Interaction between light and atoms Lecture 7-11

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1 Interaction between light and atoms Lecture 7-11 Thierry Chanelière Laboratoire Aimé Cotton, Orsay December 7, 015 Contents I Representation of the optical Bloch equations on the Bloch sphere 3 1 Optical Bloch Equations in the rotating wave approximation Reminder Including the phase of the field Bloch vector 4.1 Definition Bloch vector equations of motion Properties Navigating the Bloch sphere Rabi oscillations Arbitrary phase Particular cases Pulse area Free evolution Ramsey experiment 6 5 Homework Problem: Optical nutation Solution: Optical nutation Method Final result II Propagation through an atomic medium 9 6 Introducing the field propagation OBE including field propagation Polarization Bloch-Maxwell equations Maxwell s equations in dielectric Slowly varying envelope approximation (SVEA) Perturbative regime Steady state solutions Weak pulse transmission Linearized Bloch-Maxwell equations Definition of the Fourier transform Solution in the Fourier space Homework Problem: Rate equation limit of the Bloch equations Solution: Rate equation limit of the Bloch equations

2 III Pulse propagation: Absorption, dispersion and causality Linear susceptibility in atomic media Definition Causality and Kramers-Kronig relations Index of refraction, group index, group velocity Atomic expression of the linear susceptibility Connection with the Bloch-Maxwell model Is the complex Lorentzian causal? Homework 19.1 Problem: Two-pulse photon echo Solution: Two-pulse photon echo Propagation through a structured periodic profile Transmission of a periodically modulated profile - Optically thin case 3 0. Stimulated or three-pulse photon echo Three-pulse echo and diffraction grating Envelope propagation and group delay On resonance group delay Far-off-resonance group delay Application of the Kramers-Kronig relations Real and imaginary parts Example Hilbert transform tables Relationship with the Fourier transform Numerical calculation Homework Problem: Is the complex Lorentzian causal? Solution: Is the complex Lorentzian causal? IV Inhomogeneous broadening and propagation Warm-up: Bloch-Maxwell equations with two independent atomic species Introducing the inhomogeneous broadening Perturbative regime Weak pulse propagation Linearized Bloch-Maxwell equations Coherent propagation 0

3 Part I Representation of the optical Bloch equations on the Bloch sphere I will here introduce the Bloch vector formalism and its graphical representation on the Bloch sphere. The formalism is directly derived from the Optical Bloch Equations (OBE). There is no new physical ingredient. The Bloch sphere is an elegant graphical representation of the manipulation of atomic states when a excitation field is applied. I ll focus on the representation of the coherent interactions of wellknown problems as Rabi flopping and Ramsey experiment that I will represent graphically. 1 Optical Bloch Equations in the rotating wave approximation Starting from the OBE, I ll derive the Bloch vector formalism. I ll use the rotating wave approximation (RWA) as already presented by David Clément. The only difference I ll introduce is the phase of the field (ωt + Φ). I include Φ to eventually account for phase changes of the field between two successive pulses. In that case, the OBE for the density matrix coefficient read as t ρ ee = iω R cos(ωt + Φ) ( ρ eg ρ ge ) ρee /T 1 t ρ gg = t ρ ee = iω R cos(ωt + Φ) ( ρ eg ρ ge ) + ρee /T 1 t ρ eg = iω 0 ρ eg + iω R cos(ωt + Φ) ( ρ ee ρ gg ) ρeg /T t ρ ge = iω 0 ρ ge + iω R cos(ωt + Φ) ( ρ gg ρ ee ) ρge /T 1. Including the phase of the field (1a) (1b) (1c) (1d) I have introduced the electromagnetic field phase Φ, that can be represented by the complex amplitude Ω R cos(ωt + Φ) = 1 ( Ω exp (iωt) + Ω exp ( iωt) ) () where Ω = Ω R exp (iφ). The slowly varying terms are written as ρ gg = σ gg (3) ρ ee = σ ee (4) ρ eg = σ eg exp ( iωt) (5) ρ ge = σ ge exp (iωt) (6) The components exp (iωt) are neglected in the RWA, so I obtain the OBE in the RWA as already written by David Clément: t σ ee = i ( Ωσeg Ω σ ge ) σee /T 1 (7a) 1.1 Reminder I here consider a closed two-level system so there is no decay from the ground state. The population of the excited state decays toward the ground state at a decay rate written 1/T 1. The decay rate of the coherence is written as 1/T. The phenomenological decay rates are written as T 1 and T to follow the terminology of nuclear magnetic resonance (NMR) in which the Bloch vector formalism is extensively used. t σ gg = i ( Ω σ ge Ωσ eg ) + σee /T 1 t σ eg = iσ eg + i Ω ( σ ee σ gg ) σeg /T t σ ge = iσ ge + i Ω( σ gg σ ee ) σge /T (7b) (7c) (7d) where = ω ω 0. With respect to what you have seen before, there is no difference except that Ω is a complex amplitude (and Ω its complex conjugate). If the phase of the field 3

4 is constant, the choice of the phase is arbitrary and you can simply choose Φ = 0 thus Ω = Ω. The terms σ gg, σ ee, σ ge and σ eg of the density matrix describe the atomic state and its evolution when a electromagnetic field is applied. Bloch vector The Bloch vector formalism consists in replacing the density matrix coefficients by a three real components (U,V,W) defining a vector in 3D..1 Definition The components (U,V,W) are defined as: They are indeed real. σ gg = 1 W σ ee = 1 + W σ eg = U iv σ ge = U + iv. Bloch vector equations of motion The eqs (7) can be rewritten in terms of (U,V,W) as t U = U/T + V + Ω i W t V = U V/T Ω r W t W = Ω i U + Ω r V (1 + W)/T 1 (8) (9) (10) (11) (1a) (1b) (1c) where Ω = Ω r + iω i is decomposed into its real and imaginary parts. The components U and V are sometimes called the in-phase and out-of-phase (or in-quadrature) components of the Bloch vector. The only interest of the Bloch vector formalism is its graphical representation. I can alternatively use a fully vectorial form, familiar to the nuclear magnetic resonance (NMR) community 1 by including U B = V and β Ω r = Ω i (13) W The coherent evolution part can be represented by t B = β B (14) where is a cross or vector product as opposed to. the scalar product. β is called the control vector. I haven t written the decay terms to focus on the coherent evolution. This occurs when the evolution is faster than the different decay times..3 Properties The eq.(14) describes the precession of the Bloch vector around the control vector. This vectorial equation is encountered in different domains in Physics. If β is constant (time independent), one can show that The precession angle of B around β is constant because t ( B. β ) = 0. The norm of the Bloch vector is B ( ) is constant because t B. B = 0. The speed of precession is β The vector is contained in a sphere of radius 1, the Bloch sphere because B = U + V + W 1. The equality holds for pure states. This directly comes from the property of the density matrix Tr(ρ ) 1. This later can indeed be written as ρ gg + ρ (1 W) (1 + W) ee + ρ ge ρ eg = (U + iv)(u iv) 4 Each point on the Bloch sphere represents a pure state. Examples at t = 0: g ; 0 ( g + e ) / 0 ; 1 ( g + i e ) / ; See for example (NMR) 4

5 3 Navigating the Bloch sphere We focus again on fully coherent evolution of the atomic state, T 1 = T =, and describe two rudimentary phenomena. The general evolution of the Bloch vector is described by a precession around the control vector. (eq.14). This can be generally decomposed into rotation matrices on the sphere. I ll calculate the rotation matrices in different situations by giving a formal solution. 3.1 Rabi oscillations Rabi flopping is the coherent measurement par excellence. Its interpretation on the Bloch sphere is enlightening. The Bloch vector simply rotates. We assume the excitation to be sufficiently strong so the effect of the detuning can be neglected during the excitation: Ω Arbitrary phase First, I do not make any assumption on the phase of the field Φ. The equations of motion (eq.1) read as t U = Ω 0 sin Φ W t V = Ω 0 cos Φ W t W = Ω 0 sin Φ U + Ω 0 cos ΦV where Ω = Ω 0 exp (iφ) = Ω r + iω i The equations can be partially decoupled by doing the base change with the inverse formulas So the system becomes (15a) (15b) (15c) U = cos Φ U + sin Φ V (16) V = sin Φ U + cos Φ V (17) W = W (18) U = cos Φ U sin Φ V (19) V = sin Φ U + cos Φ V (0) W = W (1) t U = 0 (a) t V = Ω 0 W t W = Ω 0 V (b) (c) whose solution is an elemental rotation around the first coordinate axis U (t) U (0) V (t) = 0 cos (Ω 0 t) sin (Ω 0 t) V (0) (3) W (t) 0 sin (Ω 0 t) cos (Ω 0 t) W (0) Coming back to the initial coordinates gives the general evolution of the atomic state after a strong excitation of arbitrary phase Φ: U(t) cos (Φ) sin (Φ) V(t) = sin (Φ) cos (Φ) 0 0 cos (Ω 0 t) sin (Ω 0 t) W(t) sin (Ω 0 t) cos (Ω 0 t) cos (Φ) sin (Φ) 0 U(0) sin (Φ) cos (Φ) 0 V(0) (4) W(0) In the rotated frame (U,V,W ), the Bloch vector rotates around U. As expected from the general form of eq.(14), the Bloch vector precesses around a control vector which lies in the equatorial plane Particular cases We can treat particular cases with Φ = 0 and Φ = π/. The phase is zero: Φ = 0 U(t) V(t) = cos (Ω 0 t) sin (Ω 0 t) U(0) V(0) = R 0 (Ω 0 t) U(0) V(0) (5) W(t) 0 sin (Ω 0 t) cos (Ω 0 t) W(0) W(0) The evolution is represented by an elemental rotation around U. U(t) U(0) Example of a π-pulse (Ω 0 t = π): V(t) = V(0) i.e. σ eg (t) = σ ge (0). W(t) W(0) Out of phase: Φ = π/ U(t) cos (Ω 0 t) 0 sin (Ω 0 t) U(0) U(0) V(t) = V(0) = R π/ (Ω 0 t) V(0) (6) W(t) sin (Ω 0 t) 0 cos (Ω 0 t) W(0) W(0) The evolution is represented by an elemental rotation around V. 5

6 3.1.3 Pulse area So far we have assumed that the filed is switched on at t = 0 and keeps a constant value Ω 0. These result can be generalized to a pulse whose amplitude Ω(t) varies in time (with a constant phase). One can define the pulse area θ(t) = t Ω(t )dt (7) The solutions of the Rabi problem are very similar. The term Ω 0 is replaced by t θ in eqs.(15). The solution is now given by the rotation matrix R 0 (θ(t)). After the end of the pulse, the final state is given by θ = area. 3. Free evolution + Ω(t )dt, the total pulse The free evolution (no field) is also described by an elemental rotation. From leads to the final state sin (τ) 0 (31) cos (τ) t U = V (8a) t V = U t W = 0 (8b) (8c) we obtain U(t) cos (t) sin (t) 0 U(0) U(0) V(t) = sin (t) cos (t) 0 V(0) = F (t) V(0) (9) W(t) W(0) W(0) The evolution is represented by an elementary rotation around W. 4 Ramsey experiment A first strong π/-pulse excitation of the atoms initially in the ground state is followed by a free evolution during a delay τ. A second π/-pulse is applied at the 0 end. Writing the final state as R 0 (π/) F (τ) R 0 (π/) 0 with 1 R 0 (π/) = cos (τ) sin (τ) and F (τ) = sin (τ) cos (τ) 0 (30)

7 5 Homework 5.1 Problem: Optical nutation We here reconsider the Rabi problem by including the effect of the detuning. We have seen that the condition Ω gives nice Rabi oscillations. We will see how it goes when we deviate from this condition by considering an inhomogeneous sample (atoms with different detunings). So far, we have considered a single atom or an ensemble of identical atoms so the detuning and the Rabi frequency is the same for all the ensemble. We can anticipate the future lectures and consider an ensemble with a so-called inhomogeneous broadening: the detuning is not the same for all the atoms. We can calculate the ensemble emission including the inhomogeneity of the sample. Different atoms have different detunings. They are initially in the ground state. They are all excited at the instant t = 0 by a single long pulse of constant amplitude. The laser phase is zero Φ = 0 and the decay terms are neglected so the Bloch vector dynamics reads as t U = V t V = U ΩW (3a) (3b) t W = ΩV (3c) One can calculate the evolution of U(t) V(t) for the different detunings by W(t) first guessing W(t) = A cos (Ω g t) + B sin (Ω g t) + C where Ω g = Ω + and A, B&C are constant. Question 1: Justify the previous guess and derive the evolution by calculating A, B&C. As we will see later in the course, the total emission is proportional to i U(t)+ iv(t). Question : Calculate the ensemble emission. Hints: The first order Bessel function can be represented by J 0 (0) = 1 π π 0 exp (iz cos θ) dθ leading to J 0 (Ωt) = π 0 sin ( Ω + t ) d Ω Solution: Optical nutation The guess W(t) = A cos (Ω g t)+b sin (Ω g t)+c is justified because the Bloch vector precesses around the control vector β = Ω r Ω i. The speed of precession is given by the norm of β = Ω g. The evolution reads as an elemental rotation around β. In the frame of β, the motion is a linear superposition of cos (Ω g t) and sin (Ω g t). To find U(t), V(t) and W(t), one needs to come back to the initial frame by applying rotation matrices (base change). In any case, the evolution of U(t), V(t) and W(t) will be a linear superposition of cos (Ω g t) and sin (Ω g t). Thus justifying the initial guess Method Starting from W(0) = 1, one find C = 1 A. V(t) is derived from eq. (3c). V(0) = 0 gives B = 0 and finally V(t) = A Ω g Ω sin (Ω gt) Using (3a) gives the expression of U(t) = A Ω (cos (Ω gt) 1). Finally using eq. (3b), one finds A = Ω. One also verifies that Ω g = Ω + which is an a posteriori verification of the initial guess. 5.. Final result Ω U(t) Ω (cos (Ω g t) 1) g Ω V(t) = sin (Ω g t) W(t) Ω g Ω Ω cos (Ω g t) g Ω g The ensemble emission is given after integrating over of i Ω g (33) U(t)+iV(t) =. The integration of U(t) is actually zero because it is an odd function of (Ω g being an even function of ). The total emission is given by V(t) J 0 (Ωt). 7

8 The emission of the inhomogeneous ensemble is an oscillating J 0 function Ωt (see fig. 1). 1 Bessel function J 0 (θ) π π 3π 4π 5π 6π 7π 8π Figure 1: Bessel function J 0 as a function of θ = Ωt. The oscillation period is roughly 1/Ω. 8

9 Part II Propagation through an atomic medium When the action of an atomic medium is considered on light, propagation must be included. A medium can have a significant effect on light only if it is optically thick. For example, lasing is due to gain, meaning the exponential growth of the field amplitude by propagation though the medium. I will then introduce the propagation of light through an unidimensional atomic medium by using the slowly varying envelope approximation, I will here assume the medium to be homogeneous in the sense that all the atoms are identical. More precisely, they have the same atomic frequency, the same dipole and the same decay times. They are all described by the same set of OBE. I ll derive the Bloch-Maxwell equations. I ll define the socalled perturbative regime and consider the absorption (and gain) for continuous and pulsed fields. 6 Introducing the field propagation And the the slowly varying atomic quantities as ρ gg = σ gg ρ ee = σ ee ρ eg = σ eg exp ( i (ωt kz)) ρ ge = σ ge exp (i (ωt kz)) As before, we introduce the Bloch vector (U, V, W) σ gg = 1 W σ ee = 1 + W σ eg = U iv σ ge = U + iv and find the Bloch vector equations exactly as before (eqs. 1) t U = U/T + V + Ω i W t V = U V/T Ω r W (35a) (35b) (35c) (35d) (36a) (36b) (36c) (36d) (37a) (37b) I first consider the effect of a propagating field on an atomic two-level system and rederive the OBE in that case. I ll keep the Bloch vector formalism and use the Bloch sphere as a support for the graphical interpretations. 6.1 OBE including field propagation So far, we have considered that the position of the atoms was given and fixed. We now consider an ensemble of atoms distributed along z. We will look at the propagation of light through the sample. For the sake of simplicity, we include the total phase of the electric field (ωt kz + Φ) into the definition of the slowly varying atomic quantities σ eg (and U, V, W) and the field Ω. All these quantities depend on time t and now on z but slowly. The field now reads as Ω R cos(ωt kz + Φ) = 1 ( Ω exp (i (ωt kz)) + Ω exp ( i (ωt kz)) ) (34) where Ω = Ω R exp (iφ) t W = Ω i U + Ω r V (1 + W)/T 1 (37c) where the complex field amplitude reads as Ω = Ω r + iω i. As compared to eqs. 1, there is no difference. The local phase kz of the field is included in the definition of the atomic operators. Eqs (37) describes the action of the field on the atoms. To account for the back-action of the atoms on the field, we have to evaluate the polarization which is precisely the source of radiation. 6. Polarization The relevant variable to quantify the emitted field is the macroscopic polarization: ˆD atoms P = Volume = N ˆD ( = N Tr ˆρ ˆD ) = N d (ρ ge + ρ eg ) = N d U + iv exp (i (ωt kz))+c.c. (38) N is the density of dipoles. The polarization is the macroscopic quantity built from the atomic coherence. It is also slowly varying. 9

10 7 Bloch-Maxwell equations 7.1 Maxwell s equations in dielectric The field propagation is defined by the Maxwell equations which precisely include the polarization as a source of emission. The medium is supposed free charges and currents, thus defining a dielectric medium. The Maxwell s equations read as rot E = B t div B = 0 (39) rot H = D t + 0 div D = 0 (40) where the magnetizing and displacement field are defined as H = 1 µ 0 B and D = ɛ 0 E + P. I only consider the propagation in one dimension, along z. From the Maxwell s equations, one can derive the following propagation equation by calculating the (rot rot E). ze 1 c t E = 1 c ɛ 0 t P (41) 7. Slowly varying envelope approximation (SVEA) The propagation equation can be further simplified by considering the slowly varying envelope: E = E 0 (t, z) cos(ωt kz + Φ) = E 0 (t, z) exp (i (ωt kz)) + c.c. (4) The slowly varying envelope approximation applied to E 0 gives ze 1 ( c t E ik z E 0 iω ) c te 0 exp (i (ωt kz)) + c.c. (43) U and V are also slowly varying so t P N d 1 (U + iv) ω exp (i (ωt kz)) + c.c. (44) The propagation differential equation simplifies to the first order z E c te 0 = i N ωd cɛ 0 (U + iv) (45) By multiplying by d, one retrieves the complex amplitude of the Rabi frequency: z Ω + 1 c tω = i N ωd cɛ 0 (U + iv) (46) z Ω r + 1 c tω r = N ωd cɛ 0 V z Ω i + 1 c tω i = N ωd cɛ 0 U (47a) (47b) We can introduce α = N ωd T which has the dimension of an absorption coefficient. We will see later that it is indeed the absorption cɛ 0 coefficient. z Ω r + 1 c tω r = α T V z Ω i + 1 c tω i = α T U (48a) (48b) with the eqs. (37), this set is known as the Bloch-Maxwell coupled equations for an homogeneous medium. The Bloch-Maxwell equations are not analytically solvable in general. Simplifications are required. We consider the so-called perturbative regime for weak excitations. 8 Perturbative regime The term weak excitation has to be defined. It actually means that the population is unchanged. The Bloch-Maxwell equations can then be linearized and analytically solved. In this regime, I ll calculate the transmission of the continuous (steady state solutions) and a pulsed field. 8.1 Steady state solutions I ll look at the steady state solutions and specify what I mean by weak excitation. I take from now t (U, V, W) = 0 and Ω doesn t depend on time. I ll also assume Ω = Ω r for the sake of simplicity and specify the term weak excitation by looking 10

11 at the population in that case. The steady state solutions for the Bloch vector are given by (eqs. 37) For the population, we obtain 0 = U/T + V (49a) 0 = U V/T Ω r W (49b) 0 = Ω r V (1 + W)/T 1 (49c) 1 W = 1 + Ω r T1T 1+ T We see that the population are unchanged and stays in the ground state (50) W = 1 if Ω T 1 T 1 (51) The condition (51) defines the perturbative regime. The term Ω T 1 T is actually the on-resonance saturation parameter. When it is sufficiently weak, we can neglect the population in the excited state. The result is actually general even if Ω is complex. When can now look at the coherence by replacing W = 1 0 = U/T + V Ω i (5a) 0 = U V/T + Ω r (5b) The solution can be compactly written with R = σ ge = U + iv R = iω (i + 1/T ) The coherence R is proportional to Ω as compared to the population W which depend on Ω r (square of the field). It is then perfectly valid in the perturbation series to consider the first order term Ω acting on the coherence and neglect the population change (to the second order). We can now look at the field given by the propagation equation (48). The field amplitude is given by z Ω = α (53) 1/T (i + 1/T ) Ω (54) On resonance = 0, we find the Bouguer-Beer-Lambert absorption law 8. Weak pulse transmission Ω (z) = Ω (0) exp ( αz/) (55) As opposed to the steady state solution, we now consider a pulsed excitation or the transient solutions. The perturbative regime has to be defined again in that case. On the Bloch sphere, the Bloch vector is weakly perturbated if the pulse area is small. As a consequence, the population are unchanged in that case. This defines the perturbative regime for pulsed excitations. Assuming W = 1 constant, the problem can be solved in the Fourier space Linearized Bloch-Maxwell equations Staying in the ground state, the population W = 1 is assumed constant. One can linearized the Bloch-Maxwell equations t U = U/T + V Ω i t V = U V/T + Ω r Or compactly written with R = σ ge = U + iv (56a) (56b) t R = (i + 1/T ) R + iω (57) The equation is linear in Ω as opposed to terms like WΩ which are not linear (because W depends actually on Ω). The propagation equation eq. (48) is already linear z Ω + 1 c tω = i α T R (58) We obtain the linearized Bloch-Maxwell equations. They are coupled but linear. They can be solved using the Fourier transform. 8.. Definition of the Fourier transform I define the Fourier transform of a time depend function f(t) as f(ν) with the relations f(ν) = f(t) exp( iνt)dt (59a) t f(t) = 1 f(ν) exp(iνt)dν (59b) π ν 11

12 8..3 Solution in the Fourier space Eqs (57) and (58) can be written in the Fourier space R = i Ω (i + iν + 1/T ) (60) and for the propagation iν z Ω + c Ω = i α α R = T 1/T (i + iν + 1/T ) Ω = α L (T + νt ) Ω (61) where we have introduced the complex Lorentzian function L(x) = The formal solution in the Fourier space is 1 (ix + 1) Ω(z, ν) = Ω(0, ( ν) exp i ν c z α ) L (T + νt ) z (6) (63) The propagation may be difficult to solve analytically in general. It is still consider as a simple numerical problem because it involves only Fourier transform that can be calculated efficiently. It is solvable analytically if the incoming bandwidth is small as compared to 1/T, the resonance width. In that case, L (T + νt ) L (T ) and the outgoing pulse shape is simply. ( Ω(z, t) = Ω(0, t z/c) exp α ) L (T ) z (64) We actually retrieve the Bouguer-Beer-Lambert law for a weak pulse. The term z/c is the delay due to propagation through the distance z. The term L (T ) represents the response of the medium (the susceptibility). 1

13 9 Homework 9.1 Problem: Rate equation limit of the Bloch equations The so-called rate equation limit allows to calculate the evolution of the populations when the coherences are in a steady-state. This corresponds to an intermediate time-scale between T and T 1 when T T 1. It describes accurately the population dynamics when the incoming laser has a short coherence time τ. For longer time-scale than τ, the coherence can be considered in their steady-state but the population can still evolve. Rate equations describe the optical pumping process. Starting from (37), we simple assume the coherences in their steady state meaning t U = t V = 0 leading to The term Ω T is the on-resonance pumping rate as opposed to 1/T 1, the decay rate. For the steady state solution, we find W(+ ) = thus retrieving eq. (50) for an Ω complex Ω T 1T 1+ T (66) 0 = U/T + V + Ω i W (65a) 0 = U V/T Ω r W (65b) t W = Ω i U + Ω r V (1 + W)/T 1 (65c) Question 1: By introducing R = U + iv in eqs. (65a,b), express R as a function of W. Question : Use the previous expression to derive from eq. (65c) the rate equation W Solution: Rate equation limit of the Bloch equations By doing (65a) + i (65b), we obtain. R = Ω i/t W In eq. (65c), the term Ω i U + Ω r V should be written as [ I [RΩ Ω ] ] = I W = Ω T i/t 1 + T W So we obtain the rate equation: t W = Ω T 1 + T W (1 + W)/T 1 13

14 Part III Pulse propagation: Absorption, dispersion and causality I have previously described of weak pulses by eq.(63) in the Fourier space. This latter predicts the outgoing pulse shape for a given incoming one. I ll now define and discuss the group velocity and consider slow and fast light propagation. These two intriguing phenomena are due to the dispersion relation for the susceptibility. I ll talk about the causality which is hidden behind the Kramers-Kronig relations and the Bloch-Maxwell model. 10 Linear susceptibility in atomic media I ll derive the Kramers-Kronig relations. They are not very practical expect if you know very well the distribution theory. The underlying physics is the causality which is built-in the Bloch-Maxwell model Definition χ is defined as P (t) = ɛ0 χ E(t) assuming the The ideal linear susceptibility medium is instantaneously polarized in response to the applied field. More generally, taking into account the medium response function R: The causality imposes that R(t ) = 0 for t < 0. Alternatively in the Fourier space: P (t) = ɛ 0 t R(t t )E(t ) (67) P (ν) = ɛ 0 χ (ν) Ẽ(ν) (68) where χ = R is the Fourier transform of R keeping in mind R(t ) = 0 for t < Causality and Kramers-Kronig relations Kramers-Kronig relations represent the causality in the Fourier space. By definition, χ (ν) = R(ν) is the Fourier transform of R keeping in mind R(t) = 0 for t < 0 or mathematically: R(t) = h(t)r(t) (69) where h(t) is the Heaviside step function. So χ (ν) = h(t)r(t) exp( iνt)dt = h(t) 1 π t t ν χ (ν ) exp(iν t)dν exp( iνt)dt (70) = 1 χ (ν ) h(t) exp( i(ν ν )t)dtdν = 1 χ (ν ) h(ν ν )dν π ν t π ν (71) We all know the Fourier transform the Heaviside step function 3 h(ν) = πδν=0 i ν This is the hard point in the derivation because a good knowledge of the distribution theory is required. So the Kramers-Kronig relations read as (7) χ i χ (ν ) (ν) = π ν ν ν dν (73) which can be alternatively written as two relations between the real and imaginary part of the susceptibility as we will discuss later in 13. The Kramers-Kronig relations (73) are not very practical expect with a profound knowledge of the distribution theory. It is something safer to go back to the definition of causality meaning the linear impulse response R is zero at negative time Index of refraction, group index, group velocity The susceptibility allows to define the group velocity that we will discuss later in 1. I here give the definition of the group velocity from the index of refraction n n(ν) = 1 + χ r(ν) and the group index n g n g (ν) = n(ν) + ν n ν The group velocity is defined as V g (ν) = c n g (ν) 3 (74) (75) (76) 14

15 11 Atomic expression of the linear susceptibility I ll reconnect the propagation equation (63) with the definition of the susceptibility Connection with the Bloch-Maxwell model Let s go back to the Maxwell equations in dielectric where the polarization is defined: ze 1 c t E = 1 c ɛ 0 t P (77) in the Fourier space including the susceptibility zẽ + ν c Ẽ = ν ν P = χ c (ν)ẽ(ν) (78) ɛ 0 c ( So Ẽ is the sum of two terms: exp i ν ) 1 + χ (ν)z and ( c exp i ν ) 1 + χ (ν)z c Assuming χ (ν) 1, meaning the field is not affected over the length scale of a wavelength. The terms are now i ν ( χ ) (ν) 1 + z and i ν ( χ ) (ν) 1 + z c c This expression can be identify with the solution (63) of the atomic model derived in lecture 8: Ω(z, ν ) = Ω(0, ν ) exp ( i ν c z α ) L (T + ν T ) z (79) This should be done with precaution because Ω(z, t) is the slowly varying envelope of E(z, t) i.e. E(z, t) Ω(z, t) exp (i (ωt kz)) + c.c.. As a consequence Ẽ(z, ν) Ω(z, ν ω) exp ( ikz) ( = Ω(0, ν ω) exp i ν ω z α ) c L (T + νt ωt ) z ikz (80) χ (ν) = i c ν αl (T + νt ) i λ π αl (νt ω 0 T ) (81) We finally show that the susceptibility is actually given by a complex Lorentzian center on the atomic transition. 11. Is the complex Lorentzian causal? The susceptibility is proportional to the the complex Lorentzian that describes the response of the medium after propagation. The Lorentzian comes from the Bloch model. Does the Bloch model contains the causality? The answer must be yes. That is what we will verify. I will simply calculate the Fourier transform of the complex Lorentzian and show that it is zero at negative time. Starting from the result with f(t) = h(t) exp( t/t ) (8) the Fourier transform of f is calculated by a direct formal integration 4 : f(ν) = T 1 + iνt (83) The complex Lorentzian is clearly causal because f(t) = 0 for t < 0. One can also look at the real and imaginary part of the complex Lorentzian independently. [ ] R f(ν) = f S (ν) = T [ ] 1 + (νt ) and I f(ν) = f AS (ν) = Knowing the inverse Fourier transform of f(ν), one can show that νt 1 + (νt ) (84) f S (t) = exp( t /T )/ and f AS (t) = sign(t) exp( t /T )/ (85) The real part of the complex Lorentzian f S is not causal nor the imaginary part. But the sum, the complex susceptibility, is causal. This is not by chance. The Bloch model is based on partial differential equations which intrinsically respect causality. 1 Envelope propagation and group delay I ll now consider fast and slow light propagation. This is main objective of the lecture even if it was important to clarify the causality before that. 4 It is also possible to work out the inverse Fourier transform by writing: f(t) = 1 T exp(iνt)dν π ν 1 + iνt The derivation in time of the expression leads to an inhomogeneous linear ordinary differential equations. tf = δ t=0 f/t that can be solved by the method of variation of constants. We find the expression (94) for f(t). 15

16 Instead of using the expressions of 10.3, I ll go back to the definition the group delay. The envelope propagation is well defined when the dispersion profile (real part of the susceptibility) can be considered as locally linear ( ην) over the incoming bandwidth. To the first order, we can write Ω(z, ν) = Ω(0, ν) exp ( iην) gives in time Ω(z, t) = Ω(0, t η) (86) η, the slope of the dispersion profile as the dimension of a time and is actually the group delay. It represents the delay of the outgoing pulse as compared to the incoming one. When the transmission spectrum in not linear, the pulse is not only delayed, it is also distorted because of the high order spectral components (high order expansion of the susceptibility). To consider slow and fast light, we will assume the detuned by from the transition and do a first order expansion of the complex Lorentzian in (63) to obtain the group delay though the atomic medium assuming νt 1: ( ) 1 νt L (T + νt ) 1 i it + 1 it + 1 We will now consider two situations. 1.1 On resonance group delay When = 0, we have for the propagation Ω(z, ν) = Ω(0, ( ν) exp i ν c z α ) z + iα zνt And in time Ω(z, t) = Ω(0, t z/c + α zt ) exp ( α ) z The group delay is z/c α zt. It can be negative corresponding to superluminal light (fast-light) propagation. Its observation is difficult because of the exp ( α z) term. The pulse is superluminal but strongly absorbed. Both effects go together with α. It is in practice difficult to observe the delay of a very weak pulse. 1. Far-off-resonance group delay When T 1, we now obtain Ω(z, t) = Ω(0, t z/c α ( ) α z) exp i z T T (87) (88) (89) (90) The group delay is z/c+ α z corresponding to slow-light propagation. There T is no absorption apparently because we assumed T 1 but as is increased, the group delay is reduced. Off resonance excitation is required to avoid absorption but the detuning cannot be increased too much, otherwise the group delay is reduced. 13 Application of the Kramers-Kronig relations 13.1 Real and imaginary parts The relation (73) can be split into its real and imaginary parts leading to: χ r (ν) = 1 χ i (ν ) π ν ν ν dν and χ i (ν) = 1 χ r (ν ) π ν ν ν dν (91) It means that by knowing the imaginary part (absorption), one can retrieve the real part (dispersion, refractive index) of a sample and vice versa. It is due to causality, so there is no assumption on the nature of the sample. The possibility to deduce the dispersion from the absorption is from my point of view the main interest of the Kramers-Kronig relation. Measuring the absorption is the essence of spectroscopy, well mastered and sufficient to deduce the dispersion. This last step will be discussed by an example. 13. Example I d like to fully characterize a black-box by doing transmission measurements. The black-box under study has a transmission curve with an apparent periodic modulation given by 1 + ɛ cos(ντ) where ɛ 1. We will calculate the associated dispersion profile by three methods Hilbert transform tables Since ɛ 1, we can simply identify the term χ i (ν) = ɛ cos(ντ). The relations (13.1) tell us that χ r (ν) is the Hilbert transform of χ i (ν). Hilbert transforms of common functions are tabulated. The effort maid by the so-called Bateman project is remarkable in this direction. In the volume, one can recognize on p.45, the relation between the real and imaginary parts of the complex Lorentzian. On p.5, they confirm that the Hilbert transform of the cosine is the sine function. So we conclude that the susceptibility is given by χ (ν) = χ r (ν) + i. χ i (ν) = ɛ sin(ντ) + iɛ cos(ντ) = iɛ exp( iντ) 16

17 13.. Relationship with the Fourier transform A very pragmatic manner to calculate the susceptibility by applying the causality is to Fourier transform the imaginary part. The imaginary part has negative components (t < 0). To restore the causality, one can simply zero the negative components and multiply the positive ones by a factor. In the Fourier space, one multiplies by the distribution (1 + sign). Or in other words, one adds the appropriate antisymmetric function (given by sign to restore causality. Coming back to our example, the Fourier transform of the cosine function χ i (ν) is the sum of the Dirac peaks δ τ and δ τ. The Fourier transform of χ (ν) is then given by twice the positive components δ τ following our procedure. This latter is precisely exp( iντ) confirming our previous result Numerical calculation The present example is somehow trivial because the cosine function has a simple Hilbert and Fourier transform. The relationship with the Fourier transform is nevertheless fundamental and can be used to calculate the real part of the susceptibility for an arbitrary imaginary part profile. This can be implemented numerically by calculating a first Fourier transform (FFT), applying the distribution (1 + sign) in the Fourier space and a second FFT to obtain the final result. This simple algorithm exists in many scientific computing software. I ll give two examples. xhi = scipy.signal.hilbert(xhi_i) matplotlib.pyplot.plot(nu,numpy.real(xhi),label= $\chi_i$ ) matplotlib.pyplot.plot(nu,numpy.imag(xhi),label= $\chi_r$ ) matplotlib.pyplot.legend() matplotlib.pyplot.savefig( hibert_cos.eps ) matplotlib.pyplot.show() The result is plotted in fig χ i χ r Matlab/Octave code function. nu=linspace(0,3,51); xhi_i=cos(*pi*nu); xhi=hilbert(xhi_i); to calculate and plot the Hilbert transform of the cosine plot(nu,real(xhi),nu,imag(xhi), linewidth,) legend({ $\chi_i$ $\chi_r$ }, fontsize,18, Interpreter, latex ) Python code to calculate and plot the Hilbert transform of the cosine function. Figure : Numerical calculation of the cosine function using Python. This procedure applies to any experimental measurement of the absorption without analytic well-defined shape. import numpy import scipy.signal import matplotlib.pyplot nu = numpy.linspace(0,3,51) xhi_i= numpy.cos(*numpy.pi*nu) 17

18 14 Homework 14.1 Problem: Is the complex Lorentzian causal? In the equation (63) describing the propagation in the Fourier space, we have the 1 term. This latter is sometimes called the complex Lorentzian (it + iνt + 1) function. It is proportional to the susceptibility (eq. 81). The Lorentzian comes from the Bloch model.the Bloch model contains the causality because it is a set of differential equations which describes the evolution. That is what we will verify by calculating the Fourier transform of the complex Lorentzian, namely its impulse response. Let s write the complex Lorentzian f(ν) = iνt (9) Question 1: Write the impulse response f(t) defined as the inverse Fourier transform of f(ν) 5 Question : Derive in time the previous expression leading to an inhomogeneous linear ordinary differential equation. Question 3: Solve the differential equation f(t) = 1 1 exp(iνt)dν (95) π ν 1 + iνt By deriving the previous expression, one obtains the differential equation t f = δ t=0 /T f/t. This latter can be solved by the method of variation of constants leading to the expression of f(t) f(t) = h(t) exp( t/t ) T Alternatively, one can directly integrate h(t) exp( t/t ) exp(iνt)dt T t to retrieve f(ν). The complex Lorentzian is causal because its Fourier transform f(t) is zero at negative time (h(t) is the Heaviside step function). You should verify that the following function f(t) (eq. 94) is a solution. f(t) = h(t) T exp( t/t ) (94) where h(t) is the Heaviside step function. Question 4: Starting from the result of f(t) calculate f(ν) by a direct formal integration of the Fourier transform. Verify that you find eq. (9). Question 5: Is the complex Lorentzian causal? 14. Solution: Is the complex Lorentzian causal? The impulse response f(t) is given by: 5 As a reminder, the definition of the Fourier transform is: f(ν) = f(t) exp( iνt)dt t f(t) = 1 f(ν) exp(iνt)dν π ν (93a) (93b) 18

19 Part IV Inhomogeneous broadening and propagation So far, we have considered that the ensemble of atoms is homogeneous. All the atoms are identical. 1/T is called the homogeneous linewidth in that case. As an example, this is not the case for an atomic vapor at room temperature whose Doppler profile cannot be neglected. The atoms having different velocity, they have different detunings. Introducing an inhomogeneous broadening where each atoms is defined by its own detuning actually covers a wide range of situation from atomic gases to luminescent solids. 15 Warm-up: Bloch-Maxwell equations with two independent atomic species We can consider a medium as a mix of only two homogeneous media. For example, it can be a mix of two atomic species called 1 and. Both have the sample field coupling (Rabi frequency) and the same decay times (T 1 and T ) but different detunings k with k = 1,. The two species are described by two independent Bloch vectors driven by: t U k = U k /T + k V + Ω i W k t V k = k U k V k /T Ω r W k t W k = Ω i U k + Ω r V k (1 + W k )/T 1 Both dipoles contribute to the macroscopic polarization P = later can be decomposed into P = ˆD atoms Volume = ˆD atoms (96a) (96b) (96c) Volume. This ( ) U 1 + iv 1 U + iv N 1 + N 1 exp (i (ωt kz)) + c.c. (97) simply adding the two contributions where N 1 and N are the densities of the two species. Concerning the propagation thought the ensemble, the Bloch-Maxwell equations read under the SVEA as in section 7.: z Ω + 1 c tω = i ωd cɛ 0 [N 1 (U 1 + iv 1 ) + N (U + iv )] (98) This simple example can be extrapolated to a density distribution for the different detunings. 16 Introducing the inhomogeneous broadening The previous approach can be generalized by considering a continuous distribution, so called the inhomogeneous broadening described by a spectral density n: z Ω + 1 c tω = i ωd n() (U + iv ) d (99) cɛ 0 where n() is the spectral density normalized as N = n()d with N the total density. The spectral density can be replaced by a dimensionless distribution g() = n() where n(0) is the spectral density at the laser frequency (zero detuning). n(0) One defines α an absorption coefficient by α π = n(0)ωd. This latter has the cɛ 0 same unit than the absorption coefficient introduced in eqs. (48) because n(0) is a spectral density (density divided by a frequency or multiplied by a time). z Ω + 1 c tω = i α π g() (U + iv ) d (100) Each atoms dynamics is described by the Bloch equations for a given : t U = U /T + V + Ω i W t V = U V /T Ω r W t W = Ω i U + Ω r V (1 + W )/T 1 (101a) (101b) (101c) The previous two sets of equations (eqs. 100 and 101) describe the propagation in a inhomogeneous sample. 19

20 One can retrieve the homogeneous case by writing the inhomogeneous profile g as a Dirac peak g() = π T δ =0 (δ is a Dirac peak). 17 Perturbative regime 17.1 Weak pulse propagation As in 8., the perturbative regime for weak pulse means that the population is not affected to the first order so that W is constant (in time) over the inhomogeneous profile. 17. Linearized Bloch-Maxwell equations We assume W = 1, the atoms stay in the ground state. The coherence is compactly written with R = U + iv as in 8..1: and the propagation still described as t R = (i + 1/T ) R + iω (10) z Ω + 1 c tω = i α π g()r d (103) As compared to 8..1, there is no fundamental difference. The Bloch dynamics has to be evaluated for the different detunings and integrated to account for the inhomogeneous broadening in the propagation equation. 18 Coherent propagation As compared to (64), where we have assumed the incoming bandwidth smaller than 1/T, or in other words, the pulse is longer than T, we now consider the inverse situation. The pulse is much shorter than T. We will take the limit T. As in 8..3, the propagation is considered in the Fourier space: The coherences are given by R = i Ω (i + iν + 1/T ) (104) and the propagation by iν z Ω + c Ω = i α g() R d (105) π = α π Ω g() d (106) (i + iν + 1/T ) Taking the limit T should be done with precaution at this stage. It is necessary to decompose the complex Lorentzian into its real and imaginary part. 1 (i + iν + 1/T ) = T 1 + (T + νt ) + i T (T + νt ) 1 + (T + νt ) (107) If we take the limit T, the real part tends to a Dirac peak πδ = ν and the 1 imaginary part tends to. this later is a distribution defined by the Cauchy + ν principal value. The integral term in (106) read as α g() π (i + iν + 1/T ) d = α g( ν) α π i g() + ν d Concerning the second term, following the relations (13.1), if we define the real distribution g i (ν) such as g(ν) is the Hilbert transform of g i (ν), we can then write the second term as i α g i( ν). z Ω + iν c Ω = α Ω [g( ν) + ig i ( ν)] (108) and after integration over z Ω(z, ν) = Ω(0, ν) exp ( i ν c z α ) G( ν)z (109) where we have written G = g + ig i a complex distribution, which verifies the Kramers-Kronig relations (13.1) and whose real part is the inhomogeneous distribution g. The result is quite simple. The inhomogeneous distribution (with the imaginary part g i added) substitutes the homogeneous profile to define the susceptibility. As an example, if the inhomogeneous linewidth is lorentzian, G is a complex lorentzian. The solutions (64) and (109) are very similar. This means that by measuring the absorption profile in a perturbative regime (weak probe pulse), it is impossible to know if the profile is homogeneous or inhomogeneous. The two situations correspond to very different realities. Eq. (64) has been obtained for a pulse duration longer than T as opposed to eq. (109) obtained for 0

21 a pulse shorter than T. In this latter case, the propagation is said to be coherent because the experiment is done in a time shorter than the coherence time. The evolution of the atoms is fully coherent: there is no dissipation as opposed to eq. (64) (spontaneous emission or population decay). In that case, the absorption is not due to dissipation. The field disappears through the propagation because the dipoles emission is destructively interfering in the forward direction. The destructive interference is due to the inhomogeneous broadening. The dipoles oscillates until T which is much longer than the pulse duration. 1

22 19 Homework 19.1 Problem: Two-pulse photon echo We will consider the two-pulse photon echo (PE) sequence by considering the fully coherent evolution on the Bloch sphere. The PE is very similar to the Ramsey experiment discussed in 4 but with a different pulse sequence. We will considered an inhomogeneous sample (atoms have different detunings ) and look at the evolution for the different. The PE sequence is composed of a π/ pulse (uniform for all the atoms ), followed a free evolution period (duration τ). At time τ, a π pulse is sent (uniform for all the atoms ). This latter excitation is again followed by a free evolution during a time t τ. The coherent evolution on the Bloch sphere can be described by the following elemental rotation R 0 (π/) = (110) for a π/ pulse and R 0 (π) = (111) for a π pulse. The free evolution during τ reads as cos (τ) sin (τ) 0 F (τ) = sin (τ) cos (τ) 0 (11) sin ( (τ t)) F ( (t τ)) R 0 (π) R 0 (π/) F (τ) 0 = cos ( (τ t)) (113) 1 0 The total emission is i g() (U(t) + iv(t)) d = g()e i(τ t) d = g(t τ) where g is the Fourier transform of g. If g() is a gaussian, the term g(t τ) is a gaussian pulse centered at time t = τ whose duration is the inverse of the inhomegeneous linewidth. This latter is called a two-pulse photon-echo because it appears at t = τ after the first two pulses at t = 0 (π/-pulse) and t = τ (π-pulse). The atoms are all in the ground state. Question 1: Calculate the Bloch vector corresponding to the final state after the free evolution at time t τ. The π/ and π pulses are assumed to be sufficiently short and intense to cover the complete inhomogeneous broadening uniformly (The Rabi frequency is the same for all the atoms). Question : Calculate the total ensemble emission as i g() (U(t) + iv(t)) d where g() defines the inhomogeneous broadening distribution. Question 3: Why is the sequence called a two-pulse photon-echo? What is the echo duration if g() is a gaussian function? 19. Solution: Two-pulse photon echo The final state is evaluated by calculating the matrix product:

23 0 Propagation through a structured periodic profile 0.1 Transmission of a periodically modulated profile - Optically thin case We will consider a fictitious situation and supposed the inhomogeneous broadening has a small periodic modulation g() = 1 + ɛ cos(ντ) We here assume that the inhomogeneous broadening is infinite: it covers the whole spectrum. This corresponds to the situation where the incoming pulse spectrum (bandwidth) is narrower that the inhomogeneous broadening. We will look at the propagation through this structured periodic profile by using the general propagation equation (109) with g() = 1 + ɛ cos(τ). To do so, we have to determine G = g+ig i, a complex distribution, which verifies the Kramers-Kronig relations (13.1). That is precisely what we did in the example 13.. So we have: G() = g() + ig i () = 1 + ɛ exp(iτ) (114) The propagation equation (109) could be solved analytically in that case. To avoid lengthy calculations, I ll make the following assumption: the medium is optically thin. In other words, it slightly perturbs the incoming field. In practice, it means that the term exp ( α ) G( ν)z in (109) can be expanded to the first order because αz 1. Ω(z, ν) = Ω(0, ν) exp ( i ν ) c z ( i ν ) ( c z Ω(0, ν) exp Ω(0, ν) exp or in the time domain: [ Ω(z, t) = 1 + αz exp ( α ) G( ν)z 1 + αz G( ν) ) ( i ν c z ) [1 + αz/] + Ω(0, ν) exp ] Ω(0, t z c ) + [ ɛ αz (115) (116) ( i ν c z iντ ) [ɛαz/] (117) ] Ω(0, t z τ) (118) c For an optically thin medium αz 1, the dominant contribution comes from the incoming pulse Ω(0, t z c ) (delayed by the free space propagation by z/c). Because of the interaction with the medium, we observe a replica of the pulse delayed by z/c + τ. This latter is called a stimulated photon-echo. 0. Stimulated or three-pulse photon echo We have just seen that a periodic modulation of the inhomogeneous profile produce an echo. A periodic density distribution doesn t exist naturally but it can be fabricated in practice. In 4. we have seen that after a Ramsey experiment (two π/-pulses), the population as a function of the detuning is given by W = cos (τ). Let s take this final state as a new initial state of an inhomogeneous medium and consider the propagation of a weak third pulse. Eq. (10) cannot be used (perturbative regime) because precisely W 1. We have to step back to eqs. (101) and slightly modify it when the initial state of the atoms is given by W = cos (τ). t R = (i + 1/T ) R iw Ω (119) The propagation of the third pulse is still described by (103). For a sufficiently weak pulse, the populations W are not modified and stay constant. As in 18, the coherent propagation is only slightly modified The coherences are now given by R = and the propagation by iν z Ω + c Ω = α π Ω iw Ω (i + iν + 1/T ) (10) W g() d (11) (i + iν + 1/T ) We see that the term [W g()] represents an effective inhomogeneous profile because W doesn t depend on time. This latter acts as a periodic profile W = cos (τ) if g is flat over the spectrum of interest. So the third pulse generates an echo after a time τ. Assuming that W doesn t depend on time is a generalization of the perturbative regime. This is actually not obvious. Perturbative means that R and W depend on the field. This dependency is developed in a perturbative series. We have see that in the ground state (or the excited state), the coherences R go like the pulse area and populations W like the square of the area. As a consequence, when the pulse area is small, the populations are not modified to the first order. This discussion can be represented graphically on the Bloch sphere. On contrary, for a given arbitrary W ±1, the perturbative expansion is relevant only if U = V = R = 0 initially. The atomic state is not pure in that case. The total sequence is called a three-pulse echo. Two π/-pulses generates a spectral population grating and the third one is diffracted on this grating to produce an echo. From the previous discussion, after the first two pulses, R = 0 is 3