FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di Laurea Magistrale in Fisica
Tesi di laurea magistrale:
Experimental studies of the lifetimes
of Rubidium Rydberg atoms as sensors
of blackbody radiation
Candidato: Alessandro Greco Relatore: Dr. Oliver Morsch Anno Accademico 2017-2018Introduction iv
1 Theoretical background 1
1.1 Introduction on Rydberg atoms . . . 1
1.2 Interaction mechanisms . . . 2
1.3 Stark effect . . . 4
1.4 Theoretical treatment of Rydberg atom lifetimes . . . 6
1.5 Deviations from Planck’s formula . . . 9
1.6 Dressed atom . . . 10
1.7 Damping of the Rabi oscillations . . . 11
2 Experimental Apparatus 13 2.1 Magneto Optical Trap . . . 13
2.1.1 Our apparatus . . . 16
2.2 Excitation and de-excitation . . . 18
2.2.1 Excitation . . . 18
2.2.2 De-excitation . . . 20
2.2.3 Electric field ionization . . . 22
2.2.4 Timings . . . 24
3 Experimental results 25 3.1 Lifetime protocol . . . 25
3.1.1 Single-atom regime to avoid interactions . . . 26
3.2 First results . . . 28
3.2.1 Interactions . . . 29
3.3 Stray electric field . . . 29
3.4 Microwaves . . . 33
3.4.1 Simulation . . . 35
3.4.2 Shielding . . . 36
3.4.3 Planck’s formula and COMSOL . . . 38
3.5 Conclusion . . . 40
4 Single transition rates 41
4.1 Excitation of a P state . . . 41
4.1.1 Radio frequencies and Rabi oscillations . . . 43
4.2 Repopulation measurement . . . 45 4.3 Conclusion . . . 46 5 Simulations 48 5.1 ARC Package . . . 48 5.2 getStateLifetime . . . 49 5.3 My program . . . 50
5.3.1 Deviation from a pure single decay law . . . 51
5.3.2 Local fit method . . . 56
5.4 Three-level model . . . 56
5.4.1 Evaluation of the target lifetime at 0 K . . . 57
5.4.2 Evaluation of repopulation rate . . . 59
5.5 Conclusion . . . 63
6 Conclusion 64 6.1 Outlook . . . 65
A COMSOL Multiphysics 66 A.1 Computing Q-Factors and Resonant Frequencies of Cavity Resonators . . . 66
A.2 Conclusion . . . 68
B ARC 71 B.1 The basic lifetime program . . . 71
B.1.1 Enter the parameters . . . 72
B.1.2 Initializations . . . 73
B.1.3 The core of the program . . . 77
B.1.4 Graph . . . 79
B.1.5 Time elapsed . . . 80
The description of a quantum system using classical devices is an outstand-ing task which requires exponential computational power which is practically impossible to obtain [1]. The use of quantum simulators allows us to simu-late quantum systems we are interested in and it is therefore a fundamental resource. Within cold atom physics, two types of approaches are possible in order to create an interacting simulating system: to consider neutral systems and induce in them strong interactions (by using Feshbach resonances or op-tical lattices) or already interacting systems (as ions and polar molecules) and use precise geometric configurations. Rydberg atoms join these two ap-proaches [2].
Rydberg atoms are neutral atoms and, thanks to their high principal quan-tum number n, show extreme physical properties compared to atoms in the ground state.
Among these properties we point out the great sensitivity to electromag-netic fields, strong interactions and the very long lifetime: all of these are fundamental characteristics in creating controllable systems. Owing to these features, Rydberg atoms can be used in a variety of applications: they are good candidates for quantum simulators [3]; they can be used as sensors of static electric fields [4], microwaves [5] and terahertz radiation [6].
However, before implementing Rydberg atoms in complex systems, a deep basic knowledge is required. Our aim is the study of the lifetime of Rubid-ium 87 Rydberg atoms in systems as close as possible to the single atom regime.
To that end, experiments which involve Rydberg states must be carried out in a strongly controlled regime. This is an experimental challenge and it requires considerable efforts in order to minimize any source of disturbance. The subject of this thesis is the measurement of the lifetime of Rubidium 87 Rydberg atoms in controlled conditions in order to avoid any interaction effects and disturbing electromagnetic fields.
The lifetime of Rydberg atoms is determined by two processes [7] the spon-taneous emission and transitions towards neighbouring Rydberg states due to blackbody radiation (BBR). The latter becomes the leading term for Ry-dberg states above n ∼ 40 [2]. Previous works regarding RyRy-dberg atoms successfully measured lifetimes up to n = 45 [8] [9]. This means that the range of principal quantum numbers for which the blackbody radiation is the dominant decay mechanism have not been systematically investigated thus far.
In this thesis I present the experimental measurements of Rydberg atoms lifetimes for nS states with 60 . n . 100 by using the de-excitation tech-nique [10], a hybrid optical and field ionization techtech-nique. For a deeper comprehension we are interested in evaluating single transition rates be-tween two neighbouring states and for this reason we develop a protocol designed to accomplish this. Numerical simulations with which to compare the experimental results have been developed.
This thesis starts by recalling in Chapter 1 the theoretical background re-lated to the topics treated in this thesis: briefly, we introduce the interaction mechanism and the Stark effect; a more detailed discussion which regards Rydberg state lifetime and the effect of the blackbody radiation on it is done; finally we recall the dressed atom approach and the characterization of the Rabi oscillations in the presence of decoherence mechanism to describe the behaviour of atoms in the presence of radio frequency radiation.
Chapter 2 presents the experimental apparatus of our laboratory. In particular, the basic principles of the magneto optical trap; the excitation, de-excitation and field ionization processes to excite, manipulate and detect Rydberg atoms are explained. The timing of our protocol, used to measure the lifetime, is shown.
Chapter 3 presents the experimental measurements of the lifetime of Ry-dberg atoms in a wide range of nS states with principal quantum numbers 60 . n . 100 and they are compared with theoretical predictions [11]. I discuss the causes of some deviations which we found and that are related to a stray electric field in our apparatus. A further deviation from theory that we observe is related to the dimensions of our experimental appara-tus, which are close to values of the wavelengths between first neighbouring states and that can lead to a deviation of the effective blackbody radiation from Planck’s formula [12]. Consequently deviations of lifetimes from the theoretical predictions (which are based on Planck’s formula) appear.
The protocol to evaluate single transition rates between first neighbour-ing states is the subject of Chapter 4. It will allow us to study the cause of shorter lifetimes in the narrow region of nS state with 90 . n . 100.
Numerical simulations to extract Rydberg atoms lifetimes in different conditions by using a Python program based on the alkali-Rydberg-calculator
package [13] are exposed in Chapter 5. Supporting with the numerical simu-lations, I discuss the possibility of using our lifetime measurements to extract the lifetime at temperature 0 K and I evaluate the order of magnitude of the re-population rate from neighbouring states to the excited one.
Two appendices are presented. The first reports the attempt to study the resonant frequencies of our apparatus using COMSOL Multiphysics. The second contains the Python program used in Chapter 5 to perform the nu-merical simulations.
Theoretical background
1.1
Introduction on Rydberg atoms
Rydberg atoms are neutral atoms in highly excited states. First studies re-garding Rydberg atoms were performed in the 19th century. The empirical Balmer’s law (1885) was the first attempt to describe Hydrogen spectral lines by means of the observation of the spectra of stars. By including other spectroscopic measurements, three years later Johannes Rydberg general-ized Balmer’s formula and defined the Rydberg constant. The aim of the empirical Rydberg formula is the description and prediction of the transition wavelength between two states
1 λ= Ry ∗ 1 (n1− δn1)2 − 1 (n2− δn2) 2 (1.1)
where Ry∗ = Ry(1 + me/mnucleus), n1 and n2 two integer numbers (n1 <
n2) which denote a specific transition, δn1 and δn2 are two coefficients which
depend on the both specific transition (n1 ↔ n2) and atomic species.
Only after 1913, with the advent of Bohr’s atomic model, the integer numbers n1 and n2 got a more clear physical description.
The properties of Rydberg atoms can be described by using scaling laws which depend mainly on n∗ = (n − δn,l,j). The quantity δn,l,jis the quantum
defect related to the atomic species. In particular, due to their high principal quantum number n, Rydberg atoms show extreme properties.
For example we can consider that the size (orbit radius) of Rydberg atoms scales with ∼ n2. For comparison, the size of an atom in the ground state is of the order of 0.1 nm whereas the size of a Rydberg atom with n = 100 is around 1 µm. This means that in highly excited states the valence electron is, on average, far from the nucleus and, to a first approximation, the Rydberg atom can be similarly described as the Hydrogen atom:
• the valence electron can be thought as the unique electron of the Hy-drogen atom which orbits around the positively charged core;
• other particles (the N protons and the remaining N −1 electrons which result as an overall single e+ charge) can be thought as the nucleus of the Hydrogen atom.
However, the presence of internal electrons modify the Rydberg atom be-haviour from the simple model of the Hydrogen atom and this additional complexity is taken into account by using the quantum defect δn,l,j.
The binding energy of Rydberg atoms is written
En,l,j = −
Ry∗ (n − δn,l,j)2
. (1.2)
In the following sections, we will briefly discuss only those properties of Rydberg atoms that are relevant for this thesis: interaction mechanisms, the polarizability which is used in the description of the Stark effect and the long lifetime.
A further quantity of interest, which I will discuss first, is the ionization threshold of a Rydberg state. To obtain its classical ionization threshold we can consider an electron as subject to the Coulomb potential and to an electric field along the ˆz direction with norm
ε
ϕ = ke
z +
ε
z. (1.3)in which k = 1/4π0 where 0 is the vacuum permittivity. Its value at the
saddle point is given by
ϕ = 2 √
ke
ε
(1.4)thus if we equalize this, times −e, with the binding energy −eϕ = En,l,j
(Eqs. 1.4 and 1.2) we find
ε
= Ry∗2
4k(n − δn,l,j)4e3
(1.5)
which is of the order of some V/cm for states we excite (the nS states with 60 . n . 100) and manipulate in our laboratory. In Tab. 2.1 some of these values are listed.
1.2
Interaction mechanisms
The interaction mechanism between two Rydberg atoms is described as func-tion of the distance. In particular, at short range it can be expressed as the dipole-dipole interaction whereas at long distances as the van der Waals interaction mechanism.
The most evident effect of the interaction among Rydberg atoms is the dipole blockade phenomenon. The presence of a neighbouring highly excited
atom shifts out of the resonance the energy of Rydberg states with respect to the exciting laser: in this manner it cannot be excited. By controlling the dipole blockade mechanism we can inhibit the creation of other Rydberg atoms, below a certain distance from an initial excitation, reaching a satu-rated system. The shift of the energy can affect our measurements, thus we must take it into account and, as far as possible, avoid it.
I will now briefly derive the formula for dipole-dipole interactions be-tween Rydberg atoms, following the discussions presented by Cristiano Si-monelli in [14] and by Nikola ˇSibali´c in [15].
First of all, the classical potential due to a volume charge distribution is described by φ = k Z dτ ρ(~r0) |~r − ~r0|dτ (1.6)
and for r r0 we can expand
|~r − ~r0|−1 ≈ 1 r 1 +r 0 rcosθ 0 (1.7)
so, by replacing Eq. 1.7 into Eq. 1.6, we obtain by the use of the two fundamental relations, q =Rdτρ(~r0) dτ and ~p =Rdτρ(~r0) · ~r0dτ ,
φ = k q r + ~ p · ~r r3 (1.8)
However, Rydberg atoms are neutral atoms thus q = 0 and the electric field created by this electric potential, Eq. 1.8, is expressed by the relation
~
ε
= − ~∇φ = k 3(~p · ~r)~r − r2~p
r5
The interaction energy between two dipoles is simply given by
U (r)ab= − ~
ε
a· ~pb = −k3( ~pa· ~r)( ~pb· ~r) − r2( ~pa· ~pb)
r5
The corresponding quantum mechanical operator ˆU is easily obtained by replacing ~p with the dipole operator defined asd = eˆ~ˆ ~r, in which ˆ~r is the position operator. The interaction term is
U (r) = k ~ da· ~db r3 − 3 ~da· ~r ~d b· ~r r5
The Hamiltonian of a system composed of |a, ai and |b0, b00i can be writ-ten as
H =
0 U (r) U (r) ∆
in which ∆ = Eb0+ Eb00− 2Ea and its eigenvalues are
E±=
∆ ±p∆2+ 4U2(r)
2 (1.9)
The van der Waals radius rvdW in which ∆ = U (rvdW) defines two
different regimes as function of the distance:
• short distances: for short range the interaction energy U (r) is higher than the energy difference ∆, that is U (r) ∆. This condition implies that the resulting eigenenergies are E± = ±U (r) = Cr33
• large distances: for long range the interaction energy U (r) is lower than the energy difference of ∆ that is ∆ U (r). In this case the eigenenergy are described by E±= ∆ = −U (r)
2
∆ ∝
C6
r6
The blockade mechanism depends on the distance between two Rydberg atoms and it is determined by
C6
r6 b
= }Γlaser
where Γlaser is the combined laser and residual Doppler linewidth. We can
write the blockade radius explicitly as
rb = 6
r C6
}Γlaser
(1.10)
In Tab. 3.1 some C6 coefficients and blockade radii are listed.
1.3
Stark effect
The great sensitivity of Rydberg atoms to electric fields is a very useful characteristic for shifting their energies in a controlled way. On the other hand, due to this sensitivity, the presence of any stray electric field could be a serious problem and, for this reason, in our experiment we want to minimize them as far as possible.
The properties of an atom in the absence of any electric field are de-scribed by using the unperturbed Hamiltonian H0. An atom in the
eigen-state |nlji satisfies the equation H0|nlji = Enlj0 |nlji where Enlj0 is the
unperturbed energy of the state.
In the presence of an electric field the Hamiltonian is
where the term V = e
ε
z is function of the electric field ~ε
along the ˆz axis. In the case of a small electric field ~ε
we can apply perturbation theory otherwise a full diagonalization of the Hamiltonian is necessary.The new eigenenergy of the perturbed |nlji state is, up to the second order perturbation theory,
Enlj = Enlj0 + hnlj| V |nlji +
X
n0l0j06=nlj
|hn0l0j0| V |nlji|2
Enlj− En0l0j0 + · · · (1.12)
In alkali atoms the quantum defect is appreciably different from zero only for states with l ≤ 3 therefore it lifts the degeneracy and the effect of the Stark shift on the first order is null. The perturbation theory to the second order is necessary and the energy shift takes the form
∆EStark = −
1 2α
ε
2 (1.13)
where α is the polarizability of the |nljmji state which is defined as
α = 2e2 X
n0l0j06=nlj
|hn0l0j0| z |nlji|2
En0l0j0− Enlj
(1.14)
It must be noted that in the Stark shift expression Eq. 1.13 the energy value is shifted downwards. In particular, mj and m0j are not displayed in
Eq.1.14 because they must be equal according to the selection rule ∆mj = 0.
In fact, since the direction of the electric field is along the quantization axis it only couples states with ∆mj = 0.
We can easily derive the scaling law which describes the polarizability α by noting that the dipole moment scales as ∼ n2 whereas the energy difference as ∼ n−3, thus we obtain α ∼ n7.
States with l > 3 are degenerate and we can use the first order pertur-bation theory for degenerate states [16], so the Stark effect experienced by these is linear. The same behaviour can be observed in Hydrogen because the quantum defect δnlj = 0 and the states are degenerate. These states,
which are linearly shifted, compose a fan due to the different radial matrix elements
hnl| r |nl + 1i = −3n √
n2− l2
2 (1.15)
In particular, the Stark shift of the extreme states of the fan is around ±3
2n
2
ε
. The condition for which the upper extreme level of principalquan-tum number n − 1 (which is shifted upwards by ∼ +32(n − 1)2
ε
) crosses the lower extreme level of principal quantum number n (which is shifted downwards by −32(n)2ε
) can be obtained by equalizing the energies of the two states (atomic units)− 1 2(n − 1)2 + 3 2(n − 1) 2
ε
= − 1 2n2 − 3 2n 2ε
(1.16)This expression allows to find the electric field in which the crossing happens: the Inglis-Teller limit. The scaling law of the Inglis-Teller limit is
ε
∼ 1n5 (1.17)
1.4
Theoretical treatment of Rydberg atom
life-times
The long lifetime of Rydberg atoms is a fundamental feature which makes Rydberg atoms so interesting for a variety of experiments.
To first order, the effective lifetime of Rydberg atoms is determined as the inverse of the sum of all the depopulation rates. Depopulation processes do not include only spontaneous decays towards low lying states but also the transfers to neighbouring states by means of BBR-induced transitions
1 τ = Γ = Γ0+ Γ BBR= 1 τ0 + 1 τBBR (1.18)
We will discuss these two different processes separately.
The description of the spontaneous decay is performed by using the Ein-stein A coefficients which quantify spontaneous emission rates. The Anl→n0l0
rate describes the spontaneous decay rate from the nl state to the lower n0l0 state Anl→n0l0 = 4e2ω3 nl→n0l0 3}c3 lmax 2l + 1R 2(nl → n0l0) (1.19)
in which ωnl→n0l0 = |Enl− En0l0|/} is the transition frequency between the nl
state and the n0l0 state, with energies Enl and En0l0 respectively; lmax is the
larger of l and l0; R2(nl → n0l0) = |hn0l0| r |nli|2 is the radial matrix element of the electric dipole transition.
The sum over all n0l0 states yields the total spontaneous decay rate Γ0
and its inverse is the value of the spontaneous lifetime τ0
τ0= Γ−10 = " X n0l0 Anl→n0l0 #−1 (1.20)
However, by means of Eqs. 1.19 and 1.20 it is not possible to obtain a single scaling law as function of n. The two different scaling laws determine lifetime values for Rydberg states with low-l number in comparison with the circular state in which l ≈ n.
The ω3nl→n0l0 factor in 1.19 states that the higher the frequency transition
is the higher the spontaneous decay rate Anl→n0l0 is.
In the low-l Rydberg states, the highest frequency transition is towards the lowest lying state with l0 = l − 1 except for S states which are to-wards P states. It must be noted that for highly excited states this fre-quency can be approximated by a constant: frequencies towards the low-est principal quantum number are of the order of 1014− 1015Hz whereas
the energy spacing among Rydberg states is of the order of 109− 1010Hz.
For this reason, frequency variations among Rydberg states can be rea-sonably neglected. In fact, by comparing the ratio τ0(60S)/τ0(80S) ∼
0.41, instead the ratio between the two frequencies towards the 5P state is ω80S→5P/ω60S→5P ∼ 1.0007. The difference in these two lifetime values of
so highly excited states depends mainly on the dipole radial matrix element (R2(80S → 5P )/R2(60S → 5P ) ∼ 0.40). According to [2], the scaling law of this dipole matrix element is n−1.5 thus we get
τ0low−L∝const · n−3−1
= n3 (1.21) By contrast, for circular states with l ≈ n the lifetime scaling law is different. By taking into account the selection rule of the dipole operator, it allows the coupling between l and l0 = l ± 1 states only. However, we must exclude the l0 = l + 1 transition because it needs a principal quantum number at least n0 = n + 1 which corresponds to an higher energy level than the starting one. It is straightforward to see that a nl state with l = n − 1 can decay only into the n0l0 state with l0= l − 1 = n − 2 and with n0 = n − 1. The difference between two neighbouring n states is of the order of n−3, thus in this case we cannot consider the frequencies as a constant. In the end, the radial matrix element between hn − 1n − 2| r |nn − 1i ∼ n2 so for Eq. 1.19 we get τ0circular ∝ n 4 n9 −1 = n5 (1.22)
The presence of BBR-induced transitions adds other channels of decay and thus lifetime values decrease in comparison to the only spontaneous decay lifetime. In particular, for highly excited states the sum of the BBR-induced transition rates does not depend on l and for this reason the most affected lifetime values are those of the longest lived states [17].
Rydberg states, whose energies are a few eV above the ground state, are not populated thermally except at very high temperatures. The effect of the BBR does not affect the transition between the low lying and Rydberg states but it affects only the highly excited states.
The rate of the BBR-induced transition is defined by Knl→n0l0 and it is
1.19. The coefficient nω is the effective number of BBR photons per mode
nω=
1
e}ω/kT − 1 (1.23)
The expression of the BBR-induced transition rate, where nω is evaluated
at each frequency ωnl→n0l0, is
Knl→n0l0 = nωAnl→n0l0 (1.24)
The effective number of BBR photons per mode for large frequencies of transitions at T = 300 K is nω 1 and thus the effect of the BBR is low.
Among Rydberg states the transition frequencies are of the order of some GHz. Eq. 1.23 for low frequencies }ω kT becomes
nω ≈
kT
}ω (1.25)
and nω ∼ 103.
The strong sensitivity of Rydberg atoms to the BBR is related to two reasons: the first is the energy spacing among the Rydberg states which is of the order of }ω kT thus the nω, as written in Eq. 1.25, is significant;
the second is the value of the dipole matrix element which is large. For these reasons the coupling of Rydberg states to neighbouring states is remarkable and it must be taken into account to calculate the effective lifetime. For comparison, the lifetime at T = 0 K (only spontaneous decay) of the 80S Rydberg state of Rubidium is τ0(80S) = 623.54 µs whereas at T = 300 K,
taking into account BBR-induced transitions, τ300K(80S) = 209.42 µs: the
first is factor of three higher than the second.
To highlight the dependence of the lifetime due to BBR-induced transi-tions as function of the temperature T we can exploit the oscillator strength [7], which is defined as fnl→n0l0 = 2 3ωnl→n0l0 lmax 2l + 1R 2(nl → n0 l0) (1.26)
and in the regime of }ωnl→n0l0 kT we obtain
ΓBBR= 2α3kTX
n0l0
ωnl→n0l0fnl→n0l0 (1.27)
and by using the sum rule
X n0l0 ωnl→n0l0fnl→n0l0 = 2 3n2 (1.28) we find τBBR= 3n2 4α3kT (1.29)
By the comparison of the scaling laws between the two different depop-ulation rates due to spontaneous decays Γ0∼ n−3 (the inverse of Eq. 1.21)
and the BBR-induced transitions ΓBBR ∼ n−2 (the inverse of Eq. 1.29), we
observe that ΓBBR> Γ0 for n & 40. [2]
We can express ΓBBRnlj→n0l0j0, which depends on Eq. 1.19, as function of j
[18] ΓBBRnlj→n0l0j0 ∝ 4e2ω3nlj→n0l0j0 3}c3 2j0+ 1 2j + 1|j 0 r |ji| 2 (1.30)
We anticipate that Eq. 1.30 implies that ΓBBR84P
1 2 →85S1 2 is different respect to ΓBBR 84P3 2 →85S1 2
, not only for the ratio 2j2j+10+1, but also because the two dipole matrix elements |hj0| r |ji|2 can be very different.
1.5
Deviations from Planck’s formula
In the previous section we have discussed the influence on Rydberg states of the thermal radiation in free space. The expression of the overall Γ depopulation rate depends also on ΓBBR and the latter depends on the
blackbody spectrum which is calculated by Planck’s formula.
By considering a closed space, like a cavity, the previous results are still valid but, by well controlling its dimensions, it is possible to enhance or suppress precise transition rates. The presence of those spatial constraints modifies the actual frequency distribution of the BBR and this one is forced into the cavity modes. The result is that some frequencies, which match the cavity modes, are enhanced whereas the other frequencies, which do not match the cavity modes, are decreased.
In the past, some experiments studied this effect. First experiments were carried out in the eighties and exploited various devices to reveal the effect of cavities on Rydberg atoms. One of these experiments exploited the electric field to vary the energy distance between a couple of D, P states placed between two parallel plates. Their distance d cut off the wavelengths above 2d and it permitted or suppressed the specific transition [19]. An-other example used a Fabry-Perot cavity to tune in or out the resonance: it was possible to observe even an enhancement of the transition [20]. By using circular states, which can radiate by only a single dipole transition, it is possible to completely suppress or increase the decay. As before, the variation of the energy spacing between two levels, above or below the cut off frequency, affected the decay. An important deviation was found [21], where the circular state transition rate was either suppressed or enhanced up to 50% in comparison to the free space.
In our experiment the MOT is at the center of the surrounding appara-tus. In principle, due to the metallic objects which compose the apparatus,
it can be thought of as a cavity. Obviously we cannot consider it as a perfect cavity but its finite dimensions can lead, as we have said, to a de-viation of the frequency spectrum from Planck’s formula. In fact, Planck’s formula is derived starting from the hypothesis that considered wavelengths must be smaller than blackbody dimensions and its spatial variations [12]. The range of validity of Planck’s formula is 1 (λ/l)3 where l is equal to the dimensions of the apparatus (and therefore it depends also on its shape). In our case the energy spacings among neighbouring Rydberg states (λnS↔(n−1)P ∼ 6 cm for n = 90) are comparable with the dimensions of the apparatus (30 x 24 x 90 mm) so deviations from Planck’s formula can appear.
1.6
Dressed atom
The dressed atom approach [22] is useful to describe the effects of the in-teraction among atoms and strong radio frequency fields, either resonant or non-resonant.
First of all, we can consider the two systems (atom and RF quanta) as non-interacting; the interaction will be introduced later.
The Hamiltonian which describes the internal state of a two-level atom, with a ground state |gi and an excited one |ei, is HA: it gives
HA|ei = }ωA|ei HA|gi = 0 (1.31)
where ωA is the frequency corresponding to |Ee− Eg|/}.
The Hamiltonian of the radio frequency field of the cavity is written as
HRF = }ωRFa+a (1.32)
where ωRF is the frequency of the RF quanta, a+ and a the creation and
annihilation operators of the RF quantum respectively.
The Hamiltonian which describes the system composed of the atom and RF quanta is
HARF0 = HA+ HRF (1.33)
The energies of the system are
HARF0 |e, N i = }(ωA+ N ωRF) |e, N i
HARF0 |g, N + 1i = (N + 1)}ωRF|g, N + 1i
(1.34)
and their difference is ∆EARF0 = Eg,N +1− Ee,N = }(ωRF − ωA) = }δ. The
energy difference between the two states of the whole system depends on the value of the detuning ωRF − ωA= δ and they are degenerate for δ = 0.
In the presence of the interaction among the atom and the radio fre-quency field the Hamiltonian of the whole system acquires another term VARF, thus it becomes
HARF = HA+ HRF + VARF (1.35)
Because of the selection rules, VARF couples only
he, N | VARF|g, N + 1i = √ N + 1 he, 0| VARF |g, 1i = }Ω0 2 √ N + 1 = }ΩN +1 2 (1.36)
in which we have used the definitions of the Rabi frequency he, 0| VARF|g, 1i = }Ω0
2 and ΩN +1= Ω0
√ N + 1.
The energies of the eigenstates |1(N )i and |2(N )i are found by the re-sulting matrix } δ ΩN +1 2 ΩN +1 2 0 !
and their energy splitting is given by
E|1(N )i− E|2(N )i = }eΩN +1 (1.37) in which eΩN +1= q Ω2 N +1+ δ2.
In Sec.3.4.2 we will measure this energy splitting }eΩN +1 in order to
obtain the power of the radio frequency field.
1.7
Damping of the Rabi oscillations
In a more general description we want to define the dynamics of a system in the presence of a resonant radiation whose frequency is ω. The components of these two-level system are denoted as |gi for the ground state and as |ei for the excited one. By neglecting relaxation terms (no losses and no de-coherence), the system oscillates back and forth between these two levels with the Rabi frequency
Ω = d
ε
} (1.38)
where d is the dipole matrix element between the two states |gi,|ei and
ε
is the electric field of the driving radiation.In case of loss mechanisms, it is necessary to use the density matrix
ρ =ρgg ρge ρeg ρee
H = }ωg|gi hg| + }ωe|ei he| +}Ω
2 e
+iωt|gi he| +}Ω
2 e
−iωt|ei hg| (1.39)
and to describe the time evolution of the density matrix we can exploit the Schr¨odinger equation
i}d
dtρ = [H, ρ] (1.40) By writing the population difference w = ρgg− ρee it is possible to find
the optical Bloch equation for the populations [23]:
d dtw = −γ⊥ Ω2 γ2 ⊥+ δ2 w − γk(w − w0) (1.41)
in which w0 is the value of w in thermal equilibrium. The two terms γk and
γ⊥ define two different relaxing processes:
• γk is the spontaneous emission coefficient;
• γ⊥ takes into account the de-phasing of the coherence.
In the adiabatic approximation, in which the populations evolve much more slowly than the coherence γk γ⊥, Eq. 1.41 becomes
d dtw = −γ⊥ Ω2 γ2 ⊥+ δ2 w (1.42)
thus the rate to reach the stationary state is defined as
Γstationary≈ γ⊥ Ω2 γ2 ⊥+ δ2 ∼ Ω 2 γ⊥ (1.43)
Two typical regimes are possible:
• the coherent regime for Ω2 γ
⊥in which the dynamics evolves
coher-ently with the excitation probability which depends linearly on ∼ Ωt;
• the incoherent regime for Ω2 γ
⊥ in which the dynamics evolves
incoherently (classically) with the excitation probability which depend quadratically on ∼ Ω2t.
In Sec. 4.1.1 we will observe the Rabi oscillations whose dynamics can be described by using the above treatment.
Experimental Apparatus
In this chapter I will describe the experimental apparatus of our laboratory which uses ultracold Rubidium 87 atoms in order to measure the lifetime of Rydberg atoms. Further details can be found in [24].
We can divide the process of the lifetime measurement of Rydberg atoms in two fundamental parts. The first part is the cooling and the trapping of Rubidium atoms; the second part is the excitation and the manipulation of Rydberg atoms. The first goal is realized by using a magneto optical trap (MOT) whereas the second purpose is obtained by the carefully timed excitation and de-excitation pulses of two lasers.
2.1
Magneto Optical Trap
The magneto optical trap is the fundamental tool which we use to cool and to trap the thermal87Rb atoms.
The cooling and the trapping are two distinct mechanisms.
The cooling mechanism is the reduction of the dispersion in the velocity distribution ∆v: it requires a radiative force which is velocity dependent to selectively modify the velocity distribution. We exploit the Doppler effect in order to obtain a velocity dependent radiative force.
The trapping mechanism reduces the dispersion in the position distri-bution ∆x: it requires a radiative force which depends on the position of the atoms to selectively modify the position distribution. In order to real-ize the trapping, we exploit the Zeeman effect: by using an inhomogeneous magnetic field we obtain a position dependent radiative force.
The magneto optical trap exploits the velocity and the position depen-dences of the radiative force to bring about both cooling and confinement.
In our system we realize two MOTs: the first is a 2D MOT and it is used as a preparatory tool; the second one is a 3D MOT in which the experiments described later in this thesis are performed. For the sake of simplicity we explain the operating mechanism in a 1D configuration. The extension to
2D and 3D configurations is straightforward.
It is possible to change the momentum of an atom at rest by using a laser radiation.
A laser photon which moves in the −ˆx direction is characterized by an energy of }ωl and a momentum of −}klx. When the atom absorbs theˆ
laser photon its energy is enhanced and it changes its internal state from the ground state to the excited one. Furthermore, the atom changes its momentum from 0 to −}klx.ˆ
On the other hand, for times of the order of the lifetime τ , the atom decays from the excited state back into the ground state. The decay mecha-nism can occur either as stimulated emission or do spontaneous emission. In the first case, the stimulated emission, the atom emits an identical photon to the absorbed one: it releases a photon with the same energy }ωl and
identical momentum −}klx. The atom absorbs and releases in the same di-ˆ
rection the same amount of momentum, hence there is no change in the net momentum of the atom. In the second case, the spontaneous emission, the atom releases a photon with the same amount of energy }ωl and the same
momentum }kl of the absorbed one but in an isotropic process. There is
not a preferential direction and therefore the mean value of the momentum released through the spontaneous emission by the atom is zero.
This means that there may be a net momentum transfer and that this is due only to the absorption coupled with the spontaneous emission process. Now, we selectively exploit the transfer of momentum from the laser radiation with ωlfrequency to the atoms in order to reduce the velocity
dis-persion ∆v. If we consider an atom with a certain velocity v, the frequency of the laser radiation seen by the atom is modified by the Doppler effect
ωD(v) = ωl 1 ±v c (2.1)
• the - sign is used when the atom and the laser wave move in the same direction;
• the + sign is used when the atom and the laser wave move in the opposite direction.
In this way we can reach the resonant condition ωa = ωD(v) and we can
excite the atoms only for a precise velocity class. Only to this class of atoms the photons transfer momentum; the atoms which are in other velocity classes continue to move unperturbed.
To be selectively slowed down, an atom with positive velocity along the x-axis +v ˆx must absorb a photon which moves in the opposite direction with −}klx momentum. In light of this, we must consider the atom and theˆ
counter-propagating laser wave: for this reason, we consider Eq. 2.1 with the + sign. It means that we must use a laser frequency ωl < ωD(v) = ωa. By
defining the detuning δ = ωl− ωa, we need a red detuned (δ < 0) laser light
to slow down the atom. By using two counter-propagating red detuned laser beams we can equally reduce the velocity along the two opposite directions, hence we obtain the minimization of the velocity dispersion ∆v.
We obtain a viscous medium but it does not trap the atoms.
To also obtain a trapping effect, we require a radiative force which is also position dependent.
To this end, we use an inhomogeneous magnetic field. For small magnetic fields the energy shift ∆EZ, due to the Zeeman effect, is small compared
to the energy separation of the hyperfine structure states ∆Ehf s. For this
reason, as long as the following condition remains valid
∆EZ ∆Ehf s (2.2)
we can treat the Zeeman effect as a small perturbation and the F -number is still a good quantum number. The energy shift caused by the Zeeman effect can be expressed
∆EZ= −~µ · ~B = −µB· gf · mf · B (2.3)
For the sake of simplicity we can suppose that the ground state is in the hyperfine level F = 0 whereas the excited state is in the F = 1 level. In presence of a small magnetic field the degeneracy of the excited state is removed and it is split in three Zeeman sublevels with m0f = −1, 0, +1. In presence of a position dependent magnetic field B = −b0x the energy shift of the m0f Zeeman sublevels depend on the x position:
• for a |m0f = +1i state the energy level is shifted towards higher energy for x > 0 and lower energy for x < 0;
• for a |m0f = −1i the energy level is shifted towards lower energy for x > 0 and higher energy for x < 0.
In this way we can selectively excite the atoms as function of the position. The coupling of |mf = 0i → |m0f = +1i is realized by using a σ+
circu-lar pocircu-larized radiation whereas the coupling of |mf = 0i → |m0f = −1i is
obtained by using a σ− circular polarized radiation.
For example, as we can see in Fig. 2.1, if we send a σ+ circular po-larized red detuned radiation from the -ˆx direction it is resonant with only |m0f = +1i in the x < 0 region. In the x > 0 positions the state |m0f = +1i is far from the resonance and the σ+ radiation is not absorbed. Similarly, if we send a σ− circular polarized red detuned radiation from +ˆx direction it is resonant with only |m0f = −1i for the x > 0 region. In the x < 0 positions the state |m0f = −1i is far from the resonance and the radiation is not absorbed.
Figure 2.1: Schematic simplified diagram of the operating mechanism of the MOT. The ground state is F = 0; the excited state is F = 1 which is split into m0f = −1, 0, +1 by the Zeeman effect. An inhomogeneous magnetic field B = −b0x is created in the apparatus.
The result is a radiative force which depends on the position and the velocity. It continuously pushes the atoms towards x = 0, the center of the MOT.
The extension of this mechanism to a 2D and a 3D configuration is realized with two and three orthogonal pairs counter-propagating laser beam respectively.
2.1.1 Our apparatus
In our laboratory we use the Rubidium 87 (its nuclear spin is I = 32). The levels which are used in our MOT to cool and to trap the atoms are the hyperfine levels of the 5S1
2 and 5P 3
2 states, as shown in Fig. 2.2. The
hyperfine structure states of the ground state 5S1
2 are F = 1, 2 whereas the
hyperfine structure states of the 5P3 2 are F
0 = 0, 1, 2, 3.
The trapping radiation is provided by a 780 nm laser which couples the F = 2 → F0 = 3 levels. It is red detuned by δTrap = −2.9Γ5P3
2
, with Γ5P3
2
= 2π · 6.0666 MHz. From this state, the atom decays into the F = 2 level and the cycle is repeated. However, with low probability, it is possible to couple the F = 2 → F0 = 2 levels which can decay into the F = 1 state. On the other hand the F = 1 level is a dark state and the population is lost from the trapping cycle and therefore is lost from the MOT. To overcome this problem, a repump radiation is used to resonantly couple the F = 1 → F0 = 2 levels: in this way the population is inserted again in the trapping cycle.
5S½ 5P3⁄2 F=1 F=2 F’=0 F’=1 F’=2 F’=3 780.241209686(13) nm Tr apping Repump δ
Figure 2.2: Hyperfine structures states of the 5S1
2 and 5P 3
2 energy levels
of Rubidium 87. The trapping and the repump radiation are displayed. Dashed lines represent spontaneous emissions.
The inhomogeneous magnetic field is created by a pair of coils in the anti-Helmholtz configuration. The gradient of the magnetic field is of the order of ∼ 10 G/cm. According to [18], the energy distance between the hyperfine levels of the 5S1
2 level is ∆EF =1↔F =2 = h · 6.8 GHz whereas the
energy difference between the two hyperfine levels of 5P3
2 is ∆EF
0=2↔F0=3 =
h · 266 MHz. The Zeeman shift for the F = 1, 2 levels is h · 0.93 MHz/G while for the F0 = 2, 3 levels is h · 0.70 MHz/G: the resulting shifts are only a few MHz and they are negligible compared with ∆Ehf s. The condition of Eq.
2.2 is well respected.
The 2D MOT is a preparatory tool: it is confined in the yz directions while it is elongated along the x direction. It collects the atoms in the first vacuum cell (10−8mbar) just after they are released by the dispenser. By exploiting the radiation pressure produced by a pushing beam they are pushed towards the x direction. They pass through a small graphite tube towards a second vacuum cell (10−10mbar) where they are trapped in a 3D MOT. This second vacuum cell is made of quartz glass with the dimensions of 90 x 30 x 24 mm and a thickness of 3 mm.
The 3D MOT has a spherical shape, with a peak density around ∼ 1010− 1011cm−3. Typical dimensions of our 3D MOT are between 100 and
300 µm and the mean number of atoms is around 105. We use a CCD camera to collect in the xz plane the fluorescence emitted by the trapped atoms: in this way we can get the MOT dimensions and the number of atoms.
The typical temperature reached by the atoms in our MOT is around T ∼ 150 µk.
2.2
Excitation and de-excitation
To study the lifetime of Rydberg atoms we need to excite Rubidium atoms from the ground state to the Rydberg state and then to de-excite Rydberg atoms to the ground state.
To this end, we use a two-photon excitation scheme: the first step uses a blue laser λB = 421 nm whereas the second step is realized with a tunable
IR laser λIR ∼ 1011 nm. This IR laser will be used for the de-excitation
process too. In Fig. 2.3 we display both of these processes.
Note that a 421 nm photon can photoionize an atom in the 5P3 2 state.
For this reason the MOT beams are turned off before starting the excitation process so the atoms in the 5P3
2 state can decay into the ground state with
the typical lifetime τ5P3 2
∼ 26 ns.
2.2.1 Excitation
The first step drives the atoms from the 5S1
2 state to a virtual level, detuned
by δ6P ∼ +40 MHz from the 6P3
Figure 2.3: On the left the two-photon excitation scheme. The first transi-tion ω5S1
2
→6P3 2
, which is blue detuned by δ6P, is displayed with a blue arrow;
the second transition ω6P3 2
→nS1 2
is displayed by a red arrow. On the right, the green arrow represents the de-excitation radiation ωnS1
2
→6P3 2
, which is resonant, and the dashed line represents the spontaneous emission from the 6P3
2 level which has a lifetime τ6P
6P3
2 state, through the absorption of an IR photon, the atoms are coherently
excited to the nS Rydberg state.
The value of the resonant transition frequency towards the 6P3 2 state
(δ6P = 0) is obtained by photoionizing the atoms. By the absorption of two
blue photons a Rubidium atom from the ground state can be photoionized and by the application of the electric field its signal is recorded.
The choice of the value δ6P ∼ +40 MHz is due to a technical constraint
of the cat’s eye configuration used in the de-excitation process discussed in the following section.
During my thesis work I explored a wide range of nS states, with 60 ≤ n ≤ 120. This is possible because the IR laser is in the Littrow configuration: we can coarse-tune its frequency by moving the internal grating. By the use of a λ-meter we check the value of the wavelenght λIR and compare it with
the theoretical wavelength of the 6P3 2
→ nS1
2 transition.
The fine tuning of the IR laser is realized in a similar way to the blue one: we scan the frequencies near the theoretical wavelength of the 6P3
2
→ nS1 2
transition and we maximize the signal of Rydberg atoms collected by the ionization pulse.
The total Rabi frequency of the two-photon excitation process is
Ω = ΩBlueΩIR 2δ6P
(2.4)
where ΩBlue and ΩIR are the Rabi frequencies of the single transition
5S1 2 → 6P3 2 and 6P 3 2 → nS1
2 respectively, δ6P is the detuning from the 6P 3 2
state.
Typical power of the blue beam is around 20 µW and its waist is of the order of 40 µm. The power of the IR beam in the excitation process is around 0.2 mW and its waist is of the order of 90 µm. The typical value of the Rabi frequency Ω ∼ 10 MHz. The two excitation pulses are simultaneous with a typical duration of a few hundred nanosecond.
By considering the electric dipole selection rules, with this two-photon excitation scheme one can excite the S or the D states but not the P states.
2.2.2 De-excitation
Along the optical path of the IR laser there is a double-pass acousto-optic modulator system which allows us to rapidly (∼ 100 ns) change the IR fre-quency by ∼ +40 MHz.
The double-pass acousto-optic modulator system, called cat’s eye con-figuration, is presented in Fig. 2.4. Its fundamental optical elements are a polarizing beam splitter (PBS), an AOM, a λ/4 plate, a lens (L) and a mirror (M).
The AOM is composed of a piezoelectric transducer which is driven by an oscillating electric signal (typically radio frequencies supplied by a voltage
Figure 2.4: Double-pass acousto-optic modulator system called ”cat’s eye”. The red line is the incoming laser beam with ωl frequency, the blue one is
the outgoing shifted laser beam with frequency ωl+ 2ωRF.
controlled oscillator (VCO)) and which induces sound waves in the internal quartz. The periodic oscillations change the index of refraction of the ma-terial and the incoming light scatters due to the Brillouin scattering. The IR laser is diffracted into few orders: the 0 order maintains the same fre-quency ωl; the ±1 orders have a shifted frequencies ωl± ωRF respectively
and, in general, the other ±n orders have ωl± nωRF frequencies. The
an-gular deflection of the outgoing light is a function of the RF and the order of diffraction: this can understood by using the Bragg law θ = arcsinmλ0
2Λn,
where m is the order, λ0 and Λ are the wavelengths of the IR light and of
the acoustic wave respectively, n is the index of refraction of the material. We will consider only the first diffracted order because it is the most intense among the shifted orders.
The linearly polarized light of the IR laser, with ωlfrequency, is
transmit-ted through a polarizing beam splitter and enters the AOM. By considering only the +1 order, with ωl+ ωRF frequency, it passes through a λ/4 plate
which converts the linear polarization to circular polarization, then through the lens and in the end it is reflected on the mirror. The reflected beam en-ters the AOM in the opposite direction hence it is diffracted another time. The resulting frequency is ωl+ 2ωRF but its direction is aligned with the
incoming laser beam. By the two passages through the λ/4 plate the polar-ization of the IR beam is again linear, but with the direction of polarpolar-ization rotated by π/2 so it is now reflected by the polarizing beam splitter: the incoming and the outgoing shifted beams are thus divided on two different optical paths. The lens is placed in the middle of the AOM and the mirror. It is necessary to focus the beam on both the mirror and the AOM. In fact, the direction of the diffracted beam changes as function of the RF frequency. This scheme is used to avoid changes in the direction of the outgoing beam.
Due to the long optical path from the cat’s eye to the MOT, the displace-ment of the IR beam, at the position of the MOT, is checked before starting a lifetime measurement. The maximum tolerated displacement is 10 µm, which is smaller than the size of the IR beam (around 90 µm).
By switching the RF of the AOM in the cat’s eye configuration, the frequency of the IR laser is quickly increased by δ6P thus it is now resonant
with the transition nS1
2 → 6P
3
2. We can de-excite the nS Rydberg level
driving the atoms into the 6P3
2 state which quickly (∼ 120 ns) decay into
the ground state.
The detuning from the 6P3
2 state in the first excitation step δ6P
∼ +40 MHz is a technical constraint of the cat’s eye configuration, as in-troduced in the previous section. The use of higher detuning can be not convenient for two reasons: a larger displacement of the excitation and de-excitation beams; a lower diffraction efficiency (the two RF which drive the AOM would be further from the region of greater efficiency of the AOM) which leads to a lower power in the excitation and de-excitation processes.
The typical power of the IR beam in the de-excitation process is of the order of 50 mW. The usual value of the Rabi frequency in the de-excitation process is of the order of 2 MHz. The de-excitation pulse lasts 5 µs.
2.2.3 Electric field ionization
Rydberg atoms are neutral atoms and for this reason one might think that it will be hard to detect single Rydberg atoms. However, they are highly excited states hence it is possible to ionize them using a suitable electric field. The value of the electric field which is necessary to ionize a n Rydberg state is
ε
ion(n) = Ry024k(n − δn,l,j)4e3
(2.5)
in which Ry0 = Ry · M/(M + me) where M is the total mass of the nucleus,
k = 1/4π0 where 0 is the vacuum permittivity.
By using an electric field
ε
>ε
ion(n) it is possible to ionize all Rydberg states higher than the n Rydberg state. In this way the detection process is reduced to a problem of charged particles.To this end we use two pairs of copper plates which are mounted one pair on the front side and the other one on the lateral sides of the quartz cell. Two fast high voltage switches change rapidly the voltage on the plates between the two inputs: one is set to the ground, the other is set to a particular voltage. The front plates are set at voltage of +3500 V, the lateral ones are set to −1000 V: taking into account the threshold state as the 63S, it corresponds to
ε
ion(63S) = 25.30 V/cm.Classical ionization threshold
n state
ε
ion(S state) [V/cm]ε
ion(P state) [V/cm] ∆ε
ionnS−nP[V/cm]
70 16.25 15.80 0.45
80 9.31 9.08 0.23
90 5.71 5.58 0.13
100 3.70 3.62 0.08
Table 2.1: Values of the classical ionization threshold for the S and P states and their differences calculated by Eq. 2.5. For comparison, the values of ∆
ε
ionnS−(n+1)S ∼ 2 · ∆ε
ionnS−nP.
Only for the 60S lifetime measurement we increase the voltage on the frontal and lateral plates to +4000 V and −2500 V respectively.
This electric field lasts only 9 µs to avoid the electric charging of the cell and it is useful not only for the process of the ionization but also to guide the ions of Rydberg atoms towards the channel electron multiplier. When a charge hits the internal surface of the detector, some electrons are emitted by its surface. They are accelerated and they hit again the surface: an avalanche process is triggered. When the produced electrons reach the charge collector an electric signal is obtained. The signal is recorded by a fast oscilloscope and it is sent to a Labview program which counts peaks. Each of these peaks represents one detected ion.
The detection efficiency of this apparatus is η ∼ 40%.
We always present the number of revealed Rydberg atoms N . When we display the number of real Rydberg atoms we write Nreal and it takes into account the detection efficiency Nreal= N/η.
In the end we remark that in our studies we do not use the state selective field ionization (SSFI): this method exploits a high-voltage ramp to ionize each state because the different ionization threshold of Rydberg states al-lows to detect them in different arrival times on the detector (previous works which used SSFI realized lifetime measurements up to n = 45 [25][8][26]). The reason for which we do not apply the SSFI method is due to the small energy distances between the excited Rydberg states used in our experi-ments. In fact we cannot determine the different Rydberg states by using the arrival times on the detector. By considering Eq. 2.5, the estimated classical ionization threshold values for the S, P states and the differences between them are listed in Tab. 2.1: ∆
ε
ionnS−nP/ε
ion(S state) ∼ 2 − 3% which is too low to precisely determine the two different populations from the arrival times.Figure 2.5: Timing of the excitation, de-excitation and ionization pulses. The red scheme is used to collect the red data points whereas the blue one is used to collect the blue data points in a lifetime measurement.
2.2.4 Timings
The repetition rate of every our experiment is set to 4 Hz.
Each point of a frequency scan measurement which is used to find the right value of the ω5S1
2 →6P3 2 and ω6P3 2 →nS1 2
is typically obtained as an av-erage of 50 shots. Each point of the de-excitation curve is obtained as an average of 70 shots. The measurement of the lifetime of Rydberg atoms is obtained collecting 100 shots for each experimental data point.
The timings are displayed in Fig. 2.5: all the time steps of excitation, de-excitation, ionization are manually set by using a single delay generator. The time between the excitation pulse and ionization pulse is the free evolution time.
In the lifetime measurements which are discussed in the next chapter, two points are collected for each free evolution time: changing when the ionization pulse starts we change the free evolution time. The red points are collected by using the excitation and ionization pulses in the red scheme on the top of Fig. 2.5; the blue points are collected by using the excitation, de-excitation and ionization pulses as displayed in the blue scheme on the bottom of Fig. 2.5.
Experimental results
In this chapter I will present the experimental results regarding measure-ments of the lifetime of highly excited Rydberg states by means of the de-excitation technique.
Our purpose is the measurement of lifetimes of nS Rydberg states with principal quantum number n ≥ 60 to the highest n possible. The lower limit, n = 60, is set by the maximum electric field that we can produce in order to ionize Rydberg atoms, as said in Sec. 2.2.3, whereas the upper limit is caused by a stray electric field that affects our apparatus, as we will see in Sec. 3.3.
In this regime of high principal quantum numbers, the ionization thresh-olds of Rydberg states are very close to each other. By using the state se-lective field ionization, in order to distinguish the different arrival times on the detector of the different ionized Rydberg states, it does not allow us to resolve them precisely as discussed in Sec. 2.2.3. To overcome this limit, we used the de-excitation technique, which is a hybrid optical and field ioniza-tion method and allows us to measure the populaioniza-tion of the excited Rydberg state.
3.1
Lifetime protocol
We have already discussed the tools (excitation, de-excitation and ioniza-tion) used in a lifetime measurement in Sec. 2.2.
We excite a particular Rydberg state and we leave the system evolving freely. After a certain period of free evolution, we ionize Rydberg atoms collecting subsequently two kinds of point for every chosen time:
• red point: it represents the population of all Rydberg states above the critical state, called ”ensemble”;
• blue point: it represents the population of all Rydberg states above the critical state without the population of the excited state, called
”support”. The population of the excited state has been transferred under the ionization threshold by the de-excitation pulse just before the ionization pulse.
For clarity, the timings of the whole protocol, both for the red and the blue points, are depicted in Fig. 2.5.
By subtracting these two points, we obtain the green point: it represents the population of the excited state at the chosen time, called ”target”.
Repeating these steps, by varying the duration of the free evolution, it is possible to obtain the target curve up to 1500 µs and the ensemble curve up to 10000 µs. The choice of the maximum time in the target lifetime measurement is mainly due to two factors: theoretical predictions suggested that lifetimes that we are interested are of the order of few hundreds of microseconds, so this maximum time is well above them and it is not a limiting factor; the effect of gravity on the displacement of the atoms is on the order of ∼ 10 µm and may start becoming significant for the efficiency of the de-excitation pulse for longer times.
Data are fitted with a single exponential fit, by imposing the vertical offset to zero for t → ∞. It yields τ ± στ: τ is the lifetime value, στ
is its error. To test the reproducibility of this protocol we took several independent measurements for some nS states.
A typical measurement of the lifetime by using the protocol just de-scribed is displayed in Fig. 3.1: red, blue and green points are experimental values of the ensemble, support and target population respectively; the red and green line are single exponential fits of the ensemble and target popu-lation respectively.
3.1.1 Single-atom regime to avoid interactions
At large distances, the interaction mechanism between two highly excited atoms is described by the van der Waals interaction, as showed in Sec. 1.2. In order to neglect interaction effects, we want to work as close as possible to the single-atom regime. The creation and the manipulation of a single Rydberg atom ensures that we avoid any sort of interaction; however the detection process of a single atom constitutes a non-trivial effort and makes the measurements less reliable due to a low signal-to-noise ratio. Because of this limitation, we enhanced the average number of excited atoms to two.
Taking into account the detection efficiency η ∼ 40% we get a number of real Rydberg atoms which are in the system equal to NRyreal= NRy/η ∼ 5.
This number must be a good value for all the range of nS states on which we work in order to guarantee similar conditions among all experiments: this means that the mean distance among real Rydberg atoms must be always much larger than the blockade radius
1.5 1.0 0.5 0.0 <N> 3000 2500 2000 1500 1000 500 Time [µs] Ensemble Support Target
Figure 3.1: Lifetime measurement of the 70S Rydberg state. Red, blue and green points are experimental values of the ensemble, support and target population respectively. Green and red line are the single exponential fit of target and ensemble lifetime respectively. Lifetime values obtained by the fits are τ70S = 129.6 ± 21.1 µs and τensemble(70S)= 381.6 ± 23.5 µs.
This condition assure us that we are reasonably far from the beginning of the saturation regime [27] of the system and we can suppose that our atoms are still in resonance respect to excitation and de-excitation pulse.
For example, by considering a laser linewidth (the combined laser and residual Doppler linewidth) of Γlaser = 2π · 0.7 MHz [28], we get a blockade
radius for the 100S state of rb = 6
q
C6
}Γlaser = 15.3 µm. In Tab. 3.1 C6
coefficients and blockade radii rb for some nS states are listed. Calculations
are done by using ARC, a Python package [13][15] which I will introduce in Chap 5.
Typical dimensions of our 3D MOT are σx ∼ 200 µm, σz ∼ 140 µm
and σy ∼ 170 µm; the two excitation lasers are co-propagating along the ˆx
direction. To first order, the excitation volume can be modelled as a cylinder of height h = σx∼ 200 µm and radius (set by the smaller beam) r ∼ 40 µm.
Given the mean number of real Rydberg atoms
< NRyreal>= ρhπr2
ρ = < N
real Ry >
hπr2
we get a mean distance, among Rydberg atoms, of
< drealRy >= ρ−13 ∼ 58.6 µm (3.2)
thus we can reasonably affirm that the condition in Eq. 3.1 is well respected. If it were not so, the effect of strong interactions would be an excitation
C6 coefficients and blockade radii nS1 2 C6/h [GHz/µm 6] r b[µm] 60 135.298 5.60 65 358.078 6.58 70 862.693 7.62 75 1947.59 8.73 80 4161.41 9.91 85 8465.74 11.1 90 16500.9 12.5 95 30965.3 13.8 100 56170.6 15.3 105 98836.4 16.8
Table 3.1: Values of C6 coefficient and blockade radius returned by ARC for
nS1
2 states using Γlaser = 2π · 0.7 MHz.
blockade because the energy of the Rydberg state is shifted out of resonance from the exciting laser by the presence of a neighbouring Rydberg atom.
In light of the above we can state that interactions can be justifiably neglected as long as Eq. 3.1 is valid.
The validity of the discussion of the van der Waals interaction among Rydberg atoms assumes that there is no external electric field, thus the dipole moment of the atom is zero leading to a second order treatment of the interaction. Nevertheless, we do not know a priori if this hypothesis is true and I will discuss this problem in Sec. 3.3.
Nevertheless, even in case of dipole-dipole interactions due to S, P states, the interaction energy is less than one MHz because of the large distances between Rydberg atoms. For comparison, the C3 coefficient between 100S
and 99P states is of the order of ∼ 30 GHz · µm3, thus the value of the interaction energy is U (< drealRy >) ∼ h · 0.15 MHz for the mean distance given by Eq. 3.2.
Interactions can be reasonably neglected.
3.2
First results
First measurements of the lifetime are in reasonable agreement with theo-retical predictions up to n ∼ 76, as we can see in Fig. 3.2.
As soon as we study Rydberg states with n & 76, the measured values of the target lifetime decrease suddenly by two orders of magnitude and we find experimental target lifetimes of only few µs in contrast with hundreds µs that we expected from theory (Fig. 3.2a). These states also exhibit two
further important features, as the target lifetime decreases: the ensemble lifetime becomes two-three times longer (Fig. 3.2b) and the de-excitation efficiency decreases (Fig. 3.2c). In particular, the higher the principal quan-tum number, the higher the ensemble lifetime and the lower the de-excitation efficiency.
3.2.1 Interactions
As discussed above, long range interactions among neutral atoms can be quantified using the C6 coefficients listed in Tab.3.1. There is no such a
difference in trend of C6 that could explain such a sudden drop in target
lifetime for n ∼ 76. Moreover, if this drop had been caused by interactions, it would not have explained the increasing of ensemble lifetime. Interac-tions can be therefore most likely be neglected for the explanation of this deviation.
3.3
Stray electric field
Rydberg atoms in range of high principal quantum numbers are character-ized by a large value of the polarizability α, which scales as n7: for com-parison α85S ≈ h · 2140 MHz · cm2/V2 whereas α5S ≈ h · 0.079 Hz · cm2/V2
[18].
In order to check the presence and the value of the background electric field, we increase the power of exciting lasers and scan a wide range of frequencies around the resonant frequency of the target state. The aim of this scan is to check for the presence of a possible manifold (the fan composed of l > 3 states which experience the linear Stark effect described in Sec. 1.3). In87Rb the quantum defect for S states is δn,0,0.5 = 3.13, for this reason
the energy level of a nS state is slightly below the manifold of (n − 3)l states with l > 3. In fact, states with l > 3 have a quantum defect δn,l>3,l±0.5 <
0.005, thus we can consider it negligible.
We can easily evaluate the energy distance between the target n state and the manifold composed of m = n − 3 states (atomic units):
∆E(n)nS−manif old = −
1 2n∗2 + 1 2m∗2 = − 1 2(n − δn,0,0.5)2 + 1 2(n − 3 − δn,l>3,l±0.5)2 = − 1 2(m∗− 0.13)2 + 1 2(m∗)2 ' − 0.13 (m∗)3 ' − 0.13 (n∗)3 (3.3)
Taking into account that the difference in energy between two different man-ifolds is ∆E(n)(n+1)−n ∼ n∗−3, a factor of 0.13, as shown in Eq. 3.3, means
300 250 200 150 100 50 0 Target Lifetime [µs] 90 85 80 75 70 65 60
principal quantum number, S state
(a)
(a) 2000 1500 1000 500 0 Ensemble Lifetime [µs] 90 85 80 75 70 65 60principal quantum number, S state
(b)
(b) 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 De-excitation efficiency 90 85 80 75 70 65principal quantum number, S state
(c)
(c)
Figure 3.2: (a) the target lifetimes as function of the principal quantum number. Green points are the experimental values, thin line represents Beterov’s article data [11] whereas thick line shows other Beterov’s data [29] which take into account the repopulation processes. (b) the ensemble lifetimes as function of the principal quantum number. Red points are the experimental values whereas the thick dashed line shows Beterov’s data [29]. (c) the efficiences of the de-excitation process at zero time as function of the principal quantum number. Yellow points are the experimental values.
50 40 30 20 10 0 <N> -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Relative frequency [GHz]
Figure 3.3: Wide frequency scan around the 85S Rydberg state (located at relative frequency 0). Number of excited Rydberg atoms as function of the detuning from the resonant frequency for the exciting IR laser. The peaks near the principal one (the target state 85S) are clearly visible.
a distance of around few GHz between the target state and the fan of the above manifold with zero electric field.
In particular, we can use this scaling law to assess the value for the 85S (starting from the reference value ∆E(43S)(n+1)−n= h · 100.5 GHz [2]):
∆E(85S)85S−82manif old= h · 100.5 GHz · 43∗3·
−0.13
85∗3 = h · −1.51 GHz (3.4) which lies below the 82l>3manifold. Using ARC [15], by the LevelPlot class,
we get ∆E(85S)85S−82manif old= h·−1.52 GHz while, by the StarkMap class,
we get ∆E(85S)85S−82manif old = h · −1.57 GHz. These values are in good
agreement with Eq. 3.4.
In light of the above, for negligible background electric field, we expect to see, in the excitation process, one resonance only: the signal of the target state.
However, we observe a structure of energy levels which shows the pres-ence of several tens of small peaks around the principal one, Fig. 3.3
This is due to a stray electric field which is present in our apparatus: because of the Stark effect, the energy level of the target state mixes the fan of high − l states. This means that the energy level that we excite and de-excite is not a pure nS state but it is mixed with a large number of other states. When this background electric field is large enough, the target level crosses the fan mixing with other states and we measure a very short lifetime.
250 200 150 100 50 0 Lifetime 85S [µs] 200 180 160 140 120 100 Electric Field [mV/cm] 1400 1200 1000 800 600 Lifetime Ensemble [µs]
Figure 3.4: Values of target lifetime (red points) and ensemble lifetime (blue diamonds) as function of the absolute value of background electric field for the 85S Rydberg state. The Inglis-Teller limit ∼ 125 mV/cm for the 85S state is highlighted with a green dashed line. The theoretical prediction for the value of the target state is τ85S = 257.6 µs whereas the value of the
ensemble lifetime is τensemble= 719 µs.
of the values of the ensemble lifetime take place close to the first crossing of the target state with the manifold, the Inglis-Teller limit. The value of the Inglis-Teller limit depends on the principal quantum number with a scaling-law ∼ n−5, as we have seen in Sec. 1.3.
By measuring our stray electric field, we obtain a value of ∼ 215 mV/cm equivalent to the Inglis-Teller limit for n = 76, which is in correspondence to the principal quantum number where the measured lifetime values drop down.
In order to overcome this limit, at first we employed just the front plates used for the ionization to apply a compensation field, and then we added other two pairs of electrodes specially designed by us to fit well close to the cell. After doing this, we set the voltage on the different plates and electrodes by minimizing the Stark shift (Eq. 1.13). This minimization is directly related to the decrease of the absolute value of the background electric field [4].
This tool enabled us to do experiments with a reduced background field. By varying the absolute value of the background electric field and by mea-suring target and ensemble lifetimes we obtain Fig. 3.4, in which the corre-sponding Inglis-Teller limit is highlighted by the green dashed line
Note that the excited state does not mix with high − l states at first but with the P states (Sec. 1.3), then with the D states, and so on [7] : this behaviour can be used to explain the deviation of target and ensemble lifetimes.
