Decoherence by controlled spin baths D. Rossini, T. Calarco, V. - - PowerPoint PPT Presentation
Decoherence by controlled spin baths D. Rossini, T. Calarco, V. Giovannetti, S. Montangero and R. Fazio SNS - Pisa SISSA - Trieste BEC - Trento ITAMPT - Harvard cond-mat/0605051 Motivation Decoherence Paradigm models Engineered
SNS - Pisa SISSA - Trieste
ITAMPT - Harvard
BEC - Trento
cond-mat/0605051
Decoherence Paradigm models
Engineered Quantum Baths
N.V. Prokof'ev and P.C.E. Stamp,
W.H. Zurek (1982) F.M. Cucchietti, J.P Paz, and W.H. Zurek, (2005)
indipendent spins
D.V. Khveshchenko, (2003) C.M. Dawson et al., (2005)
H.T. Quan et al (2006). F.M. Cucchietti, S. Fernandez-Vidal, and J.P. Paz, (2006)
interacting spins Highly symmetric interactions and baths
System Bath
A two-level system(the quantum system) is coupled to a single spin of a one-dimensional spin-1/2 chain (the environment). J J J ε
Optical lattices as simulators
HT L = ω1|11|
HT L = −|11|σz
1
ρ10(t) = ρ10(0)D(t)
Palma, Suominen, and Ekert, (1996)
Loschmidt echo
Ising universality class XY universality class
γ = 0 γ = 0 H = −J 2
N
(1 − γ)σx
i σx i+1 + (1 + γ)σy i σy i+1 − h N
σz
i
Ising chain in a transverse field
Exact solution - free fermions
Pfeuty ‘70
1
σx = 0
λ = h J
H =
Ji
i σx i+1 + σy i σy i+1 + ∆ σz i σz i+1
Constant couplings
Critical region Ferromagnetic Antiferromagnetic
Ising chain environment
Aj,k = −J(δk,j+1 + δj,k+1) − 2(λ + j)δj,k Bj,k = −Jγ (δk,j+1 − δj,k+1) C =
B −B −A
iΨj
Ising chain environment
25 50 7
J t
0.985 0.99 0.995 1
L
0.2 0.4 0.6 0.99 1
0.5 1 1.5 2
!
0.2 0.4 0.6 0.8 1
"
0.5 1 1.5 2
!
0.985 0.99 0.995 1
L
#2
a)
I
b) c)
$
II
REGION I REGION II
LI(t) ∼ e−αt2
constant
λ > 1 λ < 1
Ising chain environment
10
10 10
1
10
2
J
0.975 0.98 0.985 0.99 0.995 1
L
50 100 500 1000 2000 N 0.978 0.98 0.982 0.984 0.986
L
c
c
!
Jt
Lc(t) = c0 (1 + c1 ln t) Lc
∞ =
l∞ 1 + β ln N
Configuration 1 Configuration 2
If the chain is not at the critical point, the decay of the coherences is independent of the number of spins in the bath
Ising chain - correlations in the bath
Ising chain - correlations in the bath
Configuration 1 Configuration 2
10 20 30 40 50 60
t
0.97 0.98 0.99 1
L
10 20 30 40 50 60
t
0.0005
δL
Ising chain environment - Universality
0.5 1 1.5 2 2,5
λ
dλα
0.5 1 1.5 2
λ
0.95 0.96 0.97 0.98 0.99 1
L∞
50 100 150 200
J t
0.95 0.96 0.97 0.98 0.99 1
L
ε
XY chain environment
0.25 0.5 0.75 1 1.25 1.5
λ
dλα
0.25 0.5 0.75 1 1.25 1.5
λ
0.4 0.5 0.6 0.7 0.8 0.9 1
L∞
200 400 600 800 1000
J t
0.5 0.75 1
L
ε
The chain is critical
50 100 200 300 500N
0.915 0.92 0.925 0.93 0.935 0.94L∞
50 100 200 300 500N
0.74 0.76 0.78 0.8 0.82 0.84L∞
l0=0.990088 β=0.0128334 l0=1.18854 β=0.0952356λ=0.25 λ=0.9
Decoherence vs interaction & entanglement in the bath
see the suggestion of
How to measure entanglement?
network
Bipartite entanglement The state of the two selected spins is mixed
| β =| 11
| α = 1 √ 2(| 00+ | 11)
β | α = 0 β | α = 1 | α =| 00
| β = 1 √ 2(| 00+ | 11)
Measure of mixed state entanglement for two spin-1/2 states
C(i, j) = max{0, λ1(i, j) − λ2(i, j) − λ(i, j) − λ4(i, j)}
R = ρ(i, j)˜ ρ(i, j) ˜ ρ . = σy ⊗ σyρ∗σy ⊗ σy
where
where s are the eigenvalues of in ascending order
λ R
Decoherence vs entanglement in the bath
0,1 0,2 0,3
C(1)
0,01 0,02 0,03 0,04 0,05 0,06
α
Δλ
0,1 0,2C(1)
λ=0 λ=0.8 λ=1 λ=2 λ=1.2
λm
Heisenberg chain environment - tDMRG
1 2 3 4
J t
0.9925 0.995 0.9975 1
L
1 2 3 4
Δ
0.002 0.004 0.006 0.008 0.01
α
critical region
a) b)
Coupling to m spins in the environment
Configuration A Configuration B
Configuration A Configuration B
0.5 1 1.5 2 2.5
λ
0.01 0.02 0.03 0.04 0.05 0.06
α /m
0.94 0.96 0.98 1 1.02 1.04 1.06λ
0.03 0.035 0.04 0.045α /m
Text Weakly dependent on m
Configuration A Configuration B
0.5 1 1.5 2 0.2 0.4 0.6 0.8 1
L!
λ
Strong dependence on the number of couplings m increases
✔ Optical lattices to simulate quantum baths ✔ Exact solution for an interacting baths ✔ Numerical t-DMRG for Heisenberg bath