Padova, June 2011mir/Casimir2011/MaiaNeto.pdf · Padova, June 2011. Collaborators LKB - ENS, Paris...

Angular momentum and the Casimir effect

Paulo A. Maia Neto Instituto de Física

Rio de Janeiro, Brazil

Padova, June 2011

Collaborators

LKB - ENS, Paris

Romain GuéroutAstrid LambrechtRiccardo MessinaSerge Reynaud

LANL, Los Alamos

Diego Dalvit

IF - UFRJ, Rio de Janeiro

Claudio CcapaAna M. Contreras-ReyesCarlos FarinaFrançois Impens Guilherme TravassosIF - UFF, Rio das Ostras

Robson Rodrigues

3

Introduction Driving quantized vortices in a BEC with

a rotating diffraction grating Dynamical dissipative Casimir torque ?

Angular momentum and the Casimir effect: outline

4

A single moving plate in empty space

Radiation pressure force (non-relativistic limit)

Barton; PAMN 1994

12x larger than scalar result of Ford+Vilenkin (1982) for scalar field

Zero temperature: electromagnetic field in the vacuum state

Sign of the force: quantum vacuum field takes energy from the plate - Casimir dissipation

Energy taken is converted into real photons - dynamical Casimir effect

no force for uniform motion: quantum vacuum does not define state of absolute motion

PAMN+LAS Machado 1996

TE TME E

HH

5

... which raises the question of the angular momentum coupling with the quantum vacuum field

Static Casimir effect: linear momentum transfer from the vacuum field to one of the plates (but net transfer to both plates vanishes): Casimir pressure = quantum average of the Maxwell stress tensor

What about angular momentum transfer from the vacuum field ...?

Introduction

Rotating sphere or disk: cannot rule out a dynamical Casimir torque even in the case of uniform angular velocity.

Casimir Torque between corrugated surfaces

Introduction

R. Rodrigues, PAMN, A. Lambrecht and S. Reynaud EPL 2006

See also: Casimir torque with anisotropic cristalsBarash 1978, van Enk 1995, Munday et al 2005, Torres-Gusman & Mochan 2006

Static Casimir effect: breaking rotational symmetry

θλ

Ly

Casimir energy dependent on some angular variable

Introduction

λ = 1.2 μm

λ = 2.4 μm

1

2πL/λ = 2.6

R1

R2

ωP

λP = 2πc/ωP

F

FCAS

L

e−κL

r1r2e−2κLE = E

F =π2

240

c

d4A

E = − π2

720

c

d3A

R. Rodrigues, PAMN, A. Lambrecht and S. Reynaud EPL 2006

Applications in NEMS..?

θλ

Ly

Casimir energy/area (k Ly >> 1)

k = 2π/λ; a1 ,a2 = corrugation amplitudesb = lateral relative displacement

atomic dimensions zA

P ∼ h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

φ= φ

r =b− a

br + a

1

8

Dynamical Casimir torque ?

Rotating sphere, radius R

Introduction

G. Barton 1996: scalar field, no dispersion nor aborption, time dependent perturbation theory (v/c<<1) in the Born approximation (|n-1|<<1) no effect for uniform rotation (Ω = cte) harmonic oscillation: Ω(t) = Ω0 cos(ω0 t): radiated power (`rotational’DCE)

Manjavacas & García de Abajo 2010: uniform rotation, arbitrary temperature, dissipative metallic medium (Drude model) electric dipole approximation (Ω R << c ?) zero temperature: power dissipated

2R


P ∼ h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

φ= φ

r =b− a

br + a

1


P = h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

φ= φ

r =b− a

br + a

1


P =h

30π2

Ω6R3

c3σ0

P = h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

φ= φ

1

9

Introduction

...we probably need a very sensitive probe of the angular momentum coupling with the vacuum field...

Bose-Einstein condensate !

Superfluidity Quantized vortices

10




Vortex generation in Bose-Einstein Condensates

Rotating SpoonK. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, PRL 2000

Cigarillo-BEC

105 87Rb atoms @80nK

BEC = Superfluid Quantized Velocity Circulation

Conditions for vortex generation in BECs

Rotating Frame Hamiltonian :

Favors emergence of states with Non-Zero Angular Momentum

Dynamic Instability (Quadrupole mode Excitation)

(1)

(2)

Rotating Anisotropic Potential:

V. Bretin, P. Rosenbusch, F. Chevy, G. V. Shlyapnikov, and J. Dalibard, PRL 2003

Vortex Nucleation(1)+(2)

x

y

E. Hodby, G. Hechenblaikner, S. A. Hopkins, O. M. Marago, and C. J. Foot, PRL 2002

BEC Vortices stirred bysmall anisotropies(≈ 2% )

Conditions for vortex generation in BECs

BEC Vortices =Sensitive Probesof potential anisotropy

Region of Vortex Nucleation

Proposed SetupSymmetric Harmonic Trap

Casimir-Polder Potentialof Rotating Grating

+

“Casimir Spoon” driving vortices

Casimir potential in the rotating frame ≈ static potential(quasi-static approximation)Gross-Pitaevskii eq. in the rotating frame


Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

φ= φ

r =b− a

br + a

r = 0 → r = a; r = b → r = b

F =π2

240

hc

L4A

=ω2x − ω2

y

ω2x + ω2

y

E = 0

E2 > 0

ρPS =force amplitude

force amplitude PFA

1

BEC below “plateau”BEC diameter < d/2

BEC confinement :

No torque in the Proximity Force

Approximation (PFA)

FPFA

TransverseQuadratic Expansionof Casimir potential

Ω

BEC width << z0

d/2


R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

φ= φ

r =b− a

br + a

r = 0 → r = a; r = b → r = b

F =π2

240

hc

L4A

=ω2x − ω2

y

ω2x + ω2

y

E = 0

1


ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

φ= φ

r =b− a

br + a

r = 0 → r = a; r = b → r = b

F =π2

240

hc

L4A

=ω2x − ω2

y

ω2x + ω2

y

1

Transverse Trap Frequency

Size of the `pancake-like’ condensate

Anisotropy:

Non Perturbative Scattering Method: Casimir Frequency


R⊥ =d

4

Rz = 0.2z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

φ= φ

r =b− a

br + a

r = 0 → r = a; r = b → r = b

F =π2

240

hc

L4A

=ω2x − ω2

y

ω2x + ω2

y

E = 0

1

+ number of atoms N

λ

zA λ

RA

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

ρF =Fexact

FPFA

a (µm)

L/R ∼ 10−3 − 10−2

1

U(RA) = h ∞

0

dξ

2πTr log

1−RS e−KzA RA e−KzA

K = diag(κ) , κ =

ξ2/c2 + k2

λ

zA λ

RA

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

ρF =Fexact

FPFA

1

U(xA, yA, zA) = h ∞

0

dξ

2πTr log


K = diag(κ) , κ =

ξ2/c2 + k2

λ

zA λ

RA

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

ρF =Fexact

FPFA

1


0

dξ

2πTr log


K = diag(κ) , κ =

ξ2/c2 + k2

λ

zA λ

RA

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

ρF =Fexact

FPFA

1

Reflection (scattering) operators

ground-state atom:

material surface:

λ

zA λ

RA

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

ρF =Fexact

FPFA

a (µm)

L/R ∼ 10−3 − 10−2

1

λ

zA λ

RS

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

ρF =Fexact

FPFA

a (µm)

L/R ∼ 10−3 − 10−2

1

zA

U(RA) = h ∞

0

dξ

2πTr log


K = diag(κ) , κ =

ξ2/c2 + k2

λ

zA λ

RA

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

ρF =Fexact

FPFA

1

Plane-wave basis: k = wavevector on the xy plane

Polarization p : TE (transverse electric) or TM (transverse magnetic)

k, p|RA|k, p = − ξ2

2κ

α(iξ)

0c2−p (k) · +

p(k)e−i(k−k)·rA .


0

dξ

2πTr log


K = diag(κ) , κ =

ξ2/c2 + k2

λ

zA λ

RA

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

1


k, p|RA|k, p = − ξ2

2κ

α(iξ)

0c2−p (k) · +



0

dξ

2πTr log


K = diag(κ) , κ =

ξ2/c2 + k2

λ

zA λ

RA

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

1


k, p|RA|k, p = − ξ2

2κ

α(iξ)

0c2−p (k) · +


U ≈ −h ∞

0

dξ

2πTr

RS e−KzA RA e−KzA


0

dξ

2πTr log


K = diag(κ) , κ =

ξ2/c2 + k2

λ

zA λ

RA

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

1

How to compute the Casimir-Polder potential for large-amplitude

corrugations?

λ

zA λ

RS

2π/λ

λ = 1.2 µm

λ = 2.4 µm

zeq/a

2πa/λ

F = 2P

cz

c = 3× 108 m/s

R = 100 nm

ρF =Fexact

FPFA

a (µm)

L/R ∼ 10−3 − 10−2

1

rA

Scattering approach

R. Messina, DAR Dalvit, PAMN, A. Lambrecht and S. Reynaud PRA2009

A. Contreras-Reyes, R. Guerout, PAMN, D. A. R. Dalvit, A. Lambrecht, and S. Reynaud, PRA 2010

a = 100 nmd = zA = 200 nm

a

Theoretical ResultsAnisotropy predictedby the exact scatteringapproach :- N =100 87Rb atoms- Gold surface

Anisotropy predictedby the pairwise sumapproach

Anisotropy sufficientto produce vortices experimentally

Generation of ‘Casimir’ Vorticesup to 3 microns

Title

the sample region and thus unable to transfer any angular

momentum to the condensate (the PFA potential is piece-

wise constant in the transverse coordinates with jumps

following the plate discontinuities.).

The macroscopic sample wave-function then satisfies the

time-dependent Gross-Pitaevskii equation (GPE) in the

rotating frame [1]

ih∂ψ

∂t=

− h2

2m∆+ U(r, t) +Ng|ψ|2 − ΩLz

ψ , (2)

where the effective quadratic potential U(r, t) becomes

anisotropic under the influence of the Casimir-Polder in-

teraction

U(r, t) =1

2mω2

⊥

(1 + )x2

+ (1− ) y2 +ω20z

ω2⊥(z − z0)

2

+ UCP(0, z),

with ω⊥ =ω20⊥ + ω2

CP/2. In Eq.(2), g is a constant

related to the s-wave atomic scattering length as by g =

4πh2as/m. The anisotropy is expressed as

= ω2CP/(2ω

20⊥ + ω2

CP) . (3)

The normal Casimir-Polder potential UCP(0, z) is not rel-evant in the following discussion. It slightly shifts the

sample vertical position, which is taken into account by

a redefinition of z0. After a stage of adiabatic approach

of the plate, the rotating frame potential U(r, t) becomes

time-independent.

In order to optimize the sensitivity of our setup, we

maximize the anisotropy for a given plate corruga-

tion and a given distance z0 between the sample cen-

ter and the plate. Accordingly, we choose the mini-

mum radial trap frequency providing a confinement satis-

fying Eqs.(1). Using the relations µ =12mω2

0⊥,zR⊥,z and

µ =12 hω

15Na/(h/mω)1/2

2/5with ω = ω2/3

0⊥ ω1/30z be-

tween the Thomas-Fermi radii R⊥,z, chemical potential µand trap frequencies ω⊥,z, one finds readily

ω0⊥(d, z0, N) =16√15h(Nas)1/2

mc1/2// c2⊥

1√z0d2

. (4)

This equation suggests to use a relatively dilute conden-

sate, for which a weak radial trapping can balance the re-

pulsive interactions and achieve the desired confinement.

The pancake-shaped samples considered is this Letter are

thus not rigorously in the Thomas-Fermi regime, i.e. their

transverse radius R⊥ is comparable to the the harmonic

trap scale Rω =

h/(mω). However, the transverse ki-

netic energy is much smaller than the interaction energy,

so that the required trapping frequency is given by Eq.(4)

with a good approximation.

Let us now calculate the anisotropies that can be at-

tained through the Casimir interaction between the con-

densate and the grating. For the low angular frequen-

cies used here, of the order of the transverse trapping

frequency and thus of a few dozens of hertz, dynamical

Casimir and non-contact quantum friction effects are neg-ligible [32]. For separation distances in the micrometer

range, the time it takes for light to travel between the

grating and the atom is of the order of 10−14

s and hence

much shorter than the time scale associated to the grat-

ing rotation. Thus, the atom interacts with the instan-

taneous angular position of the plate (on the other hand

retardation is very important as far as charge fluctuations

in both atom and grating are concerned). Therefore

in the rotating frame the potential is given by the static

potential, which we compute by following the scattering

approach [25,30]. The atom-surface potential is written in

terms of the dynamic atomic polarizability α evaluated at

imaginary frequencies iξ and of the reflection operator RS

describing non-specular diffraction by the grating [20].

To calculate RS for a periodic grating, we employ

the Rayleigh basis for the fields propagating inside the

homogenous regions corresponding to the bulk material

medium (z > 0) and to the empty space below the grating

(z < −a). The fields in the modulated region of thickness

a are obtained by solving coupled differential equations.The reflection matrix elements j,σ|RS(kx, ky, ξ)|j,σ(with kx varying in the first Brillouin zone [−π/d,π/d] andσ and j representing polarization and diffraction order re-

spectively) are then obtained by matching the expansions

in the three regions across the boundaries at z = 0 and

z = −a. The potential at position (x, y, z) is written as

UCP(x, z) =h

oc2

∞

0

dξ

2π

∞

−∞

dky2π

π/d

−π/d

dkx2π

×

j,j

ξ2

2κjα(iξ) e2πi(j−j)x/d e−(κj+κj )|z| (5)

×

σ,σ

j,σ|RS(kx, kz, ξ)|j,σ −σ (k, iξ) · +σ(k, iξ) ,

with κj =

ξ2/c2 + (kx + 2πj/d)2 + k2y. The unit vectors

±σ provide the direction of the electric field propagating

upwards/downwards with polarization σ = TE or TM.

To calculate ωCP, we first differentiate UCP(x, z) twice

with respect to x and then evaluate the resulting expres-

sion at x = 0 (middle of the plateau) numerically. We

consider a silicon rectangular (lamellar) grating with am-

plitude a = 4µm, period d = 30µm and gap width d/2(see Fig. 1). The corresponding polarizability function

α(iξ) is provided by Ref. [33], whereas the electric per-

mittivity for intrinsic silicon at the imaginary frequency

axis, required to compute the reflection matrix RS in (5),

is calculated from data at real frequencies obtained from

Ref. [34] with the help of a suitable Kramers-Kronig rela-

tion [35]. We choose a sample of N = 10087Rb atoms,

which is both sufficient to detect vortices with state-of-the

art techniques and compatible with a weak radial confine-

ment.

In Fig. 2, we plot the anisotropy as a function of

the separation distance, with ω0⊥ computed from (4)

p-3

corrugation amplitude = 4μm, period d = 30 μm

F. Impens, A.M. Contreras-Reyes, PAMN, D. Dalvit, R. Guerout, A. Lambrecht and S. Reynaud EPL 2010

20




21

True `dynamical’ non-adiabatic Casimir effect...?

Non-adiabaticity: generation of photons out of the vacuum state - dissipative torque ?

Is it possible to enhance the angular momentum coupling by taking a corrugated surface?

⇒ single rotating corrugated disk on a flat plate

rotation `easier’ than translation....


h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R3

c3σ0

P = h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

1

22

small local height h(x,y,t) - first order in h (small slope + non-relativistic limit)

perfect reflectivity - no zeroth-order effect example/main motivation: rigid rotation (polar coordinates)

h(x,y,t)Assumptions:

θ0(t)

angular velocity


Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R3

c3σ0

P = h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

1


h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R3

c3σ0

P = h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

θ = θ

1

23

example/main motivation: rigid rotation (polar coordinates)

Assumption (2):

θ0(t)

h(x,y,t) = 0 outside a disk of radius R (with Ω R << c )

angular velocity


Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R3

c3σ0

P = h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

rr(b) =1

θθ=

1

φφ

Λii =

∂xi

∂xi

1


kz = (ω2/c

2 − k2)1/2

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R

3

c3σ0

P = h (1− 1/n2)2Ω20 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

1

24

B) Reflected fields to first order in h(x,y,t)

Theoretical approach (1)


e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω

)H(k−k,ω−ω

)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R

3

c3σ0

P = h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

1

A) Angular spectrum representation for the field operators, polarization patomic dimensions zA

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω

)H(k−k,ω−ω

)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R

3

c3σ0

P = h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

1

zeroth-order:

FTh(x,y,t)

h(x,y,t)

z

k’

k


e(0)p (k

,ω)

e(1)p(k,ω) eikzz

kz = (ω2/c

2 − k2)1/2

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R

3

c3σ0

P = h (1− 1/n2)2Ω20 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

1


e(0)p (k

,ω)

e(1)p(k,ω)

kz = (ω2/c

2 − k2)1/2

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R

3

c3σ0

P = h (1− 1/n2)2Ω20 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

1


Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R3

c3σ0

P = h (1− 1/n2)2Ω2

0 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

Ω ∼ ω0⊥ ∼ 102Hz

rr =b

b− a

(r − a)2

r2

θθ = φφ =b

b− a

1

25

C) Flux density of angular momentum (z component) along the z direction

Theoretical approach (2)

D) Casimir torque on the disk of radius R atomic dimensions zA

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R

3

c3σ0

P = h (1− 1/n2)2Ω20 ×

(ω0R/c)4

2160π, ω0R/c 1

(ω0R/c)10

5103000π, ω0R/c 1

ωCP

R⊥ =d

4

Rz = 0.2 z0

1

FT...


τ (ω) = 0

ω/Ω

H(k, θ,ω) = exp

iω

Ωθ

H(k, 0,ω)

τ (ω) =R

2π

∞0

dkJ1(kR)

kχ(k,ω)

2π

0dθ∂θH(k, θ,ω)

0|Mzz(k, 0+,ω)|0 = χ(k,ω) ∂θH(k, θ,ω)

τ (ω) =R

2π

d2k

J1(kR)

k0|Mzz(k, 0

+,ω)|0

e(0)p (k

,ω)

e(1)p(k,ω)

kz = (ω2/c

2 − k2)1/2

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

1


0|Mzz(k, 0+,ω)|0 = χ(k,ω) ∂θH(k, θ,ω)

τ (ω) =R

2π

d2k

J1(kr)

k0|Mzz(k, 0

+,ω)|0

e(0)p (k

,ω)

e(1)p(k,ω)

kz = (ω2/c

2 − k2)1/2

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

h(r, θ, t) = f (r, θ − θ0(t))

P =h

30π2

Ω6R

3

c3σ0

P = h (1− 1/n2)2Ω20 ×

1


ω/Ω

H(k, θ,ω) = exp

iω

Ω

H(k, 0,ω)

τ (ω) =R

2π

∞0

dkJ1(kr)

kχ(k,ω)

2π


0|Mzz(k, 0+,ω)|0 = χ(k,ω) ∂θH(k, θ,ω)

τ (ω) =R

2π

d2k

J1(kr)

k0|Mzz(k, 0

+,ω)|0

e(0)p (k

,ω)

e(1)p(k,ω)

kz = (ω2/c

2 − k2)1/2

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

1


ω/Ω

H(k, θ,ω) = exp

iω

Ωθ

H(k, 0,ω)

τ (ω) =R

2π

∞0

dkJ1(kr)

kχ(k,ω)

2π


0|Mzz(k, 0+,ω)|0 = χ(k,ω) ∂θH(k, θ,ω)

τ (ω) =R

2π

d2k

J1(kr)

k0|Mzz(k, 0

+,ω)|0

e(0)p (k

,ω)

e(1)p(k,ω)

kz = (ω2/c

2 − k2)1/2

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

1


τ (ω) = 0

ω/Ω

H(k, θ,ω) = exp

iω

Ωθ

H(k, 0,ω)

τ (ω) =R

2π

∞0

dkJ1(kr)

kχ(k,ω)

2π


0|Mzz(k, 0+,ω)|0 = χ(k,ω) ∂θH(k, θ,ω)

τ (ω) =R

2π

d2k

J1(kr)

k0|Mzz(k, 0

+,ω)|0

e(0)p (k

,ω)

e(1)p(k,ω)

kz = (ω2/c

2 − k2)1/2

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

1


τ (ω) = 0

ω/Ω

H(k, θ,ω) = exp

iω

Ωθ

H(k, 0,ω)

τ (ω) =R

2π

∞0

dkJ1(kR)

kχ(k,ω)

2π


0|Mzz(k, 0+,ω)|0 = χ(k,ω) ∂θH(k, θ,ω)

τ (ω) =R

2π

d2k

J1(kR)

k0|Mzz(k, 0

+,ω)|0

e(0)p (k

,ω)

e(1)p(k,ω)

kz = (ω2/c

2 − k2)1/2

τ (t) =

r≤Rd2r 0|Mzz(r, z = 0+, t)|0

E(0)p(k, z,ω) = e

(0)p(k,ω)

e−ikzz (−)

p± e

ikzz (+)p

e(1)p(k,ω) =

p

d2k

(2π)2 dω

2πRpp(k,ω;k

,ω)H(k−k

,ω−ω)e(0)p(k

,ω)

Mzz = y

0ExEz +1

µ0BxBz

− x

0EyEz +1

µ0ByBz

Ω =dθ0dt

1

26

Flux density of angular momentum

Results

Dynamical Casimir torque on the disk of radius R

susceptibility does not depend on the direction of k

Example: rigid rotation with constant angular velocity Ω

reality is even

27

Conclusion:

driving vortices in a BEC: Casimir effect with a rotating plate provides a contactless transfer of angular momentum - much stronger than a trivial pairwise effect !!

no dissipative Casimir torque on a rotating corrugated plate (uniform velocity, first order)

THANK YOU !


Padova, June 2011mir/Casimir2011/MaiaNeto.pdf · Padova, June 2011. Collaborators LKB - ENS, Paris...

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Transcript of Padova, June 2011mir/Casimir2011/MaiaNeto.pdf · Padova, June 2011. Collaborators LKB - ENS, Paris...