University of Amsterdam
Report Bachelorproject Physics and Astrophysics
Number of ECT: 15, performed between 28-04-2015 and 06-07-2015
Two photon transitions in a two level
system for
87
Rb atoms on a magnetic lattice
Author:
Marieke Kral
Student number:
10195076
Institute:Van der Waals-Zeeman,
Faculty of Science
Supervisor:
dhr. dr. R.J.C. Spreeuw
Second corrector:
dr. N.J. van Druten
July 14, 2015
Abstract
This thesis describes how the decoherence time of a two level system formed by the states 52S1/2
(F =1, mF=-1)and 52S1/2(F =2, mF=1) of87Rb atoms, which are magnetically trapped by a
mag-netic chip of FePt and by a magmag-netic field of external coils, can be determined by using Ramsey spectroscopy and how the Rabi frequency of this system is influenced by a helical antenna. The de-duced decoherence time is 56± 7 ms. The transition betweens those states is made by two photons, which are radiated by a microwave (MW) antenna and a radiofrequency antenna. By replacing the MW-patch antenna with a helical antenna with the adequate parameters the Rabi frequency of the system was raised, from Ω = 2π · 384 Hz to Ω = 2π · 419 Hz. However, the significance of this increase still needs verification and further optimization may yield better results.
Samenvatting
Het lijkt een eeuwigheid geleden, maar misschien herinnert iemand zich het nog wel: de tijd dat com-puters zo traag waren, dat je ondertussen makkelijk even wat anders kon gaan doen. Tegenwoordig zijn onze computers een stuk sneller, maar soms is dit toch niet goed genoeg. Zo moeten sommige wetenschappers met de huidige computers langer dan een maand wachten voordat zij het antwoord op een berekening krijgen. Daarom zijn zij al lange tijd op zoek naar een manier om computers sneller te maken. Mogelijk kan dat door gebruik te maken van de eigenschappen van deeltjes op kleine schaal: de schaal waarin de regels van de kwantummechanica gelden.
Huidige computers werken met nullen en eentjes. Ieder signaal dat verstuurd wordt, wordt verpakt in een code van 0 en 1 en wordt vervolgens door de computer weer vertaald in een voor ons begrijpbaar signaal. Het verzenden van een 0 of 1, noemen we een bit, maar je kunt je misschien wel voorstellen dat het sneller zou zijn als we ook een combinatie van de twee kunnen verzenden. Dit kan met quantummechanica. Dan heb je dus een quantummechanische bit: een qubit.
De vraag is natuurlijk wat die nulletjes en eentjes precies moeten zijn, als dit alles zich op de kleine schaal van de quantum moet afspelen. Het blijkt dat deze nullen en enen gevormd worden door atomen. Zoals je weet bevinden zich rondom atoomkernen elektronen. Deze elektronen hebben echter niet altijd dezelfde energie. Als het heel koud is bijvoorbeeld, heeft een elektron niet veel energie, maar je kunt ook energie toevoegen: niet zoveel energie dat het elektron het atoom verlaat, maar genoeg dat het elektron anders gaat bewegen. Zo kun je dus twee toestanden cre¨eren: een met een lage energie (de 0) en een met een hogere energie (de 1). Omdat de regels van de quantumme-chanica gelden, kan dus door energie aan een atoom toe te voegen een 0, 1 of een combinatie van de twee gemaakt worden. Die energie komt van een antenna, die deze energie uitstraalt. Zo kun je dus door aan een groep atomen energie toe te voegen een qubit maken.
Het probleem is dat de omstandigheden voor deze atomen net allemaal verschillend zijn. De atomen zullen daarom ieder anders op de toevoeging van deze energie reageren. Dit hoeft geen probleem te zijn, als dit verschil maar klein is, maar hoe langer je wacht, hoe meer de atomen zich anders gaan gedragen. Op dat moment heb je geen controle meer over deze atomen, want als je dan weer energie toevoegt, zullen ze allemaal anders reageren. We zeggen dat de atomen niet meer coherent zijn, dus decoherent, en we drukken dit uit in de decoherentie tijd: hoe lang het duurt voordat het grootste deel van de atomen zich niet meer hetzelfde gedraagt.
Als je de qubits, dus de atomen, wil gebruiken om signalen te communiceren in een computer, moet dit dus gebeuren voordat decoherentietijd voorbij is. Vanaf dat punt verlies je immers de controle over de atomen. Daarom wil je dat de atomen zo snel mogelijk reageren, wanneer je energie toevoegt. In dit onderzoek is zowel de decoherentietijd gemeten als gekeken hoe je de atomen sneller kunt laten reageren. Het blijkt dat de atomen sneller reageren, als de energie door een antenna in de vorm van een helix, in plaats van een vierkant, wordt toegevoegd. Deze atomen moeten wel snel reageren, want de vastgestelde decoherentietijd is ongeveer 55 ms. Meer hierover staat in dit verslag en misschien is het mede daarom straks wel een eeuwigheid geleden dat we niet met quantumcomputers werken.
Contents
1 Introduction 4
2 Theoretical framework 5
2.1 Trapping atoms . . . 5
2.2 Two level system . . . 7
2.3 Bloch sphere and Ramsey scheme . . . 7
2.4 Contrast and Rabi oscillation . . . 11
3 Experimental set-up 13 3.1 Preparations . . . 13
3.2 Finding the magic field . . . 13
3.3 Benefits helix antenna . . . 14
3.4 Design helix antenna . . . 15
3.5 Reflection . . . 16
3.6 Emission . . . 19
3.7 Antenna in magnetic chip set-up . . . 20
4 Decoherence time 22 4.1 Rabi oscillation . . . 22
4.2 Ramsey fringes . . . 22
4.3 Contrast . . . 25
4.4 Rabi oscillation with helical antenna . . . 25
5 Discussion 27
6 Conclusion 27
1
Introduction
In the contemporary world society depends on the use of computers. This is why in the last decades many people have dedicated their work to finding ways to make computers faster. In the past this was mostly done by scaling down the existing computer chips, today a new answer might be found in the so-called quantum computer. Because this computer would be based on non-classical, quantummechanical bits (qubits), this type of computer could be faster than existing ones in many instances.
However, for this to be reality there are a number of issues. A distinct characteristic of a qubit is that instead of a classical bit, which is either a 0 or a 1, a qubit can be a linear combination of both. This creates the question how to create and control large numbers of qubits while making interactions between the qubits possible. A possible solution would be a magnetic atom chip.
This chip is made of a permanent magnetic material creating a large number of microscopic magnetic traps (microtraps) for neutral atoms. If the atoms in a trap can be controlled in a closed two level system, they act as a qubit. In this way the ground state of the atoms could be seen as the 0 of the system and the excited state as the 1. By applying a pulse which corresponds to the energy gap between the two levels, a single atom can be excited. The pulse has to be applied for a certain amount of time to excite all the atoms. If, for example, the pulse is applied only half this time, a superposition of 0 and 1 is created.
Although this system of atoms could theoretically be manipulated endlessly into the 0 state, 1 state or a superposition of both, in reality the system is not perfectly coherent. The system is damped by the dephasing of the atoms and as time passes, the state of less and less atoms will be controllable by the pulse. This is known as decoherence and is characterised by the decoherence time. More details on this process will follow later on in this thesis.
In a quantumcomputer interactions between the qubits need to take place before the decoherence time of the qubits is reached; otherwise the information is lost because the atoms can no longer be controlled. This articulates the main goal of this thesis, i.e. to raise the number of operations that can be conducted on the atoms before the decoherence time is reached. Therefore the first step is to determine the deco-herence time of the atoms in the magnetic traps. The focus will be on the execution of a method called Ramsey spectroscopy, which will be explained in this thesis, and the processing of the obtained data to determine the decoherence time.
Before the atoms are brought in the microtraps, they will be trapped in a larger magnetic trap (macro-trap) that is created with the help of an external magnetic field, forming an initial trap. This thesis will focus on establishing the decoherence time in this macrotrap and taking the first steps towards the determination of the decoherence time in the microtraps.
In this thesis the operations that can be conducted on the atoms, take shape as the excitations being
done on rubidium-87 (87Rb) atoms from the groundstate in a closed two-level system on a magnetical
lattice. Two antennas are used for the excitation: one for a microwave (MW) pulse and one for a ra-diowave pulse. These two pulses combined excite the atoms. To excite the atoms as fast as possible and to thus raise the number of operations that can be executed on the atoms before the decoherence time is reached, it would be ideal to use an antenna with a high intensity and proper polarization. Currently a rectangular microstip antenna (also known as a patch antenna) is used for the MW-antenna. The second step will therefore be to explore the possible advantages of a helix antenna in comparison to the patch antenna. The most ideal design and use of the helix antenna is therefore examined as well.
This thesis will first elaborate on the theoretical background of this experiment. The focus will be on the
question why and how the 87Rb atoms are used. An explanation of the concept of decoherence (time)
and the Ramsey spectroscopy are also included. Following this, the experimental set-up that was used will be discussed. This will include a short description of the overall experiment and an explantion of the theoretical advantages of a helix antenna. The produced results will follow and be further discussed.
2
Theoretical framework
As mentioned before, for a qubit a superposition of two states has to be created, corresponding to a superposition of the 0 and 1, known from the world of computers. The problem with using atoms to create a qubit is that often an atom can be in more than two states: it can usually be excited and decay to a number of states. If the atom ends up in another state than the two states that form a qubit system, the qubit and the information it tries to communicate are gone. Another challenge lies in the fact that atoms move around as this will lead to unwanted interactions. The qubits should only interact when this is wanted, otherwise the system of qubits is not controlable. Even if the atoms are kept at a certain position, they will create a damping on any form of operation which is being executed, because of thermal motion. This lowers the decoherence time and should therefore preferably be limited. Thus, the used atoms have to be stable and still, and can only be in two states. This implies certain conditions for the qubit sytem. Firstly, the atoms need to be trapped to keep them in a certain location. Secondly, the two atomic states have to be stable. Finally, the atoms must be cold. The first two aspects will be discussed in this section. The cooling of the atoms will follow in the experimental set-up. This section will also include a theoretical description of the process of excitation and damping of atoms using the Bloch sphere. Additionally, an explanation of the different methods used to determine the decoherence time of the system, Ramsey and Rabi spectroscopy, will be provided.
2.1
Trapping atoms
The experiment described in this thesis uses 87Rb atoms in 52S
1/2 (F =1, mF=-1) and 52S1/2 (F =2,
mF=1) states as the intended states for the qubit. One of the reasons is that these states are lowfield
seeking, and can therefore be trapped by a magnetic field. To understand what this means, the energy structure of the atom must be examined.
When describing the energy levels of an atom, the angular momentum of the system has to be taken into account, but also the fact that the atom has relativistic properties. The principal structure of an atom can be found by analysing the influence of the Coulomb interaction of the nucleus on the motion of the electron, but because of these relativistic properties and the interactions between the orbital angular momentum L and the spin S, the principal energy levels can split, creating a fine structure. However,
this fine structure is not relevant for the energy levels of the 87Rb atoms studied in this thesis, yet
the hyperfine structure is relevant. In many instances, an atom will also have a nuclear spin I which will interact/couple with the total angular momentum J. This will lead to an energy splitting as well, creating a hyperfine structure. In the end, this means that in the theoretical description terms for these couplings are added to the Hamiltonian:
H = H0+ Hrel+ (ζnl/¯h2)L · S + (ahf s/¯h2)I · J, (1)
where ahf s is the hyperfine coupling numerical constant and ζnl a positive numerical constant for the
fine structure. This will lead to the following hyperfine shift:
∆E = 1
2ahf s F (F + 1) − j(j + 1) − I(I + 1), (2)
where F is the total angular momentum,
F = S + I. (3)
In figure 1a this hyperfine splitting is illustrated for 87Rb, where I=3/2. However, these energy shifts
all take place in the absence of externally applied magnetic fields. If such a field would be applied, a
Zeeman shift would occur: the energy levels would split depending on their quantum number mF. The
Zeeman shift ∆Ez in the low B-field limit is given by:
(a)
(b)
Figure 1: a) Hyperfinesplitting of87Rb in the absence of a magnetic field for the lowest energy level for n = 5. b) A
schematic sketch of the hyperfine structure diagram for the magnetic energy of the various sublevels of5S1/2. Observed
is the Zeeman shift for these sublevels and the linear inclination for the |F = 2, mF = 1i and
F = 1, mf = −1 states in
a low magnetic field. The B-field where the slopes of the |1, −1i and |2, 1i is actually equivalent, as drawn here, is at 3.23 Gauss.
where gF is a dimensionless, positive number for the state with F = I + 1/2 and negative for the state
with F = I − 1/2. Since mF is given by:
−F ≤ mF ≤ F, (5)
figure 1b shows the energy splitting shown for 87Rb. As shown in the figure, in the low field limit
the energies of the states |F, mFi=|1, −1i and |F, mFi=|2, 1i rise linearly when the applied B-field is
stronger. This follows from Eq. (4): |1, −1i will have a negative gF since F = 1 means F = I − 1/2, but
a combination with mF = −1 leads to a rise in energy when B is raised. For |2, 1i both mF and gF will
be positive, so the same applies here.
Thus, in a non-homogeneous B-field atoms in these particular states will go to wherever the B-field is at it lowest because at these points their energies will be at their lowest. In other words, these states are lowfield-seeking states: they will seek the low points in the field and stay there. That is how the atoms are trapped with the magnetic chip: a non-homogeneous magnetic field is created above the chip with a large number of magnetic minima seperated by a certain distance. Atoms are then brought into lowfield-seeking states and trapped.
However, not just any low B-field will do. The slope of the energy as a function of the B-field, so the slope of the lines depicted in figure 1b, have to be equivalent for both states. If this is not the case, a slight variation in the B-field would mean that the energy gap between the two states no longer has the same value: there would be a different resonance frequency. Since the atoms are in a thermal distribution, the hotter atoms (the atoms with a higher energy) will already ’feel’ a higer B-field. This means that these hotter atoms will have a different resonance frequency. A schematic sketch of this is depicted in figure 2. This means that a pulse with a certain frequency will not have the same effect on every atom and this will therefore lead to more decoherence (for more on decoherence, see paragraph 2.3). In conclusion, what is wanted is a low B-field where a slight variation in B-field strength leads to the same change in energy values for both state. The B-field value in which this is the case is called the magic field and has a value of 3.23 Gauss (Lewandowski 2002).
Figure 2: Shown is the potential energy of the B-trap (UB) as a function of the position. The magnetic traps at 3.23
Gauss, which will be filled up with atoms, is shown by the bold parabolas, the dotted parabolas show the magnetic traps at a different B-value. The atoms with the higher energy will experience a higher potential energy than the colder atoms. If the atoms are excited when in the lowest dotted parabola, the resonance frequency of the hotter atoms will differ from the colder atoms (shown by the additional red arrows). This would not be the case if the atoms are in the magnetic traps shown by the bold lines, at 3.23 Gauss.
2.2
Two level system
As can be seen in figure 1b, more low-field seeking states than |1, −1i and |2, 1i are present in the
hyperfine structure of 5S1/2. This raises the question why these specific states are chosen to form the
basis of a qubit. One of the reasons is that these two states form a stable two level system. Electrons can be excited and sponteaneously decay, but not to every state. Obviously, a state cannot decay beyond
the groundstate. Since the occupied orbitals for87Rb are,
1s22s22p63s23p63d104s24p65s,
the 5S1/2 is the groundstate. This means the state |1, −1i is stable, but it can be excited to |2, 1i.
Since this forms a transition between different hyperfine levels, it is a magnetic dipole transition (M1). The selection rules for the M1, which govern the transition from one quantum state to another, yield
∆n, ∆l = 0. So a transition is possible as long as it is within the Zeeman levels of 52S
1/2, which is the
case for a transition from |1, −1i to |2, 1i.
However, the transition between |1, −1i and |2, 1i can only be made by two photons with the right
polarization since it is governed by these selection rules. For positive circularly polarized (σ+)waves
∆mF = +1 applies, for negative circularly polarized (σ−) waves ∆mF = −1 and for linearly polarized
(π) waves ∆mF = 0, so two σ+-pulses are needed to make this transition of ∆mF = +2. A spinflip is
associated with this, since the atoms are excited from F = 1 to F = 2. This can happen when the two pulses have an overall energy equal to the energy gap. This is done by applying a microwave (MW) pulse of 6832 MHz and a radiowave (RF) pulse of 2.9 MHz at the same time. As can be seen in figure 3, the MW-pulse is slightly off resonance with the |2, 0i and the |2, −1i state.
If the MW-pulse would be on resonance with these intermediate states the atoms would be able to populate those states and escape the trap as the |2, 0i and |2, −1i are not low-field seeking like the |2, 1i and |1, −1i states. This process not only leaves fewer atoms in the trap it also ends the two-level system. When the atoms are brought into the |2, 1i state, the probabilty of the atom spontaneously decaying into another state, is extremely low. The role of the spontaneous emission can therefore be neglected.
2.3
Bloch sphere and Ramsey scheme
The excitation with the MW- and RF-waves can be visualised using the Bloch sphere, as shown in figure 4. The |1, −1i is visualised by the |0i vector, pointing downward on the z-axis of the sphere. This vector can be flipped to the |1i vector by applying the right pulse, which visualises the |2, 1i state. This pulse
mF = -2 -1 0 1 2 MW: 6.8GHz RF: 2.9MHz |1 |0 F=1 (g F=-½) ∆ = 0. 7MHz F=2 (g F=+½)
Figure 3: Schematic overview of the two photon transitions in a low magnetic field between the |2, 1i and the |1, −1i state of the 5S1/2 state from87Rb (Wicke 2012). Mentioned is the g-factor gF, which is dimensionless and characterizes the
magnetic moment of the atom.
Figure 4: The Bloch sphere. State |1, −1i is visualised by red |0i vector and state |2, 1i by the blue |1i vector.
is called a π-pulse and is the equivalent of the MW- plus RF-pulse. If a pulse with twice the length of a π-pulse is applied, the atoms will be returned to |0i. An ongoing pulse would force the atoms into a Rabi-oscillation as a result of stimulated emission and excitation.
If not the entire π-pulse is applied, but for example only half of it (so a π2-pulse), the vector would not
flip from |0i to |1i but to the equator of the Bloch sphere, creating the superposition state √1
2(|0i + |1i).
In conclusion, the way to create a superposition of atomic states is to apply a pulse to the atoms shorter or longer than the length of the π-pulse.
This is exactly what is done when a Ramsey scheme is carried out, see figure 5.1 For a time τ a π2-pulse
Figure 5: A Ramsey scheme. For time τ a π/2-pulse is applied, after which a delay time T follows where there is no pulse. This followed by another π/2-pulse for a time τ .
is applied on the atoms. For a time T , the delay time, the system is left to evolve freely after which a π
2-pulse is again applied for a time τ .
From the theoretical description of this system we can derive the probability that an atom is found in the
excited state P1 after such a Ramsey scheme. Without considering dephasing this probability is given
1A Ramsey scheme can be carried out for each pulse time with Ωτ <π
2. Since this is expected to show a weaker signal, π
2 is described here. The description would not change much for another pulse: the vector wouldn’t make it back to the
by the equation P1= 2 Ω2 ω2 sin 2(ω1τ 2 ) cos2(τ ω1 2 ) 1 + cos(T δ)+ sin2(ω1τ 2 ) 1 − cos(T δ)− δ ω1 sin(ω1τ ) sin(T δ) , (6)
where δ is the detuning from resonance, Ω the Rabi frequency, τ the time of the π2-pulse, T the time
between the pulses and ω1 =
√
δ2+ Ω2 (for the derivation, see Appendix). The relation between the
Rabi frequency Ω and the pulsetime τ is
Ω = π 2
τ.
Using this relation, Eq.(6) is plotted in figure 6, for T = 22 ms, τ = 0.75 ms and δ = 3 kHz. The figure shows how the probability that an atom is excited varies strongly depending on the frequency of the pulse that is applied. These oscillations are characteristic for a Ramsey scheme, and are known as ’Ramsey fringes’. After applying this Ramsey scheme, the number of excited atoms is measured. When
Figure 6: Shown is the excitation probabilty as a function of the detuning from the resonance frequency needed to excite the atom from |1, −1i to |2, 1i. This is an example of a plot of Eq.(6) with T = 22 ms, τ = 0.75 ms and δ = 3 kHz.
T = 0 it means that a π-pulse is applied on the Bloch sphere, which excites all the atoms to |1i if there is no detuning.
However, if T 6= 0 the situation changes. To understand how this works the visualization of the Bloch sphere remains convenient. The rotation around the horizontal axis is initiated by a π- or fraction of a π-pulse, thus by the Rabi frequency Ω. However, the vector is also able to rotate around the vertical axis. This is caused by a detuning δ between the resonance frequency of the transition and the frequency
of the applied pulse. So if a π
2-pulse is applied with a detuning δ, the vector will rotate on the equator
of the Bloch sphere for a time T . If δT = (2n + 1)π the vector will be on the other side of the equator,
where the next π2 will not bring it to the north pole of the sphere but back to the south pole. In other
words: the product of δ and T will determine whether a vector will end up on the northern or southern part of the sphere.
However, it has to be taken into account that detuning does not only take place because of the frequency of the pulse, and that the system involves more than one atom. There is a group of atoms, which are in a thermal distribution. This means that the hotter atoms already experience a slight detuning from the resonance frequency of the colder atoms. This means that after applying the first π/2-pulse, these atoms will behave differently than the colder atoms. Since they experecience a slightly different detuning, they will rotate faster around the equator. Since this rotation speed will then differ per atom, they will
become out of phase with each other; dephasing. This means that the next π
2 pulse will no longer have a
coherent effect on the atoms as they will end up on diverse places on the Bloch sphere and rotate there because of their detuning. Figure 7 shows this Ramsey scheme on the Bloch sphere for one vector, which eventually ends up on the northern part of the Bloch sphere.
Figure 7: Ramsey scheme on a Bloch sphere. A π/2-pulse which excites the |0i to the equator where it can rotate freely for a time T before being subjected to a π/2 pulse again (Pla et al. 2013).
Some atoms will even end up back on the southpole because the specific δ in combination with T will equal (2n + 1)π. The time T which it takes for the coherence to drop 1/e from its initial value is char-acteristic for this process and is called the decoherence time.
This decoherence clearly has an effect on the number of measured excited atoms. Measuring the number of excited atoms means projecting the Bloch vectors of the atoms on the vertical axis. The more the atoms are dephased from each other, the smaller the sums of these projected vectors will be and the lower the number of excited atoms which can be achieved by applying a Ramsey scheme.
Thus, the atoms can no longer be uniformely controlled, which results in a loss of information. Therefore, cooling the atoms, to lower the width of the thermal distribution, is desirable to raise the decoherence time and maintain information for a longer time period. However, dephasing and therefore decoherence are not only a consequence of the initial thermal distribution. They are caused by a number of factors, for example the interaction with the given pulse. The first step towards knowing these factors and their influence is to measure the decoherence time. This time is determined by measuring the number of excited atoms after executing a Ramsey scheme. Doing this for different pulse detunings δ for a certain delay time T can lead to determining a value for the decoherence time.
This is possible because the equation for the excitation probability P1 after a Ramsey scheme with
dephasing has been deduced:
P1= 1 2 − 1 2 δ2+ Ω2cos(ω 1τ ) 2 ω4 1 −e −T γ ω3 1 Ω2sin(ω1τ )
ω1cos(δT ) sin(ω1τ ) + δ − 1 + cos(ω1τ ) sin(δT )
+δe −γTΩ2 ω4 1 − 1 + cos(ω1τ )
δ − 1 + cos(ω1τ ) cos(δT ) − ω1sin(ω1τ ) sin(δT )
!
, (7)
where γ is the decoherence rate, related to the decoherence time by 1/τc, where τcis the decoherence time
(for the derivation see Appendix). Using the specific values of an applied Ramsey scheme, so the time T ,
τ , the Rabi frequency Ω, the detuning δ and the decoherence time τc, Eq.(7) calculates the probability of
the atoms being excited after the Ramsey scheme. A plot of Eq.(7) for T = 22 ms, τ = 0.75 ms, τc=20
ms and the detuning varying from -3 kHz to 3 kHz is shown in figure 8. When comparing this plot to figure 6, the influence of the decoherence is visible: the excitation probability is damped.
In the sytem described in this thesis the decoherence time is not known and Eq.(7) was used to determine
it. After applying a Ramsey scheme τ , T and δ are known and by fitting this equation to the data τc, the
decoherence time, should be found. That is if the model for dephasing used to deduce Eq.(7) is indeed correct (see paragraph 4.2 Ramsey fringes for more on this).
Figure 8: Shown is the excitation probabilty as a function of the detuning from the resonance frequency needed to excited the atom from |1, −1i to |2, 1i. This is an example of a plot of Eq.(7) with T = 22 ms, τ = 0.75 ms, δ = −3 kHz to 3 kHz and a decoherence time of 20 ms.
2.4
Contrast and Rabi oscillation
Fitting Eq.(7) is not the only way of using a Ramsey scheme to find a decoherence time. One other way is to use the contrast. The contrast C is defined as
C = Ntop− Nmin
Ntop+ Nmin
, (8)
where Ntop is the maximum number of excited atoms and Nmin the minimum.
The dephasing of the system does not only affect the maximum amount of atoms that can be excited, but also the minimum. The atoms are no longer controlable as is desired: e.g. when the detuning of the applied pulse and the delay time T correspond to δT =π, it is expected that the number of excited atoms will drop. However, the atoms are dephased so not all atoms can be expected to be excited to the
southern part of the Bloch sphere by this detuned pulse. In other words, the effect of the second π2-pulse
can be exciting them to the nothern part of the sphere. This means that the number of excited atoms
will not drop as much as is expected from neglecting the influence of dephasing on P1. The longer the
delay time T is, the more dephasing occurs and so this phenomenon will increase as time T is raised. Since increasing time T means lowering the number of excited atoms when a maximum is expected, the contrast will be able to tell something about the decoherence. This is also visible when comparing figures 8 and 6 where the decoherence led to a change in contrast of the fringes in figure 8. In the results further examples of this will be shown.
It is also possible to determine the decoherence time by carrying out a Rabi measurement. This means that there is an ongoing pulse on resonance, which will lead to a sinusoidal movement of the number of excited atoms. This is known as Rabi flopping. These oscillations will be damped because of the
decoherence. By fitting the following equation for the excitation probability P1, the decoherence time
can be determined: P1= 0.5 − 0.5e− t τc cos(tπ tπ ), (9)
where τc is the decoherence time and tπ the duration of the π-pulse.
This measurement is expected to give a lower value of the decoherence time than when deduced by Ramsey spectroscopy. Since a Rabi measurement uses an ongoing pulse, the atoms will constantly be under the influence of this pulse, which is expected to lead to a faster dephasing. In a Ramsey scheme
there is only a pulse applied for a short amount of time so there is less decoherence to be expected.
Before carrying out a Ramsey scheme it can be convenient to determine the value of the π-pulse tπ and
therefore the Rabi frequency. For fitting Eq.(9) to the data this is not necessary, since the time of the pulse only appears as a constant in this equation. However, if two pulses would be carried out which together would not form a π-pulse, the number of excited atoms would decline. That is why the π-pulse
was indeed determined through a Rabi oscillation (tπΩ=π) before the Ramsey scheme took place. Such
Rabi oscillation measurements were also used to determine a decoherence time (see paragraph 4.1 for more details).
3
Experimental set-up
3.1
Preparations
The experimental set-up used for the trapping of87Rb atoms on the magnetic chip is described in detail
in Tauschinsky (2013). The main steps will be shortly discussed.
The magnetic chip is located inside a vacuum chamber with a dispenser of87Rb atoms. The atoms are
first cooled with laser cooling, which uses the Doppler shift due to the movement of the atoms. Wires sitting beneath the permanent magnetic atom chip together with the external coils can create a relatively broad magnetic minimum or Ioffe-Pritchard type trap. This trap is also called z-wire trap or macrotrap. The atom chip itself works in a similar way but relies on its permanent magnetic material FePt instead of currents, to make the microtraps. However as mentioned before, the macrotrap is used. The atoms are trapped in this magnetic minimum after they have been optically pumped to the |1, −1i state. The optical pumping happens after the laser cooling and is shown in figure 9. The atoms will be in
the ground state 52S
1/2, either in F =2 or in F =1. A repump laser pumps the atoms from the F =1 to
the F0=2 of the 52P3/2 state, while the atoms in the F =2 will be excited to this same state but by a
different laser using σ−polarized light (the optical pump laser). The atoms in the F0=2 will decay back
to either F=1, where they will be repumped, or to F =2 were the optical pump laser will excite them
again. Eventually the atoms will end up in the state |F0, mFi=|2, −2i, where they can decay into the
state |F, mFi=|1, −1i. If the repump laser has been shut down by that time, the atoms are trapped in
this state.
Figure 9: Optical pumping to |F, mFi=|1, −1i of the 52S1/2state. The F0= 2 belongs to the 52P3/2state. The repump
laser is not shown. Black lines show the optical pumping laser (atoms can also decay back along those lines), blue dotted lines show π-polarized decay, green lines shows σ−polarized decay, and red shows σ+decay.
The atoms are further cooled down by evaporative cooling. RF waves are used to spin-flip the hottest atoms of the thermal distribution, which then leave the trap. A new and cooler thermal distribution is then established from which again the hottest atoms are excited and removed by the laser. This process is repeated until the atoms are cooled to a few µK.
A probe laser is used to determine the number of excited atoms in the |2, 1i state. The laser excites the
atoms from |2, 1i to F0=3. Thus, the atoms absorb part of the pulse light, and the remaining light is
used the create an image of the number of excited atoms. This process is called absorption imaging and is also discussed in Tauschinsky (2013).
3.2
Finding the magic field
Between the cooling and the imaging, the two-level excitations are carried out. At this point the B-field must be at its magic value of 3.23 G. To determine if this is the case, a measurement is carried out known as a RF-knife measurement. Since the atoms are in a thermal distribution within the trap, the ’hotter’ atoms have a different energy than the colder ones and therefore a different resonance frequency. By applying a pulse which excites the hotter atoms to a non-trapped state, only the colder atoms will remain in the trap (evaporative cooling). By gradually adjusting the pulse frequency and then imaging
the remaining atoms, one can determine the resonance frequency of the coldest atoms, namely the pulse frequency at which there are no more atoms left. The relation between the frequency and the B-field is known, because the energy splitting is due to the Zeeman shift (0.7 MHz/Gauss). Therefore the value of the B-field at the trapbottom can be determined and checked against the value of the magnetic field. A frequency of 2.261 MHz, which corresponds to 3.23 Gauss, is expected if the B-field is at its magic value. Figure 10a shows such a measurement in which it has been checked whether the trapbottom was at the magic value. Since the graph drops around 2240 kHz, which is close to the value of 3.23 Gauss, the magnetic field was not adjusted. The same measurement was also used to check whether or not the atoms were too hot. The broader the graph, the warmer the group of atoms. If the width of the graph came in the range of the value in which the MW-pulse is off-resonance with the |2, 0i and |2, −1i states, (see figure 3), there would be the risk of atoms escaping to other levels resulting in the lack of a closed two level system. Since the width was at most 200 kHz, there was no need to further cool the atoms. If the B-field is not at its magic value, it must be adjusted. As the B-field that creates the macrotraps is a combination of the B-field from the magnetic chip and the external B-field, this might not be easy.
However, the Bx-field generated by the coils can indicate how the field must be adjusted to become its
magic value. By sending out a pulse of 2.261 MHz at a certain Bx value, imaging how many atoms
remain and repeating this process for the next Bx value, the magic field can be found, namely the value
where there is a sharp decline of excited atoms. In the case of figure 10b, this was about -0.25 Gauss.
2 1 0 0 2 1 5 0 2 2 0 0 2 2 5 0 2 3 0 0 2 3 5 0 2 4 0 0 2 4 5 0 2 5 0 0 0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 A to m s (N ) F r e q u e n c y ( k H z ) (a) - 0 , 4 - 0 , 2 0 , 0 0 , 2 0 , 4 0 2 0 0 0 0 4 0 0 0 0 6 0 0 0 0 8 0 0 0 0 1 0 0 0 0 0 A to m s (N ) B x - f i e l d ( G a u s s ) (b)
Figure 10: A) A RF knife measurement. The RF frequency (x-axis) is varied, to make visible when the number of excited atoms (y-axis) suddenly drops. This frequency corresponds to the B-value at the bottom of the magnetic trap. B) The number of excited atoms as a function of the external magnetic field in the x-direction. This measurement is to determine at which value the number of excited atoms drops, which provides the Bx of the external field needed for the creation of
the magic field.
3.3
Benefits helix antenna
For a working qubit the atoms would ideally react as quickly as possible to the pulse, because that means it is possible to carry out more operations on the qubits before the decoherence time is reached. This means that a high intensity pulse is preferred: in that way the atoms are subjected to a large amount of energy in a short amount of time, which means that the maximum number of excited atoms is reached quicker. This means that the duration of a π-pulse is shortened.
The question remains how to create a pulse with these characteristics. Since the pulse that is used in this magnetic chip experiment actually exists out of two pulses, one MW-pulse and one RF-pulse, improving the intensity of one of these pulses would in theory lower the duration of the π-pulse of the system. Since
the Rabi-frequency is defined as:
Ω =ΩM WΩRF
2∆ , (10)
where ΩM W and ΩRF are the Rabi frequency of the MW-pulse and the RF-pulse and ∆ is the detuning
from the intermediate |2, 0i state (see Foot (2010)). The Rabi-frequency is defined by Eq.(16). This
means that if the BM W is raised, both ΩM W and the Ω will rise. Since the intensity is proportional to
B2, raising the intensity of the MW-antenna will lead to an enhancement of the B-value and therefore
of the Rabi frequency. This means a shorter π-pulse can be achieved.
One of the ways to enhance the intensity of the MW-pulse is to change the type of antenna that is used. Currently a patch antenna is used, yet there could be some advantages to using a helical antenna. One of these advantages is that the beam coming out of the antenna is directional, creating a high intensity to wherever the antenna is pointing at. Another advantage would be its polarization. For the two-photon
excitation σ+-photons are required. A patch antenna radiates linearly polarized light, which means that
the patch antenna is able to drive different transitions depending on the orientation of the quantization
axis: π-transitions which require linearly polarized light parallel to the quantization axis and σ+ and σ−
transitions which require linearly polarized light perpendicular to the quantization axis, also known as
σ-polarized light (a linear combination of σ+ and σ− light).
A helical antenna however radiates circularly polarized light by its nature and when aimed at the target
from the right axis, σ+ photons will drive the excitation. Therefore using a helical antenna is expected
to lead to a higher intensity of σ+transitons and a lower intensity of other transitions. The polarization
of the radiation from the helix has another advantage: the loss of atoms due to the antenna driving other transitions, should be limited.
Another benefit from the helix compared to the patch antenna would be the limited sensitivity to changes in the radiated frequency; when tuned to a certain frequency, a slight detuning from this frequency will not drastically change the intensity of the pulse. This means that the helix antenna will have a high Q-factor compared to the patch antenna. However, the advantage of the patch antenna is that it can be placed very close to the vacuum chamber, which is not possible for the helix antenna.
3.4
Design helix antenna
Since the theoretical advantages of the helical antenna are known, the challenge that lies ahead is to determine how to design and use such an antenna to benefit from these advantages. As can be seen in figure 11a, there are many characteristics of the antenna which can be varied, for example the size of the groundplate, the length of the helix, the number of turns, the spacing between the turns and the connec-tion point between the antenna and the groundplate. The design of the helical antenna has therefore been varied to tune for this system. However the fact that the antenna should radiate at 6.83 GHz produced certains demands for its design. The design was made based on Silver (2011) and Weeratumanoon (2000). First, the diameter of the helix was determined. The helix spirals, but when calculating the diameter it
is approximated as a row of circles. This means that the diameter is defined by λπ. Assuming the speed
of light in vacuum, the wavelength will be 43.9 mm. This gives a diameter of 14 mm, so this became the diameter of the helix.
The spacing S between the turns is geometrically about equal to λ/4. This would give a length of about 11 mm. In the final helix the spacing became 9 mm, which was due to an error in the fabrication. The circumference of the helix, C, needs to be equal to approximately one wavelength for the antenna to radiate as is depicted in figure 11b. Such a radiation pattern would be ideal since it makes for a very directional antenna. C is therefore calculated by πD, therefore becoming 47,1 mm.
To reach the maximum level of efficiency of the antenna the number of turns needs to be between 3 and 15 (Weeratumanoon 2000). It is expected that if the number of turns varies between these numbers, the aperture would not change much. However, the beamwidth will become broader for a smaller amount of
(a) (b) Antenna.
Figure 11: A) A schematic drawning of a helical antenna and its parameters. B) The radiation pattern created by a helical antenna directed in the z-direction. Both figures originate from Weeratumanoon (2000)
turns. The number of turns in the helix is chosen to be 15 to enable shortening the helix if wanted. To reach a maximum amount of power, a maximum amount of power has to reach the antenna. The wire and the antenna will together form a voltage divider and the antenna absorbs maximum power when the impedance of the wire and antenna are equivalent. The impedance for the antenna is calculated by 140
Ω ·Cλ, which makes 150 Ω. Since the impedance of the wire is 50 Ω, the impedance must be matched.
This can be done with a piece of copper forming the transmission line between the ground plate and the antenna. This piece of copper is not shown in figure 11a but is placed between the ground plate and the antenna. This provides ’extra’ impedance, i.e. it forms an impedance transformer, therefore maximalizing the amount of power coming into the antenna. This piece of copper is also used to tune the antenna.
There has been suggested that the size of the groundplate should be between 0.8λ and 1.1λ (Silver 2011). As this was first unkown the initial antenna had a diameter of 84 mm. This information was known while making the second helical antenna, so this antenna has a groundplate with a diameter of 50 mm (so a little over 1.1λ).
Both antennas are supported by a plexiglass cylinder which is located within the empty space in their helix. This ensures more stability for the helix and is assumed to have no or minimal influence to the radiation of the antenna. The remaining characteristics of both antennas are summarized in table 1.
Parameter Value for helix antenna 1 Value for helix antenna 2
Diameter helix 14 mm 14 mm
Distance between turns 9 mm 9 mm
Number of turns 15 15
Size of the groundplate 84 mm 50 mm
Table 1: Overview of the parameters for the design of two helical antennas.
3.5
Reflection
The antennas provide a B-field and as clarified in the section before, the strength of this field is of interest. Measurements of the reflection and of the radiated signal have been executed for both helices.
(a) (b)
Figure 12: A) Set-up used for a baseline signal of a reflection measurement. A RF-signal generator generates a signal of 6845.00 MHz and -15.0 dBm, which goes through a directional coupler, after which a spectrum analyser measures reflected power. B) Set up used for a reflection measurement of the helix. After a signal of 6845.00 MHz and -15.0 dBm is generated, the signal goes through the circulator to the antenna. What part of the signal is reflected, is measured by the spectrum analyser.
generator2 provided a signal of 6845.00 MHz, while being locked to an external reference signal. This
signal went through a directional coupler/circulator, where it left at point 2 as depicted in figure 12a. It was sent through a open wire, so a wire which was not connected to anything, and is thus assumed to reflect everything. This was used as a reference power to compensate the setup losses. The signal
that was returned to the circulator was sent via point 3 through the spectrum analyser3, which was also
connected to the same external reference. This was the reference/base line measurement: it measured how strong the 6845 MHz signal was after the path in figure 12a was taken. After this, the antenna was connected to the disconnected wire as shown in 12b. Whichever part of the 6845 MHz signal now returned from point 2 indicated the losses by the antenna. The measurement with the open wire is de-fined as 100% reflection and by measuring what part of the signal is returned compared to this baseline measurement the reflected power ratio can be determined.
What makes a reflection measurement convenient is that it tells us something about the emission. Three things can happen to a signal: it can be reflected, emitted and absorbed. If it is clear what part of the signal is reflected, this tells us what part of the signal is emitted and absorbed. If the absorbtion would be zero, the percentage of emitted power would be immediately clear from this reflection measurement. However the absorbtion is rarely zero so the reflection measurement only indicates the maximum power that could be emitted/radiated.
For the experiment a high intensity of radiated MW-pulses is wanted as was explained before. Since measuring the reflection of the different antennas indicates whether a high intensity of emission is possible, the measurement described above was done for both the helical antennas and the patch antenna. For the second helical antenna, which has a smaller groundplate, a signal of 6832.00 MHz and -11.0 dBm was generated by the RF-signal generator for this measurement.
The result is shown in table 2. There is calculated what part of the signal is maximally emitted. An example: if the difference between the baseline measurement and the reflected signal is -22,6 dB, this means about 1/(2*100) of the power is reflected so 1-1/(2*100) is maximally emitted. This calculation is used for the patch antenna, but the calculation remains the same for the other antennas. The table shows that the patch antenna in theory could radiate the most power, although helical antenna 2 also could have a high intensity.
A different reflection measurement using a network analyser4was done as well for both helical antennas.
2Hittite HMC-T2100
3Rhode & Schwarz FSH8 Spectrum Analyser 4Rhode & Schwarz ZVA Vector Network Analyser
Antenna Reflected signal (dB) Maximum emission(%)
Patch antenna -22,6 99
Helix antenna 1 -12,2 94
Helix antenna 2 -16,8 98
Table 2: Reflection of multiple antennas. The difference between the baseline measurement and the measurement with the helix antenna in the setup is shown. This provides the reflected power ratio and therefore indicates the maximum percentage of the signal which is emitted. This is therefore displayed as well.
With this analyser it was possible to measure the reflection while sweeping to a whole range of frequencies creating, amongst other things, a clear view of the resonance frequency of the antenna. The reflected signal was measured in the same way as before: a signal is sent to the antenna and measured is how much of the signal is returned. Since the wires etc. will also reflect a part of the signal, first a calibration takes place. The result is shown in figures 13 and 14.
4 , 0 4 , 5 5 , 0 5 , 5 6 , 0 6 , 5 7 , 0 7 , 5 8 , 0 8 , 5 9 , 0 9 , 5 1 0 , 0 - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 R e fl e c ti o n [d B ] f r e q [ M H z ]
Figure 13: The reflection of the helix antenna with groundplate of 84 mm. The x-axis shows the frequency and the y-axis the strength of the signal in dB. The dip around 6.8 GHz shows were the resonance frequency of the antenna is. The fringes are due to noise.
5 , 0 5 , 5 6 , 0 6 , 5 7 , 0 7 , 5 8 , 0 8 , 5 9 , 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 R e fl e c ti o n (d B ) f r e q [ M H z ]
Figure 14: The reflection of the helix antenna with a groundplate of 50 mm. The x-axis shows the frequency and the y-axis the strength of the signal in dB. The dip around 6.8 GHz shows were the resonance frequency of the antenna is.
dips in both figures are broad, which indicates a low sensitivity for a change in frequency, so a low Q-factor. The only difference between the two antennas seems to be that the antenna with the larger groundplate has more oscillations within the dip. These fringes are due to the faulty calibration of the network analyser and should be regarded as noise. The measurement of figure 13 and 14 show some difference with the measurements displayed in figure 2. This is due to the fact that in the directional circulator the signal is not perfectly passed along; some of the signal ’leaks’ away.
3.6
Emission
A measurement of the radiated power was also executed. The set up is shown in figure 15a. The
(a) (b)
Figure 15: A)Set-up used for measuring the emitted signal. A RF-signal generator provides a signal of 6845.00 MHz and -15.0 dBm, which goes through an amplifier of 42 dB, and then reaches the antenna. The spectrum analyser measured the strength of the 6845.00 MHz signal, radiated by the helical antenna. The distance measured between the top of the antenna and the probe is varied. B) Set-up used for measuring the emitted signal with glass or copper coil. The set-up works the same as A, however on 2 cm distance from the top of the antenna a glass plate or a copper coil is placed, after which the distance between the probe and the antenna is varied.
impedance of all the used devices is assumed to be 50 Ω. A signal of 6845.00 MHz was generated by a RF-signal generator to an amplifier, after which the signal reached the antenna. A probe connected to a spectrum analyser intercepted parts of the radiated signal. The spectrum analyser depicted the signal in dB and thus indicated the power of the signal from the antenna in comparison with the background. This means the amplification of the signal on 6845.00 MHz was registered. The distance between the probe and the end of the antenna was varied to study the pattern of the signal radiated by the antenna. Since the antenna will eventually be used in the experiment in front of a glass vacuum chamber and between different metal objects, coils etc., this same measurement was repeated with a glass plate placed 2 cm from the antenna with a varying distance between the antenna and probe. The same was done for a copper coil placed 2 cm from the antenna. Figure 15b shows this set-up. The results of these measurements are shown in figure 16. The figure shows how in the first 4 cm measured from the top of the antenna there can be no clear prediction of where the maximum of the signal will be: this location differs per measurement. What becomes clear, is that after those four cm the further away the probe is placed, the weaker the signal becomes. The figure also indicates that from 4 cm onwards there can be now coherent answer to the question on the influence of other attributes on the strength of the signal, since the coil seems to amplify the signal but the glass plate seems to attenuate it. Since the antenna will be surrounded by other attributes within the actual experiment this measurement won’t forecast whether the signal will then be weakened or strengthened by those attributes. The experiment also shows the instability of the system: the errorbars show that the antenna do not radiate a constant signal. The stability of the signal would increase by making notches in the plexiglass in which the helix would fit. The helix would not move as much and be less fragile.
The same set-up as shown in figure 15a is used to compare both helix antennas. Figure 17 shows the result. A clear difference can be distinguished between the two helix antennas; the antenna with the smaller groundplate, antenna 2, radiates more power. It is also visible that the further the antenna is
0 2 4 6 8 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 S ig n a l re la ti v e t o n o is e (d B ) D i s t a n c e p r o b e - h e l i x ( c m ) M e a s u r e m e n t 1 M e a s u r e m e n t 2 C o i l - P r o b e 2 c m G l a s s - P r o b e : 2 c m
Figure 16: The power of the emitted signal on 6845.00 MHz of the helix antenna with a groundplate of 50 mm as a function of the distance between the antenna and the probe, relative to the noise signal. The influence of a glass plate and a metal plate placed between the probe and the antenna is also shown.
0 2 4 6 8 1 0 1 2 1 4 2 0 2 5 3 0 3 5 4 0 4 5 s ig n a l re la ti v e t o n o is e (d B ) D i s t a n c e ( c m ) H e l i x A n t e n n a 1 H e l i x A n t e n n a 2
Figure 17: The power of the emitted wave on 6845.00 MHz of the helix antenna with the groundplate of 84 mm (helix antenna 1) and of the helix antenna with the groundplate of 50 mm (helix antenna 2), relative to the noise signal, as a function of the distance between the antenna and the probe.
placed from the receiver of the signal, the weaker the signal.
As mentioned before, the influence of the number of turns on the intensifaction of the signal on 6845.00 MHz has also been investigated. The set-up shown in figure 15a was used to vary the distance between the probe and the helix, registering the amplification of the 6845.00 MHz signal and repeating this for a helix with a different number of turns. The results are shown in figure 18.
It shows there can be no clear conclusion as to the question what the ideal number of turns of the helix would be for a maximum power when varying the number of turns between 13 and 15. At some distances more turns seem to be preferable when at other distances less turns seem to be better and all the measurements are within each other’s margin of error.
3.7
Antenna in magnetic chip set-up
Antenna 2 is also placed in the experimental set-up for the magnetic chip. The antenna is not placed directly in front of the vacuum chamber with the magnetic chip, as is the case for the patch antenna, but outside of the chamber, coming in from an angle from the x-axis. After the placement in the experiment,
6 8 1 0 1 2 1 4 1 6 1 8 1 0 1 5 2 0 2 5 3 0 3 5 4 0 s ig n a l re la ti v e t o n o is e ( d B ) d i s t a n c e ( c m ) 1 3 , 5 t u r n s 1 5 t u r n s 1 3 t u r n s
Figure 18: The strength of the emitted signal on 6845.00 MHz as a function of the distance between the helix antenna 1 (groundplate 84 mm) and the probe for a helix of 13, 13.5 and 15 turns.
two types of measurements were done to compare the performance of the helical antenna to the patch. First, a measurement which drives the different possible transitions from |1, −1i to |2, −2i, |2, −1i and
|2, 0i. Since σ+microwaves can only attribute to the transition to |2, 0i, the measurement with the helix
is expected to show this transition the most prominently, compared to the other transitions and to the measurement with the patch antenna. The second measurement is a Rabi-oscillation. When comparing this to the patch antenna, one would expect a higher Rabi frequency, since the characteristic of the helix
(directional, σ+-polarization etc.) implies a higher intensity.
The result of the first measurement is shown in figure 19. Against expectation, the transitions which
Figure 19: The graph shows the number of excited atoms as a function of the frequency, radiated by either the patch or the helical antenna. Depicted is how the detuning was decreased from +6 MHz to 0 from a pulse of 6827.00 MHz. The three dips represent the transition from |1, −1i to |2, 0i, |2, −1i and |2, −2i (in that order, from left to right).
require π and σ− light are visible as well, and the depth of the peaks show little difference from the
peaks of the patch antenna. The time the atoms were exposed to the MW-pulse was shortened to exclude the posibility that there not being enough atoms would be the reason for the lack of difference in the measurements.the only reason why there is no difference visible, is that fact that there are not enough atoms. This did not show a different result.
reflection can change the polarization of the light and therefore create the possibility to drive other transitions. If this is the only cause for the result in figure 19, this problem cannot be solved easily: trying to adjust the reflection inside the vacuum chamber would influence many other factors as well.
4
Decoherence time
4.1
Rabi oscillation
Before the Ramsey scheme was carried out, the π-pulse was determined by carrying out a Rabi oscillation. The result is shown in figure 20. The π-pulse was estimated to be between 1.3 and 1.5 ms. This result was later used in the Ramsey measurement. The Rabi-oscillation was fitted by Eq.(9) with the sum of least squares method after which the decoherence time was determined to be 21 ms. The Rabi frequency was also fitted which resulted in a value of 384 Hz (so Ω = 2π · 384).
Figure 20: A Rabi oscillation with a fit. The patch antenna radiated a signal of 6.83 GHz, together with the RF-antenna creating an on resonance pulse for the transition from |1, −1i to |2, 1i. The data provided information on the number of excited atoms as a function of the detuning. Therefore, Eq.(9) times the maximum number of excited atoms, was first used to fit both the decoherence time and the maximum number of excited atoms. The latter was then used to normalize the data. The provided value of the decoherence time, 21 ms, was used to draw the shown fit line with Eq.(9).
4.2
Ramsey fringes
As mentioned, after establishing the proper field and the Rabi frequency a Ramsey scheme was carried out. In figure 21a the detuning from the resonance frequency of the RF-pulse was 3 kHz and the scheme
was carried out for the delay time T = 22 ms. The pulse time was 0.75 ms (π/2-pulse)5. In figure 21b
Eq.(6) times the largest number of excited atoms has been plotted for the same conditions. A wiser choice would be to multiply Eq.(6) by the total number of atoms present, however this could not be determined.
The figures show Ramsey fringes. The small fringes are a result of the detuning in combination with the time T . As can be most easily seen from Eq.(6), a minimum in the excitation probability occurs when T δ = (n + 1)π and a maximum when T δ = (2n)π. This causes the small fringes. This means that the number of fringes in 1 kHz is given by 1000/T since δ = 2πf . This explains one of the differences between the two figures. Since the stepsize of the measurement was 50 Hz, the system was undersampled
(a) (b)
Figure 21: A)Display of the number of excited atoms as a function of the detuning from the resonance frequency after applying a Ramsey scheme, for T = 22 ms and τ = 0.75 ms. Steps of 50 Hz were taken. B) The theoretical prediction of the plot on the number of excitated atoms as a function of the detuning for the same circumstances as A, but with stepsize of 2,5 Hz.
since each fringe is about 45 Hz wide. The fact that figure 21b uses Eq.(6) and not Eq.(7), which takes into account damping, is not of influence. As can be seen by comparing Eq.(7) to Eq.(6), the damping does not influence the width of one fringe as this is still determined by setting T δ to (n + 1)π.
The envelop of both figures however, does resemble. The width of the envelop is determined by the terms
with ω1τ . Setting this product equal to 2π and solving for the detune frequency, results in finding the
width of the envelop, in this case about 1300 Hz. Since the data have been collected in a range of 3 kHz, this envelop and even the second one (which is found by setting the product equal to 4π) is visible. The influence of the decoherence becomes visible when comparing the figures, even though Eq.(6) hasn’t been multiplied with the correct number of atoms. The correct number of atoms would only raise the top of figure 21b with a certain factor, but the overall shape of the plot would not change. In this way the data show that the dephasing results in the minima of the fringes being less deep, as mentioned in
paragraph 2.4. A part of the atoms can no longer be adressed in the way that is desired by the π2-pulse;
they have developed freely on the Bloch sphere and the pulse no longer has the effect of bringing them to the preferred state.
To further investigate this property the measurements and its steps are repeated, but the range of the detuning and the stepsize are reduced to make sure there’s no undersampling. A larger variation of delay time T is applied, namely 10, 20, 40, 60 and 80 ms. The result is shown in figure 22. Since the range no longer lies in the kHz, the envelopes are no longer visible. The fringes that are registered occur in the top of this envelope.
It becomes clear that the frequency of the fringes increases as the delay time is raised. This is as expected: once the delay time increases, δT = π is reached quicker (the vectors are faster in completing a circle on the Bloch sphere). In that case, the detuning has a larger effect on the excitation probability.
The equations for the excitation probability dictate that on resonance a maximum should occur. This makes sense since the dephasing is limited because of the limited amount of detuning. This knowledge combined with a plot such as figure 22 can tell whether the pulse, which is thought to be on resonance, indeed is: if there is no maximum when δ = 0, but e.g. 24 Hz from δ = 0, the resonance pulse should be adjusted with 24 Hz. This is exactly why the data for the T = 10 ms and T = 20 ms measurements have been shifted by 24 Hz in figure 22; to make sure a maximum occurs on resonance.
What is again visible is that the contrast is lowered when the delay times is increased and this is again as expected. The increasing of the delay time caused there to be more dephasing in the system and thus less atoms could be controlled by the pulses.
Figure 22: Ramsey fringes for delay times 10, 20, 40, 60 and 80 ms and a pulsetime τ of 0.65 ms. The data of the 10 and 20 ms delay time have been shifted for 24 Hz, since the pulse was off resonance.
To be able to put a number to this decoherence, Eq.(7) has been fitted to the data from 40, 60 and 80 ms. The data from 10 and 20 ms have not been fitted with this equation. This is because not much decoherence can be registered for these datasets. The data still closely resembles the theoretical prediction based on Eq.(6) in which no damping has been assumed. This is illustrated in figure 23 for 10 ms. It indicates that the decoherence time lies at least above 20 ms. Fitting Eq.(7) to the other data
Figure 23: The number of excited atoms as a function of the detuning after applying a Ramsey scheme for T = 10 ms and τ = 0.65 ms. The fit has been made with Eq.(6), which does not take damping into account
sets yields a decoherence time of 54±5 ms. Fitting to higher delay times, produces a higher decoherence time, which gives the upper limit on this number. The fit to 40 ms delay time provides the lower limit. The fit to T = 40 ms is depicted in figure 24a and the fit to 80 ms in figure 24b.
(a) (b)
Figure 24: A) The excitation probabilty as a function of the detuning after applying a Ramsey scheme for T = 40 ms and τ = 0.65 ms. The data provided information on the number of excited atoms as a function of the detuning. Therefore, Eq.(7) times the number of maximum excited atoms was first used to fit both the decoherence time and the number of maxium excited atoms. The latter was then used to normalize the data. The provided value of the decoherence time, 49 ms, was used to draw the shown fit line with Eq.(7). B) The excitation probabilty as a function of the detuning, after applying a Ramsey scheme for T = 80 ms and τ = 0.65 ms. The same method as with A. was applied, and the provided value of the decoherence time was 59 ms.
4.3
Contrast
As mentioned, the contrast changes with a different delay time. Therefore, the contrast has been plotted by taking the average of all the maxima and minima for each dataset (10, 20, 40, 80 ms) and using
Eq.(8). The result is shown in figure 25. An exponential of the form e−τct , where τc is the decoherence
time, has been fitted in the figure. This gives a decoherence time of 63 ms.
Figure 25: The contrast plotted as a function of the delay time. An exponential fit of the form e−τct has been made, which
provided a τcof 63 ms.
4.4
Rabi oscillation with helical antenna
Figure 26 shows Rabi oscillations with the helix antenna. It shows a higher Rabi frequency than the Rabi oscillation with the patch antenna, which is shown in figure 20. This shows a frequency of 419 Hz,
while the patch antenna showed a frequency of 384 Hz. However, the fit shows a decreased decoherence time of 14 ms compared to 21 ms of the patch antenna. The use of the helix antenna therefore provides
mixed results. Unexpectedly, it drives σ− and π-transitions as can be seen in figure 19, but it does raise
the Rabi frequency with about 9 %.
Preferably the Rabi frequency would be raised even more. An attempt to raise this could be made by decreasing the number of turns. As figure 18 showed, decreasing the number of turns to 13 or 13,5 would not change the strength of the signal on a fixed point. It has been suggested that this remains the case when further cutting down to 3 turns, but that this would widen the beamwidth of the signal (Weeratumanoon 2000). So if some turns would be cut from the antenna, while the antenna was placed closer to the vacuum chamber, more atoms would perhaps be exposed to the beamwidth of the strong B-field. Therefore the Rabi frequency could be raised even more.
Figure 26: A Rabi oscillation with a fit. The helical antenna radiated a signal of 6.83 GHz, together with the RF-antenna creating an on resonance pulse for the transition from |1, −1i to |2, 1i. The data provided information on the number of excited atoms as a function of the detuning. Therefore, Eq.(9) times the number of maximum excited atoms, was first used to fit both the decoherence time and the number of maxium excited atoms. The latter was then used to normalize the data. The provided value of the decoherence time, 14 ms, was used to draw the shown fit line with Eq.(9).
5
Discussion
The decoherence time in this thesis has been deduced using two different types of measurements: Rabi and Ramsey measurements. Theoretically deduced formulas have been fitted to the Rabi oscillations and Ramsey fringes and additionally an exponential decay has been fitted against the contrast of these Ramsey fringes. All these fits provided a number for the decoherence time, ranging from 49 to 63 ms. Two things are striking about this result. Firstly, the pattern which is found in determining the deco-herence time. Secondly, the small value of the decodeco-herence time.
First the pattern. It is interesting that when the delay time between the π
2-pulses is raised, a higher
decoherence time is deduced by the fit (see the results in paragraph 4.2 ’Ramsey fringes’). When the delay time is 40 ms the decoherence time is 49 ms, for 60 ms it is 54 ms and for 80 ms it is 59 ms. It is possible that this method of making a fit to deduce a decoherence time works better for longer delay times. Calculating the margin of error on the fit can indicate whether or not this is the case. Moreover, performing Ramsey measurements for delay times longer than 80 ms would also provide more information on this pattern; for example whether or not the deduced decoherence times will continue to rise with this pattern when the delay time is again raised by 20 ms.
Secondly, the small value of the decoherence time is striking. Previous measurements showed a deco-herence time of 110±20 (Ockeloen 2010) or even 500 ms. The first value was obtained by carrying out a Ramsey measurement, the second value by performing a Rabi measurement. Based on the results of this thesis, a Ramsey scheme would probably have provided a decoherence time reaching over 500 ms. Yet, when comparing these two previous measurements to the measurements in this thesis it is noticable that the previous ones were done using different antennas and a lower power.
It would be interesting to examine the influence of the power of the antennas on the decoherence. A Ramsey measurement in which the power of either the MW- or the RF-antenna is lower than in the measurements in this thesis, perhaps by using a smaller amplification, could be performed to investigate this possible relation.
Since the type of antenna also varies over the course of all these measurements, executing a Ramsey measurement with the helix antenna could perhaps indicate whether there is a relation between the geometry of the applied field and the decoherence. However, to know whether the change in decoherence time is due to the power or the type of antenna, the helix antenna must radiate at the same power as the previous antennas. Otherwise it would be impossible to determine which factor is decisive.
In this thesis it is assumed that the pulses themselves have no influence on the decoherence. It would be interesting to investigate to which extent this is true by performing a Ramsey measurement that includes a different duration of the pulses. Lowering the duration of the pulse beyond the current length of a
π
2-pulse would surely reduce the number of excited atoms, but this would not necessarily be a problem
as long as the remaining signal remains high enough to determine a decoherence time. If lowering the duration of the pulse would lead to a difference decoherence time it could be stated that there is a relation between the pulses and the decoherence, in which case the theoretical model in this thesis should be adjusted.
6
Conclusion
The main focus of this thesis has been on magnetically trapped atoms in a two level system between which a two photon transition is possible. The main goal was to raise the number of operations that can be conducted on the atoms before the decoherence time was reached. This unfolded into two specific questions; what the decoherence time of the system was and whether a helix antenna could be used to raise the Rabi frequency.
Applying the Ramsey spectroscopy resulted in finding a decoherence time of 56±7 ms using the macro-traps in the experiment. This raised different questions, of which the biggest one became how to raise the
decoherence time. Comparing the results to previous results led to the reasonable assumption that this value could indeed be raised. If the decoherence time is increased it would be particularly interesting to measure what happens to this value when the atoms are trapped in the microtraps in which it is desired that the atoms eventually function as a qubit system.
The use of a helical antenna with the right parameters indeed led to an increase in Rabi frequency. It raised from Ω = 2π · 384 Hz, which was the Rabi frequency when using the patch antenna, to Ω = 2π · 419 Hz . Improving the position and/or the number of turns of the antenna might lead to a further increase in Rabi frequency. However, the helix did not seem to provide all the advantages that were expected.
One of its characteristics, the σ+ polarization, was expected to lead to a decrease in the loss of atoms
due to π and σ− transitons. However, there seemed to be no visible difference between the patch and
the helix antenna when driving π, σ− or σ+ transitions.
All the above gives evidence that plenty of steps need to be taken before qubits can be controlled for a period long enough to perform a high number of operations between them. They might be able to function for a short period, but before the qubits can truly form the building block of a computer, more questions should be answered and more conclusions should be drawn.
