Open Quantum Systems

(1)

Laurent Sanchez-Palencia

Center for Theoretical Physics

Ecole Polytechnique, CNRS, Institut Polytechnique de Paris F-91128 Palaiseau, France

Open Quantum Systems

(2)

Probable Programme

1. Introduction to open quantum systems

1.1 Position of the problem 1.2 Simple approaches

1.3 The density operator (reminder ... or not)

2. Master equation : The statistical physics approach

2.1 Formal derivation of the master equation

2.2 Statistical equilibrium : The damped harmonic oscillator 2.3 Driven systems : The Optical Bloch equations

3. The quantum information approach

3.1 Kraus operators

3.2 Lindblad form of the master equation

3.2 Quantum jumps and stochastic wavefunctions

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Lecture 2 at a Glance

Formal derivation of the quantum master equation

Density operator formalism

Main assumptions :

bath / reservoir / environment (B) system (S)

^ρ_S⊗B(0)=^ρ_S(0)⊗^ρ_B(0)

(i) S⊗B is isolated with

(ii) B weakly affected from the point of view of S, ie S always « sees the same bath »

Elimination of the bath B dynamics

d ^ρ_S dt = 1

i ℏ

[

^H^{^} S(t ), ^ρ_S(t )

]

⁻ ¹

ℏ²

∫

0 ∞

d τ Tr_B

[

^H^{^} ^I^{(t ),}

^[

^H^{^} ^I⁽^{t−τ) , ^ρ}^S⁽^{t )⊗ ^ρ}^B⁽⁰⁾

^] ]

^ρ_S⊗B(t )→ ^ρ_S(t )⊗^ρ_B(t)

(iii) Separation of time scales : (« weak coupling ») → Markov process

Γ,Δ ≪ω_S∼ω_ℓ

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Lecture 2 at a Glance

Application to the damped

harmonic oscillator

bath / reservoir /

environment (B)

∣0 

∣1 

∣2 

d ^ρ_S dt = 1

i ℏ

[

^H^{^} ^S^{' , ^ρ}^S

]

^{+ ℒ}^'^[^{^ρ}^S^]

ℒ'[^{^ρ}S]^{=(Γ+ Γ}^')

(

^{c ^ρ}^{^} ^S^c^{^}^†⁻¹2 ^ρ_Sc^^†c−^ 1

2 c^^†c ^ρ^ _S

)

^+Γ^'

(

^c^{^}^†^{^ρ}^S^c−^{^} ¹2 ^ρ_Sc ^c^ ^†−1

2 c ^c^ ^†^ρ_S

)

with

Physical content Explicit form

accounts for « spontaneous emission » terms Γ

accounts for « stimulated emission and absorption » terms Γ'=⟨ Nℓ(ω_S)⟩_BE

Dynamics

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Probable Programme

1. Introduction to open quantum systems

1.1 Position of the problem 1.2 Simple approaches

1.3 The density operator (reminder ... or not)

2. Master equation : The statistical physics approach

2.1 Formal derivation of the master equation

2.2 Statistical equilibrium : The damped harmonic oscillator 2.3 Driven systems : The Optical Bloch equations

3. The quantum information approach

3.1 Kraus operators

3.2 Lindblad form of the master equation

3.2 Quantum jumps and stochastic wavefunctions

4. Decoherence, entanglement, and control

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Driven systems : The optical Bloch equation

Atom coupled to the radiation field

Elements of quantum electrodynamics

Atom at rest in a radiation field

Atom coupled to the radiation field seen as a driven open quantum system ; Optical Bloch equations ; Solutions in simple cases

Atom moving in a laser field

Emergence of radiative forces ; Radiation pressure and dipole force ; Application to laser control of atomic gases

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Quantum Electrodynamics

Atom-field interaction

Electric dipole approximation :

H^ _R=

∑

ℓ

ℏ ω_ℓ(^{^b}ℓ

†b^_ℓ+1/2)

V^ _AR=− ^⃗D⋅^⃗E_⊥( ⃗R)

Atom-laser interaction (driving)

Electric dipole approximation : V^{^} _AL=−D⋅⃗E^⃗ _L( ⃗R , t )

Rabi frequency :

δ=ω_L−ω_A

Ω_L( ⃗R)=

|

^⃗^d⋅⃗^ℰL( ⃗R )

|

^{/ ℏ}

φ_L( ⃗R )=arg

[

^⃗^d⋅⃗^ℰL( ⃗R)

]

^{+ π}

Phase : Detuning :

The optical Bloch equations (OBE) describe the evolution of the internal degrees of freedom of an atom coupled to the quantized radiation field (spontaneous emission) and to a laser field (classical)

∣e 

∣g  ℏ ω_A

ℏ Γ

ℏ ω_L ℏ δ H = ^^ H_A+ ^H_R+ ^V_AR+ ^V_AL

spont.

emission

May be represented in the vacuum :^{∣ψ }R=∣0 

Responsible for spontaneous emission

H^ _A=ℏ ω_A∣e  〈 e∣

Atom (A) and electromagnetic field (bath B)

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Driven systems : The optical Bloch equation

Atom coupled to the radiation field

Atom at rest in a radiation field

Atom moving in a laser field

(9)

Optical Bloch Equations

d ~ρ_ge

dt = −

(

^Γ2 +i δ

)

^~^ρ^ge^−iΩ^L⁽^ρ^ee^−1/2⁾

d ~ρ_eg

dt = −

(

^Γ2 −i δ

)

^~^ρêg⁺îΩ^L⁽^ρêe^−1/2⁾

d ρ_ee

dt = −Γ ρ_ee+iΩ_L

2 (^~^ρeg−~ρ_ge)^mmmi

The quantum master equation is derived as for the damped harmonic oscillator (see lecture 2), up to the substitution

ρ_gg=1−ρ_ee

~ρ_ge≡ρ_geexp [i(φ_L−ω_Lt)]1 2 c→ ^σ^ _-=∣g ⟩ 〈 e∣

c^^†→ ^σ₊=∣e ⟩ 〈 g∣

N.B. : In the derivation of the QME, we have never used any commutation rule of and , so we need not care about the fact that they are not preserved under the substituion above.

c^^† c^

It yields , ie^{d ^ρ}

dt = 1

i ℏ

[

^H^{^} A+ ^V_AL(t), ^ρ

]

^+Γ

(

^σ^{^}^- ^{^ρ ^}^σ⁺⁻¹²^{ρ ^}^{^}^σ⁺^σ^{^}^-⁻¹² ^σ^{^}⁺ ^{^σ}^- ^{^ρ}

)

with , and is included in the

definition of . Here .

Δ

ω_A δ=ω_L−ω_A

provided (hierarchy of time scales).Ω_L, Γ , δ≪ω_A∼ω_L

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One finds a relaxation of both the populations and the coherences (population transfer from to ) :

d ~ρ_ge

dt = − Γ

2 ~ρ_ge d ~ρ_eg

dt = − Γ

2 ~ρ_eg d ρ_ee

dt = −Γ ρ_eei

~ρ_ge(t) = e^−Γ^{t /2}~ρ_ge(0) ρ_ee(t ) = e^−Γ^tρ_ee(0) ρ (t) = 1−e^−Γ^tρ (0)

Non driven case

In the absence of a laser, Ω_L=0, the OBE reduce to

Here, we have (Ω_L=0 et δ=0)

Solutions of the Optical Bloch Equations

∣e 

∣g 

Γ

∣e 

∣g 

~ρ_ge≡ρ_geexp (−i ω_At)1 2

~ρ_eg≡ρ_egexp(+i ω_At)1 2

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Solutions of the Optical Bloch Equations

One find a stationary term ( ) and two oscillating terms at the angular frequencies

d ~ρ_ge

dt = −i δ~ρ_ge−i Ω_L(^ρee−1/2)

d ~ρ_eg

dt = +i δ~ρ_eg+iΩ_L(^ρee−1/2)

d ρ_ee

dt = +iΩ_L

2 (^~^ρeg−~ρ_ge)^mmmm ω₀=0

ω=

√

^Ω²^L^+δ²

For an atom initially in the ground state , and , one finds ρ_gg(0)=1

∣g 

ρ_ee(0)=ρ_ge(0)=0

ρ_ee(t)= Ω_L²

Ω_L² +δ² sin²

( √

^Ω^L²^{+ δ}²^{t /2}

⁾

⇒ Rabi oscillations (for )t≾1/ Γ

Case without dissipation

In the case where the spontaneous emission can be neglected, Γ=0, the OBE reduce to

(12)

Solutions of the Optical Bloch Equations

General case : Resolution

To solve the OBE, it is worth using the variables

u(t) = Re(~ρ_ge) = (~ρ_ge+~ρ_eg)/2 v (t) = Im(~ρ_eg) = (~ρ_ge−~ρ_eg)/2 i w (t) = ρ_ee−1/2

The OBE can then be cast into the matrix form d

dt

(

^w^u^v

)

⁼⁻

( _⏟

^{Γ /}^{+ δ}⁰² ^−Ω^{Γ /}^−δ^{2 +Ω}L Γ⁰ ^L

)

M

(

^w^u^v

)

⁻

(

^{Γ /}⁰⁰²

)

The solution read as

In the general case, for an atom at rest, one finds damped oscillations

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Solutions of the Optical Bloch Equations

ρ_gg^st = 2+s

2(1+s) ρ_ee^st= s 2 (1+s)

~ρ_ge^st =δ+i Γ /2 Ω_L

s

1+s ~ρ_eg^st =δ−i Γ /2 Ω_L

s 1+s s= Ω_L²/2

δ²+Γ²/4

General case : Stationary solutions

The stationary solutions are found by inverting the matrix M :

w_st=−1 2

1 u_st= δ 1+s

Ω_L s

1+s v_st=Γ /2 Ω_L

s 1+s

where is the staturation parameter

No population inversion, ρ_gg^st ⩾ρ_ee^st 1 2

In the large s limit, the stationary populations saturate to ρ_gg^st =ρ_ee^st≃1/2 1 2

The coherences are non monotonous and slowly vanish for large s

Main properties

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Solutions of the Optical Bloch Equations

ω / Γ

⇒ All the components of the density

γ₀/ Γ γ₁/ Γ γ₂/ Γ

Ω_L/ Γ

δ/ Γ δ/ Γ Ω_L/ Γ δ/ Γ Ω_L/ Γ

General case : Relaxation towards the stationary solutions

The dynamics towards the stationary state is given by the eigenvalues of M. There are two possible cases :

(i) The three eigenvalues are real :

(ii) One is real and the other two are complex conjugates :

u(t)=u_st+A₀e^−γ⁰^t+A₁e^−γ¹^t+A₂e^{− γ}²^t

u(t)=u_st+A₀e^−γ⁰^t+A₁e^−γ¹^tcos(ωt +ϕ)

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Driven systems : The optical Bloch equation

Atom coupled to the radiation field

Atom at rest in a radiation field

Atom moving in a laser field

(16)

Application : Radiative Forces

Dynamics of an atom in a laser field

λ_L=2 π/ k_L

λ_dB=

√

^{2 π ℏ}²^/^{m k}^B^T

V⃗

Hypotheses : (i) atom localized on a short size (classical motion),

(ii) internal state adiabatically follows the local laser field, λ_dB≪ λ_L

(p±Δ p) Γ⁻¹/m ≪λ_L

(17)

Application : Radiative Forces

Dynamics of an atom in a laser field

Within a semi-classical approach, if the atom is sufficiently slow, its internal states adiabatically adapts to the intensity and the phase of the laser locally, it is subjected to the force

⃗F ( ⃗R )=−ℏ u

⏟

_st∇ Ω⃗ _L

⃗Fdip

+ ℏ

⏟

v_stΩ_L∇ φ⃗ _L

F⃗PR

with ^ΩL( ⃗R)=

|

^⃗^d⋅⃗^ℰL( ⃗R )

|

^{/ ℏ}

φ_L( ⃗R )=arg

[

^⃗^d⋅⃗^ℰL(⃗R)

]

^{+ π}

The dissipative part is significant for and if there is phase gradient, . This is the radiation pressure.

⃗ Γ≿δ F_PR

∇ φ⃗ _L≠0

The reactive part dominates for and if there is an intensity gradient, . It is the dipole force.

Γ ≪ δ

⃗F_dip

∇ Ω⃗ _L≠0

⇒ Basic tools for cooling and manipulating neutral atoms using lasers !

u_st= δ Ω_L

s

1+s v_st=Γ /2 Ω_L

s

, and 1+s

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Radiation Pressure

Radiation pression : Dissipative part

The general form of the dissipative force is

⃗F_PR( ⃗R )=ℏ Γ s (⃗R)/2 1+s( ⃗R)

⏟

Π_e=ρ_ee^st

∇ φ⃗ _L( ⃗R)

For a laser beam of wave vector (plane wave), one finds ^⃗k_L

⃗E (⃗r , t )=ℰ⃗ _L

2 exp

[

ⁱ

⁽

^⃗^k^L^⋅⃗^R−ω^L^t

⁾ ]

⁺^c.c

φ_L( ⃗R )=⃗k⋅⃗R +π

so

⃗F_PR=Γ Π_e ℏ ⃗k_L

and

average spontaneous Recoil taken by the

∣e 

∣g 

spontaneous photon ℏ ⃗k_L

laser photon

⃗P=⃗P₀+ ℏ ⃗k_L

⃗P₀ ℏ ⃗k_L laser photon

stimulated emissionsion

δ ⃗P=0

δ ⃗P=ℏ ⃗k

(19)

Radiation Pressure

Radiation pressure and the tails of cometes

Dust tail

Yellow and wide

Pushed by the radiation pressure Curved due to dragging

Ion tail (plasma)

Blue and narrow

Created and pushed by the solar wind (also a plasma)

(20)

Radiation Pressure

Manipulation of atoms by lasers

Laser cooling by pairs of counter-propagating beams Zeeman slower

ℏ δ

∣e  ℏ δ'_left

ℏ δ'_right

∣e 

∣g 

(21)

Dipole Force

Dipole force : Reactive part

The dipole force is conservative, , and derives from the potential

V_dip(⃗R)=ℏ δ

2 ln

[

^{1+s( ⃗}^R)

]

For large detuning and weak intensity, (weak radiation pressure and weak saturation),

⃗F_dip=− ⃗∇V _dip

|δ|≫ Γ , Ω_L

V_dip(⃗R)≃ℏ Ω_L² 4 δ

∣e  ℏ δ<0

∣g 

ℏ δ>0

∣e 

∣g 

Repulsive potential

(δ>0 ; « blue detuning »)

→ barrier

Attractive potential

(δ<0 ; « red detuning »)

→ optical trap

(22)

Dipole Force

Controlled trapping potentials for quantum simulation

Periodic potentials and optical lattices Holographic traps

Disordered potentials

(23)

Lecture 3 Quantum information approach

mercredi 11 décembre 2019 21:15

Open Quantum Systems

Laurent Sanchez-Palencia

Open Quantum Systems

Probable Programme

1. Introduction to open quantum systems

2. Master equation : The statistical physics approach

3. The quantum information approach

Lecture 2 at a Glance

Formal derivation of the quantum master equation

[

]

∫

[

[

] ]

Lecture 2 at a Glance

Application to the damped

harmonic oscillator

[

]

(

)

(

)

Probable Programme

1. Introduction to open quantum systems

2. Master equation : The statistical physics approach

3. The quantum information approach

4. Decoherence, entanglement, and control

Driven systems : The optical Bloch equation

Atom coupled to the radiation field

Atom at rest in a radiation field

Atom moving in a laser field

Quantum Electrodynamics

∑

|

|

[

]

The optical Bloch equations (OBE) describe the evolution of the internal degrees of freedom of an atom coupled to the quantized radiation field (spontaneous emission) and to a laser field (classical)

Driven systems : The optical Bloch equation

Atom coupled to the radiation field

Atom at rest in a radiation field

Atom moving in a laser field

Optical Bloch Equations

(

)

(

)

[

]

(

)

Non driven case

Solutions of the Optical Bloch Equations

Solutions of the Optical Bloch Equations

√

( √

)

Case without dissipation

Solutions of the Optical Bloch Equations

General case : Resolution

(

)

( ⏟

)

(

)

(

)

Solutions of the Optical Bloch Equations

General case : Stationary solutions

Solutions of the Optical Bloch Equations

General case : Relaxation towards the stationary solutions

Driven systems : The optical Bloch equation

Atom coupled to the radiation field

Atom at rest in a radiation field

Atom moving in a laser field

Application : Radiative Forces

Dynamics of an atom in a laser field

^[

^] ]

⁾

( _⏟

⁽

⁾ ]