Bill Coish - Home | University of Waterloo | University of Waterloo · 2013. 5. 21. · Quantum Metrology with highly entangled states and realistic decoherence FQRNT Bill Coish Department

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)

Quantum Metrology

Precision measurements?

j0i H HUÁ0 1

Á = !t

Repeat N times...

P = (1 + cos!t)=2j1i+ ei!t j0i

T: total experiment time

Classical:

±!class: =

pP (1¡ P )=N

jdP=d!j =1pNTt

/ 1pN

Quantum (GHZ state): j1111:::i+ eiN!t j0000:::i

±!quant: =1

NpT t

/ 1

N

Problem! DecoherenceMarkovian (exponential) dephasing, spatially uncorrelated noise:

S. F. Huelga et al., PRL (1997)

¿dec: = 1=(°N)

P = (1 + cosN!t)=2! (1 + e¡N°t cosN!t)=2

±!opt: / 1pNT

/ ±!class:

For (Markovian) spatially correlated noise:

¿dec: / 1=N2 ) ±!quant: / const:

Even worse!

U. Dorner, New J. Phys. (2012)

With dephasing (in general):

±!opt: / 1

Np¿dec:T

What kind of decoherence?

T. Monz, ..., WAC,..., R. Blatt PRL (2011)

non-exponential (long correlation time)

¿c > ¿dec:

“Superdecoherence”(long correlation length)

²(N)=²(1) / N2F = 1¡ ²

»c > L » N

Large N: dephasing becomes local

H(t) =X

k

±hk(t)Szk

L

»c

Generalized model:

Gaussian fluctuations:

Space Time

But: L / N (incr. with N) ¿dec: decr. with N!

Features of the environment(independent of N!)

h±hk(t)±hl(0)i =­±h2(0)

®e¡jxk¡xlj=»c £ e¡t=¿c

Sz =1

2(j1i h1j ¡ j0i h0j)

Large N: Quantum advantage?

±!opt: / 1

Np¿dec:T

¿dec: /1pN

) ±! / 1

N3=4

Also see, e.g., Jones et al., Science (2009); Matsuzaki, Benjamin, Fitzsimons PRA (2011)

¿c > ¿dec:; »c < L :

Large N: Quantum advantage?

¿dec: = T1=N

I : ¿c < ¿dec:; L (» N) < »c

II : ¿c > ¿dec:; L (» N) < »c

OR ¿c < ¿dec:; L (» N) > »c

III : ¿c > ¿dec:; L (» N) > »c

If the qubit frequency fluctuates locally in space and is approximately constant in time, quantum wins.

local, non-Markovian

Model summary: Spatial and temporal correlations

¿c < ¿dec:

¿c ¿dec:

» e¡t=¿dec:

¿c > ¿dec:

¿dec: ¿c

» e¡(t=¿dec:)2

»c ¿ L

¿dec: / N¡1; ±! / N¡1=2

»c À L

¿dec: / N¡2; ±! = const:

»c À L

¿dec: / N¡1; ±! / N¡1=2

F (t) F (t)

»c ¿ L

¿dec: / N¡1=2; ±! / N¡3=4

See also: Matsuzaki, Benjamin, Fitzsimons, PRA (2011)

Scaling: Regimes

t t t t

»c » n0 = 7

¿c = 1

±!0 = 10¡4

Can we do better?

Measurement time long compared to correlation time!

1

N3=4

Jones et al., Science (2009)Matsuzaki et al., PRA (2011)

!(t)

tprepare measure tprepare measure ¿c

Parameter Estimation = “Instantaneous Measurement”

After many measurements, frequency still not precisely defined!

......!

±! = 1=pNtT !k = !0 + k¼=t

T ¿ ¿c“Instantaneous”: ½(!)

½(!) = PN+(1¡ P )N¡N+=N0

More realistic: Gaussian prior

!¾

½(!)½(!) = PN+(1¡ P )N¡N+½0(!)=N0

½0(!)

P = (1 + cos!t)=2

Measurement times t?

Advantage in performing measurements at short times, even if the standard formula suggests larger t is always better:

±! » 1ptT

Classical:

Quantum:

¢! =¼

t> ¾

¢! =¼

ntPeak spacing smaller: ¢! > ¾ ) t < 1=n¾

±!class: »1pNtT

»r

¾

NT

±!quant: »r

¾

NTThis protocol gives the same scaling! (not optimal?):

Improved measurement strategy

tMeasurement time

Measurement time t=2

Measurement time t=4

Measurement time overhead:

t0 =X

k

1

2kt = 2t

±!quant: =

p2

NptT

(GHZ)

Result:

T ¿ ¿c

±!class: =

p2p

NptT

Summary

A static random frequency can always be found with Heisenberg-limited precision using GHZ states, provided the prior distribution has finite width.

T ¿ ¿c

±!quant: »1

NpT t

This beats the ~1/N3/4 scaling found previously [Jones et al., Science (2009), Matsuzaki et al., PRA (2011)], even for Gaussian decay of P(t).

t ¿ ¿c ¿ T ?

In this regime, frequency drifts between measurements; problem still open?

Conclusions

●“Superdecoherence”: a short-term problem for ion-trap and other implementations.

●Quantum-enhanced precision measurements still possible in spite of dephasing.

How large can the quantum region be?

Physical dephasing mechanisms:

Charge traps (fluctating electric field)

Surface spins (magnetic field)

Power-law correlations in space/time

Open questions: