Complex collective states in a one-dimensional two-atom system

0.46 MB·English

by  Gonzalo Ordonez

Complex collective states in a one-dimensional two-atom system
Gonzalo Ordonez and Sungyun Kim
Center for Studies in Statistical Mechanics and Complex Systems,
The University of Texas at Austin, Austin, TX 78712 USA
(Dated: February 1, 2008)
We consider a pair of identical two-level atoms interacting with a scalar field in one dimension,
separated by a distance x21. We restrict our attention to states where one atom is excited and
the other is in the ground state, in symmetric or anti-symmetric combinations. We obtain exact
collective decaying states, belonging to a complex spectral representation of the Hamiltonian. The
imaginarypartsoftheeigenvaluesgivethedecayrates,andtherealpartsgivetheaverageenergyof
thecollectivestates. Inonedimensionthereisstronginterferencebetweenthefieldsemittedbythe
atoms, leading to long-range cooperative effects. The decay rates and the energy oscillate with the
4 distancex21. Dependingonx21,thedecayrateswilleitherdecrease,vanishorincreaseascompared
0 with the one-atom decay rate. We have sub- and super-radiance at periodic intervals. Our model
0 may be used to study two-cavity electron wave-guides. The vanishing of the collective decay rates
2 then suggests the possibility of obtaining stable configurations, where an electron is trapped inside
thetwo cavities.
n
a
PACSnumbers: 03.65.-w,32.80.-t,73.63.-b
J

8
1 I. INTRODUCTION Wewilldescribethecollectivetwo-atomstatesthrough

complex eigenstates of the Hamiltonian [11, 12, 13, 14].
1
The complex eigenstates decay exponentially in time,
v Systems of interacting atoms form collective states
breaking time-symmetry. The real and imaginary parts
7 where the atoms behave differently from isolated ones
9 [1, 2]. For atoms in their ground states the collective ofthecomplexeigenvaluesgivetheaverageenergyofthe
0 collective states and the emission rates, respectively.
effects are relatively small. They produce the van-der
1 The Hamiltonian, being a Hermitian operator, can
Waals or Casimir-Polder forces between atoms [3], asso-
0 only have complex eigenvalues if the eigenstates do not
ciated with the cloud of virtual photons surrounding the
4
belong to a Hilbert space. In one-atomsystems the non-
0 atoms.
Hilbertian nature of the complex states is manifested in
/ When the atoms are in excited states they can ex-
h theirfieldintensity,whichincludesafactorgrowingexpo-
p change real photons originated by spontaneous emis- nentially with the distance from the atom. This growth
- sion. Depending on the situation, the real photons
t in space is related to the exponential decay of the atom
n can give strong collective effects, altering the forces be- in time [15]. The exponential field inside the light-cone
a tween atoms and also their rate of spontaneous emission
of the atom represents the real emitted photons. As we
u [1,4,5]. Forexampleforidenticalatomsinthreedimen-
q will show, this field has physical effects, which can be
sions separated by small distances, the decay rate will
: seen adding a second atom.
v essentially double or vanish, depending on whether the
Outside the light-cone, the field associated with the
i initial state is symmetric or antisymmetric with respect
X complexeigenstatesgrowsexponentiallywithouttrunca-
toexchangeofatoms[4,6]. Wehave“super-radiance”or
r tion. However, including the complete set of eigenstates
a “sub-radiance,” respectively. If the atoms are not iden- in the complex representation of the Hamiltonian, the
tical [7], or if the distance between the atoms is larger
exponential field outside the light-cone is cancelled by
than their characteristicwavelengths,the effects become
renormalized field states [15]. This is consistent with
much smaller [4].
causality, because the field further away from the atom
Most studies on two-atoms (see, e.g., Refs. [1, 2, 3, 4,
is emitted earlier. Going further away, one reaches the
5, 6, 7, 8, 9, 10]) focus on three-dimensional systems. In
point corresponding to the time where the atom was ex-
thispaperwewillconsideraone-dimensionalsystem. We
cited. At this point the field stops growing.
will show that the exchange of real photons gives strong
One can introduce complex states that are truncated
collective effects even for large separations between the
outside the light-cone, using distributions dependent on
atoms.
the test functions or observables. These are considered
Our system is analogous to electron wave guides con- in Ref. [16].
sistingoftwocavitiesconnectedbyalead[17,18]. Hence In the two-atom system, the real photons emitted by
theeffectswewilldiscussmaybestudiedexperimentally. each atom are absorbed by the other atom. In d dimen-
Wewillconsiderasimplifiedmodelwherewehavetwo sions the emitted field includes a 1/rd−1 decrease factor
identical two-level atoms, with basis states where one withdistance. Hencethefieldhasastrongeffectind=1
atom is excited, while the other is in the ground state. dimensions, as compared with d=2 or 3 dimensions.
Wewillusethedipoleandrotating-waveapproximations We will show that in one dimension the decay rates of
for the interaction with the field. the collective states oscillate with the distance between
2
theatoms. IncontrasttoDicke’sstates[1],bothsymmet- L . For L finite the momenta k are discrete. In the
→∞
ricandantisymmetricstatesofthetwoatomscanbecome limit L they become continuous, i.e.,
→∞
sub-radiant and super-radiant as the distance between
2π
the two atoms is varied. For distances that are integer dk. (2)
multiples of the atom wavelength, the collective decay L →
k Z
X
rates vanish, leading to stable collective states. These
We have ab = δ = Kroenecker delta. In the limit
states can trap field energy between the atoms. a,b
h | i
L ,
In Sec. II we briefly discuss the complex representa-
→∞
tion of the Hamiltonian for a one-atom system. In Sec.
L
′
III we introduce our two-atom model and its complex δk,k′ δ(k k ). (3)
2π → −
collective eigenstates. In Secs. IV and V we discuss the
emergence of the collective states and the bouncing of The interaction term is obtained through the dipole
photons between the atoms. In Sec. VI we consider the approximation as well as the rotating-wave approxima-
decay rate and average energy of the collective states as tion. ThepotentialVk isoforderL−1/2. Forconvenience
a function of the distance between the atoms. We dis- we write
cuss super-radiance and sub-radiance, including stable
V =(2π/L)1/2v , (4)
collectivestates mentionedabove. We alsogiveaheuris- k k
tic discussion on the force between the atoms. In Sec.
where v is of order 1 in the continuous spectrum limit
k
VII we discussthe mapping ofourmodeltoa two-cavity
L . As a specific example we will assume that [19,
electronwaveguide,andshowthatthissystemallowsan → ∞
20]
approximate stable collective state.
ω1/2
v =v(ω )= k (5)
k k [1+(ω /ω )2]n
II. ONE-ATOM SYSTEM k M
with n = 1. The constant ω−1 determines the range of
M
In orderto introduce the complex spectralrepresenta- the interaction. We shall assume that the interaction is
tion of the Hamiltonian, we consider first a single two- of short range, i.e., ω ω .
M 1
≫
level atom interacting with a field in one-dimensional The state 1 is unstable if
| i
space. This is the Friedrichs-Lee model in one dimen-
sion. We briefly review the main results. More details ω > ∞ dk λ2vk2 (6)
1
can be found in Ref. [14]. Z−∞ ωk
The Hamiltonian is given by
Otherwise, it is stable [22]. Hereafter we will consider
H = H +λV (1) the unstable case.
0
In the unstable case one can construct renormalized
= ω 1 1 + ω k k +λ V (k 1 + 1 k ).
1 k k field eigenstates that diagonalize the Hamiltonian as
| ih | | ih | | ih | | ih |
k k
X X
H = φ˜± ω φ˜± (7)
where we put c = h¯ = 1. The state 1 represents the | ki kh k|
| i k
bareatominits excitedlevelwithno fieldpresent,while X
the state k represents a bare field mode (“photon”) of where
| i
momentum k together with the atom in its ground state
lim φ˜± = k (8)
(see Figure 1). λ→0| ki | i
and hereafter we use the summation over field modes in
the sense ofEq.(2). The index refers to either “in”or
±
“out” scattering eigenstates.
The explicit form of the eigenstates is given by [14]
λV λV
φ˜± = k + k 1 + l l (9)
| ki | i η±(ωk)"| i ωk ωl iǫ| i#
l − ±
X
FIG. 1: One-atom system
where ǫ is an infinitesimal positive number. The limit
ǫ 0 is taken after the limit L . In Eq. (9),
→ →∞
The energy of the ground state is chosen to be zero; λ2V2
iωs1thisetphheobtaorneeenneerrggyy. oTfhtehecoeuxpciltinedg cleovnesltaanntdλωk ≡1|kis| η±(ω) ≡ ω−ω1− k′ (ω−ωkk′′)±
X
dimensionless. Weassumeperiodicboundarycond≪itions. = ω ω 2 ∞dk′ λ2vk2′ (10)
Weputthesystemina“box”ofsizeLandtakethelimit − 1− (ω k′)±
Z0 −
3
is the inverse of Green’s function. The + (or ) super- III. TWO-ATOM SYSTEM
−
scriptinEq.(10)indicatesanalyticcontinuationfromthe
upper (or lower) half-plane of ω [14]. Using the complex In this Section we discuss the complex spectral repre-
delta-function δC [11] we can write sentation of a two-atom system with Hamiltonian
1 (ω k′)−1, Imω >0
(ω−k′)± =(cid:26) (ω−−k′)−1∓2πiδC(k′−ω), ±±Imω <0 H = ω1|1ih1|+ω2|2ih2|+Xk ωk|kihk|
(11) + λ V eikx1 1 k +e−ikx1 k 1
1 k
| ih | | ih |
When ω is real, we have Xk (cid:0) (cid:1)
+ λ V eikx2 2 k +e−ikx2 k 2 (18)
∞ λ2v2 2 k | ih | | ih |
η±(ω)≡ω−ω1−2 dk′ω k′k′ iǫ. (12) Xk (cid:0) (cid:1)
Z0 − ± Thestate 1 representsatom1initsexcitedstate,while
| i
Withourformfactorandsmallλ,Green’sfunctionhas atom 2 is in the ground state and no field is present.
one pole z in the lower half plane, i.e., η+(z )=0 for Conversely,the state 2 represents atom 2 in its excited
1 1 | i
state, while atom 1 is in the ground state and no field
z ω˜ iγ (13) is present. The state k represents a field mode k with
1 1 1
≡ − | i
both atom 1 and atom 2 are in their ground states (see
Thenegativeimaginarypart iγ1describesdecayfort> Figure2). Theatoms1and2arelocatedatthepositions
−
0. Therealpartω˜1givestheshiftedaverageenergyofthe x1 and x2, respectively. We use the potential in Eq. (5).
excited state. For the other branch we have η−(z∗)=0,
1
∗
with z describing decay for t<0.
1
Note that in the representation(7) the decayrate and
shifted energy of the excited state do not appear in the
spectrum. One can incorporate z (or z∗) into the spec-
1 1
trum by extracting the residue of Eq. (7) at the pole
ω =z (or z∗). This gives the complex spectral decom-
k 1 1
positions [11, 12, 13, 14]:
H = φ z φ˜ + φ+ ω φ˜+
| 1i 1h 1| | ki kh k|
k
X
= φ˜ z∗ φ + φ− ω φ˜− (14)
| 1i 1h 1| | ki kh k|
k
X
The state φ and its dual φ˜ are complex eigenstates
1 1
| i h |
of H. Their explicit forms are [14]
λV
φ =N1/2 1 + k k , (15)
| 1i 1 | i | i(z1 ωk)+ FIG. 2: Two-atom system
h Xk − i
λV
φ˜ =[N∗]1/2 1 + k k .
| 1i 1 h| i Xk | i(z1∗−ωk)−i We will first assume that the two atoms are at fixed
positions, so that the distance between them
The state φ+ has the same form as the state φ˜+ with
| ki | ki x = x x (19)
the replacement 21 2 1
| − |
1 1 ω z is fixed. This can happen if the atoms are heavy. A
k 1
− (16) system of two fixed atoms is analogous to a two-cavity
η+(ω ) ⇒ η+(ω )(ω z )+
k k k 1
− waveguide (see Sec. VII).
and similarly, the state φ− has the same form as the We will consider the case where the two atoms are
state φ˜− with the comp|lekxiconjugate replacement. identical,
| ki
The states in Eq. (14) form a bi-orthormal set, with
λ =λ =λ
1 2
the relations
ω =ω (20)
1 2
φ˜ φ =1, φ˜+ φ =0
h 1| 1i h k| 1i We introduce the symmetric and antisymmetric states
hφ˜+k|φ+k′i=δk,k′ (17) 1 + 2 1 2
s = | i | i, a = | i−| i (21)
and their complex-conjugate relations. | i √2 | i √2
4
which are eigenstates of the unperturbed Hamiltonian The poles z are solutions of the equation
j,n
H =H(λ=0) as
0 η+(z )=0 (33)
j j,n
H j =ω j , j =s,a (22)
0| i j| i Inthefollowingwediscussthisequationanditssolutions.
ω =ω =ω . (23)
s a 1 From Eqs. (11) and (29) we obtain
We will use the notation ∞ λ2v2
z = ω +2 dk′ k′ (1+σ cos(k′x ))
j,n 1 z k′ j 21
1, for j =s Z0 j,n−
σj ≡(cid:26) −1, for j =a, (24) − 4πiλ2[vzj,n]2(1+σjcos(zj,nx21)) (34)
Note that cosine in the last term includes the factor
With this notation we have j =(1 +σ 2 )/√2.
As for the one-atom syst|emi, we| ican dji|agionalize the exp(γj,nx21), which grows exponentially with the dis-
tance x between the atoms. Assuming weak coupling
Hamiltonian as and tak2i1ng only the pole contribution in the k′ integral
H = F˜± ω F˜± (25) we obtain the set of equations
| k i kh k |
Xk ω˜j,n ω1+2π[λv(ω˜j,n)]2σjeγj,nx21sin(ω˜j,nx21) (35)
≈
where
γ 2π[λv(ω˜ )]2[1+σ eγj,nx21cos(ω˜ x )] (36)
F˜± = k +β± s +β± a + β± k′ (26) j,n ≈ j,n j j,n 21
| k i | i sk| i ak| i Xk′ k′k| i In Fig. 3, the pole of [ηs+(ω)]−1 with real part closest to
the unperturbed frequency ω is also closest to the real
1
and axis. We callthis pole z =z . Similarly, for[η+(ω)]−1
s,0 s a
wedenote the pole closestto ω asz =z . Boththese
1 λV 1 a,0 a
β± = k eikx1 +σ eikx2 (27) poles are obtained by a perturbation expansion around
jk √2η±(ω ) j
j k λ=0. We have
(cid:0) (cid:1)
z =z ω as λ 0. (37)
j,0 j 1
βk±′k = √12ωk−λωVkk′ ±iǫjX=s,aβj±k(cid:16)e−ik′x1 +σje−ik′x(22(cid:17)8) Isfhoxw21thisatnothtetopoolleasrgzej,n→(xa2r1e∼givγe1−n1b→yor smaller), one can
z +2nπ/x +δz , for σ n>0
ηj±(ω) zj,n ≈ zjj +(2n+σ21j)π/x2j,1n+δzj,n, for σjjn<0.
=ω ω 2 ∞dk′ λ2vk2′ (1+σ cos(k′x )) (cid:26) (38)
− 1− (ω k′)± j 21
Z0 − where δz is an O(λ2) correction. The approximate
(29) j,n
valueRe(z z )predictedbythisequationagreeswith
j,n j
−
Fig. 3 (for j = s) and a similar figure for j = a, which
Following a procedure similar to one found in Ref. [14],
one can show that the new diagonalizedstates F˜± sat- we omit.
isfy the orthogonality and completeness relation|sk i We write the poles zj as
z =ω˜ iγ . (39)
F˜± F˜± = 1 1 + 2 2 + k k (30) j j − j
| k ih k | | ih | | ih | | ih |
Thepolesz ,havingthesmallestdecayratesγ ,willgive
k k j j
X X
hF˜k+|F˜k+′i=hF˜k−|F˜k−′i=δk,k′ (31) aboduonmceinsaonfttchoentfireilbdutbieotnwteoenthtehteimateoemvso.luItniotnhiasftwearya,ftehwe
Green’sfunction[η+(ω)]−1 haspolesinthe lowerhalf- complex collective states defined in Eq. (40) emerge.
j As in the one-atom system we can obtain complex
plane, and conversely, [η−(ω)]−1 has poles in the upper
j eigenstates φ of the total Hamiltonian, such that
half-plane. From now on we discuss only the + branch | ji
with poles on the lower half-plane. H φ =z φ (40)
j j j
A new feature with respect to the one-atom system is | i | i
that due to the cosine term in Eq. (29), there are many Their explicit forms are given by
poles of Green’s function, as shown in Fig. 3. We label
φ
the poles as | ji
2−1/2λV
zj,n =ω˜j,n−iγj,n (32) =Nj1/2"|ji+ (zj ωk)k+ e−ikx1 +σje−ikx2 |ki#
k −
X (cid:0) (cid:1)
where n is an integer. (41)
5
0.05 1
0
0.1
-0.05
-0.1 P1(t)
0.01
-0.15
-0.2
0.001
-0.25 0 1 2 3 4 5 6 7
0 1 2 3 4 t
FIG. 4: Log plot of thesurvival probability P1(t) (solid line)
FIG. 3: Contour plot of log(1/|ηs+(z)|). The x and y axes andcomplexcollectivestatecomponentP1,zs(t)(dashedline).
are Re(z) and Im(z), respectively. The contours concentrate The crosses indicate the decay rate 2γs1, γs1 = 0.0233, be-
around the poles zs,n of Green’s function. Parameters are tween t = 0 and t = 1 (see Sec. V). Time is in units of
x21 =29.025, ω1 =2, λ=0.05, ωM =5. x21. The distance between atoms is x21 = 29.025. Other
parameters are ω1=2, λ=0.05, ωM =5, and L=500.
where the initial field is zero (similar calculations can be done
if the initial state is a ). We will calculate the survival
λ2V2 −1 probability of state 1| ,i
N = 1+ k (1+σ coskx ) . (42) | i
j " Xk [(zj −ωk)+]2 j 21 # P1(t)=|h1|e−iHt|si|2 (46)
Before we go into details, we can guess the behavior the
For these states we have φ j as λ 0.
| ji→| i → system will show. Since the initial state is symmetric,
The dual states satisfying φ˜ H =z φ˜ are givenby
j j j the following discussion also applies with atoms 1 and 2
h | h |
exchanged. Say atom 1 is to the left of atom 2. At the
φ˜ =N1/2 j + 2−1/2λVk eikx1 +σ eikx2 k beginning, atom 1 decays and emits a field. Half of this
h j| j "h | (zj ωk)+ j h |# fieldwillberadiatedawaytotheleft,whiletheotherhalf
k −
X (cid:0) (cid:1) willreachandexciteatom2. Atom2willthendecayand
(43)
emitits ownfield, partofwhichwillbe radiatedawayto
theright,therestgoingtotheleft,backtowardsatom1.
Like in the one-atom case, we have the complex spectral
Continuing this process, we see that energy will bounce
representation
back and forth between the two atoms. As time passes,
this energy will decrease due to the outgoing radiation.
H = φ z φ˜ + F+ ω F˜+ (44)
| ji jh j| | k i kh k | Eventually both atoms will decay to the ground state.
jX=s,a Xk Noting that the time it takes for the field of one atom
to reach the other atom is t = x (with c = 1) we
where F+ hasthesameformasthestate F˜+ withthe 21
| k i | k i conclude that, as it decreases, the survival probability
replacement
should oscillate with period x .
21
1 1 ω z This behavior is shown in Fig. 4. This was obtained
k j
− (45) through a numerical solution of Schr¨odinger’s equation.
ηj+(ωk) ⇒ ηj+(ωk)(ωk−zj)+ The fieldwas discretizedinto 2501modes. The eigenval-
ues and eigenfunctions of the Hamiltonian matrix were
We have as well the complex-conjugate representation,
obtainedusing tri-diagonalizationandthe “QL”method
taking the complex-conjugates of Eqs. (44), (45).
[23]. This allowed us to calculate explicitly the operator
exp( iHt). For this and the subsequent numerical plots
−
we used the following parameters: ω = 2, λ = 0.05,
IV. EMERGENCE OF THE COMPLEX 1
ω =5. Other parameters are indicated in each figure.
COLLECTIVE STATES M
In order to calculate the survival probability we start
with Eq. (25) to obtain
Time evolution of the two atom system can be solved
by using Eq.(25) orEq.(44). As anexample we assume P (t) = 1e−iHt F˜+ F˜+ s 2
the atoms are initially in the symmetric state s and 1 |h | | k ih k | i|
| i Xk
6
2
energy and the field P(x,t) around the atoms starts to
= e−iωkt 1F˜+ F˜+ s approach the field intensity due to the collective state
(cid:12) h | k ih k | i(cid:12)
(cid:12)Xk (cid:12) given by
(cid:12) (cid:12) 2
(cid:12)1 λ2V2 (cid:12)
= (cid:12)2(cid:12) e−iωkt ηs+(ωkk)2(1(cid:12)+coskx21)(cid:12) (47) Pzs(x,t)=|hψ(x)|φsiexp(−izst)hφs|si|2 (53)
(cid:12)Xk | | (cid:12)
(cid:12) (cid:12) (see Fig. 6). The collective state decays exponentially
whereusedthe f(cid:12)actthatoddfunctions ofk vanish(cid:12)under
(cid:12) (cid:12) (Fig. 7).
the summation. For later use we define the amplitude in
Eq. (47) as
λ2V2 0.006
I(t) e−iωkt k (1+coskx ) (48)
≡ η+(ω )2 21
s k
Xk | | 0.005
ThedominantcontributiontoP (t)willcomefromthe
1
poles of Green’s function, shown in Fig. 3. The different 0.004
pole contributions should add up to give the bounces
seen in Fig. 4. But rather than computing all the pole P(x) 0.003
contributions,we will follow aneasier method in Sec. V.
Here we will focus on the pole z . As mentioned be-
s 0.002
fore, this will give the dominant contribution after some
bounces, since it gives the slowest decay rate. It is this
0.001
pole contribution that is extracted in the representa-
tion (44). Using this representation, and noting that
φ˜a s =0 we have 0 -4 -3 -2 -1 0 1 2 3 4 5
h | i x
2
P (t)= 1φ e−izst φ˜ s + 1F+ e−iωkt F˜+ s FIG. 5: Field intensity P(x,t) for t = 0.32x21. The atoms
1 (cid:12)h | si h s| i h | k i h k | i(cid:12) are located at x1 = 0 and x2 = 1. Space coordinate x is in
(cid:12) Xk (cid:12) unitsofx21 andP(x,t)isdimensionless. Theparametersare
(cid:12) (cid:12)
(cid:12) (cid:12)(49) thesame as in Fig. 4.
(cid:12) (cid:12)
The second term contains contributions from the poles
other than z of Green’s function [η+(ω)]−1 as well as
s s
contributions coming from the branch cut of this func-
tion. Neglecting all these contributions we obtain
0.006
2
P (t) P (t) 1φ e−izst φ˜ s (50)
1 1,zs s s
≈ ≡ h | i h | i 0.005
(cid:12) (cid:12)
This is represented by the(cid:12)dashed line in Fi(cid:12)g. 4. After
(cid:12) (cid:12)
a few bounces the initial state s reaches the collective 0.004
| i
state φ .
s
| i
We turn to the time evolution of the field. Defining P(x) 0.003
the state
1 0.002
ψ(x) = e−ikx k , (51)
| i (2ω L)1/2 | i
k
k
X 0.001
the intensity of the field in space-time can be written as
P(x,t)= ψ(x)e−iHt s 2 (52) 0 -4 -3 -2 -1 0 1 2 3 4 5
|h | | i| x
Again we calculated this using the numerical solution FIG. 6: Field intensity P(x,t) for t = 1.12x21 (solid line)
of Schr¨odinger’s equation. The intensity of the field is andthecomplexcollectivestatecomponentPzs(x,t)(dashed
plottedinFigs. 5-7fordifferenttimes. Atthebeginning, line). Space coordinate x is in units of x21 and P(x,t) is
both atoms emit their fields spontaneously. Each field dimensionless. Parameters are thesame as in Fig. 4.
has an exponentially growing envelope (plus corrections
due to the initial dressing processes[15]), which stops at
the light cone x x =t (Fig. 5). In summary, the atoms emit a field growing exponen-
i
| − |
Aftereachemittedfieldreachestheneighboratom,ab- tially with the distance from them, within their light-
sorptionandre-emissionoccur. Thetwoatomsexchange cones. After the field emitted from each atom reaches
7
As shown in Fig. 4, the decay rate of P (t) changes
1
0.006 abruptly at t = x . For t > x the decay rate quickly
21 21
approaches the collective decay rate γ . The wiggling
s
0.005 of the decay rate shows the absorption and re-emission
of the fields, or in other words the energy bounces. For
0 < t < x the decay rate should be close to the one-
0.004 21
particle decay rate.
To analyze the energy bounces and the decay for 0 <
P(x) 0.003 t<x we first note that
21
0.002 η+(k) η−(k)=4πiλ2v2(1+coskx ), (56)
s − s k 21
Hence in Eq. (48) we can write
0.001
∞ λ2v2
0 -4 -3 -2 -1 0 1 2 3 4 5 I(t) = 2 dke−iωkt η+(kk)2(1+coskx21)
x Z0 | s |
∞
1 1 1
FIG. 7: Field intensity P(x,t) for t = 4.02x21 (solid line) = 2πi dk η−(k) − η+(k) e−ikt (57)
andthecomplexcollectivestatecomponentPzs(x,t)(dashed Z0 (cid:18) s s (cid:19)
line). TheoutersmallerpeaksofP(x,t)comefromtheinitial
Since e−ikt vanishes in the lower infinite semi-circle of
one-atom emission. The inner, larger peaks come from the
complex k plane for t > 0, we can take the pole contri-
emissionafterthefirstexchangeofenergybetweentheatoms.
Pth(ex,att)omassy.mTphtoetyiccaolliyncaidpeprionacthheesrPegziso(nx,bt)etawseexnatphperoaatcohmess cbluotsiionngstehxetceonndtinougrtwheitkh itnhtiesgsreamtiio-cnirfcrloem. O−n∞lyt[ηo+∞(k)a]n−d1
s
(between x = 0 and x = 1). Parameters are the same as in has poles in the lower half-plane. We write η+(k) as
s
Fig. 4
∞ λ2v2 (1+cosk′x )
η+(k) = k ω 2 dk′ k′ 21
s − 1− k k′+iǫ
Z0 −
the other one, the collective state with complex energy = ηs+1(k)−∆(k) (58)
z emerges.
j where η+(k) is defined by
As we discuss now, the exponential field has a strong s1
influence on z . The field amplitude associated with
j η+(k)
the collective states is given by k φ . This amplitude s1
in turn determines zj through ihts|injtieraction with the =k ω 2 ∞dk′ λ2vk2′ (1+ 1e−ik′x21)
atoms. We have − 1− k k′+iǫ 2
Z0 −
1 ∞ λ2v2 eik′x21
z =ω + λV eikx1 +σ eikx2 k φ (54) dk′ k′ . (59)
j 1 2N k j h | ji − k k′ iǫ
j k Z0 − −
X (cid:2) (cid:3)
where we usepd Eqs. (40), (41), and (29), with ω = z . and
j
Since k φ are functions of z , this is a self-consistent
h | ji j ∆(k)= 2πiλ2v2eikx21 (60)
relation. The exponential component of the field is seen − k
in the approximateequations(35)and(36)for zj,0 =zj, Unlike [η+(k)]−1, [η+(k)]−1 has only one pole in the
which include the factor exp(γ x ). Due to the expo- s s1
j 21 lower half plane. Let this pole be
nential nature of this factor, the pole z may deviate
j
substantially from the one-atom pole z1. zs1 =ω˜s1 iγs1 (61)
In spite of the exponential factor, for increasing x −
21
the equations (35) and (35) can still have solutions if γj Thisisessentiallythe poleofone-atomGreen’sfunction,
decreases as modified by the overlap of the atomic clouds at the dis-
tance x . For x ω−1 we have
γ x−1 (55) 21 21 ≫ 1
j ∼ 21
z z . (62)
for large x21. The decrease of γj with increasing x21 is s1 ≈ 1
seen in Fig. 8.
We expand 1/η+(k) as
s
1 1
V. BOUNCES η+(k) = η+(k) ∆(k)
s s1 −
∞
∆(k)n
Inthissectionwedescribetheenergybouncesbetween = . (63)
the atoms, seenin the survivalprobability of each atom. (η+(k))n+1
n=0 s1
X
8
The expansion is possible since 0.1
η+(k)=k ω 2 ∞dk′ λ2vk2′(1+cosk′x21) 0.08
s1 − 1− P k k′
Z0 (cid:18) − (cid:19)
+2πλ2vk2sinkx21+2πiλ2vk2, (64) ma 0.06
m
a
∆(k) = Im(η+(k)) (η+(k)). (65) g 0.04
| | | s1 |≤| s1 |
Using Eq. (63), Eq. (57) is written as 0.02
1 ∞ e−ikt
I(t)= dk
2πi η−(k) 0 5 10 15 20 25 30
Z0 s x21
1 ∞ ∞ ( 2πiλ2v2)ne−ik(t−nx21)
dk − k . (66)
−2πin=0Z0 (ηs+1(k))n+1 FIG. 8: The decay rates γs (×) and γa (+) oscillating as a
X functionofx21. Thesolidlineistheone-atomdecayrateγ1=
In Eq. (66), the pole contributions come from 0.0235. γs vanishes for distances close to (2n−1)π/ω˜1, and
1/(η+(k))n+1. For n=0, e−ikt/η+(k) has a simple pole γa for distances close to 2nπ/ω˜1 with n integer. An example
s1 s1
in the lower half plane at k = z . Its effect appears for is x21 = 12.7 ≈ 8π/ω˜1, where γa vanishes, while γs is large.
s1
t > 0, when we can close the integration contour in the For large x21 thedecay rates decrease.
lower half plane. For n = 1, e−ik(t−x21)/(η+(k))2 has a
s1
doublepole. Itseffectappearsfort>x . Ingeneral,for
21
eachx21 time step there appearsa new pole effect which As shown in Appendix A we have
is smaller by λ2 order than the previous pole effect. In
this way we can explain the wiggling decay rate (Figure I˜(t)=N e−izst (73)
0 s
4).
for all t>0, where
As we discuss now, this description of the bounces is
connected to emergence of the collective state. Approxi- 1
mating (for λ 1) Ns = (74)
≪ 1−∂∆(k)/∂k(cid:12)k=zs
η+(k) k z (67) (cid:12)
s1 ≈ − s1 for weak coupling. Eq. (73) show(cid:12)s that the sum of all
the pole contributions in Eq. (66) are given by bounces gives the contribution from the collective state
φ with eigenvalue z .
I (t) −1 ∞ dk 1 e−ikt | sEiq. (73) is consistenst with zs giving the slowest expo-
0
≈ 2πiZ−∞ k−zs1−∆(k) nential decay. To see this we use Eq. (58) to write the
1 ∞ ∞ ∆(k)n equation for zs,n as
= − dk e−ikt (68)
2πinX=0Z−∞ (k−zs1)n+1 ηs+1(zs,n)−∆(zs,n)=0 (75)
Taking the residues at the pole k = z we obtain an or
s1
expression of the form
z z +∆(z ) (76)
s,n s1 s,n
∞ ≈
I0(t)= θ(t−nx21)fn(t) (69) [for x21 ≫ ω1−1 we have zs1 ≈ z1 and we recover Eqs.
n=0 (35) and (36) for j =s].
X
The function k z ∆(k) in Eq. (68) has zeroes
where s1
− −
at k = z . For t only the residue at the pole
s,n
1 ∂n → ∞
f (t)= ∆(k)ne−ikt . (70) k =zs,0 =zs remains and we get
n − n!∂kn
(cid:20) (cid:21)k=zs1 lim N−1eizstI (t)= lim N−1eizstI˜(t)=1 (77)
We have f λ2n. Note that the sum stops at n such t→∞ s 0 t→∞ s 0
n
∼
that t < nx21. Since for weak coupling the terms fn(t) which is consistent with Eq. (73).
become smaller as n increases, after a few bounces we
have
VI. DECAY RATE AND ENERGY VS.
I (t) I˜(t) (71)
0 ≈ 0 DISTANCE
where
∞ In this section we investigatethe behaviorof the com-
I˜0(t)= fn(t) (72) plexeigenvaluesoftheHamiltonianzj fordifferentvalues
of x .
n=0 21
X
9
2.06
0.008
2.04
0.007
2.02
a 0.006
g 2
e
m
o 1.98 0.005
1.96 P(x) 0.004
1.94 0.003
0 5 10 15 20 25 30 0.002
x21
0.001
FIG.9: Theenergiesω˜s (×)andω˜a (+)asafunctionofx21.
Theyoscillatewithx21. Thesolidlineistheone-atomenergy 0 -9 -7 -5 -3 -1 0 1 3 5 7 9
ω˜1 =1.985. x
FIG.11: Field intensity P(x,t) for theatoms in astationary
collective state. The field between the two atoms, located at
x=0 and x=1 remains trapped. x is in units of x21 =12.7
0.55 andP(x,t)isdimensionless. Theinitialconditionis|ai. Time
is t = 7.09x21. The wave packets on each side represent the
0.5
field emitted before the atoms formed the collective state.
0.45 Afterthis, thereis no emission (we havesub-radiance).
0.4
0.35
0.3
P1(t) 0.25 0.009
0.2 0.008
0.15 0.007
0.1
0.006
0.05
0.005
00 1 2 3 4 5 6 7 P(x)
t 0.004
FIG. 10: Survival probability P1(t) of atom 1 for symmet- 0.003
ric (dashed) and antisymmetric (solid) initial conditions and
0.002
x21 =12.7. Thesymmetricstategivesrisetoasuper-radiant,
stationarycollectivestate. Theantisymmetricstategivesrise 0.001
to a sub-radiant (stationary) collective state. Before t=x21,
P1(t) has the one-atom decay rate. Time t is in units of x21. 0 -9 -7 -5 -3 -1 0 1 3 5 7 9
P1(t) is dimensionless. x
FIG. 12: Field intensity P(x,t) for initial condition |si at
time t = 7.09x21. x is in units of x21 = 12.7 and P(x,t) is
dimensionless. The collective state has decayed. The smaller
The equation η+(z)= 0 can be solved numerically by peaks on the far sides are the field emitted individually by
j
iterationsofz =z η+(z). Theimaginaryandrealparts each atom before attaining the collective state. The larger
− j peakscorrespond tothetwo-atom collective emission (super-
of z thus obtained are shown in Figs. 8 and 9 (we used
radiance).
the same parameters as in the previous figures). The
numerical iteration was started around z = ω so, with
1
the exception of two isolated points seen in Fig. 8, the
solutions obtained are the collective eigenvalues z = z .
j become smaller or larger than the one-atom decay rate
Gaps in the graphs are points missed by the numerical
(solid line in Fig. 8). We have sub-radiance and super-
solution.
radiance,respectively. In particular,it is noticeable that
Aswesee,γj andω˜j oscillatewithx21. Theoscillation there are distances at which the decay rates γj vanish
periodis approximately2π/ω˜1 where ω˜1 is the one-atom (see Appendix B). This means that for these distances
renormalized frequency (see Appendix B). there is no outgoing radiation. The outgoing emitted
Due to the oscillations, the collective decay rate can fieldsofthe atomscancelbydestructiveinterferenceand
10
attractiveforcebetweentwoatomsbecomeslocallymax-
0
imumwhenthecollectivedecayrateislocallymaximum.
Theatomstendtoattracteachotherwhentheyemitthe
-0.1
field outwards, and tend to repel when the field remains
-0.2 trapped between them.
Also we see in Fig. 13 that there are points x0 for
21
-0.3 which vanishes. Ifd /dx <0atthesepoints,then
s s 21
any smFall displacementF∆x around x0 creates a force
-0.4 inthe opposite direction. T2h1us in this c2a1sex0 arestable
21
points[ifd /dx >0thepoints areunstable]. Theex-
-0.5 Fs 21
istence of stable points suggests the possibility of having
5 10 15 20 25 30 aone-dimensional“molecule.” Thismoleculewouldhave
x21 a lifetime of the order of γ−1.
s
FIG.13: Thegraphof Fs =−dω˜s/dx21 (blackline) and the
decayrate−γs(dottedline). Weseethatwhen−γsisalocal VII. TWO-CAVITY WAVE GUIDES
minimum Fs has also a local minimum value.
a standing field is trapped between two atoms, storing
energy. Note that both symmetric and antisymmetric
initial conditions can give rise to either sub-radiant or
super-radiant states, since both γ and γ oscillate with
s a
x .
21
Theoscillationsofthedecayrateandtheenergyshown FIG. 14: A two-cavity waveguide
in Figs. 8 and 9 are a unique feature of one-dimensional
systems. For two or three dimensions, these quantities
canchangesignificantlyonly forshortdistancesbetween As shown in Refs. [17, 18], a Hamiltonian of the form
atoms (see Appendix C). (18) can be used to describe a two-dimensional electron
As an example of sub-radiance and super-radiance we wave guide as seen in Fig. 14.
show numerical simulations with the same parameters This wave guide can be constructed by superposing
usedinthepreviousexamples,exceptwechooseL=250 two closed identical cavities and a lead. This forms the
(tohavehigherspaceresolution)andx =12.7. Forthis unperturbedsystem. Theinteractionappearsasthecav-
21
value of x , the decay rate of the antisymmetric state ities and the lead are connected.
21
vanishes while the decay rate of the symmetric state is In suitable units the horizontal dimension of the cavi-
maximum, (see Fig. 8). In Fig. 10 we show the survival ties is 1, and the vertical dimension is D. The lead has
probability of atom 1 for the antisymmetric and sym- a horizontal dimension L and a vertical dimension
→∞
metric initial conditions, showing the appearance of sta- W. We consider a non-relativistic electron, neglecting
tionary sub-radiant collective state and a super-radiant the spin.
collective state. In Figs. 11 and 12 we show the corre- Ifthe electronis placedinside aclosedcavity,its wave
sponding fields. functions correspond to discrete cavity modes. The cav-
We turn to the force between the atoms. Here we will ity modes can be labeled as m,n , where m, n are posi-
| i
onlygiveaheuristicdiscussion. Amoredetailedanalysis tiveintegersrepresentingthehorizontalandverticalwave
requires including the Casimir-Polder or van der Waals numbers. The corresponding energies are
forces between the atoms, as well as the inertia of the
ξm,n =m2+n2/D2 (79)
atoms, which we are not considering in this paper.
Since the atoms are unstable, the force between them
An electron placed inside the lead (with no cavities)
should be time-dependent [24]. We expect the force will
hasmodesthatcanbelabeledas k,l wherekL/πisthe
decay exponentially during the time scales where the | i
horizontalwave number and l the verticalwave number.
collective-state components dominate. For the depen-
The energies are
dence on x of the force, the quantity
21
E =k2/π2+l2/W2 (80)
k,l
= dω˜ /dx (78)
j j 21
F −
As L , k becomes a continuous variable. On the
can give an indication because ω˜ is the average energy → ∞
j other hand, m, n and l are always integers.
of the collective state.
Weconsideranelectronwithlowenergynarrowlycen-
As we can see in Figure 13, oscillates with x (
s 21 a tered around
F F
has a similar behavior). > 0 corresponds to a re-
s
pulsive force, and < 0Fto an attractive force. The ξ0 =ξm0,n0 (81)
s
F