AP/P387 Note2 Single- and entangled-photon sources

Transcription

1 AP/P387 Note Single- and entangled-photon sources Single-photon sources Statistic property Experimental method for realization Quantum interference Optical quantum logic gate Entangled-photon sources Bell states Parametric down-conversion Evaluation of entangled state Quantum teleportation

2 Application and motivation For quantum cryptography key distribution Secure communication method for the distribution of secret key between two distant partners using the statistics and correlation of photons. For optical quantum computer Photon has a strong potential advantage towards quantum computing because it preserves the coherence of quantum bit for long time. For Long distances quantum communication Photon works as a mediator to manipulate spins between separate quantum memory nodes.

3 Statistic property The probability distribution of the number of photons for several light sources, where the average number of photons in the mode is 1, Large number fluctuations due to black-body radiation. p ( m) n 1 = ( 1+ n ) thermal m+ 1 m+ 1 m The fluctuations is limited by the average number of photons. α α = e n α n n! 1 n n e pcoherent ( m) = e m! m! m n n = α No fluctuation due to photon number squeezing. ψ = a 0 + b1, a + b = 1 p single photon 1 = ψ = b However, currently available photon-counting detectors are limited and not to distinguish photon numbers sufficiently. How to evaluate light sources?

4 Correlation function Photon statistics of a light source is characterized numerically by the second-order correlation function. The quantized electric field becomes an operator, written as the sum of two Hermitian-conjugate terms (Refer Chap.3): ( ) ω E ( r, t) = i e a, σ () t e ε V, σ o ikl ir The quantum mechanical second-order correlation function is defined as (Refer Chap.4), g ( ) ( τ ) () () () () aˆ () t aˆ() t + + E t E t+ τ E t+ τ E t aˆ t aˆ t+ τ aˆ t+ τ aˆ t = = + E t E t The function shows the existence of next photon coming after the time delay τ. For zero time delay, the function means how much coincidence of two photons occurs. 0 or 1 photon? τ Photon-counting detectors (Ex.) Against for coherent state α : g ( ) ˆˆˆˆ α aaaaα α 0 = = = 1 α aa ˆˆα α (Ex.) Against for single-photon state n =1 : g ( ) 4 ( ) ˆˆˆˆ ˆˆ nn 1 naaaan nn 1aan = = = = 1 0 = 1 naan ˆˆ n n n ( n )

5 Correlation function The function becomes more than 1 in classical light field (thermal, coherent state, etc) and can be lower than 1 in quantum light field (sub-poisson state, number state, etc). g ( ) ( τ ) Bunched state (classical) Coherent state (classical) Anti-bunched state (quantum) τ Thermal state Photon number is randomly distributed. Coherent state Photon number is most randomly (Poisson) distributed. Single-photon state Photon number is regulated distribution.

6 Solid-state single-photon sources Setup for second-order correlation measurement, so-called Hanbury Brown and Twiss (HBT) experiment. Measured second-order correlation function CW excitation case Pulse excitation case The exponential dip width reflected the decay time of the observed single photons. Zero delay peak is suppressed when the excitation laser pulse is short enough to decay time.

7 Solid-state single-photon sources Semiconductor quantum dots in microcavity Long coherence of dipole and electron spins Large inhomogeneous broadening energy Organic molecules Works in room temperature Small inhomogeneous broadening Broadband, non stability due to photo bleaching Diamond color centers Long coherence of electron and nuclear spin, small inhomogeneous broadening Short coherence of dipole, hard to fabricate

8 Quantum interference Photon coalescence occur when two indistinguishable single photons are simultaneously enter the two ports of a beam splitter due to Bose- Einstein statistics followed by photons (Hong-Ou-Mandel interference). a c b d ˆ + ˆ + = aa a b a b 0 BS 1 ( ˆ + ˆ + )( ˆ + ˆ + ac + ad ac ad ) 0 1 { ( ˆ ) ˆ ˆ ˆ ˆ ( ˆ ) } = ac acad + acad ad 0 1 = ( 0 0 c d c d)

9 Quantum interference C. Santri et.al, Nature 419, 594 (00). It is technically difficult to realize quantum interference using solid-state single-photon sources mainly due to spectrum diffusion. First experimental demonstration has been done by semiconductor quantum dots in distributed Bragg reflector (DBR) micro cavity. InAs QDs in DBR cavity

10 Optical quantum logic gate K. Knill, R. Laflamme, & G.J. Milburn, Nature 409, 46 (001). T. C. Ralph et al., Phys. Rev. A. 65, (001). Quantum logic gates are fundamental component for constructing quantum computer to manipulate quantum bits coherently. Indistinguishable single-photon sources and conditional detections make the quantum logic gate, so called controlled-phase gate. ψ IN ψ = α 0 OUT = + β 1 C + γ ( α 0 + β 1 γ ) C C C C Phase flip occurs only when both mode including photons.

11 AP/P387 Note Single- and entangled-photon sources Single-photon sources Statistic property Experimental method for realization Interference of single photons Optical quantum logic gate Entangled-photon sources Bell states Parametric down-conversion Evaluation of entangled state Quantum teleportation

12 Bell states There are four possible states with maximum entanglement for a pair of photons with two possible orthogonal H (horizontal) and V (vertical) polarizations for optical modes 1 and. + 1 ψ = + 1 ψ = + 1 Φ = + 1 Φ = ( H V V H ) 1 1 ( H V V H ) 1 1 ( H H V V ) 1 1 ( H H V V ) 1 1,,,. Under the correlation, you could not represent the polatization states as a simple product of single-photon states like ψ ψ. 1 In this sense, the photon pair is entangled, and no longer separable, even if their spatial wavepackets are macroscopically separated. A perturbation or a measurement at the position of one photon instantaneously modifies the whole states, as at the remote position of another photon. Einstein, Podolsky, and Rosen pointed out that the instantaneous character of the quantum measurement at first sight appears inconsistent with relativistic causality, seemingly involving spooky action at-a-distance in 1935 (EPR paradox). The correlation arising from entanglement are stronger than classical correlation. Bell shows that the fact can be experimentally verified to violate Bell s inequality that arise from realistic hidden variable theory in 1965.

13 Parametric down-conversion P. G. Kwiat et al., Phys. Rev. Lett. 74, 4763 (1995). A popular method to produce polarization-entangled photon pair is by parametric down-conversion source relies on non-collinear type-ii phase matching of nonlinear crystal (Refer Chap.3 appendix). Additional birefringent crystals (C1, C) compensate the walk off between the ordinary and extraordinary photons that propagates with different velocity. The compensator crystals also change the phase φ between the two components and prepare φ=0 or π, ψ 1 ± = H V ± V H 1 1 An additional half-wave plate (HWP1) in one of the two beams change the polarization 90 o and you can prepare all four Bell states, ± 1 Φ = H H ± V V 1 1..

14 Fidelity and entropy The quality of entangle photon pairs are evaluated numerically by Fidelity and Entropy. When the general polarization state of two photons is given by, ψ = ahh + bhv + cvh + dvv, a + b + c + d = 1, The density matrix of the states becomes, HH HV VH VV * * * a a b a c a d HH * * * HV ba b bc bd ρ = ψ ψ = * * * VH ca cb c cd VV * * * da db dc d When the two photons are in polarization entangled state, the experimentally measured density matrix ρ Experiment always has some mismatch from theoretically expected ideal density matrix ρ Theory. The measure of state overlap between theory and experiment is called Fidelity and given as, { } Theory Experiment = Theory Experiment Theory F ρ, ρ Tr ρ ρ ρ. The measure of mixture degree from pure state is given by Von Neuman entropy and given as, where p i is the eigenvalues of ρ. S Tr ( ρlog ρ) = pilog ( pi), i

15 Quantum teleportation D. Bouwmeester et.al, Nature 390, 575 (1997). An application of spooky entanglement correlation is quantum teleportation of an arbitrary quantum state through macroscopic distance. Alice (Information sender) has photon 1 in a certain quantum state that she want to teleport to Bob (Information receiver). ψ α β α β = H + V + = 1, Suppose photon and 3 are shared by Alice and Bob and they are in polarization entangled state, + 1 Φ = ( 3 3). 3 H H + V V Alice then performs a joint Bell state measurement (BSM) on the photon 1 and photon, projecting them onto an entangled state of pair. After Alice send the results of her BSM as classical information to Bob, he will enable to reconstruct the initial state into photon 3. ψ α β = H + V

16 Quantum teleportation D. Bouwmeester et.al, Nature 390, 575 (1997). Actual setup for quantum teleportation of a polarization state Initial state in Alice + Φ 3 1 ψ = α + β + 13 ( H1 V1 ) ( HH3 VV 3 ) + Φ 1 Final state in Bob ( H1H VV 1 ) ( α H3 β V3) ( H1H VV 1 ) ( α H3 β V3) ( HV 1 VH 1 ) ( α V3 β H3) ( HV 1 VH 1 ) ( α V3 β H3) For example, if the result of Bell state measurement (BSM) on photon 1 and are Φ +, then the photon 3 in Bob is directly projected into initial state.

17 Books for further interests Fundamental quantum optics: The quantum theory of light / Rodney Loudon / Oxford ; New York : Oxford University Press, 000. Quantum optics / D.F. Walls, Gerard J. Milburn. / Berlin : Springer, c008. Single-photon sources: Single-photon devices and applications / Charles Santori, David Fattal, Yoshihisa Yamamoto / Wiley-VCH, 010. Entanglement and quantum information: The physics of quantum information : quantum cryptography, quantum teleportation, quantum computation / Dirk Bouwmeester, Artur K. Ekert, Anton Zeilinger (eds.) / Berlin ; New York : Springer, c000, 001 printing. Quantum computation and quantum information / Michael A. Nielsen & Isaac L. Chuang. / Cambridge, U.K. ; New York : Cambridge University Press, 000.