D E S I G N I N G & T E S T I N G A R A D I O F R E Q U E N C Y- A N D M I C R O WAV E - I N T E G R AT E D AT O M C H I P
w i n d a l m i j n
Bachelor of Science Institute of Physics
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Supervisor: Lara Torralbo Campo Professor: dr. R.J.C. Spreeuw
Bachelor Natuur- en Sterrenkunde Thesis (study load 12 EC)
A B S T R A C T
Magnetic atom chips are used as a platform for Quantum Informa-tion Science. Within this platform magnetic fields play a key role in establishing the requirements for qubit operations. However, creating a stable operating environment for qubit operations is still proving quite difficult, in particular generating substantial oscillatory mag-netic fields for the qubit rotations.
The previous setup didn’t experimentally provide the requested fields, so modifications were made. To improve this, a new prototype was de-signed for including both microwave (MW) and radiofrequency (RF) transmitters near the chip. This thesis describes the testing and de-signing of this revised prototype, featuring improved RF antennas and the MW patches rotated. These adaptions proved very viable, and will probably be included in the next chip generation.
C O N T E N T S
1 i n t r o d u c t i o n 1
1.1 Quantum Computing 1
1.2 Magnetic Atom Chips 1 1.3 Qubit Manipulation 2 1.4 This Thesis 4
2 r f a n d m w d e s i g n t h e o r y 5 2.1 Current Setup & Problems 5 2.2 Computer Simulations 6
2.3 MW Design 7
2.3.1 Micro-strip patch Design 7 2.4 RF Design 8
2.4.1 Skin Effect 8
2.4.2 Reducing Antenna Losses 10 2.4.3 Optimal Shape 10 2.4.4 Current Conclusions 12 3 e x p e r i m e n ta l s e t u p 13 3.1 Setup Overview 13 3.2 MW equipment 17 3.2.1 Transmitters 17 3.2.2 MW signal generation 17 3.2.3 Probes 18 3.2.4 MW Signal measurements 19 3.3 RF equipment 19 3.3.1 Transmitters 19 3.3.2 EM Sources 20 3.3.3 Probes 22 3.3.4 Power Measurement 22 4 e x p e r i m e n ta l t i m e l i n e & results 25 4.1 MW measurements 25 4.2 RF measurements 29
4.2.1 Used lab antennas 29
4.2.2 On-substrate wire antennas 32 4.2.3 Wire-integrated chip 34
4.2.4 On-substrate wires for new base-chip design 35 4.3 Theoretical & Experimental Values Comparison 36
4.3.1 Expected Field Values 36 4.3.2 Antenna comparison 37 5 d i s c u s s i o n 39 6 c o n c l u s i o n 41 Acknowledgements 43 Wetenschappelijke Samenvatting 45 v
A C R O N Y M S
DDS Direct Digital Synthesis EM Electromagnetic FG Function Generator MOT Magneto-Optical-Trap MW Microwave PM Power Meter RF Radiofrequency SA Spectrum Analyzer SG Signal Generator vii
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I N T R O D U C T I O N
Within 30 years the maximum number of transistors on our current classical computer chips will be reached [11]. Production will be forced to either increase chip size or stack chips [17]. Quantum computers, which feature quantum effects such as qubit entanglement and super-positioning offers an alternative with a more efficient way to process information and yields additional security as well. [4]
1.1 q ua n t u m c o m p u t i n g
Quantum computers, which rely on quantum states of single particles instead of electrons travelling through transistors, provide an alterna-tive for increasing information processing power. Quantum states can be defined using several quantum properties of particles; in our case our approach is using atom chips and cold atoms for this matter. Where classical computers use the quantum several thousands of elec-trons to store/retrieve the information stored as "bit" in a memory module, in quantum computing a single qubit will do [9]. A qubit (quantum bit) describes the quantum state of an attribute of a sin-gle particle, such as the spin, polarization or electronic excitation of an ion. This property is derived from the entanglement property of qubits, which arises from the one-to-one relation between logical and physical states. These states are either |0> or |1>.
The general advantage of using qubits over normal bits is that a sys-tem of n qubits can store information of up to 2n quantum states.
These increasing amount of usable quantum states per qubit added to the system, combined with quantum parallelism [9] enables quantum computers to solve certain problems, such as key breaking of current encryptions keys systems used online, in significantly less time than classical computers.
1.2 m a g n e t i c at o m c h i p s
In this setup a permanently magnetized film atom chip provides the foundation for the experiments. 87Rbatoms are loaded onto
micro-traps above the surface, where they’ll eventually serve as qubits. To be able to use quantum information in the setup, we require a
tice with 5-10 µm spacing for these micro-traps. The current setup achieved a spacing of around 10 µm [13], but improved fabrication techniques should improve this significantly for the next chip [7]. After the87Rbatoms are emitted by the dispensers, they are cooled in a Magneto-Optical-Trap (MOT) which combines laser cooling with magnetic trapping to cool and trap the atoms. Laser cooling is based on using lasers slightly detuned from an electronic transition fre-quency, to reduce the impulse of the individual atoms, which is ther-mally emitted and results in the atoms being cooled. In addition 2 coils located near the atom cloud in opposite direction are used, which provide a quadrupole magnetic field. The magnetic field in-creases the Zeeman shift of the atoms moving away from the center, and combined with the lasers causes the atoms to move towards the center and reduces their temperature into the µK range [13].
Figure 1: Diagram of a typical MOT setup. The green sphere in the middle represents the atom cloud. The rotational arrows indicate either the polarization (laser beams) or current direction (coils). [1]
However, this laser cooling is not enough to reach the level for quan-tum effects to become effective (T ~ nK), so further cooling techniques are required, such as evaporative cooling. Evaporative cooling is per-formed using Radiofrequency (RF) waves on the atoms in a trap. By sweeping through a frequency range (1 to 50MHz), it’s possible to extract the hottest particles from the cloud, in a similar way that one would blow over a cup of coffee to accelerate the cooling process.
1.3 q u b i t m a n i p u l at i o n
In order to use the atoms for information storage and processing af-ter they are cooled, several requirements must be met. The specific atomic structure of 87Rb, includes the magnetic levels |F,mF> :
|1,-1>, |2,1> and |2,2> for 52S1/2, which are all trappable states. Using optical pumping the atoms are prepared in either the |1,-1> or |2,2>
1.3 qubit manipulation 3
states, indicated inFigure 2by ’initial cloud’ next to these respective levels. After this step, the two atom states of interest are |1,-1> (|a>) and |2,1> (|b>), which are subject to Rabi oscillations.
Figure 2: The atomic levels used in a diagram. Important here is to distinct between the big gap between |1,-1> and |2,1> of 6.8GHz, and the hyperfine shift between |2,1> and |2,2> of 1MHz
An important aspect of quantum computing is the rate at which can be switched between these different atomic states, and life-time of an atom in a specific quantum state. In this case, the switching between states is done by introducing an external time-invariant magnetic field. A two-level atom in such a field can be described by the following Hamiltonian [8]:
H = hω0 2 ΩReiωt/2 ΩRe−iωt/2 − hω2 0 ! (1)
where ω0 is the frequency difference (due to hyperfine shift) between the two states (6.824 GHz) [13], and t is the time.
Solv-ing the time-dependent SchrödSolv-inger equation can be achieved using the rotating wave approximation, with the atom in the ground state (|a>) as initial condition. The probability of find-ing the atom in the excited state |2,1> (|b>) is defined by:
P(t) =
ΩRΩ0 R 2sin
2 Ω0 R 2t
, where ΩR0 = q Ω2R+ ∆2 and ∆ = (ω0− ω)2 (2)ΩR0 is the effective Rabi frequency, where the detunings plays a vital role in the contrast of the oscillation. Choosing a detuning with ωrf+mw, slightly above the Larmor frequency leads to ∆ = ωrf+mw− ωL, with the respective Rabi frequency defined as [16]:
ΩR0 = |gF||µB| 2 h
|Brf+mw× B|
|B| (3)
where gF is the Landé factor, µB the Bohr magneton and B is the static magnetic field (3.2G) generated by the FePt chip and
Z-wires underneath the chip. As this formula is based on the rotation of both magnetic fields, only the part of Brf+mw that is perpendicular to the static magnetic field contributes to the frequency.
The local magnetic field consists of the oscillating part provided by the RF and MW, and the static by the silver wires and the FePt. If the 4 sources (2 patches, and 2 antenna), are combined in patch-antenna pairs phasewise, the oscillating part can be rewritten in complex notation as:
Brf+mw(r) = brf+mw(r)eiωt+ b∗rf+mw(r)e−iωt (4) If this formula is rewritten in terms of left- and right-circular fields (with respect to the chip plane), and projected using local coordinates (for the static field) [16], this results in the
ampli-tudes of the circular components of the rf fields:
b±(r) = ˆe∗±(r)· brf+mw(r) (5)
Taking into account that |a> and |b> have the same gf but dif-fer in sign, they couple on difdif-ferent circular polarization com-ponents of the RF field. The Rabi frequency of these states cou-pling to their respective circular polarization is described as:
ΩR0 =|gf||µB|b+,−/ h (6)
1.4 t h i s t h e s i s
In previous experiments, only RF wires have been integrated on the atom chips, which are used to generate magnetic fields for qubit operations. As other qubit transitions also require sig-nals in the microwave range, the goal was to also implement a microwave transmitter on a prototype for a new chip setup. Both type of transmitters combined on-chip, should prove an-other step towards an improved quantum computer.
This next chapter, Chapter 2, will explain antenna design the-ories and the mathematical methods used for computer simu-lations. Chapter 3 will discuss both the previous and the im-proved setup in detail, followed by Chapter 4 with the results from the experiments. Chapter 5 and Chapter 6 will handle both the discussion and conclusion of the results, as well as improvements and suggestions for future experiments.
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R F A N D M W D E S I G N T H E O R YAs qubit rotations depends on the magnetic field provided, key is having the most efficient transmitter to obtain the desired field values on the atoms. However, as the transition frequen-cies for |1,-1> to |2,1> and |2,1> to |2,2> lie within different domains of the frequency spectrum, different antennas are re-quired to provide a solid signal for both. This thesis will con-tinue on a new prototype designed by a previous bachelor stu-dent, Floris Kegel [6], which uses an alternative design which
does allow for both radiofrequency (RF) and microwave (MW) antennas near the chip in vacuum. However, as antennas for these frequency ranges require significant different design, they will both be addressed separately.
2.1 c u r r e n t s e t u p & problems
The latest lab setup uses a cantenna for the microwave source, and a coil as a Radio-frequency source, which are both placed outside vacuum. The advantages of this are that the equipment doesn’t have to be designed for the bake-out (T ~ 150◦C) , and it significantly simplifies the wiring for our chip.
For generating microwave signals, a "Cantenna" (can-shaped antenna) has been used [16], which was placed at a distance of
approximately 12cm away from the chip. This antenna’s design was based on a WiFi transmitter, where the default transmitter frequency of 2.4GHz was replaced with the 6.8GHz needed for our experiment.
For RF the transmitter is located near the glass cell, and has a double winding with a diameter of 3.2cm. It’s thickness is 2mm, and it consists of a bundle of 8 copper wires, the wind-ing shape is held together with tie-wraps. It’s located at a 2.5cm distance from the chip placed inside the vacuum chamber, at a 45◦angle towards the chip surface. If provided with a 10W (40dBm) signal, it can achieve an approximate magnetic field of 50mG through a range of frequencies.
The latest setup introduces another problem which is the gen-erated fields are not perpendicular to the chip surface. The RF
coil antenna design concentrates the power of the signal in the center of the coil, and in direction perpendicular to the center. However, the vacuum chamber can’t provide a side view of the chip surface as this would obstruct laser bundles required to cool the atoms; this significantly reduces the effectiveness of the RF on the atoms. RF wires in vacuum near the chip have been implemented in other setups before [14], but this is not
possible in our latest setup.
2.2 c o m p u t e r s i m u l at i o n s
In order to quickly test what effect certain alterations to our de-sign would have, the FEKO Suite program was used to simulate behaviour of the electromagnetic waves on our chip. First, the chip base, including the different layers of material were pro-grammed into a FEKO model as a base for our tests. The next step was to create RF and MW transmitters close to the surface, yet not obstructing the laser beams. This eventually lead to a prototype with RF and MW transmitters on corners opposite to each other, as shown inFigure 3.
Figure 3: A 3d-image representation of Floris’ final model made in FEKO. The effectiveness of the antennas was calculated using two math-ematical models; Method of Moments (MoM) which was used most of the time, as it is faster and requires less parameters, and the Fine-Element-Method (FEM) which takes longer com-puting time.
A brief description of both methods: 1. Method Of Moments
For antenna engineering this is the most used method, in which the scattering/radiating structure gets replaced by equivalent (usually surface) currents. This current is split up in wire segments and/or surface patches, and every
2.3 mw design 7
interaction is calculated using a Green function for the problem. This solution renders an approximate current for each segment, and the results in a matrix that must be filled with complex valued entries.
2. Finite Element Method
This method is quite old and uses the partial differen-tial equation (PDE) form of Maxwells’ equations. The so-lution can be found using either variational analysis or weighted residuals. The first will find a variational func-tional which has a minimum solution of this PDE, with re-spect to certain boundary conditions. The second method uses a weighted residual error, in which Green’s theorem is used to shift one of the differentials in the PDE to the weighing functions.
An extensive explanation and comparison of methods can be found in [3].
2.3 m w d e s i g n
For the MW signal generation fortunately only the 6.8GHz fre-quency was required. The power of the antenna needs to be focused in one peak frequency, which lead to the use of micro-strip patches.
2.3.1 Micro-strip patch Design
Firstly the microwave transmitter design was based on a micro-strip patch design. This type of transmitter consists of a top conducting layer placed on a substrate. The substrate is placed on a ground plane, and the conducting layer is provided a sig-nal through a coaxial probe feed, as can be seen in Figure 4.
The resulting fringing fields on the edges of the top layer are the fields radiating outwards and serve as the microwave source for our setup. These fringing fields depend on the rand thickness of the substrate material [6]. These factors, combined with the
length and width of the patches defines the properties of the micro-strip patch. For these particular patches, silver was used, and a 1mm thick silicon with r = 11.68 was used for the sub-strate.
2.4 r f d e s i g n
Antenna design for our RF case is quite complex because of the wide range of output frequencies required. For RF frequen-cies between 1 and 50 MHz, there’s at least 5 octaves, which means a broadband antenna would be required to address such a spectrum of frequencies. Space within the vacuum chamber is limited, so this adds quite a restraint to the design process. Another issue is the direction of the field, which is highly de-pendent to the antenna’s shape.
2.4.1 Skin Effect
For all cylindrical conductors with time-variant current (such as RF wire antennas) the skin effect is applicable. This effect can be derived from the Maxwell equations and predicts that the current will be concentrated on the surface of the cylinder. This happens due to the electric and magnetic fields spawning from the variance in current. From Maxwell equations:
∇ × E = −jωB (7a) ∇ × H = J (7b)
Now substitute E = J/σ and H = B/µ to eliminate H and E from the formulae. The current density vector is assumed to only have a z component which depends on y alone. From the Biot-Savart law, there’s only an x-component for the B vector. After substitution in (7a) and (7b), we obtain:
∂Jz
∂y = −jωσBx (8a)
∂Bx
∂y = µJz (8b)
2.4 rf design 9
the first formula, this results in the following differential equa-tion:
∂2Jz
∂y2 = jωσµJz (9)
Which can be solved and results in:
Jz(y) = J1eKy+ J2e−Ky → K = (1 + j) r ωµσ 2 → k = s 2 ωσµ (10) This formula leads to the definition of the skin-depth in meters which is defined as:
k = s
2
σµω [m] (11)
Where σ is the conductivity (in S), µ the magnetic permeability and ω is the angular frequency of the time variance. As our RF wiring is annealed copper (σ = 5.8 · 107S/m, µ = µ0), this leads to the following table of skin depths:
f r e q u e n c y (mhz) s k i n d e p t h (mm) r a d j u s t 1 0.0660855 1 5 0.0295543 2.23607 10 0.0208981 3.16228 15 0.0170632 3.87298 20 0.0147772 4.47214 25 0.0132171 5 30 0.0120655 5.47723 35 0.0111705 5.91608 40 0.010449 6.32456 45 0.00985144 6.7082 50 0.0093459 7.07107
Table 1: Skin depth
The skin effect causes the current to flow on the outer 0.9µm to 66µm for the RF frequencies. Another effect of this skin effect is that it increases the resistance of the conductor in comparison to no time-invariance. This effect can be described with RAC = RDC· k ·
√
f, where RDC is the resistance at 0 frequency, k is a
is the frequency in MHz. This effect is of great importance as it requires the impedance of the antenna to be adjusted for a range of frequencies if a constant output power is desired.
2.4.2 Reducing Antenna Losses
This skin-effect can lead to power losses, especially at lower fre-quencies (<1 MHz). In order to contradict this effect, instead of using a single thick wire, an alternative is Litz-wire, a wire that consist of multiple strands of wire. Usage of Litz wire will also reduce the local proximity effect, where the individual strands affect each others output field, by being wound as a bundle [12].
There’s a limit to the number of strands that effect the resistance of the total wire positively, which can be calculated per fre-quency. In addition the diameter of individual strands should be less than the skin depth, which is quite hard to achieve for higher frequencies.
2.4.3 Optimal Shape
The previous RF antenna, designed by D. Pijn [10], was based
on a coil wire antenna, where the length, radius, and number of windings were optimized. Using Mathematica led to the cur-rent coil of 2 windings, which could generate a 50mG H-field at a 2.5cm distance for a 40dBm input power. Summarizing the ef-fect of the windings: more leads to higher output power, but it drops rapidly for higher frequencies, correlated to the number of windings. Also the H-field is larger in the origin of the loop, which results in a very directional field (mainly in plane with the chip). Furthermore the loop can’t be placed perpendicular to the chip surface due to vacuum chamber and laser pathway constraints.
Alternative design was sought after. Wires underneath chips were done before in atom chips [14], but mainly generated a
in-plane directional field (for the chip), where an perpendicular (x or y-directional) field is required. As a broad range antenna is required, the easiest solution would be to create a dipole an-tenna in either a dual hemisphere form or biconical form [2],
2.4 rf design 11
(a) Dual hemisphere (b) Biconical
Figure 5: Example broadband antenna shapes
The latter has a variable radius over its height which results in all the wavelengths being able to emit better. However, for both of these antennas to work, they need to have certain dimensions and, as with the perpendicular loop, space constraints due to the laser beams prevent both of these as an option.
Figure 6: Dual hairpin antenna. Long sides counteract, and better shielding against ground plane because of the thicker substrate
Eventually the design process lead to the dual hairpin shaped antennas (Figure 6). The long sides counteract each other, how-ever the short sides allow for a pretty similar field in two of the Cartesian coordinates. Combined with a thicker substrate based on aluminum-oxide (Al3O2), this leads to a decrease in signal loss from ground planes.
2.4.4 Current Conclusions
The current loop antennas used for RF work quite well in-plane with the chip, however for the setup the perpendicular fields are required. Increasing the number of windings has a positive effect for lower frequencies, but causes a reduction on field val-ues for higher RF frequency. A lot of shapes that might have potential as an antenna are not possible in the setup due to space constraints. Multi-stranded wire types are favoured over single wires, especially for these high frequency applications, as the skin effect causes all the current to run across the outer surface only, which can result in losses.
Therefore a new RF antenna should be placed closer to the chip, and be placed on proper insulation, to prevent signal loss. The shape should not be square or loop like, as this either cancels or concentrates the signal in a single direction. If possible a dual antenna is recommended. When combined with controllable relative phase between the transmitters this should allow alter-ing of the polarization of the magnetic fields, which can greatly improve the effectiveness of the rabi-oscillations between the qubit states.
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E X P E R I M E N TA L S E T U P3.1 s e t u p ov e r v i e w
Figure 7: The chip mount (red, left) and the translation stages with the probe (blue, right)
The experimental setup was started using a simplified ver-sion of an atom chip setup. Using a center copper base for a chip, RF and MW signals are provided using current lab equip-ment and are measured using probes and power measureequip-ment devices. At first only the center copper base was used for chip representation, but at a later stage also gold and silicon chips were put on top. The copper base and outer substrate area can be seen in detail inFigure 8:
Figure 8: The left-side view for the simplified chip setup
The outer chip area with the microstrip patches and RF wires resides on the 2 outer pillars, where the copper base which rep-resents the chip resides on a separate center pillar. The previous setup was adjusted with some improvements:
1. The rifling of the center pillar was removed, which allow the chip-base to be adjusted in height.
2. The three pillars were mounted on a rotatable base, en-abling the base to be easily rotated for both x and y direc-tion measurements.
Figure 9: The top view for the setup
The probe mount stage (visible on the right hand side of Fig-ure 7), was based on multiple translation stages. The bottom stage controlled y-direction, and on top rotated 90 degrees an-other stage was mounted to cater for the x-direction. Both these stages had a position range of 20mm. Pillars were mounted on top to equalize the probe to the chip level, on top another translation stage was mounted perpendicular to the x-direction, to allow control over the z-position of the probe over a range of 10mm. This stage was fitted with a base-mount for the alu-minum pillars, which allowed attaching of various probes. All stages allowed position control with 0.1mm accuracy.
3.1 setup overview 15
Figure 10: The new base-chip design. The atom-chip would be placed on the center base, the new wire antennas are placed on the half-circle substrate.
Eventually a new base-chip design was fabricated (Figure 10), to reduce the amount of ground plane surface area near the chip. This also allowed for the antennas to be placed closer to the chip, as well as having more insulation below to provide ad-ditional ground shielding. For the following sections Table 2 is provided, containing the abbreviations for the equipment used.
e q u i p m e n t na m e m w o r r f a b b r e v i at i o n
a n t e n na s
"Cantenna" MW [M-ANT1]
Microstrip patch MW [M-ANT2]
Coil selection RF [R-ANTS1A-E]
Lab coil antenna RF [R-ANT2]
On-substrate hairpins wires RF [R-ANT3]
On-substrate heartshaped wires RF [R-ANT4]
Chip-integrated wires RF [R-ANT5]
Big loop antenna RF [R-ANT6]
Triangular shaped antennas RF [R-ANT7]
Double hairpin antennas RF [R-ANT8]
s i g na l s o u r c e s
Analog Devices ADF4108 MW [MS1]
Analog Devices AD9959 RF [RS1]
Agilent 33250A RF [RS2]
Rohde & Schwarz SML01 RF [RS3]
s i g na l a m p l i f i e r s
RFPA RF67006900 MW [MA1]
Amplifier Research AR-25A250 RF [RA1]
Mini-Circuits ZHL-32A RF [RA2]
E&I 420LA RF [RA3]
Mini-Circuits TVA-R5-13 RF [RA4]
p r o b e s
Aaronia PBS-1 6mm RF [RP1]
Aaronia PBS-1 12mm RF [RP2]
Self-made probe MW [RP3]
s i g na l m e a s u r e m e n t s d e v i c e s
Tektronix PSM-4110 Power Meter RF and MW [PM]
Rigol DSA-815 Spectrum Ana-lyzer
RF [RSA]
3.2 mw equipment 17
3.2 m w e q u i p m e n t 3.2.1 Transmitters
Figure 11: 3D model of the Cantenna positioning towards the chip center. As measurements were done at the center of the chip, the cantenna was directed at the copper base and place 12 cm away.
Microwave transmitters were fewer in number, which consisted of the microwave patches (designed by Floris Kegel) and the Cantenna. As frequencies of microwave wavelength are more sensitive to cable-connections and proper wiring, these anten-nas were connected with the shortest possible wire and using SMA wiring instead of BNC.
3.2.2 MW signal generation
For the microwave, the setup was very basic. As a microwave source, we use a self-made device based on a Analog Devices ADF4108 [MS1] evaluation board. This board supports 3 out-puts which can be programmed using the program supplied by Analog Devices when the device is connected to a (Windows) computer. For the experiments port 1 was set a fixed frequency of 5.9GHz, and the port 2 was set to 6.8GHz. As there was only MW amplifier, only one port was used at a time. The output was connected with an SMA cable of approximately 20 cm to a computer controlled switch. From the switch there was an-other 20cm of SMA cable connected to the amplifier. This RFPA RF 67006900-10 [MA1] had a power of 10W and provided an amplification of 40dB for 6.7 to 6.9 GHz, resulting in an output power of approximately 40dBm.
Figure 12: Setup for the MW source. The function generator was connected through the amp to either the MW splitter (for the patches) or the Cantenna
3.2.3 Probes
The self-made probes were used for the MW frequencies. Re-garding these probes, a metallic-base probe was manufactured which could perform MW measurements up to 8GHz. Unfortu-nately there was no proper calibration sheet available convert-ing dBm into mG. Apart from this flaw, the probe provided very consistent measurements over time.
3.3 rf equipment 19
3.2.4 MW Signal measurements
For measuring the power picked up by the home-made probe the Tektronix PSM-4110 Power Meter [PM] was used. This de-vice could measure between 10 MHz and 8 GHz, and it’s sen-sitivity allowed measuring signals between -60 and +20 dBm. This device can be calibrated to a target frequency using the Power Meter Appliance software supplied by Tektronix. In this process, the total power measured over all frequencies will be summed, which might result in an incorrect measurement if the generated signal contains strong side-peaks. However for the source and amplifier used this was not the case.
3.3 r f e q u i p m e n t 3.3.1 Transmitters
Various RF antennas were used in the experiments. In the first stage, the on-substrate wires from previous design were tested
[R-ANT3](seeFigure 14a). Afterwards several other loop-based wire-antennas were tested with different diameter
[R-ANTS1A-E], number of windings and wire type (Figure 14b). In order to replicate the current setup, first the best performing loop-wire was used, and at a later stage the loop-loop-wire used in the current setup [R-ANT2]. For the revised prototype new loop-wire antennas were soldered using different loop-wire material. All RF antennas were connected to the source through either BNC female, BNC-to-SMA [R-ANT3&R-ANT4] or BNC to banana plug [R-ANT5] connectors.
(a) Previous prototype (b) Wire loop antennas
Figure 14: A selection of previous RF antennas used
The loop wire antennas created for the modified prototype were based on the design that a wire was required to loop back to the BNC female connector ground pin. This resulted in loops of various shapes. First a big hairpin was created the so-called "Big Wire loop" [R-ANT6] (Figure 15a), which was broad enough to
easily fit around the center base. Afterwards to reduce the size of the loop, a triangle shaped antenna was created [R-ANT7] (Figure 15b), which had the benefit of having a (triangle) side parallel to the center. As this resulted in a field mainly in the z-direction, Hans Agema from Electronics was consulted on the matter. His advice lead to smaller hairpin loops [R-ANT8], which were tested both single and in dual setup.
(a) Big loop wire (b) Dual triangular wire
(c) Dual hairpin wire
Figure 15: A selection of newly fabricated antennas.
These final hairpins (Figure 15c) were also tested with various types of wire-material and both with BNC female connectors and coaxial (RG-6) wires. The advantage of the coaxial wire was that it reduced the size of the loop as the wire had a smaller diameter than the BNC connectors and thus it was easier to place the connection point closer to the substrate. However, both the coaxial and antenna wires consisted of multiple cop-per strands and had a coating that complicated tinning of the material. These attributes impaired the ability to solder both to-gether properly, and eventually lead to the switch back to BNC connectors.
3.3.2 EM Sources
In the RF tests multiple setups were used:
1. First option (Figure 16) consists of a Direct Digital Syn-thesis (DDS) Analog Devices AD9959 evaluation board
[RS1]. The output frequency, 10KHz to 50MHz, could be set by the computer through a 10V BNC connector. The
3.3 rf equipment 21
signal continued to an Analog Devices AD835 attenuator via BNC. This device, controlled by computer, allowed re-duction of power by a maximum of 20dB through input voltage (0 to 10V). Next in the chain was the computer con-trolled Mini-Circuits ZYSWA-2-50DR switch with 2 out-puts, of which output 2 was BNC-terminated. After this the signal was sent to Mini-Circuits ZHL-32A [RA2], which provided a gain of up to +29dB for a maximum input of 0 dBm. Finally the signal was sent through Amplifier Re-search AR-25A250 [RA1], amplifier which had a rotary for specifying amplification strength, up to +44dB gain for an input power of 0dBm.
Figure 16: Setup for a single RF source used in first experiments. Here the "RF Amp 1" is [RA2]. For multiple antennas, the signal would be split with a BNC T-bridge at the current antenna position.
2. The DDS [RS1] had quite some noise on the sidebands, so at a later stage it was replaced by both the Agilent 33250A and Rohde & Schwarz SML01 devices, which pro-vided a much cleaner signal. Also due to technical issues and equipment being used for other experiments, addi-tional amplifiers were used, the E&I 420LA [RA3] and the Mini-Circuits TVA-R5-13 [RA4]. As these replacement sig-nal generators provided an easily adjustable output power, the attenuator and switch and [RA2] were removed from the setup. This setup is displayed inFigure 17.
Figure 17: Setup for a dual RF source. In this setup the [RA1] was replaced with the [RA3], and in addition [RA4] was used for the second RF source signal from [RS3].
3.3.3 Probes
To measure the fields strengths generated by the transmitters the Aaronia PBS-1 probe set [RP1&2] was used (Figure 18). This set consisted of 4 probes, featuring a 6mm, 12mm, 25mm and 50mm diameter loop for measurements. It also came with a cal-ibration sheet, which converted dBm to mG, with a different calibration per probe size.
3.3.4 Power Measurement
The power picked up by the probes was carefully measured us-ing both a Tektronix PSM-4110 [PM] power meter and a Rigol DSA-815 digital spectrum analyzer [RSA]. First the power me-ter was used, where a SMB-SMA cable was used to connect to its input. The power meter was connected to a laptop, where the power could be read out using the Power Meter Applica-tion software provided by Tektronix. The advantage of using the power meter was its broad range, 10MHz to 8GHz.
How-3.3 rf equipment 23
Figure 18: Aaronia PBS-1 probe set, in which 2nd and 3rd probe on the left are the 6mm [RP1] and 12mm [RP2] probe respectively
ever the power meter couldn’t measure an individual peak of the power, and rather integrated over the total range, leading to the Total Power [T-Power] of the signal.
If phase-related sideband harmonics [15] differ greatly
com-pared to the peak signal this is no problem, but connected to our DDS this lead to an increase of power. Afterwards the spectrum analyzer was used, as this could isolate the power of the peak signal, and had a greater power-to-noise measurement range (-100 dBm to 20 dBm versus -60 dBm to 20 dBm provided by the PM).
4
E X P E R I M E N TA L T I M E L I N E & R E S U LT SThe goal of these experiments was to measure the generated os-cillating H-field in different cartesian directions. In-plane with the chip was defined as the z-direction, the other 2 perpendicu-lar respectively x and y. As the in-plane fields get canceled be-cause of the enormous copper ground base underneath the chip the latter are of importance. The atoms get trapped quantized in the z-direction, and key to maximizing Rabi-oscillations is polarized fields in perpendicular directions.
4.1 m w m e a s u r e m e n t s
e x p e r i m e n t a n t e n na s o u r c e a m p l i f i e r p r o b e
MEXP1 M-ANT1 MS1 MA1 MP1
MEXP2 M-ANT2a
M-ANT2b
MS1 MA1 MP1
MEXP3a M-ANT2a MS1 MA1 MP1
MEXP3b M-ANT2a
M-ANT2b
MS1 MA1 MP1
Table 3: MW setups overview
As shown in Table 3, the MW antennas were connected to the same equipment throughout all the experiments. The signal coming from the [MA1] was either fed to the Cantenna directly or to a MW splitter, which divided the signal equally to both patches. To create a proper reference, first the "Cantenna" was retested for the microwave field generation. The Cantenna was placed 12 cm away from the chip as it had been before in earlier experiments [M-EXP1], directed at the center of the chip. Secondly, the MW-integrated chip was tested, [M-EXP2]. For this experiments measurements were done at the center of the chip. The fields are very alike for the different directions (around -40dBm), however values are quite different from the -20dBm
reached by the Cantenna transmitter.
d i r e c t i o n p o w e r [dbm] m-exp1 Cantenna -20 m-exp2 X-direction -41 Y-direction -43 Z-direction -40
Table 4: Results of M-EXP1 & M-EXP2. The Cantenna provides a signifi-cantly more powerful signal
Tilted patch measurements
As the signal was quite a bit weaker than the Cantenna some adjustments were made. As the FEKO simulations showed that the signal produced by the patches had a parabolic peak (see Figure 19) above the patches, theoretically tilting the patches to-wards the center of the chip would improve the field measured at the center.
Figure 19: Simulated radiation pattern (from FEKO)
As the outer area of the prototype containing the substrate and the antennas was one piece, and the design of the microstrip patches requires them to be placed on the substrate, it was not possible to tilt both patches individually towards the center. This lead to two new experiments where the entire outer area was tilted towards the copper base which represented the chip. In the first experiment both patches were aligned vertically, with the best patch placed to the center of the chip [M-EXP3a], see Figure 20. For this experiment only this patch was con-nected to the MW signal. The second experiment [M-EXP3b] featured both patches aligned horizontally (seeFigure 21), with
4.1 mw measurements 27
Figure 20: Single patch at 90◦
both patches connected, placed as close to the chip as possible. First the probe was placed above the patches to check the out-put power. Afterwards the patches were tested both single and dual at 90 degrees in y and z-direction.
pat c h m at e r i a l d i r e c t i o n p o w e r [dbm] v e r t i c a l A (above) Copper z -32(1) B (above) Copper z -23(1) B (center) Copper y -25(1) B (center) Copper z -32(1) h o r i z o n ta l
Both (center) Copper y -47(1)
Both (center) Copper z -44(1)
Both (center) Gold y -47.8(1)
Both (center) Gold z -44.7(1)
Table 5: H-field from patches in multiple directions
The results show that the vertical positioning works a lot better than the horizontal positioning. The generated field is strongly dependent on the distance of the patches to the center, hence the vertical position succeeds. Another point of interest is the fact that it barely matters whether we only use the copper base or the gold patch on top of it. The gold layer is thus not thick enough to completely shield the magnetic fields. An issue
re-Figure 21: Both patches at 90◦
mains that these patches are outperformed by the Cantenna according to these measurements. As these patches should the-oretically easily outperform the Cantenna, additional research [5] was done after this thesis, which indicated the feed-line for
the signal in these patches was faulty. After improvements by the Electronics department to these feed-lines they performed considerably better, easily surpassing the performance by the Cantenna.
4.2 rf measurements 29
4.2 r f m e a s u r e m e n t s
For RF a lot of different equipment was used. As multiple exper-iments were performed in the lab at the same time, some equip-ment was in use and therefore alternatives were gathered from other labs. There also were some issues with the [RA1] ampli-fier used, which was later replaced by the [RA3] device. In gen-eral the experiment was divided over the following phases: 1) reviewing the used lab antennas 2) reviewing the on-substrate antennas 3) reviewing the wire-integrated chip 4) testing on-substrate wires for new base-chip design. A brief overview of different setups used for the sub-experiments can be seen in Table 6.
e x p e r i m e n t a n t e n na s o u r c e a m p l i f i e r p r o b e
REXP1 R-ANTS1 RS1 RA2+RA1 RP2
REXP2 R-ANT2 RS1 RA2+RA1 RP1
REXP3 R-ANT3a R-ANT3b RS2 RS3 RA4 RA3 RP1 REXP4 R-ANT4a R-ANT4b RS2 RS3 RA4 RA3 RP1
REXP5 R-ANT5 RS2 RA3 RP1
REXP6 R-ANT6 RS2 RA3 RP1
REXP7 R-ANT7a R-ANT7b RS2 RS3 RA4 RA3 RP1 REXP8 R-ANT8a R-ANT8b RS2 RS3 RA4 RA3 RP1
Table 6: RF setups overview
4.2.1 Used lab antennas
The various RF coils were connected to the [RA2] and [RA1] with a 2x 1.8m BNC cable. The coil was placed at 2.5cm away from our probe. The probe would measure horizontal and verti-cal components, (Figure 22andFigure 23). For horizontal mea-surements the "y" (or x due to rotational symmetry of the loop) direction is measured, which is perpendicular to the loop direc-tion. Vertical measurements represent the in-plane (z) fields of the loop antenna. In our lab setup 40 dBm was used for input power, but this was varied in several experiments between 30 and 50 dBm. However in the latter case, as the amplifiers didn’t have enough power to amplify it up to 50 dBm, this would lead
to a maximum signal output of around 44dBm (=25W).
Figure 22: "Horizontal" coil measurement
Figure 23: "Vertical" coil measurement
Coil Antennas
Firstly several coils (Figure 24) used in the lab were tested with an input power of 40 dBm, a frequency of 10MHz and the 12mm Aaronia probe [RP2]. Connected to the power meter, this lead to the following results (Table 7) for the vertical fields: From these measurements it becomes apparent that not the ra-dius but the number of windings is the key factor, especially as 10 MHz is quite low in our target range (1 to 50MHz). This correlates to the findings from previous research [10].
4.2 rf measurements 31
Figure 24: All earlier coils tested, A on left, D on the right.
c o i l s: a b c d Thickness [mm] 1.5 2 4 4 Number of strands 1 10 20 20 Radius [mm] 1.75 1.75 1.8 2.5 Number of windings 1 2 1 1 h-field [mg] Above Chip 3.6 7.5 4.4 5.5 In Air 5.9 9.5 6.9 7.8
Table 7: Coil comparison
Lab coil antenna
At this stage the RF coil used in the lab was available for testing. Both y (horizontal) and z (vertical) field measurements were done for frequencies within the target range. This led to an ap-proximation for the fields could be generated within the current atom chip setup [R-EXP2].
In Figure 25there’s a large difference between the vertical and the horizontal fields in these results. This is due to the field be-ing very directional because of the loop shape of the antenna, causing the field to be become focused in the loop center.
Figure 25: H-field measurements done in y and z direction.
4.2.2 On-substrate wire antennas
Next in line were the on-substrate wire antennas [RA3] (de-signed by F. Kegel). The position of the probe over the chip would be varied in x, y and z-direction. For the perpendicular fields (x and y), most of the measurements were done along the axis of the center of the chip, with a ±1 cm variation in position. For the in-plane fields (z), the probe was adjusted in height above the chip, to check if any peaks in field could be measured there. As the chip was a thin layer above the ground plane, it was expected that the field would reduce to nearly zero just microns above the chip. However, the translation de-vices were not accurate enough to provide such distances to-wards the chip. As the atoms will be loaded on the center of
(a) Hairpin shaped wires (b) Improved Heart-Shaped wires
Figure 26: On-substrate wire comparison
the chip, this point is of great importance. The hairpin shaped antennas [R-EXP3] provided an average of 2 to 4mG field in the center for different frequencies and field orientations. As this was far from our target (50mG) they were eventually altered to Heart-shaped wires [R-EXP4], to improve the signal based on
4.2 rf measurements 33
simulations from FEKO, of which the results are displayed in Figure 27.
Figure 27: H-field measurements performed in y and z direction at 40 dBm input power with 2 RF sources
As you can see the signals for both directions seem quite simi-lar. However, taking into account this is the best possible field with 2 sources performing at the optimal phase, the field is far from our target so unfortunately these antennas are also not a candidate for the new atom chip.
(a) Silicon chip (b) Gold chip
Figure 28: Photos of the silicon and the gold chips used
The effect of different chip surfaces was also tested. Prelimi-nary tests were done using the copper base, but later tests also included silicon and gold chips on top of the copper base. The-oretically the gold would be harder to penetrate for EM-waves, so the effect of this was also studied.
Figure 29 shows the effect of different material chips is negli-gible for the frequency range, so the effect of gold on the EM-waves is minimum. These measurements were all done in the y-direction with an input power of 40dBm.
Figure 29: H-field measurements performed for different materials
4.2.3 Wire-integrated chip
The chip setup used by Rene Gerritsma’s experiment was mounted sideways, which resulted in the chip surface being 45◦angle to-wards the z-surface R-EXP5. The 6mm probe [RP1] was used to test the RF generated field in both z and y-directions for 40dBm. The RF wires integrated in a 4 windings loop underneath the chip were swept from 10 through 50 MHz, shown inFigure 30.
Figure 30: H-field measurements performed in y and z direction at 40 dBm input power
The old integrated chip wires work fine in the z-direction, how-ever in the y-direction the signal is very weak. The RF is ex-pected to quickly reduce at higher frequencies[10], however as
4.2 rf measurements 35
is far less, and the antennas did not have a coax shielded feed, meaning not only the coil is emitting the H-field. This was also noticeable, as our other lab equipment reacted to the RF sig-nal, especially around 10 to 20 MHz. During the brief measure-ments often the lasers would unlock, making this antenna not useable.
4.2.4 On-substrate wires for new base-chip design
As the RF signal needed improvement, eventually measurements were done for the new base-chip design. This design reduced the effect of the copper base by having thicker substrate and a smaller center base. The main 3 shapes tested were the "Big Loop" (a) [R-EXP6] , the "Triangle Wires" (b) [R-EXP7] and "Dual Hairpin" (c) [R-EXP8], as shown in Figure 31.
(a) Big Loop (b) Triangle wires (c) Dual Hairpin
Figure 31: Revised prototypes antennas
Big Loop was placed around the center base on the new 5mm thick Al2O3 substrate, and measurements were done at the same height without the substrate. The other loops were placed on substrate at both sides of the center base. As these used 2 RF sources which were synchronized via 10MHz reference signal, the phase difference between the antennas was set to an opti-mum. The probe was placed at the center of the copper base, and rotated for measurements in all directions. The results are displayed in the graphs (Figure 32andFigure 33) below:
It is apparent that the big loop and triangle wires work bet-ter in the z-direction. However, the Dual Hairpins work quite well in both y and z-direction and thus seem the best choice. The orientation of the antennas should cause a counteraction in the magnetic fields in the x-direction. The experimental results confirmed this, and showed a very weak field and was there-fore not included in these graphs.
Figure 32: H-field measurements performed in y-direction at 40 dBm input power
Figure 33: H-field measurements performed in z-direction at 40 dBm input power
4.3 t h e o r e t i c a l & experimental values comparison 4.3.1 Expected Field Values
Using the basic laws of electrodynamics, and simplifying the antenna to a simple wire, it’s possible to make an estimate of the field that should be possible at a distance d. First write power in terms of P = I2R → I = pP/R. An input power of 40dBm is equal to 10W, and for RF signals an impendance of 50 Ohm is used. This comes down to an input current of approxi-mately 0.45A. Taking into account our current is oscillating as a
4.3 theoretical & experimental values comparison 37
sinusoid, Irms = I/ √
2. Then using the formula for the magnetic field generated by a wire:
B = I√rmsµ0
2πd (12)
where µ0is the vacuum permeability, the field at distance d can be calculated. Here mG is the unit of preference, and filling in some values for the distance this leads to the following table:
d i s ta n c e[cm] h-field [mg] 0.1 1.27·103 0.5 255 1 127 1.5 84.8 2 63.6 2.5 50.9
Table 8: H-field possible at distance d from wire
If the distance of the wire antennas in the revised prototype is estimated at around 0.5cm from the chip center, the fields measured in the previous sections ars very close to the values predicted by this table.
4.3.2 Antenna comparison
Using (6) fromSection 1.3, it’s a matter of filling in the magnetic fields in mG to obtain the Rabi frequencies. These frequencies have a linear relation to the field strength, for 100mG this is 70kHz.
a n t e n na t y p e h-field [mg]
Y {Y, Z}rms
Lab coil 8 30
Heart Shaped wires 18 19
Wire-integrated chip 8 51
Big Loop wire 18 100
Triangle wires 9 112
Dual Hairpin wires 102 103
Table 9: Field comparison for antennas
account, the Triangle wires perform best, closely followed by the "Dual Hairpin" and "Big Loop". However, the y-component is the key factor, in which case the dual hairpins perform much better than any of the other antennas.
5
D I S C U S S I O NFinding the optimum design can prove quite a challenge, be-cause we have certain limitations while trying to mimic the real setup and measure as close to reality as possible. For example the probe performs a local perturbation in the H-field and thus affect the measured field. Also neither set of probes was prop-erly calibrated for the MW range. The Aaronia PBS-1 had a calibration up to 3GHz, where the self-made set only provided a fair response up to 8GHz. Therefore results were compared to the Cantenna, based on the dBm signal rather than the mag-netic field values.
The clarity of the radio-frequencies could not be thoroughly tested. Operating at 40 dBm, the level is quite higher than the maximum input for the Power Meter & Spectrum Analyzer (20dBm), and the attenuators available did not provide a flat signal reduction. Finally, the antennas were not checked for the setup compatibility. As the entire chip will be included in the bake-out process (T ~ 150◦), the antennas (and their materials) must be resistent to these temperatures. However, at the time of writing, this is currently being done and a solution is probably at hand.
6
C O N C L U S I O NThe proposed experimental setup for MW + RF should improve field possibilities within the next atom chip generation. This prototype has less ground plane surfaces and should also pro-vide less signal losses. If was found that the spectrum analyzer should be used for measuring the signal power, due to the na-ture of DDS frequency synthesizers and RF amplifiers.
MW measurements done in these experiments showed that the Cantenna still outperformed the micro-strip patches in field strength, even at a distance of 12cm from the chip. However fur-ther research after these experiments indicated that the signal-feed for these patches was faulty, and when corrected they per-form significantly better. As with all antennas, operating dis-tance is a key factor, so reducing this by tilting the patches 90 degrees should also improve the field strength.
As for the RF fields, experiments showed that the previous on-substrate wire type antenna could not provide a signal near the request value (>= 50 dBm). The integrated RF wires in Ger-ritsma’s previous chip worked okay in the z-direction, however caused unlocking of the lasers, and were therefore not suitable. Revising the prototype to a new smaller copper base proved very viable. It provided more surface space for the RF antennas and allowed placement closer to and (partially) underneath the chip. The dual hairpin shaped resulted in the greatest overall field, and created a quite similar field in both y and z-direction. These dual hairpins should provide a good candidate for the next generation chip when combined with the corrected and tilted micro-strip patches. Together they should create a proper atom chip setup generating sufficient RF and MW fields for qubit rotations and evaporative cooling.
A C K N O W L E D G E M E N T S
As final worlds I would like to thank some people who’ve helped during this project. I could not have done it without the Electronics staff, in particular Hans Agema, whose day to day advice on RF improvements was indispensable. Also Hugo Schlatter from the WZI technical staff, for the technical help and manufacturing of the modified prototype.
In and around the lab many people were of great assistance. Firstly, Arthur La Rooij who helped me get started quickly and all his efforts in testing the micro-strip patches at AMOLF and TU Delft. Secondly Maarten Soudijn, for the support in the lab and patience while my measurements sometimes interfered with his experiment. Julian, for advice on the target fields and help with RF measurements with the spectrum analyzer. I would also like to thank Daniel Pijn for his assistance while replicating his results, and his extensive sheet on the previous RF coil design. In addition, Floris Kegel, for all his efforts on the previous prototype and the help getting started with the FEKO program. And finally, Lara Torralbo Campo for her daily advice and help with the experiment and writing of this thesis.
W E T E N S C H A P P E L I J K E S A M E N VAT T I N G
Quantum computers worden steeds een belangrijker onderw-erp voor onderzoek. Het wordt technisch steeds lastiger om de informatie verwerking van klassieke computers op te schroeven, waarbij de afvoer van hitte en het opvoeren van het aantal tran-sistoren per vierkante centimeter de belangrijkste factoren zijn. Om deze problemen tegen te gaan zal uiteindelijk moeten wor-den overgegaan op quantum computers.
In tegenstelling tot normale bits werken quantum computers met quantum bits (qubits). Waar klassieke bits in computer-systemen worden gerepresenteerd door stromen en ladingen gevormd door tienduizenden electronen, wordt een qubit beschreven door de quantum toestand van 1 deeltje. Een quantum toestand is bijvoorbeeld de polarisatie van een photon, de spin van een deeltje of de geëxciteerde staat van een ion. Deze toestanden hebben voor elk individueel subsysteem de staat |0> of |1>. Het wordt echter interessant als de verstrengeling van qubits hi-eraan wordt toegevoegd. Aangezien logische en fysieke waarde van qubits een-op-een aan elkaar verbonden zijn, is het niet mogelijk om meerdere qubits individueel af te lezen. Deze ver-strengeling zorgt er voor dat een systeem van n qubits 2n ver-schillende waardes kan beslaan. Daarnaast is het mogelijk om parallel op meerdere qubits tegelijkerheid dezelfde operator uit te voeren. Deze laatste eigenschap zorgt er o.a. voor dat com-plexe problemen zoals encryptiesleutels eenvoudig gekraakt kun-nen worden, als men genoeg qubits tot zijn beschikking heeft. In dit onderzoek zijn qubits verschillende energieniveaus van atomen. Het verschil tussen deze toestanden worden ook gerep-resenteerd in frequenties, en de belangrijkste shifts hierin zijn in het radio en microgolf bereik. Deze toestanden kunnen wor-den gewijzigd door radiogolven uit te zenwor-den op deze frequen-ties. Om deze niveauovergangen mogelijk te maken is het van belang dat zo sterk magnetische velden op deze frequenties te creëeren. Hiervoor zijn verschillende antennes nodig.
Verschillende antennes zijn getest, en uiteindelijk is er een vernieuwd prototype ontworpen om het mogelijk te maken om deze an-tennes zo dicht mogelijk bij de atoomchip te brengen. Bij deze
antennes was het ook van belang dat de magnetische velden in de goede richting werden gemaakt, om zo zoveel mogelijk qubit overgangen mogelijk te maken. Uit de experimenten is gebleken dat de combinatie van micropatches met haarspeld antennes uiteindelijk het efficiëntst zal werken om de desgewen-ste qubit overgangen uit te voeren. Een dergelijk ontwerp zal zeer waarschijnlijk gaan worden doorgevoerd in de volgende generatie van atoom chips in onze onderzoeksgroep.
B I B L I O G R A P H Y
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