Z −∞ DR(t + ∆/2, t− ∆/2)eiω∆d∆. (5.39) Le théorème standard de fluctuation-dissipation n’est pas valide pour les systèmes hors équilibre, et par conséquent, il n’y a pas de moyen direct de relier les fonctions de corrélation aux fonctions de réponse [114, 115]. Cependant, nous montrons que deux mesures à ω et −ω peuvent être utilisées pour extraire la partie imaginaire de la transformée de Wigner de la fonction de Green retardée hors de l’équilibre.
L’expression de la transformé de Wigner de la fonction de Green retardée est donné par ∆DRW(t, ω) = (1− ch(r)) e−2γt e2i(ω+ω0)t ω + ω0+ iγ − e2i(ω−ω 0)t ω− ω0+ iγ + 2 i sh(r) cos (2ω0t)e−2γt e 2iωt ω + iγ, (5.40) où ∆DR W(t, ω) ≡ DR W(t, ω)− DR eq,W(t, ω) et DR
eq,W(t, ω) est la transformée de Wigner de la fonction de Green retardée à l’équilibre. Nous voyons dans Eq. (5.40) que la partie imaginaire de la transformée de Wigner de la fonction de Green retardée oscille dans le temps pour toutes les fréquences ω. On ne peut pas s’attendre à un tel poids spectral oscillatoire dans le cas d’un quencg thermique du bain. Ainsi, il est possible de détecter les états squeezé du phonon en sondant la fonction de Green retardée à l’aide de la spectroscopie Raman à résolution temporelle.
5.4 Conclusion générale
La dynamique du réseau hors équilibre suivant l’excitation par une pompe non courte, telle qu’observé en spectroscopie ultra-rapide, est complexe et très difficile à décrire micro-scopiquement. Dans cette thèse, nous avons exploré l’interaction entre les degrés de liberté du réseau et de la charge dans des systèmes fortement corrélés excité hors équilibre par une impulsion laser ultra-rapide. Une manifestation particulièrement intéressante des degrés de liberté du réseau dans la spectroscopie pompe-sonde, qui a reçu beaucoup d’attention expérimentalement, est la génération de phonons cohérents par la pompe. Motivé par cela, nous avons développé une théorie microscopique des phonons cohérents dislacive excité par
transfert de chaleur. La théorie microscopique capture la physique au-delà de la description phénoménologique standard, à savoir, la modification des niveaux d’énergie électronique due à l’excitation du phonon, et comment ce changement se répercute sur la dynamique du phonon. Cet effet de l’interaction électron-phonon conduit à une dynamique plus riche et à des corrections qualitatives du mouvement cohérent du phonon. En particulier, nous avons démontré que le déplacement atomique moyen oscille avec une phase dépendante du temps. Cette phase peut être interprétée comme un "chirping" à des échelles de temps courtes, alors que, à des échelles de temps longues, elle apparaît comme une phase résiduelle finie dans le signal oscillatoire. Nous avons appliqué avec succès la théorie au phonon cohérent A1g de BaFe2As2, démontrant ainsi que les données de la pompe-sonde peuvent être reliées à des quantités microscopiques et éventuellement à la physique de l’équilibre.
Un aspect important de la dynamique du réseau est la fluctuation autour du déplacement atomique moyen. Après une discussion préliminaire, nous avons développé une description microscopique, en utilisant Keldysh, de la dynamique hors équilibre des fluctuations du réseau dans les systèmes métalliques. En particulier, nous avons étudié théoriquement le comportement oscillatoire de la variance du déplacement atomique entraîné par l’impulsion de la pompe. Nous avons trouvé que lorsqu’un phonon est squeezé par une pompe impulsive, la variance du déplacement atomique oscille à 2ω0. Cependant, nous avons proposé deux autres mécanismes qui peuvent aussi exciter de telles oscillations de la variance du déplace-ment atomique à 2ω0, sans squeezer l’état du phonon. Nous avons montré que la variance du déplacement atomique d’un phonon couplé à un bain thermique avec une température quenché oscille à 2ω0. Nous expliquons aussi comment l’excitation d’un phonon cohérent pour lequel des termes cubiques anharmoniques sont autorisés peut aussi exciter de telles oscillations à 2ω0 dans le bruit. Nous affirmons que les oscillations de 2ω0 sont plutôt une conséquence de la brisure soudaine de la symétrie de translation temporelle par la pompe. Ainsi, nous insistons sur le fait que ces oscillations à 2ω0sont nécessaires mais pas suffisantes pour conclure que le phonon est dans un état squeezé.
La variance du déplacement atomique a qualitativement les mêmes propriétés dans le cas d’un quench de la température du bain thermique et d’un phonon squeezé. Ainsi, il n’est pas possible de détecter un squeezing de phonon seulement en se basant sur la mesure de la variance hors équilibre. Cependant, les fonctions de Green retardée dans les deux mécanismes sont en fait très différentes. Par conséquent, ces deux mécanismes peuvent être distingués en sondant la fonction de Green retardé du phonon, puisque ce n’est que lorsque le phonon est squeezé que la fonction de Green retardé brise la symétrie de translation temporelle. En particulier, nous avons montré que la partie imaginaire de la transformée de Wigner de la fonction de Green retardée oscille dans le temps si le phonon est dans un état
squeezé, et qu’on ne peut s’attendre à une telle oscillation du poids spectral dans le cas d’un quench thermique du bain. Ainsi, il est possible de détecter les états squeezé du phonon en sondant la transformée de Wigner de la fonction de Green retardée. Nous avons montré qu’en comparant le poids spectral à deux fréquences ω et −ω dans la spectroscopie Raman à résolution temporelle, on peut extraire la fonction de Green retardée hors équilibre.
La description microscopique du bruit hors équilibre fournit de nouvelles indications sur la façon d’interpréter les données expérimentales en spectroscopie pompe-sonde. Nous avons proposé une expérience concrète pour détecter le squeezing des phonons en utilisant des techniques de spectroscopie Raman à résolution temporelle. Nous espérons que nos résultats stimuleront d’autres recherches expérimentales, et que le protocole expérimental que nous proposons sera mis en œuvre dans le futur.
Appendices
Appendix A
Details of the Calculation for DECP
A.1 Structure of the retarded response function out of
equilibrium
The response function used in the main text is defined by
ΠT (t)(t− t′)≡ iθ(t − t′)hhO(tˆ ′), ˆO(t)iiH0,T (t), (A.1) where ˆO ≡ (λ/N )Pk,a,b,σC(k)abc†kaσckbσ is the weighted electron density operator, H0 ≡ H(λ = 0), and the Hamiltonian H is given by Eq. (1) in the main text. In equilibrium ΠT (t)(t− t′) is a function of (t− t′) only, but this is no longer the case out-of-equilibrium. Here, we discuss the t and t′ dependencies of ΠT (t)(t− t′). We write the response function in the Lehmann representation where the time structure can be made explicit
ΠT (t)(t− t′) = iθ(t− t′)X
n,m
|hn| ˆO|mi|2ei(t−t′)(En−Em)
× e−β(t)Em− e−β(t)En , (A.2) where En and |ni are respectively a complete set of eigenenergies and eigenstates of the Hamiltonian H0. We see from (A.2) that the response function is a function of the time difference (t −t′), and that the explicit time t dependence enters only through the electronic temperature T (t). We can then define the Fourier transform of the response function with
respect to the time difference (t − t′) evaluated at the electronic temperature T (t) ΠT (t)(Ω) =X n,m |hn| ˆO|mi|2 1 En− Em− Ω + iη × e−β(t)En − e−β(t)Em , (A.3) where η is an arbitrarily small positive constant that ensures the convergence of the Fourier transform. By inspection, we see that the response function in frequency domain (A.3) is the equilibrium retarded phonon self-energy evaluated to second order in the electron-phonon interaction (λ2) at temperature T (t) = 1/β(t).
A.2 Solution of the equation of motion
As discussed in the main text, if the instantaneous out-of-equilibrium force f(t) is evaluated to second order in electron-phonon interaction the theory captures the modification of the electronic dispersion due to the coherent phonon excitation, and how that feeds back upon the dynamics of the phonon itself. Taking this feedback into account the differential equation governing the atomic displacement u(t) is given by [see Eqs. (4) and (6) in main text]
∂t2+ 2γ∂t+ ω02u = f (t) = ω0 X1e−t/τ1+uX2e−t/τ2
. (A.4) The parameters (ω0, X1, X2, γ, τ1/2) are defined in the main text. Here we discuss the solution of the above differential equation subject to the initial conditions u(0) = 0 and ∂tu(0) = 0, and in the experimentally relevant limit of [γ/ω0, 1/(ω0τ1/2)] → 0. The equation of motion (A.4) is linear, the solution is then the sum of the homogeneous and particular solution u(t) = yh(t) + yp(t). We first discuss the homogeneous solution, then following the same method we give the particular solution. We start from the following ansatz for the homogeneous solution yh(t) = ∞ X n=0 aneknt+ cc, (A.5) with kn = iω1 − γ − n/τ2, and ω1 = p(ω0)2− γ2. We insert (A.5) into the homogeneous equation, and obtain an equation for the coefficients an
an(k2
n+ 2γ0kn+ (ω0)2) = ω0X2an−1,
a0(k02+ 2γ0k0+ (ω0)2) = 0. (A.6) 114
Since k0 satisfies the equation (k2
0 + 2γ0k0 + (ω0)2) = 0, a0 is then an arbitrary complex constant. We solve the coupled equation (A.6), and obtain for an
an= a0(ω0X2)n n Y m=1 1 k2 m+ 2γkm+ ω2 0 = a0 n! iX2τ2 2ω1 n n Y m=1 1 1 + (2i)(m/τ2ω0) ≈ an!0 iX2τ2 2ω0 n , (A.7)
where in the last step we took the limit [γ/ω0, 1/(ω0τ2)] → 0, the homogeneous solution then reads yh(t) = a0e(iω0−γ)t ∞ X n 1 n! iX2τ1 2ω0 e−t/τ2 n + cc = a0e−γt eiω0+iX2τ22ω0 e−t/τ2 + cc, (A.8) We replace a0 = 12Aeiψ and finally obtain for the homogeneous solution
yh(t) = Ae−γtcos ω0t + X2τ2 2ω0 e−t/τ2 + ψ , (A.9) where (A, ψ) are arbitrary constants to be determined from the initial conditions. We follow the same method to find the particular solution, we start from the ansatz
yp(t) =
∞
X
n=0
bneαnt, (A.10) with αn = −1/τ1 − n/τ2. We insert (A.10) into the equation of motion (A.4), and get an equation for the coefficients bn
bn(αn2 + 2γ0αn+ (ω0)2) = ω0X2an−1, b0(α20+ 2γ0α0+ (ω0)2) = ω0X1.
(A.11) We solve the coupled equations and obtain
bn = ω0X1(ω0X2) n (1/τ1)2− 2γ(1/τ1) + (ω0)2 n Y m=1 1 α2 m+ 2γαm+ ω2 0 ≈ X1 ω0 X2 ω0 n , (A.12)
where in the last step we took the limit [γ/ω0, 1/(ω0τ1/2)]→ 0, the particular solution then reads yp(t) = X1 ω0e −t/τ1 ∞ X n X2e−t/τ2 ω0 n = X1 ω0− X2e−t/τ2e−t/τ1. (A.13) We use the initial conditions u(0) = 0 and ∂tu(0) = 0 to calculate the arbitrary constants (A, ψ). The solution in the limit [γ/ω0, 1/(ω0τ1/2)]→ 0 reads
u(t) = X1 ω0− X2e−t/τ2e−t/τ1 − ω X1 0− X2 e−γtcos ω0t + X2τ2 2ω0 e −t/τ2 − 1 , (A.14) where we finally recognize Eq (8) of the main text.
Appendix B
Green’s Function of a Squeezed Phonon
Coupled to a Thermal Bath
Here, we evaluate the retarded phonon Green’s function defined by 1 2ω0 h(t)∂t2+ h′(t)∂t+ 2γ∂t+ ω20+ g(t)ω0 DR(t, t′) = δ(t− t′), (B.1) see Eq. (4.34b) of the main text. The Dirac delta function δ(t − t′) is vanishing for t6= t′, hence, for time time t 6= t′ the Green’s function satisfies the homogeneous equation
h(t)∂t2+ h′(t)∂t+ 2γ∂t+ ω02+ g(t)ω0
DR(t, t′) = 0. (B.2) The retarded Green’s function has a causal structure, i.e. it vanishes for t < t′. Since the retarded Green’s function satisfies a second order differential equation, it is defined up to two arbitrary constants. Thus, we need two further equations to compute the arbitrary constants of the Green’s function. The first condition is obtained from the definition of the retarded Green’s function at equal time
DR(t+, t) =−ihhX(tˆ +), ˆX(t)ii = 0, (B.3) with t+ = t + 0+. The second one is obtained from the jump condition imposed by the delta function in Eq. (4.34b), where one can show that the first derivative of the retarded Green’s function must satisfy
∂tDR(t+, t) =−2ω0. (B.4) Thus, the retarded Green’s function is the solution of the homogeneous equation Eq. (B.2), which boundary conditions satisfy the continuity Eq. (B.3) and jump Eq. (B.4) equations.
To solve the homogeneous equation of the retarded Green’s function Eq. (B.2), we replace it by a set of two coupled first order equations and write
∂tDR(t, s) = [ω0 − g(t)] K(t, s), (B.5a) ∂tK(t, s) =− [ω0 + g(t)] DR(t, s)− 2γ
ω0
∂tDR(t, s), (B.5b) where K(t, s) is a function that we introduce as a mathematical tool to solve Eq. (B.2). Notice that for g(t) = 0, the equation of the retarded Green’s function Eq. (B.5) is that of a damped harmonic oscillator. Therefore, we propose the following ansatz for the solution
DR(t, s) = X0(t, s) cos(Ωt)e−γt+ 1 Ω[ω0P0(t, s) + γX0(t, s)] sin(Ωt)e −γt, (B.6a) K(t, s) = P0(t, s) cos(Ωt)e−γt− 1 Ω[ω0X0(t, s) + γP0(t, s)] sin(Ωt)e −γt, (B.6b) where Ω2 = ω2
0−γ2. This form of the solution ensures that for constant X0(t, s) and P0(t, s), the functions DR(t, s) and K(t, s) satisfy the equation of motion of a damped harmonic oscillator. The fact that the phonon is a well defined excitation implies that γ ≪ ω0. Thus, in this limit the equations of motion for X0(t, s) and P0(t, s) read
" ∂tX0(t, s) ∂tP0(t, s) # =−g(t) " − sin (2Ωt) cos (2Ωt) cos (2Ωt) sin (2Ωt) # " X0(t, s) P0(t, s) # . (B.7) We take the limit where the width of the pump-pulse goes to zero τp → 0 and recover the Dirac delta function g(t) = rδ(t). Using the properties of the Dirac distribution, we have that
sin(Ωt)g(t) = sin(0)g(t) = 0, (B.8a) cos(Ωt)g(t) = cos(0)g(t) = g(t), (B.8b) hence Eq. (B.7) further simplifies, and we obtain
" ∂tX0(t, s) ∂tP0(t, s) # =−g(t) " 0 1 1 0 # " X0(t, s) P0(t, s) # . (B.9) The above equation can be solved analytically, and one can show that X0(t) and P0(t) are
given by X0(t, s) = ch[F (t, s)]K1(s)− sh[F (t, s)]K2(s), (B.10a) P0(t, s) = ch[F (t, s)]K2(s)− sh[F (t, s)]K1(s), (B.10b) with F (t, s) = Z t s g(t′)dt′, (B.11) where ch(x) and sh(x) denote the cosine and sine hyperbolic functions, respectively. K1(s) and K2(s) are arbitrary functions independent of time t, that are to be calculated using the boundary conditions, see Eq. (B.3) and Eq. (B.4). Finally, using Eq. (B.6a), the retarded phonon Green’s function in the limit γ ≪ ω0 is given by
DR(t, s) = ch[F (t, s)] K1(s) cos (Ωt) + K2(s) sin (Ωt) e−γt − sh[F (t, s)] K2(s) cos (Ωt) + K1(s) sin (Ωt) e−γt. (B.12) We evaluation the arbitrary functions K1(s), K2(s) using the boundary conditions. For clarity, we define the arbitrary functions A(s) and φ(s) such as
K1(s) =−A(s) sin Ωs + φ(s)eγs, (B.13a) K2(s) = A(s) cos Ωs + φ(s)eγs, (B.13b) and write the propagator in a more convenient form
DR(t, s) = ch[F (t, s)] sin [Ω(t− s) − φ(s)]) − sh[F (t, s)] cos [Ω(t + s) + φ(s)] A(s)e−γ(t−s). (B.14) Using the continuity condition Eq. (B.3) of the retarded Green’s function DR(t, t′) = 0, we get
ch[F (t, t)] sin [−φ(t)] − sh[F (t, t)] cos [2Ωt + φ(t)]
A(t) = 0. (B.15) Following the definition of F (t, s) Eq. (B.11) we have that F (t, t) = 0, hence, using ch(0) = 1 and sh(0) = 0 we obtain
φ(t) = 0. (B.16) The function A(s) can be calculated using the jump condition Eq. (B.4). We evaluate the
derivative of the phonon propagator ∂tDR(t, s) and get ∂tDR(t, s) = g(t) sh[F (t, s)] sin [Ω(t− s)]) − ch[F (t, s)] cos [Ω(t + s)] A(s)e−γ(t−s) + Ω ch[F (t, s)] cos [Ω(t− s)]) + sh[F (t, s)] sin [Ω(t + s)] A(s)e−γ(t−s), (B.17) where we took the limit γ ≪ ω0. We use F (t, t) = 0 and write the derivative of the Green’s function as ∂tDRsq(t+, t) = A(t) Ω− g(t) cos (2Ωt) . (B.18) In the limit g(t) = rδ(t), for time t 6= 0 the function A(t) is given by
A(t) =−2. (B.19) Finally, we replace in Eq. (B.14) and write the phonon Green’s function as
DR(t, s) = θ(t− s)
ch[F (t, s)] sin [Ω(t− s)]) − sh[F (t, s)] cos [Ω(t + s)]
e−γ(t−s), (B.20) where for a delta shaped pump-pulse g(t) = rδ(t), we have
F (t, s) = r Z t
s
δ(t′)dt′ = r sign(t)6= sign(s) (B.21a) F (t, s) = r
Z t s
δ(t′)dt′ = 0 sign(t) = sign(s) (B.21b) with sign(t) the sign function. Thus, the retarded phonon Green’s function is defined piece-wise and reads
DR(t, s) = DR eq(t, s) sign(t) = sign(s), DR sq(t, s) sign(t)6= sign(s), (B.22) with
DReq(t, s) =−2θ(t − s)e−γ(t−s)sin [Ω(t− s)], (B.23a) DRsq(t, s) =−2e−γ(t−s)θ(t− s) ch(r) sin [Ω(t− s)]) − sh(r) cos [Ω(t + s)] , (B.23b) where DR eq(t, s) and DR
sq(t, s) stand for the equilibrium and squeezed retarded Green’s func-tion. The fact that the squeezed propagator is not a function of the the time difference DR
sq(t, s)6= DR
sq(t− s) is a consequence of breaking time translational symmetry. It is inter-120
esting to see that the pump-pulse modifies the retarded Green’s function DR(t, s) only for times sign(t) 6= sign(s). This can be understood from the following intuitive argument. The retarded Green’s function DR(t, s) describes the evolution of the phonon between the time interval s to t. On the other hand, the action of the light on the phonon is instantaneous at time t = 0. Thus, a phonon propagating before t, s < 0 or after t, s > 0 the pump-pulse does not feel the action of the pump.
We now calculate the Wigner transform of the retarded Green’s function defined in Eq. (B.23). The out of equilibrium part of the Wigner transform is defined as
∆DWR(t, ω)≡ DRW(t, ω)− DeqR(ω), = +∞ Z −∞ DR(t + ∆/2, t− ∆/2) − DReq(∆) eiω∆d∆, (B.24) where DR
eq(ω) is the equilibrium retarded Green’s function. Note, in equilibrium the Wigner transform of a two time correlation function coincides with its Fourier transform. The retarded Green’s function DR(t + ∆/2, t − ∆/2) of a squeezed phonon is different from equilibrium only for times where Sign(t + ∆/2) 6= Sign(t − ∆/2). Therefore, the integral in Eq.(B.24) does not vanish only for times ∆ > 2t, where the Wigner can be written as
∆DRW(t, ω) = +∞ Z 2t DRsq(t + ∆/2, t− ∆/2) − DR eq(∆) eiω∆d∆, =−2 ch(r) − 1 +∞ Z 2t
sin(2ω0∆)e−2γteiω∆d∆ + 2 sh(r) cos(2ω0t)
+∞
Z
2t
e−2γteiωt. (B.25) We write the sine function in complex notation sin(2ω0t) = eiω0t−eiω0t
2i , and get ∆DRW(t, ω) = i ch(r)− 1 +∞ Z 2t ei(ω+ω0+iγ) − ei(ω−ω0+iγ)∆ d∆ + 2 sh(r) cos(2ω0t) +∞ Z 2t e−2γteiωt, (B.26) we integrate over time ∆ and find
∆DWR(t, ω) = 1− ch(r)e−2γt e2i(ω+ω0)t ω + ω0+ iγ − e2i(ω−ω 0)∆ ω− ω0+ iγ + 2i sh(r) cos(2ω0t)e−2γt e iωt ω + iγ. (B.27)
The Wigner transform of a squeezed phonon oscillates in time for all frequencies ω.
Appendix C
Calculation of Noise in Temperature
Quench Model
In this appendix, we study the pole structure of the variance of the atomic displacement of a phonon coupled to a thermal bath. In the main text, we showed that Eq. (4.77) the variance of the atomic displacement in frequency space is given by
∆2X(Ω) = γ 2π(Ω/2 + iγ) +∞ Z −∞ A [Ω + 2ω0, Te(τ )]− A [Ω − 2ω0, Te(τ )] eiΩτdτ (C.1) so that A(Ω + 2ω0, Te)≡ +∞ Z −∞ x coth [x]− |x| x2− [(Ω + 2ω0+ i2γ)/4Te]2dx. (C.2) Let us first discuss the different energy scales of the problem involved in pump-probe setups. Typically, the electronic subsystem system is pumped to a high temperature TH ∼ 800K [38], for which we expect TH ≫ γ. On the other hand, the system can be pumped from an arbitrarily low temperature where γ ≪ TL. Therefore, we shall discuss the behavior of the function A(z, Te) in the two regimes γ ≫ Te and γ ≪ Te.
Consider the low temperature regime where we have γ ≫ Te(τ ). The integral defined in Eq. (4.78) is quickly converging, and decays exponentially for x ≫ 1. Therefore in the low temperature limit we have that
Ω/2+iγ±ω0
2Te
approximated by A(Ω + 2ω0, Te)≈ − 4T 2 e (Ω/2 + iγ± ω0)2 +∞ Z −∞ (x coth [x]− x) dx, ≈ −π 2 12 4T2 e (Ω/2 + iγ + ω0)2, (C.3) where we used +∞R −∞ (x coth [x]− 1) dx = π2
6. The variance in frequency space has a poles at Ω =±2ω0− i2γ, therefore it oscillates at frequency 2ω0 in time domain. However since this result is valid for small temperature Te/γ ≪ 1, such oscillations are expected to be small.
We now consider the high temperature regime γ ≫ Te, which is more relevant for pump-probe experiments, and write
A(Ω + 2ω0, Te) = +∞ Z −∞ x coth [x]− |x| x2− [(Ω + 2ω0+ i2γ)/4Te]2dx, (C.4) we split the denominator and get
A(Ω+2ω0, Te) = 2Te(τ ) Ω + 2ω0+ i2γ +∞ Z −∞ x coth [x]− |x| x− (Ω + 2ω0+ i2γ)/4Te− x coth [x]− |x| x + (Ω + 2ω0+ i2γ)/4Te dx. (C.5) We see that the function has a simple pole at Ω ≈ −2ω0 − 2iγ, and hence, we expect the variance to oscillate at 2ω0 in time domain. Since x coth (x) − |x| is a smooth function, we find that the for Ω ≈ −2ω0 the function A(Ω + 2ω0, Te) is asymptotically given by
A(z1/2Te(τ ))≈ 2Te(τ ) Ω + 2ω0+ i2γ +∞ Z −∞ (x coth [x]− |x|) 1 x− iγ/2Te − 1 x + iγ/2Te dx, ≈ Ω + 2ω2Te(τ ) 0+ i2γ +∞ Z −∞ (x coth [x]− |x|)x22i(γ/2T+ (γ/2Te) e)2dx. (C.6) We take the limit γ/Te → 0 and obtain
A(z1/2Te(τ ))≈ 2iπ 2Te(τ ) Ω + 2ω0+ i2γ +∞ Z −∞ (x coth [x]− |x|)δ(x)dx, ≈ 2iπ 2Te(τ ) Ω + 2iγ + 2ω0 , (C.7) 124
where we used lim
x→0x coth(x) = 1. In conclusion, the variance ∆2X(Ω) of the atomic dis-placement in frequency space can be written as
∆2X(Ω) = γ 2π(Ω/2 + iγ) +∞ Z −∞ A [Ω + 2ω0, Te(τ )]− A [Ω − 2ω0, Te(τ )] eiΩτdτ (C.8) where the function A [Ω ± 2ω0, Te(τ )] has a pole at Ω≈ ±2ω0− iγ, which is asymptotically given by A(Ω± 2ω0)≈ 2iπ 2Te(τ ) Ω + 2iγ ± 2ω0 γ ≪ Te, (C.9a) A(Ω± 2ω0)≈ −π 2 12 4T2 e(τ ) (Ω/2 + iγ± ω0)2 γ ≫ Te, (C.9b) with γ and ω0 the phonon damping and frequency, respectively, and Te is the electronic temperature. The poles at Ω = ±2ω0 − i2γ, translates into oscillations at 2ω in time domain.
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