice is not exactly the same when taking each

21

Figure 11: Upper: Ratio of the number of atoms in the ±2~k to the total number of atoms

in the sample as a function of time. Lower: Total number of atoms in the ±2~k peaks.

Lattice depth is 8.36, 8.38 and 8.57 ER at input voltage 2.54V

measurement point [1]. Fig 12 shows the averaged population of fig 11 along with a fit in

the form of eq. 44.

It is then the frequency, ω, that we use to find the lattice depth since it is directly related

to the band gap between the ground state and second exited state, as we saw in eq. 43.

This band gap in turn is a function of lattice depth, see fig 8. Finding the lattice depth is

found by solving for the solving for the lattice depth as a function of the bandgap which in

turn is found by solving for the eigenvalues in eq 30 at q = 0. When solving for eigenvalues

the sum is ussually evaluated from -2..2, i.e. a 5x5 matrix, since evaluating from −∞..∞ is

infeasible. Note that we can only do this if the lattice depth is sufficiently shallow. If it is

too deep we will have to take other bands and momentum orders into account, since these

start to a non negligible population (se fig. 6 a and b).

In calibrating the lattice the goal is to find the lattice depth as a function of the input voltage

22

Figure 12: Plot of the averaged population of |m| = 1 (blue) along with the |m| = 0

population (red). Plottet along with these are a fit of the |m| = 1 population which is used

to find the lattice depth.

of the laser, which is something that can be controlled. Using the theory from section the

potential, more specifically, eq. 21 we have that the depth is linearly dependent on the

power of the laser, which in turn is a linear function of the input voltage. That is, the depth

is linearly dependent on the voltage. Hence a linear fit is made to obtain the lattice depth

as function of the input voltage. Note that we mostly work with relatively shallow lattices

due to interference from higher band and momentum orders at higher depths. We can

though, due to the linearity between depth and power, extrapolate to deeper lattices which

we wouldn’t otherwise be able to determine by the method used for the calibration. Fig. 13

shows the linear fits of the lattice depth against power in mW (on the left) and input voltage

in V (on the right) along with the theoretical expectations. The theoretical expectations

have been calculated with eq. 21, where the polarizability α was obtained from [5]. On the

plot of the lattice depth vs. power the curves have been forced through the origin due to the

23

0 200 400 600 800 10000

2

4

6

8

10

12

14

16

18

20

Power [mW]

Lat

tice

Dep

th [

ER

]

Arm A measuredArm A Expected

0 2 4 6 8 100

5

10

15

20

25

30

Input Voltage [V]

Lat

tice

Dep

th [

ER

]

Arm A measuredArm A Expected

0 200 400 600 800 10000

2

4

6

8

10

12

14

16

18

20

Power [mW]

Lat

tice

Dep

th [

ER

]

Arm B measuredArm B Expected

0 2 4 6 8 100

5

10

15

20

25

30

Input Voltage [V]

Lat

tice

Dep

th [

ER

]

Arm B measuredArm B Expected

0 200 400 600 800 10000

2

4

6

8

10

12

14

16

18

20

Power [mW]

Lat

tice

Dep

th [

ER

]

Arm C measuredArm C Expected

0 2 4 6 8 100

5

10

15

20

25

30

Input Voltage [V]

Lat

tice

Dep

th [

ER

]

Arm C measuredArm C Expected

Figure 13: Calibration of the lattice depth as a function of the power in mW (the left) and

input voltage (the right). The dashed line is the expected value lattice depth as a function

of mW and voltage respectively. Top: Arm A, Middle: Arm B, Bottom: Arm C.

.24

Arm Measured slope ER/mW Expected slope Pct of exp.

A 35.8± 0.4983 · 10−3 54.8 · 10−3 65%

B 17.0± 0.4577 · 10−3 29.5 · 10−3 57%

C 27.9± 0.7716 · 10−3 33.9 · 10−3 82%

Table 1: Lattice depth by arm as a function of power

Arm Slope ER/V y-intercept ER/V Expected Pct of exp. (slope)

A 3.9742± 0.1629 0.0397± 0.3725 6.02x-0.57 66%

B 1.9998± 0.2456 −0.6845± 1.0478 3.25x-0.22 62%

C 3.9196± 0.4172 −1.2028± 0.9966 4.65x-1.15 84%

Table 2: Lattice depth by arm as a function of voltage

assumption of no lattice depth when the laser power is zero. The uncertainties of the lattice

depths, given by the errorbars, are found by using the upper and lower confidence bound

of the frequency, and calculating the corresponding latticedepth. Due to the non-linear

relation between band gap and lattice depth, these uncertainties come out as asymmetric.

The slopes of the fits are given in table 1 and 2. These are the calibration of each arm of

the lattice as a function of power in mW in table 1 and voltage in table 2.

From these tables we can see that arm C performs well since it reaches depth that are

around 80% of the expected. The other two perform less well with arm B doing the worst

at around 60%. Note however that after the measurements for arm B were taken, it was

discovered that the AOM was damaged and therefore it will be recalibrated later.

To further estimate the quality of our results, i.e. how confident we are in our measurements

25

of the lattice depth, we return to the fit and look at some of the other parameters used in

the fit. From eq. 40 we know that the population in the respective momentum groups have

both an offset and an amplitude. These both depend on the lattice depth, and as such can

be used to give an indication of the quality of the lattice depth calibration. Fig 14 a and 14

b shows the theoretical amplitude and offset as a function of the lattice depth in terms of

the recoil energy ER. The points shown in the plots are found by plotting the fitted offset

and amplitude (denoted by A and B in eq. 40) against the lattice depth calibration obtained

from the frequency of the population oscilllations. For the amplitude arm A and B show

good agreement with what we would expect since they follow the curve and more or less

fall within the errorbars. Arm C however seems to indicate some random noise indicating

discrepancy between theory and result. As for the offset, there seems to be a tendency in

the points that seems follow the theoretical expetations. Taking this into account one would

be inclined to have greater certainty in the lattice depths calculations that have been made,

and as such the quality of the calibration.

Conclusion

We have throughout the paper reviewed the theory necessary to understand trapping of

ultra cold atoms in an optical lattice consisting of three counter propagating laser beams.

This included the basic theory of a Bose-Einstein condensate, how to cool atoms to achieve

the condensate, the interaction of the BEC with the light field and the derivation of the

population in band and momentum orders. This was then used to calibrate the three lattice

arms in use in the experiment all in all making a 3D lattice. Here Arm C showed to perform

especially well obtaining around 80% of the expected value, while arm A and B perform less

well at around 65% and 60% repsectively.

26

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Lattice Depth [ER

]

Am

plitu

de [

ER

]

Amplitude vs. DepthArm AArm BArm C

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Lattice Depth [ER

]

Off

set E

R

Offset vs. DepthArm AArm BArm C

Figure 14: Top: Relative population of band 0,2 and 4 as a function of lattice depth s [ER]

at q=0. Red is the zeroth band (i.e. sum over b0,0(m)), green the second band and blue the

fourth band, Bottom: The measured offset from the fit of the oscillating (points) along with

the theorectical expected offset as a function of the lattice depth

27

References

[1] Phillips, W. D., 2002: A Bose-Einstein Condensate in an Optical Lattice

[2] Andersen, H. K., 2008: Bose-Einstein Condensates in optical lattices

[3] Grimm, R and Weidemller, M, 1999: Optical Dipole Traps for Neutral Atoms

[4] Smith, H and Pethick, C.J, 2008: Bose-Einstein Condensation in Dilute Gases, Cam-

bridge University Press

[5] Mickelson, P.G, 2010: Trapping and Evaporation of 87Sr and 88Sr Mixtures

[6] Morsch, O and Oberthaler, M: Dynamics of Bose-Einstein condensates in optical lattices,

Review of Modern Physics, Volume 78, January 2006

[7] Griffiths, D.J: Introduction to Electrodynamics, Prentice Hall, 1999

[8] Foot, C.J: Atomic Physics, Oxford University Press, 2005

[9] Schroeder, D.V: An Introduction to Thermal Physics, Addison-Wesley, 2000

[10] Harris M.L, 2008: Realisation of a Cold Mixture of Rubidium and Caesium

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