OPTICAL QUANTUM DOTS FOR QUANTUM INFORMATION Tom Reinecke Naval - - PowerPoint PPT Presentation

OPTICAL QUANTUM DOTS FOR QUANTUM INFORMATION

Tom Reinecke Naval Research Laboratory Washington, DC , USA reinecke@nrl.navy.mil

• utline
• single spin qubits
• two qubit gates
• entanglement in QD molecules
• utline
• single spin qubits
• two qubit gates
• entanglement in QD molecules

NRL theory: S. Economou, D. Solenov, T. Reinecke NRL expt: D. Gammon, A. Bracker, S. Carter

• U. Wuerzburg: fabrication, expt
• U. Dortmund: expt

support: DARPA, NSA and ONR

Optical quantum dots

spin in QD Quantum dots

• spin natural qubit
• ‘long’ coherence
• fast optical manipulations
• integrate into semiconductor

technology

• potentially scalable
• fixed locations

E1 E2

~5-10 nm

θ φ

z x

α β Ψ = ↑ + ↓

HH LH

• ptical properties, control (trion)

substrate GaAs: small lattice constant deposited Material InAs: large lattice constant

WL QDs

Stranski-Krastanov ‘self

• rganized’ QDs

Quantum dots

Electric Field (kV/cm)

20.1 33.6 47.0 60.4 73.8 87.2 1254 1257 1260 1263 1266 1269 1272

PL energy (meV)

X+ X° X- X2- X3-

S-shell

Intensity

trion X+ X° X- X2- X3-

spins in low states in QDs

• 1V

QD EF V.B. C.B. e- 0V

• No Raman transitions |1/2> |-1/2>
• Cannot make arbitrary spin rotation

Spin initialization, manipulation

(‘z basis’ - growth direction/optical axis)

|1/2>-|-1/2> |1/2>-|-1/2> ≡ |-x> |-x>

σ- σ+

|3/2 3/2> |-3/2> |-3/2>

σ- σ+

Bx ≠ 0

|1/2>+ |1/2>+|- |-1/2> 1/2> ≡ |+x> |+x>

• Voigt field mixes spin states &

enables Raman transitions

• optical pumping gives initialization

(‘x basis’)

x z

σ- σ+

|3/2> |3/2> |-3/2> |-3/2> |1/2> |1/2> |-1/2>

• 1/2>

B = 0

Useful feature of 2-level systems

|e> |g>

Δ = detuning β = pulse bandwidth

sec ( ) i t

ge

V h t e β

Δ

= Ω

analytical solution

t

I

β−1

|g> |e>

*

( ) ( ) ~ ( ) ( )

ge ge e

V t H t V t E E ⎛ ⎞ ⎜ ⎟ − ⎝ ⎠

Δ

properties: (ι) Ω = σ 2π, no excitation left in |e> (ii) pulse area indep of detuning (iii) |g> acquires phase

2 2

2 tan( ) β φ β Δ = Δ −

Δ = 0 (Rosen & Zener, Phys. Rev., 1932)

Spin rotations around one axis (z)

*

( ) ~ ( ) ( )

B i t B i t T

H t t e t e

ω ω

ω ω ε

−

⎛ ⎞ ⎜ ⎟ Ω ⎜ ⎟ ⎜ ⎟ Ω ⎝ ⎠

{ }

, , z z T

‘z-basis’

|1/2> |1/2>

σ+

|3/2 3/2> |-3/2> |-3/2> |-1/2> |-1/2>

B Bx

(fast pulses: β>>ωB)

2arctan( / ) φ β = Δ

θ φ

z x

Properties: (i) 2π pulses return to spin subspace with no trion excitation (ii) 2π pulses give phase φ of |z> wrt |z’> equivalent to a z rotation

sec ( ) i t

ge

V h t e β

Δ

= Ω

Pulses:

Rotations around another axis (x)

• single, broad band, linearly

polarized pulse couples both transitions

• different phase |x> and |x’>
• pulse area independent of detuning
• gives rotation (φ1-φ2)

H

|3/2> - |3/2> - |-3/2>

• 3/2>

|+ |+x> x>

|3/2> |3/2> + + |-3/2 |-3/2>

H

|-x> |-x>

z x

general rotations from those about z and x πx polarizations

general rotations, full solutions

( ) ( )

( ) / 2 ( ) / 2

y z x z

R R R R φ π φ π

+

=

parameters for InAs QDs Fidelity 99.28% fast gates ~ 60 psec

• density matrix calculations

including mixing, losses

• arbitrary rotation from

rotations around two axes

Economou & Reinecke, PRL (2007)

z rotation x rotation z rotation pulses

Greilich et al, Nature Physics 5, 262 (2009)

Spin rotations – φ vs Δ

2 arctan σ φ ⎛ ⎞ = ⎜ ⎟ Δ ⎝ ⎠

control

general rotations demonstrated by varying precession Sz

fast, distant, frequency selective coupling and architectures

Cavity coupling

laser ~ μm

m

V f e g

2

4πκε =

2 2

( ) / 4

c e p c e

g ω ω γ γ

± −

= ± − +

strong coupling (coherent) weak coupling (Purcell)

cavity Cavity development

100 nm

In0.3Ga0.7 As

• gex-p ~ 100-200 meV
• 40 pillars from several wafers

with gex-p > 50 meV

• confirmed by intensity and linewidths
• ne dot spectra

Strong coupling

Reithmaier et al, Nature 432, 197 (2005)

1322.5 1323.0 1323.5

Intensity (arb units)

25 K 10 K C C X X

rC=0.75 μm P=2 μW Temperature γC=0.18 meV γX<0.05 meV

Q= 7350

Energy (meV)

1.3260 1.3265 1.3270 1.3275 1.3280 1.3285 500 1000 1500 2000 2500 3000 3500 4000

Intensity (a.u.)

Temperature (K)

d050318 10K 40K

• ne dot spectra

Requirements for two qubit (entangling) gate:

• (Long-range) physical interaction
• Dynamical control of interaction
• Conditionality (control-U operation)
• Scalability

ω cavity mode qubit 1 qubit 2

InAs dot GaAs

e−

InAs dot

e− Magnetic Field growth direction photonic crystal membrane

qubit 1 qubit 2 If “1” Add “‐” (π phase) entangled

Two qubit entangling gate (theory)

compatible with one qubit operations

( )( )

† † † ,

. .

e c n n n n n

H g t c t c h c a a

− ↑ ↑ ↓ ↓

= + + +

( ) ( )

† † , † † e t n e e n n n n n h n h n n n n

H c c c c t t t t ω ω ε ω ε ω

− ↑ ↑ ↓ ↓ ↑ ↑ ↓ ↓

= − + + − + +

† c

H a a ω =

( )

† ( ') ' ' , 1,2

( ) ( )cos . .

p p p n n p n

V t t t t t c h c

ξξ ξ ξ ξξ

ω

=↑ ↓ =

= Ω − +

∑

System System Pulses Pulses

( 1,2) n =

system

|↑〉 |↓〉 |⇑〉 |⇓〉

E฀ E฀

⊥

E

z

B r

Magnetic field perpendicular to the growth axis (z)

(1) (2)

2 2

e e qubits z z

H ω ω σ σ = +

spectra

trion 2 trion 1 qubit states trion 1+2 qubits + photon trion 1 + photon trion 2 + photon qubits + 2 photons

ω

Energy

ω

Energy

(cavity mode frequency) relevant optical transitions

|↑⇑〉 |↑⇓〉 |↓⇓〉 |↓⇑〉 |⇑↑〉 |⇓↑〉 |⇓↓〉 |⇑↓〉 |⇑⇑〉 |⇓⇓〉 |⇑⇓〉 |⇓⇑〉

• effects of cavity-induced splitting in
• ne-particle sector negligible for

short pulses

• single qubit operations same as

previously

• need single and two-qubit
• perations for complete set of
• perations

Single qubit operations: compatibility

|↑〉 |↓〉

trion

|↑⇑〉 |↑⇓〉 |↓⇓〉 |↓⇑〉 |⇑↑〉 |⇓↑〉 |⇓↓〉 |⇑↓〉 |↑↑〉 |↑↓〉 |↓↓〉 |↓↑〉 2 |↑↑〉 |↑↓〉

fast pulses

CNOT

• ne‐qubit

gates

U

universal

CZ

g

1 1 1 1 U ⎛ ⎞ ⎜ ⎟ − ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

| | | | | ψ α β δ γ 〉 = ↑↑〉 + ↑↓〉 + ↓↑〉 + ↓↓〉 | | | | | ψ α β δ γ 〉 = ↑↑〉 − ↑↓〉 + ↓↑〉 + ↓↓〉

H H

=

Two-qubit gates: CZ gate

Pulse A Pulse A (0 (0‐ ‐4 and 1 4 and 1‐ ‐6) 6) Pulse Pulse B B (4 (4‐ ‐16) 16)

ω

Energy

|↑⇑〉 |↑⇓〉 |↓⇓〉 |↓⇑〉 |⇑↑〉 |⇓↑〉 |⇓↓〉 |⇑↓〉 |⇑⇑〉 |⇓⇓〉 |⇑⇓〉 |⇓⇑〉

CZ gate

| | | | ↑↑〉 ↑↓〉 ↓↑〉 ↓↓〉

• ne‐

particle subspace two‐particle subspace qubit subspace

• ne trion

two trions (one in each dot)

population inversion (π pulse), Pulse A phase change (2π pulse), Pulse B ‐1 +1

i − i − 1 − ( ) ( 1) ( ) i i − × − × − ( ) ( ) i i − × −

cavity

Γ

tr

Γ

tr

Γ

Fidelity

Solenov, Economou and Reinecke, to be published

• 1. losses due to unwanted coherent

dynamics of off-resonant transitions

• 2. losses due to carrier recombination and

cavity losses

• - can be improved through e.g., pulse

shaping

Scalability

• gates for unequal dot

excitation energies allow coupling between multiple dots

• multiple confined

photon exist modes in cavities

2 electron spins in two dots

Energy (1 division =25 μeV) triplet singlet

ΔT/T

• Tunnel coupled QDs
• Exchange interaction always on
• Can access optically single qubit and two-qubit

states

S T

T X S

kinetic exchange

Δee Initialization

Energy diagram

Short vs. Long optical pulses

Short pulses Long pulses

Time domain:

• Acts on single spin state because

faster than exchange interaction Time domain:

• Acts on joint spin state because slower

than exchange interaction Frequency domain: Frequency domain: S T X Acts on S + T T X S Acts just on S

Optical 2-qubit phase gate

pulse 1 (1 qubit) pulse 2 (1 qubit, variable delay) 2π control pulse 2 qubit, fixed delay

Ramsey fringes with control pulse Pulse sequence

Can vary phase change from -180° to +180° with pulse detuning

( )

2 for , π ϕ

φ

= ↑ ↓ − ↓ ↑ ⇒ ↑↓ ↑ ↓ − ↓ ↑ ⇒ ↑ ↓ − ↓ ↑ i ei

gate SWAP

& 2qubit pulse & 2qubit pulse

Kim et al, Nat. Physics, 7, 223 (2011)

Alternative qubit: hole spin

Ramsey fringes of single hole qubit Coupled hole qubits B=0T T S

SUMMARY

• single qubit operations
• single qubit operations
• Two qubit phase gate
• Two qubit phase gate
• Spins in quantum dots
• Spins in quantum dots
• entanglement in QD

molecules

• entanglement in QD

molecules

g

1 1 1 1 U ⎛ ⎞ ⎜ ⎟ − ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

S T

NRL Sophia Economou Dmitry Solenov Dan Gammon Sam Carter Allan Bracker Dortmund University Manfred Bayer group Alex Greilich Wuerzburg University Alfred Forchel group Support: ONR DARPA NSA

Acknowledgements