Table Of ContentTransient properties of modified reservoir-induced transparency
D. G. Angelakis, E. Paspalakis and P. L. Knight
Optics Section, Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom
(February 1, 2008)
We investigate the transient response of a Λ-type system with one transition decaying to a
modified radiation reservoir with an inverse square-root singular density of modes at threshold,
underconditionsoftransparency. Wecalculatethetimeevolutionofthelinearsusceptibilityforthe
probe laser field and show that, dependingon thestrength of thecoupling to themodified vacuum
andthebackgrounddecay,theprobetransmissioncanexhibitbehaviourrangingfromunderdamped
to overdamped oscillations. Transient gain without population inversion is also possible depending
on thesystem’s parameters.
PACS: 42.50.Gy,42.70.Qs
0
0
0 It has been now well documented that quantum co- andthe strengthofthe coupling to the modifiedvacuum
2 herence and interference effects can modify the absorp- modes. We also find that, under certain conditions, the
tion and dispersion properties of an atomic system [1,2]. system can exhibit transient gain without inversion.
n
In the most common situation, that of a Λ-type three- Theatomicsystemunderconsiderationisshowninfig-
a
J levelsystem,themediumbecomestransparenttoaprobe ure 1. It consists of three atomic levels in a Λ-type con-
1 laser field near an otherwise absorbing resonant transi- figuration. The atom is assumed to be initially in state
2 tion. This is achieved via the application of a second 0 . Thetransition 1 2 istakentobenear-resonant
| i | i↔| i
laser field coupling to the linked transition. In addi- with a frequency-dependent reservoir, while the transi-
2 tion to steady state studies, considerable work has been tion 0 1 is assumed to be far away from the gap
v | i ↔ | i
done on the transient properties of coherent phenomena and is treated as a free space transition. The dynamics
8
such as, for example, electromagnetically induced trans- of the system can be described using a probability am-
1
parency[3,4],gain(orlasing)withoutinversion[5–7]and plitude approach. The Hamiltonianofthe system,inthe
0
9 coherent population trapping [8,9]. interactionpictureandtherotatingwaveapproximation,
0 As has been recently shown [10], transparencycan oc- is given by (we use units such that h¯ =1),
9
cur in the steady state absorption of a Λ-type system
/9 when one of the atomic transitions is coupled to a mod- H = Ωeiδt 0 1 + gk,λe−i(ωk−ω12)t 1 2ak,λ
h ified radiation reservoir having a threshold with an in- (cid:20) | ih | Xk,λ | ih |
p
verse square-root dependence of the density of modes,
γ
-
nt ρ(ω)=Θ(ω−ωg)/ π√ω−ωg ,withΘbeingtheHeavi- +H.c.(cid:21)−i2|1ih1|. (1)
a sidestepfunctionan(cid:0)dωg being(cid:1)thegapfrequency. Sucha
u density ofmodes canbe found near thresholds in waveg- Here, Ω = µ01 ǫE is the Rabi frequency, with µnm
q uides [11,12], in microcavities [13,14], and near the edge being the dip−ole m· atrix element of the n m tran-
: of a photonic band gap material which is described by sition. The unit polarizationvector and|thie↔ele|ctriic field
v
i an isotropic model [15–19]. We also note that there is amplitude of the probe laser field are denoted by ǫand
X current interest in coherent phenomena which occur in E respectively. Also, δ = ω ω is the laser detun-
10
r modified reservoirs having relatively weak modal densi- ing from resonance with the 0− 1 transition, where
a ties where the Born and Markov approximations can be ω = ω ω and ω is t|hei e↔ne|rgiy of state n and
nm n m n
applied [20,21]. ω is the pro−be laser field angular frequency. In ad|diition,
It is known that coherence effects can take a consid- γ denotes the background decay to all other states of
erable time to be set up [22], and the purpose of the the atom. It is assumed that these states are situated
present work is to investigate this question when struc- far from the gap so that such background decay can be
tured radiation reservoirs are employed. In this article treated as a Markovian process. We note that we are
we study the transient behaviour of the absorption of interested in the perturbative behaviour of the system
a Λ-type system, similar to the one used in ref. [10], to the probe laser pulse, therefore γ can also account
where transparency in the steady state absorption spec- for the radiative decay of state 1 to state 0 Finally,
| i | i
trum of the system was predicted. In our system, one gk,λ = i 2πωk/V ǫk,λ µ12 where V is the quantiza-
− ·
of the atomic transitions is spontaneously coupled to a tion volumpe, ǫk,λ is the unit polarization vector, ak,λ is
frequency-dependent reservoir which displays the above the photon annihilation operator and ωk is the angular
mentioned inverse square-root behaviour in its density frequency of the k,λ mode of the modified radiation
{ }
of modes. Solving the equation of motion for all times, reservoir vacuum field.
we show that the rate at which the atomic medium be- The wavefunction of the system, at a specific time t,
comestransparenttotheprobefielddependscruciallyon canbe expandedin terms of the ‘bare’eigenvectorssuch
boththebackgrounddecayrateoftheupperatomiclevel that
1
ψ(t) =b (t)0, 0 +b (t)e−iδt 1, 0 where ǫ is a real number chosen so that s=ǫ lies to the
0 1
| i | { }i | { }i
rightofallthesingularities(polesandbranchcutpoints)
+ bk,λ(t)|2,{k,λ}i, (2) of function ˜b (s).
Xk,λ 1
Forthecaseofaninversesquare-rootsingularityinthe
and b0(t = 0) = 1, b1(t = 0) = 0, bk,λ(t = 0) = 0. frequency-dependent reservoir density of modes K˜(s) =
We substitute Eqs. (1) and (2) into the time-dependent β3/2e−iπ/4/ s+i(δg δ) with δg = ωg ω12, the in-
− −
Schr¨odinger equation and obtain the time evolution of verse Laplacpe transform of Eq. (9) yields
the probability amplitudes as
5
ib˙0(t)=Ωb1(t), (3) b1(t)= αi(xi+yi)ex2it
γ Xi=1
ib˙ (t)=Ωb (t) δ+i b (t)
1 i 0tdt′−K(cid:16)(t t′)2b(cid:17)1(t1′), (4) −Xi=51αiyi(cid:20)1−erf(qx2it)(cid:21)ex2it, (11)
− Z −
0
ib˙k,λ(t)=gk,λei(ωk−ω12−δ)tb1(t), (5) where yi = x2i and xi are the roots of the equation
p
with the kernel x5+c x3+c x2+c x+c =0. (12)
3 2 1 0
K(t−t′)=Xk,λgk2,λe−i(ωk−ω12−δ)(t−t′) cHe=rec3K=δγ′,/2δ−′ =i(δδg+δ′δ),,Kc2 ==−iβiK3/02,ec−1iπ=/4−aδn′d(δger+fiiγs/t2h)e,
0 0 g 0
− −
β3/2 dωρ(ω)e−i(ω−ω12−δ)(t−t′), (6) error function [23]. The roots of this equation are de-
≈ Z termined numerically. The expansion coefficients αi are
given by
and β being the atom-modified reservoir resonant cou-
pling constant. All the coupling constants (gk,λ, β, Ω) iΩxi
α = , (13)
are assumed to be real, for simplicity. i (x x )(x x )(x x )(x x )
i j i k i l i m
The time evolution of the absorption and dispersion − − − −
properties of the system are determined by, respectively, with i,j,k,l,m = 1,2,3,4,5. Also, if Re(x ) > 0 we have
i
the imaginary and real parts of the time-dependent lin- y = x , while if Re(x ) < 0 we have y = x , in order
i i i i i
−
earsusceptibilityχ(t). Inourcase,thesusceptibilitycan to keep the phase angle of x2 between π and π [23]. In
be expressed as [3] addition, if x = 0 then α =i 0. There−fore, at least two
i i
roots and at most three roots contribute on the solution
4π µ01 2 ∗ (11) depending on the system parameters.
χ(t)= N| | b (t)b (t), (7)
− Ω(z,t) 0 1 Within our perturbative approach, Eq. (7) yields
∗
χ(t) b (t), where b (t) is given by Eq. (11). As
with being the atomic density. The solution of Eqs. ∼ − 1 1
has been shown in ref. [10], steady state transparency
N
(3) and (4) is obtained by means of time-dependent per-
occurs for the case that δ = δ . This is the case that
g
turbation theory [5,10]. We assume that the laser-atom
alsointerests us here. In figure 2 we plot the time evolu-
interaction is very weak (Ω β,γ) so that b (t) 1 for
0 tion of the imaginary part of the linear, time-dependent
≪ ≈
all times. Then, Eqs. (3) and (4) reduce to
susceptibility[ Im(χ(t))]fordifferentvaluesoftheback-
−
ground decay γ and with δ = δ = 0. In the case that
γ t g
ib˙ (t) Ω δ+i b (t) i dt′K(t t′)b (t′). (8) γ β, the susceptibility is always positive (which de-
1 ≈ −(cid:16) 2(cid:17) 1 − Z0 − 1 no≫tesabsorptioninourconvention),hasamaximumand
the steady state value is reached adiabatically. If γ =β,
WefurtherassumethatΩ(z,t)isapproximatelyconstant
thesystemexhibitsonlyabsorption,howeversmalloscil-
inthemediumandwiththeuseoftheLaplacetransform
lationsarevisible atthe beginning. As γ decreases,then
we obtain from Eq. (8)
these oscillations become more pronounced, and small
gain (or lasing) without the presence of population in-
Ω
˜b (s)= , (9) version between 1 and 0 , shown by negative values
1
s δ+iγ/2+iK˜(s)+is of the time-depen|dient lin|eair susceptibility, is found. If
h i
the background decay decreases further and reaches the
where ˜b (s) = ∞e−stb (t)dt, K˜(s) = ∞e−stK(t)dt. regime that γ β the oscillations increase further, the
1 0 1 0 ≪
TheamplitudebR1(t)isgivenbytheinverseRLaplacetrans- gainwithoutinversionincreases,theinteractionbecomes
form morenon-adiabaticandthesteadystatevalueisreached
for very large times.
b (t)= 1 ǫ+i∞est˜b (s)ds, (10) The behaviour displayed in the previous figure can be
1 2πiZǫ−i∞ 1 understoodif the time evolutionofthe populationofthe
2
excited state 1 is examined. As can be seen from fig- Foundation (SSF) and the European Commission TMR
| i
ure 3, after an initial weak absorption the population Network on Microlasers and Cavity QED.
of the state 1 can either decay smoothly to zero (for
| i
the case γ β) or evolve by undergoing damped Rabi
≫
oscillations between states 1 and 2 due to reversible
| i | i
decay which arises via the interaction with the modi-
fied reservoir [15,16]. These oscillations increase as the
background decay decreases compared to the coupling
strength to the frequency-dependent radiation reservoir. [1] M.O. Scully and M.S. Zubairy, Quantum Optics (Cam-
Insuchawayatime-dependentcoherencebetweenstates bridge University Press, Cambridge, 1997), Chapter 7.
1 and 2 is created which is responsible for the phe- [2] J.P. Marangos, J. Mod. Opt.45, 471 (1998).
|noimenon| iof transient gain without inversion shown in [3] Y.-Q. Li and M. Xiao, Opt.Lett. 20, 1489 (1995).
[4] H.X. Chen, A.V. Durrant, J.P. Marangos and J.A. Vac-
figure 2.
caro, Phys. Rev.A 58, 1545 (1998).
This behaviourofthe systemis relatedto the onepre-
[5] S.E. Harris and J.J. Macklin, Phys. Rev. A 40, 4135
dicted [3] and experimentally observed [4] in a typical
(1989).
three level Λ-type atomic system which exhibits electro-
[6] E.S. Fry, X.F. Li, D.E. Nikonov, G.G. Padmabandu,
magnetically induced transparency through the applica-
M.O. Scully, A.V. Smith, F.K. Tittel, C. Wang, S.R.
tion of a coupling laser field. The difference in our case, Wilkinson and S.-Y. Zhu Phys. Rev. Lett. 70, 3235
is that the transparency and the transient gain without
(1993).
inversion occur due to the coupling to a radiation reser- [7] Y. Zhu,Phys. Rev.A 55, 4568 (1997).
voirwithaninversesquare-rootsingularityofthedensity
[8] P.L. Knight, M.A. Lauder and B.J. Dalton, Phys. Rep.
ofmodesatthresholdandarenotinducedbyanexternal 190, 1 (1990).
laser field. [9] E.A. Korsunsky, W. Maichen and L. Windholz, Phys.
In summary, we have discussed the transient proper- Rev. A 56, 3908 (1997).
ties of the transparency in a Λ-type atom in which one [10] E. Paspalakis, N.J.Kylstra and P.L.Knight,Phys. Rev.
transition spontaneously decays to a specific frequency- A 60, R33 (1999).
dependent radiationreservoir. The time evolutionofthe [11] D. Kleppner,Phys. Rev.Lett. 47, 233 (1981).
absorption and thus the way that the steady state is [12] M. Lewenstein, J. Zakrzewski, T.W. Mossberg and J.
reached depends crucially on the background decay rate Mostowski, J. Phys.B 21, L9 (1988).
andthestrengthofthecouplingtothemodifiedreservoir. [13] M.LewensteinM.andT.W.Mossberg, Phys.Rev.A 37
Transient gain without population inversion is found to 2048 (1988).
[14] M.A. Rippin and P.L. Knight, J. Mod. Opt. 43, 807
exist if the coupling strength to the modified reservoir
(1996).
is larger than the background decay rate. We have only
[15] S.JohnandT.Quang,Phys.Rev.A50,1764(1994);T.
been concerned with the time evolution of the linear ab-
Quang, M. Woldeyohannes, S. John and G.S. Agarwal,
sorption properties of the medium. The time evolution
Phys. Rev.Lett. 79, 5238 (1997).
of the dispersive properties of the system, which is an-
[16] A.G.Kofman,G.KurizkiandB.Sherman,J.Mod.Opt.
other topic of interest [10], will be discussed separately.
41, 353 (1994).
In such a study the simple relationship between the real [17] S.-Y. Zhu, H. Chen and H. Huang, Phys. Rev. Lett. 79,
part of the susceptibility and the group velocity cannot
205 (1997).
be applied (as it holds only for the steady state), and a
[18] S. Bay, P. Lambropoulos and K. Mølmer, Phys. Rev.
different approachneeds to be implemented. Lett. 79, 2654 (1997).
[19] E. Paspalakis, D.G. Angelakis and P.L. Knight, LANL
e-print quant-ph/9906093.
ACKNOWLEDGMENTS [20] O. Kocharovskaya, P. Mandel and M.O. Scully, Phys.
Rev. Lett.74, 2451 (1995).
[21] M. Erhard and C.H. Keitel, submitted for publication
E.P.thanksNielsKylstraforhelpfuldiscussionsinthe (1999).
subject. We wouldliketoacknowledgethe financialsup- [22] B.J.DaltonandP.L.Knight,J.Phys.B15,3997(1982).
port of the UK Engineering and Physical Sciences Re- [23] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Se-
search Council (EPSRC), the Hellenic State Scholarship ries and Products (Academic Press, New York,1980).
3
j1i
(cid:0)(cid:14)
j2i
j0i
FIG. 1. The system under consideration. The solid line
denotestheprobelasercoupling,thethickdashedlinedenotes
thecouplingtothemodifiedradiationreservoirandfinallythe
thin dashed line denotesthe background decay.
n
o 0.8
i
s
s
i 0.6
m
s
n 0.4
a
r
T 0.2
e
b
o 0.0
r
P
-0.2
5 10 15 20 25 30 35 40
Time
FIG. 2. The time evolution of the imaginary part of the
time-dependent linear susceptibility [−Im(χ(t))] (in arbi-
trary units). In our notation positive (negative) values de-
note probe absorption (gain). The parameters used were
δ = δg = 0 and γ = 5 (shot dashed curve), γ = 1 (long
dashed curve), γ =0.5 (thin solid curve), and γ =0.2 (thick
solid curve). Time and γ are measured in units of β.
> 0.0025
1
|
f 0.0020
o
n 0.0015
o
i
t
a 0.0010
l
u
p 0.0005
o
P
0.0000
5 10 15 20 25 30 35 40
Time
FIG.3. The time evolution of the population of state |1i.
The parameters and the units used are the same as in figure
2.
4
