Characterization and compensation of stray electric fields in cold Rydberg atom experiments

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Universit`

a di Pisa

FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea Magistrale in Fisica

Tesi di laurea magistrale:

Characterization and compensation

of stray electric fields

in cold Rydberg atom experiments

Candidato:

Lucia Di Virgilio

Relatore:

Dr. Oliver Morsch

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Introduction

Atoms with at least one highly excited electron are known as Rydberg atoms. The interest in Rydberg atoms is principally due to their weakly bound elec-tron which makes them both easy to describe by hydrogen-like wavefunctions and to experimentally manipulate.

In fact, laser technology, supported by the availability of trapped atoms, has allowed controllable excitations to specific Rydberg states. In particular, this capability accomplished the study of a wide range of their properties. Rydberg atom properties scale strongly with the principal quantum num-ber n and useful scaling laws can be deduced [1]. For example, the binding energy scales as n−2 and lifetimes scale as n3_{. Those properties make}

Ry-dberg atoms extremely interesting for a variety of experimental applications. The first property allows to easily ionize them and then detect the produced ions, the second one makes Rydberg states an excellent resource for practical applications.

For example, the strong interaction between Rydberg atoms, which is described by the van der Waals mechanism, can induce an energy shift and block the excitation of multiple Rydberg atoms within a finite volume, an effect known as Rydberg blockade. By taking advantage of the strong inter-actions between these neutral atoms the first implementation of a controlled-NOT quantum gate using the Rydberg blockade was realized in [2]. Finally, the large extent of the electronic wavefunction, which scales as n2, and the large polarizability of Rydberg atoms, which scales as n7, make them ex-tremely sensitive to electric fields. Rydberg atoms are used in such diverse areas as quantum information [3], quantum optics [4] and electric field sens-ing [5].

The subject of my thesis is the investigation of cold trapped 87Rb Ry-dberg atoms in the presence of electric fields. The extreme sensitivity of high-lying Rydberg states to electric fields can be exploited to perform spec-troscopic measurements and to detect small fields [6][7][8].

In fact, electric fields shift unperturbed Rydberg energies and cause the iv

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INTRODUCTION v new eigenstates to be a linear combination of the unperturbed states due to the Stark effect. When the electric field is smaller than the ionization threshold, but high enough to strongly mix energy levels of different n, the excitation of the Rydberg state is strongly suppressed. The higher the prin-cipal quantum number is, the more sensitive the Rydberg state is. Therefore, a possible limitation of experiments which involve Rydberg atoms can be such a sensitivity to uncontrolled background electric fields.

A technique based on the Stark effect can be applied to compensate a stray electric field in the experimental volume [9] and reduce the residual field.

In my thesis the observation of the energy level of the 85S state and its overlap with the upper linear shifted Stark states is reported. The meas-urement of the energy difference between the 85S resonance and the energy position of the most down-shifted line of the upper manifold has been com-pared with theoretical Stark map. This allows us to evaluate the stray electric field in the vacuum cell in which the atomic cloud is formed. Theor-etical Stark maps are obtained by using a library written in Python, Alkali Rydberg Calculator (ARC) [10].

To compensate this uncontrolled electric field, we applied controllable electric fields in one spatial direction and measured the Stark shift of the nS state. Because the nS state is non degenerate, its energy exhibits a quad-ratic Stark shift. Therefore, by the representation of energies as function of the applied fields, we obtained a parabolic curve whose apex corresponded to the minimum electric field experienced by the atoms. Finally, a simple model has been developed to convert the applied voltage in the applied elec-tric field. From the value of the initial uncontrolled field and the maximum compensating one, we estimated the residual electric field at the position of the atoms. The application of further electrodes allowed us to compensate the background electric field in the other directions and led to a residual electric field around 30 mV/cm, which means that ∼ 85% of the background electric field could be compensated. These lower electric fields allowed us to measure Rydberg lifetimes in a wide range of principal quantum numbers 60 . n . 100. In particular, the good control of the electric field in the experiment was used to investigate the behavior of the measurement of the lifetime as function of the applied field.

The thesis starts by recalling in Chapter 1 the properties of Rydberg atoms, focusing on the effect of the electric field on alkali metal nondegener-ate (s, p, d, f) stnondegener-ates and on degenernondegener-ate (l > 3) ones [11]. The description of the experimental apparatus is given in Chapter 2, where in particular Ry-dberg atom excitation and detection techniques are discussed. In Chapter 3 I will present a simulation study to evaluate the direction and amplitude

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INTRODUCTION vi of the applied electric fields at the position of the atoms. The simulation is done using COMSOL Multiphysics software, which allows one to solve physics problems by modelling geometry, materials and initial conditions [12]. Chapter 4 reports the use of the ARC package and in particular of the StarkMap Class, which returns the theoretical Stark map of a specific unperturbed state[10]. The experimental results of characterization and compensation of stray electric fields in Rydberg atom experiments, men-tioned above, are discussed in Chapter 5. Finally, Chapter 6 is devoted to the description of the measurement of the lifetime of Rydberg atoms in a wide range of principal quantum numbers at different fixed electric fields. Also the measurement of a specific S state lifetime as function of the residual electric field in the cell is studied.

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Chapter 1

Rydberg physics

1.1 Introduction to Rydberg atoms

Atoms with an electron in an energy level with high principal quantum num-ber n are known as Rydnum-berg atoms. Rydnum-berg atoms were originally identified in the Balmer’s spectral series of Hydrogen. Their name is due to Johannes Robert Rydberg who described the distribution of spectral lines in spectra of atoms, notably alkali atoms. Although the first spectroscopic studies re-lated to Rydberg atoms began before the advent of Quantum Mechanics, the interest in them emerged in the nineteen seventies when laser technolo-gies made it possible to excite well defined Rydberg states in order to study their properties. Rydberg states have properties that scale strongly with the principal quantum number n. They are atoms with extreme properties[1].

We have to discriminate hydrogenic Rydberg states from nonhydrogenic ones. In fact for Hydrogen we have a 1/r Coulomb potential, but for non-hydrogenic atoms this potential is no longer applicable. More particularly, if the valence electron is far from the electronic core, we expect properties similar to Hydrogen. On the other hand, when the electron is close to the electronic core the charge distribution of the core plays a role. The electron which enters the electronic cloud around the alkali nucleus, is exposed to a more unshielded positive charge and its energy increases. This effect is described by introducing a quantum defect δn,l,j which mostly depends on

the angular momentum l and slightly on the principal quantum number n and on the total angular momentum j [13]. Therefore, there are interesting differences of properties between low l and high l states of alkali atoms. In the higher l states the electron is subject to the centrifugal term and it does not sample small r, so the quantum defect is small. In Tab.1.1 we show δn,l,j

for fixed n and different l, whereas in Tab.1.2 δn,l,j for l = 0 as function of

n is given. The effect of the l-dependent quantum defect is that the binding energy of the atom is given by:

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CHAPTER 1. RYDBERG PHYSICS 2 Quantum defect δn,l,j State δn,l,0.5 85S 3.1312 85P 2.6549 85D 1.3464 85F 0.0165 85G 0.0041

Table 1.1: Quantum defect for87_{Rb 85l states obtained by using Alkali Rydberg}

Calculator, ARC [10]. Quantum defect δn,l,j nS State δn,0,0.5 40S 3.13131 60S 3.13124 80S 3.13121 100S 3.13120

Table 1.2: Quantum defect for 87_{Rb nS states obtained by using ARC.}

0 = −

Ry0 (n − δn,l,j)2

(1.1) where Ry0 = Ry∞× _m_eM_+M is the Rydberg constant of the specific element

which depends on Ry∞' 13.6 eV and on the nuclear mass M . It follows that

alkali energy levels are lower than those of Hydrogen for the same principal quantum number. Another interesting consequence is that the degeneracy is lifted for low l states (s, p, d, f), whereas the hydrogenic behavior is maintained for l > 3 states.

Also the value of radial matrix elements is crucial for the calculation of many properties of Rydberg atoms. In fact, we will show that by the combination of radial matrix elements and the energy difference between Rydberg states, it is possible to infer the n scaling of many properties of Rydberg atoms[1].

1.2 Lifetime of Rydberg atoms

The radiative lifetime of a nl state is given by the Einstein coefficient An0_l0_,nl,

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CHAPTER 1. RYDBERG PHYSICS 3 An0_l0_,nl = 4e2ω3_n,l,n0_,l0 3~c3 lmax 2l + 1|n 0_{, l}0 rn, l|2 (1.2) Therefore, A value depends on ω3 and on the radial matrix elements between the nl Rydberg state and the low-lying n0l0 states. In particu-larly, for high nl states Rydberg energy levels approach each other and we can consider ω as a constant. This means that the value only depends on | hn0_l0_{|r|n, li|}2

which scales as n−3[1]. The radiative lifetime is simply the inverse of the total radiative decay rate:

1 Γ0 = τ0= X n0_l0 An0_l0_,nl −1 (1.3)

Thus, the lifetime of Rydberg states scales as n3.

For Rydberg states with n ∼ l this formula does not apply. The n, l = n − 1 Rydberg state can allow the transition to the n0 = n − 1, l0 = l − 2 state. The energy difference between these two states is given by :

δ0 = − Ry0 (n − δn,l,j)2 + Ry 0 (n − 1 − δn0_,l,j)2 ∼ 1 n3 (1.4)

Since | hn − 1, n − 2|r|n, n − 1i|2 ∼ n4 _{and ω}3

n0_,l0_,n,l ∼ n−9, so τn,n−1 scales

as n5_.

However, experiments with cold Rydberg atoms are usually performed at room temperature when blackbody radiation (BBR) induces depopulation of the initially populated Rydberg state towards higher and lower lying states [14]. The reason is given by two facts. First, energies of high-lying Rydberg states are very close and their spacing is smaller than kBT at a

room temperature of 300K: ∆nSnP₀ ∼ 10−5_{eV < k}

BT ∼ 1/40 eV. Second,

dipole matrix elements of the transitions between Rydberg states are large [1]. Room-temperature measurements of lifetime of Rubidium nS, nP and nD Rydberg atoms have been performed for 26 < n < 45 [15] [16]. The rate of BBR-induced transitions is expressed as a function of the number of BBR photons per mode ¯n:

Kn0_l0_,nl= ¯nA_n0_l0_,nl (1.5)

where the photon occupation number ¯n is given by: ¯ n = 1 e hν kB T − 1 (1.6)

A typical transition frequency from the ground state of an atom is around ν ∼ 3 × 1014Hz, whereas a transition between two Rydberg states is around ν ∼ 3 × 1011Hz. If we recall that the frequency 1 eV/h ∼ 2.417 × 1014Hz,

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CHAPTER 1. RYDBERG PHYSICS 4

Figure 1.1: Spontaneous decays (red bins) and BBR-induced transitions (green bins) for 30S1/2 to nP1/2,3/2states at 300 K [10].

then hν kBT , where hν is the energy difference of the transition between

the two Rydberg states and kBT is the thermal energy at room temperature.

Therefore ¯n ≈ kBT

hν is large. In this case the effect of the BBR radiation is

significant because a large ¯n means a large ΓBBR given by:

ΓBBR=

X

n0_l0

An0_l0_,nln¯ (1.7)

On the other hand, transitions with frequency ν ∼ 1014Hz from the ground state imply hν > kBT and ¯n 1, which means that blackbody radiations

are unimportant. Finally, the effective lifetime of a Rydberg state is determ-ined by the sum of the radiative and BBR-induced rate:

1 τef f = Γ0+ ΓBBR= 1 τ0 + 1 τBBR (1.8) The Fig.1.1 shows the spontaneous and BBR-induced decay for the Rubid-ium 30S1/2 state for the environment temperature of 300 K calculated by

using ARC [10]. Green bins highlight the importance of the BBR redistri-bution of the 30S_1/2 state among the nearby P states.

1.3 Electric field effect

Rydberg atoms have a large spatial extent and as the principal quantum number increases the size of the orbit of the valence electron increases as n2. Also the binding energy is related to n by 0 ∝ n−2, and the more n

increases the less bound the electron is. This weakly bound orbit makes Rydberg atoms easily sensitive and manipulated by electric fields.

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CHAPTER 1. RYDBERG PHYSICS 5 An alkali-metal atom valence electron in the presence of an electric field can be described by the single particle Hamiltonian:

H0 = H0+ V (1.9)

where H0 is the unperturbed Hamiltonian and V = eE ˆz is the Stark poten-tial due to an electric applied field along the ˆz direction. Effects of electron and nuclear spin as well as the fine structure are now neglected. To find the energy of the atom in the presence of an electric field, approximation meth-ods as the perturbation theory can be suited but it is valid if hVi H0_,

otherwise the exact diagonalisation of the Hamiltonian H0 is necessary. In our experiments the maximum electric field experienced by atoms is around ∼ 200 mV/cm, so for small electric fields the perturbation theory is applic-able as we will estimate below. The derivation of the perturbation theory presented in this chapter follows from the discussion presented by K. Konishi and G. Paffuti in [17].

1.3.1 Perturbation theory for non degenerate states

To understand how eigenvalues and eigenvectors change with the presence of a perturbation, we can develop the following discussion valid for non degenerate states, in our case for nonhydrogenic atoms with δn,l,j 6= 0 which

lifts the degeneracy.

In the presence of a small perturbation we can assume that the new Schr¨odinger equation, that is:

H0|ψi = |ψi (1.10)

owns a solution similar to the unperturbed one. Therefore the |ψi state and the eigenvalue can be written as:

|ψi = |ii + |ψ(1)i + |ψ(2)i + .. (1.11) = 0+ δ(1)+ δ(2)+ .. (1.12)

where |ii is the initially unperturbed state which satisfies the unperturbed equation H0|ii = ₀|ii, whereas δ(n) _{and |ψ}(n)_{i are the nth correction to}

the eigenvalue and to the eigenvector respectively. Moreover, because the unperturbed states are a complete set, |ψi takes the form:

|ψi = |ii +X

k6=i

ak|ki ≡ ai|ii + |φ⊥i (1.13)

By replacing Eq.1.11 and Eq.1.12 in the new Schr¨odinger equation Eq.1.10, we obtain:

(H0+ V)( |ii + |ψ(1)i + |ψ(2)i + . . .)

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CHAPTER 1. RYDBERG PHYSICS 6 By equating the terms which contain (explicit and implicit) powers of V, we have a system of equations:

H0|ii = ₀|ii (1.15a)

V |ii + H0|ψ(1)i = 0|ψ(1)i + δ(1)|ii (1.15b)

V |ψ(1)i + H0|ψ(2)i = 0|ψ(2)i + δ(2)|ii + δ(1)|ψ(1)i (1.15c)

. . . = . . . (1.15d)

The first equation is true because it is the unperturbed Schr¨odinger equation for the |ii state. For the second, if we apply the dot product with |ii, we obtain that:

δ(1) = hi|V|ii (1.17) If we apply the dot product between Eq.1.15b and an unperturbed state |ki, where k 6= i, we find: |ψ(1)i =X k6=i hk|V|ii 0,i− 0,k |ki (1.18)

The explicit steps to get this solution are given below. From the projection of the |ki state, the equation Eq.1.15b becomes:

hk|V|ii + hk|H0|ψ(1)i = hk|0|ψ(1)i + hk|δ(1)|ii (1.19)

However, the last term is zero for orthogonality and we can write:

hk|V|ii + hk|H0_|ψ(1)_{i =}

0hk|ψ(1)i (1.20)

(0,i− 0,k) hk|ψ(1)i = hk|V|ψ(1)i (1.21)

The projection of |ki on Eq.1.13 makes explicit that ak = hk|ψ(1)i but it

is also true that hk|ψ(1)i = ₍hk|V|ψ(1)i

0,i−0,k) from the Eq.1.21, therefore by using

Eq.1.13, we obtain: |ψ(1)_{i =}X k6=i hk|V|ii 0,i− 0,k |ki (1.22)

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CHAPTER 1. RYDBERG PHYSICS 7 Proceeding by iteration, the corrections for the second order of the energy and the eigenstate are respectively:

δ(2) = hi|V|ψ(1)i =X k6=i | hk|ez|ii|2 0,i− 0,k E2 (1.23) |ψ(2)i =X k6=i X k0_6=i hk|V|k0i hk0|V|ii (0,i− 0,k)(0,i− 0,k0) |ki (1.24)

In particular, in our case we will deal with87Rb unperturbed non degen-erate states (so with non zero quantum defect) which have distinct parity, so δ(1) = hi|V|ii = 0. The energy shift caused by the perturbation is quadratic in the electric field:

δ(2)= hi|V|ψ(1)i =X k6=i | hk|ez|ii|2 0,i− 0,k E2 = −1 2αE 2 _(1.25)

where α is the polarizability. Because α is proportional to the square of the dipole element which scales as n4 and the inverse of the energy difference which scales as n3, the polarizability scaling is as n7. The absence of the linear term in the electric field means that the atom has no intrinsic dipole and the effect of the electric field is the deformation of the charge distribution and so we have an electric induced dipole moment.

In our experiments we have worked with nS states with principal quantum numbers around n = 85 and maximum electric field ∼ 200 mV/cm, so we obtain that the perturbation theory is a good approximation:

|δ(2)| = 1 2αE

2_{∼ 40 MHz |}

0| ∼ 490 GHz (1.26)

where α ∼ 2140 MHzcm_V22 for the 85S state and it has been calculated by

using ARC.

1.3.2 Perturbation theory for degenerate states

The physics of alkali metal Stark maps is completely determined by the quantum defect [11], in fact for l > 3 states δn,l,j ∼ 0 does not lift the

degeneracy of the binding energy Eq.1.1 and we have to consider them de-generate states.

The previous discussion is not applicable because some of the denomin-ators in energy shifts would be zero. In this situation we have to consider the perturbation theory for degenerate states [17].

We name E0 the vector space related to the degenerate eigenvalue of the

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CHAPTER 1. RYDBERG PHYSICS 8 their sum is the complete space of the states. The eigenstate |ψi can be written: |ψi =X α cα|αi + X k dk|ki (1.27)

where |αi are a basis of vectors in E0 and |ki a basis in E⊥. In this case the

Schr¨odinger equation H0|ψi = |ψi is given by: X α (H0+ V − )cα|αi + X k (H0+ V − )dk|ki = 0 (1.28)

By projecting this equation on a state |βi 6= |αi ∈ E0 and by considering

only the terms of the first order in V, we have:

((0− )δαβ + Vαβ)cα= 0 (1.29)

This means that at the first order in perturbation theory the Stark shift is given by the diagonalization of the perturbation on the vector space E0

related to the degenerate eigenvalue.

This discussion is applicable to Hydrogen atom because nl states have the same binding energy and to the degenerate l > 3 states of alkali metals. This means that all these states shift their energies as hα|eˆz|βi E because here dipole matrix elements are not zero. In fact nl states are eigenstates of the parity because spherical harmonics have definite parity, so an unperturbed state |αi can be coupled to all the |βi states of opposite parity. Linear Stark shift means Stark states have permanent electric dipole moment.

The Fig.1.2 shows the Stark structure of Hydrogen. The unperturbed 9S state (whose contribution is in dark red which corresponds to 0.2%) is mixed due to the electric field. This Stark map has been obtained by using ARC and it represents |n = 9, li states (manifold with dark red and gold lines) which crosses with the |n = 10, li states (upper gold manifold). The reason for which in the (n = 9, l) manifold dark red lines alternate with the golden ones is because only the matrix elements h9S|eˆz|βi with |βi of the opposite parity respect to the S state are non zero.

1.3.3 Inglis-Teller limit

The Stark shift is not large if the radial matrix element is small [1], in particular for Hydrogen in atomic units:

hn, l|r|n, l + 1i = −3n √

n2_{− l}2

2 (1.30)

This explains that for l ∼ n the Stark shift is negligible, so the states in the centre of the manifold are high l states. On the other hand low l states are the most down(up)-shifted lines and the previous equation gives a value around ±3n₂2.

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Figure 1.2: Stark map of the Hydrogen 9S state which is highlighted in the dark red colour. The n = 9 manifold crosses with the n = 10 fan (upper gold manifold). Also the lines of the n = 11 manifold are visible from ∼ 10500 V/cm.

In the case of Hydrogen, the electric field which corresponds to the in-tersection of the n + 1 most down-shifted state with the n most up-shifted one is given by the equality of the two energies, expressed here in atomic units (~ = e = me= 4πε0= 1): − 1 (n + 1)2 − 3E(n + 1)2 2 = − 1 n2 + 3En2 2 (1.31)

By resolving it one obtains the crossing field: E ∼ 1

3n5 (1.32)

which is called Inglis-Teller limit.

1.3.4 Nonhydrogenic atoms

Rydberg states of alkali metals have essentially similar characteristics to the Hydrogen atom in an electric field, but the effect of a finite core leads to important differences.

In fact to describe the Hamiltonian of a nonhydrogenic atom, we have to include within H0 a term Vd(r) which is the difference between the alkali

atom potential and the Coulomb potential [1]. Consequently this new term is non zero for low l states. The diagonalization of the new H0shows two new terms: a diagonal matrix element of Vd(r) which shifts further the energy

and an off-diagonal term hn, l, m|Vd(r)|n0, l0, m0i which lifts the degeneracy

of two crossing energies in a non zero field. This phenomenon is called avoided crossing and the amount of the split is given by:

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∆ = 2 hn, l, m|Vd(r)|n0, l0, m0i (1.33)

Therefore, an important difference between alkali metals Rydberg states and Hydrogen ones is that energy levels of different n do not cross as shown in Hydrogen Fig.1.2, but exhibit avoided crossings. When an avoided crossing is large enough that the resolution of the laser is finer than its size, it can be measured as was done in [11]. At the anticrossing the eigenstates are the symmetric and antisymmetric combination of the crossing states. For example in [18] the 28S − 24k manifold anticrossing signal is detected as a decrease in the 28S field ionization signal. Where, 28 and 24 are the principal quantum numbers n and k is the angular momentum of the zero field state which is connected to the Stark state [18].

1.3.5 Rubidium

For heavy alkali metals the fine-structure interaction has to be taken into account because it is proportional to the Rydberg constant of the specific element, that is Ry0 = Ry∞×_m_eM_+M and so it depends on its nuclear mass

[17]. The addition of the ξL · s term implies that the unperturbed basis to describe the problem has to be |n, j, mji and mj rather than mlis a ”good”

quantum number. This basis is used in ARC Stark maps, as we will show in Sec.4.2. The effect of the fine structure is visible in particular in the splitting of the P states as shown in Fig.1.3. Also 9F and 10D states and l > 3 states experience the splitting but it is smaller and not visible in the scale of the Fig.1.3. A comparison between Fig.1.2 and Fig.1.3 makes clear that alkali metal Stark maps are completely determined by the quantum defect.

Finally by resolving better the energy levels we show in Fig.1.4 the avoided crossing between the 9F and 12S state. We can notice that one of the lines due to the fine structure of the 9F energy exhibits a larger avoided crossing than the other one. This can be explained by observing that for low fields ml and ms can be considered almost ”good” quantum

numbers and due to the selection rules the interaction with the ml = 0 level

is bigger as has been observed in [18].

1.4 Ionization threshold

High n states are difficult to detect via fluorescence because the decay rate to the ground state decreases as n increases. However, the great sensitivity of Rydberg atoms to electric fields allows one to use field ionization and to detect easily the produced ions. An order of magnitude of the field required for the ionization is given by writing the Coulomb-Stark potential[1]:

Vp= −

ke2

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Figure 1.3: Stark map of the87_{Rb 12S state which is highlighted in blue.}

Figure 1.4: Avoided crossing between87_{Rb 12S state and 9F}

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Property (n∗)x Rb(5S)[19] Rb(85S) Binding Energy (n∗)−2 4.18 eV 2.06 meV Level spacing (n∗)−3 2.50 eV(5S − 6S) 48.7 µeV

Lifetime (n∗)3 _(5P

3/2− 5S1/2)26.2 ns 234.88 µs

Polarizability [MHzcm_V22] (n∗)7 79.4 × 10−9 ∼ 2143

Orbit radius [µm] (n∗)2 ∼ 29.8 × 10−5 0.532

Table 1.3: Properties of Rydberg atoms as function of the n∗ = n − δn,l,j

de-pendence. Values for the 5S ground state are tabulated in [19], those for the 85S Rydberg state are calculated by using ARC.

Where k = _4πε1

0 and ε0 is the vacuum permittivity. The equation Eq.1.34

has its saddle point at z = − q

ke

E where the potential Vp = −2e

√

keE. By equating Vp = 0, where 0 is the binding energy in Eq.1.1, we obtain the

classical field for ionization: Eion =

Ry02 4ke3_{(n − δ}

nlj)4

(1.35) Finally we show in Tab.1.3 a summary of some interesting properties of Rydberg atoms and their scaling as function of n∗ = n − δnlj.

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Chapter 2

Description of the

experimental apparatus

The main purpose of the experimental apparatus is the cooling and trapping of a cloud of atoms and their excitation to Rydberg states. In this chapter I will present a brief overview of the physical processes and experimental techniques which provide the manipulation of Rydberg atoms. Since the performed experiments involve a wide range of principal quantum numbers, I will explain how to change the Rydberg state. Finally, I will discuss the applied detection technique of Rydberg atoms and the role of electric fields in our experiments.

2.1 Magneto-optical trap (MOT)

Trapping atoms provides a well-controlled system to perform accurate meas-urements of atomic properties. The starting point of our experiments is based on the cooling and trapping of Rubidium atoms in the magneto-optical trap. This technique employs both optical and magnetic fields [20].

For simplicity, I describe the principle in one dimension, then I will extend it to three dimensions.

We consider an atom whose energy difference between the excited level and the ground one is ~ωA.

If an atom is at rest and absorbs a photon with momentum +~kL and

energy ~ωL= ~ωA, it is excited into the higher energy level. Within a time

scale of the order of the lifetime of the excited state, the atom can re-emit the photon in two ways: by stimulated emission or by spontaneous emission. In the first process, the emitted photon has the same energy and the same amplitude of the momentum along the same direction of the absorbed photon.

In the absorption process, the atom increases its momentum of +~kL

but in the stimulated emission process it releases a photon with momentum 13

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CHAPTER 2. EXPERIMENTAL APPARATUS 14 of +~kL; therefore, after this cycle, the momentum-energy balance is null.

In the second process, the emitted photon has the same energy and the same momentum, but it can be re-emitted in a random direction.

In fact, in the absorption process the atom increases its momentum of +~kL, but in the spontaneous emission it releases a photon with momentum

of ~|kL| in any direction; therefore, after this cycle, along the ˆkL direction

the momentum balance is different from zero.

This means that we can induce a transfer of momentum to the atom by using the cycle of the absorption process and the spontaneous emission. On various cycles of absorption and spontaneous emission, this transfer of the momentum is more clear because the average over numerous repetitions of the spontaneous emissions gives a null contribution to the atom motion. Thus, only the absorption process contributes to determine the final mo-mentum of the atom.

To selectively slow down the atoms, we need a force that acts on them which depends on their velocity and minimizes their velocity dispersion in order to cool the ensemble of the atoms. To this end, we use two counter-propagating lasers.

If the two counter-propagating laser waves have frequency ωL= ωA, the

absorption of the counter-propagating photons has the same rate, therefore the net force on the atom is null.

We suppose we want to slow down an atom with vˆz > 0 velocity along the ˆz direction. If we use two counter-propagating laser waves with frequency ωL < ωA on the moving atom, due to the Doppler effect, the

counter-propagating wave is shifted closer to the resonance than the co-counter-propagating wave, which is shifted away from the resonance, according to:

ωA= ωL 1 ±v c (2.1) By using red detuned lasers δ = ωL − ωA < 0, the absorption of the

counter-propagating photon is more frequent than the absorption of the co-propagating photon for an atom with vˆz > 0. Therefore, the transfer of the momentum to the atom, due to the absorption coupled to the spon-taneous emission of the first process, is larger than the transfer due to the second process. This allows one to use laser Doppler cooling to obtain an induced imbalance between two opposite radiation pressure forces.

This technique can be generalized to three dimensions by using three pairs of counter-propagating laser beams in the three orthogonal directions in order to provide the cooling of atoms down to the Doppler temperature TD = _2k~Γ_B, where Γ represents the natural line width of the excited state.

However, in order to also trap the atoms, it is necessary to have a position dependent force. This is realized through an inhomogeneous magnetic field. We consider a one-dimensional system in which an atom has only two energy levels where the angular momenta are Jg = 0 and Je= 1 for the ground and

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CHAPTER 2. EXPERIMENTAL APPARATUS 15

Figure 2.1: (a) Principle of the magneto-optical trap (MOT) in 1D [20]. Because of the Zeeman effect an atom in an inhomogeneous magnetic field shifts the atomic energies in z 6= 0. The atom in the +z position is closer to the resonance with the σ− laser beam than with the σ+ beam. The opposite situation happens for

an atom in the −z position. The gradient of the magnetic field causes atoms to feel a restoring force towards the 0 position. (b) 3D configuration of the magneto-optical trap composed of three pairs of counter-propagating laser beams σ+and σ−

polarized in the three orthogonal directions and a pair of coils in anti-Helmholtz configuration.

the excited state respectively. The Zeeman effect, due to the inhomogeneous magnetic field (Bz = −bz), splits the energy level of the excited state into

three levels given by m = 1, m = 0 and m = −1 in z 6= 0, as shown in Fig.2.1. For an atom in a small magnetic field B such that −µ · B Ahf s,

where µ is the magnetic dipole of the atom and Ahf sthe hyperfine splitting,

F is a good quantum number and the energy levels are given by:

= −gµBmFB = gµBmFbz (2.2)

To trap the atoms it is necessary to use two counter-propagating laser waves σ+ and σ− polarized, which have the same ωL frequency such that δ =

ωL− ωA< 0. The σ− laser beam allows the transition toward the m = −1

level, the σ+ beam toward the m = +1 level. The principle of the MOT

is that if the atom is in the z > 0 position, the σ− laser beam is closer

to the resonance than the the σ+ laser beam. Thus, the allowed transition

is towards the m = −1 excited state. For an atom localized in the z < 0 position the opposite phenomenon occurs and the allowed transition is towards the m = +1 excited state.

An atom in the presence of a gradient of magnetic field and in interaction with a pair of red-detuned counter-propagating laser beams experiences a force that can be expanded around the 0 position as:

Fz = −αvz− κz with κ = 2kLµzbs

−δΓ δ2₊Γ2

4

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CHAPTER 2. EXPERIMENTAL APPARATUS 16 where µz is the magnetic moment of the atom, δ < 0 is the red detuning, Γ

the spontaneous decay rate of the excited state and s the saturation para-meter which depends on the Rabi frequency, δ and Γ. This equation shows that in addition to the first friction term, which cools the atomic cloud, the second one causes a force on the atoms which points towards the 0 position, where they are trapped. The magnetic gradient is produced by a pair of coils in anti-Helmholtz configuration, Fig.2.1.

2.1.1 Experimental MOT setup

Our apparatus uses two quartz cells: in the first one two pairs of counter-propagating laser beams and one pair of coils in anti-Helmholtz configuration produce a 2D MOT. This one induces a continuous flux of atoms into the second cell where three counter-propagating laser beams and one pair of coils in anti-Helmholtz configuration produce a 3D MOT. The vacuum pressure inside the two quartz cells is 10−8mbar and 10−10mbar respectively.

The generalization of the principle of the MOT that we have described for a two-level atom requires some care. In fact, the Zeeman splitting of the ground and excited state of 87Rb atoms is more complex and displayed in Fig.2.2. In our experiments we use a magnetic gradient of ∼ 10 G/cm. Because the magnetic sublevels (F levels) of the 5S_1/2 and 5P_3/2states shift 0.7 MHz/G and 0.93 MHz/G respectively and their hyperfine structure is around at least 100 MHz [21], we obtain that the −µ · B Ahf scondition

is satisfied, so F is a good quantum number.

We use the transition between 5S_1/2, F = 2 →

5P_3/2, F0 = 3 to cool and trap the atoms. Atoms in the5P3/2, F0 = 3 state spontaneously decay

onto the5S_1/2, F = 2 state remaining in the trap cycle.

However, it is possible that the atoms in the ground state are off-resonantly excited to 5P3/2, F0 = 2. As a result of such an off resonance excitation,

the atoms can decay onto the5S_1/2, F = 2 state or onto the

5S_1/2, F = 1 state. In the latter case they can not be coupled again to the trap cycle be-cause of the selective rules. Thus, we need a repumping beam which couples

5S_1/2, F = 1 →

5P_3/2, F0 = 2 so that no atoms escape from the trap. In fact the 5P_3/2, F0 = 2 state remains in the cycle both it decays onto the

5S1/2, F = 2 state and onto the

5S1/2, F = 1 state. The Fig.2.2 shows

the trapping and repumping transitions for87Rb atoms.

The final temperature of atoms in our experiment is ∼ 150 µK. The density and the dimension of the trapped atomic cloud is observed through the fluorescent signal. We use a high resolution camera CCD, whose res-olution has been estimated to be ∼ 3.3 µm, to obtain the image and the dimensions of the atomic cloud. The camera captures the xz plane of the MOT, where x is the direction of excitation laser beams and z the vertical axis. From the integrated Gaussian profiles, we obtain the dimensions of the MOT in the two directions and an estimate of the number of atoms with an

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CHAPTER 2. EXPERIMENTAL APPARATUS 17 5S ½ 5P 3⁄2 7 8 0.2 41(2 1)nm Tr ap p ing R e p ump δ F = 1 F = 2 F ’ = 0 F ’ = 1 F ’ = 2 F ’ = 3 6.3 8 4 (6 8 ) GH z

Figure 2.2: Level scheme of the trap and repumping processes of 87_{Rb atoms.}

The dashed lines are the channels of decay. The red detuning is δ = −2.9Γ, where Γ = 2π × 6.0666(18) MHz is the linewidth of the 5P3/2 state [21]. The repumping

beam is required to trap and cool the atoms that have decayed into the lower hyperfine level of the 5S1/2 state.

error around 20%. The dimensions are σx ∼ σz ∼ 200 µm, the number of

atoms is around 105 with a peak density of 1010− 1011_atoms/cm3_.

2.2 Excitation into Rydberg states

The excitation into Rydberg states is possible by two-photon and sometimes by three-photon transitions. Also one-step excitation scheme of Rubidium atoms is possible involving a UV laser at 297 nm [22], but it does not allow to access the S-states. In our experiment the excitation into nS Rydberg states is accomplished by a two-photon excitation scheme.

We use lasers at 421 nm and 1012 nm for the 5S1/2 → 6P3/2 → nS/nD

scheme as shown in Fig.2.3. In all the experiments we detune the blue laser from the resonance of the 6P_3/2 state to avoid the population in this state in order to work with an effective two level system. In doing so, we choose a blue detuning δ6P = 40 MHz from the upper hyperfine sub-level of

the 6P_3/2 state, that is 6P_3/2, F = 3. This detuning is sufficiently large to avoid populating the intermediate state, on the other hand small enough to provide two-photon transition strength. The two-photon Rabi frequency is given by:

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ΩR=

ΩblueΩIR

2δ6P

(2.4) where Ωblue and ΩIR are the Rabi frequencies of the 421 nm and 1012 nm

laser radiations [23]. To obtain coherent coupling between the ground state and the Rydberg state we use ∼ 25 µW of the 421 nm laser power and ∼ 0.5 mW of the 1012 nm laser. In fact for the second step, the dipole matrix element is a thousand times smaller requiring more laser power for a comparable coupling strength.

In some of the experiments described in this thesis more power of the blue laser is necessary to observe the Rydberg signal. This is discussed in Sec.5.6.1 when the Stark structure of Rubidium atoms is reported. Owing to the higher power (∼ 0.35 mW) of the 421 nm laser, in those experiments a small population is excited in the intermediate state. By using a 1 µs pulse of the blue laser with a detuning of 40 MHz from the 6P_3/2 state, we detect a number of ions coming from the 6P3/2 state, that is N6P ∼ 0.85. In fact,

the atoms in the 6P3/2 state can absorb a 421 nm photon and be ionized.

Therefore, in those experiments, when both 421 nm and 1012 nm lasers are switched on, we reveal both ionized Rydberg atoms and ions coming from the 6P3/2 state. In these experiments we scan the frequency of the 1012 nm

laser, so when the number of detected ions is equal to the number of the only ions coming from the populated 6P_3/2 state (obtained by swtching off the 1012 nm laser), we can deduce that we excite no Rydberg atoms at those frequencies.

For the purposes of this thesis, we needed to excite Rydberg atoms in a wide range of principal quantum numbers. This is possible because the 1012 nm laser is an external-cavity diode laser which incorporates a diffrac-tion grating as a wavelength-selective element. Thus, an adjustment of the grating angle provides a wide coarse wavelength tuning range. Then, by the use of a wavemeter we select the transition for a well-determined Rydberg state.

2.2.1 Overlap of laser beams

We now present a discussion of the overlap of excitation laser beams in space and in time.

The blue and IR laser beams are co-propagating along the ˆx direction. The volume of the excited Rydberg states can be evaluated by considering a cylindrical volume whose dimensions are the MOT dimensions and the laser waists. In fact, the region of interest is given by the overlap of the two laser beams and the Rubidium atomic cloud. The blue and IR laser waists are around 40 µm and the 90 µm respectively. Therefore, if we take into account the waist of the blue laser wb = 40 µm and the σx MOT dimension,

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5S

1/2

6P

3/2

nS

1/2 δ 6P

Figure 2.3: Two-photon excitation scheme for the Rydberg excitation, via a vir-tual level obtained by the blue detuning δ6P from the 6P3/2 state. The atoms in

the ground state absorb a 421 nm photon and a 1012 nm photon to reach the nS Rydberg state.

V = πw2_bσx

Also a good timing and fast switching of the lasers are essential in these experiments. The fast switching is given by the use of acousto-optical mod-ulators (AOM) along the optical paths of the laser beams. An AOM is a device which can control both quickly and easily the frequency and direction of a laser beam, by means of a RF signal. In an AOM the incoming light is diffracted and its frequency is shifted by using the acousto-optic effect. A piezoelectric transducer is attached to a crystal which vibrates when a RF signal drives the transducer. These vibrations produce travelling sound waves which periodically modulate the index of refraction of the crystal. The change of the refractive index can then be used to diffract the laser beam if the Bragg condition is satisfied. When the incoming laser beam hits the AOM a scattering photon-acoustic wave happens with the absorption or emission of a phonon. In the case of the absorption of a phonon, the photon in output has frequency ωf in = ωin− ωRF, in the case of emission

ωf in = ωin+ ωRF. The use of a diffracted order as beam which arrives on

the atoms provides to switch on the laser radiation with very fast switching times of about 100 ns. The repetition rate of the experiment is 4 Hz. At the beginning of each cycle the MOT beams are turned off to prevent pop-ulation in the 5P3/2 level during the Rydberg excitation, which could lead

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Classical ionization threshold nS state Eion[V /cm] ∆E_ion(n+1)−n[V /cm]

70S 16.25 0.94 75S 12.18 0.66 80S 9.31 0.47 85S 7.23 0.34 90S 5.71 0.26 95S 4.56 0.19 100S 3.70 0.15

Table 2.1: Classical ionization threshold for some nS states. ∆E_ion(n+1)−nis the dif-ference of the ionization threshold between the nS and the (n + 1)S state expressed in V /cm.

photoionization of atoms in that level by absorption of a photon at 421 nm. In the experiments discussed in this thesis the excitation pulses are typically ≤ 1 µs. The measurements reported in this thesis present statistical aver-ages over typically 50 repetitions of the experimental cycle. Measurements of the lifetime of Rydberg atoms are performed by recording 100 repetitions.

2.3 Detection of Rydberg atoms

After excitation, we want to detect Rydberg atoms. An efficient method is the electric-field ionization and the ions detection with a channel electron multiplier (channeltron). An estimate of the classical ionization threshold can be given by using the equation [1], as discussed in Sec.1.4:

Eion =

Ry02 4ke3_{(n − δ}

nlj)4

(2.5) We have indicated the mass-corrected Rydberg constant Ry0 = Ry∞ ×

M

M +me, where M is the atomic mass of the nucleus of Rubidium, methe mass

of the electron and Ry∞' 13.6 eV. For example, the value of the ionization

threshold for the 85S state is 7.23 V/cm. The difference between neigh-bour nS states is ∼ 5%. Instead, the difference of the ionization threshold between S and P states of the same principal quantum number is around ∼ 2%. In Tab.2.1 we give some examples of classical ionization thresholds for the nS states and the difference between their values and the (n + 1)S value. In Tab.2.2 there are the classical ionization thresholds for the nS and nP states and the difference between their values.

To produce ionizing electric field pulses we use two front plates and two lateral plates external to the cell. The spatial arrangement is displayed in Fig.2.5. The ionized atoms are guided through a rectangular electrode

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Classical ionization threshold

n state Eion(S state)[V /cm] Eion(P state)[V /cm] ∆EionnS−nP[V /cm]

70 16.25 15.80 0.45 75 12.18 11.86 0.32 80 9.31 9.08 0.23 85 7.23 7.07 0.16 90 5.71 5.58 0.13 95 4.56 4.47 0.09 100 3.70 3.62 0.08

Table 2.2: Classical ionization threshold for some nS and nP states. ∆EnS−nP_ion is the difference of the ionization threshold between the nS and the nP state expressed in V /cm.

to the channeltron, which is positioned about 10 cm higher than the cell in order to not intercept the laser beams, and 15 cm from the centre of the cell [24]. The voltage on the front plates is +3500 V, the one on the lateral plates −1000 V and the rectangular electrode voltage is −1050 V. The channeltron is a vacuum-tube structure that multiplies incident charges, in this case ions. When an ion hits the internal part of this tube, which is covered by an emissive material, it triggers an avalanche of electrons that are accelerated by a potential voltage towards a metal anode at the end of the tube. Hence, the multiplied electrons are collected and detected as a current signal. In our experiment the applied voltage between the two extremities of the channeltron tube is Vchanneltron = −2500 V. The signal is observed on an

oscilloscope and appears as a series of peaks as shown in Fig.2.4. Each peak corresponds to the arrival of an ion at the channeltron and its height depends on the amplification in the channeltron. The number of peaks is counted by a peak finding routine. The overall detection efficiency of our system is around η = 40% [24]. It depends on the conversion of ions into current and on the geometrical structure: some charged particles cannot enter into the acceptance cone of the detector. The applied voltages are chosen to accelerate the particles in order to maximize the number of detected ions. In the remainder of this thesis we will always refer to the number of ions as to the detected ones. The number of real ions in the excitation volume is given by Nreal= N/η.

2.4 Role of the electric field

In our experiments the plates used to ionize Rydberg atoms are located outside the quartz cell. A continuous electric field application causes elec-trical charges on the cell walls [25] that we avoid by applying short ionizing

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Figure 2.4: Signal of ions recorded by the channeltron and shown on the oscil-loscope. Each peak corresponds to a detected ion. The horizontal scale indicates the time of flight of the ions (200 ns/div), the vertical scale the ion signal voltage (10 mV/div).

Figure 2.5: Scheme of the experimental setup. The excitation laser beams in-tercept the atomic cloud created in the quartz cell. The external front and lateral plates ionize the Rydberg atoms and the rectangular electrode guides them towards the channetron where the ions are collected. The four external plates close to the atoms are used to compensate the background electric field in the cell before the ionization pulse.

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CHAPTER 2. EXPERIMENTAL APPARATUS 23 electric pulses of 9 µs. However, we detected a background electric field, as discussed in the chapter on experimental results. The cause of this electric field is unknown and some hypotheses and estimates are given in the follow-ing chapter in which we simulated the electric fields in the experiment. We reduced the electric field in the cell by compensating the background elec-tric field first by using the front plates and then by installing four further electrodes, shown in Fig.2.5. The compensating electric field is switched on 200 µs before the excitation and it is switched off 1500 µs after it. With the application of the compensation pulse we prevent electrical charges on the cell walls and we provide to perform the experiments in condition of the minimum electric field.

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Chapter 3

Simulation study of the

electric field in the

experiment

The presence of an uncontrolled electric field in our cell made it necessary to apply controllable electric fields to compensate it. In this chapter I will present simulations in order to evaluate the direction and amplitude of the applied electric fields at the position of the atoms. The electric fields which act on the atoms are generated by plates external to the cell, as shown in Fig.3.1. The simulation is done by using COMSOL Multiphysics software, more simply referred to as COMSOL. Particularly, I will present the results for the compensation on the front plates, already present in our apparatus, and for the compensation on them and on the further four electrodes, which we have placed later.

3.1 About COMSOL

COMSOL is an interactive environment for solving and modelling problems of physics in many application areas which are categorized according to Modules[12]. Our interest is the simulation of electric fields and this is possible by using the AC/DC Module[26]. In this thesis I will not deal with of the specific use of COMSOL but just give the main ideas of how I used this software in order to present the results.

The modelling procedure starts by defining the geometry of the problem. In our case the geometry is given by the MOT apparatus, shown in Fig.3.1. In particular, the creation of the geometry model has been discussed in the thesis of Matteo Archimi[27], my contribution has been the addition of the four further electrodes and the studies that I will describe in this chapter. The geometry defines the domains under study, but it is necessary to as-sign the material properties to each domain. COMSOL has got predefined

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CHAPTER 3. SIMULATION STUDY 25

Figure 3.1: Model of the MOT apparatus. The cell is mounted on a central metallic structure and is composed of a cylindrical quartz cell and a parallelepiped one. In the second one a pair of coils provides the gradient to produce the MOT. Front and lateral plates are placed on the sides of the cell in order to create the electric fields needed for field ionization. Therefore, a rectangular electrode, placed inside the cylindrical quartz cell, guides the ions along the x direction towards the channeltron where they are collected for detection. The x coordinate is along the major axis of the cell, the z direction is parallel to the coils axis and the y direction is along the minor axis of the cell. The figure shows four further electrodes near the MOT position. We placed these to compensate the uncontrolled electric field in our apparatus.

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CHAPTER 3. SIMULATION STUDY 26 material data available to build models. In this situation, I have defined: copper for the external front and lateral plates, the coils, the rectangular electrode and the channeltron; steel for the cylindrical metallic structure and silica glass for both cells. The vacuum inside the cell is defined by using the function for adding user-defined materials. It is air with a pressure of 10−8Pa. The specific material parameters are used for the physics of the model. In particular, the whole of the system is modelled in air.

3.2 First Simulation

The first simulation studies the electric field at the position of the atoms when high voltage plates external to the quartz cell are used to ionize Ry-dberg atoms. This simulation does not include the four further electrodes because I want to compare these results to the experiments in which we had not yet installed those electrodes. Our Rydberg detection scheme is based on the ionization of Rydberg atoms by electric field pulses. In particular, the use of short electric field pulses is explained in [25], where the screening of the electric field was measured on Rydberg excitation spectra. In order to avoid electrical charges on the cell walls, now we use short electric pulses with a rise time of a few ns and 9 µs duration on the front and lateral plates. Then, the produced ions are guided to the channeltron through an high voltage rectangular electrode whose electric pulse duration is 30 µs.

To create the model for the electric pulses, I used the Heaviside function, defined in COMSOL by

y = f lc1hs(x, scale) = (

0 if x < scale

1 if x > scale (3.1) In the interval −scale < x < scale, f lc1hs(x, scale) is a smoothed Heav-iside function. Thus, by writing the equations below for front and lateral plates respectively, one obtains electric pulses of 9 µs.

1 3 5 0 0 ( ( f l c 1 h s ( t [1/ s ] -1 e -6 ,0.5 e -6) ) -( f l c 1 h s ( t [1/ s ] -9 e -6 ,0.5 e -6) ) )

2 -1000(( f l c 1 h s ( t [1/ s ] -1 e -6 ,0.5 e -6) ) -( f l c 1 h s ( t [1/ s ] -9 e -6 ,0.5 e -6) )

)

Similarly, one can define the electric pulse of 30 µs on the rectangular elec-trode at −1500 V. The three pulses start together. We want to simulate the electric field in the vacuum cell, in particular at the position of the atoms, when the field ionization is applied and after its application.

The atomic cloud is a small sphere with diameter of ∼ 200 µm at the centre of the cell which corresponds to the centre of the coils. Thus, it can be thought of as a point around 33.5 mm from the cell wall. This point is displayed with a pink or black spot in the plots in order to indicate its position.

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3.2.1 Basic steps

For simulating electromagnetic fields, COMSOL uses the AC/DC Module which allows one to model both stationary and time dependent problems.

The definition of the AC/DC Module imposes the setting of initial con-ditions of the electric potential (the default value is 0 V) and the definition of objects which satisfy the charge conservation condition. The initial values set initial conditions to solve Laplace’s equation for the electric potential. The rectangular electrode and front and lateral plates are defined as ter-minals by the Heaviside function. The cylindrical structure, on which the cell is mounted, is defined at ground. The coils in the experiment are at a small potential difference which is smaller than 1 V, so to simplify the sim-ulation I modelled the coils with a constant potential, and so zero potential difference, which is computed during the study.

These conditions are applied to the whole of the simulations. Before solving the problem, the following step is the discretization of the domains. This is achieved by meshing them. In fact, COMSOL is based on FEM (Finite Element Method Procedure) which allows to pass from the resolution of a problem defined on a continuous space to the resolution of this problem on a discrete space. Then COMSOL interpolates the computed function at the boundaries of each domain. This introduces some discontinuities in the graphs as in Fig.3.2. Finally, a time dependent study is chosen to obtain the solution. By the definition of a step time and a final time, COMSOL solves the electrostatic problems in the predefined range for each step time. The discussion of the results is done below.

3.2.2 Results

From this simulation it appears that at the end of the ionizing pulses the electric field in the cell is very small around 10−4mV/cm. Experimentally we observe a larger electric field around 200 mV/cm, which we measured via Rydberg spectroscopy as I will show in the following chapter. This uncontrolled electric field could be due to the accumulation of charges at cell surfaces. The electrical conductivity of glass walls which are exposed to caesium vapors has been studied in [28], where the authors show that the conductivity of cell walls is due to the adsorbed alkali atoms.

Another possible cause of stray electric fields is dust on the outside of the vacuum cell in [29], the authors measured the charge value of a particle of dust and found a mean charge ∼ 106e, where e is the elementary charge. By a simple estimate in which one supposes Q = 106e the charge of a particle of dust, one can calculate the distance at which that charge produces an electric field of ∼ 200 mV/cm. This distance is around 8mm. Thus, the presence of a single dust particle on the vacuum cell would be enough to cause the measured electric field.

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Figure 3.2: Electric field norm as function of the x coordinate. The first figure is taken when the front and lateral plates are ionizing the Rubidium atoms. The second one is taken when the only high voltage on the rectangular electrode is present. The lower figure shows the electric field after 20 µs from the high voltage pulses. The pink spot indicates the position of the atomic cloud.

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CHAPTER 3. SIMULATION STUDY 29 1µs 50µs Compensation Pulse -30V Front Plates High Voltage +3500V 50µs 59µs Lateral Plates High Voltage -1000V 50µs 59µs Grid High Voltage -1050V 50µs 80µs

Figure 3.3: The temporal sequence of the electric field pulses.

3.3 First compensation

The presence of the uncontrolled electric field made it necessary to use a controllable electric field in order to reduce the electric field in the cell. This has been obtained by the use of the front plates. We observed that the application of around −30V on the front plates reduced around 50% the electric field in the cell. In this second simulation, I modified the previous electric pulses by adding an electrical compensation pulse before the ioniz-ation in order to get the compensated electric field at the position of the atoms. The temporal sequence of the electric pulses and the application of voltages to the plates and to the accelerating rectangular electrode is shown in Fig.3.3. The results of the calculated electric field along the x direction is around −180 V/m, this is a factor ten bigger than the compensating electric field which we deduced from the experimental Stark maps, as shown in the following chapter. This disagreement can be due to the formation of wall charges (adsorbed Rubidium) with a screening of an external electric field [25]. Because of the symmetry of the problem, the applied electric field is zero along the z and y directions. I show in Fig.3.4 (blue line) the electric field along the x direction when the front plates are at −30V.

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3.4 Final compensation

The use of the front plates provides to compensate the electric field only along the x direction, to compensate it in the other two directions it is ne-cessary to install additional electrodes. I have modelled in COMSOL the electrodes which were specially designed and created by ourselves. Their dimensions have been chosen to adapt to the structure. Actually their thick-ness is 0.05 mm, but I could not use this dimension in the simulations. Even if the necessary meshing was possible, my computer could not run the calcu-lation. Therefore, in these simulations, the thickness is taken the minimum possible to be computed, that is 0.3 mm. Apart from the thickness, the design is the same. The distance between the electrodes and the cell walls is taken 1 mm according to the real distance.

The first simulation presents the same initial conditions which are dis-cussed at the beginning of this chapter, but now the compensation pulse is −30 V on the front plates and 0 V on the four electrodes. We observed in the experiments that the addition of the new electrodes changes the back-ground field from ∼ 215 mV/cm to ∼ 180 mV/cm, as I will discuss in the next chapter.

The green line in Fig.3.4 shows that also in the simulation we observe a decrease of the compensating electric field: the green line is higher than the blue one which refers to the simulation in the absence of new electrodes. We have to notice that in these simulations we have no background electric field. However, if we suppose that i is a generic direction and that when we com-pensate EBGi− EiApplied = 0, so the compensation along i gives information

about the background electric field along the i direction. In other words, in the the presence of the four electrodes the compensating electric field is reduced, so we can suppose that also the the background field is reduced because we need smaller applied field to reach the compensation.

The second simulation is performed as the previous one except for the voltages on the four new electrodes. I used −1.31 V on the bottom left elec-trode, +0.17 V on the bottom right one, and 0 V on the other ones. This choice of the applied voltages is made because experimentally this configur-ation allowed us to reach the smallest residual electric fields experienced by atoms. We obtained around 30 mV/cm, as shown in the next chapter.

The red line in Fig.3.4 is the electric field along the x direction when the voltages are applied on the electrodes. We can notice that the com-pensating electric field is smaller than the previous situation. However, this configuration allows field compensation also along the y direction Fig.3.5. This means that even if EBGx− ExAppliedis higher, the residual electric field

Eres

Eres=

q

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CHAPTER 3. SIMULATION STUDY 31 -1000 -800 -600 -400 -200 0

Electrical field along the x direction [V/m]

100 80 60 40 20 0 x-coordinate[mm]

Figure 3.4: Electric field along the x direction as function of the x coordinate. The blue line shows the ExApplied value when we compensate only on the front

plates by applying −30 V before the installation of the four electrodes. The green line shows the ExApplied value when we compensate only on the front plates by

applying −30 V after the installation of the four electrodes which are at 0 V. The red line shows the case in which we apply −30 V on the front plates, −1.31 V on the bottom left electrode, +0.17 V on the bottom right one, and 0 V on the other ones. The black spots show the position of the atoms.

can be smaller then in the previous situation.

This is more visible in Fig.3.7, where the electric field in the x direction is plotted as function of the y coordinate. When the four electrodes are applied, the atoms do not experience the maximum compensating electric field along the x direction (red line). This also shows that the voltage on the four electrodes affects the electric field along the x direction.

The following Figures 3.4, 3.5 and 3.6 compare the electric field in each direction as function of the x direction for the three simulations.

3.5 Conclusions

The compensation procedure has to be repeated once every few days in the laboratory. These simulations can be used to evaluate the direction of the compensating fields in the system. In this way one can monitor any change of the electric field due to the presence of dust particles on the vacuum cell or charge which is released from the trap and associated to rubidium atoms adsorbed on the cell walls.

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Figure 3.5: Electric field along the y direction as function of the x coordinate. The blue line shows the EyApplied value when we compensate only on the front

plates by applying −30 V before the installation of the four electrodes. The green line shows the EyApplied value when we compensate only on the front plates by

applying −30 V after the installation of the four electrodes which are at 0 V. The red line shows the case in which we apply −30 V on the front plates, −1.31 V on the bottom left electrode, +0.17 V on the bottom right one, and 0 V on the other ones. The black spots show the position of the atoms.

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CHAPTER 3. SIMULATION STUDY 33 15 10 5 0 -5

Electrical field along the z direction [V/m]

100 80 60 40 20 0 x-coordinate[mm]

Figure 3.6: Electric field along the z direction as function of the x coordinate. The blue line shows the EzApplied value when we compensate only on the front

plates by applying −30 V before the installation of the four electrodes. The green line shows the EzApplied value when we compensate only on the front plates by

applying −30 V after the installation of the four electrodes which are at 0 V. The red line shows the case in which we apply −30V on the front plates, −1.31 V on the bottom left electrode, +0.17 V on the bottom right one, and 0 V on the other ones. The black spots show the position of the atoms.

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CHAPTER 3. SIMULATION STUDY 34 -100 -80 -60 -40 -20 0

Electrical field along the x direction [V/m]

20 10 0 -10 -20 y-coordinate [mm]

Figure 3.7: Electric field along the x direction as function of the y coordinate. Both lines are referred after the installation of the four electrodes. The green line displays when we compensate only on the front plates by applying −30 V and the four electrodes are at 0 V. The red line shows the case in which we apply −30 V on the front plates, −1.31 V on the bottom left electrode, +0.17 V on the bottom right one, and 0 V on the other ones. The black spots show the position of the atoms.

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Chapter 4

Stark map calculations for

atoms in external fields

Almost all the Rydberg state properties are derived from the knowledge of Rydberg atom energy levels and matrix elements. As shown in Chap 1, Ry-dberg states are extremely sensitive to external electric fields, in particular, they are affected by the Stark shift. This behavior is shown in Stark maps. The calculation of Stark maps requires the evaluation of electric dipole mat-rix elements which appear both in energy shift and in the expression of the eigenstates of the system in the presence of the electric field. In order to know how the electric field mixes the bare atomic energy levels by interaction matrix elements, I used the Alkali Rydberg Calculator (ARC) [10]. ARC is a library, written in Python, which uses object-oriented programming which provides reusable classes and methods for performing calculations of single-atom and two-atoms properties of Rydberg states. In this chapter I will discuss the use of the ARC package and I will focus on the StarkMap Class that I used to study theoretically Rydberg states in external electric fields. In fact, the comparison of the calculated Stark maps of high-lying Rydberg states with the experimental ones, presented in Chap.5, has been fundamental to extrapolate the value of the electric field in our apparatus. In addition, in interactive mode the plot of Stark map can be clicked to ob-tain the contribution of the target unperturbed state within each eigenstate. This interactive mode to see the state composition has been used in Chap.6, where the lifetime of Rydberg states is studied as function of the electric fields.

4.1 About Stark map calculator

The calculation of Stark maps is a useful tool for deducing the electric field for experiments of electrometry. In fact, the electric field can be obtained by the observation of Stark energies of Rydberg states.

Characterization and compensation of stray electric fields in cold Rydberg atom experiments

Universit`

a di Pisa

Characterization and compensation

of stray electric fields

in cold Rydberg atom experiments

Contents

Introduction

Chapter 1

Rydberg physics

1.1

Introduction to Rydberg atoms

1.2

Lifetime of Rydberg atoms

1.3

Electric field effect

1.4

Ionization threshold

Chapter 2

Description of the

experimental apparatus

2.1

Magneto-optical trap (MOT)

2.2

Excitation into Rydberg states

5S

6P

nS

2.3

Detection of Rydberg atoms

2.4

Role of the electric field

Chapter 3

Simulation study of the

electric field in the

experiment

3.1

About COMSOL

3.2

First Simulation

3.3

First compensation

3.4

Final compensation

3.5

Conclusions

Chapter 4

Stark map calculations for

atoms in external fields

4.1

About Stark map calculator