Abstract Investigation and application of an ecient population transfer technique to a multi-level atomic system is presented. Adiabatic passage in a

Transcription

1 Ecient Population Transfer in a Multi-level Atom by Grace D. Chern Advisor: Professor Dmitry Budker An undergraduate thesis submitted to the Department of Physics at the UNIVERSITY of CALIFORNIA at BERKELEY 2000

2 Abstract Investigation and application of an ecient population transfer technique to a multi-level atomic system is presented. Adiabatic passage in a two-level system is investigated both theoretically and experimentally. By changing the detuning of a light eld frequency from an atomic resonance and satisfying adiabatic conditions, one can achieve 100% population transfer in a two-level system. Furthermore, we investigate a generalization of our ecient population transfer technique to a two- and three-step process. This approach is applied to our ongoing atomic parity nonconservation search in dysprosium by sending an atomic beam across diverging laser beams. We achieve ecient transfer in all transverse atomic velocity groups to obtain an increase in statistical sensitivity.

3 Acknowledgements I would like to give my sincere thanks to my advisor Professor D. Budker and graduate student A.T. Nguyen for their patience, advice, and support. I would also like to thank fellow group members D. Kimball and D. Brown for their valuable assistance. i

4 Contents Acknowledgements i 1 Introduction 1 2 Theory of Adiabatic Passage Analogy to Spin in Magnetic Fields Application to Optical Systems Dressed Basis Model Bloch Vector Model Adiabatic Criteria Magnetic Sublevels Experimental Set-up Overview Population Scheme Atomic Beam Laser Beams Lenses Calculations Gaussian Laser Beams ii

5 4.2 Two-Level Steady-State Populations Application of the Adiabatic Criteria Density Matrix Calculations Experimental Data and Analysis First Transition: State G! e Second Transition: State e! f Conclusions 39 Appendix 40 Atomic Parity Nonconservation References 42 iii

6 List of Figures 2.1 Magnetic elds in a) lab frame and b) rotating frame B e for dierent values of! Atomic beam crossing adiverging laser beam Adiabatic eigenenergies and dressed state energies vs. laser detuning Bloch vector and unit vector ^n in u ; v ; w space Experimental setup (picture not to scale): a) atomic beam produced by an oven source at T=1500 K b) atomic beam collimators c) set of cylindrical lenses to diverge the laser beams d) spherical mirror to improve light collection eciency e) interference lter(s) f) auxiliary laser-atomic beam interaction region Energy diagram of population scheme. Two-laser excitation of the G! e and e! f transitions. Spontaneous emission for f! B transition. F values are given for 163 Dy Iodine absorption lines near the 669-nm transition. The 669-nm transition lies near the (X =6 J =15)! (B =4 J = 14) I 2 absorption line. and J are the vibrational and rotational quantum numbers, respectively Population density of states a, b, c, and d as a function of time. Typical parameter values were taken: 833-nm light power at 50 mw, 669-nm light power at 30 mw, H 1 = 0:3 cm, H 2 = 0:2 cm, d 1 = 0:1 ea 0, d 2 = 0:05 ea 0, = 0:02, b = 0:01 MHz, c = 0:4 MHz, v = cm/s, and 1div = 2div =0:1 rad. The atom intersects the axis of the 833-nm laser beam at time t = nm peak uorescence dependence upon 833-nm laser power. Circles indicate zero laser beam divergence (without lenses) and squares indicate a beam divergence of 0.06 rad iv

7 nm uorescence curves for w/o (circles) and w/ (squares) lenses in 833- nm laser beam. The 833-nm laser power is 50 mw. The divergence is 0.05 rad. The 833-nm laser frequency is locked on resonance. 669-nm laser beam is collimated and low-powered ( 5 mw). Fluorescence detected as 669-nm laser frequency is scanned. Solid curve represents transverse Doppler distribution of atoms Peak uorescence in the 743-nm channel dependence upon 833-nm laser power for the case w/o (circles) and w/ (squares) divergence in the 833-nm laser beam. For the case w/ lenses, the 833-nm laser divergence is 0.10 rad. The 669-nm laser beam is used as a probe, i.e. it is collimated and low-powered ( 5 mw) Peak uorescence in the 743-nm channel dependence upon 669-nm laser divergence. 833-nm laser power at 50 mw and 669-nm laser power at 30 mw. 833-nm laser beam divergences are 0.06 rad (circles) and 0.04 rad (squares). 37 v

8 Chapter 1 Introduction In our experiment, we attempt to increase statistical sensitivity in a measurement of atomic parity nonconservation (PNC) by improving the counting rate [1]. (See Appendix for discussion of PNC.) To prepare atoms in the excited state of a two-level system, we send an atomic beam across a cw laser beam so the axes of the two beams are orthogonal and tune the light eld frequency to that of the atomic transition. We approximate the transverse atomic velocities as following Maxwell's distribution [2]. If the cw laser beam is collimated, the light eld interacts with only a narrow band of atomic velocities in the distribution. Assuming the line width of the laser beam is narrow compared to the natural line width of the upper state, ; 0, the width of atoms excited is dominated by the latter line width. The fraction of the atoms excited is therefore ; 0 =2; D, where ; D is the Doppler width of the transverse atomic velocity distribution. The factor of 1/2 is due to the fact that applying a light eld directly on resonance with a desired atomic transition resultsinamaximum excitation of 50% of the atoms. The same light eld which excites the atoms from the lower to upper state will also stimulate transitions from the upper to lower 1

9 state at the same rate [3]. On average, the number of atoms in the lower state is equal to that in the upper state. To excite all transverse velocity groups, we diverge the laser beam in one-dimension using a set of cylindrical lenses. When the divergence of the laser beam is matched to that of the atomic beam, almost all atoms in the distribution will see resonant light at some point during their journey through the laser beam. This increases the fraction of atoms excited by a factor of ; D =; 0. Incidentally, sending an atom across a diverging laser beam provides the necessary conditions for adiabatic passage. As the atom traverses the laser beam, it experiences a change in light frequency due to the Doppler shift. If this change is slow, or adiabatic, and the upper state lifetime is long enough (according to the conditions discussed in Section 2.3), the atoms will be excited to the upper state with a maximum probability of 100%. This doubles the fraction of atoms excited, giving an overall population enhancement factor of 2; D =; 0. The result of our population technique is an improvement of the statistical sensitivity in a PNC experiment by 2-3 orders of magnitude over the previous PNC result [1]. 2

10 Chapter 2 Theory of Adiabatic Passage 2.1 Analogy to Spin in Magnetic Fields To understand adiabatic passage in an optical system, it is useful to describe the analogy to a spin-magnetic eld system [4, 5]. Adiabatic passage is often applied to the latter system to ip the direction of the spin. Let us consider the spin-1/2 system in a static magnetic eld, B 0 = B 0^e z, where B 0 =! 0 =,! 0 is the energy separation of the spin-up and spin-down states created by the static magnetic eld, and is the gyromagnetic ratio. (Throughout this paper, we will use units of h =1.) The spin S, when placed in the magnetic eld, will feel a torque due to the eld and precess around it. The precession of the spin is described by ds dt = S B 0: (2.1) We then apply an auxiliary rotating magnetic eld, B 1 = B 1 cos!t^e x ; B 1 sin!t^e y, with a magnitude jb 1 jjb 0 j and frequency!, in the plane perpendicular to the static magnetic eld B 0. We again examine the time evolution of the spin vector: ds dt = S (B 0 + B 1 ): (2.2) 3

11 Figure 2.1: Magnetic elds in a) lab frame and b) rotating frame. To see how the spin vector changes direction, we transform into the rotating frame in which B 1 is static [Fig. 2.1]. The basis vectors in the rotating frame are now ^e 0 x = ^e x cos!t ; ^e y sin!t (2.3) ^e 0 y = ^e x sin!t + ^e y cos!t (2.4) ^e 0 z = ^e z : (2.5) The time evolution of the spin vector in the rotated frame becomes ds 0 dt = ds dt ;! S = S [(B 0 + B 1 )+!] (2.6) where! = ;!^e 0 z is the angular velocity vector of the rotating frame. The eective magnetic eld producing a torque on the spin vector is B e = B 0 + B 1 +! = 1 (! 0 ;!)^e 0 z + B 1^e 0 x : (2.7) If!! 0, then according to Eq. (2.7), B e points mainly in the +^e z direction. Now if instead,!! 0,thenB e will point inthe;^e z direction [Fig. 2.2]. Therefore, by sweeping! from far below to far above! 0, one can ip the direction of the eective magnetic eld. 4

12 Figure 2.2: B e for dierent values of!. The precession frequency of S around B e is = q (B 1 ) 2 +(! 0 ;!) 2 : (2.8) If S precesses about B e much faster than! is swept, S will adiabatically follow B e. That is, if the adiabatic condition jj d!=dt jj (2.9) is satised, the spin vector will follow the eective magnetic eld and thereby ip its direction. 2.2 Application to Optical Systems Adiabatic passage applied to an optical system allows an atom initially in the lower state to be excited to the upper state with 100% eciency. Consider the two-level atomic system of the ground and excited state, with transition frequency! 0 and transition dipole matrix element d. We then apply a light eld, which is analogous to the rotating magnetic eld, with frequency! L and electric eld amplitude E. After applying the rotating-wave approx- 5

13 Figure 2.3: Atomic beam crossing a diverging laser beam. imation, we see that an atom initially in the ground state will oscillate between the ground q and excited state, at the frequency = (de) 2 + 2, where de is called the Rabi frequency on resonance and =! L ;! 0 is the detuning of the light eld frequency. Now, let us sweep! L from! L! 0 to! L! 0 (or vice versa). Analogous to sweeping! across! 0 in the previous system to ip the spin direction, sweeping the detuning across zero can completely transfer an atom initially in the ground state to the excited state. In an optical system, sweeping can be done either by changing the atomic transition frequency (for example, by Stark shifting) and leaving the light eld tuned to! 0, or by leaving the atomic transition alone and sweeping the light eld frequency through! 0. In our experiment, we apply the latter method by sending an atomic beam perpendicularly across a diverging laser beam, with the light eld frequency tuned to! 0 [Fig. 2.3]. As an atom traverses the laser, it experiences a change in laser frequency due to the Doppler shift given by! 0 =! 0 (1 ; v c ^n) (2.10) where v is the atomic velocity vector, c is the speed of light in vacuum, and ^n is the unit 6

14 vector in the direction of light propagation (which diers for dierent parts of the diverging laser beam). Before the atom crosses resonant light at point B where! 0 =! 0 (see Figure 3), the atom encounters blue-shifted light. For example, at point A, the atom \sees" light of frequency! 0 =! 0 (1 + v=2c), assuming the laser full divergence angle is small. After it goes through resonant light, the atom leaves the laser experiencing red-shifted light. At point C,it \sees" light offrequency! 0 =! 0 (1 ; v=2c). Under certain criteria (discussed more thoroughly in Section 2.3), an atom in the ground state can be adiabatically transferred to the excited state. One condition, analogous to Eq. (2.9) for a spin-magnetic eld system, can be written approximately as jj d=dt : (2.11) jj The above expression states that the light frequency an atom experiences must change slowly compared to the time scale of Rabi oscillations at frequency Dressed Basis Model Adiabatic passage can also be described using the dressed basis model. The Hamiltonian which usually describes a light eld interacting with a two-level atomic system, the ground state jgi and the excited state jei, is 0 H = de de (2.12)! 0 where d is the transition dipole matrix element, E is the electric eld of the light, and! 0 is the energy separation between the two atomic states. Now let us instead, introduce another set of basis states (the unperturbed or uncoupled 7

15 states) which includes not only the atomic but also photon states. The unperturbed states are labeled by the atomic level (in this case, g or e) and the number of available photons in the light eld. In the case of a two-level atom and one available photon light eld, the uncoupled ground state is jg 1i and the uncoupled excited state, which is formed when the atom absorbs the photon, is je 0i. (The following presentation can similarly be applied to multi-photon light elds [6].) The Hamiltonian in this new basis is then 0 de H = de ; (2.13) where is the detuning dened as =! L ;! 0,and! L is the energy of the light eld. The eigenstates of this Hamiltonian, or adiabatic eigenstates, j1i and j2i are 1 j1i = p 1 j2i = p (2dE) 2 +(;+ 0 ) 2 (2dE) 2 +(;; 0 ) 2 2dE ;+ 0 (2.14) 2dE ; ; 0 (2.15) where 0 = q 2 +(2dE) 2. The eigenstates are written in vector form where the top component is the probability amplitude for the atom to be in jg 1i and the bottom component is the probability amplitude for the atom to be in je 0i. The corresponding eigenenergies 1 and 2 are q 1 = ;+ 2 +(2dE) 2 (2.16) 2 q 2 = ; ; 2 +(2dE) 2 : (2.17) 2 Note here that in the E! 0 limit, the adiabatic eigenstates and eigenenergies become the uncoupled states and energies j1i! j2i! 1 0 1! 0 (2.18) 0 ;1 2!;: (2.19) 8

16 Figure 2.4: Adiabatic eigenenergies and dressed state energies vs. laser detuning. As shown in Fig. 2.4, the adiabatic eigenstates correspond to the uncoupled states when the light eld is far detuned from the atomic resonance. (For example, when! L! 0, j1i corresponds to jg 1i and j2i corresponds to je 0i.) Now let us consider an atom initially prepared in jg 1i. As detuning is changed slowly from! L! 0 to! L! 0, so that the condition (2.11) is fullled, the atom follows adiabatically along its eigenstate j1i. In the uncoupled basis, this corresponds to an atom initially in the ground state being transferred to the excited state Bloch Vector Model One can also model adiabatic passage by applying Bloch's spin vector formalism to our optical two-level system [7]. Again, states a and b represent the atomic ground and excited state, respectively,! 0 is the transition frequency, d is the transition dipole matrix element, 9

17 ! L is the applied light eld frequency, and E is the electric eld amplitude of the light. In general, atoms can be described using the density matrix. For a two-level system, it is represented by aa = ab (2.20) ba bb where the quantities aa and bb are the population of the states a and b, and ab and ba are the coherences between them. The evolution of the atomic states are governed by the following Liouville equation, i d dt =[H ] ; i f; g (2.21) 2 where H is the Hamiltonian Eq. (2.12) and ; is the relaxation matrix. The square brackets denote the commutator of H and, dened as (H; H), and the curly brackets represent the anti-commutator of ; and, dened as (; + ;). Neglecting relaxation, Eq. (2.21) reduces to i d dt =[H ]: (2.22) Using Eqs. (2.12, 2.22), we solve for the time evolution of the components of and obtain the following density matrix equations for a two-level atom d dt ^ aa = ; i 2 de(^ ab ; ^ ba ) (2.23) d dt ^ bb = i 2 de(^ ab ; ^ ba ) (2.24) d dt ^ ab = ; i 2 de(^ aa ; ^ bb ) ; i^ ab (2.25) d dt ^ ba = i 2 de(^ aa ; ^ bb )+i^ ba (2.26) 10

18 where =! L ;! 0 is the detuning of the light eld, and the following variables have been introduced to suppress explicit time-dependence of the coecients in Eqs. ( ), ^ ab = ab e ;i! Lt (2.27) ^ ba = ba e i! Lt (2.28) ^ aa = aa ^ bb = bb : (2.29) Using the density matrix formalism, one can make an analogy to a spin vector in a magnetic eld. First, we create a \pseudospin" or Bloch vector =(u v w), where is written such that u, v, and w are the components of in the ^e x, ^e y,and^e z directions, respectively. The variables u, v, and w are related to the elements of the density matrix by u = 1 2 (^ ab +^ ba ) (2.30) v = 1 2i (^ ab ; ^ ba ) (2.31) w = 1 2 (^ bb ; ^ aa ): (2.32) To determine the time evolution of the Bloch vector, and thereby the time evolution of the density matrix elements, we introduce the vector (;de 0 ;): (2.33) Analogous to a magnetic eld's eect on a spin vector, produces atorqueon such that precesses about. The equation of motion of the Bloch vector, similar to Eq. (2.1), is given by d dt = (2.34) and the precession frequency is = q (de) : (2.35) 11

19 Solving for the time evolution of the Bloch vector components, we obtain what are known as the optical Bloch equations: _u =v (2.36) _v = ;u + dew (2.37) _w = ;dev: (2.38) Eqs. ( ) and the normalization condition ^ aa +^ bb = 1, are equivalent to the density matrix equations, Eqs. ( ). The advantage of using the Bloch vector model is that it easily demonstrates how adiabatic passage eciently transfers an atom from the ground state to the excited state. If an atom is prepared in the ground state, no coherences exist between the states. The Bloch vector is therefore initially = (0 0 ;1=2). Now, suppose that! L is changed from! L! 0 to! L! 0. Then, according to Eq. (2.33), goes from pointing mainly in the ;^e z to +^e z direction. If is swept slowly compared to the rate at which precesses around it, as expressed mathematically in Eq. (2.11), will adiabatically follow the vector. At the end of the sweep, = (0 0 +1=2), which indicates that the atom has been completely transferred to the excited state. One clearly sees how the analogy to adiabatic passage in nuclear magnetic resonance can be done using the Bloch model. 2.3 Adiabatic Criteria As described in Section 2.2, adiabatic population inversion can occur by sweeping the light frequency through atomic resonance, provided certain requirements are met. One condition, 12

20 Figure 2.5: Bloch vector and unit vector ^n in u ; v ; w space. which can be derived using the Bloch vector model described in Section 2.2.2, is that the vector =(;de 0 ;), given in Eq. (2.33), must satisfy the following criterion, d=dt jj ^n jj (2.39) where ^n =(; 0 de)=jj is the unit vector perpendicular to (both ^n and are in the u ; w plane [Fig. 2.5]). The rst vector on the left side of Eq. (2.39) is dotted into ^n to give the change of in the direction perpendicular to. Changes to in the direction parallel to are negligible here since they aect only the magnitude of, and not the direction. Substituting in for ^n and, _ one can obtain the following expression from Eq. (2.39), = d _E ; de _ where a dot represents a derivative with respect to time. jj 3 1 (2.40) Another condition for adiabatic inversion is that the lifetime of the upper state must be suciently longer than the time T it takes for inversion to occur. Adiabatic passage is often referred to as adiabatic fast passage due to this requirement. Analogous to the nuclear 13

21 magnetic resonance technique [8], the upper state lifetime restriction can be expressed as, T ' (0) _ : (2.41) Magnetic Sublevels For transitions involving states with multiple magnetic sublevels, it is convenient to consider a basis jjmi in which the direction of light polarization is taken as the quantization axis. In this basis, light couples only sublevels with the same m. As a result, one has a set of two-level systems, in each of which adiabatic passage occurs independently. For each of the two-level systems there may be a dierent adiabatic criterion, which depends on the dipole matrix element connecting the two sublevels. This dipole matrix element is given by the Wigner-Eckart Theorem [9], hj 0 mjdjjmi = hjm10jj 0 mi p 2J 0 +1 hj 0 kdkji (2.42) where hjm10jj 0 mi denotes the appropriate Clebsch-Gordan coecient and hj 0 kdkji is the reduced dipole matrix element. 14

22 Chapter 3 Experimental Set-up 3.1 Overview An atomic beam is sent through two sets of collimators, then across two counter-propagating diverging laser beams [Fig. 3.1]. One of the lasers is at 833-nm and the other is at 669- nm wavelength. Each laser beam is diverged with a separate set of cylindrical lenses. At the laser-atomic beam interaction region, uorescence is detected by a photomultiplier tube (PMT) after passing through interference lter(s). To improve detection eciency, a spherical mirror is placed opposing the PMT. The 833-nm laser beam is chopped at 500 Hz and the uorescence signal from the PMT is detected using alock-in technique. 3.2 Population Scheme In our PNC experiment, we would like to eciently populate the odd parity state B. We do so using the three-step population scheme shown in Fig The rst transition from the ground state G! e at 833-nm wavelength is excited using a narrow-band cw laser (laser 1). 15

23 Figure 3.1: Experimental setup (picture not to scale): a) atomic beam produced by an oven source at T=1500 K b) atomic beam collimators c) set of cylindrical lenses to diverge the laser beams d) spherical mirror to improve light collection eciency e) interference lter(s) f) auxiliary laser-atomic beam interaction region. Then another narrow-band cw laser (laser 2) is applied to excite the second transition from state e! f at 669-nm wavelength. Finally, we rely on spontaneous emission at 1397-nm for the third transition from state f to our desired odd parity state B. The branching ratio for this transition was previously measured as.30(9) [10]. The lifetimes of states e, f and B are, respectively, e = 16:5(2:6) sec, f = 432(5) nsec, and B > 200 sec [10]. We use the isotope 164 Dy for its large natural abundance (28.2%) [11] and well-resolved 833-nm transition spectrum [10]. 16

24 Figure 3.2: Energy diagram of population scheme. Two-laser excitation of the G! e and e! f transitions. Spontaneous emission for f! B transition. F values are given for 163 Dy. 3.3 Atomic Beam The atomic beam is produced by an eusive oven with a multi-slit nozzle array at an operating temperature of T 1500 K. (See Ref. [1] for a more detailed description of the atomic beam source.) At the laser-atomic beam interaction region, the density of atoms is atoms/cm 3. In order to maintain this high overall density over a large volume, the oven multislit nozzle array and the external collimators were designed to provide only weak collimation of the atomic beam. The characteristic longitudinal atomic speed corresponding to the oven temperature T, given by v 0 = q 2kT=m (3.1) is cm/s, where m is the mass of the atom and k is the Boltzmann constant. For the 833-nm transition, the transverse Doppler width (FWHM) ; D, was measured to be 65 17

25 MHz. The atomic beam full divergence angle is related to ; D by sin = ; D =v 0 (3.2) where is the wavelength of the laser. The corresponding atomic beam divergence angle is therefore 0:14 rad. 3.4 Laser Beams To excite the 833-nm transition, we use a single-frequency diode laser (Sanyo DL8032) with an external cavity in the Littrow conguration. The laser beam has a 0.2 cm 0.3 cm elliptical cross-section and an output power of about 80 mw. A small fraction of the laser beam is sent to a wavemeter and a Fabry-Perot interferometer to monitor the frequency spectrum. The power reaching the main interaction region (having accounted for reections o of mirror and lens surfaces) is 65 mw. We split o another portion of the light and send it through an auxiliary laser-atomic beam interaction region located 20 cm downstream from the main interaction region [Fig. 3.1]. In this area, we detect 833-nm uorescence with a PMT and interference lter. We use this signal in conjunction with a feedback loop system to lock thediode laser on resonance. To excite the second transition, we use a Coherent 599 dye laser (with DCM dye) pumped by a 5 W Ar+ laser. Typical stable output power is about 100 mw. The available power at the main laser-atomic beam interaction region is 35 mw. The laser beam has a circular cross-section with a diameter of 0.2 cm. Although the laser is locked to a reference cavity, it suers from frequency instabilities, which eectively broadens the laser line width from the 18

26 specied < 1MHzto 8MHz. A fraction of this laser beam is sent through an iodine vapor cell. The cylindrical cell, with a 5 cm diameter and 8 cm length, is heated to 120 o C. By referencing I 2 absorption lines from the electronic ground state (X 1 + g ) to the excited state (B 3 + ou) transition, we can easily locate the 669-nm transition. We use the ( = 6 J = 15)! ( =4 J = 14) absorption line, which lies near the 669-nm transition, as our reference signal [Fig. 3.3]. Here, and J represent the vibrational and rotational quantum numbers, respectively. The values of and J corresponding to each I 2 absorption line near the 669-nm transition are determined by using data from Ref. [12] and calculating transition energies between vibrational and rotational sublevels of the ground and excited states. 3.5 Lenses For each of the two laser beams, we use a pair of cylindrical lenses with focal lengths 1.27 cm and 10.0 cm. For independent control of divergences and eective focal positions of each lens system, we slide the lenses along rigid tracks. Prior to the experiment, we determined the divergence angle for each lens system as a function of lens position using a beam proler placed at the main laser-atomic beam interaction region. 19

27 Figure 3.3: Iodine absorption lines near the 669-nm transition. The 669-nm transition lies near the (X = 6 J = 15)! (B =4 J = 14) I 2 absorption line. and J are the vibrational and rotational quantum numbers, respectively. 20

28 Chapter 4 Calculations 4.1 Gaussian Laser Beams We approximate the intensity prole of the laser beams as a Gaussian. In the case of a laser beam diverged with a spherical lens and propagating in the ^e x direction, the electric eld is given by [13] E(x y z t) =E 0 w 0 w(x) e;(y2 +z2 )=w2 (x) cos(!t ; ) (4.1) where is dened as =kx + k (y2 + z 2 ) 2R(x) ; arctan(x=x 0 ): (4.2) The eld amplitude E 0 at the waist depends on the total time-averaged laser power k =2= is the wavenumber, where is the wavelength! is the light frequency R(x) is the radius of curvature of the wavefront x 0 is the Rayleigh range and w(x) is the spot size. The parameter w(x)isknown as the spot size because if the laser beam is projected onto a screen, one would see a circular spot of radius w(x). The spot size varies with the distance of propagation and is dened as s w(x) =w 0 1+ x2 (4.3) x

29 where w 0 = = tan(=2) is the spot size at the beam waist, and is the far-eld full divergence angle. The Rayleigh range, which characterizes a Gaussian beam, is given by x 0 = w2 0 : (4.4) At this distance of propagation, as measured from the beam waist, the radius of curvature of the wavefront R(x), dened as R(x) =x + x2 0 x (4.5) is minimal. At x x 0, R approaches innity like that of a plane wave. As the wavefront propagates away from the waist, it begins to bend and reaches a minimum curvature at x 0. It then increases to nite values and at x x 0, R(x) x, resembling a spherical wave centered at the beam waist. In Eq. (4.2), the rst term is the plane-wave phase shift and the second term gives the phase shift variation at positions o the optical axis. The last term is known as the Gouy phase shift [14] and takes into account the acquired phase shift when a wave comes from ;1, passes through the beam waist (at x = 0), and expands to 1. If instead, the laser beam is diverged with a cylindrical lens only in the ^e y direction, the electric eld becomes E(x y z t) =E 0 s w0 w(x) e;y2 =w2 (x) cos(!t ; ) (4.6) with =kx + k y2 2R ; 1 2 arctan(x=x 0): (4.7) 22

30 To determine E 0 as a function of the total time-averaged laser power P, we integrate the time-averaged intensity, che 2 i=4, over the ^e y -^e z plane. Using Eq. (4.6) for the electric eld, the laser power is then P = c Z he 2 idzdy = c 4 8 E2 0 H w Z +1 0 w(x) ;1 e;2y2 =w2 (x) dy = chw 0 8 p 2 E2 0 (4.8) where H is the height ofthebeam in the ^e z direction. Solving for E 0, we obtain E 0 = vu u t8 p 2P chw 0 : (4.9) 4.2 Two-Level Steady-State Populations Consider a two-level atom of the ground state j1i and excited state j2i, with spontaneous decay back to state j1i at a rate ; 0. A collimated atomic beam, with atomic velocity v, intersects a collimated laser beam of width l, such that the axes of the beams are perpendicular. Let R(!) denote the stimulated absorption and emission rate as a function of light frequency!. To determine the rate equations for an atomic beam system, one must consider not only the rates due to ; 0 and R(!), but also the rate ; t = v=l at which atoms enter and exit the laser-atomic beam interaction region. ; t, known as the transit width, is dened as the inverse of the atomic transit time across the laser beam. We can now express the rate equations for the fraction of atoms n i in state jii as dn 1 dt = R(!)(n 2 ; n 1 )+; 0 n 2 +; t (1 ; n 1 ) (4.10) dn 2 dt = R(!)(n 1 ; n 2 ) ; ; 0 n 2 ; ; t n 2 : (4.11) In Eq. (4.10), the pumping term ; t is included to ensure that n 1 = 1 and n 2 = 0 in the absence of a light eld (R(!) = 0). 23

31 We would like to examine the steady-state case where dn 1 =dt = dn 2 =dt = 0. Solving Eqs. (4.10) and (4.11) for n 1 and n 2, one obtains n 1 = 1 2+ (4.12) 2 1+ n 2 = 1 (4.13) 2 1+ where is the frequency-dependent saturation parameter dened as = P P 0 = 2R(!) ; 0 +; t : (4.14) Here, P denotes the time-averaged light power and P 0 is the saturation power. In the limit that R(!)! 1 (or P P 0 ), one can see from Eqs. ( ) that both n 1 and n 2 approach 1/2. The stimulated emission or absorption rate for monochromatic light R(!) can be expressed by [15] R(!) = (de)2 4 ; 0 (4.15) 2 +; 2 0 =4 where E is the electric eld amplitude of the light, d is the transition dipole matrix element, and is the frequency detuning. However, one must also take into account the nite laser line width ; L. Assume that the laser intensity distribution, centered about frequency! 0,is I(! ;! 0 ; L =2 )=I 0 (! ;! 0 (4.16) ) 2 +; 2 L =4 where I 0 = R 1 ;1 I(!)d! is the total time-averaged light intensity. The resultant stimulated emission or absorption rate is then a convolution of Eq. (4.16) with Eq. (4.15), both of which are Lorentzian functions. 24

32 In general, the convolution of two functions f(x) and g(x) is dened by the integral [16] f g = 1 p 2 Z 1 ;1 and, in terms of the Fourier transform of the functions, f(x ; )g()d (4.17) f g = F ;1 [F [f ]F [g ] ]: (4.18) Here, F [h ] denotes the Fourier transform of the function h(x) dened as F [h ] = 1 p 2 Z 1 and F ;1 [F [h ] x] =h(x) is the inverse Fourier transform dened as h(x) = 1 p 2 Z 1 ;1 ;1 h(x)e ix dx (4.19) F [h ]e ;ix d: (4.20) In the case of a Lorentzian function, given in the form h(x) =1=(a 2 + x 2 ) with width a, the Fourier transform is F [h ] = q =2e ;ajj =a. From this it is straightforward to show that taking the convolution of two Lorentzians produces another Lorentzian, but with a width equal to the sum of the widths of the initial two Lorentzians. The convolution of Eqs. (4.16, 4.15), both being Lorentzians, then gives the following stimulated emission or absorption rate with width ; 0 +; L : R(!) = (de)2 4 Using this expression for R(!), Eq. (4.14) becomes = (de)2 2 ;0 +; L ; 0 +; t (; 0 +; L ) 2 +(; 0 +; L ) 2 =4 : (4.21) 1 2 +(; 0 +; L ) 2 =4 : (4.22) One must also consider the fact that due to the Doppler eect, an atom experiences a detuning which depends on its transverse velocity component. To account for this, for example, 25

33 when determining the fraction of atoms in the upper state, one takes the convolution of n 2 [Eq. (4.13)] with the transverse atomic velocity distribution. Assume this distribution to be a normalized Gaussian D(v T ): D(v T )dv T = 1 p e ;(v T =v 0T ) 2 dvt (4.23) v 0T where v T is the transverse atomic velocity and v 0T is the characteristic transverse atomic velocity. Written instead as a function of frequency, D(! ;! 0 )d! = 2 ; D s ln 2 e;4ln2(!;!0 =; D )2 d! (4.24) where ; D, as dened in Eq. (3.2), is the Doppler width measured at FWHM. The convolution of D(! ;! 0 ) with n 2, using Eq. (4.17), then gives the overall fraction of excited atoms as s ~n 2 = (de)2 ln 2 ;0 +; Z L 2; D ; 0 +; t where ; 0 is the power broadened line width dened as e ;4ln2(!;!0 =; D ) 2 (! 0 ;! 0 ) 2 +; 02 =4 d!0 (4.25) ; 0 =(; 0 +; L ) p 1+: (4.26) Since Eq. (4.25) can not be integrated analytically, let us examine the case when ; 0 ; D. We can now approximate D(! 0 ) as being constant near! 0 =! 0, and therefore take the exponential out of the integral. The result is the following simplied expression for the fraction of atoms in the excited state,! ; 0 p ~n 2 = n 2 ln 2 e ;4ln2(=; 2 D) : (4.27) ; D To see how ~n 2 relates to the saturation parameter in this limit, we substitute ; 0 =(; 0 + ; L ) p 1+ [Eq. (4.26)] and n 2 = 1 =(1 + ) [Eq. (4.13)] into the above equation to obtain 2 ~n 2 () / p 1+ : (4.28) 26

34 In contrast to the Doppler-free case [Eq. (4.13)], ~n 2 () never reaches a constant value in the limit that 1 (as long as ; 0 ; D ), but instead, continues to increase as p due to power broadening. 4.3 Application of the Adiabatic Criteria In this section, we calculate the necessary parameters for adiabatic passage specic to our experiment. Let an atom, at time t = 0, perpendicularly intersect the center of a diverging laser beam at a distance X (cm) from the focal spot. Using Eqs. (4.6) and (4.7) in the limit that X x 0, and making the substitutions w(x) X div =2 and y = vt, the electric eld amplitude (V/cm) and detuning (MHz) the atom experiences is given respectively as E(t) = :85 s P e ;2 ln2 (vt=x div ) 2 (4.29) HX div (t) = ; d dt = ; v 2 t X (4.30) where div (rad) is the divergence angle (FWHM) and v (cm/s) is the atomic velocity. For this geometry, one can verify that (see Eq. (2.40)) is maximal at the center of the beam (t =0): max = _ = (de) 2 v 2 H div P(:85 d) : (4.31) 2 Notice that, as a consequence of using a cylindrical lens, this expression does not depend upon X. The power needed to satisfy the adiabatic criterion can be estimated by setting max = 1. For example, taking typical parameters for d = 0:1 ea 0, v = cm/s, H = 0:3 cm, and div = 0:1 rad, one nds that about 50 mw of 833-nm light is required. These estimates are conrmed by experimental observations and density matrix calculations (discussed in Section 4.4). 27

35 As for the upper state lifetime restriction in this geometry, Eq. (2.41) can be expressed as, T ' (0) _ s = 0:85d PX : (4.32) v 2 H div Unlike Eq. (4.31), this constraint does depend on X and sets an upper limit on the laser power P. If the laser power is too high, then the atom spends too much time in the upper state, which gives it a higher probability to decay back to the ground state. Using the same values for the parameters above and letting P =50mWandX =1cm, T 0:2sec. 4.4 Density Matrix Calculations As in Section 2.2.2, we use the density matrix formalism to describe our atomic system. Let states a, b and c represent state G, e, and f, respectively the rst transition is a! b and the second transition is b! c. For a three-level system, the density matrix is represented by = 0 aa ab ac ba bb bc ca cb cc 1 C A (4.33) where the diagonal elements are the populations of the states labeled by the subscripts, and the o-diagonal elements are the coherences between the states. We again use the Liouville equation, given by Eq. (2.21), to describe the time evolution of the density matrix, i d dt =[H ] ; i f; g: (4.34) 2 The Hamiltonian H for the system is H = 0 0 d 1 E 1 0 d 1 E 1! 1 d 2 E 2 0 d 2 E 2! 1 +! 2 1 C A (4.35) 28

36 where! i is the frequency and d i is the averaged dipole matrix element for the i th transition and E j is the electric eld of the j th laser. The electric elds (see Eq. (4.6) and (4.9)), written in complex notation, are E 1 = E 2 = vu u t 8p 2P 1 ch 1 w 1 (x) e;y2 1 =w2(x) 1 e ;i(! L1t; 1(t)) = 1 e ;i(! L1t; 1(t)) (4.36) d 1 vu u t 8p 2P 2 ch 2 w 2 (x) e;y2 2 =w2 2 (x) e ;i(! L2t; 2(t)) = 2 d 2 e ;i(! L2t; 2(t)) (4.37) where the subscript j is used to denote that the variable corresponds to the j th laser. (In the frame of the atom, the parameters x, y and are also functions of time.) The relaxation matrix ; is dened by ;= b c 1 C A (4.38) where b =1= b and c =1= c are, respectively, the natural line widths of state b and c, and b and c are the lifetimes. Similar to Section 2.2.2, we make a change of variables to suppress time dependence in the coecients of the density matrix equations, ^ ab = ab e i(! L1t; 1(t)) ^ ba = ba e ;i(! L1t; 1(t)) (4.39) ^ bc = bc e i(! L2t; 2(t)) ^ cb = cb e ;i(! L2t; 2(t)) (4.40) ^ ac = ac ^ ca = ca ^ aa = aa ^ bb = bb ^ cc = cc : (4.41) By solving Eq. (4.34) and applying the rotating wave approximation [17], one obtains the following density matrix equations for a three-level atom: _^ aa = i 2 1(^ ab ; ^ ba )+ b^ bb (4.42) _^ bb = ; i 2 1(^ ab ; ^ ba )+ i 2 2(^ bc ; ^ cb ) ; b^ bb + c ^ cc (4.43) 29

37 _^ cc = ; i 2 2(^ bc ; ^ cb ) ; c^ cc (4.44) _^ ab = i 2 1(^ aa ; ^ bb ) ; i( 1 ; _ 1 )^ ab ; b 2 ^ ab (4.45) _^ ba = ; i 2 1(^ aa ; ^ bb )+i( 1 ; _ 1 )^ ba ; b 2 ^ ba (4.46) _^ bc = i 2 2(^ bb ; ^ cc ) ; i( 2 ; _ 2 )^ bc ; 1 2 ( b + c )^ bc (4.47) _^ cb = ; i 2 2(^ bb ; ^ cc )+i( 2 ; _ 2 )^ cb ; 1 2 ( b + c )^ cb (4.48) _^ ac = i(! 1 +! 2 )^ ac ; c 2 ^ ac (4.49) _^ ca = ;i(! 1 +! 2 )^ ca ; c 2 ^ ca (4.50) _^ dd = c (1 ; )^ cc (4.51) where j =! Lj ;! i is the detuning from the atomic resonance at! i of laser j. The last term in Eqs. (4.42) and (4.43) is added explicitly to account for, respectively, decay into state a from b and decay into state b from c (the former has a unity branching ratio the latter has branching ratio ). Eq. (4.51) is also introduced to allow the atoms in state c, which do not decay into state b, to decay into an auxiliary state d at the rate c (1 ; ). Consider an atom at the center of the transverse atomic velocity distribution crossing the two diverging cw laser beams, which are both on resonance. The laser beams are oset by 0.1 cm in the direction of the atomic beam axis. The atom intersects the 833-nm and 669-nm laser beams at 1 cm from the laser beam waists. Taking the typical parameters for 833-nm light power at 50 mw and 669-nm light power at 30 mw, the calculated population of states a, b, c, and d as the atom traverses the two lasers is shown in Fig One can see from this Figure that at the approximated minimum 833-nm light power required of 50 mw, the atom transfers to state b with almost 100% probability. Due to decay from state b! a and 30

38 Figure 4.1: Population density of states a, b, c, and d as a function of time. Typical parameter values were taken: 833-nm light power at 50 mw, 669-nm light power at 30 mw, H 1 =0:3 cm, H 2 = 0:2 cm, d 1 = 0:1 ea 0, d 2 = 0:05 ea 0, = 0:02, b = 0:01 MHz, c = 0:4 MHz, v = cm/s, and 1div = 2div = 0:1 rad. The atom intersects the axis of the 833-nm laser beam at time t =0. from state c! b, and insucient 669-nm light power of 30 mw (minimum 669-nm light power is 200 mw [Eq. (4.31)]), the population density of state d does not quite reach 1. It should be noted here that this calculation does not model our physical system exactly. It does not take into account the longitudinal or transverse atomic velocity distribution nor does it include the nite line width of the lasers. We use the density matrix calculations mainly for qualitative, not quantitative, comparisons with our experimental results. 31

39 Chapter 5 Experimental Data and Analysis 5.1 First Transition: State G! e The reduced dipole matrix element hekdkgi = :55 ea 0 is determined from the upper state lifetime (16:5(2:6) s) and the fact that state e decays mostly back to state G [18, 19]. For both the collimated and the diverged 833-nm laser beam case, uorescence at 833 nm is detected as the 833-nm laser frequency scans across resonance. The peak of the signal represents the portion of the transverse atomic velocity distribution excited to state e. Fig. 5.1 shows peak uorescence as a function of light power for a collimated and diverged laser beam. By tting the signal peak data for a collimated laser beam to Eq. (4.28), an eective saturation power of P 0 0:3 mw is obtained. We also analyze the corresponding widths of the uorescence curves to determine the laser line width. By comparing this to the width of the distribution given in Eq. (4.25) as a function of laser line width, we extract a laser line width of 2.0(5) MHz [20]. At 65 mw of 833-nm laser power, the fraction of atoms in 32

40 Figure 5.1: 833-nm peak uorescence dependence upon 833-nm laser power. Circles indicate zero laser beam divergence (without lenses) and squares indicate a beam divergence of 0.06 rad. state e (without lenses), inferred from our data and using Eq. (4.25), is ~n 2 = 15(5)%. Using cylindrical lenses, the laser beam divergence is increased to 0.06 rad. Due to limited 833-nm laser power, this angle, and not that of the atomic beam ( 0:14 rad), corresponds to the maximum signal peak. The light power reaching the interaction region decreases by about 15 mw to 50 mw due to reections o of the surfaces of the cylindrical lenses. At this power, increasing the laser beam divergence beyond 0.06 rad reduces the power of the light on resonance an atom experiences such that not all adiabatic criteria are met. The atomic beam crosses the laser beam at 1 cm from the focal spot. Compared to the collimated laser beam case, the signal peak size increases by a factor of about 5 when using a diverging 33

41 laser beam. This implies that the fraction of atoms transferred into state e is >50%. We attempted to investigate more accurately the population eciency in the rst step by using the 669-nm transition to probe the population of atoms in state e. Collimated, lowpower ( 5 mw) 669-nm light interacts with atoms 0.1 cm downstream from where the atomic beam intersects the 833-nm laser beam. The 833-nm laser power is kept at 50 mw and the light frequency is locked on resonance. Fluorescence from state f in the 743-nm channel [Fig. 3.2] is detected as the 669-nm light frequency is scanned across resonance. The obtained uorescence curve [Fig. 5.2] represents the transverse velocity distribution of atoms that are excited by a collimated and a diverged 833-nm laser beam. The population eciency for the central velocity group approximately doubles when the 833-nm laser beam is diverged. This is a denite signature of adiabatic passage for atoms near the center of the transverse velocity distribution. Overall, the fraction of atoms excited to state e is 50(20)%, which is limited only by insucient 833-nm light power. At 50 mw of laser power, the 743-nm signal is maximized at a 833-nm laser divergence near 0.05 rad, so only about half of the entire transverse atomic velocity distribution is covered. Fig. 5.3 shows the 743-nm peak uorescence as a function of 833-nm laser power. It clearly shows that we are currently limited by insucient 833-nm light power. Numerical density matrix calculations indicate that one would need to increase the 833-nm light power by a factor of 2(to100 mw). 34

42 Figure 5.2: 743-nm uorescence curves for w/o (circles) and w/ (squares) lenses in 833- nm laser beam. The 833-nm laser power is 50 mw. The divergence is 0.05 rad. The 833-nm laser frequency is locked on resonance. 669-nm laser beam is collimated and lowpowered ( 5 mw). Fluorescence detected as 669-nm laser frequency is scanned. Solid curve represents transverse Doppler distribution of atoms. 35

43 Figure 5.3: Peak uorescence in the 743-nm channel dependence upon 833-nm laser power for the case w/o (circles) and w/ (squares) divergence in the 833-nm laser beam. For the case w/ lenses, the 833-nm laser divergence is 0.10 rad. The 669-nm laser beam is used as a probe, i.e. it is collimated and low-powered ( 5 mw). 36

44 Figure 5.4: Peak uorescence in the 743-nm channel dependence upon 669-nm laser divergence. 833-nm laser power at 50 mw and 669-nm laser power at 30 mw. 833-nm laser beam divergences are 0.06 rad (circles) and 0.04 rad (squares). 5.2 Second Transition: State e! f Although the 669-nm laser beam is also diverged, not all criteria for adiabatic passage in the second transition are fully satised. Unlike the rst transition, the lifetime of the upper state f (432(5) ns) is comparable to the inversion time (T 100 ns see Eq. (4.32)). However, the desired state B is still eciently populated since atoms excited to state f decay into state B with a high probability (.30(9) branching ratio). Fig. 5.4 shows 743-nm peak uorescence with lenses in the 833-nm as a function of the 669-nm laser beam divergence. The maximum peak uorescence is a factor of 2 larger 37

45 than for the case of zero divergence (without lenses), which suggests that the eciency of the second step 1 is nearly 100%. The fractional increase of the peak uorescence can be estimated as a function of 1 as (5.1) where 1 is the width of the distribution of atoms excited by the diverging 669-nm laser beam ( 40 MHz [Fig. 5.2]), 2 is the power broadened line width of the 669-nm transition without lenses ( 20 MHz), and 2 is the eciency of excitation with a collimated 669-nm laser beam. At maximum peak uorescence, the 669-nm laser divergence angle is matched to that of the 833-nm laser beam, so 1 is determined by the width of the atomic distribution excited by 833-nm light. The eciency 2 is determined to be about 80% using steady-state rate equations for a 4-level system, similar to those of a 2-level system [Sec. 4.2]. The eciency is limited only by 669-nm laser power. Using the data shown in Fig. 5.4 and the given values for 1, 2, and 2, the eciency 1 for a diverging 669-nm laser beam is determined to be the same as that for the collimated 669-nm beam: about 80%. 38

46 Chapter 6 Conclusions Utilizing diverging cw laser beams, we populate state B, the odd parity state of interest for a PNC search in atomic dysprosium. We achieve ecient population of the transverse velocity distribution and an overall population eciency of state B of 12% (this is the product of the eciencies of all three transitions: 50% for the 833-nm transition, 80% for the 669-nm transition, and 30% for the spontaneous decay from f to B determined by the branching ratio). For the 833-nm transition, a substantial portion of the transverse velocity distribution underwent adiabatic passage. With collimated laser beams (no lenses), the total eciency would have been<0.5%. The population eciency of B can easily be improved by increasing the power of the lasers used, especially that of the 833-nm laser, and by introducing another laser at wavelength 1397-nm to stimulate the f! B transition. Although, the population eciency of B is 2 smaller than in the previous PNC experiment using collimated pulsed lasers, the 10 4 increase in duty cycle gives roughly 100 times greater statistical sensitivity. With 20 hours data taking time (10 greater than in the previous experiment), we hope to reach a sensitivity to H w (the weak matrix element) of 10 mhz. 39

47 Appendix Atomic Parity Nonconservation A parity operation is a reection of a coordinate system through the origin, r!;r. Let ^P represent the parity operator. If j (r)i is an eigenstate of ^P, then ^P j (r)i = j (;r)i = j (r)i (1) where is the eigenvalue of ^P corresponding to the state j (r)i. Applying ^P once more to the eigenstate j (r)i gives the result ^P (j (;r)i) = 2 j (r)i = j (r)i: (2) The possible values of are, therefore, +1 and -1. The state j (r)i is said to be a state of denite parity, which is either even or odd depending on the corresponding eigenvalue +1 or -1, respectively. In an atom, the dominant electromagnetic interactions conserve parity. That is, the Hamiltonian H 0 describing the interactions commutes with the parity operator ^P, meaning eigenstates of H 0 are simultaneously eigenstates of ^P. One can therefore categorize the atomic 0 states j n i according to their parity. However, there are smaller weak interactions between a valence electron and the nucleus which do not conserve parity. Since these interactions 40

48 are much smaller than those due to the electromagnetic force, perturbation theory can be used to include their eects. The new Hamiltonian is then H = H 0 + H w, where H w is the purely imaginary weak Hamiltonian describing the perturbation. Since the weak interaction is parity-odd and time-even, H w should be purely imaginary to prevent the emergence of a permanent electric dipole moment (EDM) and time reversal invariance violation. To rst order, the eigenstates of H are j 0 ni + X m 0 0 h m jh w j n i j En 0 ; Em 0 0 mi (3) where E 0 n and E 0 m are the energies of the old eigenstates j 0 n i and j 0 m i, respectively. In our parity nonconservation (PNC) experiment in atomic dysprosium (Dy), we attempt to measure the matrix element of the weak Hamiltonian H w. We use dysprosium because it has a pair of nearly degenerate opposite parity states (states A and B in Fig. 3.2). The energy separation is particularly small (3.1 MHz) between F = 10:5 hyperne sublevels of the isotope 163 Dy, and it can be further reduced for specic Zeeman sublevels by applying an external magnetic eld [11]. As seen in the last term of Eq. (3), mixing is inversely proportional to the dierence in energies of the opposite parity states. Therefore, the extraordinarily small energy separation between the states A and B greatly enhances the mixing between them. A previous dysprosium PNC experiment using pulsed lasers reported jh w j = j2 3j Hz [1]. This disagrees with the theoretical prediction H w = Hz [21]. Our experiment attempts to improve upon the previous one by employing cw lasers and the adiabatic passage technique to increase statistical sensitivity topncmeasurement by a factor of 10 2 [10]. 41