Light and Matter Waves - USPfep.if.usp.br/.../transparencias/11_3_M_Carolina_Nemes.pdf · 2008. 2....

Light and Matter Waves

λ =

h

p

M.C. Nemes

Escola Jorge Andre Swieca de Optica Quantica

February 11th 2008

SummaryPart I Part II Part III

M.C.Nemes 2 / 28

I. De Broglie: λ = h/p

◆ Questions raised

◆ Situation of answers today


M.C.Nemes 2 / 28




II. Two aspects to consider:

◆ Interference: Classical Light waves ⇔ Quantum wavesCan we measure a difference?


M.C.Nemes 2 / 28






◆ The Gouy phase:Classical Ligth Waves: well known (since 1890) in Optics

vsMatter Waves: is there an analog? An experiment?


M.C.Nemes 2 / 28






◆ The Gouy phase:Classical Ligth Waves: well known (since 1890) in Optics

vsMatter Waves: is there an analog? An experiment?

III. Conclusions

Part I

Part I Part II Part III

IntroductionPart I Part II Part III

M.C.Nemes 4 / 28

λ =h

pL. de Broglie

Questions raised immediately:


M.C.Nemes 4 / 28

λ =h

pL. de Broglie


■ is it true?Electrons diffraction


M.C.Nemes 4 / 28

λ =h

pL. de Broglie


■ why don’t we observe such phenomena with macroscopic objects?


M.C.Nemes 4 / 28

λ =h

pL. de Broglie


■ why don’t we observe such phenomena with macroscopic objects?Bohr: Correspondence Principle

Quantum −→ Classical

Large quantum numbers


M.C.Nemes 4 / 28

λ =h

pL. de Broglie


■ why don’t we observe such phenomena with macroscopic objects?Bohr: Correspondence Principle

Quantum −→ Classical

Large quantum numbers

... How large?

Experimental setupPart I Part II Part III

M.C.Nemes 5 / 28

Markus Arndt, Olaf Nairz, Julian Vos-Andreae, Claudia Keller, Gerbrand van der Zouw and

Anton Zeilinger, Nature 401, 680 (1999).

Experimental dataPart I Part II Part III

M.C.Nemes 6 / 28

Mass M ≈ 1.2 × 10−24 kgRadius R ≈ 3.5 × 10−10 mTemperature ΘF ≈ 900 KEnv. temperature ΘA ≈ 300 KAverage wave lenght λ ≈ 2.5 × 10−12 mAverage time of flight T ≈ 6 × 10−3 sGrating–screen distance L ≈ 1.25 mCollimator aperture a = 10−5 mEffective slit width b ≈ 3.6 × 10−8 mSlit spacing d = 10−7 m



Experimental dataPart I Part II Part III

M.C.Nemes 6 / 28

Mass M ≈ 1.2 × 10−24 kgRadius R ≈ 3.5 × 10−10 mTemperature ΘF ≈ 900 KEnv. temperature ΘA ≈ 300 KAverage wave lenght λ ≈ 2.5 × 10−12 mAverage time of flight T ≈ 6 × 10−3 sGrating–screen distance L ≈ 1.25 mCollimator aperture a = 10−5 mEffective slit width b ≈ 3.6 × 10−8 mSlit spacing d = 10−7 m

Note that M ≈ 107 me.



ResultsPart I Part II Part III

M.C.Nemes 7 / 28

I(x) =I02

sinc2

(

πbx

λL

)[

1 + cos

(

2πdx

λL

)]

sincx ≡ sinx

x




M.C.Nemes 8 / 28

λ =h

pL. de Broglie


■ How is the quantum–classical transition ?


M.C.Nemes 8 / 28

λ =h

pL. de Broglie


■ How is the quantum–classical transition ?

“The transition occurs because of the action of the environment on theisolated quantum system under observation.”

D. Zeh, W. Zurek, J. P. Paz, ...

Losing coherencePart I Part II Part III

M.C.Nemes 9 / 28

Decoherence Diffusion factor, Λ (m2/s)

Scatteringthermal photons Λesp

ph ≈ 2,4 × 102

air molecules Λair . 3,2 × 1015

Photon emissionblack body radiation Λbb

ph ≈ 2,5 × 109

excitation decay Λvibph . 5,0 × 1013

Global effects Λ . 3,3 × 1015

Viale, A., Vicari, M. and Zanghi, N. Phys. Rev. A 68, 063610 (2003).


M.C.Nemes 10 / 28

As function of the distance:



M.C.Nemes 10 / 28

As function of the pressure:



M.C.Nemes 10 / 28

As function of the mass:



M.C.Nemes 11 / 28

λ =h

pL. de Broglie


■ Finally: “what is this wave?” (P. Debye)


M.C.Nemes 11 / 28

λ =h

pL. de Broglie



Schrodinger equation (free particle):

i~∂

∂tΨ(~r, t) = − ~

2

2m∇2Ψ(~r, t)

⊕Interpretation


M.C.Nemes 11 / 28

λ =h

pL. de Broglie




i~∂

∂tΨ(~r, t) = − ~

2

2m∇2Ψ(~r, t)

⊕Interpretation

Light wave in vacuum:

1

c2∂2

∂t2E(~r, t) = ∇2E(~r, t).


M.C.Nemes 11 / 28

λ =h

pL. de Broglie




i~∂

∂tΨ(~r, t) = − ~

2

2m∇2Ψ(~r, t)

⊕Interpretation


1

c2∂2

∂t2E(~r, t) = ∇2E(~r, t).

What are the similarities and differences between these waves?


M.C.Nemes 11 / 28

λ =h

pL. de Broglie




i~∂

∂tΨ(~r, t) = − ~

2

2m∇2Ψ(~r, t)

⊕Interpretation


1

c2∂2

∂t2E(~r, t) = ∇2E(~r, t).

What are the similarities and differences between these waves?Can one tell them apart experimentally?

Part II


Light vs matter wavesPart I Part II Part III

M.C.Nemes 13 / 28

It is well known the diffraction of a plane wave:

laserz


M.C.Nemes 13 / 28

How is that for a matter wave?

t = z/v

C60

σ0


M.C.Nemes 13 / 28

How is that for a matter wave?

t = z/v

C60

σ0

In principle, by changing σ0 one could change the interference pattern!

Simple ModelPart I Part II Part III

M.C.Nemes 14 / 28

û

3

x

z

L l

2σ0

2b

T t

d


M.C.Nemes 14 / 28

û

3

x

z

L l

2σ0

2b

T t

d

Ψ(x, 0) =

(

1

σ0√π

)1/2

exp

(

− x2

2σ20

)


M.C.Nemes 14 / 28

û

3

x

z

L l

2σ0

2b

T t

d

Ψ(x, T ) =

(

1

B(T )√π

)1/2

exp

[

− x2

2B2(T )

(

1 − iT

τ0

)

+ iφ(T )

]


M.C.Nemes 14 / 28

û

3

x

z

L l

2σ0

2b

T t

d

Ψ(x, T ) =

(

1

B(T )√π

)1/2

exp

[

− x2

2B2(T )

(

1 − iT

τ0

)

+ iφ(T )

]

τ0 = mσ20/~: matter wave characteristic!


M.C.Nemes 14 / 28

û

3

x

z

L l

2σ0

2b

T t

d

Ψ(x, T ) =

(

1

B(T )√π

)1/2

exp

[

− x2

2B2(T )

(

1 − iT

τ0

)

+ iφ(T )

]

B(T ) = σ0[1 + (T/τ)2]1/2


M.C.Nemes 14 / 28

û

3

x

z

L l

2σ0

2b

T t

d

Ψ(x, T ) =

(

1

B(T )√π

)1/2

exp

[

− x2

2B2(T )

(

1 − iT

τ0

)

+ iφ(T )

]

φ(T ) = arctan(T/τ)

“Aging” of the wave packetPart I Part II Part III

M.C.Nemes 15 / 28

We keep B2(T )σ2

0

fixed and artificially decrease σ0:


M.C.Nemes 15 / 28

We keep B2(T )σ2

0


Intensity

Position, x (µm)-40 -20 0 20 40

σ0 = 6 µm


M.C.Nemes 15 / 28

We keep B2(T )σ2

0


Intensity

Position, x (µm)-40 -20 0 20 40

σ0 = 0.02 µm


M.C.Nemes 15 / 28

We keep B2(T )σ2

0


Intensity

Position, x (µm)-40 -20 0 20 40

σ0 = 0.0175 µm


M.C.Nemes 15 / 28

We keep B2(T )σ2

0


Intensity

Position, x (µm)-40 -20 0 20 40

σ0 = 0.013 µm

The Gouy phasePart I Part II Part III

M.C.Nemes 16 / 28

Classical waves: in 1890, C.R. Gouy showed that a focused electromagneticbeam will acquire an additional π phase shift with respect to a plane wave asit evolves through its focus.


M.C.Nemes 16 / 28


Best known in the case of paraxial beams.


M.C.Nemes 16 / 28


Best known in the case of paraxial beams.

What is a paraxial beam?

Wave equationPart I Part II Part III

M.C.Nemes 17 / 28

Helmholtz Equation:(

∇2 + k2)

E(~r) = 0 .


M.C.Nemes 17 / 28


∇2 + k2)

E(~r) = 0 .

Electric Field:E(~r) = A(~r) eikz

A(~r) varies slowly within one laser wavelength λ = 2π/k.


M.C.Nemes 17 / 28


∇2 + k2)

E(~r) = 0 .

Electric Field:E(~r) = A(~r) eikz

A(~r) varies slowly within one laser wavelength λ = 2π/k. Then

(

∂2A

∂x2+∂2A

∂y2

)

+∂2A

∂z2+ i2k

∂A

∂z= 0 .

Paraxial approximationPart I Part II Part III

M.C.Nemes 18 / 28

As A(~r) varies slowly in the direction of propagation z, then:

∂2A

∂z2� k

∂A

∂z


M.C.Nemes 18 / 28


∂2A

∂z2� k

∂A

∂z

Thus we find(

∂2

∂x2+

∂2

∂y2+ i4π

1

λL

∂

∂z

)

A(x, y, z) = 0 .


M.C.Nemes 18 / 28


∂2A

∂z2� k

∂A

∂z

Thus we find(

∂2

∂x2+

∂2

∂y2+ i4π

1

λL

∂

∂z

)

A(x, y, z) = 0 .

Compare with the Schrodinger equation for a free particle in 2D:

(

∂2

∂x2+

∂2

∂y2+ 2i

m

~

∂

∂t

)

ψ(x, y, t) = 0 .


M.C.Nemes 18 / 28


∂2A

∂z2� k

∂A

∂z

Thus we find(

∂2

∂x2+

∂2

∂y2+ i4π

1

λL

∂

∂z

)

A(x, y, z) = 0 .


OR


M.C.Nemes 18 / 28


∂2A

∂z2� k

∂A

∂z

Thus we find(

∂2

∂x2+

∂2

∂y2+ i4π

1

λL

∂

∂z

)

A(x, y, z) = 0 .


(

∂2

∂x2+

∂2

∂y2+ i4π

1

λp

∂

∂z

)

ψ(x, y, t = z/vz) = 0 .

Formally identical !

Guassian beamPart I Part II Part III

M.C.Nemes 19 / 28

Monocromatic beam in z = 0:

E(x′, y′, 0) = A0 exp

[

−x′2 + y′2

w20

]


M.C.Nemes 19 / 28


E(x′, y′, 0) = A0 exp

[

−x′2 + y′2

w20

]

Propagating the field:

E(x, y, z) = A0w0

w(z)exp

[

−x2 + y2

w(z)2

]

exp

[

ikz + ik(x2 + y2)

2R(z)

]

exp[iζ(z)] .


M.C.Nemes 19 / 28


E(x′, y′, 0) = A0 exp

[

−x′2 + y′2

w20

]


E(x, y, z) = A0w0

w(z)exp

[

−x2 + y2

w(z)2

]

exp

[

ikz + ik(x2 + y2)

2R(z)

]

exp[iζ(z)] .

where

w(z) = w0

[

1 +

(

z

z0

)2]1/2

and R(z) = z

[

1 +(z0z

)2]

ζ(z) = arctanz

z0is the Gouy phase and z0 =

kw20

2.


M.C.Nemes 19 / 28


E(x′, y′, 0) = A0 exp

[

−x′2 + y′2

w20

]


E(x, y, z) = A0w0

w(z)exp

[

−x2 + y2

w(z)2

]

exp

[

ikz + ik(x2 + y2)

2R(z)

]

exp[iζ(z)] .

Recall the solution of the Schrodinger equation in 2D:

Ψ(x, y, T ) =1

B(T )√π

exp

[

−x2 + y2

B2(T )

]

exp

[

im(x2 + y2)

2~R(T )+ iφ(T )

]

AnalogiesPart I Part II Part III

M.C.Nemes 20 / 28

w(z) → B(T ) = σ0

[

1 +

(

T

τ0

)2]1/2

,


M.C.Nemes 20 / 28

w(z) → B(T ) = σ0

[

1 +

(

T

τ0

)2]1/2

,

R(z) → R(T ) = T

[

1 +

(

τ0T

)2]

,


M.C.Nemes 20 / 28

w(z) → B(T ) = σ0

[

1 +

(

T

τ0

)2]1/2

,

R(z) → R(T ) = T

[

1 +

(

τ0T

)2]

,

ζ(z) → ϕ(T ) = arctan(T/τ0) ,


M.C.Nemes 20 / 28

w(z) → B(T ) = σ0

[

1 +

(

T

τ0

)2]1/2

,

R(z) → R(T ) = T

[

1 +

(

τ0T

)2]

,


z0 → τ0 = mσ20/~ .


M.C.Nemes 20 / 28

w(z) → B(T ) = σ0

[

1 +

(

T

τ0

)2]1/2

,

R(z) → R(T ) = T

[

1 +

(

τ0T

)2]

,


z0 → τ0 = mσ20/~ .

Rayleigh length: “Aging” time:

z0 =πw2

0

λL⇔ τ0 =

mσ20

~


M.C.Nemes 20 / 28

w(z) → B(T ) = σ0

[

1 +

(

T

τ0

)2]1/2

,

R(z) → R(T ) = T

[

1 +

(

τ0T

)2]

,


z0 → τ0 = mσ20/~ .

Rayleigh length: “Aging” time:

z0 =πw2

0

λL⇔ τ0 =

mσ20

~

What is the physical meaning ?

Uncertainty principlePart I Part II Part III

M.C.Nemes 21 / 28

■ Matter waves:

◆ Heisenberg’s: ∆x∆p ≥ ~

2.


M.C.Nemes 21 / 28

■ Matter waves:


2.

◆ Schrodinger-Robertson’s generalization:

Σ =

(

σMxx

12σ

Mxp

12σ

Mxp σM

pp

)


M.C.Nemes 21 / 28

■ Matter waves:


2.


Σ =

(

σMxx

12σ

Mxp

12σ

Mxp σM

pp

)

σMxx =

⟨

(x− 〈x〉)2⟩

, σMpp =

⟨

(p− 〈p〉)2⟩

,

σMxp =

⟨

(x− 〈x〉)(p− 〈p〉) + (p− 〈p〉)(x− 〈x〉)⟩


M.C.Nemes 21 / 28

■ Matter waves:


2.


detΣ = σMxxσ

Mpp − 1

4(σM

xp)2=

~2

4

quadratic evolution for any initial state!


M.C.Nemes 21 / 28

■ Matter waves:


2.


detΣ = σMxxσ

Mpp − 1

4(σM

xp)2=

~2

4


■ Light waves: in analagous way we obtain

σLxxσ

Lkk − 1

4(σL

xk)2 =

1

4


M.C.Nemes 21 / 28

■ Matter waves:


2.


detΣ = σMxxσ

Mpp − 1

4(σM

xp)2=

~2

4


■ Light waves: in analagous way we obtain

σLxxσ

Lkk − 1

4(σL

xk)2 =

1

4

σMxp =

T

τ0⇐⇒ σL

xk =z

z0

ThereforePart I Part II Part III

M.C.Nemes 22 / 28

By measuring σLxx and σL

kk we can obtain:

σLxk = ±

√

1

2

( w

w0

)2− 1


M.C.Nemes 22 / 28


kk we can obtain:

σLxk = ±

√

1

2

( w

w0

)2− 1

... and thenζ(z) = arctan(σL

xk)


M.C.Nemes 22 / 28


kk we can obtain:

σLxk = ±

√

1

2

( w

w0

)2− 1


xk)

Varying beam waist:

wai

st(m

m)

z (mm)

Cor

rela

tion

σx

k

z − zc (mm)


M.C.Nemes 22 / 28


kk we can obtain:

σLxk = ±

√

1

2

( w

w0

)2− 1


xk)

Varying beam waist:

Gouy’

sphas

e(r

ad)

z − zc (mm)

Matter wave diffractionPart I Part II Part III

M.C.Nemes 23 / 28

The apparatus:

Nairz, O., Arndt, M. and Zelinger, A. Phys. Rev. A 65, 032109 (2002).


M.C.Nemes 23 / 28

Experimental data:


M.C.Nemes 23 / 28

Experimental data: Theoretical predictions:

Part III


ConclusionsPart I Part II Part III

M.C.Nemes 25 / 28

λ =h

p


M.C.Nemes 25 / 28

λ =h

p

■ tell them apart in an interfer-ence experiment


M.C.Nemes 25 / 28

λ =h

p


■ collimation dependence of thefinal pattern


M.C.Nemes 25 / 28

λ =h

p



■ not measured yet


M.C.Nemes 25 / 28

λ =h

p




■ learn from their similarity in theparaxial approximation


M.C.Nemes 25 / 28

λ =h

p





■ Gouy’s phase for matter waves


M.C.Nemes 25 / 28

λ =h

p





■ Gouy’s phase for matter waves

■ indirect indication of consis-tency

In progress:Part I Part II Part III

M.C.Nemes 26 / 28

Focusing lenses for matter interferometer:

EndingPart I Part II Part III

M.C.Nemes 27 / 28

λ =

h

p

Wave–particle duality

⇒ still a source of several fundamental questions

simple, nontrivial, intriguing physics!

CollaboratorsPart I Part II Part III

M.C.Nemes 28 / 28

■ I. G. da Paz

■ J.G. Peixoto de Faria

■ C. H. Monken

■ S. Padua

Light and Matter Waves - USPfep.if.usp.br/.../transparencias/11_3_M_Carolina_Nemes.pdf · 2008. 2....

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Transcript of Light and Matter Waves - USPfep.if.usp.br/.../transparencias/11_3_M_Carolina_Nemes.pdf · 2008. 2....