Coherence and entanglement dynamicsof exciton qubits

Martijn Wubs, NBI/DTU

AMOLF, Oct. 26, 2009

x1 x2 x3

x

Qb 1 Qb 2 Qb 3

in collaboration withR. Doll, S. Kohler, and P. Hänggi (Augsburg)

intro: quantum coherence

wave function for isolated systems

|ψ⟩ = |0⟩+ |1⟩p2

density matrix for open quantum systems

ρ(0) = |ψ⟩⟨ψ| = 1

2|0⟩⟨0|+ 1

2|1⟩⟨1|+ 1

2|0⟩⟨1|+ 1

2|1⟩⟨0|

|0>

|1>

relaxation

pure dephasing

intro: search for quantum coherence

Light harvesting Quantum information processing

Outline

recent experiments:exciton spectra in 3D and on 1D substrates

1 qubit:coherence dynamics under dephasing (in 1D, 3D)

2 qubits:entanglement dynamics, finite separation (in 1D, 3D)

N qubits:benchmarking master equations with exact results

conclusions

Quantum dots and substrates

from: group Rosenauer, Bremen

quantum dots = artificial atoms

self-assembled, eg.InGaAs on GaAs

size 2−10nm0D: discrete energy levels

long-lived optical excitations

useful as:

single-photon sources

quantum memory? Coherence?

Qubit in a quantum dot

two-level system |0⟩, |X⟩: exciton confined to q-dot

Zrenner et al., Nature (2002).

experiments: exciton qubits in carbon nanotubes

1

2

31300 4000 6700

cts/sec

1 2 3 4 5

1

2

3

(a)

200 nm

0 500 1000 15000

10

N

Length (nm)

(b)

<L> = 520 nm

units: nm

77 K

em= 885 nm

0

em= 872 nm

X - translation (µm)

Y-

transla

tion

(µ

m)

(c)

= 15 nm

3000 6000

0 250 5000.0

0.4

0.8

Heig

ht

0

Högele et al., PRL (2008)

experiments: excitons in nanotubes as single-photon emitters

0

10

20

0

20

40

G(2

) ()

-30 -20 -10 0 10 20 30

0

30

60

Time delay, (ns)

0

100

200

0

30

60

PL

Inte

nsity (

cts

/se

c)

850 860 870 880

0

30

60

Wavelength (nm)

4.2 K

10 K

25 K 0.15

0.03

0.03

= 4.4 meV

9.1 meV

9.5 meV

Energy (eV)

1.459 1.4091.433(a) (b)

Högele et al., PRL (2008)

“. . . intrinsic phonon-induced pure dephasing is 2 orders of magnitude larger

than the lifetime broadening . . .”, (Galland et al. (2008))

1 qubit: modeling pure dephasing due to substrate

qubit-bath Hamiltonian

H =ħΩσz +∑

kħωkb†

kbk +Hq−b

interaction

Hq−b =ħσz∑

k

(gkbk +g∗

k b†k

)initial state

R(0) = ρq(0)⊗ρeq.b

reduced qubit state

ρq(t) = Trb[R(t)] =(

p0 ρ01(0)c(t)ρ∗

01(0)c∗(t) 1−p0

),

1-qubit coherence function c(t)

1 qubit: interaction with substrate

deformation-potential coupling to acoustic phonons, gk ∝pk

bath spectral density J(ω) =∑k |gk|2δ(ω−ωk)

linear substrate (1D) → J(ω) ∝ω1 ohmicbulk substrate (3D) → J(ω) ∝ω3 superohmic

model spectral density with cutoff ωc = vphon/`dot

J(ω) =αωc (ω/ωc)d e−ω/ωc

1.42 1.43 1.44 1.45

101

102

103

104

0 1 2

0.1

1

Energy (eV)

|(t

)|

Time (ps)

100

5 meV

0

4

250

PL in

tensity

(a.u

.)

d = 1 d = 3

T = 5K

T=0.1K

Counts

T=6K

1 qubit: exact coherence for arbitrary substrate dimension d

reduced qubit state

ρq(t) = Trb[R(t)] =(

p0 ρ01(0)c(t)ρ∗

01(0)c∗(t) 1−p0

),

1-qubit coherence function c(t)

exact 1-qubit coherence cd(t) = exp[−λd(t)], with (sorry)

λd(t) = 8αs(−θ)d−1[

Fd−1(θ)−ReFd−1(θ[1+ iωct])]

+4αdΓ(d−1)

(cos[(d−1)arctan(ωct)]

(1+ω2c t2)(d−1)/2

−1

),

Γ(z) is Euler’s Gamma function,

F(z) = logΓ(z), and Fn(z) is its n-th derivative

1 qubit: incomplete pure dephasing in bulk substrate

0.94

0.96

0.98

1

exp−

Λ(3

)0,1(t)

0 5 10 15τ

kBT = 0.015 ~ωc

kBT = 0.15 ~ωc

kBT = 0.3 ~ωc

c(3)(τ→∞) → exp(−4α) (θ = kBT

ħωc→ 0)

α= 0.01, time scale for fast decay t∗ 'ω−1c

Vagov et al. PRB 2004

1 qubit: complete pure dephasing in a nanotube

0

0.25

0.5

0.75

1

coheren

ce

0 100 200 300 400 500

ωct

c(t)

cME(t)

r cME(t)

α = 0.05, kBT = 0.005hωc

LIN

0.1

0.2

0.5

1

coheren

ce

0 100 200 300 400 500

ωct

α = 0.05, kBT = 0.005hωc

LOG

master equation:

cME (t) = exp(−t/T2),T−1

2 = 4παkBT/ħ

exact coherence :

fast initial decay of c(t)time scale ħ/kBTamplitude

r =[

2πθ2θ−1

Γ2(θ)

]4α

' (2πθ)4α (θ¿ 1).

1 qubit: complete pure dephasing in a nanotube

“The non-Markovian nature of this decoherence mechanism may have adverse

consequences for applications of one-dimensional systems in quantum information

processing.” (Galland et al., (2008))

0

0.25

0.5

0.75

1

coheren

ce

0 100 200 300 400 500

ωct

c(t)

cME(t)

r cME(t)

α = 0.05, kBT = 0.005hωc

LIN

Q: 1D substrates less ideal?A: compare quantum errorcorrection rates

1 qubit: temperature dependence of QEC rates

1

10

100

1000QEC

rate

ωqec[√

ǫωc]

10−5 10−4 10−3 10−2 10−1 100 101 102 103

Temperature T [~ωc/kB]

ǫ = 10−4

ǫ = 10−3

ǫ = 10−2

αs = 0.1 s = 1s = 2

s = 3

0.01

0.1

1

10

10−4 10−2 100 102

αs = 0.001

1 qubit: coupling dependence of QEC rates

0.1

1

10

100

1000

QEC

rate

ωqec[ω

c]

10−5 10−4 10−3 10−2 10−1

coupling strength αs

ǫ = 10−4

ǫ = 10−3

ǫ = 10−2

kBT = 0.01ħωc

2 qubits: robust Bell states

Now: 2 qubits at distance x12, coupled to same phonon bath.

x1 x2

x

Qb 1 Qb 2

Known: 2 qubits coupled to same bath at same position:

|ψrobust⟩ =|01⟩+ |10⟩p

2

robust: decoherence-free subspace

Entanglement measure:Concurrence C[ρ] = max0,

√λ1 −

√λ2 −

√λ3 −

√λ4,

λi are ordered eigenvalues of ρσ1yσ2yρ∗σ1yσ2y.

2 qubits: entanglement of robust Bell state

0.8

0.9

1

C(3

)−

(τ)

0 5 10 15 20τ

τ12 = 0τ12 = 2τ12 = 10|ψ−〉

(c)

0

0.5

1

C(1

)−

(τ)

0 200 400 600 800τ

τ12 = 0τ12 = 200τ12 = 600

|ψ−〉

(a)

decoherence-poor subspace

upper = 3Dlower = 1D

coupling α= 0.01,

scaled temp.θ = kBT/ħωc = 0.015,

scaled distanceτ12 =ωcx12/vphon,

scaled time τ=ωct.

2 qubits: final concurrence of robust Bell state

0

0.25

0.5

0.75

1

C(∞

)

10−2 100 102 104 106 108

ωct12

kBT = 10−6 ~ωc

kBT = 10−5 ~ωc

kBT = 10−4 ~ωc

kBT = 10−3 ~ωc

C(1)− (τ→∞) = (1+τ2

12)4α∣∣∣∣Γ[θ(1− iτ12)]

Γ(θ)

∣∣∣∣16α

→ 1

(1+τ212)4α

(θτ12 → 0)

N qubits: entanglement dynamics

x1 x2 x3

x

Qb 1 Qb 2 Qb 3

W states:

|WN ⟩ = 1pN

(|100. . .0⟩+ |010. . .0⟩+ . . .+|000. . .1⟩)

N-qubit generalization of |W2⟩ = |ψrobust⟩, robust Bell state

Entanglement measure for N qubits: problematic

Here: fidelity F(t) = Trρ(t)ρ(0)

N qubits: fidelity dynamics and master equations

0

0.25

0.5

0.75

1

F(t)

0 5 10 15 20 25 30

ωct [104]

(b) kBT/~ωc = 10−3

N = 3

N = 6

0.5

1

100

103

106

ωct

N = 3

0

0.25

0.5

0.75

1

F(t)

(a) kBT/~ωc = 10−4

N = 3

N = 6

usual master equation

FME(t) = 1 ∀ t (bad)

causal master equation

d

dtρjj′ (t) =−8απkBT

ħ[

1−Θ(t − tjj′ )]ρjj′ (t)

Two assumptions for QEC violated1 exponential decay2 qubits in uncorrelated baths

N qubits: scaling of final fidelity

0

0.25

0.5

0.75

1

F(∞

)

1 10 20 30 40

N

exact, α = 0.001approx.exact, α = 0.01approx.

F(∞) '[Γ( 1

2 −4α)

Γ( 32 −4α)

− 1

1−4α

](Nωct12)−8α→ (Nωct12)−8α (α< 0.005)

conclusions

experiments:i. exciton spectra in 3D and on 1D substratesii. non-exponential dephasing challenge for QIP

1 qubit:i. fast initial dephasingii. stabilization in 3D, exponential decay in 1Diii. lowest error correction rates for 1D

2 qubits:i. entanglement stabilization in 3Dii. and in 1D for the robust state

N qubits:i. Incomplete pure dephasing of W states, exact dynamicsii. Master equation very inaccurate. We improved it.

thanks to . . .

Roland DollSigmund Kohler (Augsburg)Peter Hänggi

Thanks for your attention!

1 Two-qubit entanglement, EPL 76, 547 (2006),

2 Incomplete pure dephasing of N qubits, PRB 76,

045317 (2007),

3 Quantum error correction rates, EPJ B 68, 523 (2009).

4 Exact results from approximate master equations,

Chem. Phys. 347, 243 (2008). www.martijnwubs.nl

2 qubits: entanglement of fragile Bell state

0.8

0.9

1

C(3

)+

(τ)

0 5 10 15 20τ

τ12 = 0τ12 = 2τ12 = 10

|ψ+〉

(d)

0

0.5

1

C(1

)+

(τ)

0 200 400 600 800τ

τ12 = 0τ12 = 200τ12 = 600

|ψ+〉

(b)

upper = 3D, superohmiclower = 1D, ohmic

coupling α= 0.01,scaled temp.θ = kBT/ħωc = 0.015,scaled distanceτ12 =ωcx12/vphon,scaled time τ=ωct.

general identity

C(d)+ (t)C(d)

− (t) = |c(d)(t)|4

Deformation-potential coupling dominates

Pure dephasing due to deformation-potential coupling:

Pure dephasing due to piezo-electric coupling:

Krummheuer et al., PRB (2002)

1 qubit: Non-Markovian dephasing and lineshape

Krummheuer et al. (2002)