# Neutral mercury from a frequency standards point of view

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## Optical frequency standards

The very basic and simple scheme of a frequency standard is shown on fig. 1.1. It consists of a local oscillator, a reference and a servo sys-tem. The frequency of the local oscillator (t) is being compared to the frequency of a reference 0. Then the result of the comparison (t) = (t) 0 is fed into the servo system. Based on (t) servo sys-tem generates a correction signal, which introduces a shift of the local oscillator frequency such that (t) ! 0. Thereby the frequency of the local oscillator is actively locked to the frequency of the reference.
An ideal local oscillator after being locked to a reference would pro-vide exactly the frequency 0 of the reference, but it is not the case in reality. Frequency of a real device will fluctuate with time to some ex-tent. Therefore all frequency standards are characterized by two main quantities. First of all ”good” frequency standards should have high frequency stability, which means that a set of temporal measurements of local oscillator’s frequency should be well concentrated around some averaged frequency avg, i.e. deviation of the measured frequency from avg should be small. The second criteria is the systematic uncertainty, which reflects how well avg matches the frequency of unperturbed atomic transition, the closer avg to the frequency of the transition the better the clock. It is also common and useful to characterize a clock by its precision. This can be defined as the difference between the actual output frequency and the nominal value with respect to a pri-mary frequency standard. If there is no absolute reference or it is less accurate than the frequency standard that is being compared, then the precision of the standard is defined as the limits of the confidence interval of the measured quantity [13]. In order to be able to compare completely different frequency standards relative stability and relative accuracy are often used, which are the stability and accuracy divided by a carrier frequency.
Along with advances in science and technology the stability and accuracy of time and frequency standards are increasing, demanding more and more accurate and stable references. It became apparent that atoms are very well isolated and robust against external perturba-tions and able to provide stable, reproducible and universal frequency reference. In 1950s it was shown that using the appropriate transi-tion between two energy levels in caesium (Cs) atom it is possible to reach relative frequency uncertainty Δνν = 1 × 10−9 [14], where Δν is frequency deviation and ν is the central frequency of the transition. This led to the fact that in the 1967 the second was redefined as du-ration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom [15]. This stimulated fast development, evolution as well as implementation of atomic clocks in industry. The research of microwave atomic clocks was advancing for more than 50 years increasing the accuracy with an average speed of one order of mag-nitude per decade. Today best atomic Cs clocks, which are called Cs fountains, because in this clock atoms at micro kelvin temperature are being launched vertically as in fountains, has relative frequency uncertainty of ∼ 2 × 10−16 and stability 1.6 × 10−14τ−1/2. This is the limit that can not be surpassed by microwave clocks, because of sta-bility issues due to cold atom collisions and distributed cavity phase shift [16].
The way to reach even higher relative frequency uncertainty is to switch to the optical frequency domain. Optical frequencies are sev-eral orders of magnitude higher than the microwave and the quality factor Q = is as well higher in optical domain. For comparison cur-rently for our mercury optical clock the quality factor is Q 1014, while for the Cs fountain clock it is only Q 1010. Moreover, given level of uncertainty can be reached within much shorter time with optical fre-quencies comparing to microwave since the stability of a clock limited by quantum projection noise is proportional to Q1 here y is the Allan deviation (see ch.5.4), which is commonly used to characterize stability of a clock , Tc is the cycle time, is the time scale at which stability is characterized and P is the fundamental noise of atomic signal that comes from the indeterminacy of quantum mechanics and known as quantum projection noise (QPN). In case of p an ensemble of Nat atoms QPN is proportional to 1= Nat [17], [18]. With the development of laser cooling and trapping techniques [19] it became possible to confine neutral atoms in a certain area of space in vacuum. Movement of such confined atoms with a good level of approximation can be described as a movement of an oscillator in a harmonic potential. We will assume that an atom oscillates in the trap with frequency !t. Then the radiation field that the atom experiences will be phase modulated
E(t) = E0 cos(!0t + cos(!tt)) (1.2)
where !0 is the frequency of the radiation field and is the modula-tion index, which equals to the maximum phase difference between the modulated and unmodulated radiation field. In the frequency do-main such phase modulated radiation field will be represented by a pure unbroadened carrier !0 and an infinite series of sidebands with frequencies !0 n!t, where 1 < n < 1. It was calculated [20] that if an atom is confined to a space, which is less than half wavelength of the radiation field the sidebands become very weak and mostly the radiation at the carrier frequency will be absorbed and emitted by the atom. Therefore confinement of an atom in the area smaller than the half wavelength of the probe radiation removes the first order Doppler broadening. Such condition is called Lamb-Dicke regime.
Intense development during 30 years led to the creation of two main types of optical atomic clocks, which today have outstanding accuracy and stability and competing with each other for the right to redefine the SI second and become a new primary time and frequency standard.

### Optical clocks based on ions

The first type are the optical clocks that are using ions as a fre-quency reference – optical ion clocks. The main advantage of using ions is their strong interaction with electric fields, that allows to rela-tively easily trap and confine them [21]. A single ion can be stored in a trap for months (virtually forever) and probed well into the Lamb-Dicke regime. Putting trapped ions in an ultra-high vacuum environment largely reduces collisions with other atomic species and coupling to the outside world. All this allows modern optical ion clocks to achieve very high accuracy (see e.g. [22]).
However there are also drawbacks in using ions as a frequency reference for an optical clock. The main problem of ion clocks comes from the very same fact the advantage does: ions have a charge. Due to this fact several ions placed in the same potential well are pushed by their mutual Coulomb repulsion from the center of a quadrupole trap to area where trapping field is stronger. This leads to higher amplitude motion of ions in the trap that induces a frequency shift and may inhibit laser cooling [23]. This limits accuracy of a clock. There should be only one ion in a potential well, but the signal from a single ion has big quantum-projection noise (QPN) [17] and a correspondingly high stability limit. With low signal to noise ratio optical ion clocks are limited in short-term stability. Today several atomic species are being used for the optical ion clocks. The accuracy record belongs to the ion clocks made at the NIST laboratories in the USA based on single ions of Al+. They have reached relative uncertainties at the level of several of 1018 and stability at 1015 level [22]. Ion optical clocks based on Sr+ and Yb+ had reported accuracy at the level of therefore improve short-term stability different types of ion traps (for instance linear Pauli trap [28], surface traps [29]) are being used that are capable to trap several ions in the same time, keeping each ion in a separated potential well [30], [31]. It was found that interaction between separated ions in those traps couldn’t be fully suppressed and still gives significant contribution to the loss of accuracy. The way to increase stability without sacrificing the accuracy is to use neutral atoms instead of ions.

#### Optical clocks based on neutral atoms

Compared to ions, neutral atoms do not have a charge and there-fore their mutual interaction is weaker. More than 106 neutral atoms can be confined in the same trap providing much better QPN limit and therefore better short-term stability in comparison to ion clocks. At the same time neutral atoms have low sensitivity to electromagnetic fields which makes them difficult to trap. In order to trap neutral atoms at room temperature in a deep enough potential well one would have to generate fields with an unreasonably high amplitude. Nevertheless, with the discovery and development of laser cooling techniques it be-came possible to cool neutral atoms down to temperatures of order of tens of K (see chapter 3.2) and drastically reduce their kinetic energy. The lower the kinetic energy of an atom, the smaller the minimal depth of a potential well that is required to trap it.
It will be shown in section 2.3.2 that an electromagnetic wave shifts the energy levels in atom due to ac-Stark effect. Two coherent counter propagating and overlapped laser beams with the same wavelength l form a standing wave with intensity nodes separated by l=2. Such intensity pattern will create a spatially modulated Stark shift of the energy levels in atom. As the result an atom placed in the standing wave will experience optical potential where z is the axis along the direction of propagation of the light, w(z) is the laser beam waist, r is the radial coordinate in the transverse direction, kl is the wave vector of the light. The depth of the trap U0 is where (!l) is the polarizability (see sec.2.3.2). It was found that the depth of the trap created by two laser beams of reasonable power was big enough to trap atoms cooled down to K temperatures. Such traps are called dipole or lattice traps [32]. In principle, lattice traps are not suitable for atomic clocks due to the fact that the energy shifts induced by the trap light of the ground and excited states of a clock transition are different. This causes a frequency shift of the clock transition that depends on the intensity of the lattice light and worsens the accuracy and stability of the clock. The situation changed with the proposal of the magic wavelength by H. Katori [33]. The idea is to use in lattice traps light at a special so-called ”magic” wavelength, for which the energy shifts of the ground and excited states are exactly the same. When this condition is satisfied no frequency shift due to lattice trap occurs (see more in sec.2.3.2). This enabled the use of lattice traps in clocks based on neutral atoms and such clocks are called optical lattice clocks.
Today several atomic species are being used in different laborato-ries all around the world in optical lattice clock experiments. However not every atom, as well as not every ion, have the right properties to be a reference for an accurate ultrastable clock. The first and the most important criteria is imposed on the clock transition. Highly accurate and stable clocks require atomic species with very high Q-factor. In other words only transitions with sufficiently narrow natural linewidth can provide desirable stability and accuracy. In general allowed dipole transitions have too large linewidths and can not be used as a clock transition. However, one can find some atomic species in which nor-mally forbidden intercombination transitions ( S 6= 0) are weakly al-lowed due to fine and hyperfine mixing of energy levels (see chapters 2.2 and 2.3). These transitions have very narrow linewidth suitable for optical clock applications. It is also favourable if the excited state is not connected to any other transitions except the clock transition, which means no losses of excited atoms due to spontaneous decay to other levels.
The second criteria is the existence of a suitable transition (or com-bination of transitions) for laser cooling. Laser cooling technique al-is called Doppler temperature limit [13]. In case of lattice clocks this transition should be relatively narrow, so the atoms could be cooled down to 100 K, but in the same time intensive enough to have good cooling efficiency. However, in some cases sub-Doppler cooling can exist [34].
Several neutral atoms do fit both criteria and were chosen for lattice clock experiments in different laboratories around the world: Yb ([8], [35]), Sr ([36],[9],[37],[38]), Hg ([39]), Mg ([40]). Each of these atomic species has different properties, advantages and disadvantages from the frequency standard point of view, which makes them interesting to study. In chapter 2 detailed description of mercury properties is given and comparison of mercury with other atomic species for each important characteristic is done.
There are two fermions and four bosons among isotopes. All bosons have nuclear spin equal to 0, which means that clock transition is nor-mally forbidden for them. However, it was shown in [43] that, due to Zeeman effect, a small static magnetic field applied to alkaline-earth-like atoms can mimic hyperfine mixing of states and induce coupling to the forbidden 1S0 !3 P0 transition. This effect is called magnetic quenching of a transition. It is remarkable that quenching of a clock transition also allows to control its linewidth by magnetic field. Thereby in spite the fact, that transition 1S0 !3 P0 is forbidden in bosons they still can be used for lattice clock purposes. However, fermions were chosen to be used in optical lattice clocks because the clock transition in fermions is weakly allowed (lifetime > 1 s) without additional complications and limits connected with quenching meth-ods. Also, according to the Pauli exclusion principle two fermions can not share state with a given energy in the same time and in the same space. This means that for spin polarized sample of 199Hg atoms each fermion will occupy different motional state of a trap and therefore the atoms are less prone to collide (see more in sec.5.5.1). All the experi-ments presented in this thesis were performed with 199Hg isotope.
Another remarkable property of mercury is that its boiling and melting points are at 356.73 C and -38.83 C respectively at 1 bar pressure. Therefore mercury is liquid at room temperature at normal pressure. From figure 2.1 one can see that in order to have reason-able vapour pressure Sr and Yb, which are mainly used in optical lattice clocks (OLC), have to be heated up to 300-400 C or more. Mercury has vapour pressure at room temperature 14 orders of mag-nitude higher than Yb and Sr and has to be on contrary cooled down to not over flood the vacuum chamber. This is another advantage of mercury – absence of an oven or other hot devices helps to control the background temperature with better accuracy and less effort. It also enables the use of efficient sources of slow atoms, such as 2D Magneto-optical trap.

Introduction
1 Optical frequency standards
1.1 Optical clocks based on ions
1.2 Optical clocks based on neutral atoms
2 Neutral mercury from a frequency standards point of view
2.1 General properties of mercury
2.2 1S0 – 3P1 cooling transition
2.3 1S0 – 3P0 clock transition
2.3.2 Light shift and the magic wavelength
3 Experimental setup
3.1 Cooling light laser source at 253.7 nm
3.1.1 Yb:YAG thin-disk laser
3.1.2 Frequency doubling stages
3.1.3 Frequency lock on saturated absorption
3.2 Magneto-optical trapping of neutral mercury
3.2.1 Vacuum system
3.2.2 Detection scheme
3.2.3 Experiment operation cycle
3.3 Ultra-stable light laser source at 265.6 nm
3.3.1 Probe beam spatial filtering
3.4 Lattice trap light laser source at 362.5 nm
4 Improvements of the experimental setup
4.1 Cooling laser system improvements
4 CONTENTS
4.1.1 New cooling laser system
4.1.2 New doubling stage at 254 nm
4.2 New lattice trap
4.2.1 Lattice trap parabolic approximation
4.2.2 Test of the new lattice trap mirrors
4.2.3 Implementation of the new lattice trap
4.2.4 Characterisation of the new lattice trap
5 Spectroscopy experiments on 1S0 – 3P0 transition
5.1 Spectroscopy of 1S0 – 3P0 transition
5.2 Measurement of magnetic field at trapped atoms site
5.3 State selection and dark background
5.4 Frequency stability measurements
5.4.1 Interleaved lock
5.5 Systematic shifts
5.5.1 Collision shift
5.5.2 Lattice light shift estimation
5.5.3 Second order Zeeman shift
6 Conclusion

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