interações dispersivas e a ótica quântica - if.ufrj.brtgrappoport/davidovich/maianeto.pdf ·...

Interações dispersivas e a ótica quânticaPaulo A. Maia Neto

Fatos e Fótons

Uma homenagem aos 67 anos de Luiz Davidovich

Rio de Janeiro, agosto 2013

http://fatosefotons.if.ufrj.br/

http://fatosefotons.if.ufrj.br/

PUC-Rio nos anos 80...

83-84: Diretas Já

1987: eletrodinâmica quântica em cavidadesM Brune, J-M Raymond, S. Haroche

átomo ressonante + 1 modo intracavidade do campo electromagnético

micromaser

+ recente: cavidade aberta cavidade supercondutora:

fóton num dado modo da cavidade durante ~ 0.1 s

~ 109 idas-e-voltas !

Átomo + campo da cavidade

soma sobre modos da cavidade

Estado de mais baixa energia do campo electromagnético: nenhum fóton - vácuo quânticoEvac = energia associada às flutuações de ponto zero do cpo electromagnético

Problema: Evac é sempre infinito !

Interações dispersivas e a ótica quântica

A princípio todos os modos normais da cavidade (índice α) são relevantes

1948 – Casimir shows how to extract a finite, physical quantity out of the vacuum energy !

Does the vacuum energy have a physical meaning ?W. Pauli (1933) : “..here it is more consistent, in contrast with the material oscillator, to not introduce a zero-point energy (ħω/2) per degree of freedom. On one hand, this energy would lead to an infinitely large energy density, and on the other hand it would not be observable since it cannot be emitted, absorbed, or scattered, and hence cannot be contained between walls and, as evident from common experience, does not produce any gravitational effect.”

W. PauliHendrik BG Casimir


Simplest example: two neutral metallic plates in vacuum. Frequencies depend on separation L !

λ = L

L L

λ = 2L

L


1

θ

E ∼ 1

L3

E ∼ 1

L2

E = − π2

720

�cL3

A

F = − π2

240

�cL4

A

kz → iκ

eikzL

r1r2 e2ikzL = 1

λT = 7.6µm

U (1)(xA, zA) = h0 g(kc, zA) sin(kcxA)

U (1)(xA, zA) ≈ g(0, zA)h(xA)

h(x) = h0 sin(kcx)

kc = 2π/λ

Fps = 2πREpp

A

U

UPP|

E = −0.0245ωP�A2πL2

b (µm)

U = −C4

z4

bigskip

1

θ

E ∼ 1

L3

F = −dE(L)

dL

E ∼ 1

L2

E = − π2

720

�cL3

A

F = − π2

240

�cL4

A

kz → iκ

eikzL

r1r2 e2ikzL = 1

λT = 7.6µm


U (1)(xA, zA) ≈ g(0, zA)h(xA)

h(x) = h0 sin(kcx)

kc = 2π/λ

Fps = 2πREpp

A

U

UPP|

E = −0.0245ωP�A2πL2

b (µm)

U = −C4

z4

Casimir force between two perfect plane metallic plates (zero temperature) - 1948

Zero-point (vacuum) field energy depends on the cavity length L

(usually) attractive force between neutral plates

A

Lconnection with van der Waals dispersive force

F ~ 1/L3 ....... ????


...but for short separation distances finite conductivity effects are important !

E. M. Lifshitz 1956: Casimir with real metals or dielectric plates

Casimir considered the ideal perfectly-reflecting

model for metals.... A

E. M. Lifshitz

Dispersive Interactions: introductionInterações dispersivas e a ótica quântica

Lifshitz formula - from the point-of-view of Quantum Optics/cavity QED:

density of modes modified by the mirrors reflection coefficients as seen by the

intracavity field intracavity mode: defined by condition of

constructive interference

Sum over polarizations

Dispersive Interactions: introduction

1

θ

r1r2 e2ikzL = 1

λT = 7.6µm


U (1)(xA, zA) ≈ g(0, zA)h(xA)

h(x) = h0 sin(kcx)

kc = 2π/λ

FPFA = 2πRUplane

A

U

UPP|

U = −0.0245ωP�A2πL2

b (µm)

U = −C4

z4

bigskip

δEPP(a.u.)

λC/Ly

2λC

Ly

λC

Ly

1

θ

kz → iκ

eikzL

r1r2 e2ikzL = 1

λT = 7.6µm


U (1)(xA, zA) ≈ g(0, zA)h(xA)

h(x) = h0 sin(kcx)

kc = 2π/λ

FPFA = 2πRUplane

A

U

UPP|

U = −0.0245ωP�A2πL2

b (µm)

U = −C4

z4

bigskip

δEPP(a.u.)

λC/Ly

2λC

Ly

1

e−κL

r1r2e−2κLE = E

F =π2

240

�c

d4A

E = − π2

720

�c

d3A

c/L� ωP

c/L� ωP

EPP(L) = −0.0245 ωP�cA

2πL2

EPP(L) = − π2

720

�cA

L3

FPP

A= − 1

A

dEPP(L)

dL= − π2

240

�c

L4

1 2Closed loops

r1

r2

1

L

e−κL

r1r2e−2κLE = E

F =π2

240

�c

d4A

E = − π2

720

�c

d3A

c/L� ωP

c/L� ωP

EPP(L) = −0.0245 ωP�cA

2πL2

EPP(L) = − π2

720

�cA

L3

FPP

A= − 1

A

dEPP(L)

dL= − π2

240

�c

L4

1

θ

eikzL

r1r2 e2ikzL = 1

λT = 7.6µm


U (1)(xA, zA) ≈ g(0, zA)h(xA)

h(x) = h0 sin(kcx)

kc = 2π/λ

FPFA = 2πRUplane

A

U

UPP|

U = −0.0245ωP�A2πL2

b (µm)

U = −C4

z4

bigskip

δEPP(a.u.)

λC/Ly

2λC

Ly

1

θ

eikzL

r1r2 e2ikzL = 1

λT = 7.6µm


U (1)(xA, zA) ≈ g(0, zA)h(xA)

h(x) = h0 sin(kcx)

kc = 2π/λ

FPFA = 2πRUplane

A

U

UPP|

U = −0.0245ωP�A2πL2

b (µm)

U = −C4

z4

bigskip

δEPP(a.u.)

λC/Ly

2λC

Ly

z

k

k

Ep(k)

r2;p(k)Ep(k)

ω → iξWick rotation: integrate over imaginary freqs.


Numerical example: two metallic mirrors described by the plasma model

Power law modification: from van der Waals to

Casimir

quantum fluctuating surface plasmons

Geometry and the Casimir effect - theory

1

ωP

λP = 2πc/ωP

F

FCAS

L

e−κL

r1r2e−2κLE = E

F =π2

240

�c

d4A

E = − π2

720

�c

d3A

c/L� ωP

c/L� ωP

zero frequency limit of reflection coefficients r[0] = -1 ⇒Perfect reflectors

1

θ

E = − π2

720

�cL3

A

kz → iκ

eikzL

r1r2 e2ikzL = 1

λT = 7.6µm


U (1)(xA, zA) ≈ g(0, zA)h(xA)

h(x) = h0 sin(kcx)

kc = 2π/λ

FPFA = 2πRUplane

A

U

UPP|

U = −0.0245ωP�A2πL2

b (µm)

U = −C4

z4

bigskip

δEPP(a.u.)

λC/Ly

1

θ

E = − π2

720

�cL3

A

kz → iκ

eikzL

r1r2 e2ikzL = 1

λT = 7.6µm


U (1)(xA, zA) ≈ g(0, zA)h(xA)

h(x) = h0 sin(kcx)

kc = 2π/λ

FPFA = 2πRUplane

A

U

UPP|

E = −0.0245ωP�A2πL2

b (µm)

U = −C4

z4

bigskip

δEPP(a.u.)

λC/Ly

L+++--

+++

--


van Kampen, Nijboer, and Schram 1968

Casimir 1948

Lamoreaux - U. Washington → Yale (1997 - ...)

Mohideen et al - Riverside (1998 - ...) – AFM

Decca et al - IUPUI - Indianapolis – measurement of force gradient (frequency shift)

....

Capasso et al - Harvard (2009) - repulsive Casimir force in the retarded regime

AFM setup

Radius ~ 100 µm

d ~ 100 nm

F ~ 200 pN


Modern experiments



Enhancing the distance between predictions with/without dissipation: ⇒ Hg microspheres

Measuring the Casimir force at the LPO - UFRJ with optical tweezers !

*** tunable stiffness ***

Hg+ethanol+polystyrene: cross-over from attraction to repulsion

polystyrene sphere radius = 2 μm


Atom - surface dispersive interaction


zA

atom-surface dispersive interaction

R. Messina, D. Dalvit, PAMN, A. Lambrecht and S. Reynaud, Phys Rev. A 2009

Casimir-Polder (1948):metallic surface, long-distance limit (atom fast, field slow)

atomic dimensions � zA

F (0)CP(zA) = − 3hcα(0)

8π2�0 z5A

RS = R(0) +R

(1) +O(h2)

→Ep (k, ω) =

� d2k�

(2π)2

�

p��k, p|RS|k

�, p��←Ep� (k�, ω)

�k, p|RA|k�, p�� = − ξ2

2κ

α(iξ)

�0c2�−p (k) · �+

p�(k�)e−i(k−k�)·rA .

U ≈ −h� ∞

0

dξ

2πTr

�RS e−KzA RA e−KzA

�

U(xA, yA, zA) = 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

1

zA/c � 1/ωat

kx,y/P (pNµm−1mW−1)

a = 0.268µm

a = 0.376µm

a = 0.527µm

ky/P (pNµm−1mW−1)

a (µm)

d (µm)

z0 (µm)

cV/aP

(c/a)V/P

1

10−1

100

10−2 10−1 100 101

ηF

zA (µm)

atomic dimensions � zA

ηF ≡F (0)

F (0)CP

g(k, z) = ρ(k, z) ηF F (0)CP ≈ ρperf

CP (kz) ηF F (0)CP

ρ(k, zA) ≈ ρperfCP (kzA) = e−kzA

�

1 + kzA +16(kzA)2

45+

(kzA)3

45

�

ρ =g(k, zA)

F (0)(zA)

g(k, zA) ≈ g(0, zA) = U (0)�(zA) ∼ 1/z4A

λ = 2π/k

F (0)CP(zA) = − 3hcα(0)

8π2�0 z5A

RS = R(0) +R

(1) +O(h2)

→Ep (k, ω) =

� d2k�

(2π)2

�

p��k, p|RS|k

�, p��←Ep� (k�, ω)

�k, p|RA|k�, p�� = − ξ2

2κ

α(iξ)

�0c2�−p (k) · �+

p�(k�)e−i(k−k�)·rA .

U ≈ −h� ∞

0

dξ

2πTr

�RS e−KzA RA e−KzA

�

U(xA, yA, zA) = h� ∞

0

dξ

2πTr log

�1−RS e−KzA RA e−KzA

�

K = diag(κ) , κ =�

ξ2/c2 + k2

λ

zA � λ

1

ex: Rb atomsAu

Si

Measuring the atom-surface interaction with atom interferometers

J. D. Perreault and A. D. Cronin, PRL 2005 S. Lepoutre et al, EPL 2009

.....only in the short-distance van der Waals regime so far

Non-planar geometries: atom-surface interaction atom-surface dispersive interaction

Mach-Zender interferometer - surface interaction in one of the arms

k = 2

k = 1z1(t)

zA/c � 1/ωat


a = 0.268µm

a = 0.376µm

a = 0.527µm


a (µm)

d (µm)

z0 (µm)

cV/aP

1

t = 0

t = T

z1(t)

zA/c � 1/ωat


a = 0.268µm

a = 0.376µm

a = 0.527µm


a (µm)

d (µm)

1

t = 0

t = T

z1(t)

zA/c � 1/ωat


a = 0.268µm

a = 0.376µm

a = 0.527µm


a (µm)

d (µm)

1

ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

t = T

z1(t)

zA/c � 1/ωat


a = 0.268µm

a = 0.376µm

a = 0.527µm


1

Casimir (or van der Waals) phase for narrow wavepackets

Taking the atomic motion into account....our goal : dynamical correction to the Casimir phase


ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

t = T

z1(t)

zA/c � 1/ωat


a = 0.268µm

a = 0.376µm

a = 0.527µm


1

k = 1t = 0 t = Tz1(t)

zA/c � 1/ωat


a = 0.268µm

a = 0.376µm

a = 0.527µm


a (µm)

d (µm)

z0 (µm)

cV/aP

1

only in the quasi-static regime!!

position of atomic center of mass

Electric field

Metallic plate

Hamiltonian in the electric dipole approx.

ra

d

ra

E(r)H = - d.E(ra)

Full quantum theory of Casimir interferometers


Atomic center-of-mass as an open quantum system :coupling with electromagnetic field and atomic dipole

dipole moment: internal atomic

degrees of freedom

F Impens, R Behunin, C Ccapa -Ttira and PAMN, EPL 2013



We trace over dipole and field degrees of freedom


focus on the CM

Two types of contributions:- single path: associated to individual paths- double path: associated to pairs of paths

Effect of reservoir (dipole + field) captured by the influence functional

{ϕSPk ≈ −1

h

� T

0dt U(zk(t))

ϕSPk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}� G (rk(t), t; rk(t

�), t�) .

ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

t = T

z1(t)

zA/c � 1/ωat


a = 0.268µm

1



F Impens, R Behunin, C Ccapa-Ttira and PAMN, EPL 2013

- single path phase (path k) for short distances

ϕSPk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}� G (rk(t), t; rk(t

�), t�) .

ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

t = T

z1(t)

zA/c � 1/ωat


a = 0.268µm

a = 0.376µm

a = 0.527µm

1

rk(t’)rk(t)

rIk(t’)

{electric field Green function: field of a point dipole at rk(t’) propagated to rk(t) after one reflection

dipole fluctuations: symmetric correlation function

long time T, quasi-static limit:

{φDPjk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}�

�G (rj(t), t; rk(t

�), t�)−G (rk(t), t; rj(t�), t�)

�

ϕSPk ≈ −1

h

� T

0dt U(zk(t))

ϕSPk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}� G (rk(t), t; rk(t

�), t�) .

ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

t = T

z1(t)

zA/c � 1/ωat


1



F Impens, R Behunin, C Ccapa-Ttira and PAMN, EPL 2013

- double path phase for short distances (arms j and k)

field of a point dipole at rk(t’) propagated to rj(t) after one reflection

r2(t)

rI1(t’)

r1

rI2

rI1

(b) (c)

rI2(t’)

r1(t) r1

rI2

rI1

S

S

example: k =1, j =2φDPjk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}�


�), t�)−G (rk(t), t; rj(t�), t�)

�

φDP21 < 0

ϕSPk ≈ −1

h

� T

0dt U(zk(t))

ϕSPk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}� G (rk(t), t; rk(t

�), t�) .

ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

t = T

z1(t)

zA/c � 1/ωat

1

because

φDPjk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}�


�), t�)−G (rk(t), t; rj(t�), t�)

�

z2 > z1

φDP21 < 0

ϕSPk ≈ −1

h

� T

0dt U(zk(t))

ϕSPk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}� G (rk(t), t; rk(t

�), t�) .

ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

t = T

z1(t)

zA/c � 1/ωat

1




- double path phase for short distances (arms j and k): harmonic oscillator model for the internal degrees of freedom

4

j to k and vice-versa, which is brought into play by the fi-

nite speed of light and the vertical motions of each packet.

In order to derive an explicit analytical result from

(14), we assume that the different atomic paths are in

the same vertical plane and share the same velocity com-

ponent parallel to the plate. On the other hand, we take

arbitrary non-relativistic motions along the perpendic-

ular direction, which correspond to the functions zk(t),under the short-distance condition ω0zk/c � 1. Neglect-

ing as before terms of order (zk/c)2, we derive from (14)

φDPjk = 3

ω0α(0)

4π�0c

� T

0dt

zk(t)− zj(t)

(zj(t) + zk(t))3(15)

Note that this phase is independent of the velocity com-

ponent parallel to the conductor plane. This follows from

translational invariance parallel to the plate and from the

condition of perfect conductivity. Because it depends

linearly on the speed of each trajectory, φDPjk is invari-

ant under time dilatation zj → zj(t) ≡ zj(Λt), j = 1, 2,T → T/Λ, with Λ arbitrary.

We now stress the main point of this letter: the

double-path phase φDPjk as given by (15) is non-additive,

since the denominator in its r.-h.-s. does not allow one

to isolate separate contributions from paths j and k,a signature of the non-local nature of φDP

jk . This non-

additivity is enhanced when considering a geometry for

which the third path is much further away from the plate

than the first and second paths (see Fig. 1): we take

z3(t) � z1(t), z2(t) and assume that the differences in

vertical atomic velocities are of the same order of magni-

tude z1(t)−z2(t) ∼ z2(t)−z3(t). It then follows from (15)

that φDP13 + φDP

32 � φDP12 : the non-additivity is maximal

in this case.

One can actually use the non-additivity in order to iso-

late the non-local dynamical corrections from the other

phase contributions. For the three-arm interferometer

shown in Fig. 1, we propose to measure separately the

three independent phase coherences appearing in Eq. (7),

φjk ≡ φ(0)jk +

1�Re [SIF[rj , rk]] with j, k = 1, 2, 3, j �= k,

by performing interferometric measurements between the

different pairs of arms. Using (8), we find that the (max-

imal) violation of phase additivity gives the desired non-

local double-path shift φDP12 :

φDP12 ≈ φ12 − (φ13 + φ32). (16)

This approach removes all the additive phases, leav-

ing only the non-local dynamical correction to the vdW

phase. We have studied [5] its amplitude for 87Rb atoms

close to a metallic plate, taking path 1 to be parallel

to the plate at a constant distance z1 = 20nm, similar

to the experimental value reported in Ref. [3]. In this

configuration, the integrated double-path phase (15) de-

pends only on the end-point positions of path 2. We take

z2(0) = z1 and z2(T ) � z1 to find φDP12 � 3 × 10−7 rad.

This is beyond the state of the art in atom interferom-

etry, but still bigger than systematics considered in the

best atom interferometers [2].

To conclude, we have used an open system theory of

atom interferometers to derive dynamical corrections to

the standard van der Waals phase shifts. We have shown

that the interplay between field retardation effects and

the external atomic dynamics generates first-order dy-

namical corrections. The local corrections, associated to

individual paths, turn out to be equivalent to coarse-

graining the vdW potential over a time scale correspond-

ing to the round-trip travel time of the atom-surface in-

teraction. The non-local phase corrections are associated

to pairs of interferometer paths rather than to individual

ones, and are of the same order of magnitude of the local

corrections. More importantly, they are generally non-

additive, a distinctive characteristic associated to non-

locality. We have proposed a method to isolate them

from other phase shifts in a three-path atom interferom-

eter. These results show that coupling with the envi-

ronment may induce, in addition to decoherence, phase

shifts with unusual properties in atom optics.

Acknowledgments

The authors are grateful to Ryan O. Behunin and

Reinaldo de Melo e Souza for stimulating discussions.

This work was partially funded by CNRS (France),

CNPq, FAPERJ and CAPES (Brasil).

[1] A. D. Cronin, J. Schmiedmayer and D. E. Pritchard, Rev.Modern Phys. 81, 1051 (2009) and references therein.

[2] J. M Hogan, D. M. S. Johnson, M. A. Kasevich, in Proc.Int. School of Physics Enrico Fermi (2007) and referencestherein.

[3] J. D. Perreault and A. D. Cronin, Phys. Rev. Lett. 95,133201 (2005); S. Lepoutre, H. Jelassi, V. P. A. Lonij, G.Trenec, M. Buchner, A. D. Cronin and J. Vigue, Euro-phys. Lett. 88, 20002 (2009).

[4] Peter Wolf, Pierre Lemonde, Astrid Lambrecht,Sebastien Bize, Arnaud Landragin, and Andre Cla-iron, Phys. Rev. A 75, 063608 (2007); Sophie Pelisson,

Riccardo Messina, Marie-Christine Angonin, and PeterWolf, Phys. Rev. A 86, 013614 (2012).

[5] F. Impens, R. O. Behunin, C. Ccapa Ttira, and P. A.Maia Neto, Europhysics Lett. 101, 60006 (2013).

[6] R. O. Behunin, and B.-L. Hu, J. Phys. A: Math. Theor.43, 012001 (2010); Phys. Rev. A 82, 022507 (2010);Phys. Rev. A 84, 012902 (2011).

[7] A. Stern, Y. Aharonov, and Y. Imry, Phys. Rev. A 41,3436 (1990); J. R. Anglin and W. H. Zurek, in Dark Mat-ter in Cosmology, Quantum Measurements, ExperimentalGravitation, pp. 263-270, edited by R. Ansari, Y. Giraud-Heraud and J. Tran Tranh Van (Editions Frontieres, Gif-

k = 2

k = 1

t = 0

t = T

z1(t)

zA/c � 1/ωat


a = 0.268µm

a = 0.376µm

a = 0.527µm


a (µm)

d (µm)

1

t = 0

t = T

z1(t)

zA/c � 1/ωat


a = 0.268µm

a = 0.376µm

a = 0.527µm


a (µm)

d (µm)

1

φDPjk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}�


�), t�)−G (rk(t), t; rj(t�), t�)

�

φDP12 = 3.5× 10−7

z2 > z1

φDP21 < 0

ϕSPk ≈ −1

h

� T

0dt U(zk(t))

ϕSPk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}� G (rk(t), t; rk(t

�), t�) .

ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

t = T

z1(t)

1

numerical example:Rb atomz0 = 20 nmvertical displ. >> z0

z0

2

We first present the standard analysis of this atom in-

terferometer by means of an instantaneous vdW poten-

tial VvdW(r). Provided that the vdW potential on the

atoms is weak enough so as to make dispersion effectsnegligible, an excellent approximation in the experimen-

tal conditions of Ref. [? ], one can apply the ABCD

propagation method [? ] for atomic waves in quadratic

potentials: at any time t > 0, each atomic wave-packet

is given by

|ψk(t)� = |χk(t)�ei[ϕ(0)k (t)+ϕ(vdW)

k (t)](1)

with a time-dependent Gaussian �r|χk(t)� =

wp(r, rk(t),pk(t),wk(t)). The precise value of the

width vector wk(t) is not important for the coming

discussion. The average atomic position rk(t) and

momentum pk(t) follow the classical equations of motion

with the initial conditions rk(0) = r0 k and pk(0) = p0 k

associated with the central trajectory corresponding

to path k (k = 1, 2, 3). More important for our

discussion are the phase contributions in Eq. (??).

The phase ϕ(0)k

(t) collects the free propagation and

external potential effects, whereas ϕ(vdW)k

(t) accounts

for the dispersive atom-surface interaction. From now

on, we focus on the phase accumulated between the

instants t = 0 and t = T , omitting explicit reference

to time T to alleviate notations. The phase ϕ(0)k

is

given by the following integral along the trajectory k:

ϕ(0)k

=1��T

0 dt

�p2

k(t)2m − E(t)− Vext(rk(t))

�, where E(t)

is the internal atomic energy at time t. In this standard

approach, the atom-surface interaction simply yields

an additional phase shift given by the integration of

the vdW potential VvdW(z) taken at the instantaneous

atomic positions along the path k :

ϕ(vdW)k

= −1

�

�T

0dt VvdW(zk(t)) (2)

The density matrix corresponding to the atomic state

at time T computed within the standard ABCD ap-

proach is then given by

ρ(T ) = ρdiag(T ) +1

3

3�

j<k

|χj(T )��χk(T )|eiφstjk +H.c.

(3)

with ρdiag(T ) ≡ 13

�k|χk(T )��χk(T )| and H.c. represent-

ing the Hermitian conjugate. We focus here on the stan-

dard phase coherences φstjk,

φstjk

= φ(0)jk

+ ϕ(vdW)j

− ϕ(vdW)k

, (4)

φ(0)jk

≡ ϕ(0)j

− ϕ(0)k

. (5)

They (obviously) satisfy additivity:

φstjk

= φstj� + φst

�k (6)

for any j, k, � = 1, 2, 3, since they originate from phases

associated to individual paths in Eq. (??).

We now analyse the multiple-path atom interferom-

eter as an open quantum system, building on our re-

cent work [? ], and show that the additivity condition

(??) no longer holds. We start from the full quantum

system, whose dynamics is described by the Hamilto-

nian H = HE + HD + HF + HAF , including the exter-

nal (HE), internal (HD) and electromagnetic field (HF )

d.o.f.s. The interaction Hamiltonian, which reads in the

electric dipole approximation HAF = −d · E(ra), couplesthe atomic center-of-mass ra to the internal dipole d and

the electric field E.

The external atomic waves are described by the re-

duced atomic density matrix obtained after coarse-

graining over the field and internal atomic d.o.f.s.

These play the role of an environment, whose effecton the atomic waves is captured by an influence phase

SIF[rj , rk] [? ]:

ρ(T ) = ρdiag(T ) + (7)

1

3

3�

j<k

|χj(T )��χk(T )|ei(φ(0)jk + 1

�SIF[rj ,rk]) + H.c.

The complex influence phase1�SIF[rj , rk], evaluated

along the central atomic trajectories j and k (a valid

approximation for narrow wave-packets), describes com-

pletely the atom-surface interaction effects. Its imaginary

part corresponds to the plate-induced decoherence, and

its real part gives the atomic phase shift arising from sur-

face interactions. This phase contains local contributions

involving a single path (SP) at a time, and a non-local

double-path (DP) contribution involving simultaneously

two paths:

1

�Re [SIF[rj , rk]] = ϕSPj

− ϕSPk

+ φDPjk

. (8)

In this letter, we provide explicit analytical results for

the single and double-path phase contributions in the

short-distance van der Waals limit ω0zk/c � 1, which

yields larger phase shifts and matches the conditions of

the experiments performed so far [? ]. In this regime,

the dominant contribution comes from the Hadamard (or

symmetric) dipole correlation function (d is any Carte-

sian component of the vector operator d)

GH

d(t, t

�) ≡ 1

� �{d(t), d(t�)}�, (9)

which contains the information about the quantum dipole

fluctuations, whereas the Hadamard electric field corre-

lation function yields a negligible contribution.

In the short-distance limit, the relevant field correla-

tion function is the retarded Green’s function represent-

ing the electric field linear response susceptibility to the

fluctuating dipole source:

GR

E(r, t; r�, t�) ≡ i

�θ(t− t�)

�

η=x,y,z

�[Eη(r, t), Eη(r�, t

�)]�,

(10)

φDP12 ≈ φ12 − (φ13 + φ32).

than the first and second paths (see Fig. 1): we take

z3(t) � z1(t), z2(t) and assume that the di

vertical atomic velocities are of the same order of magni-

φstjk = ϕst

j − ϕstk

betaprime (Drude) log2 log3 log4 log5L=7 microns, y=-10 −0.00349059 0.158024 0.185601 0.628486L=2.6 microns, y=-11 −0.0721841 0.154223 0.200146 1.07548

k�

ε0R �2 V 2rms/L

4

ε0RV 2rms/L

P patchpp (L) ≈ ε0 V 2

rms

L2

P patchpp (L) ≈ 3ζ(3)

π

ε0C[0]

L4≈ 0.90

ε0 �2 V 2rms

L4

C[k = 0] =�d2r C(r) =

1

4π�2C(0) =

1

4π�2 V 2

rms (1)

r

1

Double-path phase is non-additive !!

`Standard’ phase:

F Impens, C Ccapa-Ttira and PAMN, arxiv 2013


Multiple-path interferometer

...is additive:

Total phase: φjk = φstjk + φDP

jk

φDPjk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}�


�), t�)−G (rk(t), t; rj(t�), t�)

�

φDP12 = 3.5× 10−7

z2 > z1

φDP21 < 0

ϕSPk ≈ −1

h

� T

0dt U(zk(t))

ϕSPk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}� G (rk(t), t; rk(t

�), t�) .

ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

t = T

1

...is non-additive:

example:φjk − (φj� + φ�k) = φDP

jk − (φDPj� + φDP

�k ) �= 0

φjk = φstjk + φDP

jk

φDPjk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}�


�), t�)−G (rk(t), t; rj(t�), t�)

�

φDP12 = 3.5× 10−7

z2 > z1

φDP21 < 0

ϕSPk ≈ −1

h

� T

0dt U(zk(t))

ϕSPk =

1

4h

� T

0dt

� T

0dt� �{d(t), d(t�)}� G (rk(t), t; rk(t

�), t�) .

ϕ(vdW)1 = −1

h

� T

0dt U(z1(t))

U(zA) =h

�0c2� ∞0

dξ

2πξ2α(iξ)

� d2k

(2π)2e−2κzA

2κ

rTE(k, iξ)− (1 +2c2k2

ξ2)rTM(k, iξ)

t = 0

1

25

Conclusion

Recent experimental and theoretical developments opens the way for several applications and connections...

Casimir Physics

Quantum Opticsscattering of vacuum fluctuations

cavity QED

Atomic Physicsatom interferometry, cold atoms, BEC, atom chips Condensed Matter

patch potentials, dielectric models, dissipation

Statistical Physicsnon-equilibrium, open quantum

systems

Cosmology, Astrophysicsvacuum flucuations, dark energy

Colaboradores

UFRJ - Rio de JaneiroYareni Ayala Claudio CcapaDiney Ether JrReinaldo de MeloH. Moysés NussenzveigFelipe S. RosaNathan Viana

Parabéns Luiz !

LANL - Los Alamos Ryan BehuninDiego Dalvit

LKB - Paris Antoine Canaguier-DurandAstrid LambrechtRicardo MessinaSerge ReynaudObs. Cote d’Azur - Nice

François ImpensIFRJ - Rio de JaneiroRafael de Sousa Dutra

interações dispersivas e a ótica quântica - if.ufrj.brtgrappoport/davidovich/maianeto.pdf ·...

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