atom optics vs photon optics: specific characters of atom

ATOM OPTICS vs PHOTON OPTICS:specific characters of atom optics processes

Laboratoire de Physique des LasersUniversité Paris Nord and CNRS

Villetaneuse, France

M. Hamamda, PhD in progressJ. Grucker (PhD’07)F. Perales

G. DutierC. MainosJ. Baudon

Martial Ducloy

FRISNO-10, Ein Gedi, Israel, 9 February 2009

Collaborations : M. Boustimi (Mekkah University, Saudi Arabia)V. Bocvarski (Institute of Physics, Belgrade, Serbia)

Optics with atoms

Atoms are…• massive• neutral• composite particles:

many internal degrees of freedom

Applications:• Atom interferometry

- Gravimetry, forces, acceleration, rotation…- Atomic interactions, surface interactions- Precision spectroscopy - Foundation of Quantum Mechanics, Relativity

• Atom lithography• Sensors, communications, quantum computing• Novel sources of atom beams

Atom Optics: manipulation of atom beams/atom waves

Realisation of Atom Optics Components

Optical elements for atoms are based on:

• Diffraction from microstructures, nanostructures or crystals• Interaction with static inhomogeneous (electric, magnetic) fields• Interaction with near-resonant electromagnetic fields

Forces on Atoms:should derive from a inhomogeneous potential V(r)

Functions already realised in the field of atom optics

• atom diffraction• atom mirrors• beam splitters

→ atom interferometry

• atom laser• quantum reflection• atom holography• atom quantum statistics (Hanbury Brown and Twiss)…

Can we perform all the photon optics operations in atom optics?

What are the specific characteristics of atom optical processes?

Two examples of novel functions using magnetic internal structure of atoms

OUTLINE

- Ultra narrow non-diffracting atom beam

- Negative-index meta-medium for atom optics

Ultra-thin non-diffracting beam

The basic idea is to use a Stern-Gerlach interferometer as a spatial filter for a He metastable (23S1) atom beam

What is a Stern-Gerlach interferometer ?

β β’

S-G S-G

a0

Phase object

B-profile

He* (m0 = 0)

Majorana zones : B is small and rotates quickly (β), the spin remains at rest → linear combination of m’s

Adiabatic evolution:m accumulates the phase mφ with φ = ∫

.traj

B Bdsv

gh

μ

Final amplitudea0 = cc’ – ss’ cos φwhere c = cos β, c’= cos β’

s = sin β, s’ = sin β’

A magnetic quadrupole is an atomic axicon

Constant radial gradient G, equivalent to a matter wave index linear in the distance ρ to z axis

In an interferometer, the phase shift is proportional to ρ → annular fringes

Example with Ar* atoms (v = 1650 m/s),G = 1.66 mGauss/cm (experiment / calculation)

z

1000 2000 3000 4000 5000

2 10-9

4 10-9

6 10-9

8 10-9

1 10-8

rB. Viaris et al, EPJD 23, 25 (2003)

A phase object aimed at producing an ultra narrow profile

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15 sqrt ((G x)2 + b2)

- G x+ G x

phas

e (x

cst

)

x

GGxbxG 2/)( 222 −+Ω=φ

Two opposite quadrupoles Q1, Q2 + a longitudinal field b

sudden (diabatic) passage

Total phase shift:

b is fixed to get φ(0) = π

The stronger the gradient G, the narrower the peak; width : 3b/(2G)

d

Ω = 2g µBG d/( v)h

x

b+G -G zx

Perales et al, Europhys. Lett.78, 60003 (2007)

What is our goal ?

A central narrow BRIGHT fringe (a0 is max at x = 0)surrounded by a wide DARK fringe (a0 = 0 at large x)

The final amplitude is a0 = cos β cos β’ – sin β sin β’ cos φ

One solution : β + β’ = π/2 → a0 = cos β cos β’ (1 – cos φ)

max for β = β’ = π/4

OK, but not enough to predict the profile of the emerging atom beam

Basic tool : the Huygens – Kirchhof integral

Integral operator HK transforming (for free propagation) amplitude

a(z1, ρ) into a(z2, ρ) : a(z2, ρ) ≈ (ik/2π) 'dSR

e)',z(aikR

1R 2

−

ρ∫∫

3 steps: a(0, ρ) → a(εD – d, ρ) → a(εD + d, ρ) → a(D, ρ)

free index n(ρ) free

n(ρ)

M

SP2d

ε D (1-ε) D

N

R R’Source (σ) Observation screen

Results

2) When G and b ≠ 0, one defines a « saturation parameter » by:

ζ = [(2σ2)-1 + ik/D] [2 (1 – ε)2 Ω2]-1.

If │ζ│>> 1 (easily accessed), then

1) When G = 0, b = 0, the outgoing amplitude at plane D is (as expected) Gaussian:

G(ρ) = N exp [- ρ2 ((δ2/2s) + iδ)/2 ]

N is a normalisation constant, δ = k/D, s = [(2σ2)-1 + iδ]/2

a(ρ) ≈ G(ρ) [1 + (q Ω ρ)2)]-1/2(q = 0.4157)

Ω = 2gµBG d /( v)- The interferometer does not alter the amplitude at center

- This « quasi square root of a Lorentzian » form (much narrower than G) is indeed very simple. Moreover it will provide us with a very interesting property of the beam

h


The beam is almost not diffracting

The evolution of the amplitude beyond the interferometer is given again by the Huygens-Kirchhof integral :

a(D + Z, ρ) = HK * a(D, ρ)

It turns out that the width of the profile increases with Z much slower than that of a standard beam diffracting from an aperture of a comparable size, even for values of Z as large as several cm.

Comparison with ordinary diaphragms (log-scale)

Interferometer exit

present

Circular holeGaussian

0 20 40 60 80 1000,01

0,1

1

10

100

( Z - L ) mm

R

wid

thin

µm

( Z - L ) mm

0 20 40 60 80 1000,0

0,2

0,4

0,6

0,8

1,0

50 nm

R: intensity ratio


This makes this beam the atomic counterpart of « Bessel beams », well known in light optics (Durnin et al JOSA A 1987, PRL 1987)

OUTLINE

- Ultra narrow non-diffracting atom beam

- Negative-index meta-medium for atom optics

Negative-index optical material:

→ Hypothetical material with ε<0 and μ<0 (Veselago, 1969)Same physics as for ε>0 and μ>0 ?

Refractive index:

v=c/n → negative phase velocityPoynting vector opposite to the wave vector k :

Light runs backward in time!

εμεμ ±=⇒= nn2

Snell-Descartes Law

( Poynting vectors )

cnk ω

= ; n1

2sinsin θθ =

n = 1θ1 θ1

n > 0

Rays Wavevectors k

θ2θ2

n = 1θ1 θ1

n < 0

Rays Wavevectors k( Poynting vectors )

θ2 θ2

Planar Meta-lens

n = 1

Planar n = -1 lens :

• Does not focus parallel rays

• No optical axis

• Aberration-free

•Amplification of evanescent fields

n = -1Object plane

Image plane

: kinetic energy of the atom in absence of potential

n(r): real positive or purely imaginary

Solution: position and time-varying potential

[ ] 2/10/)(1)( ErVrn −=

An inhomogeneous static potential V(r) is equivalent toan optical index n(r):

0E

Meta-medium for Atom OpticsQuestion : How to realize a group velocity opposite to the phase

velocity ?

A « meta-medium » for matter waves is necessarily different from a meta-material for light waves

Light source

Pulsed atomsource

k

R

The Poynting vector R is outwards

The wave vector k is reversed

The group velocity* vg is transiently reversed

The wave vector k remains outwards

kvg

(*) vg = ⎜Ψ⎜-2 J , where J is the standard current density of probability flux

x

Λ

Meta-medium for Atom OpticsHow to realize such a transient reversal of the group velocity ?

One of the simplest answer is a COMOVING potential V(t, x), e.g., for atoms with spin, a comoving magnetic field B(t, x)

Transverse magnetic field B moving along x, at an (adjustable) velocity u = ν Λ

cos (2πν (t-t0)) cos (2π x/Λ) = [cos (2π (ν(t-t0) – x/Λ) + cos (2π (ν(t-t0) + x/Λ)] /2

A continuous frequency spectrum H(ν) can be used as well:V(t, x) = S(t) cos (2π x / Λ) → comoving magnetic pulse

atoms

Total phase of a k-component of the atomic wave packet:

ϕ(t, k) is the phase shift induced by co-moving potential V(t, x) :

),()2/( 2 kttkmxk ϕ+−=Φ h

⎟⎠⎞

⎜⎝⎛

Λπ

−=ϕ ∫−

m'tk2cos)'t(S'dt)k,t( t

01 h

h

The motion of the wave packet centre is derived from the stationary phase condition : ∂k Φ = 0 →

The group velocity is then :

By a proper choice of S(t), it can be made (transiently) negative

0])/()( 0 kktmktx ϕ∂−= h

⎟⎠⎞

⎜⎝⎛

Λπ

Λπ−= −

mtk2sin)t(St)m(2m/k)k,t(v 01

00gxh

h

Baudon et al, arXiv:0811.2479

First property of “meta-media” :

negative refraction and « meta » lenses

-0.02 -0.01 0.01 0.02 0.03 0.04

0.001

0.002

0.003

0.004

0.005

0.006

SZ (m)

X (m)

0.005

0.040

Comovingpulse

S is a point-like source. The incident angle θ ranges from 0 to 0.12 rad

Ar*(3P2, M = 2) atoms, velocity 20 m/s (λdB = 0.56nm), Bmax = 400 Gauss

A 3D-view of a cylindrical « meta »- lens

-0.005

-0.0025

0

0.0025

0.005

-0.02

0

0.02

0

0.001

0.002

0.003

-0.005

-0.0025

0

0.0025

0.005

0

0.001

0.002

0.003


Generalisation to a 2D-potential

)m/k,m/k,'t(V'dt)k,t( yxt

01 hhh

r∫−−=ϕ

[ ] 0)k,t()m2/(tkr.k0k

2k =ϕ+−∇

rh

rrr

V(t, x, y) such that variables x, y are separable, e.g.

V(t, x, y) = S(t) [cos (2πx/Λ) + cos (2πy/Λ)]

The phase shift is now :

The stationary-phase condition is :

and the induced shift is :

∫ πν+πνΛπ−=δ − t

0 yyxx1 ]u)'t2sin(u)'t2[sin()'t(S't'dt)m(2r

r

→ Spherical meta-lens

Effect of the 2D potential on an atomic beam

Atom beam aperture = 0.10 rad, Ar*(3P2, M = 2) atoms, velocity 20 m/s, Bmax = 400 G

m

comovingfield

Re-focussing of a diverging matter wave

by use of three subsequent pulses of comoving potential

0.025 0.05 0.075 0.1 0.125

-0.001

-0.0008

-0.0006

-0.0004

-0.0002

0.0002

0.0004

0.001 0.002 0.003 0.004

-1 ´ 10 - 2 4

-8 ´ 10 - 2 5

-6 ´ 10 - 2 5

-4 ´ 10 - 2 5

-2 ´ 10 - 2 5

Z (m)

X (m)t (s)

Focus better than 25 µm


Second property :

Enhancement of evanescent waves

- Evanescent atomic wave packets are generated at a (magnetic) potential barrierV0(x) higher than their energy distribution.

- The evolution in t and x of the matter evanescent wave is very different fromthat of a light evanescent wave. In particular the factorisation in t and x no longer holds.

1 µs 150 nmv = 1 m/s

λdB = 11.2 nm

V0 = 1.005 E0

Atom wavepacket incident at t=0

Effect of a single comoving pulse

Atomic velocity: 1 m/s (λdB = 11.2 nm); V0 = 1.005 E0

Spatial period of the comoving potential: Λ = 2 µm

S(t) = 2 g µB B [ε /(t + ε)]2 exp(-t/τ) for 0 < t < τ1 ; = 0 elsewhere,

with B = 40 Gauss, ε = 7.4 µs, τ = 0.37 µs, τ1 = 1.80 µs

1.5 µs150 nm

Atom wavepacket

Effect of two subsequent pulses : S(t) and –1.5 S(t – 0.8 µs)

0.8 µs

2d pulse 150 nm

Atom wavepacket

Collapse and revival of an evanescent matter waveThree subsequent comoving pulses

Spatial confinement, extension in time : Surface matter wave ?

2.5 µs 150 nm

Conclusion - Prospects

• Ultra-narrow metastable atom source for atom optics and atom surface interaction studies: Operation of a non-diffracting coherent “nano-beam” of metastableHelium is in progress

• Negative-index “meta-medium” for atom waves in the nanometre range has been proposed and its properties are under study: Is “sub-de Broglie wavelength” focusing possible?Can one excite surface matter waves?

• Future prospect: Laguerre-Gauss-like atom beams?

atom optics vs photon optics: specific characters of atom

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