Optical properties of atomic Mott insulators: From slow light to dynamical Casimir effects

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light to dynamical Casimir effects

Iacopo Carusotto, Mauro Antezza, Francesco Bariani, Simone de Liberato, Cristiano Ciuti

To cite this version:

Iacopo Carusotto, Mauro Antezza, Francesco Bariani, Simone de Liberato, Cristiano Ciuti. Optical

properties of atomic Mott insulators: From slow light to dynamical Casimir effects. Physical Review A,

American Physical Society, 2008, 77 (6), pp.063621. �10.1103/PhysRevA.77.063621�. �hal-02964258�

Optical properties of atomic Mott insulators: From slow light to dynamical Casimir effects

Iacopo Carusotto, ^1, * Mauro Antezza, ¹ Francesco Bariani, ¹ Simone De Liberato, ^2,3 and Cristiano Ciuti ²

1 CNR-INFM BEC Center and Dipartimento di Fisica, Università di Trento, via Sommarive 14, I-38050 Povo, Italy

2 Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot-Paris 7 et CNRS, UMR 7162, 75013 Paris, France

3 Laboratoire Pierre Aigrain, École Normale Supérieure, 24 rue Lhomond, 75005 Paris, France

共

Received 26 November 2007; published 23 June 2008

兲

We theoretically study the optical properties of a gas of ultracold, coherently dressed three-level atoms in a Mott insulator phase of an optical lattice. The vacuum state, the band dispersion and the absorption spectrum of the polariton field can be controlled in real time by varying the amplitude and the frequency of the dressing beam. In the weak dressing regime, the system shows unique ultraslow-light propagation properties without absorption. In the presence of a fast time modulation of the dressing amplitude, we predict a significant emission of photon pairs by parametric amplification of the polaritonic zero-point fluctuations. Quantitative considerations on the experimental observability of such a dynamical Casimir effect are presented for the most promising atomic species and level schemes.

DOI: 10.1103/PhysRevA.77.063621 PACS number

共

s

兲

: 03.75.Lm, 42.50.Nn, 71.36. ⫹ c, 03.70. ⫹ k

I. INTRODUCTION

Most of the recent advances in the field of nonlinear and quantum optics were made possible by the development of optical media with unprecedented properties. On the one hand, the optical response of carriers in solid-state materials can be controlled and enhanced by confining the carrier mo- tion and/or the photon mode in suitably grown nanostruc- tures

关1–5兴. On the other hand, systems of ultracold atoms

appear as very promising in view of all those applications which require long coherence times, e.g., quantum- information processing.

Even though the low density of an atomic gas limits the absolute strength of the light-matter coupling, still these sys- tems have the advantage of being almost immune from dis- order and decoupled from the environment. Furthermore, they offer the possibility of a precise control and wide tun- ability of the system parameters in real time by optical and/or magnetic means. In particular, Mott

关6,7兴 共as well as

band

关

8

兴兲

insulator states have been realized: Already for a moderately strong lattice potential a few times higher than the atomic recoil energy, a constant and integer number of atoms are trapped in the extremely regular potential of an optical lattice with negligible quantum fluctuations in the atom number. Such systems then constitute an almost perfect realization of the Fano-Hopfield model of resonant dielec- trics

关9兴.

In the present paper we present a theoretical study of the classical and quantum optical properties of an atomic Mott insulator. Recently, the case of two-level atoms was investi- gated in

关10,11兴. Here we extend the Fano-Hopfield model to

the case of a three-level system in the presence of a coherent dressing field. The rich potential of three-level configurations has already been demonstrated with the observation of a va- riety of remarkable effects, such as the quenching of resonant absorption by the so-called electromagnetically induced transparency

共EIT兲

effect

关12,13兴, the light propagation at

ultraslow group velocities in the m/s range

关14,15兴, and the

coherent stopping and storing of light pulses

关16兴. Here we

show how the peculiarities of atomic Mott insulator states can lead to further improvements of these experiments and, even more remarkably, open the way to studies of more subtle quantum optical effects.

Even in its ground state, the electromagnetic

共e.m.兲

field possesses in fact zero-point fluctuations, whose properties are nontrivially affected by the presence of dielectric and/or metallic bodies. One of the most celebrated consequence is the

共static兲

Casimir effect, i.e., the appearance of a force between macroscopic objects due to the zero-point energy of the electromagnetic field

关17,18兴. In the last decades, this

force has been the object of intense experimental and theo- retical studies in a number of different systems and its main properties can nowadays be considered as reasonably well understood.

The situation is completely different for what concerns the so-called dynamical Casimir effect

共DCE兲 关19,20兴, i.e.,

the observable radiation that is emitted by the parametric excitation of the quantum vacuum when the boundary con- ditions and/or the propagation constants of the electromag- netic field are modulated in time on a very fast time scale. In spite of a wide theoretical literature having addressed this effect for a variety of systems and excitation schemes

关21–26兴, no experimental observation has been reported yet,

mainly because of the difficulty of modulating the system parameters at a high enough speed and the presence of com- peting spurious effects.

In the second part of this paper we show how Mott insu- lators of coherently dressed three-level atoms are very prom- ising candidates for an experimental observation of the dy- namical Casimir effect. As the atomic response to e.m. fields strongly depends on the amplitude and the frequency of the dressing field, a significant time modulation of the optical properties of the atomic Mott insulator can be induced on a very fast time scale by modulating the dressing parameters via standard pulse manipulation techniques

关27兴. On the

other hand, the cleanness of atomic Mott insulator systems allows one to squeeze linewidths down to the spontaneous radiative level and hence to avoid those inhomogeneous

* [email protected]

1050-2947/2008/77

共

6

兲

/063621

共

16

兲

063621-1 ©2008 The American Physical Society

broadening mechanisms that have so far limited the perfor- mances of slow- and stopped-light experiments. Moreover, as the dynamical Casimir radiation is collected in the optical domain and no relaxation processes are involved in the modulation process, no difficulties are expected to appear as a consequence of thermal blackbody radiation or incoherent luminescence from photoexcited carriers

关24兴.

An ab initio model is developed to confirm these expec- tations in a quantitative way: Taking inspiration from recent works

关

28–30

兴

, we build a microscopic theory of the dy- namical Casimir effect in atomic Mott insulators which ex- plicitly includes the matter degrees of freedom. Thanks to the simple form of the resulting parametric Hamiltonian and to the relative weakness of the light-matter coupling constant, expressions for the emission intensity are obtained in closed analytical form. As expected, the most favorable frequency region appears to be the middle polariton branch

共the so-

called dark-state polariton of

关31兴兲, which shows a strong

resonant coupling of light with the matter degrees of free- dom, as well as a still acceptable amount of absorption losses. Advantages and disadvantages of the atomic Mott in- sulator system over previously studied solid state systems

关29,30兴

will then be pointed out, as well as the criteria for the choice of the atomic levels to be used. Quantitative estima- tions of the dynamical Casimir intensity in realistic systems appear as very promising in view of the experimental obser- vation of this still elusive effect. Generalization of the results to experimentally less demanding atomic states, e.g., Bose- condensed clouds or thermal gases is finally discussed.

The structure of the paper is the following. In Sec. II we introduce the physical system and the model used for its theoretical description. In Sec. III we discuss in a systematic way the static properties of the system, such as the dispersion and lifetime of the elementary excitations of the system, the so-called polaritons. A calculation of the dynamical Casimir emission in a spatially infinite, bulk system is presented in Sec. IV, and then extended to experimentally more relevant finite-size geometries in Sec. V. A quantitative discussion of the emission is presented in Sec. VI using realistic param- eters of state-of-the-art samples. Conclusions are finally drawn in Sec. VII.

II. PHYSICAL SYSTEM AND THE THEORETICAL MODEL

A. The physical system

We consider a gas of bosonic atoms trapped in the peri- odic potential of a three-dimensional optical lattice with simple cubic geometry of lattice spacing a _L . Unless other- wise specified, the system is assumed to be spatially homo- geneous with periodic boundary conditions in all three di- mensions. The box sizes are equal to L _x,y,z , and the total volume of the system is V = L x L y L z . For a strong enough lat- tice potential V ₀ and commensurate filling, the ground state of the system corresponds to a Mott insulator state, with an integer number n of atoms at each lattice site

关6,7兴

and al- most negligible fluctuations: The probability of having mul- tiple or zero occupancy of a site decreases in fact proportion- ally to

共a

L /a sc

兲

² exp共−4

冑

s兲, where the dimensionless

parameter s is defined as the ratio s = V ₀ / E R of the lattice potential height V ₀ and the atomic recoil energy E R

= ប ² ␲ ² /2ma L 2

关32兴, and

a sc is the atom-atom scattering length. In what follows we focus our attention on the n = 1 case in which the atoms are spatially separated and do not interact but via the electromagnetic field. The total number of atoms in the system is thus N = L x L y L z /a L 3 and the average density n at = 1/ a _L ³ . The temperature of the system is assumed to be low enough for the zero temperature approximation to hold. Generalization to the strongly correlated

关32兴

and finite temperature cases is postponed to future work.

The internal atomic dynamics takes place among three internal levels organized in either a ⌳ or a ladder structure

关12兴

as sketched in Figs. 1共a兲 and 1共b兲

关the case of a strongly

asymmetric ⌳ scheme of Fig. 1共c兲 will be discussed in Sec.

VI

兴

. The atoms are initially prepared in their internal ground state g, which is connected to an excited state e by an al- lowed optical transition of dipole matrix element d eg at a frequency ␻ e . For notational simplicity, the energy zero is set in a way to have ␻ g = 0. A coupling laser of frequency ␻ C

dresses the atoms by driving the transition between the ex- cited e level and a third, initially empty, state m of energy

␻ m . In the ⌳ case, the lifetime of the m state can be very long, much longer than the free-space radiative lifetime of the e state.

In terms of the local amplitude E _C

共

R

兲

of the dressing electric field at the atomic position R, the

共complex兲

Rabi frequency of the coupling is ប⍀ C

共R兲

= d em E C

共R兲. In the fol-

lowing we consider the case where ⍀ C is spatially uniform

关

33

兴

; the discussion of more complex cases is postponed to future works. The direct transition g → m is assumed to be optically inactive d gm = 0.

Provided the lattice potential is strong enough to fulfill the Lamb-Dicke condition and has the same effect on the atoms irrespectively of their internal state, the external degrees of freedom can be decoupled from the internal dynamics and remain frozen in the motional ground state of each lattice site

关34,35兴.

B. The Three-level Fano-Hopfield model

A quantitative description of the many-atom system inter- acting with the electromagnetic field can be developed by generalizing the Fano-Hopfield model of a resonant dielec- tric

关9兴

to the present case of three-level atoms. In this ap- proach, both the atomic electric dipole polarization and the radiation field are described as a collection of coupled har-

|e>

|m>

|g> |g>

|e>

|m>

Ω _C , ω _C Ω _C , ω _C

(a) Λ scheme (b) Ladder scheme

Ω _C , ω _C

|m>

|e>

|g>

(c) Strongly asymmetric

Λ scheme

FIG. 1.

共

Color online

兲

Sketch of the level schemes under

consideration.

monic oscillators. Neglecting for simplicity all higher-lying photonic bands and assuming the different light polarization to be decoupled, the vector potential operator has a simple

“scalar” expression in terms of the photon

共ph兲

creation and annihilation operators aˆ _ph,k ^† and aˆ _ph,k ,

A ˆ

共R兲

= _k

_苸FBZ

兺 冑 ^{2 ␲} kV ^cប

共aˆ

ph,k e ^ik·R + aˆ ^† _ph,k e ^−ik·R

兲, 共1兲

where the sum over k vectors is limited to the first Brillouin zone

共FBZ兲

of the lattice. This approximation holds under the assumption of an isotropic atomic response and in the limit of a small lattice spacing ␻ e a L /c Ⰶ 1

关51兴.

In our specific case of three-level atoms, two material degrees of freedom are associated to each atom, which cor- respond to its excitation from the g state to, respectively, the e and m states. As usual, raising and lowering operators for the excited e state of the j atom are defined as aˆ _e,j ^†

兩g典

j =

兩e典

j

and aˆ _e,j

兩

e

典

j =

兩

g

典

j , and analogously the aˆ _m,j ^† and aˆ _m,j for the m state. In the spirit of the harmonic oscillator model of

关9兴,

these operators can be extended as creation and annihilation operators satisfying the usual Bose commutation rules. This bosonic description is accurate under the assumption that the probability for a given atom to be in an excited state is small:

In this limit the higher-lying atomic states are not involved in the physics under consideration and the dynamics is re- stricted to the subspace spanned by the three

兩共g

, e, m兲典 states for which the harmonic oscillator description is exact. Gen- erally speaking, this assumption is expected to be accurate as long as the number of excitations present in the system is much smaller than the number of atoms

关52兴.

Reabsorbing the phase of the dressing field ⍀ C and its time dependence at ␻ C into the definition of the aˆ _m,j and aˆ _m,j ^† operators, the internal dynamics of the j atom is described by the following time-independent Hamiltonian:

H _at ^j = ប ␻ e aˆ _e,j ^† aˆ _e,j + ប ␻ ^˜ m aˆ _m,j ^† aˆ _m,j + ប⍀ C

共

aˆ _e,j ^† aˆ _m,j + aˆ _m,j ^† aˆ _e,j

兲 共2兲

with a real ⍀ C and a renormalized ␻ ^˜ m = ␻ m ⫾ ␻ C , the ⫾ signs referring to, respectively, the ⌳ and the ladder configuration

共see Fig.

1兲.

The bosonic Hamiltonian

共

2

兲

can be rewritten in terms of the position and momentum operators of two fictitious e,m particles of mass M harmonically bound at frequencies, re- spectively, ␻ e and ␻ ^˜ m , and mechanically coupled by the ⍀ C

dressing field

H _at ^j = M ␻ e 2

2 X ˆ

e,j

2 + 1

2M P ˆ

e,j 2 + M ␻ ^˜ m

2

2 X ˆ

m,j

2 + 1

2M P ˆ

m,j 2

+ M⍀ C

冑

_␻ e ␻ ^˜ m X ˆ

e,j X ˆ

m,j + ⍀ C

M

冑

␻ e ␻ ^˜ m

P ˆ

e,j P ˆ

m,j ;

共3兲

the position and momentum operators are defined as usual as

X ˆ

e,j = 冑 2M ^ប ␻ e

共aˆ

e,j + aˆ _e,j ^†

兲 共4兲

P ˆ

e,j = i 冑 ^បM 2 ^{␻ e}

共aˆ

_e,j ^† − aˆ _e,j

兲, 共5兲

X ˆ

m,j = 冑 2M ^ប ␻ ^˜ m

共aˆ

m,j + aˆ _m,j ^†

兲, 共6兲

P ˆ

m,j = i 冑 ^បM 2 ^{␻ ˜ m}

共aˆ

_m,j ^† − aˆ _m,j

兲. 共7兲

The electric-dipole coupling of the g → e transition to the transverse electromagnetic field

关53兴

is included by giving a charge q to the fictitious particle e and then performing the standard minimal coupling replacement P ˆ

e,j → P ˆ

e,j

− qA ˆ

共

R _j

兲

/ c in the Hamiltonian

共

3

兲

; the vector potential A ˆ is evaluated by Eq.

共1兲

at the position R j of the atom. As the g → m transition is not optically active, the m particle must be left neutral, and no minimal coupling replacement must be made on the P ˆ

m,j operator.

The charge q and the mass M of the harmonic oscillator model are to be chosen in such a way that the model cor- rectly reproduces the actual optical properties of the atomic system under examination: to this purpose, it is enough that the electric dipole matrix element between the ground and the first excited state of the harmonic oscillator model be equal to the dipole moment of the actual atomic transition.

This imposes the condition d _eg =

冑

_ប q ² / 2M ␻ e : In what fol- lows, we shall see that all observable physical quantities are a function of d eg only, and do not separately involve the q and M parameters of the model.

To take full advantage of the translational symmetry of the system, it is useful to introduce the collective atomic operators

aˆ

_共

^† _e,m

_兲

_,k = 1

冑

N 兺 _j ^aˆ

^共

^{† e,m}

^兲

^{,j e ik·R j}

^共

⁸

^兲

which create a delocalized atomic excitation with a wave vector k belonging to the first Brillouin zone of the lattice.

Analogously to their localized counterparts aˆ

_共

_e,m

_兲,j

and aˆ

_共

^† _e,m

_兲

_,j , the aˆ

_共

_e,m

_兲

_,k and aˆ

_共

^† _e,m

_兲

_,k satisfy Bose commutation rules.

Straightforward manipulations lead to the final form of the total light-matter Hamiltonian,

H = 兺 _k

^关H

^{ph,k + H at,k + H int,k}

^{兴, 共9兲}

where

H _ph,k = បck 冉 ^{aˆ ph,k † aˆ ph,k + 1} 2 冊 ^,

^共10兲

H _at,k = ប ␻ e aˆ _e,k ^† aˆ _e,k + ប ␻ ^˜ m aˆ _m,k ^† aˆ _m,k + ប⍀ C

共aˆ

_e,k ^† aˆ _m,k + aˆ _m,k ^† aˆ _e,k

兲, 共11兲

H _int,k = − iបC k

共aˆ

_e,−k ^† − aˆ _e,k

兲共aˆ

ph,−k + aˆ ^† _ph,k

兲

+ បD k

共aˆ

ph,k + aˆ _ph,−k ^†

兲共aˆ

ph,−k + aˆ _ph,k ^†

兲

+ iបC k ⍀ C

␻ e

共aˆ

m,k − aˆ _m,−k ^†

兲共aˆ

ph,−k + aˆ _ph,k ^†

兲. 共12兲

All terms consist of quadratic forms in the creation and an-

nihilation operators of the electromagnetic or the matter po- larization fields. The first term H _ph,k is the free e.m. field Hamiltonian. The second term H _at,k describes the internal dynamics of the dressed atoms. The three lines of the third term H _int,k , respectively, account for

共

i

兲

the dipole coupling of the g →e transition to the e.m. field,

共ii兲

the photon renor- malization due to the squared vector potential term, and

共iii兲

a coupling between the photon quantum field and the m ex- citation as a result of the dressing field

关54兴.

The coupling constant C _k is equal to

C _k = 冑 ^{2 ␲␻} _ប ck ^{e 2 n at} d _eg ,

共

13

兲

and D k = C _k ² / ␻ e . As expected, these expressions involve the model parameters q and M only via their physical combina- tion d eg .

Introducing the vector ␣ ^ˆ k of bosonic operators

␣ ^ˆ k =

共aˆ

ph,k ,aˆ _e,k ,aˆ _m,k ,aˆ _ph,−k ^† ,aˆ _e,−k ^† ,aˆ _m,−k ^†

兲

^T ,

共14兲

and the Bogoliubov metric ␩ ^{= diag}

关

1 , 1 , 1 , −1 , −1 , −1

兴

, the Hamiltonian

共

9

兲

can be recast in a simple matricial form,

H = ប 2 兺

k

␣ ^ˆ k

† ␩ H k ␣ ^ˆ k + E ₀

共15兲

in terms of a 6 ⫻ 6 ␩ ^—Hermitian

共H

^† ␩ ⁼ ␩ H兲 Hamiltonian matrix H k of the form

H k = 冉 ^{− K Q k} k ^{Q k}

† − K k

T 冊 ^.

^共16兲

where K k and Q k are 3 ⫻ 3 matrices. The constant E ₀ fixes the energy zero: As it has no consequences in what follows, it will be neglected from now on.

The Hermitian matrix

K k = 冢 ^{− ck i⍀ − + 2D C iC C k k / ␻ k e iC ⍀ ␻ e C k i⍀ C ⍀ ␻ ˜ C m C k / ␻ e} 冣

^共17兲

takes into account the free field, the internal atomic dynam- ics including the dressing beam, as well as the light-matter interaction terms at the level of the so-called rotating wave approximation

共RWA兲: Whenever a radiative photon is ab-

sorbed

共emitted兲, an atomic excitation is created共destroyed兲

at its place

关35兴.

The symmetric matrix

Q k = 冢 ^{− i⍀ − 2D C iC C k k k / ␻ e − iC 0 0 k − i⍀ C 0 0 C k / ␻ e} 冣

^共18兲

corresponds instead to those additional terms which describe anti-RWA, off-shell processes where a photon and an atomic excitation are simultaneously destroyed or created.

The relative importance of the RWA K k and the anti-RWA Q k terms is quantified by the ratio C ¯ / ␻ e of the radiation- matter coupling strength C ¯ = C _k=␻

e / c and the excitation fre- quency ␻ e . For most atomic systems of actual experimental

interest, this parameter is generally quite small: As a simple example, consider the D ₂ line of ⁸⁷ Rb atoms at ␭ e

= 2 ␲ c/ ␻ e

⯝

780 nm. The electric dipole moment of the tran- sition is d _eg

⯝

4.2ea _Bohr

关

40

兴

and a typical value of lattice spacing is a _L = 300 nm. For a unit filling factor n = 1, the light-matter coupling parameter is then C ¯ / ␻ e

⯝

1.7⫻ 10 ⁻⁴ . Although the condition C ¯ / ␻ e Ⰶ 1 rules out the possibility of observing the so-called ultrastrong coupling regime

关29,30兴

in such dilute atomic systems, the anti-RWA terms in the Hamiltonian can still have interesting observable conse- quences as we shall see in what follows.

III. STATIONARY STATE: GROUND STATE AND POLARITON EXCITATIONS

We begin the study of the optical properties of the system from the simplest case where the dressing parameters ␻ C and

⍀ C are kept fixed in time. For each value of them, the qua- dratic structure of the Fano-Hopfield Hamiltonian

共15兲

guar- antees that this can be set into the canonical form

H = 兺

k,r

ប ␻ r,k pˆ _r,k ^† pˆ _r,k + E ₀ ⬘

共

19

兲

by means of a Hopfield-Bogoliubov transformation

关9兴. As in

Eq.

共15兲, the constant

E ₀ ⬘ is the zero-point energy and will be neglected from now on. For each wave vector k, the frequen- cies ␻ r,k of the elementary excitations are given by the ei- genvalues corresponding to the positive-norm

共in the

␩ ^met- ric

兲

eigenvectors of the Hopfield-Bogoliubov matrix

M k =

共H

k

兲

^T .

共20兲

In the system under consideration here, the elementary exci- tations are the lower

共r

= LP兲, middle

共r= MP, often also

called dark-state polariton, e.g., in

关

31

兴兲

, and upper

共

r= UP

兲

polariton modes. All of these modes are linear superposition of light and matter excitations. The polaritonic annihilation operators pˆ _r,k can be written in terms of the eigenvectors w ជ r,k

as

pˆ _r,k = 兺

␭=1 6

w _r,k ^␭ ␣ ^ˆ k ␭ .

共21兲

The index ␭ runs over the six components of the eigenvector w ជ r,k of the Hopfield-Bogoliubov matrix

共20兲

and of the op- erator vector ␣ ^ˆ k defined in Eq.

共

14

兲

. An analogous expres- sion holds for the creation operators pˆ _r,k ^† .

Grouping the pˆ _r,k ’s in the operator vector

␲ ^ˆ k =

共pˆ

LP,k ,pˆ _MP,k ,pˆ _UP,k , pˆ _LP,−k ^† ,pˆ _MP,−k ^† pˆ _UP,−k ^†

兲

^T ,

共22兲

the transformation

共21兲

to the polaritonic basis can be cast in the simple matricial form ␲ ^ˆ k = W k ␣ ^ˆ k . The first three lines of W _k correspond to the w ជ r,k eigenvectors

共

21

兲

.

The orthonormality condition of the eigenvectors in the

Bogoliubov metric ␩ corresponds to the ␩ -unitary condition

W _k ⁻¹ = ␩ ^W k † ␩ ^,

共23兲

and guarantees that the operators pˆ _q,k satisfy the standard

Bose commutation rules. In the ␲ ^ˆ basis, the Hamiltonian

matrix has the simple diagonal form

H k ⬘ ^{= W} k H k W _k ⁻¹

= diag

关

␻ LP,k , ␻ MP,k , ␻ UP,k ,− ␻ LP,k ,− ␻ MP,k ,− ␻ UP,k

兴

.

共24兲

In view of the following developments, it is useful to give the explicit form of the r=

兵LP, MP, UP其

polariton operators in terms of the photonic

共ph兲

and matter

共e,m兲

excitation ones,

pˆ _r,k = u _r,k ^ph ␣ ^ˆ ph,k + u _r,k ^e aˆ _e,k + u _r,k ^m aˆ _m,k + v _r,k ^ph aˆ _ph,−k ^† + v _r,k ^e aˆ _e,−k ^†

+ v _r,k ^m aˆ _m,−k ^†

共

25

兲

as well as the inverse transformation

共

j =

兵

ph , e,m

其兲

, aˆ _j,k = u _LP,k ^{j ⴱ} pˆ _LP,k + u _MP,k ^{j ⴱ} pˆ _MP,k + u _UP,k ^{j ⴱ} pˆ _UP,k − v _LP,k ^j pˆ _LP,−k ^†

− v _MP,k ^j pˆ _MP,−k ^† − v _UP,k ^j pˆ _UP,−k ^† .

共26兲

The u and v Hopfield coefficients characterize, respectively, the normal and anomalous weights of the different ph , e,m components of the LP, MP, UP polaritons.

A. Polariton vacuum

The vacuum state of the system corresponds to the ground state

兩G典

of the Hamiltonian

共19兲, and is defined by the

vacuum condition

pˆ _r,k

兩G典

= 0

共27兲

for all polariton modes r=

兵LP, MP, UP其.

As both annihilation and creation operators are involved in the Bogoliubov transformation

共

26

兲

, the ground state

兩

G

典

corresponds, in the original aˆ

_共

_ph,e,m

_兲

basis, to a squeezed vacuum state with a nonvanishing expectation value of the photon and atomic excitation numbers:

N

_共

^G _ph,e,m

_兲

=

具G兩aˆ_共

^† _ph,e,m

_兲,k

aˆ

_共

_ph,e,m

_兲,k兩G典

= 兺

r=

兵

LP,MP,UP

其兩v

_r,k

^共

^ph,e,m

^兲兩

² .

共

28

兲

As the atomic systems under consideration here are far from the ultrastrong coupling regime

关

4,29,30

兴

, an accurate esti- mation of N

_共

^G _ph,e,m

_兲

can be obtained by means of a suitable perturbation theory in the light-matter coupling strength C ¯ / ␻ e Ⰶ 1. The dressing amplitude ⍀ C is assumed to be at most of the order of C ¯ .

A zeroth-order approximation w ជ _q,k ⁰ of the eigenvector can be obtained by diagonalizing the block diagonal matrix

M k

0 = 冉 ^{K 0 k T − 0 K k} 冊 ^.

^共29兲

This provides expressions for the eigenvalues ␻ _␭ 0 ,k and the eigenvectors w ជ _␭,k ⁰ that are correct upto the zeroth-order in C ¯ / ␻ e . The eigenvectors have the form

w ជ _␭,k ⁰ =

共u

_␭,k ^ph,0 ,u _␭,k ^e,0 ,u _␭,k ^m,0 ,0,0,0兲 ^T

共␭

= 1,2,3兲,

共30兲

w ជ _␭ ⁰ _,k =

共0,0,0,u

_␭ ^ph,0ⴱ _−3,k ,u _␭ ^e,0ⴱ _−3,k ,u _␭ ^m,0ⴱ _−3,k

兲

^T

共␭

= 4,5,6兲.

共31兲

Up to this level of approximation the virtual occupation

共28兲

is then rigorously vanishing.

When evaluating the first-order correction, attention must be paid to the nonpositive nature of the ␩ ^metric,

w ជ _␭ 1 ,k = 兺

␭ ⬘ ⁼

^兵

^{1,. . .,6}

^其

␭ ⬘ ^⫽␭

␧ _{␭ ⬘} w ជ _␭ ^0† _{⬘ ,k} ␩ ␦ M k w ជ _␭,k ⁰

␻ _␭,k ^{0 −} ␻ _␭ ⁰ _{⬘ ,k} ^w ជ _␭ ⁰ _{⬘ ,k} ^,

共

32

兲

where the sign ␧ _␭ is +1 for ␭ = 1, 2, 3, and −1 for ␭

= 4 , 5 , 6. The perturbation matrix ␦ M k = M k − M k 0 is of order C ¯ / ␻ e and has nonzero entries only in the off-diagonal 3 ⫻ 3 blocks. Keeping in Eq.

共

32

兲

only the lowest-order terms in C ¯ / ␻ e , one gets to the first-order corrections,

v _r,k ^j,1 = 兺

r ⬘

− iC k

␻ r,k 0 + ␻ r ⁰ ⬘ ^,k

共u

_r ^ph,0 _{⬘ ,k} u _r,k ^e,0 + u _r ^e,0 _{⬘ ,k} u _r,k ^ph,0

兲u

_r ^j,0ⴱ _{⬘ ,k} ,

共33兲

where the sum runs over the three polariton branches r ⬘

=

兵LP, MP, UP其. Note that the off-diagonal terms due to the

squared vector potential give a contribution to Eq.

共33兲

whose amplitude is of the order of D k / ␻ e =

共C

k / ␻ e

兲

² and have been therefore neglected. The same for the off-diagonal terms due to the dressing field, whose contribution is of the order of C k ⍀ C / ␻ e

2 .

The virtual population in the ground state is then of the order of

共C

k / ␻ e

兲

² : Apart from a small region around k = 0 where C _k diverges, this population is therefore very small for all polariton branches. This fact provides an a posteriori jus- tification of the use of perturbation theory.

In the most significant resonance region k⯝ k e = ␻ e / c, the frequency denominator of Eq.

共

33

兲

can be approximated by 2 ␻ e in a sort of degenerate polariton approximation, which gives

v _r,k ^ph,1

⯝

− iC ¯ 2 ␻ e

u _r,k ^e,0 ,

共34兲

v _r,k ^e,1

⯝

− iC ¯ 2 ␻ e

u _r,k ^ph,0 ,

共

35

兲

v _r,k ^m,1

⯝

0.

共36兲

Compact formulas for the fully resonant point ␻ e = ␻ ^˜ m , k

= k _e of the MP are immediately obtained by inserting in Eq.

共34兲–共36兲

the explicit form of the RWA eigenvectors of the zeroth-order Hopfield-Bogoliubov matrix

共29兲,

u _MP,k

e

ph,0 = ⍀ C

共C

¯ ² + ⍀ C

2

兲

^1/2 ,

共37兲

u _MP,k

e

e,0 = 0,

共38兲

u _MP,k

e

m,0 = − iC ¯

共C

¯ ² + ⍀ C

2

兲

^{1 / 2} ,

共

39

兲

which leads to anomalous amplitudes v _MP,k

e ph,1

⯝

v _MP,k

e

m,1

⯝

0,

共

40

兲

v _MP,k

e

e,1

⯝

− iC ¯ ⍀ C

2 ␻ e

共C

¯ ² + ⍀ C

2

兲

^1/2 .

共41兲

The accurateness of these analytical approximations is vis- ible in Figs. 2共a兲 and 2共b兲 where the Hopfield u and v coef- ficients are plotted for the fully resonant k= k e point of the MP as a function of the dressing amplitude: On the scale of the figure the analytical approximations are undistinguish- able from the exact calculations.

As long as the system parameters are kept constant in time, the virtual populations

共28兲

are intrinsically bound to the system ground state and cannot be revealed by a standard photodetector based on absorption processes

关29,30兴. On the

other hand, the dependence of the anomalous amplitude

共41兲

on ⍀ C

关see Fig.

2共b兲兴 suggests that the zero-point fluctua- tions in the ground state can be externally controlled by varying ⍀ C in time. In particular, if the time modulation of

⍀ C is sufficiently fast, the system is not able to adiabatically follow the time-dependent vacuum state. As a consequence, real polaritons are created in the system and then emitted as radiative photons into the surrounding free space where they can be detected by a photodetector. This will be the subject of Secs. IV–VI.

B. Polariton dispersion

Examples of polariton dispersion are shown in Fig. 3 for the most significant cases. The shape of the three polariton

branches changes in a substantial way depending on the dressing parameters ␻ C and ⍀ C , which provides a simple way to externally vary the optical properties of the system in real time. An application of atomic Mott insulators as dy- namic photonic structures was indeed proposed in

关36兴.

As long as linear optical properties are considered, it is important to note that the predictions of the quantum model are indistinguishable from the solution of the Maxwell equa- tions including the semiclassical expression for the local di- electric polarizability of three-level atoms

关12兴. In particular,

this is the case of the band dispersions shown in Fig. 3.

As ␻ e and ␻ ^˜ m have no spatial dispersion, the three bands have no spectral overlap and are separated by energy gaps in which the radiation cannot propagate. For the systems under consideration here, the gaps between the bands are however very narrow, on the order of C ¯ ² / ␻ e , and therefore almost invisible on the scale of the figure.

When the dressing is on resonance with the m→ e transi- tion

共

␻ e = ␻ ^˜ m

兲, the atomic resonance is split into a symmetric

Autler-Townes doublet at ␻ e ⫾⍀ C and the photonic mode anticrosses each of the two components. Two subcases are to be distinguished: For a strong dressing ⍀ C ⲏ C ¯

关panel 共c兲兴,

the two anticrossings are almost completely separated and have a half-width equal to C ¯ /

冑

2; in between the two anti- crossings

共

i.e., in the neighborhood of ␻ e

兲

, the middle polar- iton

共MP兲

almost coincides with the photon branch and has a group velocity v _MP,k ^gr = d ␻ MP,k / dk close to c. On the other hand, for a weak dressing ⍀ C Ⰶ C ¯ , the two anticrossings FIG. 2.

共

Color online

兲

Hopfield u and v coefficients

共

a,b

兲

, group velocity

共

c

兲

, and absorption coefficient

共

d

兲

of the middle polariton

共

MP

兲

as a function of dressing amplitude ⍀ C in the resonant

共␻

e = ␻ ˜ _m

兲

case. On the scale of the figure, the analytical approximations

共

42

兲

,

共

37

兲

–

共

39

兲

, and

共

41

兲

are undistinguishable from the exact calculations. Vertical lines indicate the dressing amplitude values used in the next figures. The system parameters correspond to the D ₂ line of a Mott insulator system of ⁸⁷ Rb atoms with filling factor n= 1 in a a _L

= 300 nm lattice, C ¯ / ␻ e = 1.7 ⫻ 10 ⁻⁴ , ␭ e = 2 ␲ / k _e = 2 ␲ c / ␻ e = 780 nm.

overlap, which results in a strong mutual distortion

关

panel

共a兲兴: The MP dispersion is strongly flattened, and its group

velocity becomes orders of magnitude slower than c

关inset of

panel

共a兲兴.

An approximate, yet quantitatively very accurate expres- sion for the group velocity v _MP,k

e

gr of the MP around reso- nance k= k _e = ␻ e / c is easily obtained from the expression

共37兲

of the MP eigenvector of the RWA Hopfield-Bogoliubov matrix

共29兲,

v _MP,k

e

gr = c兩u _MP,k

e

ph,0

兩

² = c ⍀ C 2

⍀ C 2 + C ¯ ² .

共42兲

As one can see in Fig. 2

共

c

兲

, arbitrarily slow values of v _MP,k

e gr

can be obtained by simply reducing the amplitude ⍀ C of the dressing field: Thanks to the trapping of atoms at lattice sites, no lower bound to the group velocity appears as a conse- quence of the atomic recoil in absorbing and emitting pho- tons

关41兴. For the specific case of Rb atoms considered in the

figures, a quite conservative value ⍀ C / 2 ␲ ^{= 12 MHz}

共

i.e., 2 times the radiative lifetime of the e state兲 already leads to v _M,k

e

gr = 11 m / s.

In the case of a nonresonant dressing

关panel共e兲兴, the os-

cillator strength of the optical transition is shared in an asym- metric way by the two components of the Autler-Townes doublet. This fact is responsible for the anticrossing to be wider for the component having a larger e state weight

共the

lower one in the figure兲. Because of the optical Stark effect, the position of this main anticrossing is slightly shifted by

⌬ ␻ OSE = ⍀ C

2 /共 ␻ e − ␻ ^˜ m

兲

from its bare position at ␻ e .

C. Effect of losses

Losses from both the e and the m states are responsible for the finite lifetime of polaritonic excitations. For typical systems, both decay rates ␥

_共

e,m

兲

are much smaller than the radiation-matter coupling C ¯ and the energy splitting ␻ q,k

− ␻ q ⬘ ^,k between different polariton bands q ⫽ q ⬘ ^{at a given}

-0.002 -0.001 0 0.001 0.002

-4×10 ^-5 0 4×10 ^-5

ω/ω e -1

10 ^-10 10 ^-9 10 ^-8 10 ^-7 10 ^-6 10 ^-5 10 ^-4 10 ^-3 10 ^-2 10 ^-1 10 ⁰ -4×10 ^-5

0 4×10 ^-5

ω/ω e -1

-0.002 -0.001 0 0.001 0.002

-4×10 ^-4 0 4×10 ^-4

ω/ω e -1

10 ^-10 10 ^-9 10 ^-8 10 ^-7 10 ^-6 10 ^-5 10 ^-4 10 ^-3 10 ^-2 10 ^-1 10 ⁰ -4×10 ^-4

0 4×10 ^-4

ω/ω e -1

-0.002 -0.001 0 0.001 0.002

k / k _e - 1

-4×10 ^-4 0 4×10 ^-4 8×10 ^-4

ω/ω e -1

10 ^-10 10 ^-9 10 ^-8 10 ^-7 10 ^-6 10 ^-5 10 ^-4 10 ^-3 10 ^-2 10 ^-1 10 ⁰

κ ^abs / k _e

-4×10 ^-4 0 4×10 ^-4 8×10 ^-4

ω/ω e -1

-0.002 0 0.002

-2e-10 0 2e-10

10 ^-7 10 ^-6 10 ^-5 10 ^-4 -2×10 ^-10

0 2×10 ^-10

0.1 1

-2×10 ^-9 0 2×10 ^-9

(b)

(d)

(e) (c)

(f) (a)

LP MP UP

MP

UP

MP

LP

UP

MP

LP

FIG. 3.

共

Color online

兲

Panels

共

a,c,e

兲

: Dispersion relation ␻共 k

兲

of the three polariton modes. Panels

共

b,d,f

兲

: Corresponding absorption spectrum ␬ ^abs as a function of the polariton frequency ␻ . Same system parameters as in Fig. 2. Black, red, and green lines refer to the lower polariton

共

LP

兲

, the middle polariton

共

MP

兲

, the upper polariton

共

UP

兲

, respectively. The blue dashed line in

共

a,c,e

兲

is the free photon dispersion

␻ =ck. Panels

共

a,b

兲

: Resonant ␻ e = ␻ ˜ _m , weak dressing ⍀ C / C ¯ = 1.86 ⫻ 10 ⁻⁴ Ⰶ 1 case. Insets in

共

a,b

兲

: Magnified views of the most significant

parts of the MP branch. Panels

共

c,d

兲

: Resonant ␻ e = ␻ ˜ _m , strong dressing ⍀ C / C ¯ = 2 case. Panels

共

e,f

兲

: Nonresonant ␻ ˜ _m − ␻ e = 5C ¯ , strong

dressing ⍀ C / C ¯ = 2 case. In the absorption

共

b,d,f

兲

panels: ⌳ level scheme with ␥ e / 2 ␲ = 6 MHz and ␥ m / 2 ␲ = 10 Hz

共

solid lines

兲

; ladder level

scheme with exchanged ␥ e / 2 ␲ = 10 Hz and ␥ m / 2 ␲ = 6 MHz values

共

dotted lines

兲

.

wave vector k. In optics, this regime goes often under the name of strong coupling regime

关2,3兴

and is characterized by losses not being able to effectively mix the different polar- iton branches, which retain their individuality.

The Fermi golden rule then provides an accurate predic- tion for the decay rate ␥ r,k of the plane-wave polariton state of wave vector k on the r branch,

␥ r,k = ␥ e

兩u

_r,k ^e

兩

² + ␥ m

兩u

_r,k ^m

兩

² ;

共43兲

physically, this decay rate is the relevant one in a light stop- ping experiment where light is stored in the system during a macroscopically long time

关16,31兴. In a propagation geom-

etry, a more relevant quantity is instead the absorption length

关42兴,

ᐉ r,k =

共

␬ r,k

abs

兲

⁻¹ = v _r,k ^gr / ␥ r,k .

共44兲

The relative value of the ␥

_共

e,m

兲

loss rates depends on the specific level scheme under consideration.

In a ⌳ configuration, ␥ m is mostly given by nonradiative effects, and generally has a very small value. As no intrinsic effect is expected to significantly contribute to ␥ m in Mott insulator states, it can in principle be suppressed to arbi- trarily small values by means of a careful experimental setup.

As a consequence of translational symmetry, energy ex- change can only take place between discrete states at the same k, so irreversible radiative decay from the e state to the g state is forbidden. This remarkable fact was discussed at length in the pioneering paper

关

9

兴

, where it was pointed out that polaritons in a rigid lattice of two-level atoms are not subject to dissipation and can propagate with a slow group velocity along macroscopic distances. It is important to note that this fact is strictly related to the Lamb-Dicke freezing of the atomic motion in the ground state of each site, which guarantees, e.g., that atoms cannot decay from the e state into motionally excited states of the g level as instead hap- pens for free atoms in the absence of a lattice

关55兴.

On the other hand, radiative decay from the e state into the m state can occur by spontaneous emission of a photon into a spatial mode different from the one of the coherent dressing field. The contribution of such process to ␥ e is equal to the e→ m radiative decay rate of an isolated atom in free space

关56兴. This can only be reduced by choosing a suitably

weak e ↔ m optical transition to dress the system.

In a ladder configuration, the roles of ␥ e and ␥ m are ex- changed. Spontaneous emission processes into spatial modes different from the dressing one can now occur for the m

→ e transition, which provides a significant contribution to

␥ m . On the other hand, no irreversible radiative decay pro- cess on the e→ g transition can contribute to ␥ e : Provided no other decay channel is available for the e state, a careful design of the experimental setup may then lead to a reduc- tion of ␥ e to arbitrarily small values.

Plots of the prediction

共44兲

for the absorption coefficient for the three polariton branches are shown in Figs. 3共b兲, 3共d兲, and 3共f兲 for, respectively, the ⌳

共solid lines兲

and the ladder

共dashed lines兲

three-level schemes

关57兴. Again, these curves

are in perfect agreement with the solution of the Maxwell equations using the semiclassical expression for the dielec- tric polarizability of three-level atoms

关12兴. In both cases of

⌳ and ladder configuration, absorption is strongly peaked in the anticrossing regions where polaritonic bands have the largest weight of matter excitations and the slowest group velocity. The only exception is the dip that is visible exactly on resonance with the two-photon Raman transition ␻ ⁼ ␻ ^˜ m

in the case of a ⌳ configuration. In optics, this effect goes under the name of electromagnetic induced transparency

共EIT兲

effect

关12,13兴: In the vicinity of the resonance, quan-

tum interference suppresses the weight of e excitation in the MP mode. Consequently, the absorption rate is quenched to the nonradiative one, ␥ MP,k _e

⯝

␥ m Ⰶ ␥ e . This effect is most dramatic in the case of a weak dressing ⍀ C Ⰶ C ¯

关see in par-

ticular the left-hand inset of Fig. 3共b兲兴, where the minimum of the spatial absorption coefficient ␬ MP,k abs _e around resonance remains remarkably deep in spite of the very slow group velocity v _MP,k

e

gr Ⰶ c. This allows for MP polaritons to propa- gate at ultraslow velocities for macroscopic distances with- out being appreciably absorbed

关

58

兴

: As one can see in Figs.

2共c兲 and 2共d兲, a group velocity as slow as v _MP,k

e

gr = 15 m /s still corresponds to an absorption length as large as ᐉ MP,k _e

⯝

16 cm.

To conclude this section, it is useful to summarize the main advantages of using a Mott insulator state for slow- light experiments:

共

1

兲

Atomic Mott insulators constitute an almost ideal re- alization of the two-level Fano-Hopfield model

关

9–11

兴

: Pro- vided one is in the Lamb-Dicke trapping regime, resonant light can propagate at slow group velocities without being absorbed nor scattered as instead happens in homogeneous gases in the absence of the lattice.

共2兲

The typical features of light propagating in systems of dressed three-level atoms such as EIT and ultraslow group velocities without absorption are further improved. As the atoms interact with each other only via the electromagnetic field, no intrinsic decoherence effect can contribute to the ␥ m

rate in a ⌳ configuration nor to the ␥ e one in a ladder con- figuration.

共3兲

The presence of a single atom at each lattice site elimi- nates the inhomogeneous broadening of the transitions that originates e.g., from the spatially varying density profile of a Bose-Einstein condensate.

共4兲

The trapping of atoms at the lattice sites eliminates the lower bound to the group velocity that atomic recoil would impose in the case of a homogeneous, untrapped gas

关

41

兴

.

IV. DYNAMICAL CASIMIR EMISSION IN THE PRESENCE OF A TIME MODULATION

When the dressing parameters ␻ C and ⍀ C are modulated in time, the quadratic form of the Hamiltonian is preserved, yet with a time-dependent Hamiltonian matrix H k

共t兲=

H k

+ ␦ ^H k

共t兲. At each time

t, a Bogoliubov transformation diago-

nalizing the instantaneous Hamiltonian can still be found, but

the transformation matrix W k

共t兲, as well as the polariton

bands ␻ r,k and the expression of the polariton operators pˆ _r,k

in terms of the original aˆ _j,k ones are now varying in time, as

well as the vacuum state

兩G共t兲典

of the system. While for slow

modulations the system is able to adiabatically follow the