Entanglement and Decoherence in Coined Quantum Walks

Transcription

1 Entanglement and Decoherence in Coined Quantum Walks Peter Knight, Viv Kendon, Ben Tregenna, Ivens Carneiro, Mathieu Girerd, Meng Loo, Xibai Xu (IC) & Barry Sanders, Steve Bartlett (Macquarie), Eugenio Roldan (Valencia) &John Sipe (Toronto) Random walk as a quincunx early mention of a quincunx: Caesar's De Bello Gallico, Book VII: (I m impressed that the Roman Empire set up its own web-site) Asterix: for the other side of the coin, translation to English

2 Ben Tregenna Viv Kendon Steve Bartlett Barry Sanders Also John Sipe, Eugenio Roldan, I Carneiro, Mathieu Girerd, Meng Loo &Xibai Xu UK Australia/Canada Why are these walks interesting? Motivation:quantum interferences relate to algorithmic speed-up, can engineer coin decoherence and study transition to classical,.

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4 Francis Galton & the Quincunx Galton s quincunx demonstrates random walk. Each peg is like the coin, and the ball goes L or R. Distribution of balls follows normal distribution with variance proportional to number of rows of pegs. How to design a quantum quincunx to show quantum walk?

5 Reminder of usual 1D classical random walk Particle confined to motion along a regular lattice in 1D, with points indexed by integers i (integer time) Decide on whether to go forward or backward according to whether an unbiased coin reveals heads (+ ) or tails (-). The distribution P(i) is binomial and approaches a Gaussian distribution after many steps. The average distance from the origin increases according to s t 1200 Sqrt(time) Dependence (St.Dev.)^ (St.Dev.)^2

6 Also Julia Kempe, Dorit Aharanov, and many people here have contributed to math description of walks. Julia Kempe review: Contemp Phys 2003/ quant-ph/ but realisations? -Decoherence?

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8 Quantum coin for Quantum Quincunx Coin replaced by quantum two-level system: Quantum coin toss - Hadamard transformation:

9 Quantum Quincunx Head for L, Tail for R: Quantum superpositions with coin flip: Superposition of heads and tails goes to superposition of left and right: interferences! Chose symmetric initial state

10 Conditional Translation of Position The particle s translation depends on the state of the coin, and the entire quantum random walk is deterministic (in a wave sense) over the joint particlecoin Hilbert space. Quantum walk is not a Random Walk.. Is it always quantum? -- see later

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12 walk this way.

13 Symmetric walk

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16 Square lattice: Hadamard coin

17 Square lattice: Grover coin

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23 Decoherence: walk on line (Kendon & Tregenna

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27 Entropy for walk on line

28 Entropy depends on initial conditions:approaches asymptote faster if more symmetric

29 Entropy for symmetric but biased coin

30 Entropy for asymmetric ics oscillates between a max of 1 and a min of 0

31 Bouwmeester, Schleich et al- already observed quincunx in classical optical experiment. Described in terms of many Landau-Zener crossings Ions: Travaglione and Milburn; Lattices: Kendon and Briegel etc

32 Its not random. Is it quantum? Knight, Roldan & Sipe (quantph /Opt Comm in press) optical implementation of walk for polarization or position cebits Bouwmeester et al already implemented in their Galton Board experiment But look at resources: can compute entanglement in terms of entropy using Schmidt basis - 4- component=4 propagating spin waves entanglement!

33 node Scheme of Paternostro et al k=0 k=+1 T 1 The proposed setup k=-1 T 2 T 2 T 2 D -4 dynamic lines T 2 T 2 T 2 D -2 j=0 j=1 T 2 T 2 T 2 D +2 D 0 D +4

34 When the walker is a single-photon state: 4 th step Position Np 1 s T 1 T 2 50 th step T 2 T 2 T Position Np T 2 T 2 D -2 D -4 classical distribution quantum distribution T 2 T 2 T 2 D 0 D +2 D +4

35 Simulating QRW with any state of light! (1) The proposed optical set-up realizes the following unitary evolution of the state of the walker after N steps = Uˆ...Uˆ Uˆ Tˆ Φ N T(j= N) T(j= 2) T(j= 1) 1(j= 0) Φ 0 Uˆ N QW Φ 0 For an input coherent state, this is easily explicitly calculated: ˆ N U QW Φ0 = α Φ = χ α χ α... χ α N N 2N with χ i that depend on the structure of the set-up. For an arbitrary state, in the Glauber diagonal P-representation ρ 0 = P( α) α α d 2 Uˆ N α QW ρ N = Uˆ N QW ρ 0 Uˆ N + QW = P( α) χ α 1 1 χ α 1 χ 2 α 2 χ 2 α... χ N α 2N χ N α d 2 α

36 Introducing randomness: decoherence in QRW Let s suppose we introduce an additional phase shift between two successive operations T 2 in the proposed set-up. We shift the state of the walker, before each T 2, by 2πl, where l is randomly chosen from a Gaussian distribution centred at 1 and with standard deviation σ ps. In practice, the random phase shift (RPS) is as follows: α T 1 T 2 T 2 Equal random shift 2πl Equal random shift 2πl T 2 T 2 Equal random shift 2πl T 2 l,l,l are taken from σ ps 1

37 Quantum Quincunx in a Cavity:Phys Rev A67 (2003) A two-level atom is the coin: a conditional Stark term shifts the phase of the cavity field clockwise or counterclockwise depending on the state of the atom. Sequence of atoms for repeated random phase shifts: classical walk. The cavity field initially has a sharp phase distribution (NB domain is a circle, not a line). variance grows linearly with number of atoms: phase diffusion

38 Recycling the Coin: quantum walk The quantum coin corresponds to the two levels of the atom, and this coin can be recycled to give a quantum walk. The cavity evolution is interrupted by periodically spaced Hadamard transformations, which flip the coin: F(ϕ) operates on the atom-cavity system in between these Hadamard coin flips. It is conceivable to apply (FH) 15 within the timescale of an experiment (Paris group parameters: J M Raimond, private communication). For small ϕ, the results are similar to 1D random walk, but large ϕ is also possible: random walk on a circle.

39 Quantum Quincunx Use a single atom (recycle the coin) Use π/2 pulse to implement quantum coin flip Quantum phase diffusion = quantum quincunx Phase spreads quadratically faster Need open cavity (Haroche)

40 Conditional phase shifts? Conditional phase-shift operator ( ) ( + ϕ = exp iϕa aσ ) Cavity prepared in a coherent state and the atom in ± iϕ / 2 either the + or state: F( ϕ) α ± = e α ± The phase of the cavity coherent state undergoes a random walk in discrete steps of ϕ. Ideally consider the (un-normalized) phase state F z F ( ) ϕ φ ± = φ ± ϕ ±, for φ n= 0 e inφ n

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42 Wigner functions, α=3, phase step 2π/6? Note fringes as well as displacements -see via quadrature measurements T=4 T=3 T=1 T=2

43 Quadrature variances

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45 conclusions Quantum version of Galton s quincunx shows differences from classical walks: interferences, quantum spreading Feasible: Haroche group developed new cavity allowing Hadamards during transit of atom (Yamaguchi quant ph-02) Quadratic enhancement of phase diffusion: quantum speed up; quantum algorithms Can engineer decoherence: observe transition from quantum to classical, suppression of interferences Funding: