Quantum Metrology with highly entangled states and realistic decoherence FQRNT Bill Coish Department of Physics, McGill University, Montréal QC Work done with Maxime Hardy (co-op Université de Sherbrooke) 2012 June 11; IQC Colloquium, Waterloo, ON Collaboration with Innsbruck: R. Blatt, T. Monz, P. Schindler, J. T. Barreiro, ...
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Quantum Metrology with highly entangled states and realistic decoherence
FQRNT
Bill CoishDepartment of Physics, McGill University, Montréal QC
Work done with Maxime Hardy (co-op Université de Sherbrooke)
2012 June 11; IQC Colloquium, Waterloo, ON
Collaboration with Innsbruck: R. Blatt, T. Monz, P. Schindler, J. T. Barreiro, ...
General Philosophy
Abstract models are excellent for fast progress, well defined questions...
BUT: too many to choose from.
Physical considerations often show the way to go.
True dynamics/decoherence often more complex than initial models suggest.
Goal: Quantum Information Processing
Ujªouti
j0i0j0i1
j0iN¡1j0iN
0 1
Initialization Arbitrary unitary Readout
0 1
0 1
0 1
Ujªouti
j0i0j0i1
j0iN¡1j0iN
0 1
Initialization Arbitrary unitary Readout
0 1
0 1
0 1
Physical Implementation:
U = T exp½¡iZ t
0
dt0H(t0)
¾H 2 HS
Goal: Quantum Information Processing
Reality: Imperfections
Ujªouti
j0i0j0i1
j0iN¡1j0iN
0 1
Initialization Arbitrary unitary Readout
0 1
0 1
0 1
~U = T exp½¡iZ t
0
dt0 (H(t0) + ±H(t0))
¾±H 2 HS HE
Qubit encoding: Single ions (40Ca+)
V(r)
r
¢E » 10eV
» 10¡10m
encoding:
R. Blatt and D. Wineland, Nature (2008)
jsi ! j0ijdi ! j1i
Collective dephasing
jÃ(0)i = 1p2(j0000:::i+ j1111:::i) (GHZ)
N qubits
N = 1
N = 2
N = 3N = 4
N = 6
deco
here
n ce
rate
1/T
2
Number of qubits, N
Uncorrelated noise
1
T2/ N
Expectation
deco
here
n ce
rate
1/T
2
Number of qubits, N
Correlated noise: “Superdecoherence”
1
T2/ N2
G. Palma et al., Proc. Roy. Soc. Lond. A (1996)
F (t) =DjhÃ(0)jÃ(t)ij2
Eav:
T. Monz et al., PRL (2011)
Collective dephasing
jÃ(0)i = 1p2(j0000:::i+ j1111:::i) (GHZ)
N qubits
N = 1
N = 2
N = 3N = 4
N = 6
deco
here
n ce
rate
1/T
2
Number of qubits, N
1
T2/ N
1
T2/ N2
Fits
something in between?
F (t) =DjhÃ(0)jÃ(t)ij2
Eav:
T. Monz et al., PRL (2011)
Dephasing: General
!(t)
j1i
j0i
_½ = ¡i [H(t); ½]H(t) = !(t)¾z=2
h¾+(t)i = eiÁ(t) h¾+(0)i Á(t) =
Z t
0
dt0!(t0)
!(t)
tprepare measure tprepare measure
Average over noise realizations:
h¾+(t)iav: =DeiÁ(t)
Eav:
h¾+(0)i
Dephasing: Generalh¾+(t)iav: =
DeiÁ(t)
Eav:
h¾+(0)i = e¡12 hÁ2(t)iav: h¾+(0)i
h±!(t)±!(0)iav:t
¿c
(Gaussian, stationary)Á2(t)
®av:=
Z t
0
dt0(t¡ t0) h±!(t0)±!(0)iav:
Reh¾+(t)i a
v:
¿c < ¿dec:
¿c ¿dec:
» e¡t=¿dec:
(Markovian)
Reh¾+(t)iav: ¿c > ¿dec:
¿dec: ¿c
» e¡(t=¿dec:)2
(Non-Markovian)
Sources of dephasing in ion traps
●Fluctuating global phase reference (laser stability, also slow)
●Global magnetic field fluctuations (slow)
s= j0i = j1i
d(orbital Zeeman)
AMO Physics: Usually assume fast, local noise.
Gaussian dephasing model:
Szk = (j0i h0jk ¡ j1i h1jk) =2
jÃ(0)i = 1p2(j0000:::i+ j1111:::i)
h±B(t)±B(0)i =±B2
®e¡t=¿c
(GHZ)
N qubits
H(t) = ±B(t)X
k
Szk
Gaussian dephasing model:
Szk = (j0i h0jk ¡ j1i h1jk) =2
jÃ(0)i = 1p2(j0000:::i+ j1111:::i)
h±B(t)±B(0)i =±B2
®e¡t=¿c
(GHZ)
N qubits
²(N; t) =N2
2
Z t
0
d¿(t ¡ ¿) h±B(t)±B(0)i
“Superdecoherence”
H(t) = ±B(t)X
k
Szk
F (t) =j hÃ(0)j Ã(t)i j2
®av:=1
2(1 + exp [¡2²(N; t)]) ' 1¡ ²(N; t)
N = 1
N = 2
N = 6
N = 3
N = 4
Revised noise model, accounting for a finite correlation time (non-Markovian)
Dominant noise source (B-field) identified; N extended to 14 qubits!
² » N2
T. Monz, ... WAC, ..., R. Blatt, PRL (2011)
Quantum MetrologyFrequency standards
Fundamental tests of gravitation
Mueller, Peters, Chu, Nature (2010)
Parameter estimation for Qm. Inf. Proc.
e.g., Rafal Demkowicz-Dobrzanski et al., arXiv (2012)