Spin-BosonModel · Bloch-Equations OpenQuantumSystem Rewiev:Densitymatrixformalism Example Pure ji...
Transcript of Spin-BosonModel · Bloch-Equations OpenQuantumSystem Rewiev:Densitymatrixformalism Example Pure ji...
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
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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
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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
)
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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
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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).
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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)
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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Bloch-Equations
Transversal Noise
Animation (transversal noise)
Averaging over random realizations of transversal δB(t)
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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
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Classical Dissipations
Outline
1 Bloch-Equations
2 Classical Dissipations
3 Spin-Boson Modell
4 Summary
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Classical Dissipations
Langevin equation
Include environment in our model
Classical Hamiltonian → Langevin equations
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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
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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)
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Classical Dissipations
Langevin Equation
Summary:
Classical fluctuations: Dissipation (no decoherence).
Correlators ∝ Temperature, mode coupling λk
Now: do it quantum mechanically!
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Spin-Boson Modell
Outline
1 Bloch-Equations
2 Classical Dissipations
3 Spin-Boson Modell
4 Summary
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Spin-Boson Modell
A review of the spin model
Any two state system (TSS) can be modeled by the spin formalism.
eE(t)
g
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Spin-Boson Modell
A review of the spin model
Unperturbed TSS
H =
(E1 00 E2
)eigenv. { |g〉, |e〉}
e
g
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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)
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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)
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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
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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)
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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)
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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)
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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
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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
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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!
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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)
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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)
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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.
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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
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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
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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)
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