Ernestinthesee@gmail.com 463 Gerwers Benedikt Text

Master Thesis in Physics Kennard-Stepanov Relation for Dense Molecular Gases written by Benedikt Gerwers at the Institut für Angewandte Physik submitted to the Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn August 2015 I hereby declare that this thesis was formulated by myself and that no sources or tools other than those cited were used. Bonn, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Date Signature 1. Gutachter: Prof. Dr. Martin Weitz 2. Gutachterin: Priv.-Doz. Dr. Elizabeth Soergel Acknowledgements I would like to thank Prof. Dr. Martin Weitz for giving me the opportunity to work on my master thesis in his research group and giving me an interesting topic to work with and also for all the support and knowledge given throughout this year. Priv.-Doz. Dr. Elizabeth Soergel for assuming the position of the second supervisor of this thesis. Prof. Dr. Stefan Linden for coming and listening to my master colloquium, substituting for Priv.-Doz. Dr. Elizabeth Soergel. Dr. Stavros Christopoulos for all the help and explanations received during my master thesis. I couldn’t have had a better advisor during this year. Also thanks for suggesting me for the DPG conference, which gave me an excellent opportunity to further my knowledge in this re- search area. Dr. Frank Vewinger for all the help and the suggestions. Your knowledge in physics seems to be endless. The team of the underground lab (current and former) : you have been really great lab colleagues and a lot of you seem to have a really good taste in music! It has been fun working together with you, especially the team of the redistributional laser cooling experiment. All the members of the research group : you have been great colleagues, always helping if required and also fun to be with outside work, be it at the mensa or on occasions like the end-of-the-year party. Lastly and especially I would like to thank my family , who supported me throughout my entire studies and made the whole thing possible in the first place. I really appreciate also the opportunity to study abroad 2 semesters. v Contents 1 Introduction 1 2 Theory 5 2.1 Redistributional Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Line Shape Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Pressure Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Saturation Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.3 Redistribution of Fluorescence . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Molecular Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Kennard-Stepanov-Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Rubidium Dimer Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Experimental Setup and Methods 17 3.1 High Pressure Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Thermal Deflection Spectroscopy Setup . . . . . . . . . . . . . . . . . . . . . 18 3.3 Transverse Thermal Deflection Spectroscopy Setup . . . . . . . . . . . . . . . 19 3.4 Transverse Thermal Deflection Spectroscopy . . . . . . . . . . . . . . . . . . 20 3.5 Kennard-Stepanov Relation Setup . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5.1 Setup for the Fluorescence Measurement . . . . . . . . . . . . . . . . 23 3.5.2 Setup for the Absorption Measurement . . . . . . . . . . . . . . . . . 24 4 Kennard-Stepanov Relation for Dense Molecular Gases 25 4.1 Kennard-Stepanov Relation Analysis on Rubidium Dimers . . . . . . . . . . . 25 4.1.1 Energy Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1.2 KS Study of Rubidium Dimer X !E Transition . . . . . . . . . . . . 26 4.1.3 KS Study of Rubidium Dimer X !D Transition . . . . . . . . . . . . 29 4.2 Discussion on Aggregated Kennard-Stepanov Results . . . . . . . . . . . . . . 32 5 Conclusion and Outlook 33 A Preperation of the Used High Pressure Cells 35 A.1 Cleaning Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 A.2 Reassembling and Sealing Test . . . . . . . . . . . . . . . . . . . . . . . . . . 35 A.3 Filling the high pressure cell with rubidium . . . . . . . . . . . . . . . . . . . 36 vii Bibliography 37 viii CHAPTER 1 Introduction The study of light has been an active scientific area for a many years but it was not until the early 20thcentury that the cooling of matter using light was proposed [1]. In 1929 Pring- sheim explicates in „Zwei Bemerkungen über den Unterschied von Lumineszenz- und Tem- peraturstrahlung” that cooling of a gas through emission of light is not in contradiction with the second law of thermodynamics [1]. Further experimental investigation into the matter began after the proposal of coherent light in 1958 [2] and the invention of the laser in 1960 [3]. Nowadays, one of the most commonly used cooling techniques is the Doppler-cooling of di- lute atomic gases, which was first suggested in 1975 [4]. Laser light, red detuned to the atomic resonances, propagates into opposite direction than that of the atoms, so that photons can be absorbed because of the so-called Doppler-shift. The photon momentum is then transfered to the atom, slowing it down. Subsequent reemission of the photon is in a random direction, thus averaging out over many cycles. From this follows that the sample is cooled down to the dop- pler limit which is~ 2kB, where is the natural linewidth of the atomic resonance. Using this technique in combination with more refined techniques (e.g. Sisiphus-cooling and at a later stage evaporative cooling) temperatures in the nanokelvin regime can be reached [5]. These low temperatures gave rise to the experimental realization of a Bose-Einstein condensate [6, 7], which was first realized in 1995 [5, 8]. Cooling techniques using light are not restricted to dilute gases, but there are also techniques applicable on solid state systems. Such is the Anti-Stokes cooling technique, which was first realized in 1981 [9], and was used by Epstein in 1995 to demonstrate cooling of a heavy-metal fluoride glass [10]. In the case of Anti-Stokes cooling in solids the energy is extracted because of thermalization processes within the solid. The solid is excited with a frequency corres- ponding to the energy di erence between the top of a ground state manifold to the bottom of an excited state manifold. Hence, both manifolds are led to an out-of-equilibrium distribution, since the bottom levels of each manifold are now populated more than they would be in thermal equilibrium. Photon interactions distribute the population according to the temperature of the solid and so the mean fluorescence is more energetic than the used optical excitation frequency leading to energy extraction. Redistributional laser cooling, the only cooling technique presently known to bo applicable in ultradense gases, was first proposed by Berman and Stenholm in 1978 [11]. It was then first realized in a proof of principle experiment in 2009 [12], where a cooling by 66 K was achieved. 1 Chapter 1 Introduction Redistributional laser cooling uses the collisionally perturbed potential curves of an electronic ground and excited state in order to achieve absorption of red detuned photons. Subsequent radiative decay occurs close to the unperturbed atomic resonances leading to in average higher energetic photons than the incident ones. The energy di erence comes from the kinetic en- ergy of the atom, thus lowering the temperature of the sample in the macroscopic case. The removed energy for this method per cycle is in the same order of magnitude as the width of the pressure broadened lineshapes: kBT, and is thus multiple orders of magnitudes greater than the eciency of the Doppler cooling method. In order to determine the temperature of the cooled samples, di erent methods have been de- veloped. The thermal deflection method is commonly used in the determination of temperature variations in gases [13, 14]. The technique is based on the fact that a gradient in temperat- ure induces a gradient in density which in turn, leads to a gradient in the refractive index. This refractive index gradient causes a deflection of a non-resonant probe beam, which is then meas- ured to determine the temperature. In the past this method has been successfully used in order to determine the temperature in samples cooled with redistributional laser cooling [12, 15]. Redistributional laser cooling can be explained in detail within a framework of rubidium-argon quasi molecules. In the hot rubidium vapor created in these experiments, the presence of rubidium dimers of- fers an opportunity to study them under high pressures and at high temperatures. The study of molecular gases through spectroscopic methods is of great interest in order to determine the fundamental thermalization properties of the investigated molecular manifolds. In this thesis, molecular manifolds are being investigated and a more accurate method to determine the tem- perature is being investigated. Especially for the gas phase this spectroscopic study is not perturbed through other physical phenomena like phonons and other excitations. As described by Kennard [16] and Stepanov [17], in the case of fully thermalized electronic ground and excited state manifolds the frequency dependence of absorption and fluorescence spectra are related by a Boltzman-like scaling. This relation has been studied for di erent cases, e.g. in dye molecules in liquid solutions [18, 19] or in dense atomic gases [20]. It can be understood to be a consequence of the detailed balance condition [21]. The Kennard-Stepanov relation has also been used in order to investigate the thermalization of photon Bose-Einstein condensates [22]. Other examples where the Kennard-Stepanov relation has been tested for are quantum wells [23] and in photoactive biomolecules [18, 24]. In this thesis spectroscopic measurements are carried out, in order to determine the thermal- ization status of rubidium dimer manifolds in high pressure argon environment. For that both fluorescence and absorption measurements using high pressure cells are performed on the ru- bidium dimer X!D and X!E transitions. Using the Kennard-Stepanov relation as an alternative to the thermal deflection method, the temperature of the gaseous ensemble can be determined. Also, initial experiments on a thermal deflection based technique are performed in order to produce a two-dimensional thermal profile in a hot rubidium-argon gas mixture. Initial investigations of the Kennard-Stepanov relation for the X !E rubidium dimer transition 2 at di erent temperatures have been done by Roberto Cota [25]. In this thesis this transition has been investigated with an improved setup and more accurate results have been obtained. Fur- thermore the Kennard-Stepanov relation applied to the X !D rubidium dimer transition has been investigated for the first time. Finally, the Kennard-Stepanov relation for the X !E cesium dimer transition has been investigated. Chapter 2 describes the theoretical background for redistributional laser cooling and pressure and saturation broadening of the lineshapes. Furthermore the thermalization caused by the frequent collisions in the system leading to the Kennard-Stepanov relation is discussed, and a detailed description of the rubidium dimers in a high pressure argon environment is given. The experimental setup is elucidated in chapter 3. Here the two-dimensional deflection setup is shown and compared to the thermal deflection spectroscopy. The setups for the fluorescence and absorption measurements are also presented. Finally, the high pressure cell used in the experiments is presented, too. Furthermore the two-dimensional thermal profile of a rubidium- argon mixture obtained using thermal deflection spectroscopy is derived and discussed. The obtained experimental results are presented and discussed in chapter 4. More specific- ally, the Kennard-Stepanov relation analysis for the X !D and the X!E molecular rubidium transition is presented in detail. Further results with di erent temperatures as well as results us- ing cesium dimers are presented. With these results the implications for the Kennard-Stepanov relation on molecular gases are discussed. Finally, in chapter 5, conclusions are given and future perspectives are discussed. 3 CHAPTER 2 Theory In this chapter, the theoretical background for the experiments carried through within this work is explained. Redistributional laser cooling is explained and the broadening of the lineshapes is discussed. The Kennard-Stepanov relation is derived and a detailed description of the rubidium dimers in a high pressure argon environment is given. 2.1 Redistributional Laser Cooling The concept of redistributional laser cooling is explained in detail in the following section. As can be seen in figure 2.1, the cooling cycle takes place between the 5S and the 5P state of rubidium in a high pressure rubidium-argon gas mixture. In the figure the potential curves as a function of the distance between a rubidium atom and an argon atom can be seen. In order to achieve cooling, red detuned laser light is irradiated into the sample. Since the energy of the irradiated photons of this laser light is less than the energy gap in the unperturbed case, the Figure 2.1: Cooling principle shown using the potential curves for the Rb – Ar quasi molecules 5 Chapter 2 Theory photons can only be absorbed during the collision. Because of the high pressure of the argon bu er gas a high collision rate is secured and ensuring the absorption of the photons. Since the lifetime of the 5P state is 27 ns and the collision time is in the order of ps, the atoms seperate before the photon is emitted again. The emitted photon has Edif=~!more energy than the absorbed photon, thus energy is extracted. This energy comes from the kinetic energy of the atom. Macroscopically the reduced kinetic energy of the atoms leads to a lower temperature. The cooling power can then be calculated through Pcool=Plaser (!laser)¯!fluorescence!laser !laser(2.1) where a maximum cooling power of 100 mW has been approximated through the cooling vis- ible at the window for a measurement of a cell at 570 K and with an excitation frequency of 365 THz [12]. In order to measure the change in temperature, thermal deflection spectroscopy is used. This methods utilizes the fact that the cooling zone created through the laser induces a gradient in temperature, which leads to a gradient in density, which in turn leads to a gradient in refractive index. Thus from the deflection of a non-resonant laser beam, the temperature can be calculated. The angle , of the deflected beam, as shown in figure 3.4, is given through tan =dj~rj dz(2.2) =1 n0Zdn drdz (2.3) =1 n0Zdn dTdT drdz (2.4) Here we only consider small deflections, which is a good approximation as experiments have shown. Using the Lorentz-Lorentz equation[26] 1  n21 n2+2! =const (2.5) we can derive for ideal gases and a refractive index n1: dn dTn1 T(2.6) which gives us together with eq. 2.4 and the assumption that n01 =Zn1 TdT drdz (2.7) In order to further simplify this equation, we make the assumption thatn1 Tis constant. We note that this assumption is only valid in the case of small temperature changes. Then we can write =n1 TZdT drdz (2.8) 6 2.2 Line Shape Broadening Assuming furthermore a gaussian profile for our laser beam, giving us an absorption induced temperature change of
T(r;z)=
T(r;0) exp( z) (2.9) By reversing eq. 2.8 and additionally using eq. 2.9 and furthermore considering the absorption coecient to be temperature independent, which is a good approximation for small temper- ature changes, we get
T(r;z)=T n1
exp( z) 1exp( L)Z (r0)dr0(2.10) This equation now can be used in order to calculate the temperature change
Tin our sample, given the ambient temperature Tof the sample, the absorption coe cient of the sample and the deflection angle in dependence on the distance of the cooling beam. The refractive index can be estimated using the ideal gas equation pV=NkBT (2.11) and assuming a constant particle number, thus giving us for the refractive index in dependence of the standard conditions (n1)=(n01)pT0 p0T(2.12) where n0=1:000199 the refractive index for argon under standard conditions, p0=1 bar the standard pressure and T0=273 K the standard temperature. The equation 2.10 thus gives us a good approximation, but as mentioned above, is limited to cases of small temperature changes. Thermal deflection spectroscopy determination of large temperature changes requires simula- tions of the deflection with a temperature dependent absorption coe cient [27]. Alternatively, di erent approaches to temperature determination, such as the Kennard-Stepanov relation, can be utilized. 2.2 Line Shape Broadening 2.2.1 Pressure Broadening One of the earliest model for the pressure broadening of line shapes is based on Michelson [28]. The model considers all collisions as hard collisions, which interrupt the fluorescence, thus interupting the wavetrain and creating a phase jump. Therefore wavetrains of the length c are created, where is the time between two collisions and cis the speed of light. The fluorescence can then be described through [29]: f(t)=Re f0exp(i!0t) t 2  (2.13) 7 Chapter 2 Theory Figure 2.2: (a) Potential curves in the quasistatic approximation of an atom A as a function of the distance R between two atoms A and B (b) Lineshape broadening, visible is the unperturbed case with emission centered at the unperturbed frequency !0and the perturbed case centered at !(R). From [27] In order to obtain the power spectrum, the Fourier transform has to be taken and squared. This leads to an intensity I(!) given through I(!)/ sin( (!!0)=2) (!!0)=2!2 (2.14) If we now consider a probability distribution for the time between collisions given through P( )=exp c!d c(2.15) where P( ) is the probability of collision to occur in the time interval between andd and cis the mean time inbetween collisions, we can then average Eq. 2.14 weighted through Eq. 2.15 in order to obtain the integral I(!)/Z1 0 sin( (!!0)=2) (!!0)=2!2 exp c! d (2.16) which leads to I(!)/1 (!!0)2+ 1 c2(2.17) The quasistatic approximation for pressure broadening assumes a slow deviation of the in- teratomic distances with time. in order to understand pressure broadening in the quasi static 8 2.2 Line Shape Broadening approximation, let us consider an atom A which enters the vicinity of an atom or molecule B. In this case, the potential curves of the electronic states of atom A are shifted, according to the interaction between A and B. This shift can be be either positive or negative. If we now assume a transition between two electronic states, as shown in figure 2.2 a), the energy di erence is given through
E=hi f=jEf(R)Ei(R)j (2.18) where EfandEiare the energies of the electronic final and initial state, respectively, which are dependent on the distance Rbetween A and B. Assuming a statistical distribution of the radii with a mean value of ¯R, we get a statistical distribution of the frequencies with a mean value ¯ . This mean value ¯ is in general not equal to unperturbed frequency 0, thus leading to a shift of=0¯. 2.2.2 Saturation Broadening One additional e ect of line-broadening which has to be considered in our experiments as well is saturation broadening. This e ect takes place when the used laser power is especially high and gets into the saturation regime of our system. In order to describe it mathmatically, let us first consider the rate equations for a two-level system [30]: dN1 dt=dN2 dt=P(!)N1R1(!)N1+P(!)N2+R2(!)N2 (2.19) with N1being the occupation number of the ground state, N2the occupation number of the excited state (coupled through N1+N2=N=const, the total number of states), P(!) the pump rate and R1andR2the relaxation rates of the two states. This can be used to determine the occupation number in dependence of the pump rate and the relaxation rates: N1=N(P(!)+R2(!)) 2P(!)+R1(!)+R2(!)(2.20) If we now consider the case without pumping, e.g. P=0, we get the occupation numbers for the thermal equilibrium N1;0=N
R2(!) R1(!)+R2(!)and N2;0=N
R1(!) R1(!)+R2(!)(2.21) Using the di erence in the occupation numbers
N=N1N2and
N0=N1;0N2;0we get:
N=
N0 1+2P(!)=(R1(!)+R2(!))=
N0 1+S(!)(2.22) Through this equation we define the saturation Sas follows: S(!)=2P(!) R1(!)+R2(!)(2.23) 9 Chapter 2 Theory If we further assume a homogeneous beam profile, we can write the saturation as a Lorentz profile: S(!)=S0 22 (!!0)2+ 22with S0=S(!0) (2.24) In addition to this, we can see that the absorption in dependence of the frequency !can be described with the saturation as well: (!)= 0 1+S(!)(2.25) Through this we can then use eq. 2.24 in order to write: (!)= (!0) 22 (!!0)2+ 22(2.26) where
!S=
!0p 1+S0 (2.27) thus giving us a Lorentz shaped absorption coe cient. 2.2.3 Redistribution of Fluorescence In order to better understand the principle of fluorescence redistribution, let us consider an atomic transition between a ground state with energy Eiand an excited state with energy Ef, with an energy di erence of E=~!0. If we ignore line shape broadening e ects for the moment, the absorption and fluorescence spectra is a -function at !=!0. Thus only photons with frequency !0are absorbed and can be emitted. Because of line shape broadening, this is not observable in nature. Instead we have a distribution around !=!(R), as can be seen in figure 2.2 b). Thus it is possible that photons with energy E=~!awith!a,!0are absorbed. The emitted photons with energy E=~!bare then again centered around !(R) and in general !e,!a. The reason for this redistribution of the fluorescence is that, especially in high pressure systems, the atoms thermalize through collisions and the kinetic energy is absorbed in such a way that the thermal shape of the fluorescence is restored. In the case of !a