Beam Splitters, Interferometers and Hong-Ou-Mandel Effect for Interacting Bosonic and Fermionic Walkers in a Lattice

Transcription

1 Beam Splitters, Interferometers and Hong-Ou-Mandel Effect for Interacting Bosonic and Fermionic Walkers in a Lattice Leonardo Banchi University College London, UK

2 Aim of the work Implement linear optics operations in an optical lattice Particles in tight binding lattices may be either static or mobile depending on the potential barriers Static particles to store information Mobile particles to process information Focus on distant particles for scalability

3 Recent experimental advances T. Fukuhara et al., Nature Physics (2013) T. Fukuhara et al., Nature (2013)

4 Content of this talk Transmission via hopping Fundamental operations (beam splitter, phase shifter) via suitably inserted impurities Hong-Ou-Mandel effect Mach-Zehnder interferometry Crossover from bosonic to fermionic behavior close to the superfluid-mott transition Discussion towards KLM quantum computation in a lattice

5 Fundamental operation: beam splitter 1 L β S (t) H = J 2 L 1 [ ] a j a j+1 + h.c. + j=1 L j=1 [ ] U 2 n j(n j 1) µ j n j µ j = µ + Jβδ j,l/2

6 Fundamental operation: beam splitter Transmission and reflection coefficients T(t) = 0 a L e ith a 1 0 R(t) = 0 a 1 e ith a 1 0 Effective Beam Splitter operator (scattering matrix) at the transmission time t L ( ) ( ) R(t S(t ) = ) T(t β i ) T(t ) R(t i+β i+β D + O(L 1 ), ) i i+β β i+β S=S(t )/D is the unitary beam splitter operation. D=O(L 1/3 ) is a damping factor

7 Fundamental operation: beam splitter L = 51, β = 0.95

8 Two particles: Hong-Ou-Mandel effect Boson U/J=0 Boson U/J=0.78 Fermion & HCB Probability P ij to measure one particle in position i and the other in position j. Fermion/Hard-core-boson behavior when U (U/J 10)

9 Bunching Probability Transition from bunching to anti-bunching close to the Mott phase transition point Stable bunching effect for U < 0.2J

10 Beam splitter and Phase shifter 1 L β S(t)

11 Beam splitter and Phase shifter 1 L β γ S(t) γ γ

12 Routes to perfection: engineered lattice Christandl et al., PRL, 2004 Albanese et al., PRL, J l = (l + 1)(N l) Mirror inversion after a certain time t ψ(x, t + t ) = ψ(n x, t)

13 Routes to perfection: minimal engineering Banchi, Apollaro, Cuccoli, Vaia, Verrucchi, NJP 2011 Banchi, Bayat, Verrucchi, Bose, PRL 2011 With optimally tuned J 0 Dynamics generated by excitations with almost linear dispersion relation NOT an effective two-site Hamiltonian, J 0 is comparable with J Ballistic coherent dynamics (t L)

14 Optimal dynamics with the XX model Dispersion relation (J 1) Density of excitations ω k = cos k ρ(k) cot 2 k = J2 0 2 J 2 0 peaked around the inflection point k = π/2 with width Group velocity of the wave-packet close to the inflection point [ ( 2 v k = k ω k const ) ] k 2 + O(k 4 ) NJ0 6 2

15 Application: Mach-Zender interference BS γ γ BS γ γ S matrix for the process (50/50 BS) S(t ) D S(t ) S φ 1 ( ) 1 ie iφ 2 ie iφ e 2iφ With fully engineered chain D = when L = 51

16 Application: Mach-Zender interference Different output ports depending on the phase shift

17 Imperfections Non-local impurity: β j =β 50/50 exp[ (N+1 j) 2 /σ 2 ] unaffected when FWHM 0.66a small deviations ( 5% with renormalized β 50/50 ) when FWHM 8a Open ends via a strong impurities β wall at the boundaries deviations < 5% when β walls 3 Curvature µ eff j = µ Jω 2 j 2 /2 No deviations from the ideal case when ω 0.03

18 Discussion towards KLM on a lattice Knill-Laflamme-Milburn quantum computer Linear optics quantum computer The fundamental non-linear interaction (non-linear phase shift) implemented with linear optics, auxiliary photons and measurements

19 Discussion towards KLM on a lattice

20 Discussion towards KLM on a lattice

21 Quantum gates from particle scattering Can we explioit natural interactions to implement quantum gates?

22 Flying qubits in one dimension Lieb-Liniger Model H = Uδ(x x1 2 x2 2 1 x 2 ), Scattering matrix for bosons/fermions (p 2 > p 1 ) S(p 2, p 1 ) = (p 2 p 1 ) ± iu SWAP 12, p 2 p 1 + iu Integrability (Bethe-Ansatz) from S

23 Quantum gate between flying qubits S B/F A B = e iφ B/F A B S B/F A B = e iφ B/F A B S B/F A B = p A+B A B iu A B p A+B + iu S B/F A B = p A+B A B iu A B p A+B + iu Maximal entanglement when p A+B = U

24 Gaussian wave-packet Mean U(1 + δ) Variance Uη

25 Implementations H= j,α J [ ] j a 2 j,αa j+1,α + h.c. + U αβ j 2 n j,αn j,β, j,α,β

26 Concluding remarks Linear optics on a lattice Operations via local optical potentials Considered applications: beam splitters, phase shifters, Hong-Ou-Mandel effect, Mach-Zender interferometry Transition from bosonic to fermionic behavior. Critical point? Main possible application: KLM quantum computer on a lattice, generation of non-classical states Quantum gate via scattering Natural interactions can generate maximal entanglement for physical parameters

27 Linear optics on a lattice E. Compagno, LB, S. Bose, arxiv: Quantum gate between flying qubits LB, E. Compagno, V. Korepin, S. Bose, soon in arxiv