1 0 AFM
3.3 Detection techniques
Throughout this thesis, we investigate the equilibrium physics of the Hubbard model. After loading the atomic cloud into the optical lattices. We allow for a variable equilibration time before increasing the lattice depths of all three optical lattices within a timeτmuch faster than the characteristic tunnelling time,i.e. τ h/t. In this way, the atomic density distribution is said to befrozen, meaning that the tunnelling dynamics happens at a much slower time, which can be neglected in our subsequent detection phase. In this final section, we introduce the necessary experimental protocols for coherent control and manipulation of the atomic ensemble during the detection.
3.3.1 State manipulation
Fundamentals of a two-level system We first consider a single atom under a coherent electric fieldE(t) = E0cos(ωt). Given an appropriately chosen driving frequencyω, the atomic energy levels can be considered as a two-level system involving only the relevant energy states. The full time-evolution of such a two-level system can be written as a state-vector
|ψi=ag(t) |gi+ae(t) |ei. (3.14)
Here|giand|eiare the ground and excited state of the atom, withag(t)andae(t)being complex and time-dependent coefficients. The Schrödinger equation for this two-level system in the rotating frame can be written as
d dt
ag(t) ae(t)
=
0 12Ω0(t)
12Ω0(t) ∆(t)
ag(t) ae(t)
. (3.15)
The off-diagonal terms contain the so-called Rabi frequency Ω0 ∝ |E0|2, which describes the light-matter interaction between the two energy levels. Here, we defined∆(t) =ω0−ω(t)as the detuning of the electric field with respect to the two-level transition energyω0. The populations of ground and excited state are given by|ag(t)|2and|ae(t)|2, respectively.
Rabi oscillation The solution of Eq. (3.15) results in a coherent oscillation between the ground state and excited state population. For two-level systems initially fully populated in the ground state, i.e. |ag(t=0)| =1, the time evolution of the ground and excited state populations are given by
Pg(t)= |ag(t)|2=cos2 1
2
∫ t
0 Ω(t0)dt0
,
Pe(t)=|ae(t)|2=sin2 1
2
∫ t 0
Ω(t0)dt0
.
(3.16)
The integral in Eq. (3.16) is the pulse area andΩ(t) = q
Ω20(t)+∆(t). For∫t
0 Ω(t0)dt0 = π, the driving field is said to realise aπ-pulse, at which the probability of occupying the excited state is unity. And for∫t
t=0Ω(t0)dt0=π/2, we have aπ/2-pulseat which the state is in a equal superposition of the ground and excited states.
Landau-Zener sweep Another way to transfer an atomic state coherently from one to another is to use a frequency sweep of the driving fieldE(t). When the rate of the frequency sweep is slow with respect to the energy gap between two states, the population can be transferred between the two dressed states with high fidelity. The driving induces an avoided crossing to make the adiabatic transfer possible. In the limit of infinitely-slow sweep,i.e. |d∆/dt| →0, known as the Landau-Zener (LZ) limit, the following transfer probability from state|aito|biis given by
Pa→b=1−exp − π2Ω20 2|d∆(t)/dt|
!
. (3.17)
One example of a common LZ sweep used in the detection phase is the coherent transfer between magnetic hyperfine states. We take the energy states of40Kin theF =9/2 manifold as an example.
At a magnetic field of 189(G), typical energy difference between hyperfine states ranges from
∆E/h=45 MHz to∆E/h=52 MHz. The LZ sweep can thus be achieved using a radio-frequency (RF) pulse. We linearly sweep the detuning of the RF pulse across the resonant frequency, with a window of 175 kHz in a duration of 2 ms. Here, the amplitude of the RF pulse follows a smoothed box envelope to prevent non-adiabatic projection. Further details of pulse shaping and coherent manipulation can be found in [55,72].
3.3.2 Single-plane tomography
One critical aspect in investigating a system with multiple two-dimensional layers is to ensure that the system being probed only contains information from one single two-dimensional layer. This is in particular important in local measurements such as density and correlations. Simultaneously probing more than one uncorrelated two-dimensional planes would result in the undesirable, uncorrelated background, which is hard to remove in the analysis.
We deploy a tomographic approach in resolving a single layer of our entire atomic ensemble. The basic idea is to utilise a vertical magnetic field gradient, such that the resonant transition frequencies are shifted due to the Zeeman effect. Then, a specifically shaped RF pulse is shone onto the system to address one single layer. In order to achieve this addressing, the RF pulse needs to fulfil a more stringent requirement compared to a global LZ sweep discussed before. For one thing, we require the frequency spread of the RF pulse being sufficiently narrow with respect to the transition energy difference between neighbouring two-dimensional layers. At the maximum gradient strength that the fast Feshbach coils can reliably produce without overheating (dB/dz≈33.3 G/cm), the frequency shift is approximately 641(14)Hz per plane for the transition between|F =9/2,mF =−5/2iand
|F =9/2,mF =−3/2i.
However, it is also beneficial to address the single layer with the highest fidelity as well as stability.
Therefore, this motivates the use of RF pulse with a flat-top frequency feature. We deploy the so-called hyperbolic-secant (HS1) pulse, which can be generated with a combination of amplitude and frequency modulation,
EHS1(t)=E0sech "
Ctrunc 2t Tpulse−1
! # ! ,
∆HS1(t)= ∆0 2 tanh
"
Ctrunc 2t Tpulse−1
! # ! .
(3.18)
−0.004
−0.003
−0.002
−0.001
0.000
0.001
0.002
0.003 νSlicing(MHz)
0 1000 2000 3000 4000 5000 6000 Atom Number
With superlattice evaporation Without superlattice evaporation Vertical magnetic
field gradient
Figure 3.12:Single-plane tomography. Tomography spectra with and without the superlattice evaporation described in Sec.3.8. A magnetic field gradientdB/dz=33.3 G/cm provides the relative Zeeman shifts in transition frequency between hyperfine states. Solid lines are fit with multiple Gaussian peaks overlapped, from which we extract a frequency spacing between neighbouring planes of∆ν=641(14)Hz.
We optimise the final shape of the frequency profile of the HS1 pulse using a truncation parameter Ctrunc=4.5 and a pulse windowTpulse=7 ms.3 In Fig.3.12, we show the tomography spectra of our two-dimensional gases. With the requirement of addressing a two-dimensional layer, one powerful utility of the vertical superlattice can be fully justified. Without emptying one of the sublattices as shown in Fig.3.8, the slicing spectra typically consist of overlapping resonance peaks. To ensure data quality, only data points at the peak are selected for post-analysis in all of the experiments. With the additional evaporation in the optical superlattice, the “effective” plane separation is doubled, therefore, allowing the use of RF pulse with wider windows for improved data quality and data acquisition rate.
3.3.3 High resolution in-situ imaging
At last, we are in the position to discuss the final stage of the experiment, namely thein-situimaging of the atomic distribution. We deploy conventional absorption imaging technique. We first outline the modified Beer-Lambert law used to deduce the optical density of the sample.
3With an upgraded arbitrary waveform generator, we can now generate flat-top pulse in larger time window of Tpulse=14 ms for sharper frequency response.
The modified Beer-Lambert law To image the atoms, we make use of theD2transition of with σ−-polarised light (see Fig.3.2b). The Beer-Lambert law describes the degradation of light intensity when propagating through a dilute medium of densityn(x,y,z)is given by [89],
dI(x,y,z)
dz =−σ0n(x,y,z)I(x,y,z). (3.19)
Here,σ0is the scattering cross section of two-level atoms in the vicinity of theD2transition andzis the along the light propagating direction. Integrating Eq. (3.19) along thezdirection results in an optical density based on logarithm of fraction of absorbed light intensity,i.e.
σ0n(x,y)=−ln
If(x,y) Ii(x,y)
. (3.20)
However, due to saturation effects in Eq. (3.21), the photon scattering cannot take place efficiently in the case of high imaging intensity, resulting in the intensity-dependent scattering cross section
σ(I)= σ0 1+(2∆
Γ)2+s. (3.21)
Similar to the notations used in Sec. 3.3.1,∆=ω−ω0is the detuning of imaging light with respect to the transition frequencyω0. Γis the natural linewidth of theD2line and the saturation parameter s = I/Isat is the ratio of the imaging intensity to the saturation intensity. The relevant properties of the D2transition are provided in Table. 3.1. Eq. (3.19), therefore, needs to be modified with intensity-dependent scattering cross section which takes into account the effect of saturation. With this in mind, the modified Beer-Lambert law reads [90]
σ0n(x,y)=−αln
If(x,y) Ii(x,y)
+ Ii(x,y) −If(x,y)
Isat0 . (3.22)
The calibration procedure for obtaining effective camera counts corresponding toIsat0 can be found in [73]. To this end, we deploy an imaging pulse of 10µs, with intensityI ≈1.5Isat. The imaging setup along the vertical direction is shown in Fig.3.13.
Frame transfer scheme To record multiple images in a single experimental run within a short period of time, we deploy the fast kinetics mode of the Andor iXon888 camera. The camera contains a CCD sensor with resolution of 1024 pixel×2048 pixel. Only an area of 1024 pixel×512 pixel is illuminated, where the rest of the chip serves as storage area as shown in Fig.3.13b. Upon triggering, the CCD is exposed to imaging light for a timeTexp. After this, the image acquisition pauses, and the recorded charges are shifted vertically by 512 pixels such that a new image can be recorded.
This process repeats twice, giving access to two additional images. To this end, three consecutive exposures are taken.
Before recording the images, we have the to-be-recorded atomic cloud typically residing in the two lowest hyperfine states|F =9/2,mF =−9/2iand|F=9/2,mF =−7/2i. A microwave pulse is first shone onto the cloud to transfer the atoms in the|F =9/2,mF =−9/2istate to theF =7/2 manifold, which is insensitive to the imaging light used later. Next, the atoms in the|F =9/2,mF =−7/2istate is transferred to the|F =9/2,mF =−9/2ivia a LZ sweep. Subsequently, the first exposure records the atomic ensemble in the|F=9/2,mF =−9/2iwith an imaging pulse duration ofτimg=10µs.
z
y x
f = 200 mm To CCD camera
f = 350 mm f = 8 mm
1024 pixels
2048 pixels pixels
Exposure 1 Exposure 2 Exposure 3
masked area
CCD chip
chip cleaning transfer transfer readout
MW amplitude
RF amplitude CCD exposure imaging light
a
b c
512
Figure 3.13:Vertical imaging setup and fast kinetics scheme. aMain imaging setup along the z-direction.
bOutline of the CCD chip of the Andor iXon888 camera.cDetails on the fast kinetics scheme. After each exposure, the charges accumulated are shifted vertically down in 6.5µs into the masked region for storage. b andcare adapted from [73].
After that, atoms in theF=7/2 manifold are transferred back to a|F =9/2,mF =−9/2i, followed by the second exposure. Finally, a third image is taken with no atoms remained in the system. This serves a “bright” image for computing the optical density using Eq. (3.22).4
4In addition the bright image, we typically record a series of images (up to∼1000 reproductions) at which no light is shone onto the camera. This so-called dark image is then used in the post-analysis to eliminate the effect of stray charges accumulated during the image acquisition.