Spin-Boson ModelA simple Open Quantum System

M. Miller F. Tschirsich

Quantum Mechanics on Macroscopic ScalesTheory of Condensed Matter

July 2012

Outline

1 Bloch-Equations

2 Classical Dissipations

3 Spin-Boson Modell

4 Summary

2 / 45

Bloch-Equations

Outline

1 Bloch-Equations

2 Classical Dissipations

3 Spin-Boson Modell

4 Summary

3 / 45

Bloch-Equations

Open Quantum System

No ideal separation: system! environment

Environment: heat bath, statistics, random

Decoherence: quantum interference → classical mixture

Dissipation: energy transfer to environment

4 / 45

Bloch-Equations

Open Quantum System

Rewiev: Density matrix formalism

ExamplePure

|−〉= |0〉− |1〉

ρpure = |−〉〈−|= 12

(1 −1−1 1

)ExampleMixed

|0〉 or |1〉

ρmixed =12

(1 00 1

)

5 / 45

Bloch-Equations

Spin Precession

Two level System: Spin-1/2

External Magnetic field: B≡ (0,0,Bz ) + fluctuations δB(t)

Semiclassical: no interplay spin ←9 field

Fluctuations: irreversible dynamics, Decoherence

6 / 45

Bloch-Equations

Spin precession

Hamiltonian (no fluctuations)

H0 =−Bx

2σx

Classical Magnetization M

M(t) ∝ S(t) := 〈σ〉σ = (σx ,σy ,σz ) spin-vector (Pauli operators).

7 / 45

Bloch-Equations

Spin precession

Rotating Frame:

σ (t) = e i H0t σe−i H0t

Heisenberg:

dσ (t)

dt=−i

[H0, σ (t)

]=−B× σ (t)

Ehrenfest-Theorem

M(t) = 〈↑x |σ (t) | ↑x 〉= M(0)cos(Bx t)

Rabi oscillations: reversible, unitary evolution8 / 45

Bloch-Equations

Fluctuations

Hamiltonian (with noise)

H =−Bx

2σx −

δB(t)

2· σ

random noise |δB(t)| � Bx

Density matrix ↔ Bloch-sphere

ρ (t) =12

(I+S(t) ·σ)

9 / 45

Bloch-Equations

Fluctuations

Hamiltonian (with noise)

H =−Bx

2σx −

δB(t)

2· σ

random noise |δB(t)| � Bx

Density matrix ↔ Bloch-sphere

ρ (t) =12

(I+S(t) ·σ)

9 / 45

Bloch-Equations

Fluctuations

Rotating frame (noise dynamics only):

ρ′ (t) = e i H0tρ (t)e−i H0t

Von Neumann eq.:

dρ ′

dt= −i

[H ′ (t) ,ρ ′ (t)

](1)

Integrate, 2nd order recursion:

ρ′ (t) = ρ

′ (0)− i∫ t

0ds[H ′ (s) ,ρ ′ (s)

]dρ ′

dt(1)=−i

[H ′ (t) ,ρ ′ (0)

]−∫ t

0ds[H ′ (t) ,

[H ′ (s) ,ρ ′ (s)

]]⟨

dρ ′

dt

⟩δB≈−

∫ t

0ds⟨[

H ′ (t) ,[H ′ (s) ,ρ ′ (t)

]]⟩δB

10 / 45

Bloch-Equations

Fluctuations

Rotating frame (noise dynamics only):

ρ′ (t) = e i H0tρ (t)e−i H0t

Von Neumann eq.:

dρ ′

dt= −i

[H ′ (t) ,ρ ′ (t)

](1)

Integrate, 2nd order recursion:

ρ′ (t) = ρ

′ (0)− i∫ t

0ds[H ′ (s) ,ρ ′ (s)

]dρ ′

dt(1)=−i

[H ′ (t) ,ρ ′ (0)

]−∫ t

0ds[H ′ (t) ,

[H ′ (s) ,ρ ′ (s)

]]⟨

dρ ′

dt

⟩δB≈−

∫ t

0ds⟨[

H ′ (t) ,[H ′ (s) ,ρ ′ (t)

]]⟩δB

10 / 45

Bloch-Equations

Fluctuations

Rotating frame (noise dynamics only):

ρ′ (t) = e i H0tρ (t)e−i H0t

Von Neumann eq.:

dρ ′

dt= −i

[H ′ (t) ,ρ ′ (t)

](1)

Integrate, 2nd order recursion:

ρ′ (t) = ρ

′ (0)− i∫ t

0ds[H ′ (s) ,ρ ′ (s)

]dρ ′

dt(1)=−i

[H ′ (t) ,ρ ′ (0)

]−∫ t

0ds[H ′ (t) ,

[H ′ (s) ,ρ ′ (s)

]]⟨

dρ ′

dt

⟩δB≈−

∫ t

0ds⟨[

H ′ (t) ,[H ′ (s) ,ρ ′ (t)

]]⟩δB

10 / 45

Bloch-Equations

Fluctuations

Rotating frame (noise dynamics only):

ρ′ (t) = e i H0tρ (t)e−i H0t

Von Neumann eq.:

dρ ′

dt= −i

[H ′ (t) ,ρ ′ (t)

](1)

Integrate, 2nd order recursion:

ρ′ (t) = ρ

′ (0)− i∫ t

0ds[H ′ (s) ,ρ ′ (s)

]dρ ′

dt(1)=−i

[H ′ (t) ,ρ ′ (0)

]−∫ t

0ds[H ′ (t) ,

[H ′ (s) ,ρ ′ (s)

]]⟨

dρ ′

dt

⟩δB≈−

∫ t

0ds⟨[

H ′ (t) ,[H ′ (s) ,ρ ′ (t)

]]⟩δB

10 / 45

Bloch-Equations

Longitudinal Noise

Hamiltonian (longitudinal noise)

H ′ (t) =δBx

2σx

δB(t) = (δBx (t) ,0,0) along precession axis noisy Rabi frequency⟨dρ ′

dt

⟩δB≈−

∫ t

0ds⟨

δBx (t)δBx (s)

4

[σx ,[σx ,ρ

′ ( 6 s → t)]]⟩

δB

=−Γ∗2

(0 ρ ′01 (t)

ρ ′10 (t) 0

)Environment correlation time: � System timescale

〈δBx (t)δBx (s)〉δBx

peaked around s = t

Exponential dephasing: ρ ′01 (t) = ρ ′01 (0)eΓ∗2t

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Bloch-Equations

Longitudinal Noise

Hamiltonian (longitudinal noise)

H ′ (t) =δBx

2σx

δB(t) = (δBx (t) ,0,0) along precession axis noisy Rabi frequency⟨dρ ′

dt

⟩δB≈−

∫ t

0ds⟨

δBx (t)δBx (s)

4

[σx ,[σx ,ρ

′ ( 6 s → t)]]⟩

δB

=−Γ∗2

(0 ρ ′01 (t)

ρ ′10 (t) 0

)Environment correlation time: � System timescale

〈δBx (t)δBx (s)〉δBx

peaked around s = t

Exponential dephasing: ρ ′01 (t) = ρ ′01 (0)eΓ∗2t

11 / 45

Bloch-Equations

Longitudinal Noise

Hamiltonian (longitudinal noise)

H ′ (t) =δBx

2σx

δB(t) = (δBx (t) ,0,0) along precession axis noisy Rabi frequency⟨dρ ′

dt

⟩δB≈−

∫ t

0ds⟨

δBx (t)δBx (s)

4

[σx ,[σx ,ρ

′ ( 6 s → t)]]⟩

δB

=−Γ∗2

(0 ρ ′01 (t)

ρ ′10 (t) 0

)Environment correlation time: � System timescale

〈δBx (t)δBx (s)〉δBx

peaked around s = t

Exponential dephasing: ρ ′01 (t) = ρ ′01 (0)eΓ∗2t

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Bloch-Equations

Longitudinal Noise

Animation (longitudinal noise)

Averaging over random realizations of longitudinal δB(t)

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Bloch-Equations

Transversal Noise

Hamiltonian (transversal noise)

H ′ (t) =δBz

2σz (t)

δB(t) = (0,0,δBz ) perpendicular; σz (t) = e−Bx σy t rotating frame:⟨dρ ′

dt

⟩δB≈−

∫ t

0ds⟨

δBz (t)δBz (s)

4

[σz (t) ,

[σz (s) ,ρ ′ (t)

]]⟩δB

RWA≈ Γ1

2

(ρ ′00 (t)−ρ ′11 (t) ρ ′01 (t)

ρ ′10 (t) ρ ′11 (t)−ρ ′00 (t)

)where

Γ1 =12

∫ t

−tds eiBx s 〈δBz (0)δBz (s)〉

δBz

Exponential dephasing rate Γ12 + state gets mixed:

ρ′00 (t)−ρ

′11 (t) =

(ρ′00 (0)−ρ

′11 (0)

)e−Γ1t

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Bloch-Equations

Transversal Noise

Hamiltonian (transversal noise)

H ′ (t) =δBz

2σz (t)

δB(t) = (0,0,δBz ) perpendicular; σz (t) = e−Bx σy t rotating frame:⟨dρ ′

dt

⟩δB≈−

∫ t

0ds⟨

δBz (t)δBz (s)

4

[σz (t) ,

[σz (s) ,ρ ′ (t)

]]⟩δB

RWA≈ Γ1

2

(ρ ′00 (t)−ρ ′11 (t) ρ ′01 (t)

ρ ′10 (t) ρ ′11 (t)−ρ ′00 (t)

)where

Γ1 =12

∫ t

−tds eiBx s 〈δBz (0)δBz (s)〉

δBz

Exponential dephasing rate Γ12 + state gets mixed:

ρ′00 (t)−ρ

′11 (t) =

(ρ′00 (0)−ρ

′11 (0)

)e−Γ1t

13 / 45

Bloch-Equations

Transversal Noise

Animation (transversal noise)

Averaging over random realizations of transversal δB(t)

14 / 45

Bloch-Equations

Bloch Equations

Summary:Rabi oscillations + fluctuations in Bloch-sphere representation

dSdt

=−B×S︸ ︷︷ ︸coherent

−Γ∗2Szz︸ ︷︷ ︸mixing

−(

Γ∗2 +Γ1

2

)(Sxx+Syy)︸ ︷︷ ︸

dephasing

Bloch (1964): Phenomenological description of dissipation in NMR

No prediction of the rates Γ

15 / 45

Classical Dissipations

Outline

1 Bloch-Equations

2 Classical Dissipations

3 Spin-Boson Modell

4 Summary

16 / 45

Classical Dissipations

Langevin equation

Include environment in our model

Classical Hamiltonian → Langevin equations

17 / 45

Classical Dissipations

Langevin equation

Hamiltonian

H =p2

2M+V (q)︸ ︷︷ ︸

System

+12

N

∑k=1

p2k

mk+mkω

2k(xk − x equ

k (q))2︸ ︷︷ ︸

coupling

V(q)

System: Particle (p,q) in Potential V (q)Environment: Bath of harmonic oscillators (pk ,xk)Weak Interaction: linear coupling from equilibrium positions

x equk (q) =

ck

mkω2kq

Interaction potential VI = q ∑k xkλk , weighted modes (λk)18 / 45

Classical Dissipations

Langevin Equation

Coupled System - Environment Dynamics:

Mq +∑k

(λ

2k /mkω

2k)q +Vq (q) = ∑

kλkxk

mk xk +mkω2k xk = λkq

(q = ∂H∂p p =− ∂H

∂q ; xk = ∂H∂pk

pk =− ∂H∂qk

)

Solve inhomogenous system

Eliminate environment by resubstitution

Initial conditions: Thermal equilibrium

19 / 45

Classical Dissipations

Langevin Equation

Langevin equation

Mq (t) +Mt∫

0

ds γ (t− s) q (s) +Vq (q) = ξ (t)

Memory friction kernel

γ (t) = Θ(t)κ (t)

κ (t) =1M ∑

k

λ 2k

mkω2kcos(ωkt)

ξ (t): fluctuating force

〈ξ (t)〉= 0, 〈ξ (t)ξ (0)〉= MkBTκ (t)

20 / 45

Classical Dissipations

Langevin Equation

Summary:

Classical fluctuations: Dissipation (no decoherence).

Correlators ∝ Temperature, mode coupling λk

Now: do it quantum mechanically!

21 / 45

Spin-Boson Modell

Outline

1 Bloch-Equations

2 Classical Dissipations

3 Spin-Boson Modell

4 Summary

22 / 45

Spin-Boson Modell

A review of the spin model

Any two state system (TSS) can be modeled by the spin formalism.

eE(t)

g

23 / 45

Spin-Boson Modell

A review of the spin model

Unperturbed TSS

H =

(E1 00 E2

)eigenv. { |g〉, |e〉}

e

g

24 / 45

Spin-Boson Modell

A review of the spin model

Unperturbed TSS

H =

(E1 00 E2

)eigenv. { |g〉, |e〉}

e

g

Perturbed TSS - tunneling terms appear

eB(t)

gH =

(E1 W12

W21 E2

)eigenv. { |+〉, |−〉}

General Hamiltonian of a TSS

HTSS = εσz + ∆σx ε =12

(E1−E2) ∆ =12

(W12−W21)25 / 45

Spin-Boson Modell

Environment

model environment as a bath of bosons behaving as oscillators

TSS

Environment

Total Hamiltonian given by H = HTSS +HB +Hint

HB = ∑k

p2k

2mk+12mkω

2k x2

k = ∑k

hωka†kak (drop zero point energy)

26 / 45

Spin-Boson Modell

Interaction env. and TSS

can be modeled asHint = σz ∑

kλkxk(t)

xk(t) ∝ (a†k +ak) is the position of the k-th harmonic oscillator

λk is the coupling strength between oscillator and spin

can contain term proportional to σz (describing spin’s energy) or σx(describing spin flips)

The λk ’s are given by the environment’s spectral density

J(ω) = ∑i

λ 2i

2miωiδ (ω−ωi) (2)

27 / 45

Spin-Boson Modell

Exact Solution

Hamiltonian

H = HS + HE + HI

=ω0

2+∑

kωk a†

k ak + σz ∑k

λk

(a†k + ak

)Interaction picture evolution (unitary)

U (t) = T←e−i∫

dsHI (s) = ϕ (t) · V (t)

Initial state: thermal equilibrium

ρ (0) = ρS (0)⊗ρE

ρE =1

ZEe−HE /kBT

28 / 45

Spin-Boson Modell

Exact Solution

Density matrix ρ (t)

ρij (t) = 〈i |trB{V (t)ρ (0)V−1 (t)

}|j〉

ρ10 (t) = ρ∗01 (t) =: ρ10 (0)exp{Γ(t)}

with Decoherence function

Γ(t) = ln trB {ρ10 (t)}=−∑k

4λ 2k

ωkcoth(ωk/2kBT )(1−cosωkt)

Continous limit: Spectral density of environment (D (ω): density ofmodes)

J (ω) = 4D (ω)λ2k

Γ(t) =−∫

ω

0dω

J (ω)

ω2 coth(ω/2kBT )(1−cosωt)

29 / 45

Spin-Boson Modell

Exact Solution

Density matrix ρ (t)

ρij (t) = 〈i |trB{V (t)ρ (0)V−1 (t)

}|j〉

ρ10 (t) = ρ∗01 (t) =: ρ10 (0)exp{Γ(t)}

with Decoherence function

Γ(t) = ln trB {ρ10 (t)}=−∑k

4λ 2k

ωkcoth(ωk/2kBT )(1−cosωkt)

Continous limit: Spectral density of environment (D (ω): density ofmodes)

J (ω) = 4D (ω)λ2k

Γ(t) =−∫

ω

0dω

J (ω)

ω2 coth(ω/2kBT )(1−cosωt)

29 / 45

Spin-Boson Modell

Master equation for open quantum systems

“master equation" (for derivation Zurek or Petruccione) describesbehavior of ρ = trenv ρT

master eq. in Lindblad form

ρ(t) =− ih

[Hs ,ρ] +∑k

γk(AkρA†k −

12A†

kAkρ− 12

ρA†kAk)

30 / 45

Spin-Boson Modell

Master equation for open quantum systems

“master equation" (for derivation Zurek or Petruccione) describesbehavior of ρ = trenv ρT

master eq. in Lindblad form

ρ(t) =− ih

[Hs ,ρ] +∑k

γk(AkρA†k −

12A†

kAkρ− 12

ρA†kAk)

initially ρT (0) = ρ(0)⊗ρenv (0)

Markovianity, i.e. env. is memoryless

γk ≥ 0

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Spin-Boson Modell

Master eq. and SBM

e

gσz

Consider SBM-Hamiltonian

H =hω0

2σz︸ ︷︷ ︸

Hs

+∑k

hωka†kak + σz ∑

kλk(a†

k +ak)︸ ︷︷ ︸Hint

prepare initial state

|ψ(0)〉=1√2

(|g〉+ |e〉) → ρ(0) =12

(1 11 1

)32 / 45

Spin-Boson Modell

Master eq. and SBM

ρ =− ih

[Hs ,ρ] + γσzρσ†z −

γ

2σ

†z σzρ− γ

2ρσ

†z σz

Information about Spectrum contained in γ ≥ 0

Lindblad operators A = A† = σz

σ2z = I and Hs = ω0

2 σz

master equation

ρ =− iω0

s[σz ,ρ] + γσzρσz − γρ

33 / 45

Spin-Boson Modell

Master eq. and SBM

Evaluating 〈i |ρ|j〉 i , j ∈ {e,g} leads to this set of equations:

˙ρgg = 0

˙ρee = 0

˙ρeg = (−iω0−2γ) ·ρeg

˙ρge = (iω0−2γ) ·ρge

34 / 45

Spin-Boson Modell

Master eq. and SBM

Evaluating 〈i |ρ|j〉 i , j ∈ {e,g} leads to this set of equations:

˙ρgg = 0

˙ρee = 0

˙ρeg = (−iω0−2γ) ·ρeg

˙ρge = (iω0−2γ) ·ρge

Leads to time evolution

ρ(t) =12

(1 e−iω0te−2γt

e iω0te−2γt 1

)

Trace is preserved but interference terms decay! Interaction withEnvironment turns ρ(t) into statistical mixture, we observe decoherence!

35 / 45

Spin-Boson Modell

Master eq. and SBM

e

gσx

Consider a diffierent interaction term

Hint = σx ∑k

λkqk = (σ+ + σ

−)∑k

λk(a†k +ak)

36 / 45

Spin-Boson Modell

Master eq. and SBM

e

gσx

Consider a diffierent interaction term

Hint = σx ∑k

λkqk = (σ+ + σ

−)∑k

λk(a†k +ak)

under rotating wave approximation (RWA) we neglect the termsσ+a†

k + σ−ak and obtain

H =hω0

2σz +∑

khωka†

kak +∑k

λk(σ−a†

k + σ+ak) (3)

37 / 45

Spin-Boson Modell

Master eq. and SBM

We prepare the TSS in the exited state

|ψ(0)〉= |e〉 → ρ(0) =

(1 00 0

)Lindblad Operators are A = σ− and A† = σ+

master equation for σx coupling

ρ =− iω0

s[σz ,ρ] + γσ

−ρσ

+− γ

2σ

+σ−

ρ− γ

2ρ σ

+σ−

38 / 45

Spin-Boson Modell

Master eq. and SBM

Evaluating 〈g|ρ|g〉 and 〈e|ρ|e〉 :

˙ρgg = γ ·ρee

˙ρee = −γ ·ρee

(4)

solving leads to

ρ(t) =

(e−γt ρegρge 1−e−γt

)

Excited state decays to ground state over time through interaction withenvironment! We observe spontanious emission.

39 / 45

Summary

Outline

1 Bloch-Equations

2 Classical Dissipations

3 Spin-Boson Modell

4 Summary

40 / 45

Summary

Summary

Open Quantum systems:Dissipation (classical)Decoherence

Spin Boson Modell: Exactly solveable

General approach: Master-equations

41 / 45

Summary

Experiments with Spin Boson Model

investigate a cooled 1D Coulomb crystal with N = 50 ions

focus laser on a central ion

assume linear spectral density J(ω) ∝ ω, high temperature regime

Plot 〈σz 〉(t), we observe effects quantum such as quantum revivals

initial spin relaxation propagates along chain and reflects at boundary

42 / 45

Summary

Experiments with Spin Boson Model

Figure: Porras, Cirac et al. 200843 / 45

Summary

Spin Boson Model

thank you for your attention

44 / 45

Appendix For Further Reading

For Further Reading I

H.-P. Breuer, F. Petruccione:The theory of open quantum systems.Oxford University Press, 2002.

U. Weiss:Quantum Dissipative SystemsWorld Scientific, 1999.

F. Bloch:Nuclear InductionPhysical Review 70, 460-473 (1946)

45 / 45