OPTICAL QUANTUM DOTS FOR QUANTUM INFORMATION Tom Reinecke Naval Research Laboratory Washington, DC , USA reinecke@nrl.navy.mil outline outline • single spin qubits • single spin qubits • two qubit gates • two qubit gates • entanglement in QD molecules • 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 z θ spin in QD φ E 2 x E 1 Ψ = α ↑ + β ↓ ~5-10 nm Quantum dots optical properties, control (trion) • spin natural qubit • ‘long’ coherence • fast optical manipulations • integrate into semiconductor technology • potentially scalable • fixed locations HH LH
Quantum dots -1V QD C.B. E F 0V e - Stranski-Krastanov ‘self V.B. organized’ QDs spins in low states in QDs substrate GaAs : small lattice constant deposited Material InAs : large lattice constant X + X 2- X 3- X ° X - QDs WL trion 1272 X + 1269 Intensity PL energy (meV) X ° 1266 1263 X - 1260 X 3- X 2- 1257 S-shell 1254 20.1 33.6 47.0 60.4 73.8 87.2 Electric Field (kV/cm)
z Spin initialization, manipulation x B x ≠ 0 B = 0 | 3/2 3/2 > |-3/2> |-3/2> |3/2> |3/2> |-3/2> |-3/2> σ + σ + σ - σ - σ + σ - |1/2>+ |1/2>+|- |-1/2> 1/2> ≡ |+x> |+x> |1/2>-|-1/2> |1/2>-|-1/2> ≡ |1/2> |1/2> |-1/2> -1/2> |-x> |-x> • Voigt field mixes spin states & • No Raman transitions |1/2> � |-1/2> enables Raman transitions • Cannot make arbitrary spin rotation • optical pumping gives initialization (‘z basis’ - growth direction/optical axis) (‘x basis’)
Useful feature of 2-level systems ⎛ ⎞ 0 ( ) V t ge ⎜ ⎟ ( ) ~ H t |e> − * ( ) ( ) V t E E ⎝ ⎠ Δ 0 ge e I analytical solution β −1 = Ω β Δ ) i t sec ( V h t e t |g> ge Δ = detuning β = pulse bandwidth |e> properties: Δ = 0 Ω = σ � 2 π , no excitation left in |e> (ι) (ii) pulse area indep of detuning (iii) |g> acquires phase β Δ 2 φ = Δ − tan( ) β 2 2 |g> (Rosen & Zener, Phys. Rev., 1932)
Spin rotations around one axis (z) | 3/2 3/2 > |-3/2> |-3/2> ω ⎛ ⎞ 0 0 B ⎜ ⎟ ω Ω ω i t ( ) ~ 0 ( ) H t t e 0 ⎜ ⎟ σ + B ⎜ ⎟ − ω Ω ε * i t ⎝ 0 ( ) ⎠ t e 0 T B { } ‘z-basis’ , , z z T |1/2> |1/2> |-1/2> |-1/2> Pulses: Δ = Ω β ) i t sec ( V h t e B x ge (fast pulses: β >> ω B ) Properties: z (i) 2 π pulses return to spin subspace with no trion excitation θ (ii) 2 π pulses give phase φ of |z> wrt |z’> equivalent to a z rotation φ φ = β Δ 2arctan( / ) x
Rotations around another axis (x) |3/2> - |3/2> - |-3/2> -3/2> • single, broad band, linearly |3/2> |3/2> + + |-3/2 |-3/2> polarized pulse couples both transitions H H • different phase |x> and |x’> • pulse area independent of detuning • gives rotation ( φ 1 - φ 2 ) |+ |+x> x> |-x> |-x> z π x polarizations x general rotations from those about z and x
general rotations, full solutions •density matrix calculations including mixing, losses pulses z rotation z rotation •arbitrary rotation from rotations around two axes x rotation ( ) ( ) + φ = π φ π ( ) / 2 ( ) / 2 R R R R y z x z Fidelity 99.28% fast gates ~ 60 psec parameters for InAs QDs Economou & Reinecke, PRL (2007)
Spin rotations – φ vs Δ S z control σ ⎛ ⎞ φ = 2 arctan ⎜ ⎟ Δ ⎝ ⎠ general rotations demonstrated by varying precession Greilich et al , Nature Physics 5 , 262 (2009)
Cavity coupling Cavity development fast, distant, frequency selective coupling and architectures laser cavity ~ μ m strong coupling (coherent) weak coupling (Purcell) In 0.3 Ga 0.7 As 100 nm ω = ω ± − γ + γ 2 2 ( ) / 4 g ± − c e p c e 2 e f = g 4 πκε V 0 m
Strong coupling one dot spectra one dot spectra γ C =0.18 meV r C =0.75 μ m C 4000 40K γ X <0.05 meV P=2 μ W X Temperature Q= 7350 3500 25 K 3000 Intensity (arb units) 2500 Intensity (a.u.) 2000 1500 1000 C 500 X 10 K 10K 0 1.3260 1.3265 1.3270 1.3275 1.3280 1.3285 1322.5 1323.0 1323.5 Energy (meV) Temperature (K) d050318 • g ex-p ~ 100-200 meV • 40 pillars from several wafers Reithmaier et al, Nature 432 , 197 (2005) with g ex-p > 50 meV • confirmed by intensity and linewidths
Two qubit entangling gate (theory) Requirements for two qubit (entangling) gate: • (Long-range) physical interaction • Dynamical control of interaction • Conditionality (control-U operation) qubit 2 • Scalability ω 0 cavity mode qubit 1 qubit 1 InAs dot InAs dot entangled e − e − GaAs If “1” growth qubit 2 Add “ ‐ ” direction ( π phase) Magnetic Field photonic crystal membrane compatible with one qubit operations
system n = System System ( 1,2) = − ω + ω † † H c c c c − ↑ ↑ ↓ ↓ e t n , e e n n n n ( ) ( ) + ε − ω + ε + ω † † Magnetic field perpendicular t t t t ↑ ↑ ↓ ↓ n h n h n n n n to the growth axis (z) = ω † H a a | ⇓〉 z c 0 | ⇑〉 ( )( ) = + + + † † † . . H g t c t c h c a a r E − ↑ ↑ ↓ ↓ E E e c n , n n n n ⊥ B Pulses Pulses ( ) ∑ | ↓〉 = Ω − ω + † ( ) ( )cos . . | ↑〉 V t t t t t c h c ξξ ξ ξ ( ') ' p p p n n p ω ω ξξ =↑ ↓ ' , = σ + σ (1) (2) e e = H 1,2 n qubits z z 2 2
spectra | ⇓⇓〉 Energy qubits + | ⇓⇑〉 2 photons trion 2 + | ⇑⇓〉 photon | ⇑⇑〉 ω trion 1 + 0 Energy photon (cavity mode frequency) trion 1+2 ω 0 qubits + photon | ↓⇓〉 | ↓⇑〉 trion 2 | ↑⇓〉 | ↑⇑〉 trion 1 | ⇓↓〉 | ⇑↓〉 | ⇓↑〉 | ⇑↑〉 qubit states relevant optical transitions
Single qubit operations: compatibility • effects of cavity-induced splitting in one-particle sector negligible for short pulses • single qubit operations same as previously | ↓⇓〉 | ↓⇑〉 • need single and two-qubit operations for complete set of | ↑⇓〉 | ↑⇑〉 operations | ⇓↓〉 trion | ⇑↓〉 | ⇓↑〉 | ⇑↑〉 | ↓↓〉 | ↑↓〉 | ↓〉 | ↑↑〉 | ↑↓〉 | ↓↑〉 | ↑〉 | ↑↑〉 2 0 fast pulses
Two-qubit gates: CZ gate universal CNOT one ‐ qubit gates U = H H CZ ⎛ ⎞ ψ 〉 = α ↑↑〉 + β ↑↓〉 + δ ↓↑〉 + γ ↓↓〉 1 0 0 0 | | | | | ⎜ ⎟ − 0 1 0 0 ⎜ ⎟ = ⎜ U ⎟ g 0 0 1 0 ψ 〉 = α ↑↑〉 − β ↑↓〉 + δ ↓↑〉 + γ ↓↓〉 | | | | | ⎜ ⎟ ⎝ 0 0 0 1 ⎠
CZ gate | ⇓⇓〉 | ⇓⇑〉 | ⇑⇓〉 | ⇑⇑〉 Pulse Pulse B B Energy (4 ‐ ‐ 16) 16) (4 ω 0 two ‐ particle two trions (one in each dot) | ↓⇓〉 subspace | ↓⇑〉 phase change (2 π pulse), Pulse B − 1 one ‐ | ↑⇓〉 particle one trion | ↑⇑〉 subspace population inversion ( π pulse), Pulse A − − i i | ⇓↓〉 qubit | ⇑↓〉 subspace | ⇓↑〉 ↑↑〉 ↑↓〉 ↓↑〉 ↓↓〉 | | | | Pulse A | ⇑↑〉 Pulse A (0 ‐ ‐ 4 and 1 4 and 1 ‐ ‐ 6) 6) (0 − × − × − − × − ( ) ( 1) ( ) ( ) ( ) i i i i +1 ‐ 1
Fidelity Γ cavity 1. losses due to unwanted coherent Γ tr dynamics of off-resonant transitions Γ tr 2. losses due to carrier recombination and cavity losses -- can be improved through e.g., pulse shaping Solenov, Economou and Reinecke, to be published
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 • Tunnel coupled QDs • Exchange interaction always on • Can access optically single qubit and two-qubit states Energy diagram S T Initialization triplet X singlet Δ T/T Energy (1 division =25 μ eV) kinetic T exchange Δ ee S
Short vs. Long optical pulses Short pulses Long pulses Time domain: Time domain: • Acts on single spin state because • Acts on joint spin state because slower faster than exchange interaction than exchange interaction Frequency domain: Frequency domain: X X T T S S Acts just on S Acts on S + T
Optical 2-qubit phase gate ( ) ↑ ↓ − ↓ ↑ ⇒ φ ↑ ↓ − ↓ ↑ e i SWAP gate ↑↓ ⇒ ↑ ↓ − ↓ ↑ ϕ = π , for 2 i Pulse sequence pulse 1 pulse 2 2 π control pulse (1 qubit) (1 qubit, 2 qubit, fixed delay variable delay) & 2qubit pulse Ramsey fringes with control pulse & 2qubit pulse Can vary phase change from -180 ° to Kim et al , Nat. Physics, 7, 223 (2011) +180 ° with pulse detuning
Alternative qubit: hole spin Ramsey fringes of single hole qubit Coupled hole qubits B=0T T S
Recommend
More recommend
