We propose a scheme for slowing down decay into a continuum. We make use of a
sequence of ultrashort $2\pi$-pulses applied on an auxiliary transition of the
system so that there is a destructive interference between the two transition
amplitudes - one before the application of the pulse and the other after the
application of the pulse. We give explicit results for a structured continuum.
Our scheme can also inhibit unwanted transitions.Comment: 11 pages and 4 figures, submitted to Physical Review Letter

A. G. Kofman

A. Imamoglu

B. Misra

C. J. Myatt

C. Search

D. B. Horoshko

D. Home

D. P. Divincenzo

D. Vitali

E. Paspalakis

E. Yablonovitch

G. C. Hegerfeldt

G. M. Palma

G. S. Agarwal

H. Walther

J. M. Raimond

J. S. Waugh

L. Viola

M. B. Plenio

M. Brune

M. Namiki

M. O. Scully

P. Lambropoulos

P. Zhu

S. Bay

S. John

S.-Y. Zhu

T. Quang

T. W. Mossberg

W. M. Itano

English

arXiv

Crossref

Physical Review Letters

Inhibition of Decoherence due to Decay in a Continuum

We propose a scheme for slowing down decay into a continuum. We make use of a sequence of ultrashort 2π pulses applied on an auxiliary transition of the system so that there is a destructive interference between the two transition amplitudes - one before the application of the pulse and the other after the application of the pulse. We give explicit results for a structured continuum. Our scheme can also inhibit unwanted transitions

Agarwal, G. S.

Scully, M. O.

Walther, H.

Publications of the IAS Fellows

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Inhibition of Decoherence due to Decay in a Continuum
G.S. Agarwal1,2, M.O. Scully 2,3 and H. Walther 2
1. Physical Research Laboratory, Navrangpura, Ahmedabad- 380009, India
2. Max-Planck-Institut Fu¨r Quantenoptik, D-85748 Garching, Germany.
3. Texas A&M, Department of Physics, College Station, TX77843-4242, USA
(February 1, 2008)
Abstract
We propose a scheme for slowing down decay into a continuum. We make use
of a sequence of ultrashort 2pi-pulses applied on an auxiliary transition of the
system so that there is a destructive interference between the two transition
amplitudes - one before the application of the pulse and the other after the
application of the pulse. We give explicit results for a structured continuum.
Our scheme can also inhibit unwanted transitions.
PACS Number(s): 03.65. Bz, 03.67 - a, 42.50. Md
Typeset using REVTEX
1
One of the fundamental causes of noise at the optical frequencies is the intrinsic spon-
taneous emission. It is well known that the performance of many systems is limited by
spontaneous emission. For example the noise figure of an amplifier is determined by spon-
taneous emission. One of the challenges therefore is to find ways in which spontaneous
emission noise can be reduced if not totally inhibited. Several proposals exist in literature
for reducing spontaneous emission noise. These include - placing of atoms in a photonic
bandgap structure [1,2], use of external fields [3] and quantum interferences [4,5]. As is well
known, spontaneous emission noise arises from considerations based on the interaction of
an atom with the vacuum of the electromagnetic field. The vacuum acts like a zero tem-
perature bath. It may be noted that the question of inhibiting the effects of a heat bath
is being extensively studied and several new proposals exist for such an inhibition [6–10].
For example, for a spin system interacting with a bath the decoherence can be slowed [6] by
applying a sequence of pi pulses applied at invervals of a short period τ which is less than
the bath correlation time.
In this letter we consider the case of spontaneous emission from an excited atom. One
has considerably more freedom with atoms than with spins as in the former case we could
use a different transition to control spontaneous emission. We propose a scheme to suppress
spontaneous emission on (say) the emission from the state |e〉 to |g〉 by using an auxiliary
transition |g〉 to |f〉. We apply a sequence of ultrashort 2pi-pulses separated by an interval
τ . We demonstrate how a destructive interference between the evolution from to to to + τ
and to + τ to to + 2τ can lead to suppression of decoherence. It should be borne in mind
that the decoherence time scale is generally proportional to the decay time. The destructive
interference is related to the very remarkable property that after the application of a 2pi
pulse the state |g〉 acquires a phase shift of pi. An important by-product of our investigation
is the possibility of suppressing undesirable weak transitions.
We first consider a simple case which is adequate to describe the main idea and which
by itself is very relevant to the subject of quantum computation [11]. Consider a quantum
system which can make an unwanted weak transition from the state |g〉 to |e〉 as a result of
2
a perturbation v. Let δ be the detuning i.e. the perturbation need not be resonant with the
transition |g〉 ↔ |e〉. The interaction Hamiltonian in the interaction picture is
H1(t) = h¯v|e〉〈g|e
−iδt +H.c. (1)
The perturbation theory leads to the probability of transition
peg = |v|
2 sin
2(δt/2)
(δ/2)2
. (2)
We next demonstrate how this unwanted transition could be inhibited by applying a sequence
of very short 2pi-pulses [12] on the transition |g〉 ↔ |l〉 [Fig.1]. We thus divide the total time
interval into a large number 2N of short intervals τ .The system evolves under v from to
to to + τ . At to + τ we apply an ultra-short 2pi-pulse on the transition |g〉 ↔ |l〉. The
system evolves from to+ τ to to+2τ under v. At the instant to+2τ the 2pi-pulse is applied
again. This process is repeated N number of times. The system then evolves as follows:-
For to < t < to + τ, we have
|ψ(t)〉 ∼ |g〉 − iv|e〉 (X(t− to)/(−iδ)) e
−iδto ;X(t) = (e−iδt − 1). (3)
At time to + τ after the application of 2pi-pulse, the state of the system is
|ψ〉 = −|g〉 − iv|e〉X(τ)e−iδto/(−iδ). (4)
At a time to+ 2τ just before the application of the second pulse the state would evolve into
|ψ〉 = −|g〉 − iv|e〉(X(τ)/(−iδ))e−iδto + iv|e〉(X(τ)/(−iδ))e−iδ(to+τ) + 0(v2), (5)
which on application of the second 2pi-pulse changes to
|ψ〉 ≡ |g〉+ iv|e〉(X(τ))2e−iδto/(−iδ) + 0(v2). (6)
The transition amplitude χeg at the end of one cycle consisting of evolution of the system
from to to to + 2τ but with 2pi -pulses applied at to + τ and to + 2τ will be
χeg = iv(X(τ))
2e−iδto/(−iδ) + 0(v2). (7)
3
The transition amplitude at the end of N such cycles will be
χeg = iv
(X(τ))2
(−iδ)
N−1∑
p=0
e−2iδτp−iδto , (8)
which leads to the following result for the net transition probability
p˜eg = |v|
2 tan2
(
δτ
2
)
sin2
(
δ
2
(2τN)
)
(δ/2)2
. (9)
On using (2) we have one of our key results
p˜eg = tan
2
(
δτ
2
)
peg. (10)
We have thus proved that the application of a sequence of 2pi-pulses on an auxiliary transition
leads to the suppression of an unwanted transition provided that the small interval and the
detuning δ are chosen such that
tan2(
δτ
2
) ≪ 1. (11)
The suppression arises from a destructive interference of the transition amplitudes (second
and third terms in Eq.(5)). This distructive interference is due to a phase change of the
state |g〉 (and not of |e〉) by pi due to the application of the 2pi-pulse. This also explains our
choice of an auxiliary transition for the application of the 2pi-pulse as we selectively want
to produce a phase change so that the interference can occur. One might think that the
procedure we describe is just the quantum Zeno effect [13]. We emphasize that it is not
as we do not carry out repeated measurements. However it can be considered at best an
analog of Zeno effect in the spirit of [14], though collapse of the state is considered to be an
essential ingredient for quantum Zeno effect [15].
We next demonstrate how the above procedure can be used to possibly inhibit sponta-
neous emission. We show the procedure schematically in Fig 2. The atom makes a transition
from the excited state |e〉 to the ground state |g〉 by emitting a photon. The photon can be
emitted in any mode ωk of the vacuum of the radiation field. The polarization of the emitted
photon will be determined by the direction of the dipole matrix element. The interaction
Hamiltonian in the interaction picture is [4].
4
H1(t) = h¯
∑
k
|e〉〈g|gkake
−iδkt +H.c; (12)
δk = ωk − ωeg.
For brevity, we do not display the polarization “s” and vectorial indices of the mode. Thus
k really stands for (k, s). In (12), ak is the annihilation operator for the mode k of the
radiation field. We now follow the precedure leading to Eq.(9). We quote the results of
calculations for the transition probabilities with (p˜ge) and without (pge) the application of
2pi-pulses
p˜ge =
∑
k
|gk|
2 tan2
(
δkτ
2
)
sin2
(
δk
2
2τN
)
(δk/2)
2 , (13)
pge =
∑
k
|gk|
2
sin2
(
δk
2
2τN
)
(δk/2)
2 . (14)
We first note how (14) leads to the standard result and how the Einstien A-coefficient
emerges. If t = 2Nτ , then
∂pge/∂t =
∑
k
2|gk|
2 sin δkt
δk
(15)
which, under the assumption that the observation time t is large compared to the width of
(δk) values (which is of the order of ωeg for spontaneous emission in free space), reduces to
the standard expression for the Einstein A-coefficient
∂pge/∂t = 2pi
∑
|gk|
2δ (ωk − ωeg) ≡ A. (16)
We next examine the conditions under which the presence of the quantum interference term
tan2 (δkτ/2) in Eq. (13) can lead to the suppression of spontaneous emission. Let us consider
a kind of one dimensional model in which we can replace (13) by
p˜ge ≡
∫ ∞
−ωeg
dx tan2
(
xτ
2
)
sin2(x
2
2τN)
(x/2)2
ρ(x), (17)
where ρ(x) is the density of states for the one dimensional vacuum. For ρ(x), we can choose
any of the functions like the Lorentzian or the exponential
5
ρ(x) =
ρoΓ/pi
(x2 + Γ2)
or, ρ(x) =
ρo
2Γ
e−|x|/Γ (18)
with Γ≪ ωeg and where ρo is related to the square of the dipole matrix element. Note that
these choices correspond to what are known as the “structured” vacuua [16]. Thus we are
essentially asking the question - under what conditions the emission into structured vacuua
can be inhibited. We use the exponential model in (17) and show a typical result in Fig.3.
For comparison we also show the result pge. On comparing the two results we see that the
use of a sequence of 2pi-pulses applied in the manner shown in Fig.2 can suppress to a large
extent, the decay into a structured continuum.
We next consider briefly the question of spontaneous emission in free space. We convert
(13) into an integral using the standard expressions for gk and by letting the quantization
volume go to infinity. The transition probabity p˜ge depends on the following integral
I =
∫ ∞
−1
dx(x+ 1)3 tan2
(
xτ
2
)
sin2(x
2
2τN)
(x/2)2
, (19)
x = (ω − ωeg)/ωeg, τ → ωegτ.
Note that the sine function is sharply peaked at x = 0 if 2Nτ is large. This is what
enabled us to simplify the expression for pge leading to (16). However, we now have the
interference factor tan2 (xτ/2) which starts growing as xτ/2 → pi/2. Thus the idea of 2pi-
pulse induced slowing down of decoherence will obviously work if the bath has a cut-off such
that tan2(xτ/2) remains much smaller than unity. In order to examine further the question
of slowing down of the decoherence we analyse the integrand I˜ in (19). We can rewrite the
integrand in the form
I˜ ≡ (x+ 1)3
16 sin4(xτ/2)
x2
[
sin2(xτN)
sin2(xτ)
]
. (20)
Note that the integrand oscillates very rapidly due to the term sin2(xτN). Though it has no
singularities as the term in the square bracket is well behaved, it still grows as N2 whenever
xτ = npi. Such a growth problem can be avoided if the vacuum mode frequencies are such
that xτ ≪ pi. If x is chosen to be of the order of unity, then the pulses have to be applied at
6
intervals much smaller than ω−1eg , which is a difficult task in the optical domain though not
for Rydberg transitions. Note that with pulses applied at intervals less than ω−1eg , the details
of spontaneous emission become sensitive to the structure of the bath as shown in the Fig.4.
We note that Vitali and Tombesi [7] have examined the damping of a harmonic oscillator
interacting with a broad band bath and have reached similar conclusions. The nature of
the integrand in (19) also suggests that if the interval τ > ω−1eg , then we have the possibility
of accelerating spontaneous emission (Fig.4; curves for ωegτ = pi) which is suggestive of an
effect that is an analog of recently discovered quantum anti-Zeno effect [18].
In conclusion, we have shown how a sequence of 2pi-pulses applied on an auxiliary tran-
sition in a system can slow down considerably the decay into a continuum [17]. The same
scheme also enables one to inhibit unwanted transitions.
The authors thank K. Kapale and S. Menon for preparing the figures.
7
REFERENCES
[1] E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987); S. John and T. Quang, Phys. Rev.
A50, 1764 (1994).
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(1997); S. -Y. Zhu, H. Chen and H. Huang, ibid. 79, 205 (1997); S. Bay, P. Lambropoulus
and K. Mølmer, ibid. 79, 2654 (1997).
[3] G. S. Agarwal, W. Lange, and H. Walther, Phys. Rev. A 48, 4555 (1993); T. W. Moss-
berg and M. Lewenstein, in Cavity Quantum Electrodynamics edited by P. R. Berman
(Academic, New York, 1994), p.171.
[4] G. S. Agarwal, Quantum Optics, Springer Tracts in Modern Physics, (Springer-Verlag,
Berlin 1974), pp.92-97.
[5] S.-Y. Zhu and M. O. Scully, Phys. Rev. Lett. 76, 388 (1996); M. O. Scully and S. -Y.
Zhu, Science, 281, 1973 (1998); E. Paspalakis and P. L. Knight, Phys. Rev. Lett. 81, 293
(1998); P. Zhu and S. Swain, ibid. 77, 3995 (1996); ibid 78, 832 (1997); A. Imamoglu,
Phys. Rev. A40, 2835 (1989).
[6] L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 (1998); L. Viola, E. Knill and S. Lloyd,
Phys. Rev. Lett. 82, 2417 (1999); C. Search and P. R. Berman, Phys. Rev. Lett. 85,
2268 (2000). Note that pulses were very successfully used [J.S. Waugh, L.M. Huber and
U. Haeberlen, Phys. Rev. Lett. 20, 180 (1968)] to inhibit decoherence due to dipole -
dipole interaction in the context of magnetic resonance.
[7] D. Vitali and P. Tombesi Phys. Rev. A 59, 4178 (1999).
[8] D. Vitali, P. Tombesi and G.J. Milburn, Phys. Rev. Lett. 79, 2442 (1997); J. Mod. Opt.
44, 2033 (1997); D. B. Horoshko and S. Ya. Kilin, Phys. Rev. Lett. 78, 840 (1997); J.
Mod. Opt. 44, 2043 (1997).
[9] G. S. Agarwal, Phys. Rev. A61, 013809, (2000).
8
[10] For measurements of decoherence times, see M. Brune, E. Hagley, J. Dreyer, X. Maitre,
A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 77, 4887
(1996); J. M. Raimond, M. Brune, and S. Haroche, ibid. 79, 1964 (1997); C. J. Myatt,
B. E. King, Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe and
D. J. Wineland, Nature 403, 269 (2000).
[11] M. B. Plenio and P.L. Knight Proc. R. Soc. London, Ser. A 453, 2017 (1997); G. M.
Palma, K. A. Suominen, and A. K. Ekert, ibid, 452, 567 (1996); D. P. Divincenzo,
Science, 270, 255 (1995).
[12] M. O. Scully, to be published.
[13] B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977); D. Home and M. A.
B. Whitaker, Ann. Phys. 258, 237 (1997).
[14] W.M. Itano, D.J. Heinzen, J.J. Bollinger and D.J. Wineland, Phys. Rev. A 41, 2295
(1990).
[15] M. Namiki, S. Pascazio and H. Nakazato, Decoherence and Quantum Measurements,
World Scientific, 1977 sec. 3.1. the references cited in this section.
[16] For a review on Quantum Optics in structured reservoirs, see P. Lambropoulos, G. M.
Nikolopoulos, T. R. Nielsen and S. Bay, Rep. Prog. Phys. 63, 455 (2000).
[17] We have used for simplicity a Taylor series solution of the Schro¨dinger equation. Similar
results are obtained by using (say) the full evolution operator or by density matrix
methods.
[18] A. G. Kofman and G. Kurizki, Nature 405, 546 (2000).
9
FIGURES
v
e
 g
l
t t t t
  N times
0 0 0 0+ τ + 2τ + 2Ντ
v v v v
FIG. 1. Scheme for suppression of an unwanted transition |g〉 → |e〉 caused by a weak
off-resonant perturbation v. The bold arrow represents the 2pi-pulse on the transition |g〉 ↔ |l〉.
The crosses denote the times, when an ultrashort 2pi-pulse is applied. In between the pulses the
system evolves under v.
10
 k
g
 k }{ g k }{ g k }{ g k }{
g f
e
 g
t t t t
  N times
2pi −  pulse
0 0 0 0+ τ + 2τ + 2Ντ
FIG. 2. Scheme for suppression of a decay into continuum with the atom going from |e〉 to |g〉
by the emission of a photon, other specification same as in Fig.1.
11
10 20 30 40 50
0.05
0.07
0.09
0.11
0.13
0
10
20
30
40
50
~p
eg
p
eg
N
FIG. 3. A comparison of p˜ge and pge (in units of A/Γ) as a function of N . The short interval
τ is taken to be one half of the bath correlation time.
10 30 50 70 90
N
0
2000
4000
6000
I
0
1000
2000
3000
4000
I0
ω
egτ = pi
ω
egτ = 1
FIG. 4. A comparison of I (Solid) (Eq.(19)) and Io (dashed) obtained by dropping tan function
in the integrand of Eq.(19) when we introduce a cut-off at x = 1. We show results for ωegτ = 1
and pi.
12


Inhibition of decoherence due to decay in a continuum

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Inhibition of Decoherence due to Decay in a Continuum

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