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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
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
Interações dispersivas e a ótica quântica
Simplest example: two neutral metallic plates in vacuum. Frequencies depend on separation L !
λ = L
L L
λ = 2L
L
Interações dispersivas e a ótica quântica
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) = 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
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 ....... ????
Interações dispersivas e a ótica quântica
...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) = h0 g(kc, zA) sin(kcxA)
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) = h0 g(kc, zA) sin(kcxA)
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) = h0 g(kc, zA) sin(kcxA)
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) = h0 g(kc, zA) sin(kcxA)
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.
Interações dispersivas e a ótica quântica
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) = h0 g(kc, zA) sin(kcxA)
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) = h0 g(kc, zA) sin(kcxA)
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+++--
+++
--
Interações dispersivas e a ótica quântica
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
Geometry and the Casimir effect - theory
Modern experiments
Interações dispersivas e a ótica quântica
Geometry and the Casimir effect - theory
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
Interações dispersivas e a ótica quântica
Atom - surface dispersive interaction
Interações dispersivas e a ótica quântica
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
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
1
t = 0
t = T
z1(t)
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)
1
t = 0
t = T
z1(t)
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)
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
kx,y/P (pNµm−1mW−1)
a = 0.268µm
a = 0.376µm
a = 0.527µm
ky/P (pNµm−1mW−1)
1
Casimir (or van der Waals) phase for narrow wavepackets
Taking the atomic motion into account....our goal : dynamical correction to the Casimir phase
Non-planar geometries: atom-surface interaction atom-surface dispersive interaction
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
kx,y/P (pNµm−1mW−1)
a = 0.268µm
a = 0.376µm
a = 0.527µm
ky/P (pNµm−1mW−1)
1
k = 1t = 0 t = Tz1(t)
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
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
Non-planar geometries: atom-surface interaction atom-surface dispersive interaction
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
Full quantum theory of Casimir interferometers
Non-planar geometries: atom-surface interaction atom-surface dispersive interaction
We trace over dipole and field degrees of freedom
F Impens, R Behunin, C Ccapa -Ttira and PAMN, EPL 2013
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
kx,y/P (pNµm−1mW−1)
a = 0.268µm
1
Full quantum theory of Casimir interferometers
Non-planar geometries: atom-surface interaction atom-surface dispersive interaction
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
kx,y/P (pNµm−1mW−1)
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
kx,y/P (pNµm−1mW−1)
1
Full quantum theory of Casimir interferometers
Non-planar geometries: atom-surface interaction atom-surface dispersive interaction
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�)}�
�G (rj(t), t; rk(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�)}�
�G (rj(t), t; rk(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
Full quantum theory of Casimir interferometers
Non-planar geometries: atom-surface interaction atom-surface dispersive interaction
F Impens, R Behunin, C Ccapa -Ttira and PAMN, EPL 2013
- 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
kx,y/P (pNµm−1mW−1)
a = 0.268µm
a = 0.376µm
a = 0.527µm
ky/P (pNµm−1mW−1)
a (µm)
d (µm)
1
t = 0
t = T
z1(t)
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)
1
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�)
�
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
Non-planar geometries: atom-surface interaction atom-surface dispersive interaction
Multiple-path interferometer
...is additive:
Total phase: φjk = φstjk + φDP
jk
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�)
�
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�)}�
�G (rj(t), t; rk(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
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