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?
YET Bloch-Redfield (perturb.) remains most popular theory for solid-state qubits! b.
Minor improvement to perturb. method More powerful Less user-friendly
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urb
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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
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??
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
Spectral universality and random matrices. Dynam- ical universality, energy diffusion, and dimensionless parameters characterising dynamics. Introduction of ...
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I wish to thank Michael J. W. Hall, Ariel Caticha, Gerald A. Goldin and Heinz-Dietrich. Doebner for their comments and suggestions regarding the work presented ...
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