Digital filter cancelling mechanical resonances

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On-chip transport of atoms

The first magnetic trap

The first magnetic trap deserves a few more words, as it determines the available atom number and phase-space density for later manipulation in magnetic traps. As I mentioned, the first magnetic trap should be a harmonic trap, which is most easily realised by a dimple trap. We use the maximum current in the crossed wires (stripline and dimple) together with high current in the macro-I, trying to form the biggest possible trap. A dimple trap is by design anisotropic (elongated in ~x). We can in principle increase the longitudinal (~x) confinement by employing more wires parallel to the dimple, but the potential improvement is marginal as they are also far apart.
In practice, the accessible dimple trap cannot perfectly match the molasses and its for-mation cannot be instantaneous. Therefore the molasses is inevitably perturbed, and some oscillatory dynamics can be observed if we hold it longer in this trap. It turned out better to immediately modify the trap for the next step.
As will be detailed in the next section, the trap for the main transport is formed by the central circular wire (dubbed Omega). To simplify the sequence and to gain in overall efficiency, we directly form the first magnetic trap with the Omega wire, which is parallel to and locally resembles the stripline at the MOT site. With a strong confinement along ~x, it can form a dimple trap with a negligible effect from the curvature. It suffices to modify a bit the compressed-MOT to form the molasses aligned with the Omega.
Current ramp-up From an experimental point of view, even for the ideally instantaneous first magnetic trap, it is practically better to ramp the fields in the presence of electric inductance. Any change in current too abruptly should be avoided, to minimise residual oscillations and risks in high transient fields. A linear ramp is generally not optimal as the second derivative at the beginning and the end of the ramp is still “infinite”.
We most often utilise a “turnOn” function, the lowest order polynomial function with vanishing first and second order derivatives at the beginning and at the end. That is, from t = 0 to the end of the ramp t = T , the field A varies as A(τ) = Ai + (Af − Ai) • (6τ5 − 15τ4 + 10τ3) (2.1) with τ = t/T between 0 and 1. Ai and Af are the initial and final values, respectively.
I shall note that without special notice, all transitions between different traps are realised by ramping consisting fields in this “turnOn” manner. It is possible that the intermediate properties of the trap are not optimal, but it turned out to work well in most situations.
Atom number stability The first magnetic trap is also crucial for obtaining a stable number of atoms. The loading efficiency is strongly affected by the spatial overlap between the trap and the molasses, while the latter is determined by the power balance of the cooling beams. In the long term, power drift of the cooling beams slowly changes the molasses position and hence the loading efficiency, which directly results in a reduction in the final atom number. Therefore in practice it is often the beam balance that is tuned to optimise the final atom number, especially in a daily or weekly practice.

Cold atom preparation and interrogation 

In short term (shot-to-shot) however, the origin of the position fluctuation of molasses is more complicated and its effect on the final atom number more subtle. We haven’t yet thoroughly studied this issue. Another major source of atom number fluctuation comes from the MOT loading. We observe a strong atom number fluctuation in the molasses (up to 10% peak-to-peak), most often related to the temperature in the lab. Luckily, the atom number in the molasses is perfectly correlated with the fluorescence signal at the end of the MOT, which allows us to control the MOT loading time to obtain a stable fluorescence. However, for the atom number in the final magnetic trap, correlation with the fluorescence persists but is not perfect, most likely due to the intervention of the position fluctuation of the molasses.
In the end, the shot-to-shot fluctuation in the final atom number remains considerable (typically 4 ∼ 5% in standard deviation) and cannot be improved by the fluorescence feed-back. This is worse than TACC-1, but we suspect that it is due to the 2D-MOT loading scheme and the increased complexity in the first magnetic trap.

Rotation in a quadrupole trap

The main transport, bridging 1 cm distance between the MOT site and the cavity, is done by trapping and moving the cloud along the circumference of the Omega. Current in the Omega plus a bias field forms a quadruple trap, which can be understood by regarding the Omega as a deformed U. The trap centre is close to the point on the circle to which the bias field direction is perpendicular. One can therefore move the trap along the circle by turning the bias field. A rotation angle θ is achieved by varying Bx and By as Bx = −B0 sin θ, By = B0 cos θ (2.2) where B0 is some field amplitude. Concerning the dynamics, θ is varied in time using the turnOn function (Eq. 2.1). It turns out to be important to have no angular velocity and acceleration at the beginning and the end of the rotation.
Between quadrupole and dimple To switch from the first magnetic trap (a dimple trap), to the quadrupole trap for rotation, one needs to ramp off the dimple current and Bx, because the rotation trap has no bias field in ~x in its initial position (otherwise it will simply rotate). However, during the transition, before Bx is completely off, the trap centre is displaced along ~x (imagine a quadruple trap experiencing a rotation of the total bias field). We can observe centre-of-mass oscillations of the cloud along ~x after the transition.
This is also true when the rotation trap is transformed back into dimple, on the cavity side. It might be possible to find an optimal ramp of the B-field that minimises this “rotation” effect. But in the end, we adopted another approach:
• From the first magnetic trap to the rotation trap, we ramp off Bx and dimple as quickly as possible. It turned out that the residual oscillation is smaller for a faster ramp.
• After the rotation to the cavity side, instead of transforming the quadrupole trap to a dimple trap on the Omega, we first move the trap to the adjacent wire (S3), but we keep it quadrupole using two perpendicular wires (B2 and B4). There is no Bx during the process. Now as S3 is not bended, it is straight forward to ramp into a dimple trap.
Cloud temperature According to calculations [108], the tangential confinement of the rotation trap is determined by the curvature of the wire and by the bias field. With our accessible bias field, the tangential trap depth is about 500 µK. One does have to consider the initial temperature of the cloud to prevent excessive loss during the transport.
One strategy is to do some evaporative cooling before the transport. But we abandoned this approach after realising that performing two evaporations will cost too many atoms and too much time. Therefore we had a guideline that is to perform the transport as quickly as possible. The initial temperature once loaded in the rotation trap, about 50 µK, also turned out not to suffer from high losses.

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“Parallel parking”

As we can see from the chip layout (Fig. 2.1), between the Omega and the stripline there are S3 and S4 in parallel, symmetric on both sides. The cloud is elongated along these wires, therefore moving the cloud sideways was dubbed parallel parking. It is the principal confinement of the trap that is switched between adjacent wires. It turned out to be sufficient to ramp down and up the current in the adjacent wires simultaneously in a “turnOn” manner. As there is a small variation in ~z of the trap centre during the transition, residual oscillation along ~z occurs and depends on the ramping speed. A small delay between the ramp-down and ramp-up can be tuned to minimise this oscillation. The longitudinal (dimple) confinement is kept constant and sufficiently large. The parallel parking is almost lossless between two dimple traps.
However, as I pointed out earlier, these parkings after the rotation from the Omega to the stripline need to be tailored for the transition between quadrupole and dimple. Specifically, we have the first parking from Omega (S2) to S3 in quadrupole traps, simply ramping up current in S3 and B2+B4 (a U-type trap). The second parking to S4 transforms the quadrupole trap to a dimple trap by ramping up the dimple current. Finally the last parking to the stripline is done in dimple traps.
Once the cloud is trapped on the stripline we start the evaporation process, in which the trap is strongly compressed. However, the efficiency of this compression is hard to directly assess in terms of atom loss and adiabaticity, since the atoms in this trap are too hot to be imaged with the time-of-flight (TOF) technique. Therefore, we optimise the compression by measuring the atom number when the cloud is again decompressed into the clock trap. We also observe that the efficiency of the evaporative cooling (assessing the atom number and temperature in the clock trap) depends on the trap of the parallel parking, especially on the longitudinal confinement (dimple current and Bx).

Summary and future improvements

In summary, I give an example of the overall transport efficiency in Table 2.2. There is still room for improvement, notably in the initial loading into the rotation trap, and in the parallel parking process. For the latter, we are in a compromise because a tighter longitudinal confinement seems to be better for the pre-evaporation compression, while a weaker confinement works better for the transition between quadrupole and dimple traps, but a better clever trap ramp is certainly possible.

Evaporative cooling

With atoms in the highly-compressed evaporation trap, we perform forced evaporative cool-ing to reach ultracold temperatures, even to BEC. The key is to achieve high collision rate for thermalisation and a well adapted RF radiation to remove the hot atoms in time. With atoms in |1, −1i, the RF radiation is blue-detuned from the Zeeman splitting between |1, −1i and the untrappable |1, 0i, capable of expelling atoms with higher energies, effectively reducing the trap depth.
By sweeping the RF frequency closer to the Zeeman splitting, while allowing sufficient time for the atoms left in the trap to reach thermal equilibrium, the atoms are cooled down. It can be shown that the optimal frequency sweep is very close to an exponential function in time [113]. In addition, the power of the RF radiation is ramped down at the end of the frequency ramp. Previous optimisations arrived at a two-step linear ramp-down of the RF power. The RF radiation is provided by a SRS DS345, using its internal exponential frequency sweep (external trigger) and analogue amplitude modulation to control the power.
In TACC-2, due the reduced lifetime, we tried to further increase the collision rate by using an even tighter evaporation trap (typical trap frequencies {ωx, ωy, ωz} ∼ 2π • {0.08, 1.8, 1.8} kHz), reducing the evaporation time from previously 3 s to 1.25 s. The fi-nal atom number and temperature are not too much compromised.

Table of contents :

Introduction 
1 Spin squeezing for metrology 
1.1 Atomic clocks and clock stability
1.1.1 Basic notions
1.1.2 Trapped-atom clocks and TACC
1.1.3 Clock stability
1.2 Concepts of spin squeezing
1.2.1 Surpassing SQL with spin-squeezed state
1.2.2 Overview of spin-squeezing generation
1.3 Spin squeezing in cavity-quantum electrodynamics
1.3.1 Real-world cavities
1.3.2 Cavity-QED in the dispersive regime
1.3.3 Squeezing by QND measurement
1.3.4 Squeezing by cavity feedback
2 Experimental methods 
2.1 Setup
2.1.1 Atom chip assembly
2.1.2 Vacuum system
2.1.3 Optical system
2.1.4 Magnetic fields
2.2 Cold atom preparation and interrogation
2.2.1 Laser cooling and optical pumping
2.2.2 On-chip transport of atoms
2.2.3 Evaporative cooling
2.2.4 The clock trap
2.2.5 Interrogation photons
2.2.6 Interrogation pulse tuning
2.2.7 Absorption imaging
2.3 Cavity probing and stabilisation
2.3.1 Cavity parameters
2.3.2 Laser scheme and setup
2.3.3 PDH lock with minimum intra-cavity power
2.3.4 Digital filter cancelling mechanical resonances
2.3.5 Feed-forward targeting the thermal drift
2.3.6 Locking “without” light
3 A highly stable cavity-QED platform 
3.1 Clock stability analysis
3.1.1 Detection noise
3.1.2 Dick effect
3.1.3 Atom number fluctuation
3.1.4 Magnetic and temperature fluctuations
3.1.5 Ramsey Contrast
3.1.6 Preliminary stability results
3.1.7 Prospects with spin-squeezed states
3.2 Characterisation of the atom-cavity coupling
3.2.1 Vacuum-Rabi splitting
3.2.2 Cavity shift in the dispersive regime
3.2.3 Atom-cavity alignment
3.2.4 Intra-cavity optical lattice
4 Spin squeezing by measurement 
4.1 Inhomogeneous coupling and decoherence
4.1.1 Phase shift by cavity probe
4.1.2 Monte-Carlo simulations
4.1.3 Contrast and phase measurements
4.2 Composite measurements
4.2.1 Spin echo
4.2.2 Composite pulse
4.2.3 Coherence measurements
4.3 Conditional spin squeezing
4.3.1 Measurement uncertainty
4.3.2 Spin noise estimation
4.4 Squeezing by cavity feedback
4.5 Outlook
4.5.1 Squeezing lifetime
4.5.2 Alternative inhomogeneous-light-shift compensation
5 Quantum amplification by ISRE 
5.1 Identical spin rotation effect (ISRE)
5.1.1 Basic principles
5.1.2 An intuitive picture with atoms in two energy classes
5.1.3 Experimental signatures
5.1.4 Mean-field kinetic equation
5.1.5 Observation of ISRE via motional energy
5.2 Interplay between ISRE and cavity measurements
5.2.1 Origin: inhomogeneous coupling
5.2.2 Cavity shift in a continuous probing
5.2.3 ISRE triggered by a probe pulse
5.2.4 Dynamics in motional energy sensed by cavity shift
5.3 Amplification of quantum fluctuations
5.3.1 Experimental observations
5.3.2 Simple model with two energy classes
5.3.3 Simulation using classical spins
5.3.4 Circumventing the amplification
5.3.5 Future work
Conclusion and outlook

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