Teleportation of electronic many- qubit states via single photons

Transcription

1 (*) NanoScience Technology Center and Dept. of Physics, University of Central Florida, homepage: Teleportation of electronic many- qubit states via single photons Michael N. Leuenberger (*) and Michael E. Flatté University of Iowa David D. Awschalom University of California, Santa Barbara Phys. Rev. Lett.. 94, 1741 (5) Supported by DARPA/ARO, DARPA/ONR, and US NSF

2 Teleportation of single photon states via EPR pairs Theory: Bennett et al., PRL 7, 1895 (1993) Experiment: Bouwmeester et al., Nature 39, 575 (1997) Initial state: 1 = α 1 + β 1 ( ± ) Ψ 3 = ± / Bell states: ( ) ( ± ) Φ 3 = ( ± ) / ( ) Shared EPR state: Ψ = ( ) 3 / Teleportation by means of Bell measurement (correlation) of photon 1 and. ( ) 3-photon state: Ψ 13 = 1 Ψ3 1 ( ) ( + ) Ψ = α β α β Ψ + Ψ + ( ) ( + ) +Φ1 ( β 3 + α 3 ) +Φ1 ( β 3 + α 3 )

3 Quantum teleportation across the river Danube Ursin et al., Nature 43, 849 (4) c fiber = 3 c vacuum The quantum channel (optical fiber F) rests in a sewage-pipe tunnel below the Danube in Vienna with a total length of 8 m. Fidelity>8%.

4 Teleportation of electronic many-qubit states via single photons GHZ teleportation method: - GHZ state of spin-photon-spin system - No Bell measurements required! - fully scalable to arbitrarily many qubits

5 Greenberger-Horne-Zeilinger states (GHZ) ( i ) Ψ GHZ = +e ϕ / Preparation of GHZ states by means of interaction Let us assume Alice measures in z direction (5% probability): Then Bob measures in z direction (1% probability): quantum correlation (entanglement) quantum correlation (entanglement) Then Carol measures in z direction (1% probability):

6 me m k = dk = de me Quantum dots d d k 1 n= f( E( k)) d f( E) = ( E μ )/ e ( π ) kbt dk kdk m me = = 3 3 de π π ( π ) Bayer et al., Nature 45, 93 (): dk kdk m = = de π π ( π ) dk m = de π π me

7 Classical Faraday effect (circular dichroism, circular birefringence) Faraday rotation angle: β = ν Bd Verdet constant is proportional to the difference in index of refraction for right (σ + ) and left (σ - ) circularly polarized light, which is proportional to the magnetization of the material: ν η η M σ + Horizontal linear polarization ( ): Vertical linear polarization ( ): σ 1 e = e = 1 Wavevector of light: k = = k z π / λ Right circular polarization (σ + ): 1 e = i Left circular polarization (σ - ): 1 e = i

8 Optical selection rules single-photon Faraday rotation entanglement of photon and spin σ + ( z) σ ( z) virtual process involving a heavy hole virtual process involving a light hole σ ( z) σ + ( z) virtual process involving a heavy hole Spherical dots 5% selection rules for nonresonant interaction virtual process involving a light hole Phase shift of photon for virtual process involving heavy holes is 3 times larger than for virtual process involving light holes net Faraday rotation of polarization of photon Phase shift depends on spin state entanglement of photon with spin

9 Optical selection rules single-photon Faraday rotation Leuenberger, Flatte, Awschalom, Phys. Rev. Lett. 94, 1741 (5) S z =1/ S z =-1/ S z =-1/ S z =1/ } ωd { σ + σ ( z) ( z) J z =3/ J z =-3/ J z =3/ J z =-3/ σ + ( z) right circularly polarized photon σ ( z) left circularly polarized photon Non-spherical dots (strong anisotropy splits heavy and light holes) 1% selection rules for nonresonant interaction accelerated Faraday rotation Phase shift of photon for virtual process depends on spin state Faraday rotation of linear polarization of photon Direction of Faraday rotation depends on spin state entanglement of photon with spin

10 Single-qubit teleportation Spin state of excess electron in quantum dot of destination: Incoming photon is linearly polarized in, e.g. x direction: After interaction of photon with quantum dot: ( ) pe t A = hh lh is ( t + T) = e + e is pe A hh lh + hh ( σ ( z) σ + Process involving heavy/light hole: = + ( z) ) / = ( σ z + σ z ) Conditional phase shift for maximum entanglement: The photon is now sent to origin D. Photon at D and spin at D are in a Bell state (EPR pair): pe S S S ( + ) hh lh = = π / ( t + T) = / A ( ) 1 e = = + lh / (similar to distribution of EPR pair produced by e.g. parametric down-conversion in traditional EPR teleportation) Preparation of spin at D: e = α + β When teleporting a qubit from D to D is desired, photon at D interacts with spin at D: epe ( α α ) ( tc ) = + + β + β / Formation of GHZ state of spin at D, photon, and spin at D

11 Traditional teleportation: Theory: Bennett et al., PRL 7, 1895 (1993) Collapse of the GHZ state Experiment: Bouwmeester et al., Nature 39, 575 (1997) Measurement of linear polarization of photon: ( ) ee 1 t D α β = + ( ) ee t D α β = + Single-spin detection of D in x direction (possible with a single photon) or { ( ) e1 = + t D α β ( ) e = t D α β ( + ) ( α + β ) ( α β ) ( α β ) ( β α ) ( β α ) } / ( ) / epe tc + T = α β + = No interaction of spin at D and photon - Bell measurement of spin at D and photon Teleportation scheme here: Single-particle measurement of spin at D and photon collapse of GHZ state Knowledge of detection outcome teleportation from D to D complete or or or ( ) e3 t D β α = + ( ) e4 = t D β α

12 Nonperturbative calculation of the singlephoton Faraday rotation Full Hamiltonian: H = HL + V Luttinger-Kohn Hamiltonian for spherical dots: H L e Interaction Hamiltonian: V = Ap mc Vector potential of electromagnetic field of photon: e Vhh = I; V g; i = A pcv = 3Vlh = 5 μev mc Full Hamiltonian in rotating frame: H 1.5 mev = ω Γ= 1 μev Basis states: Initial state: E V V hh E 5 p γ = ( γ1 + γ) pj m m ( ) iωt + iωt ( ) A() t = A () t ε a e a e hh d = E Vlh + ( z) ( z) + ω σ ;, hh-exciton ; σ ;,lh-exciton + ( ) ( ) = = σ + σ / t A z z V / ω 1 σ and σ remain fully populated. + lh d ( z) ( z) After interaction time T: Interaction frequencies: phase shifts are different: d V E σ + ω lh d qσ qσ qσ i ET i T i T A + = ( z) + ( z) σ σ Vhh 3V V lh lh Ω hh = = Ω lh = ω ω ω Ωhh + Ωlh ( t T) e e e / S =Ω hh hh d T d S =Ω lh lh d T E gap T =1 ns = ( ω + ω ) = 1.4 ev Faraday rotation! d

13 Single-photon Faraday rotation Incoming photon is linearly polarized in, e.g. x direction: ( ep t ) A ( α β ) Spin state of excess electron in quantum dot of origin: Faraday rotation angle: S / = π /4 After interaction of photon with quantum dot: Process involving heavy/light hole: ep ( A e = α + β = + hh lh is ( t + T) = e + e is ep A hh lh + ( z z ) + ( z z ) = α σ + β σ hh / = α σ + β σ lh / Conditional phase shift for maximum entanglement: S S S π hh lh = = / Faraday rotation of photon polarization: t + T) = α + β Measurement of spin in x direction photon projected onto optospintronic link between photons and electron spins t B ( ) p = α ± β

14 Enclosing dot in a microcavity permits precise control of the interaction time Accumulated phase shift during interaction time T: S = S S = Ω Ω T=ΩT ( ) hh lh hh lh π 9 1 volume cavity Ω = 1 s and T = 1 ns S = π / maximum entanglement T controlled by active Q-switching with 1 ps resolution phase error 1 ps/ 1 ns =.1% 7 T / τ 8 T / τ Q= 1 1 e = 4% Q = τ = 43 ns escape probability 1 e = % 6 T / τ Q= 1 1 e = 37% μm

15 Enclosing dot in a microcavity permits precise control of the interaction time 3.4 μm volume cavity T = 5 ps S = π / Naruse et al., Appl. Phys. Lett. 83, 4869 (3): maximum entanglement T controlled by active Q-switching with 1 fs resolution phase error 1 fs/ 5 ps = 1-3 ΓphotonT /4 Γphoton 4 Phase error due to bandwidth of photon: = = ωdt /4 ωd In D photonic crystal: Lodahl et al., Nature 43, 654 (4): τ = ns escape probability 1 e =.6 1 T / τ 3 τ = 7 ns 1 e =.7 1 T / τ 4

16 Two-qubit teleportation Two-qubit state in quantum dots of D : Incoming photons linearly polarized: After interaction of photons with qubits: () t A = e ( ) t A = pe ( ) Both photon interactions involve heavy holes: hh hh lh ( ) lh is is S is ( t T) e e + + = + + e () () () () pe A hh hh,lh lh ( z z z z z z z z ) σ σ σ σ σ σ σ σ () hh = /4 One photon interaction involves a heavy hole, the other photon interaction a light hole: ( z z z z )( ) ( z z z z )( ) σ σ σ σ σ σ σ σ () hh,lh = /4 Both photon interactions involve light holes: ( + z z z z a + z z a z z ) σ σ σ σ σ σ σ σ () lh = /4 Conditional phase shift for maximum entanglement: S S S π hh lh = = / Faraday rotation of photon polarization: ( + ) () ta T ( ) pe + = + + /

17 Formation of two entangled GHZ states Two-qubit state in quantum dots of origin: Each photon interacts with each spin at D separately: () e ( tc ) a a a a = ( ) ( a a a a ) ( a + a + a + a ) ( a a a + a ) ( ) = / () epe tc a a a a ( ) 1 After interaction: ( t + T) = Measurement of linear polarization of the two photons: () () () () () epe C ee 1 ee ee 3 ee 4 ( t ) = a a a + a () ee 1 D ( t ) = a + a a a () ee D ( t ) = a a + a a () ee 3 D ( t ) = a + a + a + a () ee 4 D / / / a = a = a = a = 1 production of four-qubit Bell states between D and D

18 Collapse of the GHZ states ( ) ee 1 t = a a a + a () D = + ( a a a + a ) + ( a + a a a ) ( a a + a a ) + ( a + a + a + a ) ( ) ee t = a + a a a () D = ( ) ee 3 t = a a + a a () D = ( ) ee 4 t = a + a + a + a () D = ( a + a a a ) + ( a + a + a a ) ( a + a + a + a ) ( a + a a + a ) + + ( a a + a a ) + ( a + a + a + a ) ( a + a + a a ) ( a a + a a ) ( a + a + a + a ) + ( a + a a + a ) ( a a + a + a ) + ( a a a + a ) + Knowledge of detection outcome teleportation from D to D complete Generalization to n qubits is straightforward. Complexity of n-qubit GHZ teleportation scales linearly with n

19 Production of entangled photons Independent nonresonant interaction of two photons with two quantum dots: () ( ) ep ta T a a a a + = Change to S x representation: 1 ( ta + T) = Measuring spins in x direction: ( ) () () () () () ep p1 p p3 p4 () p1 = a + a + a + a or or () p = a a + a a () p3 = a + a a a see also Leuenberger, Flatte, Awschalom, Europhys. Lett. 71, 387 (5). or () p4 = a a a + a Knowledge of detection outcome direct teleportation from spins to photons 1 ( ± ) a = a =, a = a = production of Bell states Ψ p = ( ± ) / 1, ( ± ) a = a = a = a = production of Bell states Φ p = ( ± ) /

20 Conclusions The many-qubit state of one quantum dot system (origin) can be teleported to a second quantum dot system (destination) via nonresonant interaction of single photons with the quantum dots (producing qubit-photon-qubit entanglement, i.e. GHZ states). Single-photon measurements are sufficient to teleport not only a single-qubit state but also a many-qubit state (no Bell measurements are required). Reason: GHZ entanglement The state of the excess electron selects the circular polarization of the photons that gets entangled with the many-qubit state via conditional Faraday rotation. GHZ teleportation scheme opens up the possibility to build a Quantum Dynamic RAM (QDRAM) where the quantum error corrections are applied to the manyqubit state (short spin decoherence time of electrons).

21 Parametric down-conversion Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko, Shih, PRL 75, 4337 (1995) Noncollinear type-ii phase matching: Nonlinear crystal: β-bariumborate [β-ba(b O 4 ) or β-ba 3 (B 3 O 6 ) ] 1 i Output state on overlapping cones: = 1 ( + e ϕ 1 ) Ψ Ordinary and extraordinary polarization: vertical and horizontal 1 1 ϕ is controlled by birefringent phase shifter or by rotating the crystal. 1 i λ/ plate for one photon produces: 1 ( e ϕ ± = ) Φ 1 Efficiency: photons out of 1 6 are entangled. Energy conservation: ω = ω1+ ω Momentum conservation: k = k1+ k Degenerate case: ω = ω = ω / 1 ±

22 Quantum mechanics of PDC i kir ω Incident classical pump light field (UV): Vr (,) t = Ve Interaction Hamiltonian in the interaction picture: ( t) 3 ˆ dx () [( 1 ) ( 1 ) ] 3 (, 1, ) i k k k ir ω ω ω t I = χijl ω ω ω ( εk ) ( ) 11 s ε s l 11 s s + H.C. L i k V j k k v k1, s1 k, s H t a a e Initial state of the quantum field: t i () exp ' ˆ t = (') 1 dt HI t vac i 1 = χ ( ω, ω, ω ) 1 1 ( t = ) = 1 vac i 1 ( εk ) ( εk ) V ak ak e vac 3 ijl 1 s s l s s L i j k, s k, s sin[ ( 1 ) ] i k k k mlm sin[ ( ( 1 ) ] 1 ) ω ω ω t ω ω ω t e 1 1 m= 1 ( k k1 k) m ( ω ω1 ω) t ( k k k ) ir k k V 1s1, s + O ( ) R is the midpoint of the nonlinear medium which has the shape of a rectangular parallelopiped of sides l 1,l,l 3. Type-II phase matching: center of and cones or not collinear.