Evaporative Cooling of 6-Li Atoms in a Spatially Modulated Dipole Trap

Master in Chemistry

Track Molecular Sciences

Minor Project

E

VAPORATIVE COOLING OF

6

_L

_{I ATOMS IN A}

SPATIALLY MODULATED DIPOLE TRAP

by

Juan Diego Arias Espinoza

11840307

July 2018

Supervisor:

Dr. Thomas Feldker

Examiner:

Dr. Rene Gerritsma

Reviewer:

Prof. Wybren Jan

Buma

Evaporative cooling of

6

Li atoms in a spatially

modulated dipole trap

Juan Diego Arias Espinoza

Abstract

In this project we have studied the formation of ultra-cold6Li atoms from an spatial modulated optical dipole trap, with the motivation to maximize the number of atoms trapped and reduce their temperature. The optical dipole trap potentials were modified by modulating in time and space a 1070 nm laser with an acusto-optical modulator (AOM). Numerical simulations were performed to estimate the shape and depth of the modulated trap potentials and the e↵ect of the modulation on the dipole trap potential was verified by Ramsey spec-troscopy.

By increasing the trap width along two directions, we were able to capture from the magnetic optical trap approximately two times more atoms (Natoms= 783_{± 71 ⇥ 10}3_{) than for the unmodulated case (N}

atoms= 446± 17 ⇥ 103), while obtaining similar atom temperatures of_{⇠ 400µK. Furthermore, we have reached} temperatures below 1 µK after evaporative cooling and phase space densities of 0.3(4), close to the threshold for Fermi gas formation.

Contents

1 Ultracold atom-ion mixtures 1 2 Theoretical background 3

2.1 Trapping of atoms . . . 3

2.2 Optical Dipole Trap: dipole potential . . . 3

2.3 Focused-beam trap . . . 5

3 Experimental 6 3.1 Optical Dipole Trap characterization . . . 6

3.1.1 Number of atoms . . . 6

3.1.2 Trap frequencies . . . 7

3.1.3 Temperature and size . . . 7

3.2 Modulation of dipole trap . . . 9

4 Results 11 4.1 TOF measurements . . . 11

4.1.1 Magnetic field frequencies . . . 11

4.2 Evaporative cooling . . . 13

4.2.1 Collision rates . . . 15

4.3 Spatial modulation of optical dipole trap . . . 16

4.3.1 Numerical calculations of trap potentials . . . 16

4.3.2 Ramsey spin echo experiments . . . 17

4.3.3 Optimization of dipole trap modulation . . . 18

4.4 Evaporative cooling with modulated dipole trap . . . 20 5 Conclusion and Outlook 21 6 Appendix I: Cloud expansion in an anti-trapping potential 24 7 Appendix II: Evaporation ramp used for modulated dipole trap 25

1

Ultracold atom-ion mixtures

This project has been done at the group Hybrid atom-ion quantum systems of the Institute of Physics of the University of Amsterdam. The main objective of this groupd is to study quantum phenomena of mixtures of ultra-cold atoms and and trapped ions. One of the end goals of the group is to build a quantum simulator, which intrinsically follow the laws of quantum mechanics, in order to study the dynamics of the quantum Hamiltonian of interest. This would allow studying systems which are impossible to simulate in classical computers.

Currently, the mixture under study is Li/Yb+[1] in a system consisting of Paul trap for ion trapping, and a combination of a magnetic optical trap and dipole trap for trapping and cooling of the atoms. Ions are routinely trapped and laser cooled to the mK regime, whereas the atoms can be cooled to below

Figure 1: Artistic illustration of the combined Paul and optical dipole trap used in these experiments

a µK using evaporative cooling. The aim is to reach ultra-cold temperatures to studied di↵erent emerging quantum phenomena[2, 3]. Unfortunately the atomic sample is rather small (about 104 _{atoms) and the experiment would} benefit significantly from obtaining a larger sample of ultracold atoms. For instance, owing to the large mass ratio, many atom-ion collisions are required to observe backaction on the ions. Since atom-ion collisions at high collision energies will result in the loss of the former, large atomic samples are needed to avoid depletion of atoms. Furthermore, the signal to-noise ratio in our atomic detection scheme would benefit greatly from larger atom numbers. Finally, the reliability of the experimental sequence would improve significantly.

In this report we will present the results of trapping and cooling of atoms from a temporospatial modulated optical dipole trap (ODT). In the first sec-tion, we present a short theoretical background on optical dipole traps. In the experimental section we will describe how to characterize optical dipole traps in order to obtain its trapping characteristics such as number of atoms, tem-perature and trapping frequencies. Also we describe our experimental approach for modulation of an ODT. In the result section, we will first present the char-acteriztion via time-of-flight measurements of the dipole traps and the results of evaporative cooling experiments of a reference (unmodulated) dipole trap. Then we will discuss the simulations of the temporospatial modulation of our ODT, the characterization of this modulated traps via Ramsey spectroscopy and the experimental optimization of the modulation parameters. Finally, we will present the results of evaporative cooling from a modulated dipole trap.

2

Theoretical background

2.1

Trapping of atoms

The confinement and generation of atoms and particles at ultra-low tem-peratures has been instrumental for the study of the quantum regime of many physical and chemical phenomema. For neutral atoms it has become common to produce ensembles with sub-miliKelvin temperatures. These ensembles are con-fined on the basis of mainly three di↵erent manifestations of dipole interactions with an electromagnetic field. Shortly, these three type of traps are:

• Radiation-pressure traps which use lasers which are nearly resonant with an atomic transtion of the specie of interest. Atoms moving against the direction of the light experience a force contrary to their propagation which slows them down, due to photons transmitting a momentum of_{~k to the} atoms. Making use of these forces a dissipative potential can be created in which atoms are cooled. However photon recoil and light-assisted inelastic collisions limits the achievable temperatures and densities. For trapping neutral atoms an additional magnetic field is used for this type of traps. • Magnetic traps which employs the magnetic dipole moment of atoms to

trap them in an inhomogeneous field. A conservative potential is thus created, which is however dependent of the electronic sub-state of the atom. This restricts the possibility of studying the internal dynamics of atoms. Furthermore trapping geometries are limited by the necessity of using coils or magnets. However this traps can be very deep by using intense magnetic fields.

• Optical dipole traps rely on the electric dipole interaction between the induced dipole in an atom and far-detuned laser light. Although this interaction is the weakest of all three, optical dipole traps are not a↵ected by the light-induced mechanisms of radiation pressure traps. Moreover many trapping geometries can be created due to the freedom available on the arrangement of di↵erent laser beams.

In our experimental setup both radiation pressure, magnetic trapping and optical dipole trapping were used in sequence to cool-down and trap Li atoms, however our experiments were focused on the last trapping mechanism. There-fore we will use this section to give a better description of the principles of an optical dipole trap and its experimental implementation.

2.2

Optical Dipole Trap: dipole potential

The optical dipole force arises due to the interaction of the induced dipole moment of the atom and the gradient of the intensity of the light field. An atom placed in an electric field E, such as the one of a laser beam, will manifest an induced dipole moment p oscillating at the field frequency !. The amplitude of the dipole moment (p) is related to the field amplitude (E) by

p = ↵(!)E (1) Here ↵ is frequency dependant polarizability. The interaction potential on the atomic dipole moment b in the light field is given by

Udip= 1

2hpEi = 1 2✏0c

Re(↵)I, (2) where the brackets indicate the time average over the rapid field oscillations and the real part of the polarizability describes the in-phase component of the dipole oscillation. The dipole force results from the gradient of this potential,

Fdip(r) = rUdip(r) = 1 2✏0c

Re(↵)rI (3) To estimate the polarizability ↵, the atom is considered as classical oscillator in the Lorentz’s model. An electron of mass meand charge e bounded elastically to a core, oscillates with an eigenfrequency !0 which corresponds to the optical transition of the atom. The oscillation is damped at a rate ( !) resulting from the classical dipole radiation of the the accelerating electron. The polarizability can now be obtained by solving the equation of motions and the results is

↵(!) = 6⇡✏0c3 !2

0(!20 !2 i(!3/!02) )

, (4)

where = (!0/!)2 ! is the on-resonance damping rate and ! is defined by Larmor’s formula

!=

e2_!2 6⇡✏0mec3

(5) In the semiclassical approach, where the atom is considered as a two-level quan-tum system, the damping rate is calculated otherwise by considering the dipole matrix element of the gorund and excited states,

= ! 3 0 3⇡✏0~c3 he|µ|gi 2 (6) For many atoms with strong dipole allowed transitions the classical expression provides a very good approximation, e.g. the D lines of the alkali atoms.

With the derived expression for the polarizability, an explicit expression for the dipole potential can be calulated in the case of large detunings and low saturation Udip(r) = 3⇡c2 2!3 0 ✓ !0 ! +!0+ ! ◆ I(r) (7) The expression show two resonances, one at ! = !0 and the other at ! =

!0. In most experiments the detuning = ! !0 is small, and the counter-rotating term (resonant at ! = !0) can be safely neglected in the rotating-wave approximation. Therefore the expression for the potential becomes

Udip(r) = 3⇡c2 2!3 0 ✓ ◆ I(r) (8)

The potential becomes negative when the laser frequency is below resonance ( < 0,“red” detuning), and therefore are attracted to regions of higher inten-sity of the light field. For frequencies above the transition frequency ( > 0, “blue” detuning), the potential is positive and atoms are repelled from volumes with high field intensities.

2.3

Focused-beam trap

One of the most common methods to generate optical dipole traps is by focusing a Gaussian laser beam, which is enough to provide 3D confinement. In our case a crossed-beam configuration is used, where two laser beams prop-agating in an horizontal plane cross at the center of the trap with an angle ↵. The horizontal alignment of the beams is necessary to compensate gravity. The potential of the trap for one of the beams propagating in the axial direction (z) can be calculated from the spatial intensity distribution of the beam in both the axial and radial (r) coordinates

I(r, z) = 2P ⇡W2_(z)exp 2 r2 W2_(z) ! I(r) (9) where P is the power of the laser and W(z) denote the waist of the beam at the position z, which depends on the Rayleigh length (zr = ⇡!02/ ) and the minimum radio of the beam (W0)

W (z) = W0 " 1 + ✓ _z zR ◆2#2 (10) In Figure 2, a cross-section of the optical dipole trap and its location within the ion trap is shown. A 1070 nm laser beam is introduced through one of the apertures of the ion trap and focused in the center of the trap. The beam exits the second aperture, where two mirrors and two lenses reflect and focus the beam back to the trap. The calculated values for the waist minimum (W0) and Rayleigh length (zR) for this beam are⇠36.8 µm and 3.98 mm respectively and the angles between the beams (↵) is 10 . Using equation 8 and 9 and the damping rate for lithium ( Li = 2pP i 5.872⇥ 106), the potentials can be calculated for di↵erent laser powers.

Figure 2: Illustration of the crossed beam optical dipole trap in our experiment. The top image shows an axial cross-section of the ion Paul trap. Extracted partially from [4]

3

Experimental

3.1

Optical Dipole Trap characterization

3.1.1 Number of atoms

Di↵erent methods exist for the estimation of the number of atoms, including calibrated measurement of fluorescence or absorption of the sample. The chosen method in our experiments is absorption imaging with near-resonance imaging at the D2 line of6_{Li. The atomic cloud is heated during imaging by the scattered} photons, increasing its temperature, therefore a new cloud is prepared for every measurement.

The images are then integrated along either the axial or a radial axis, and to the result a Gaussian curve is fitted to obtain the optical density along one axis:

OD(x) = a exp(x x0) 2

2 2 (11)

From the area under this fit (Agauss) divided by the atomic cross-section ( 0) and a correction factor corresponding to the area imaged by one pixel (Apixel), the number of atoms (N ) is calculated.

Agauss= Z x OD = pa⇢ 2⇡ (12) Apixel= A0 M2 (13) N = Agauss 0 Apixel (14) (15)

where A0 is the size of the imaging pixel and M is the magnification. 3.1.2 Trap frequencies

The total potential inside experienced by our atoms inside the dipole trap is a combination of the potential due to the dipole laser and the curvature of the magnetic field used to tune the hyperfine levels and scattering length of6Li:

Utot(r) = Udip(r) + Umag(r) (16) If the potentials can be considered harmonic, then the previous expression can be rewritten in term of trap frequencies:

1 2m! 2 totr2= 1 2m! 2 dipr2+ 1 2m! 2 magr2 (17)

Furthermore, the spatial density distribution can be both approximated by a Gaussian distribution and a Boltzmann factor:

⇢(r) = ⇢0exph Utot(r) kBT i = ⇢0exph m! 2_r2 2kBT i ⌘ ⇢0exph r 2 2 2 0 i (18) From these last three equations it is possible to express the trap frequency in terms of temperature (T) and the cloud size ( 0), such that !tot= 01

p kbT /m. To obtain the values for the temperature and the cloud-size, time-of-flight can be used, as described in next section.

3.1.3 Temperature and size

During time-of-flight measurements, the optical dipole trap is turned o↵ fast enough to allow a non-adiabatic expansion of the atom cloud. In absence of a force field, atoms will fly freely and ballistically. However in our system, the magnetic field of the Feshbach coils induce confining and anti-confining potentials which a↵ect the expansion of the atom cloud. Nevertheless, we can image the spatial distribution of the atoms after a predetermined time interval t, and use the evolution of the spatial distribution to estimate the initial velocity

of the atoms in the cloud at the time of release and the potentials present during the cloud expansion.

By considering that the initial cloud is in thermal equilibrium, the initial density distribution of the cloud in phase space is:

⇢0(x0, v0) = 1 p 2⇡ 0 exph x 2 0 2 2 0 ir m 2⇡kbT exph v 2 0m 2kbT i (19) where x0,v0 are the initial position and velocities of the atoms, m their mass and 0 is the size of the cloud.

The movement of the atom will be governed by the classical equation of motion, ¨x = !2_{x for a trapping potential of frequency ! (See Appendix I for} the anti-trapping case). The movement can be solved analytically for any initial condition ~q0= (x0, v0): ~ q(t) = ✓ x(t) v(t) ◆ = cos !t sin !t ! ! sin !t cos !t ! ✓ x0 v0 ◆ = M ~q0 (20) Then according to the solution of the time evolution for the density function reported by Weisman et. al. [5], the evolution of the atomic cloud at any time t is ⇢(x, v, t) =⇢0(x0(x, t), v0(v, t)) det ⇣ d~q0 d~q ⌘ (21) =⇢0(x0(x, t), v0(v, t)) det M 1 = ⇢0(x0(x, t), v0(v, t)) (22) using the fact that det M 1_{= 1. Then by using (20) in (19), we obtain:}

⇢(x, v, t) = _p1 2⇡ m p 2⇡kbT ⇥ exp h _x2 2 ⇣ cos2_!t 2 0 +! 2_m kT sin 2_!t⌘i ⇥ exph v 2 2 ⇣ sin2_!t !2 2 0 + m kT cos 2_!t⌘i ⇥ exph xv₂ sin !t cos !t⇣ 2!m

kT 2 ! 2 0 ⌘i (23) To obtain the width of the distribution ( (t)), we calculate the uncertainty of the coordinate x: (t) = q hx2_{i hxi}2 ₍₂₄₎ with:Dx2E= Z v Z x x2⇢(x, v, t) dxdv (25) hxi = Z v Z x x⇢(x, v, t) dxdv (26)

After solving the integrals we obtain the expression: (t) = s kbT sin2!t m!2 + 2 0cos2!t (27) Similar derivation, for an anti-trapping potential with frequency !anti, and the result for the time evolution of the cloud size is:

(t) = s kbT sinh2!antit m!2 anti + 2 0cosh 2_! antit (28) Projection of the images of the atom cloud into the trapping direction after di↵erent times of flight t, can be fitted to this expression and from the fits the temperature T , the trap frequency ! and initial trap width 0can be obtained. The frequency obtained corresponds to only to the frequency of the magnetic trapping potential !mag, because the dipole potential is absent during TOF. From the initial temperature and width, the total trap frequency (!tot) can be calculated and from equation 17 the dipole trap frequency (!dip) can be extracted:

!dip= q

!2

tot± !mag2 (29)

Figure 3: Scheme of laser system for the crossed beam optical dipole trap. The

modulation of the 1070 nm laser (IPG, black) is done with an acusto-optic modulator (AOM, red) controlled by our custom electronics

3.2

Modulation of dipole trap

To created a temporospatial modulated optical dipole trap, the beam of a 1070 laser1 _{is displaced along the horizontal plane of the system by an} acusto-optic modulator2_{fed by a custom drive signal (Figure 5). By choosing between} a triangular or sinusoidal waveform we can influence the shape of the modulate

1_{IPG YLR-200-WC 200 W Ytterbium Fiber Laser} 2_{Gooch & Housego 3110-191}

dipole potential. The modulation depth (fdepth) will determine the width of the trap and the modulation frequency (fmod) will a↵ect the dynamics of trapping. We will discuss these points in more detail in the results section.

VCO

ZX95-850W-S+ @10V

VCO

ZX95-850W-S+ @15V

Mixer

ZEM-2B+ Attenuator ZX73-2500-S+ 10-2500 mhZ

Switch

ZYSWA-2-5DR Preamp ZFL-500LN+ 10-500 MHz Amplifier ZHL-1-2W-S+ 5-500 MHz Signal Generator 0-4 Mhz Filter ZFL-500LN+ 10-500 MHz AOM

Figure 4: Diagram of controller circuit for AOM used for spatial modulation of

dipole trap. All components were obtained from Mini-circuits.

0.0 0.1 0.2 0.3 0.4 0.5 106 108 110 112 114 Time[μs] fAOM [MHz ] 1/fmod fcen fdepth

Figure 5: The two modulated signals used to drive the AOM in our experiments

This drive signal was generated by the system shown in Figure 4. Two high frequency voltage controlled oscillators3_{were used to generate the carrier signal}

at the operating frequency of the AOM (fcen = 110 MHz). This combination allowed the generate very accurately the required frequency and a wide tuning range by combining there outputs in a mixer. Furthermore, the noise in the signal is reduced in this combination which prevents the prescence of frequencies that could lead to unwanted e↵ects in the trapping of the atoms.

For the modulation of the central frequency, the control voltage of 15 V of one the VCO’s is controlled by a frequency generator (Rigol DG4162). In our experiments a sinusoidal and a triangular waveforms were used with an amplitude fdepthand a frequency fmod. A low-pass filter is used to reduce the lower harmonics in the signal and then the signal is amplified before the AOM.

4

Results

4.1

TOF measurements

x z y 0ms 3,5 ms x z y 0ms 3,5 ms

Figure 6: Expansion of atomic cloud during TOF observed from a side view (top

row) and and fittings of projections of cloud intensity in the z (orange, axial) and y (blue, radial) axis

The first results obtained corresponded to the characterization by time-of-flight (TOF) of the atomic cloud in an unmodulated dipole trap. In Figure 6 we can observe the expansion of the cloud after the dipole trap is turned o↵. Expansion occurs faster along the radial (y) axis of the cloud, due to a residual negative curvature of the magnetic field, as we will discuss later. This measurement has been performed for a sample which has not been evaporatively cooled, thefore we can use the properties of this cloud as reference values for our further evaporation experiments. This initial atom cloud is not thermalized due to the very small scattering length at zero magnetic field, therefore we have to estimate two ortogonal temperatures for the cloud.

From the fit to the projection of the images and the time evolution of the cloud size, the number of atoms and temperatures can be obtained as described in 3.1. Table 1 shows the most important values obtained from this fittings. 4.1.1 Magnetic field frequencies

During the expansion of the cloud in a time-of-flight experiment, high mag-netic fields are mantained as swithching them o↵ would lead to acceleration of

Table 1: Sizes, temperatures, number of atoms and collision rate for initial atom cloud

y(µm) z(µm) Ty(µK) Tz(µK) NA (⇥103) ela(103 s 1) 25.7±0.9 172±1 297±10 252±12 452±35 13±7

Figure 7: Time-of-flight images of one atom cloud expanding under the influence

of the magnetic field of the Feshbach coils. Top row images correspond to the cloud seen through its axial direction, and the corresponding lateral images are show in the bottom row

the atoms. Although the fields are very homogeneous, a small curvature is still present due to imperfections in the Fesbach coils. To analyze the evolution of the cloud under influence of these fields curvatures, pictures of the cloud are taken along its axial and a radial direction (Figure 7).

0 1 2 3 4 5 0 50 100 150 200 ttof[ms] σ [μ m ] 0 1 2 3 4 0 50 100 150 200 250 300 350 ttof[ms] σ [μ m ]

Figure 8: Evolution of cloud size (red: y-axis, blue: x-axis, pink: z-axis) during

time-of-flight experiments and corresponding fittings from a lateral (left) and axial (right) absorption images

When analyzing the evolution of the cloud size, oscillations are observed in the x axis (radial), an indication of positive curvature (confinement) of the magnetic field in this axis. In the second radial axis (y), the cloud expands continously due to a negative curvature (anti-confinement). This behavior can be clearly distinguished in Figure 8. For the expansion in the axial direction (z-axis), a weaker confining potential is present. The frequencies of this residual magnetic potentials is obtained as described in Section 3.1.2 and the results shown in Figure 9 for di↵erent dipole trap powers. The frequencies for each cloud axis are constant as expected, confirming the adecuacy of our fitting model. The lower confining strength of the potential along the axial direction

(pink) is reflected on a trapping frequency approximately half of the radial trapping potential (blue).

0.2 0.5 1 2 0 20 40 60 80 100 120 Pdip[W] fmag [Hz ]

Figure 9: Magnetic field frequencies for the trapping axes (blue: x-axis, pink: z-axis), anti-trapping (red:y-axis)

4.2

Evaporative cooling

After characterizing our initial cloud, we studied the evaporative cooling with di↵erent evaporation ramps. Evaporation is performed at a magnetic field of 780 G, where the scattering length of 6Li is of around 6000 Bohr radii [6] between the two trapped states, which allows for rapid re-thermalization during evaporation. The ramp sequence used is shown below:

Figure 11 shows the number of atoms, temperature and final phase space density for the di↵erent dipole trap powers of the last evaporation step and Table 2 summarizes the values for the lowest power. We report only one tem-perature for the atoms, as the atoms are thermalized after evaporation. Both the number of atoms and temperature decreases as expected, however we ob-serve that their values change faster at lower dipole powers. This behavior could be explained if the truncation parameter(⌘ = Udip/kbT ) of our evaporation is reduced at lower powers, which could accelerate the evaporation process [8]. The truncation parameter defines the fraction of atoms than can leave the trap, exp( ⌘), therefore smaller values indicates a more favorable evaporation. In our results (Figure 11), ⌘ actually increases at lower powers. However we need to take into account that at these powers the potential depth due to the dipole trap is very shallow and therefore other forces, such the ones due to the mag-netic fields and gravity start to take a role in trapping of the atoms. Therefore these higher values of ⌘ might be in reality much lower and could help explain the acceleration of evaporation at lower powers.

The quantity of interest in evaporative cooling is the phase space density (PSD) and the efficiency parameter . In our case the maximum value achieved

49 ms 1.0 ms 0.2 ms

671 nm

a.i. a.i. a.i. 1070 nm dipole trap Ilfc cool o.p. 900 ms 400 ms 450 ms 700 ms 0.2 ms 100 ms dipole trap capture to 4.0 W to 30 W to 8.3 W to Pdip ttof 100 W Iufc IMOT 0 A 35 A -200 A 250 A 259.5 A (mI = 1), 228.5 A (mI = 0) 170 A (mI = 1), 155 A (mI = 0) 164 A 122 A 3x50 μs (a.i) + 2x50 ms

Figure 10: Overview of steps and settings during evaporative cooling for the

mag-netic field coil currents (Imot, Iuf c, Ilf c), the 671 nm laser used for cooling, pumping

and imaging and the power at the dipole trap. Extracted with permission from [7] Table 2: Temperatures, number of atoms, phase space density and efficiency param-eters of evaporated atom cloud at a dipole trap power of 0.13 W

T (µK) NA (⇥103) P SD

0.18±0.5 9.4±2.8 0.31±0.42 3.4

was 0.31_{± 0.42. This phase space density (PSD) is close than the maximum} value for a Fermi gas (=1). This could lead to the formation of molecules (e.g. a Li2), which due to their bosonic nature could eventually condense in a Bose-Einstein condensate. Such molecules have been already seen in our trap.

The efficiency parameter , defined as d(ln(PSD))/d(ln(Natoms)), was ob-tained by a linear fit to the data in Figure 11 d. The value obob-tained is 3.4, which means that for every order of magnitude of atoms evaporated, 3.4 orders of magnitude are won in the phase-space density. At the end of the evapo-ration, less than ten thousand atoms remain. About one order of magnitude more atoms are preferred in our system, otherwise the efficiency of symphatetic cooling of the heavier Yb+ is limited due to the lower number of collisions. Furthermore, if reaching lower temperatures is desired, e.g. to enter the s-wave scattering regime of Li/Yb, a higher number of atoms is needed to continue cooling by evaporative cooling. Finally, practical aspects as the signal to noise ratio of measurements will be improved for increasing number of atoms.

For these reasons, increasing the initial phase space density either by re-ducing the initial cloud temperature or increasing the initial atom density is desired. This could be achieved by precooling by free evaporation of the start-ing cloud, or by modifystart-ing the potential of our trap, such that more atoms can

(a) (b) 0.1 0.5 1 5 10 50 100 0.5 1 5 10 50 100 Pdip[W] T [μ K ] (c) _(d) 1 ×104 _{5 ×10}4_{1 ×10}5 _{5 ×10}5 10-6 10-4 0.01 1 Natoms PSD

Figure 11: a) Final number of atoms, b) cloud temperature, c) truncation parameter, and d) phase space density versus atom number after evaporative cooling for di↵erent

final laser powers. The starting cloud contained_{⇠ 450.000}6Li atoms with an initial

temperature of 450 µK

be captured from the MOT. This second approach is discussed in Section 4.3, where we discuss spatial modulation of the optical dipole trap potential. 4.2.1 Collision rates

The rate of elastic collisions ( ela) in our atom cloud is another parameter that can be used to optimize the evaporation. High values are preferred as they allow fast rethermalization of the cloud during evaporation. For a cloud of Li atoms, the goal would be to evaporate in a regime where the collision rate is constant or increases during evaporation, also known as evaporation in the runaway regime[9, 10]. The collision rate can be calculated from the scattering length (a), the atom temperature (T), the peak atom density (n0) and the atomic mass (m) of Li in our cloud:

ela= 4⇡a2n0 r

kBT

m (30)

For the chosen evaporation conditions, we see that the collision rate (Figure 12) drops after the second evaporation ramp from the initial value of (13± 7) ⇥

103_s 1_{(see Table 1). It remains approximately constant down to the last ramp} for values higher than 1 W and then the rate plummets down.

0.1 0.5 1 5 10 50 100 1000 2000 5000 1 ×104 2 ×104 Pdip[W] Γela [s -1 ]

Figure 12: Elastic collision rate at di↵erent dipole trap powers.

In order to improve evaporative cooling, a option is to keep the truncation parameter ⌘ constant during evaporation. This ensures that the lowering rate of the potential depth slows as the collision rate decreases, which allows efficient evaporation at lower temperatures, To achieve this the laser intensity has to be reduced from its initial power (Pi) as follows [11]:

P (t) = Pi

(1 + t/⌧ ) (31) where and ⌧ depend on the desired truncation rate and the initial elastic collision rate.

4.3

Spatial modulation of optical dipole trap

4.3.1 Numerical calculations of trap potentials

The results of the simulation of the time averaged dipole trap potentials are shown in Figure 13 a. In first instance a sinusoidal modulation (red plot) was considered due to the simplicity of generation of such a signal at the required frequencies. For a modulation depth of 10 MHz a double well potential develops in the radial direction, with a lower potential depth at the center of the trap. This situation is expected to be unfavorable during trapping of atoms from the MOT, as the atom cloud density is the highest at the center of the latest. In contrast, a triangular modulation (blue plot) creates a flat potential at the center of the trap while approximately 1 mK deeper, although slightly narrower. For a lower modulation depth of 5 MHz (Figure 13 b), the radial width is reduced for both input waveforms as expected, and the triangular modulated potential generates again a deeper potential.

(a) Radial potential 10 MHz (b) Radial potential 5 MHz

(c) Axial potential 10 MHz (d) Axial potential 5 MHz

Figure 13: Time-averaged dipole trap potential in the axial and radial direction for a sinusoidal (red) and a triangular (blue) modulation at a frequency of 4 MHz with a modulation depth of 10 MHz (a,c) and 5 MHz (b,d). The unmodulated potentials are shown in orange

In the case of the axial potential (Figure 13 c,d), the profile becomes assy-metric due to the overlap of incoming and the retroreflected beam in the dipole trap configuration. The center of the potential shifts further to the entry port of the incoming beam for the sinusoidal and triangular modulation by approxi-mately 230 µm and 270 µm for a 10 MHz modulation and 90 µm and 140 µmfor 5 MHz modulation depth. For comparison, the size of the MOT cloud before trapping is between 390-510 µm.

4.3.2 Ramsey spin echo experiments

To measure the change of the potential depth of the optical dipole trap due to modulation, the shift of the levels of171Yb+ ion due to the electric field of the trap can be measured. This e↵ect is also known as di↵erential AC Stark shift, and occurs due to the interaction of the electric field of the dipole trap with the electric dipole moment of the ionic states.

The measurement is done by loading the 171Yb+ ions at the center of the dipole trap and measuring the transition frequency from the levels

2_S

1/2|F = 0, mF = 0i to 2S1/2|F = 1, mF = 0i with microwave Ramsey spec-troscopy with fluorescent detection. The sequence of pulses used in the experi-ment are shown in Figure 14. A first ⇡/2 pulse is used to prepare the coherent superposition state 1/p2(_{|0, 0i + |1, 0i). The optical dipole trap is then turned} on which induces a light shift leading to accumulation of a relative phase

di↵er-2ms

Dipole trap 369nm Laser

π/2 π π/2 _Microwave12.6 GHz

Figure 14: Pulse sequence used during the Ramsey spectroscopy measurements

of the di↵erential AC Stark shift of the 171Yb+ transitions. Detection is done by

measuring the fluorescence from the 2P1/2|F = 0, mF = 0i state after excitation of

the2_S

1/2|F = 1, mF = 0i with a 369 nm laser pulse

ence in the ion state:

= p1

2(|0, 0i + e

ı ac_{|1, 0i) (32)}

ac= ( E_|1,0i E_|0,0i)tdip

~ (33)

Finally a sequence of a ⇡ abd ⇡/2 pulse are applied to prevent dephasing of the superposition state, and the phase of the last pulse is scanned through a full period to obtaing Ramsey fringes.

The results of the Ramsey photon echo experiments are shown in Figure 15. From the measured phase shifts ( ) the calculated frequency shifts ( !) were 24± 2 Hz and 12 ± 2 Hz for the ion in the unmodulated and modulated dipole trap, respectively. The reduction in the Stark shift after modulation indicates a decrease of the electric field experienced by the atom, as expected from the reduction of the potential depth obtained in our calculations. Although the detection of a Stark shift confirms that the ions experience the electric field of the dipole trap, it is possible the center of the dipole trap it is not rightly aligned with the center of the ion trap. To obtain a better indication of the location and shape of the trap more experiments will need to be performed, i.e. spectroscopy at di↵erent ion positions, or calculations of shifts for di↵erent potential depths. 4.3.3 Optimization of dipole trap modulation

After choosing a triangular modulation for our dipole trap, we evaluated the e↵ect of the modulation depth (fdepth) and modulation frequency (fmod) on the atom number and temperature of the initial cloud for two di↵erent dipole laser powers. As we have seen in Section 4.3, for higher modulation depths the trap becomes wider and shallower. The widening of the trap is reflected on a increasing inital number of atoms when going from a modulation of 0 MHz up

dipole laser on

dipole laser modulated no dipole laser 0 π 2 π 3 π 2 2 π 0.0 0.5 1.0 1.5 phase(rad) det . rat e (kHz ) p h o to n c o u n ts (a .u .)

Figure 15: Detection rate of fluorescent emission of an171Yb+ion in the center of

the dipole trap. The maximum detection rate occured for a phase shift of the second

Ramsey pulse of 1.4±1 rad (dipole trap without modulation) and 0.73±0.1 rad (dipole

trap with modulation)

to⇠6 MHz. The highest number of atoms obtained was (7.9 ± 0.9)⇥ 105_{. At a} modulation depth of 12 MHz, the number of atoms suddenly drops, which most likely occurs due to cropping of the beam for high displacements which reduces the width of the trap. Another explanation is that the shift of the center of trap a↵ects the collection of atoms from the MOT, as pointed out earlier. For a laser power of 70 W, only a slight increase followed by a reduction on the number of atoms is observed. The lower trap depths for this laser power will limit trapping hot atoms from the MOT. Therefore increasing the size of the trap by modulation (which reduces the depth) will unlikely help increasing the number of captured atoms.

Trapping of atoms from the MOT by the unmodulated dipole trap (Modu-lation depth = 0 MHz), leads always to heating due to the poor mode matching between the magnetic and dipole potentials. This leads to heating of the atoms upon capture, from a temperature between the 100-200 µK at the MOT [7] to⇠ 400 µK for the 70 W and⇠ 800 µK for the 150 W dipole trap. For this highest laser power, the potential is steeper which worsens the mode matching. But by modulating the trap, the potential becomes less steep and heating of atoms is reduced, as observed by the reducing atom temperature for increasing mod-ulation depth. For the dipole trap of 70 W, the temperature reduction due to modulation is less pronounced as the mode match is better at this laser powers. Finally, we observe that the temperatures for the unmodulated 70 W dipole trap and 150 W dipole trap modulated at 12 MHz are similar. This indicates we have achieved similar trap depth for both configurations.

70 W 150 W 0 2 4 6 8 10 12 300 400 500 600 700 800 Depth [MHz] A toms (x1000 ) 70 W 150 W 0 2 4 6 8 10 12 0 200 400 600 800 Depth [Hz] Tx [μ K ]

Figure 16: Average number of atoms and temperature in initial atom cloud for

di↵erent modulation depths (fdepth) and two laser powers. The average is calculated

from the values obtained for modulation frequencies (fmod) of 4,2,1 and 0.5 MHz.

We have also studied the e↵ect of modulation frequency (fmod) on the num-ber of atoms and temperature, as seen in Figure 17. The numnum-ber of atoms generally drops down at the lowest modulation frequencies. A time average potential requires that the modulation frequency is much higher than the trap frequency of the atoms, such that the atoms cannot follow the fast modulation. The highest trapping frequencies at the beginning of the evaporation ramp are a few tenths of KHz, so we aim for modulation frequencies much higher than 100 kHz. However, parameteic excitation may still occur for low modulation frequencies

No e↵ect is seen for the temperature of the atoms, as this last one will be mainly determined by the trap depth which is insensitive to the modulation frequency. 0 MHz 4.7 MHz 6.2 MHz 12 MHz 0 1 2 3 4 300 400 500 600 700 800 900 Frequency [MHz] # A toms (x1000 ) 0 1 2 3 4 200 400 600 800 1000 Frequency [MHz] T [μ K ]

Figure 17: Initial number of atoms and temperature for di↵erent modulation

fre-quencies and modulations depths at the maximum dipole laser power

4.4

Evaporative cooling with modulated dipole trap

After optimizing the loading parameters of the dipole drap, we have per-formed evaporative cooling of the initial cloud from two traps with similar depth, 150 W with fdepth = 12 MHz, fmod = 4 MHz and 75 W unmodulated. Modulation is ramped down in the first evaporation step (See Appendix II for

No modulation With modulation 3.0 3.5 4.0 4.5 0 10 000 20 000 30 000 40 000 Pdip[W] A toms Modulation Frequency: 4 MHz No modulation With modulation 3.0 3.5 4.0 4.5 0.1 0.5 1 5 Pdip[W] T [μ K ] Modulation Frequency: 4 MHz

Figure 18: Number of atoms and temperature after evaporation with and without modulation for di↵erent end points of the evaporation ramp

evaporation ramp), to compress the atom cloud and increase density and the atom collision rate (See Section 4.2.1).

We see that increasing the initial number of atoms by modulation leads to ⇠ 2 times more cold-atoms with similar temperatures as for the unmodulated initial cloud. Furthermore, the values are similar to the ones obtained for evap-oration down to 0.13 W for an unmodulated dipole trap at 100 W (See Table 2). Further evaporation to trap powers below 1 W of the initial modulated trap could lead then to even colder atoms and Bose-Einstein condensation could be achieved.

Table 3: Initial and final cloud condtions during evaporation. Final values

corre-spond to a dipole trap power of 2.8 W

Ni(⇥1000) Ti(µK) Nf(⇥1000) Tf(µK) Modulated 783_±71 432_±20 10_±3 0.16_±0.07 Not Modulated 446_±17 412_±19 3_±2 0.12_±0.13

5

Conclusion and Outlook

We have demonstrated that through spatial modulation of an optical dipole trap we can capture two times more atoms from the MOT without a significant increase of initial temperature. This was the primary goal of the project, as this paves the way down for colder temperatures in the combined ion-atom system. Evidently, a further improvement would be to also modulate the dipole trap orthogonally to our current modulation, but first space constraints in the setup would need to be solved.

Furthermore, we have improved our fitting model such that we are able to obtain cloud temperatures out of time-of-flight experiments in the presence of an inhomogeneous magnetic field. From the information obtained from this

fittings we get more insight on the process of evaporative cooling, by being able to estiamte collision rates and efficiency parameters. The efficiency parameters of 3.4 found for the current evaporation ramp is reasonable, however we have presented some suggestions to improve the ramp scheme.

We have also determined the type of confinement due to the residual curva-tures of the magnetic fields in our experiment. From the confinement frequencies obtained and compensation scheme could be implemented to cancel the field cur-vatures. This could be the usage of another electric field from a laser beam to eliminate the anti-trapping potential along the radial axis. This would allow to observe BEC condensation during time-of-fligt experiments, by slowing down the expansion of the cloud.

References

1_{M. Tomza, K. Jachymski, R. Gerritsma, A. Negretti, T. Calarco, Z. Idziaszek,} and P. S. Julienne, “Cold hybrid ion-atom systems”, 1–60 (2017).

2_{R. Grimm, Ultracold Fermi gases in the BEC-BCS crossover: a review from} the Innsbruck perspective, April 2007 (2007).

3_{C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach resonances in} ultracold gases”, Reviews of Modern Physics 82, 1225–1286 (2010).

4_{J. Jannis, “Cold Li–Yb + mixtures in a hybrid trap setup”, PhD thesis} (Uni-versitetit van Amsterdam, 2018).

5_{R. Weisman, M. Majji, and K. T. Alfriend, “Solution of Liouville ’ s Equation} for Uncertainty Characterization of the Main Problem in Satellite”, CMES 111, 269–304 (2016).

6_{G. Z¨urn, T. Lompe, A. N. Wenz, S. Jochim, P. S. Julienne, and J. M. Hutson,} “Precise Characterization of Li 6 Feshbach Resonances Using Trap-Sideband-Resolved RF Spectroscopy of Weakly Bound Molecules”, Physical Review Letters 110, 135301 (2013).

7_{H. F¨urst, “Atom-Ion Quantum Systems”, PhD thesis (Universiteit van} Ams-terdam, 2018).

8_{M. Kohnen, “Ultracold Fermi Mixtures in an Optical Dipole Trap”, PhD thesis} (University of Heidelberg, 2008), p. 89.

9_{F. Schreck, “Mixtures of ultracold gases: Fermi Sea and Bose-Einstein} con-densate of Lithium isotopes”, PhD thesis (Universit´e Paris VI, 2002), p. 195. 10_{W. Ketterle, and N. V. Druten, Evaporative Cooling of Trapped Atoms, Vol. 37}

(1996), pp. 181–236.

11_{S. R. Granade, M. E. Gehm, K. M. O’Hara, and J. E. Thomas, “All-Optical} Production of a Degenerate Fermi Gas”, Physical Review Letters 88, 4 (2002).

6

Appendix I: Cloud expansion in an anti-trapping

potential

For an anti-trapping potential, the movement of the atoms is described by ¨

x = !anti2 x. Solving analytically for any initial condition ~q0= (x0, v0): ~ q(t) = ✓ x(t) v(t) ◆ = cosh !antit sinh !antit !anti

! sinh !antit cosh !t ! ✓

x0 v0 ◆

= M ~q0 (34)

The evolution of the atomic cloud at any time t is

⇢(x, v, t) =⇢0(x0(x, t), v0(v, t)) det ⇣ d~q0 d~q ⌘ (35) =⇢0(x0(x, t), v0(v, t)) det M 1 = ⇢0(x0(x, t), v0(v, t)) (36) because det M 1_{= 1. Then by using (2) in (1), we obtain:}

⇢(x, v, t) = p1 2⇡ m p 2⇡kbT ⇥ exp h _x2 2 ⇣ cosh2 !antit 2 0 +! 2 antim kT sinh 2 !antit ⌘i ⇥ exph v 2 2 ⇣ sinh2_! antit !2 2 0 + m kT cosh 2 !antit ⌘i ⇥ exph xv₂ sinh !antit cos !antit⇣ 2!anti

m kT

2 !anti 02

⌘i To obtain the width of the distribution, we calculate the uncertainity in x:

(t) = q hx2_{i hxi}2 ₍₃₇₎ with:Dx2E= Z v Z x x2⇢(x, v, t) dxdv (38) hxi = Z v Z x x⇢(x, v, t) dxdv (39) And after calculating the integrals:

(t) = s kbT sinh2!antit m!2 anti + 2 0cosh 2_! antit (40)

7

Appendix II: Evaporation ramp used for

mod-ulated dipole trap

49 ms 1070 nm dipole trap f depth 600 ms 400 ms 350 ms 700 ms dipole trap capture to 4.0 W to 30 W to 8.3 W to Pdip 100 W 12 MHz 0 MHz 1070 nm dipole trap

Figure 19: Ramps for dipole trap power and modulation depth used during evapo-rative cooling of a modulated dipole trap