How to considera time evolvingsystemunder a perturbation

Hierarchy of representations:

Shroedinger 𝜓(𝑡) Evolution of a pure state

Liouville – Von Neumann [𝐻, 𝜌] Evolution of a statistic ensemble Liouville 𝐿 , Γ Evolution including damping

How to consider a time evolving system under a perturbation

𝑖ℏ 𝑑|𝜓

(𝑡) >

𝑑𝑡 = 𝐻|𝜓

(𝑡) >

𝜓 r, t = φ 𝑟 𝑒

ω =

; 𝐻𝜑 = 𝐸𝜑

N.B.: As any wave, 𝜓 is a function of position and time. Time dependence is an oscillating constant term that does not change the probability 𝜓². Frequency of oscillation 𝜔 results from the solution of the stationary Shroedinger equation.

We can define a new operator to cancel out the time dependence on 𝜓 and let the function propagates free over time:

𝜓 t = 𝑒

𝜓 𝑡

𝑈 𝑡, 𝑡 _F = 𝑒 ^9:C

^(<9<

^)/ℏ

Free evolution operator

or

propagator

The system doesn’t change under the static hamiltonian H₀, it only translates in time.

Properties:

• Hermitian operator

• Composition property:

𝑈 𝑡, 𝑡

= 𝑈 𝑡, 𝑡

𝑈 𝑡

,𝑡

𝑡

→ 𝑡

→ 𝑡

• Time reversibility:

𝑈

𝑡

,𝑡 = 𝑈(𝑡, 𝑡

)

< 𝐴L 𝑡 > = < 𝜓 𝑡 𝐴L𝑠 𝜓 𝑡 >

< 𝐴L 𝑡 > = < 𝜓 0 𝑈^O𝐴L𝑠 𝑈 𝜓 0 >

< 𝐴L 𝑡 > = < 𝜓 0 𝐴L_𝐻 (𝑡) 𝜓 0 >

Shroedinger

Heisenberg: time dependent operator

What if H is time dependent:

𝑈 𝑡, 𝑡_F = 𝑒𝑥𝑝 −𝑖

ℏS 𝐻T 𝑡^U 𝑑𝑡′

<

<_D

𝑒^9W = 1 − 𝑥 + 1

2!𝑥^\ − 1

3!𝑥^{^} + ⋯

𝑈 𝑡, 𝑡_F

= 1 − 𝑖

ℏ S 𝐻T 𝜏 𝑑𝜏^<

<_D

+ − 𝑖 ℏ

\

S 𝑑𝜏 S 𝑑𝜏′𝐻T 𝜏 𝐻T 𝜏′^a

<_D

<

<_D

+ − 𝑖 ℏ

^

S 𝑑𝜏 S 𝑑𝜏′ S 𝑑𝜏^{aUU UU}𝐻T 𝜏 𝐻T 𝜏′ 𝐻T 𝜏′′ + ⋯

<_D a

<_D

<

<_D

𝑡 > 𝜏 > 𝜏

> 𝜏

> ⋯ > 𝑡

factorial terms omitted because of time ordering permutating terms not possible

Interaction with EM field: to describe an evolving system under a perturbation we need time evolution on both𝜓 𝑎𝑛𝑑 𝐻:

Perturbation theory at 1st order

𝐻 𝑡 = 𝐻

+ 𝑉(𝑡) 𝜓(𝑡)

A small perturbation allows for conservation of the basis set |n>

of the unperturbed hamiltonian

to divide the stationary part of hamiltonian from following EM perturbations on the system, we define 𝜓_I :

INTERACTION PICTURE:

If we apply the free-propagator, we can rewrite 𝜓_S so that we are solidal to its constant oscillating part.

|𝜓

𝑡 > = 𝑈

𝑡, 𝑡

|𝜓

(𝑡

) >

𝑒

𝑒

Time dependence is only due to the perturbating part V(t)

Time evolution in the interaction picture:

𝑖ℏ𝑑|𝜓₁ >

𝑑𝑡 = 𝐻(𝑡)|𝜓₁ >

𝑖ℏ 𝑑

𝑑𝑡 𝑈_F(𝑡, 𝑡_F)|𝜓_f > = 𝐻_F+ 𝑉(𝑡) 𝑈_F(𝑡, 𝑡_F)|𝜓_f >

𝑑𝑈_F

𝑑𝑡 |𝜓_f > +𝑑|𝜓_f >

𝑑𝑡 𝑈_F = −𝑖

ℏ 𝐻_F+ 𝑉(𝑡) 𝑈_F|𝜓_f >

𝑈_F 𝑡, 𝑡_F = 𝑒^{9:CD(<9<D)/ℏ}

− 𝑖

ℏ𝐻_F𝑈_F|𝜓_f > +𝑑|𝜓_f >

𝑑𝑡 𝑈_F = − 𝑖

ℏ 𝐻_F+ 𝑉(𝑡) 𝑈_F|𝜓_f >

𝑑|𝜓_f >

𝑑𝑡 = − 𝑖

ℏ𝑈_F^O𝑉(𝑡)𝑈_F|𝜓_f >

𝑖ℏ𝑑|𝜓_f >

𝑑𝑡 = 𝑉𝐼(𝑡) |𝜓_f >

Formally indentical

to shroedinger time

dependent equation

𝑈 𝑡, 𝑡_F = 𝑈₀ 𝑡, 𝑡_F + h − 𝑖 ℏ

j i ikH

S 𝑑𝜏^< _i

<_D

S 𝑑𝜏_i9H… S 𝑑𝜏^a\ _H𝑈_F 𝑡,𝜏_i 𝑉 𝜏_i 𝑈_F 𝜏_i𝜏_i9H 𝑉 𝜏_i9H …

<_D a_i

<_D

V(t)

H₀

… 𝑈_F 𝜏_\, 𝜏_H 𝑉(𝜏_H)𝑈_F(𝜏_H,𝑡_F)

𝑡

→ 𝜏

→ 𝜏

→ ⋯ → 𝜏

→ 𝜏

→ 𝑡

V

= EM field interactions

That evolution operator can be applied on a pure state 𝜓 as well as on a statistics ensemble 𝜌.

During the free-evolution propagator U

the system rearranges and relaxes.

𝝍:

𝝆:

|𝜓 𝑡 >= |𝜓 𝑡₀ > + h − 𝑖 ℏ

j i ikH

S 𝑑𝜏^< _i

<_D

S 𝑑𝜏_i9H… S 𝑑𝜏^a\ _H𝑈_F 𝑡,𝜏_i 𝑉 𝜏_i 𝑈_F 𝜏_i𝜏_i9H 𝑉 𝜏_i9H …

<_D a_i

<_D

… 𝑈_F 𝜏_\,𝜏_H 𝑉(𝜏_H)𝑈_F(𝜏_H,𝑡_F)|𝜓 𝑡₀ >

𝜓(𝑡) >< 𝜓(𝑡) = 𝑈_F(𝑡, 𝑡_F) 𝜓_𝐼(𝑡) >< 𝜓_𝐼(𝑡) 𝑈 ^†_F (𝑡,𝑡_F) 𝜌(𝑡) = 𝑈_F(𝑡, 𝑡_F) q 𝜌_𝐼(𝑡) q 𝑈^†_F (𝑡, 𝑡_F)

𝜕𝜌s_𝐼

𝜕𝑡 = − 𝑖

ℏ[𝐻t,𝜌_𝐼 u ]_𝐼

𝜌 𝑡 = 𝜌(𝑡_F) + h −𝑖 ℏ

j i ikH

S 𝑑𝜏^< _i

<_D

S 𝑑𝜏_i9H… S 𝑑𝜏^a\ _H𝑈_F 𝑡, 𝑡_F q 𝑉_𝐼 𝜏_i ,… 𝑉_𝐼 𝜏_H ,𝜌(𝑡_F) q 𝑈_F^O(𝑡, 𝑡_F)

<_D a_i

<_D

𝑉

= 𝜇

q 𝐸

𝑃 𝑡 = 𝑇𝑟 𝜇𝜌

Infinite time evolution: t0 = -∞ Σ ⟶ ∫

𝑃

𝑡 = − 𝑖 ℏ

i

S 𝑑𝜏

S 𝑑𝜏

… S 𝑑𝜏

𝐸(𝜏

)𝐸(𝜏

) …

a_} 9j a_~

9j

<

9j

𝑇𝑟 𝜇

𝜌

0 t

t

t

𝜏

𝜏

𝜏

−∞ t

The non-linear response function S⁽ⁿ⁾ is the convolution of N electric fields with the non-liner response R⁽ⁿ⁾ of the system:

𝑆

(𝑡) = 𝐸

𝐸

… 𝐸

⨂ 𝑅

(𝑡) 𝑅

(𝑡) = 𝜇

𝜇

, … 𝜇

,𝜌(−∞)

… 𝐸(𝜏

) 𝜇

𝜇(𝜏

), … 𝜇(𝜏

), 𝜌(−∞)

Notice: 𝜇 is still in the interaction picture, It contains free-evolutions

Time domain

Frequency domain

𝑃‚ 𝑡 = 𝑃

𝑡 + 𝑃

𝑡 + 𝑃

𝑡 + 𝑃

𝑡 + ⋯

𝑃‚ 𝜔 = 𝜒𝐸 + 𝜒

𝐸𝐸 + 𝜒

𝐸𝐸𝐸 + ⋯

before perturbation

Take notice:

𝑨𝒏𝒂𝒍𝒚𝒛𝒊𝒏𝒈 𝒕𝒉𝒆 𝒔𝒚𝒔𝒕𝒆𝒎 𝒓𝒆𝒔𝒑𝒐𝒏𝒔𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝑹

(𝒕):

𝑅

(𝑡) = 𝜇

𝜇

, … 𝜇

,𝜌(−∞)

t =

𝑅

(𝑡) = 𝜇

𝜇

, 𝜌

= 𝜇

𝜇

𝜌

− 𝜇

𝜌

𝜇

= 𝜇

𝜇

𝜌

− 𝜌

𝜇

𝜇

= 𝜇

𝜇

𝜌

− 𝜇

𝜇

𝜌

= 𝑅

− 𝑅

𝑅

− 𝑅

= 𝜇

𝜇

𝜌

− 𝜇

𝜇

𝜌

𝑭𝒆𝒚𝒏𝒎𝒂𝒏 diagrams _:

|𝜓 > < 𝜓|

ket bra

• Two linear terms in the response function

• One operates on the bra and one on the ket

• Both define the same process

Only the process with the emission from the ket is considered !!

𝑇𝑟 𝜇_<𝜇_H|𝜓 >< 𝜓|

t₁

E

t₀ t

• Time evolution is represented by vertical arrows from bottom to top

• side arrows represent interactions with the electric field

• The last arrow rises from the system and represents the signal: it restores a population state

• Between interactions there is free evolution of the system described by super-operator 𝐺¨ 𝑡 = 𝑈𝐴𝑈^O

𝐸

= 𝑒

Phase matching condition Resonance condition

|0 >< 0|

|1 >< 0|

|0 >< 0|

<bra|

|ket>

E

E*

absorption stim.emission

E*

E

𝐸

= 𝑒

𝐸

= 𝑒

+𝑘, +𝜔

−𝑘, −𝜔

Third order term:

𝑻𝒉𝒆 𝒑𝒖𝒎𝒑 − 𝒑𝒓𝒐𝒃𝒆

𝐹𝑒𝑦𝑛𝑚𝑎𝑛 𝑎𝑛𝑑 𝑙𝑎𝑑𝑑𝑒𝑟 diagrams _:

𝑅

(𝑡) = 𝜇

𝜇

, 𝜇

, 𝜇

,𝜌

+ 𝜇

𝜇

𝜇

𝜇

𝜌

⇒ 𝑅

+ 𝜇

𝜇

𝜌

𝜇

𝜇

⇒ 𝑅

+ 𝜇

𝜇

𝜌

𝜇

𝜇

⇒ 𝑅

+ 𝜇

𝜇

𝜌

𝜇

𝜇

⇒ 𝑅

- 𝜇

𝜇

𝜇

𝜇

𝜌

⇒ 𝑅

- 𝜇

𝜇

𝜌

𝜇

𝜇

⇒ 𝑅

- 𝜇

𝜇

𝜌

𝜇

𝜇

⇒ 𝑅

- 𝜇

𝜇

𝜌

𝜇

𝜇

⇒ 𝑅

2

𝑡𝑒𝑟𝑚𝑠 and 2

𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑡𝑒𝑟𝑚𝑠

|0 > < 0|

|1 >< 0|

|0 >< 0|

|1 >< 0|

|0 >< 0|

E₁ E₂

E₃

𝑘

= 𝑘

− 𝑘

+ 𝑘

E_sig

𝜔

= 𝜔

− 𝜔

+ 𝜔

E₁ E₂ E₃ E_sig

t

|0 >

|1 >