Which theory for dissipation in quantum systems (such as qubits) ?

Quantum optics: decay of excited atomic state. • Chemical ... Only a microscopic theory can answer certain questions: ... dependence on environ. spectrum?
807KB taille 3 téléchargements 173 vues
Institut Laue Langevin, Grenoble, France. Grenoble, Avril 2008

Which theory for dissipation in quantum systems (such as qubits) ? Robert S Whitney us

ro go

pert urb.

ri

Discussions: J. Siewert (Regensburg) Y. Gefen (Weizmann)

A. Shnirman (Karlsruhe)

S. Stenholm (Stockholm)

M. Clusel (NYU)

D. O’Dell (McMasters)

M. Hall (Australian Nat. Univ.)

Reference: R.W., J. Phys. A: Math. Theor. 41, 175304 (2008)

Overview — Two theories of dissipation [1] Phenomenological method Lindblad (1976) — rigorous [2] Microscopic method Bloch-Redfield (1957) — perturbative RESULTS DISAGREE Dumcke-Spohn (1980) pert

urb.

us

ri

ro go

rigorous perturb.

YET Bloch-Redfield (perturb.) remains most popular theory for solid-state qubits! b.

Minor improvement to perturb. method More powerful Less user-friendly

pe pert

urb

r rtu

us

ro

o rig

rigorous

Why dissipative quantum mechanics? No quantum system is isolated ⇔ energy exchange Dissipation: common in quantum world as in classical world

• Quantum optics: decay of excited atomic state • Chemical physics: most reactions 2Na2 (s)+2HCl(aq)→2NaCl(aq)+H2 (g) • Statistical physics: what is equilibrium? • Solid-state: Resistance in nanoscale circuits • Quantum information: Decoherence of qubits • Philosophy: No Schrodinger ¨ cats in everyday life density matrix "=

Summary of previous works

Phenomenological method • Know nothing about environ. • Know system dynamics are physical ⇒ Probabilities are real, positive and sum to one + rigorous Microscopic method • Know everything about environ. ˆ universe ⇒ know H + typically perturbative

pert

urb.

us ro

o rig

system

system

???

environ "universe"

Road-map of previous works MICROSCOPIC

PHENOMENOLOGICAL

(mostly perturbative)

(rigourous)

Bloch (1957) Redfield (1957) Nakajima (1958) Zwanzig (1960)

Lindblad (1976) Kraus (1983)

Scholler−Schon (1994) diagrams Spin−boson model −Leggett et al (1987)

Chemical Physics

Solid state

Quantum Optics

Mathematical Physics

QUBITS

+ EXACTLY SOLUBLE MODELS (non-generic)

Why use the perturbative method?

rigorous

...but rigorous method is phenomenological

perturb.

cf. superconductor: Landau-Ginzberg vs. BCS

Only a microscopic theory can answer certain questions: • dependence on environ. temperature? • dependence on environ. spectrum? • How do we engineer system to minimize decoherence? Perturbative method usually gives “plausible” results but it sometimes generates negative probabilities ...so can we really trust it??

Density-matrix and Bloch-sphere ! " ˆ ˆ Observable: $O%t = tr O ρˆ(t) # $ d ˆ Evolution: dt ρˆ(t) = −i H, ρˆ(t) + dissipation Any two-level system ≡ spin-half % & 1+"ˆ σz #t "ˆ σx #t −i"ˆ σy #t ρˆ(t) = 12 "ˆ σx #t +i"ˆ σy #t

B−field z−axis

1−"ˆ σz #t

final state

1.0 0.5

"ˆ σx,y #t

⇒ plot $ˆ σx %t on x-axis ... etc

0.0 −0.5 −1.0 0

x−axis

initial state

"ˆ σz #t t/T2 1

2

3

4

Lindblad’s master equation ! " d ˆ ρˆ(t) − L[ˆ ρ ˆ (t) = −i H, ρ(t)] dt ˆ n s., For set of “orthogonal” and “normalized” operators, L

L[ˆ ρ(t)] ≡

(

ˆ † ˆ ˆ(t) n λn Ln Ln ρ

'

+

ˆ †n L ˆn ρˆ(t)L



ˆ n ρˆ(t)L ˆ †n 2L

)

with no negative λn s

Lindblad proved: All other master equation are unphysical — negative probabilities Rigorous proof based on following postulates: • Evolution continuous in time • Eqn. translationally invarient in time • ··· • physical ≡ “complete positivity”

Understanding Lindblad eqn.

Eqn. is remarkable simple!! “Markovian” # — evolution is function of ρˆ(t) not $ d ˆ ρˆ(t) − L[ˆ ρˆ(t) = −i H, ρ(t)] dt

*

ˆ1 = σ Example: spin-half with one env.-coupling L ˆz % ( ) 0 L[ˆ ρ(t)] ≡ λ 2ˆ ρ(t) − 2ˆ σz ρˆ(t)ˆ σz = 2λ

dt% ρˆ(t% )(...)

"ˆ σx #−i"ˆ σy #

"ˆ σx #+i"ˆ σy #

0

&

In General: ˆn For λn > 0 — decay in all directions ⊥ to L ... but for λn < 0 — growth

Positivity and complete positivity environ

Positivity

environ EVOLVUTION

system

Complete positivity

environ

environ

system

MEASURE

MEASURE

0

t

environ

environ

system

environ

system

MEASURE

witness

time

EVOLVUTION

EVOLV. system

system

EVOLV. witness

witness

system MEASURE EVOLVUTION witness

t

0

time

All completely positive dynamics are also positive Ocassionally the two are equivalent – as for my 2-level system model M. Hall, arXiv:0802.0606

Bloch-Redfield’s master equation I Hamiltonian: ˆ univ = H ˆ sys + cΓ ˆX ˆ +H ˆ env H

system

c

^ !

perturbation

environ

^ X

"universe"

ˆ univ t] ρˆ(0) exp[iH ˆ univ t] Evolution: ρˆ(t) = exp[−iH "( 0)

ρˆ(t) = |ψ(t)%$ψ(t)|

"( t)

R

A

"( 0)

"( t) time

t

0

Second-order perturbation theory: ˆ univ t] 2nd order in exp[−iH A

ˆ univ t] 2nd order in exp[iH

R

A

t’#$ t’

0

t

1st order in both

R

A

t’#$ t’

0

t

0

R

t’#$ t’

t

Bloch-Redfield’s master equation II

*



Excite env mode, %

t’#$

f( $) t’#$

t’

% −distribution

f( $)

f( $) t’

t’#$

t’

t’

memory time

"Fourier"

%

$

ˆ = Γ ˆ Ξ(t) = Dissipative part of

L[ˆ ρ(t)] = c

2

(

f( $)

*t 0



t’#$

t’

d ˆ(t) dt ρ

ˆ Ξ(t)ˆ ˆ ρ(t) + ρˆ(t)Ξ ˆ † (t)Γ ˆ † − Ξ(t)ˆ ˆ ρ(t)Γ ˆ † − Γˆ ˆ ρ(t)Ξ ˆ † (t) Γ

)

Bloch-Redfield in Lindblad’s form

[spin-1/2 – Dumcke-Spohn (1979)]

d Dissipative part of dt ρˆ(t) ( ) 2 ˆˆ † † † † ˆ (t)Γ ˆ − Ξ(t)ˆ ˆ ρ(t)Γ ˆ − Γˆ ˆ ρ(t)Ξ ˆ (t) L[ˆ ρ(t)] = c ΓΞ(t)ˆ ρ(t) + ρˆ(t)Ξ

• Operators not orthogonal (unlike Lindblad) ⇒ ORTHOGONALIZE • Cross coupling ⇒ DIAGONALIZE ( ) ' † † † ˆ L ˆ ˆ(t) + ρˆ(t)L ˆ L ˆ ˆ ˆ(t)L ˆ L[ˆ ρ(t)] = n=1,2 λn L n nρ n n − 2Ln ρ n

...same as rigorous eqn. with λ2 ∝ −c2 ⇐ always negative

2-level system ˆ∝σ with H ˆz ˆ=σ &Γ ˆx

z

(a)

(b)

^

Hsys ^

P2

^

L1

w ^

x

P1

^

L2

^

^

y

L2

L1

Obituary for perturbative method 1957-1979 Rigorous theory says “negative λ means negative probs.”

rigorous perturb.

We have λ2 < 0 for any c ⇒ perturbative method unphysical Numerics confirm negative probabilities - Gaspard & co-workers

Resurrection of perturbative method

ˆ is time-dependent Forgot Ξ(t) • Invalidates rigorous proof !! Assumption that eqn. is time-indep. • Numerics change with t-depend. – Gaspard & co-workers OPEN QUESTION: Does perturb. method avoid negative probs. ??

Time-dependence of parameters $ d ˆ ρˆ(t) − L[ˆ ˆ(t) = −i H, ρ(t), t] dt ρ + t ˆ = Time-dependent L[ˆ ρ(t), t] since Ξ(t) dτ

#

0

0

tm

f( $) t’#$

t’

system timescale

time−dependent coupling

dissipative timescale time, t

short times long times

Analogy: without matrix structure d dt y(t) = (ih − F (t))y(t) where F (t) → f for t - memory time Approx:

d dt y(t)

= (ih − f )y(t) Trivial to solve, but incorrect for t ∼ memory time

Proving positivity for short memory-times d dt y(t)

... continue analogy

[I] Short-times t + 1/G(t) # iht

y(t) , e



= (ih − G(t))y(t)

*t 0

% ih(t−t! )

dt e

%

−iht

G(t )e

+O[G2 ]

$

y(0)

[II] Long-times t > t0 - memory-time d ⇒ y(t) , e(ih−g)(t−t0 ) y(t0 )+O[G−g] dt y(t) , (ih − g)y(t) Large overlap of regimes if memory time