# Theory of EIT in an Ideal Three-Level System

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1 Chapter 3 Theory of EIT in an Ideal Three-Level System 3.1 Three-Level Systems So far the interaction of a two-level system (or series of two-level systems) with a near resonant, monochromatic field has been considered. This leads to the Doppler-broadened spectra and saturation spectra as shown in chapter. The three-level system interacting with two monochromatic fields offers a vast array of new physics to be investigated. It is the development of monochromatic tuneable laser sources that has allowed such investigations to be undertaken. There are essentially three different types of three-level system: Λ; cascade and V. These are shown in Fig (i) a (ii) (iii) c (i) (ii) (iii) b c a c b b a Figure 3.1: The three different three-level systems, with the optical transitions shown by the red arrows. (i) A Λ system, (ii) a cascade system and (iii) a V system. 43

2 Chapter 3. Theory of 3-Level Systems Coherent Population Trapping The application of two continuous-wave (cw) fields to a three-level atomic system puts the atom into a coherent superposition of states. This superposition of states is stable against absorption from the radiation field. Hence this phenomena is referred to as coherent population trapping (CPT). The Hamiltonian for a three-level system, as shown in Fig. 3.1, is given by, H = H + V 1 + V. (3.1) H is the bare atom Hamiltonian; V 1 is the potential due to the field with Rabi frequency Ω 1 and angular frequency ω 1 ; and V is the potential due to the field with Rabi frequency Ω and angular frequency ω. The Rabi frequencies are assumed to be real. Here, H = ω a a a + ω b b b + ω c c c, (3.) V 1 = Ω 1e iω1t b a + Ω 1e +iω1t a b, (3.3) and V = Ω e iωt c a + Ω e +iωt a c. (3.4) Considering the interaction of this Hamiltonian with the three-level system provides a simple one-atom model of CPT. It is apparent that the eigenstates of H ( a, b and c ) are not eigenstates of H. Instead the eigenstates of H are linear combinations of a, b and c. In the case that the two fields are on resonance, (ω 1 = ω a ω b and ω = ω a ω c ), the three eigenstates of H are, C 1 = 1 ( a + C = 1 ( a + Ω 1 b + Ω 1 + Ω Ω 1 b + Ω 1 + Ω Ω c Ω 1 + Ω ) Ω c Ω 1 + Ω ), (3.5), (3.6) NC = Ω Ω b 1 c. (3.7) Ω 1 + Ω Ω 1 + Ω NC contains no component of a and hence there is no coupling between these two states. Thus any population in NC is trapped in that state. Both C 1 and C contain components of a and hence couple to it.

3 Chapter 3. Theory of 3-Level Systems 45 Over a period of time, dependent upon the rate of spontaneous emission from the excited state, all of the population of the system will build up in NC. Hence all of the population becomes coherently trapped in a dark state. The first experimental realization of CPT was made by Alzetta et al., [45]. In this work the fluorescence of Na atoms was observed. An inhomogeneous magnetic field was applied along the axis of the Na cell. At certain points along the length of the cell the fluorescence disappeared. These black lines were due to CPT taking place, in Λ systems, at positions where the magnetic field was such that the CPT resonance was met. A theoretical analysis of the coherence phenomena that lead to these effects was presented by Arimondo and Orriols, [46]. At the same time and independently Whitley and Stroud Jr., [47], produced a theoretical treatment of CPT in a three-level cascade system Applications of CPT CPT has many potential areas of application. Ground states are generally much longer lived than excited states. Measuring the line width of a ground state would potentially increase the resolution of any device, which relied on the width of a spectral feature, by several orders of magnitude. Considering the interactions of two fields with three atomic levels, it is possible to measure a transition with line width orders of magnitude less than the excited state line width. The increase in interest in CPT was largely fuelled by applications in metrology, [48]. More recently there has been significant interest in the field of atomic clocks, [49, 5, 51], where the use of CPT has allowed all optical miniaturised atomic clocks to be produced. Velocity selective coherent population trapping (VSCPT) is another application of CPT. The technique developed by Aspect et al., [5], requires the formation of a Λ system comprising two degenerate Zeeman ground states with counterpropagating perpendicularly polarized beams. This results in only zero velocity atoms being resonant with the two beams, and thus only zero velocity atoms become trapped in the dark state. Atoms that are not in the correct velocity range for coherent population trapping to take place rely on momentum redistribution, due to optical pumping followed by spontaneous emission, to fall into the correct velocity range for the dark state. This cooling mechanism provides

4 Chapter 3. Theory of 3-Level Systems 46 for laser cooling to below both the Doppler limit and the single-photon recoil energy. A number of review articles have been written on the subject, including the comprehensive review by Arimondo, [53], and the more recent article by Wynands and Nagel, []. 3.3 Electromagnetically Induced Transparency Electromagnetically induced transparency (EIT), [54, 55], is the cancellation of the absorption of a weak probe field. The cancellation is due to the application of a pump field, resonant with one of the levels of the original transition and a third level. The immediate similarity to CPT is apparent. The difference between EIT and CPT being that CPT tends to refer to two fields of approximately equal Rabi frequency. EIT refers to the regime where one of the fields (pump) is much stronger than the other (probe). In the case of EIT the interest is in how the pump modifies the medium experienced by the probe. As the only difference between EIT and CPT is the strength of the beams, then the brief theoretical treatment of a three-level system presented above can be extended to the case where, Ω 1 Ω. It follows from equations 3.5, 3.6 and 3.7 that the coupling and non-coupling states become, C = 1 C C, (3.8) = Ω c, (3.9) Ω 1 + Ω c, (3.1) NC = Ω b, Ω 1 + Ω (3.11) b. (3.1) Throughout this work the three-level systems under investigation are exclusively Λ-systems. The Λ-system with pump and probe fields is shown in Fig. 3.. The induced transparency is caused by the interference of coherences in the

5 Chapter 3. Theory of 3-Level Systems 47 a δ pr δ pu ω pr, Ω pr ω ab Γ a ω ac ω pu, Ω pu b c Γ b Γ c Figure 3.: A three-level Λ-system. The three levels are linked by two fields, probe and pump. The two fields have angular frequency ω pr and ω pu and Rabi-frequency Ω pr and Ω pu, respectively. The probe field has a detuning from resonance of δ pr, and the pump field a detuning δ pu. Each of the levels i has a line width (FWHM) Γ i. atomic system. Alongside the modification of the absorption is the concomitant modification of the dispersion, [1, 56], encapsulated in the Kramers-Kronig relations,.3.. This is what makes EIT of such interest, the steep dispersion is coincident with a transparent window in the medium; normally such steep dispersion would be coincident with significant absorption. Harris et al., [54], demonstrated theoretically the possibility of rendering a medium transparent on a given allowed transition from a ground state to an excited state. This required the application of a pump beam resonant with a third state and the excited state. The first experimental realization of EIT was by Boller et al., [55]. EIT was observed on a Λ system in strontium vapour. The transmission of a probe was increased from e to e 1. Xiao et al., [57], measured the dispersion due to EIT directly. This was accomplished using a Mach-Zehnder interferometer. The measurements were carried out, on a cascade system, in Rb vapour with the probe resonant with the D transition and the pump resonant with 5P 3/ 5D 5/ transition. The narrowest recorded EIT resonances in atomic vapours are of 3 Hz in a medium where the excited state line width is 6 MHz, [58]. This means that the transparent window is a factor of 1 5 narrower than the Doppler-free

6 Chapter 3. Theory of 3-Level Systems 48 transition in the absence of the pump beam. In recent years there has been a vast amount of work done in the field of EIT. Some of the more important advances and potential applications are presented in There are a number of good review articles covering this rapidly growing field, [11, 59, 6]. 3.4 EIT via Optical-Bloch Equations Consider an idealized three-level Λ-system as in Fig. 3. on the preceding page. There are obvious similarities to the idealized two-level system of Fig..1 on page 1. In the same way that the complex susceptibility, and hence absorption and dispersion, of the two-level medium was calculated in chapter, it is possible to calculate the complex susceptibility of the three-level Λ-system. The Hamiltonian for this system is given by equation C.1, H = H A + V pr + V pu ; The interactions of the atoms with the probe and pump laser fields are given by equations C.8 and C.9, V pr = Ω pr V pu = Ω pu ( a b e iω prt + b a e iωprt), ( a c e iω put + c a e iωput), where the rotating wave approximation has been made. The Hamiltonian can be rewritten in matrix form, equation C.1, ω a ( Ω pr /) e iωprt ( Ω pu /) e iωput H = ( Ω pr /) e iωprt ω b. ( Ω pu /) e iωput ω c The density matrix for the three-level system is given by equation C.11, ρ aa ρ ab ρ ac ρ = ρ ba ρ bb ρ bc. ρ ca ρ cb ρ cc

7 Chapter 3. Theory of 3-Level Systems 49 Using the Liouville equation (equation.1), ρ = i [ρ, H ] γρ, the equations of motion for the populations and coherences are derived in appendix C. The equations for the populations are given by equations C.8, C.9, and C.3 on page 163, ρ aa = iω pr ( ρ ab ρ ba ) + iω pu ( ρ ac ρ ca ) Γ a ρ aa, ρ bb = iω pr ( ρ ba ρ ab ) Γ b ρ bb + Γ a ρ aa, ρ cc = iω pu ( ρ ca ρ ac ) Γ c ρ cc + Γ a ρ aa. Similarly the equations of motion for the coherences are given by equations C.33, C.34, and C.35 on page 164, ρ ab = (γ ab iδ pr ) ρ ab + iω pr (ρ aa ρ bb ) iω pu ρ cb, ρ ac = (γ ac iδ pu ) ρ ac + iω pu (ρ aa ρ cc ) iω pr ρ bc, ρ cb = (γ cb i (δ pr δ pu )) ρ cb + iω pr ρ ca iω pu ρ ab. In the steady state the rate of change of the coherences are given by, ρ ab = ρ ac, = ρ cb, (3.13) =. Likewise the rates of change of populations in this regime are given by, ρ aa = ρ bb, = ρ cc, (3.14) =.

8 Chapter 3. Theory of 3-Level Systems 5 Furthermore, in the steady state population will be trapped in the non-coupling state. In the case of EIT, where Ω pu Ω pr, the dark state, NC is equivalent to the ground state of the probe transition, b. Hence, ρ aa ρ cc, (3.15), ρ bb 1. (3.16) Substituting from equations 3.13, 3.15, and 3.16 into the equations of motion for the coherences (equations C.33, C.34, and C.35) leads to iω pu ρ cb (γ ab iδ pr ) ρ ab iω pr, (3.17) iω pr ρ bc (γ ac iδ pu ) ρ ac, (3.18) iω pu ρ ab (γ cb i (δ pr δ pu )) ρ cb + iω pr ρ ca. (3.19) In the case that the populations of two levels is small, then the coherence between those two levels will be correspondingly small. This in conjunction with the fact that Ω pr Ω pu, then the term (iω pr /) ρ ca in equation 3.19 can be neglected. The absorption and dispersion experienced by a probe beam scanned across the transition b a is proportional to the imaginary and real components of the coherence ρ ab, as shown in chapter. To determine ρ ab, substitute equation 3.17 into equation 3.19 multiplied by iω pu /. This leads to, ( Ωpu ) ( ) iωpr ρ ab = (γ cb i (δ pr δ pu)) + (γ ab iδ pr ) ρ ab, (3.) ( ) (Ω pu /) = ρ ab γ cb i (δ pr δ pu ) + (γ ab iδ pr ) = iω pr, (3.1) ρ ab = iω pr 1 / ( (γ ab iδ pr ) + ) (Ω pu /) γ cb i (δ pr δ pu ). (3.)

9 Chapter 3. Theory of 3-Level Systems 51 From equation.64 on page the complex refractive index of the medium is given by χ = Nd ba ɛ Ω pr ρ ab. (3.3) Utilising the Bloch vector notation (equation.65 on page and equations.85,.86 on page 1 and equation.94 on page 3) the absorption coefficient, α, and dispersion, n R, can be derived, In this case, / ( u = Re { i Ω 1 pr α = Nd ba ɛ k v pr, (3.4) Ω pr n R = 1 Nd ba ɛ u. (3.5) Ω pr (γ ab iδ pr ) + )} (Ω pu /) γ cb i (δ pr δ pu ) / ( )} v = Im { i Ω 1 (Ω pu /) (γ ab iδ pr ) + pr γ cb i (δ pr δ pu ), (3.6). (3.7) Plots are made of the components of the Bloch vector as given in equations 3.6 and 3.7. Fig. 3.3 shows v/(ω pr /γ ab ) and u/(ω pr /γ ab ) as a functions of Ω pu /γ ab and δ pr /γ ab, where γ cb /γ ab = 1 6, and δ pu =. Fig. 3.4 shows v/(ω pr /γ ab ) and v/(ω pr /γ ab ) as functions of Ω pu /γ ab and δ pr /γ ab, where γ cb /γ ab = 1 6, and δ pu =. As can be seen clearly from Fig. 3.3 (iii) and (iv), the initial increase in the Rabi frequency of the pump field from causes the appearance of the induced transparency and with it the concomitant modification to the dispersion profile. Fig. 3.3 (i) and (ii) show that the continued escalation of the Rabi frequency of the pump beam lead the amplitude of the modification, to both u and v, to saturate. As the amplitude increases the width of the resonance does too. Once the amplitude has saturated, at v = and u = ±.5 respectively, continued inflation of the Rabi frequency of the pump simply broadens the EIT resonance.

10 Chapter 3. Theory of 3-Level Systems 5 (i) (iii) Probe Detuning ( ) -5 δpr/γ ab Probe Detuning ( x 1 ) δpr 1 5 /γ ab ab 5 ab ab v ( pr / ab ) Pump Rabi Frequency ( ) Ω pu γ ab ab Ω pu 1 3 /γ ab vγ ab Ω pr v ( pr / ab ) vγ ab Ω pr Pump Rabi Frequency ( x 1 ) (ii) u ( pr / ab ) uγ ab Ω pr (iv) Pump Rabi Frequency ( ) u ( pr / ab ) uγ ab Ω pr ab ab Probe Detuning ( ) -4 Ω pu γ ab Probe Detuning ( x 1 ) Pump Rabi Frequency ( x 1 ) Ωpu 1 3 /γ ab ab δ pr /γ ab.8 1. ab δ pr 1 5 /γ ab Figure 3.3: Three dimensional plots of the Bloch vector components (i) v and (ii) u, as a function of the Rabi frequency of the pump, Ω pu and the detuning of the probe field from resonance, δ pr. Both Ω pu and δ pr are plotted in units of the probe transition coherence decay rate, γ ab. The ground state coherence decay rate, γ cb is set to 1 6 γ ab, and the pump beam detuning, δ pu =. (i) and (ii) show the progression of v and u respectively until the amplitude of the EIT features saturate at and ±.5 respectively. Further increase in Ω pu leads solely to broadening of the feature. (iii) and (iv) show the initial growth of the EIT feature in the narrow central region of the probe transition for small Rabi frequencies of the pump beam.

11 Chapter 3. Theory of 3-Level Systems 53 (i) v ( / ) vγ ab Ω pr pr ab..4 γ cb /γ ab Ground State Coherence ( ) 1 ab Probe Detuning ( ) δ pr /γ ab -1 ab - (ii). ab Ground State Coherence ( ) u ( / ) uγ ab Ω pr pr ab.4 γ cb /γ ab Probe Detuning ( ) δ pr /γ ab -1 ab - Figure 3.4: Three dimensional plots of the Bloch vector components (i) v and (ii) u, as a function of the ground state coherence decay rate, γ cb and the detuning of the probe field from resonance, δ pr. Both γ cb and δ pr are plotted in units of the probe transition coherence decay rate, γ ab. The pump field has a Rabi frequency, Ω pu =.3 γ ab, and is on resonance, δ pu =. As can be clearly seen in both plots an increase in the decay rate of the ground state coherence leads to significant reduction in the amplitude of the EIT resonance, such that when γ ab Ω pu, the EIT signal is indistinguishable from the background. Fig. 3.4 (i) and (ii) show v/(ω pr /γ ab ) and u/(ω pr /γ ab ) respectively as a function of γ cb /γ ab and δ pr /γ ab, where Ω pu /γ ab =.3, and δ pu =. As is clearly shown in the two figures, the EIT feature is at its narrowest and largest amplitude when the coherence decay rate between the ground states is minimized. As the ground state coherence decay rate approaches the value of the Rabi frequency of the pump the EIT signal becomes indistinguishable from the background.

13 Chapter 3. Theory of 3-Level Systems Magnetic Field Sensitivity Three-level systems can be between three different hyperfine states or between Zeeman sublevels within two hyperfine states. As Zeeman levels are magnetically sensitive, it follows that resonances with Zeeman levels are subject to magnetic broadening in an inhomogeneous magnetic field. Zeeman Effect Every hyperfine state F has F + 1 Zeeman sub-levels, denoted m F, where F m F F. These Zeeman sub-levels are given by the projection of the total angular momentum F onto an external magnetic field B. The energy shift due to the Zeeman effect is E Z. Fig. 3.5 shows a Λ system with two ground states that are Zeeman sublevels of the same hyperfine state. It follows that the energy difference between the Zeeman sub-levels varies with B, and in the region of interest, where E Z E HFS (where E HFS is the hyperfine splitting), it can be taken to vary linearly with the magnitude of the magnetic field, B. The Zeeman shift is E Z = g F µ B Bm F. (3.8) g F is an effective Landé g-value, given by: F (F + 1) + J(J + 1) I(I + 1) g F = g J F (F + 1) g I F (F + 1) J(J + 1) + I(I + 1) F (F + 1), (3.9) where, g J = J(J + 1) + L(L + 1) S(S + 1) J(J + 1) J(J + 1) L(L + 1) + S(S + 1) + g S J(J + 1), (3.3) and, g S, (3.31) = g J = 3J(J + 1) L(L + 1) + S(S + 1). J(J + 1) (3.3)

14 Chapter 3. Theory of 3-Level Systems 56 As g I m/m, where m is the electron mass and M is the proton mass, then g I g J. Hence, g F = 3J(J + 1) L(L + 1) + S(S + 1) J(J + 1) F (F + 1) + J(J + 1) I(I + 1) F (F + 1). (3.33) The notation used above is following that used in Elementary Atomic Structure, [61]. The values of g F are calculated for all of the hyperfine states of the D-lines of 85 Rb and 87 Rb, and are shown in table 3.1 and 3., respectively. There is no g F for F =. Term F S 1/ 1/3 1/3 P 1/ 1/9 1/9 P 3/ 1 1/9 7/18 1/ Table 3.1: g F for the hyperfine states of 85 Rb. Term F 1 3 S 1/ 1/ 1/ P 1/ 1/6 1/6 P 3/ /3 /3 /3 Table 3.: g F for the hyperfine states of 87 Rb. In the following chapters a Λ system made up of σ + and σ beams with the states b and c being F, m F = 1 and F, m F = +1 will be considered. The linear frequency shift in the position of such a resonance is calculated and shown in tables 3.3 and 3.4.

15 Chapter 3. Theory of 3-Level Systems 57 E z δ pr δ pu a σ σ + c Figure 3.5: E z b m F = 1 m F = m F = +1 E z E z is the magnitude of the shift of the levels due to the Zeeman effect, in the absence of the B-field both σ + and σ would be on resonance with the states shown. Term F S 1/ P 1/ P 3/ Table 3.3: Ratio of the Zeeman shift to m F (MHz/G) for the hyperfine states of 85 Rb. Term F 1 3 S 1/ P 1/ P 3/ Table 3.4: Ratio of the Zeeman shift to m F (MHz/G) for the hyperfine states of 87 Rb.

16 Chapter 3. Theory of 3-Level Systems 58 Magnetic Field Broadening If the levels in an EIT system are magnetically sensitive, then it follows that a non-uniform magnetic field can cause position-dependent shifts in the frequency of the resonance. Specifically, consider the case of a Λ system where both ground states are different Zeeman sub-levels of the same hyperfine level (see Fig. 3.5). Any non-uniformity in magnetic field, B along the axis of the pump and probe beam, will lead to a broadening of any feature by an amount ν. From equation 3.8, ν = g Fµ B B m F h, (3.34) where m F is the difference in m F values between the ground states Probe and Pump Beam Crossing Angle When pump and probe fields were discussed previously it was in relation to saturation spectroscopy,.6.1 on page 39, and counter-propagating beams were required. In order to determine the appropriate orientation for pump and probe beams for EIT in a Λ system the following analysis is required. Consider a scheme in which a pump and a probe beam propagate at an angle θ to one another, Fig In addition consider an atom propagating at an angle α to the bisector of the pump and probe, with a velocity, v. The atom s velocity can be resolved into two components, v, parallel to the bisector of the pump and probe and v, perpendicular to the bisector. This causes a change in the detuning of both the probe and pump from the resonances, b a and c a, respectively, δ pr δ pr k pr v, (3.35) δ pu δ pu k pu v. (3.36) Thus the coherence, ρ ab (equation 3.) that leads to the probe absorption and

17 Chapter 3. Theory of 3-Level Systems 59 dispersion, becomes, ρ ab = iω pr (3.37) (Ω pu /) 1 (γ ab i (δ pr k pr v)) +. γ cb i (δ pr δ pu + k pu v k pr v) A shift in frequency of the two-photon resonance, δ v, will result. For a Λ system this will take the form, k pu θ/ θ/ α v atom v v k pr Figure 3.6: The diagram shows the crossing angle of the pump and probe beams, θ. The direction of propagation of the atoms in the medium is given by α. The reference coordinates are v, the bisector of the pump and probe beams, and v the perpendicular to the bisector of the pump and probe beams. δ v = (k pu k pr ) v, (3.38) ( ) ( ) θ θ = k pu v cos α k pr v cos + α, = [k pu (cos θ cos α + sin θ ) sin α ( k pr cos θ cos α sin θ )] sin α v. (3.39) Re-writing k pu = k pr + δk, (3.4) then δ v = [k pr sin θ ( )] θ sin α + δk cos α v. (3.41) For a Λ system where both ground states are Zeeman sub-levels of the same hyperfine state, δk pr is given by the Zeeman splitting. In the case of the Rb D line, k pr m 1. The Zeeman splitting between the two Λ groundstates

18 Chapter 3. Theory of 3-Level Systems 6 in the hyperfine state F = 1 of 87 Rb for a magnetic field of 1 G is 1.4 MHz (see table 3.4 on page 57) this gives δk = m 1, leading to: δk k pr 1 9. (3.4) Resolving the velocity of the atom into its two components, v and v, where, v = v cos α, (3.43) v = v sin α. (3.44) In the case that δk k pr, and by considering the velocity in terms the two perpendicular components equation 3.41 reduces to, δ v = k pr sin(θ/)v. (3.45) The single-photon detuning due to the Doppler shift of the probe is given by, ( ) θ k pr v = k pr v cos + α, (3.46) = k pr v pr. (3.47) v pr is the component of the atom s velocity parallel to k pr. It follows that equation 3.37 can be written, ρ ab = iω pr ( (γ ab i (δ pr k pr v pr )) + (3.48) ) (Ω pu /) 1. γ cb i (δ pr δ pu + k pr sin(θ/)v ) In a vapour cell the velocity distribution of the atoms is homogeneous and isotropic and is given by the Maxwellian distribution. From.3.4 and equation.96 on page 4, the Maxwellian velocity distribution is given by, N (v z ) dv z = N [ ( ) ] m m π k B T exp v z dv z. k B T v z is the velocity in one dimension. As the velocity is isotropic in the vapour cell, then the distribution of v z is the same as the distribution of v and v pr. It follows that the absorption coefficient of the medium is given by, α Im { ρ ab } (v pr, v )N (v pr ) N (v ) dv pr dv. (3.49)

19 Chapter 3. Theory of 3-Level Systems 61 Using Mathematica, the absorption of such a medium has been calculated. The normalized absorption coefficient is given by, α norm = α α (δ pr =, Ω pu = ). (3.5) A plot of the line-centre normalized absorption coefficient is shown in Fig As can be seen the EIT resonance dies away rapidly as the pump and probe move away from copropagating. As the temperature of the atomic ensemble increases, so does the rate at which the amplitude of the EIT resonance dies away with increasing angle between the pump and probe. Normalized Absorption Coefficient !/" (1 #3 Radians) Figure 3.7: The normalized absorption coefficient for a Λ system is plotted as a function of angle between the pump and probe beam from θ = to θ = 1 3 π. The red line shows the absorption for Ω pu =, whereas the blue line shows the absorption for Ω pu =.4γ ab. The velocity distribution of the atomic ensemble is that of 87 Rb at room temperature, 93 K. For the purpose of this plot δ pr =, δ pu =, and γ cb = 1 6 γ ab. An experimental verification of this has been provided by Carvalho et al., [6]. Ideally in the three-level Λ and V type systems, the pump and probe should be perfectly co-propagating to cancel the Doppler broadening of the two-photon resonance (due to component of the atoms velocity perpendicular to the bisector of the pump and probe beams) to as high a degree as possible.

21 Chapter 3. Theory of 3-Level Systems 63 lifetime is much longer than the transit time of the atoms through the beams; the excited state lifetime is much less than the transit time of the atoms through the beam; and where the pump and probe powers are small enough that they do not contribute to the broadening, then the expected line shape is a cusp function, with a FWHM, Γ EIT, given by, Γ EIT = ln v d. (3.51) d is the 1/e beam diameter and v is the thermal speed of the atoms, v = kb T m, (3.5) k B is the Boltzmann constant, T is the temperature of the atoms and m is the atomic mass. A cusp function being a back-to-back exponential of the general form, ( exp ω ω ω ), (3.53) where ω is the 1/e half-width, ω is the frequency and ω is the frequency of the centre of the resonance. An experimental study of this has been made by Knappe et al., [67]. Knappe et al. show that for narrow beam the cusp line shape gives a better fit than the standard Lorentzian model. However at line centre the cusp model does not fit the data, this is explained as being due to other broadening effects. As the width of the beam is increased, the cusp model becomes a less good fit, whilst the Lorenzian model accurately describes the resonance line shape Beam Profile Effect on Line Shape In the previous subsection the case of transit-time broadening was addressed for beams with Gaussian intensity profiles. In the work that is presented in this thesis, the profiles of the probe and pump beams can not always be described by a Gaussian function. The pump and probe beams are expected to have a Gaussian profile when an optical fibre patchcord is used to ensure that they are overlapped. In all other cases the profile will not necessarily be uniform, nor For a cusp function the 1/e half width is equal to FWHM/( ln ).

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