PHYSICAL REVIEW A 70, 063409 (2004)
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Effect of dephasing on stimulated Raman adiabatic passage
1
2
P. A. Ivanov,1 N. V. Vitanov,1,2 and K. Bergmann3
Department of Physics, Sofia University, James Bourchier 5 blvd, 1164 Sofia, Bulgaria
Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko chaussée 72, 1784 Sofia, Bulgaria
3
Fachbereich Physik der Universität, Erwin-Schrödinger-Str., D-67653 Kaiserslautern, Germany
(Received 19 August 2004; published 15 December 2004)
This work explores the effect of phase relaxation on the population transfer efficiency in stimulated Raman
adiabatic passage (STIRAP). The study is based on the Liouville equation, which is solved analytically in the
adiabatic limit. The transfer efficiency of STIRAP is found to decrease exponentially with the dephasing rate;
this effect is stronger for shorter pulse delays and weaker for larger delays, since the transition time is found
to be inversely proportional to the pulse delay. Moreover, it is found that the transfer efficiency of STIRAP in
the presence of dephasing does not depend on the peak Rabi frequencies at all, as long as they are sufficiently
large to enforce adiabatic evolution; hence increasing the field intensity cannot reduce the dephasing losses. It
is shown also that for any dephasing rate, the final populations of the initial state and the intermediate state are
1
equal. For strong dephasing all three populations tend to 3 .
DOI: 10.1103/PhysRevA.70.063409
PACS number(s): 32.80.Bx, 33.80.Be, 34.70.⫹e, 42.50.Vk
I. INTRODUCTION
Stimulated Raman adiabatic passage (STIRAP) [1–4] is a
simple and powerful technique for complete and robust
population transfer in three-state quantum systems. In this
technique, the population is transferred adiabatically from an
initially populated state 兩␺1典 to a target state 兩␺3典, which are
coupled via an intermediate state 兩␺2典 by two pulsed fields,
pump and Stokes. A unique and very useful feature of
STIRAP is that the intermediate state 兩␺2典, whose presence is
crucial for providing two strongly coupled single-photon
transitions, never gets populated, even transiently. The reason is that throughout the adiabatic evolution of the system
the population remains trapped in an adiabatic dark state
兩␾0共t兲典, which is a superposition of states 兩␺1典 and 兩␺3典 only
and does not involve the intermediate state 兩␺2典. Such a dark
state is formed by maintaining a two-photon resonance between 兩␺1典 and 兩␺3典 during the interaction. If the pulses are
ordered counterintuitively, the Stokes before the pump, then
the dark state is associated with state 兩␺1典 initially and state
兩␺3典 in the end; thus providing an adiabatic route from 兩␺1典 to
兩␺3典.
The robustness of STIRAP against variations in the experimental parameters derives from the adiabatic nature of
the interaction, which is enforced by making the pulse areas
sufficiently large (typically larger than 10␲). The accompanying absence of population in the intermediate state, which
may decay strongly, makes STIRAP largely insensitive to the
properties of this state, which has led to numerous applications of this technique in a variety of fields across quantum
physics, recently reviewed [3,4].
Since the existence of the dark state 兩␾0共t兲典 is vital for
STIRAP, and since this dark state is a coherent superposition
of states 兩␺1典 and 兩␺3典, maintaining coherence is crucial for
STIRAP. In the adiabatic regime, population relaxation (and
the ensuing phase relaxation) occurring from the intermediate state 兩␺2典 as spontaneous emission within the system or
irreversible population loss from 兩␺2典 to levels outside the
1050-2947/2004/70(6)/063409(8)/$22.50
system is not detrimental since state 兩␺2典 never gets populated. However, phase relaxation occurring through other
mechanisms, such as elastic collisions or laser phase fluctuations, may affect STIRAP adversely because such processes
lead to dephasing between all states, including 兩␺1典 and 兩␺3典.
In the adiabatic picture, which we shall use below, this
dephasing shows up as population loss from the dark state.
From another viewpoint, the dephasing reduces the accessible dynamical Hilbert space (as measured by the Bloch
vector length [5]) and thus reduces the maximum possible
population of the target state 兩␺3典.
In the present paper, we study the effect of dephasing
processes, proceeding at a constant rate, on STIRAP. We
derive the adiabatic solution of the Liouville equation in a
very simple analytic form by using various approximations
to reduce the tremendously complex problem to a single
equation for the dynamics of the dark state. Thus this work
complements several earlier theoretical studies of decoherence effects in STIRAP [6–8], which will be discussed in
Sec. V.
This paper is organized as follows. In Sec. II we provide
some background knowledge of STIRAP and define the
problem. We derive the adiabatic solution of the Liouville
equation in Sec. III. We then compare our analytic solution
to numeric simulations in Sec. IV. In Sec. V we discuss the
relation of our results to earlier studies. A summary is presented in Sec. VI.
II. GENERAL BACKGROUND
A. Three-state system
The model three-state ⌳ system is shown in Fig. 1. The
initial state 兩␺1典 and the intermediate state 兩␺2典 are coupled
by the pump laser pulse ⍀ p共t兲 while state 兩␺2典 and the target
final state 兩␺3典 are coupled by the Stokes laser pulse ⍀s共t兲.
The direct transition between states 兩␺1典 and 兩␺3典 is electricdipole forbidden. Two-photon resonance between states 兩␺1典
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©2004 The American Physical Society
PHYSICAL REVIEW A 70, 063409 (2004)
IVANOV, VITANOV, AND BERGMANN
兩␾+共t兲典 = sin ␽共t兲 sin ␸共t兲兩␺1典 + cos ␸共t兲兩␺2典
+ cos ␽共t兲 sin ␸共t兲兩␺3典,
共3a兲
兩␾0共t兲典 = cos ␽共t兲兩␺1典 − sin ␽共t兲兩␺3典,
共3b兲
兩␾−共t兲典 = sin ␽共t兲 cos ␸共t兲兩␺1典 − sin ␸共t兲兩␺2典
+ cos ␽共t兲 cos ␸共t兲兩␺3典,
共3c兲
where the mixing angles ␽共t兲 and ␸共t兲 are defined by
tan ␽共t兲 =
FIG. 1. The three-state ⌳ system studied in this paper. States
兩␺1典 and 兩␺2典 are coupled by the pump laser pulse ⍀ p共t兲 while states
兩␺2典 and 兩␺3典 are coupled by the Stokes laser pulse ⍀s共t兲. The direct
transition between states 兩␺1典 and 兩␺3典 is electric-dipole forbidden.
Two-photon resonance between states 兩␺1典 and 兩␺3典 is maintained
while state 兩␺2典 can be off resonance by a certain detuning ⌬. The
wavy lines depict dephasing between each pair of states.
冤
0
1
H共t兲 = ប ⍀ p共t兲
2
0
1
⍀ p共t兲
2
⌬共t兲
1
⍀s共t兲
2
0
冥
1
⍀s共t兲 .
2
0
⍀共t兲
,
⌬共t兲
共4b兲
⍀共t兲 = 冑⍀2p共t兲 + ⍀s2共t兲.
共4c兲
The eigenvalues of H共t兲 (the adiabatic energies) are
共1兲
Here the column vector c共t兲 = 关c1共t兲 , c2共t兲 , c3共t兲兴T comprises
the probability amplitudes of the three states and the respective populations are 兩cn共t兲兩2 共n = 1 , 2 , 3兲. In the rotating-wave
approximation, the Hamiltonian is given by [1–4]
共4a兲
tan 2␸共t兲 =
and 兩␺3典 is maintained while state 兩␺2典 can be off resonance
by a certain detuning ⌬. When the pulse durations are short
compared with the relaxation times in the system the quantum dynamics of the ⌳ system is described by the
Schrödinger equation [5]
d
iប c共t兲 = H共t兲c共t兲.
dt
⍀ p共t兲
,
⍀s共t兲
ប
ប␧+共t兲 = ⍀共t兲 cot ␸共t兲,
2
共5a兲
ប␧0共t兲 = 0,
共5b兲
ប
ប␧−共t兲 = − ⍀共t兲 tan ␸共t兲.
2
共5c兲
The eigenvectors (3) form the orthogonal 共W−1 = WT兲 rotation matrix
W共␽, ␸兲 =
冤
sin ␽ sin ␸
cos ␽
sin ␽ cos ␸
cos ␸
0
− sin ␸
cos ␽ sin ␸ − sin ␽ cos ␽ cos ␸
冥
,
共6兲
which diagonalizes the Hamiltonian (2),
WTHW = diag共ប␧+,ប␧0,ប␧−兲 = Ha .
共2兲
The functions ⍀ p共t兲 and ⍀s共t兲 represent the Rabi frequencies
of the pump and Stokes pulses, respectively, and each of
them is proportional to the electric-field amplitude of the
respective laser field and the corresponding transition dipole
moment, ⍀ p共t兲 = −d12 · E p共t兲 / ប and ⍀s共t兲 = −d32 · Es共t兲 / ប.
Without loss of generality both ⍀ p共t兲 and ⍀s共t兲 will be assumed positive as the populations do not depend on their
signs.
Hereafter for simplicity we drop the time variable, unless
confusion is to be avoided.
In terms of W the relations between the diabatic and adiabatic states read (m = 1 , 2 , 3; ␮ = + , 0 , −)
兩 ␺ m典 =
兺␮ Wm␮共␽, ␸兲兩␾␮典,
共8a兲
兩 ␾ ␮典 =
兺m Wm␮共␽, ␸兲兩␺m典.
共8b兲
In STIRAP the pulses are applied in the counterintuitive
order, that is the Stokes pulse ⍀s共t兲 precedes the pump pulse
⍀ p共t兲, although they overlap partly. Hence
B. STIRAP in the absence of decoherence
In the completely coherent regime, the population transfer
mechanism of STIRAP is revealed by studying the instantaneous eigenstates of H共t兲—the adiabatic states—which are
time-dependent superpositions of the bare (unperturbed)
states [3,4,9],
共7兲
⍀ p共t兲
= 0,
t→−⬁ ⍀s共t兲
lim
⍀ p共t兲
= ⬁,
t→+⬁ ⍀s共t兲
lim
共9兲
which implies that ␽共−⬁兲 = 0, ␽共+⬁兲 = ␲ / 2. STIRAP leans
on the adiabatic state 兩␾0共t兲典, Eq. (3b), which involves the
bare states 兩␺1典 and 兩␺3典 only; hence the term dark (or
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EFFECT OF DEPHASING ON STIMULATED RAMAN…
trapped) state. For the STIRAP pulse sequence (9), the dark
state 兩␾0共t兲典 is equal to state 兩␺1典 as t → −⬁ and to state −兩␺3典
as t → + ⬁. For adiabatic excitation the system remains in
兩␾0共t兲典 at all times and the population is transferred completely to state 兩␺3典.
Quite a remarkable and unique feature of STIRAP is that
during the population transfer the intermediate state 兩␺2典 remains completely unpopulated because the dark state 兩␾0共t兲典
does not involve it. Hence its properties, including population loss from this state or single-photon detuning, have no
effect on the population transfer, as long as adiabatic conditions are maintained.
It has been shown, though, that the intermediate-state detuning and loss rate can affect STIRAP because they affect
the adiabaticity condition [10,11]. However, as long as the
laser fields are sufficiently intense to make the pulse areas
sufficiently large, adiabatic evolution can be enforced upon
the system. We shall therefore assume for simplicity a singlephoton resonance,
⌬ = 0;
III. STIRAP AMIDST DEPHASING
A. Liouville equation
1. Diabatic basis
The effect of dephasing processes on quantum dynamics
is modeled by introducing phenomenological decay terms
into the Liouville equation,
d
iប ␳ = 关H, ␳兴 + D,
dt
共11兲
冤
冥
␥12␳12 ␥13␳13
0
␥23␳23 ,
0
␥32␳32
共12兲
with ␥mn = ␥nm being the constant dephasing rates. Here ␳mn
= 具␺m兩␳ˆ 兩␺n典, where ␳ˆ is the density operator. Equation (11) is
solved with the initial conditions
␳11共− ⬁兲 = 1,
␳mn共− ⬁兲 = 0
␳a = WT␳W.
共mn ⫽ 11兲,
共13兲
and the objective is to derive the populations ␳mm共t兲, and
particularly their values in the end, ␳mm共+⬁兲.
共14兲
Here ␳ = 兵␳␮␯其 with ␳␮␯ = 具␾␮兩␳ˆ 兩␾␯典 共␮ , ␯ = + , 0 , −兲. The
Liouville equation in the adiabatic basis reads
a
iប␳˙ a = 关Ha, ␳a兴 − iប关WTẆ, ␳a兴 + WTDW,
共15兲
a
where the overdots denote time derivatives and H is given
by Eq. (7). Equation (15) is solved with the initial conditions
␳00共− ⬁兲 = 1,
␳␮␯共− ⬁兲 = 0
共␮␯ ⫽ 00兲.
共16兲
B. Approximations
1. Adiabatic evolution
Since we seek the effect of dephasing on otherwise perfect STIRAP, we assume that the evolution in the absence of
dephasing is perfectly adiabatic, the conditions for which are
derived in the Appendix. We hence neglect the second term
on the right-hand side of Eq. (15), −iប关WTẆ , ␳a兴, which
contains the nonadiabatic couplings, and obtain the reduced
Liouville equation
iប␳˙ a = 关Ha, ␳a兴 + WTDW.
共17兲
This equation still involves nine coupled differential equations, whose exact analytical solution is impossible. One can
reduce the number of equations by using some properties of
the density matrix (e.g., Tr ␳a = 1 and ␳+0 = ␳0−, which can be
shown easily [12]), but the problem remains still unsoluble.
The adiabatic representation, however, allows to make some
approximations, which reduce the problem considerably.
2. Weak dephasing
In addition to adiabatic evolution, we shall also assume
that the dephasing rates ␥mn are much smaller than the peak
Rabi frequency ⍀0,
␥mn Ⰶ ⍀0
where the matrix D is responsible for the dephasing,
D = − iប ␥21␳21
␥31␳31
It is convenient to work in the adiabatic basis (3) where
the loss of transfer efficiency due to dephasing shows up as
population decay from the dark state, while in the original
diabatic basis the treatment is more complex. The density
matrices in the diabatic and adiabatic bases ␳ and ␳a are
connected by the rotation matrix (6) as
共10兲
then ␸ = ␲ / 4, which simplifies much of the algebra.
In contrast to its robustness against population loss from
state 兩␺2典, STIRAP is expected to be vulnerable to decoherence, because the dark state, which is the population transfer
vehicle, is a coherent superposition of states 兩␺1典 and 兩␺3典.
Decoherence may depopulate this state and reduce the transfer efficiency.
In the next section, we present a quantitative analysis of
the effect of decoherence on STIRAP in the form of pure
dephasing by deriving the adiabatic solution to the Liouville
equation.
0
2. Adiabatic basis
共m,n = 1,2,3;m ⫽ n兲.
共18兲
This assumption is well justified: if ␥mn ⲏ ⍀0 then, since
⍀0T Ⰷ 1 [the adiabatic condition (A4)], the relation ␥mnT
Ⰷ 1 will apply, which means that the dynamics will be completely incoherent and governed by rate equations, with
maximum transfer efficiency of 31 [5].
With the weak-dephasing condition (18), the adiabatic basis has a major advantage: the coherences ␳0+, ␳0−, and ␳+−
between the adiabatic states remain negligible throughout the
excitation and may therefore be neglected. In contrast, for
instance, in the original diabatic basis the coherence ␳13 may
reach significant values during the adiabatic population
transfer (as will be shown in Sec. IV) and has to be accounted for.
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PHYSICAL REVIEW A 70, 063409 (2004)
IVANOV, VITANOV, AND BERGMANN
In order to justify this assertion, we first point out that in
the absence of dephasing, the adiabatic coherences ␳␮␯共t兲 in
the adiabatic limit remain zero at all times, merely because
the adiabatic solution reads ␳00共t兲 = 1. With dephasing, these
coherences are nonzero but remain very small if condition
(18) holds. Indeed, since
冤
0
ប
关Ha, ␳a兴 = ⍀ − ␳0+
2
− 2␳−+
冥
␳+0 2␳+−
0
␳0− ,
− ␳−0 0
1
␳˙ 00 = ␥13 sin2 2␽共1 − 3␳00兲.
4
C. Adiabatic solution
1. Populations in the adiabatic basis
In order to determine the populations of the adiabatic
states, we shall first show that
Since w+−共−⬁兲 = 0, which follows from Eq. (16), Eq. (21) has
only the trivial solution w+−共t兲 = 0, which gives immediately
Eq. (20). In fact, this relation remains valid even without the
weak-dephasing condition (18) if ␥12 = ␥23.
Since Tr ␳ = ␳00 + ␳++ + ␳−− = 1, it is sufficient to derive just
one adiabatic population and we choose the dark-state one,
␳00. From the Liouville equation (17), using Eqs. (6), (8), and
(12), we find
␳˙ 00 = ␥13 sin 2␽ Re ␳13 .
␳++共t兲 = ␳−−共t兲 =
1
1
Re ␳13 = sin 2␽共1 − 3␳00兲 + sin 2␽ Re ␳+−
4
2
+
1
冑2 cos 2␽ Re共␳+0 + ␳0−兲.
␩共t兲 =
共25b兲
3
4
冕
t
sin2 2␽共t⬘兲dt⬘ .
共26兲
−⬁
Equations (25a) and (25b) provide the time-dependent
adiabatic solution for the populations of the adiabatic states
兩␾0典, 兩␾+典, 兩␾−典.
2. Populations in the original basis
The populations of the original, bare states 兩␺1典, 兩␺2典, 兩␺3典
can be determined from the populations of the adiabatic
states by using Eq. (14). After neglecting for consistency the
coherences between the adiabatic states, the populations of
the original states are
␳11共t兲 = ␳++共t兲 sin2 ␽共t兲 + ␳00共t兲 cos2 ␽共t兲,
共27a兲
␳22共t兲 = ␳++共t兲,
共27b兲
␳33共t兲 = ␳++共t兲 cos2 ␽共t兲 + ␳00共t兲 sin2 ␽共t兲.
共27c兲
One can find also the coherence ␳13共t兲 from Eq. (23) [it can
be shown that Im ␳13共t兲 = 0)],
␳13共t兲 = −
1
sin 2␽共t兲e−␥13␩共t兲 .
2
共27d兲
After the interaction, since ␽共⬁兲 = ␲ / 2, the populations are
␳11共⬁兲 = ␳22共⬁兲 = ␳++共⬁兲, ␳33共⬁兲 = ␳00共⬁兲, or
␳11共⬁兲 = ␳22共⬁兲 =
␳33共⬁兲 =
1 1 −␥ ␩共⬁兲
− e 13 ,
3 3
1 2 −␥ ␩共⬁兲
+ e 13 ,
3 3
共28a兲
共28b兲
and ␳13共⬁兲 = 0. Hence in the adiabatic limit the populations in
STIRAP depend only on the dephasing rate ␥13 between
states 兩␺1典 and 兩␺3典 but not on ␥12 and ␥23.
共23兲
As explained in Sec. III B, the coherences between the adiabatic states remain negligibly small at all times; we therefore
1 1 −␥ ␩共t兲
− e 13 ,
3 3
where
共22兲
Hence the population loss from the dark state occurs due to
the dephasing between states 兩␺1典 and 兩␺3典. By using Eqs.
(14) and (6) we can express Re ␳13 in terms of the adiabatic
density matrix as
共25a兲
and from here the solution for ␳++共t兲 and ␳−−共t兲 is
共20兲
共21兲
1 2 −␥ ␩共t兲
+ e 13 ,
3 3
␳00共t兲 =
at all times. Indeed, from the Liouville equation (17) we
derive the equation for the inversion w+−共t兲 = ␳++共t兲 − ␳−−共t兲,
ẇ+− = − 共␥12 sin2 ␽ + ␥23 cos2 ␽兲w+− .
共24兲
The solution for the initial condition (16) reads
共19兲
in the equation for each ␳˙ ␮␯ 共␮ ⫽ ␯兲 that derives from the
Liouville equation (17), the coefficient multiplying ␳␮␯ is
± 21 ⍀ or ±⍀. All other coefficients in this respective equation
for ␳˙ ␮␯ derive from the second term WTDW in Eq. (17) and
are proportional to and of the order of ␥mn 共m , n = 1 , 2 , 3兲, as
is evident from Eqs. (12) and (6). Since ⍀ is large compared
to both the pulse bandwidth 1 / T and the dephasing rates
␥mn 共m , n = 1 , 2 , 3兲, and since the coherences are zero initially, we conclude that their values remain negligible
throughout the interaction. From a mathematical point of
view, this situation is the same as for a state, which is weakly
coupled to other states by far off-resonant fields: the transition probability to and from this state is negligibly small.
␳++共t兲 = ␳−−共t兲
neglect the terms with ␳+−, ␳+0 and ␳0− in Eq. (23). Hence
Eq. (22) reduces to
D. Limiting cases
It is easy to verify that the adiabatic solution (28) has the
correct limits for weak and strong dephasing. Indeed, in the
completely coherent case, Eqs. (28) reduce to
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EFFECT OF DEPHASING ON STIMULATED RAMAN…
␳11共⬁兲 = ␳22共⬁兲 ——→ 0,
␳33共⬁兲 ——→ 1,
␥13→0
共29兲
␥13→0
and we recover the result for STIRAP. In the incoherent limit
we have
␳mm共⬁兲 ——→
␥13→⬁
1
3
共m = 1,2,3兲,
共30兲
which corresponds to the rate-equations result [5], with a
null Bloch vector.
E. Example: Gaussian pulses
For Gaussian pulses, with characteristic widths T, peak
Rabi frequencies ⍀0, and delay ␶,
⍀ p共t兲 = ⍀0e−共t − ␶/2兲
2/T2
,
⍀s共t兲 = ⍀0e−共t + ␶/2兲
2/T2
, 共31兲
increases), it is not important here since we have assumed
that the evolution is perfectly adiabatic, i.e., that ⍀0 is sufficiently large. Third, the least obvious result in Eqs. (34) is
the dependence on the pulse width T and the pulse delay ␶.
Whereas it is expected that the populations will decay faster
towards the incoherent limit (30) as the pulse duration T
increases (because then the dephasing acts for a longer time),
the quadratic dependence on T is not obvious. More puzzling
is the inverse dependence on the pulse delay ␶: one may
naively think that as ␶ increases, the dephasing will have
more time to destroy coherence since the interaction time
increases; instead, as ␶ increases, the populations tend to
their STIRAP values (29), rather than to the incoherent limit
(30).
The key to understand this feature is the transition time
for STIRAP, i.e., the time it takes for the transition to be
completed, which is calculated below.
the mixing angle ␽共t兲 is
2. Transition time in STIRAP
2
␽共t兲 = arctan e2␶t/T .
The integral (26) can be calculated exactly,
␩共t兲 =
3
4
冕
t
sech2
−⬁
冉
共32兲
冊
2␶t⬘
3T2
2␶t
dt
=
tanh 2 + 1 ,
⬘
2
T
8␶
T
As evident from Eqs. (3b) and (32), in the adiabatic limit
and in the absence of dephasing the population of state 兩␺3典
evolves (for Gaussian pulses) as
␳33共t兲 = sin2 ␽共t兲 =
共33a兲
␩共⬁兲 =
2
3T
.
4␶
共33b兲
␳33共⬁兲 =
1 1 −3␥ T2/4␶
− e 13
,
3 3
1 2 −3␥ T2/4␶
+ e 13
.
3 3
1 + e−4␶t/T
TSTIRAP = t0.9 − t0.1 =
共34a兲
共34b兲
Equations (34) provide the adiabatic solution for the populations of the original diabatic states for Gaussian pulses (31)
at the end of the interaction. Equations (27), along with Eqs.
(25a), (25b), (32), and (33), provide even the time evolutions
of these populations.
共35兲
.
T2
ln 3.
␶
共36兲
Had we defined the transition time differently [e.g., as the
time during which ␳33共t兲 rises from ⑀ to 1 − ⑀, ⑀ being a small
number], the result would be the same, apart from a different
numerical factor. This result defies the naive expectation that
the transition time is proportional to the interaction time,
which is ␶ + 2T, and emphasizes the difference between interaction time and transition time.
We can write the adiabatic solution for Gaussian pulses
(34) as
␳11共⬁兲 = ␳22共⬁兲 =
F. Discussion
␳33共⬁兲 =
1. General observations
The adiabatic solution (34) for Gaussian pulses (31), as
well as the general solution (28), reveal several interesting
features. First of all, the populations depend only on the
dephasing rate ␥13 between the initial and final states 兩␺1典
and 兩␺3典, but not on ␥12 and ␥23, as long as condition (18) is
satisfied. The populations depend on ␥13 in an exponential
fashion, which is indeed usually expected to be the case. The
extent of this dependence (the factor ␩ multiplying ␥13),
however, is not obvious. Second, the adiabatic solution does
not depend on the Rabi frequency ⍀0 at all, which may be a
little unexpected. However, while ⍀0 is instrumental in
achieving adiabatic evolution (adiabaticity increases as ⍀0
2
We define the transition time TSTIRAP as the time during
which ␳33共t兲 rises from 0.1 to 0.9; this gives
The adiabatic solution (28) is therefore
␳11共⬁兲 = ␳22共⬁兲 =
1
1 1 −␣␥ T
− e 13 STIRAP ,
3 3
1 2 −␣␥ T
+ e 13 STIRAP ,
3 3
共37a兲
共37b兲
where ␣ = 3 / 共4 ln 3兲 ⬇ 0.683. Displayed in this form, the
adiabatic solution confirms the conventional wisdom that the
loss of efficiency should depend exponentially on the
dephasing rate and the time during which it acts, i.e., the
transition time TSTIRAP.
IV. COMPARISON WITH NUMERICAL RESULTS
We have examined the validity of the adiabatic solutions
(25a), (25b), and (27) by comparison with exact numerical
solution of the Liouville equation (11) for Gaussian pulse
shapes (31).
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IVANOV, VITANOV, AND BERGMANN
FIG. 2. Time evolution of the elements of the density matrix in
the adiabatic (upper frame) and diabatic (lower frame) bases for
Gaussian pulse shapes (31). The dephasing rate is ␥12 = ␥23 = ␥13
= 2T−1, the peak Rabi frequency ⍀0 = 50T−1 and the delay ␶ = 1.5T.
The solid curves derive from numeric solution of the Liouville
equation (11), whereas dashed curves (barely discernible) show the
adiabatic solutions (25) and (27).
In Fig. 2 the time evolutions of the elements of the density
matrix ␳ in the adiabatic (upper frame) and diabatic (lower
frame) bases are plotted for interaction parameter values
which ensure nearly perfect adiabatic conditions. The respective adiabatic solutions (25) and (27) coincide almost completely with the numerical results. The figure also shows that
the adiabatic coherences remain very small, whereas the diabatic coherence ␳13 can reach significant values.
Figure 3 shows the final populations of the three bare
states against the dephasing rate ␥13 for interaction parameters chosen to ensure adiabatic evolution. The adiabatic solution (34) is indescernible from the exact values. As predicted, for ␥13 → 0 the population is transferred entirely to
state 兩␺3典: there is perfect STIRAP. As ␥13 increases the
FIG. 3. Final populations of the diabatic states against the
dephasing rate ␥ = ␥12 = ␥23 = ␥13 for Gaussian pulse shapes (31)
with peak Rabi frequency ⍀0 = 50T−1 and delay ␶ = 1.5T. The dots
show numeric results and the solid curves the adiabatic solution
(34).
FIG. 4. Final population of state 兩␺3典 plotted against the pulse
delay ␶ for several values of the peak Rabi frequency ⍀0 (denoted
on the respective curves with numbers in units T−1) for Gaussian
pulse shapes (31). The dephasing rates are equal, ␥12 = ␥23 = ␥13 = ␥,
with ␥ = 0 (upper frame) and ␥ = T−1 (lower frame). The full curves
are numeric simulatons and the dashed curve in the lower frame
shows the adiabatic solution (34). The small arrows show the upper
bounds on the pulse delay (A7), which are imposed by the adiabatic
condition (A1) and which depend on the peak Rabi frequency ⍀0
(denoted nearby each mark). The lower bound on the delay (A5),
␶ ⲏ 0.41T, is the same for all curves and is not marked.
populations depart from their STIRAP values and tend to 31
already when ␥13 is equal to just a few inverse pulse widths;
hence the weak-dephasing condition (18) is not very restrictive because it is always satisfied in the region of physical
interest, i.e., away from the incoherent limit (30). As predicted by the adiabatic solution (34), the final populations of
states 兩␺1典 and 兩␺2典 in the adiabatic limit remain equal for
any ␥13.
In Fig. 4 the final population of state 兩␺3典 is plotted as a
function of the pulse delay ␶ for several different values of
the peak Rabi frequency ⍀0 in two cases: without (upper
frame) and with dephasing (lower frame). As predicted by
the adiabatic solution (34), in the presence of dephasing the
population ␳33 increases as ␶ increases, as long as adiabaticity is maintained. As the delay ␶ increases beyond certain
values adiabaticity breaks down and ␳33 decreases rapidly,
departing from the adiabatic solution (34). Insofar as adiabaticity depends on both ⍀0 and ␶, the adiabaticity range is
different for the different values of ⍀0 and increases with ⍀0.
The lower and upper bounds on the pulse delay (A5) and
(A7) imposed by the adiabatic condition (A1) and derived in
the Appendix, are seen to describe well the adiabatic region.
Within the adiabatic range there is a very good agreement
between the adiabatic solution (34) and the numerical results.
In Fig. 5 the final populations are plotted against the peak
Rabi frequency ⍀0. As ⍀0 increases the adiabaticity improves and the populations eventually reach steady values
described very accurately by the adiabatic solution (34); this
feature is reminiscent to the steady approach of perfect efficiency in the absence of dephasing (upper frame). Hence the
adiabatic solution in the presence of dephasing indeed does
not depend on the peak Rabi frequency ⍀0. Note that, ac-
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EFFECT OF DEPHASING ON STIMULATED RAMAN…
VI. CONCLUSIONS
FIG. 5. Final populations plotted against the peak Rabi frequency ⍀0 for delay ␶ = T for Gaussian pulse shapes (31). The
dephasing rates are equal, ␥12 = ␥23 = ␥13 = ␥, with ␥ = 0 (upper
frame) and ␥ = T−1 (lower frame). The full curves are numeric simulatons and the dashed lines in the lower frame show the adiabatic
solution (34).
cording to the adiabatic condition (A6), the adiabatic regime
appears for ⍀0T ⲏ 4, in good agreement with Fig. 5.
V. COMPARISON WITH EARLIER STUDIES
This work complements three earlier theoretical studies of
decoherence effects in STIRAP, which have been focused on
the modeling of decoherence and have presented mainly numerical simulations. Demirplak and Rice [7] have considered
the prospect of using STIRAP in liquid solutions by assuming that the three-state system is coupled to a clasical bath.
They have used stochastic simulations with extensive averaging over solutions of the Schrödinger equation. Later, Shi
and Geva [8] assumed that the bath is quantum and carried
out numerical simulations using a quantum master equation.
Yatsenko et al. [6] have considered the effect of correlated
laser phase fluctuations, when the pump and Stokes pulses
derive from the same laser. Staying within the Schrödinger
equation treatment, they have shown that these fluctuations
introduce an effective two-photon detuning, which reduces
the transfer efficiency. Along with numerical simulations
they have derived some analytic formulas using a perturbative approach. The latter have been derived under the assumption that the intermediate state 兩␺2典 decays irreversibly
outside the system, which has enabled various approximations.
We emphasize that while all these earlier studies have
confirmed the detrimental effect of dephasing on STIRAP,
the present work is the first to derive the adiabatic solution of
the Liouville equation, to retrieve correctly the incoherent
limit (30), to predict the inverse dependence of the dephasing
losses on the pulse delay, and to find the independence of
these losses on the peak Rabi frequency. All these features
have been confirmed numerically, in excellent agreement
with the analytical solution.
In this paper we have explored the effect of dephasing on
the population dynamics in STIRAP. We have derived the
adiabatic solution of the Liouville equation, which has been
verified by comparison with numerical simulations to provide a very accurate approximation to the populations. In the
adiabatic limit the populations depend only on the dephasing
rate ␥13 between states 兩␺1典 and 兩␺3典 but not on ␥12 and ␥23.
The population of the final state 兩␺3典, which is unity in the
adiabatic limit in the absence of dephasing, decreases exponentially as the dephasing rate increases. In the limit of very
strong dephasing, all three populations tend to 31 , forming a
completely incoherent superposition with a zero Bloch vector length. Interestingly, for any dephasing rate, the populations of the initial state 兩␺1典 and the intermediate state 兩␺2典
remain equal in the adiabatic limit.
Our adiabatic solution reveals some other interesting and
probably unexpected features. It shows that the dephasing
losses decrease as the pulse delay increases. This has been
explained by the fact that the transition time TSTIRAP is inversely proportional to the pulse delay. Furthermore, the
adiabatic solution does not depend on the Rabi frequencies
of the pump and Stokes pulses at all. Hence the population
transfer efficiency cannot be improved by increasing the
pulse intensity, since the dephasing losses depend only on the
dephasing rate and the transition time TSTIRAP. One can reduce these losses by reducing the transition time TSTIRAP,
either by reducing the pulse width T or/and increasing the
pulse delay ␶, as long as adiabaticity is maintained.
ACKNOWLEDGMENTS
This work has been supported by the EU Research and
Training Network QUACS, Contract No. HPRN-CT-200200309 and by Deutsche Forschungsgemeinschaft. P.A.I. acknowledges support from the EU Marie Curie Training Site
Project No. HPMT-CT-2001-00294. N.V.V. acknowledges financial support from the Alexander von Humboldt Foundation.
APPENDIX: ADIABATIC CONDITION
For the sake of convenience we summarize below the
restrictions on the interaction parameters required to enforce
adiabatic evolution in STIRAP. The condition for adiabatic
evolution for single-photon resonance, in the absence of decoherence, is [1–4]
1 ˙
1
冑2 兩␽共t兲兩 Ⰶ 2 ⍀共t兲.
共A1兲
For Gaussian pulse shapes (31) it reads
1
␶
−␶2/4T2−t2/T2冑
cosh 2␶t/T2 .
2
2 Ⰶ ⍀ 0e
T cosh 2␶t/T
共A2兲
1. Global adiabatic condition
The global adiabatic condition is obtained by integrating
condition (A2), giving
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PHYSICAL REVIEW A 70, 063409 (2004)
IVANOV, VITANOV, AND BERGMANN
1
␲
Ⰶ
冑2 ⍀0Tf共␶/T兲,
2
共A3兲
where
f共␶/T兲 =
冕
⬁
冑e−2共x − ␶/2T兲
2
2
+ e−2共x + ␶/2T兲 dx
−⬁
is a slowly changing function, increasing monotonically
from 冑␲ for ␶ / T = 0 to 冑2␲ for ␶ / T Ⰷ 1. Hence the adiabaticity condition can be written in the usual form
⍀0T Ⰷ 1.
共A4兲
2. Local adiabaticity condition
batic coupling (exponentially). If T␽˙ ⬎ T⍀, i.e., if ␶ ⬍ T共冑2
− 1兲, then there will be two regions, at early and late times,
where there is an appreciable nonadiabatic coupling and
small eigenenergy splitting. This combination can give rise
to nonadiabatic transitions in these two regions, and the intereference between them will lead to partial oscillations in
the populations. Hence we obtain the following rough lower
bound for the pulse delay:
␶ ⲏ T共冑2 − 1兲.
共A5兲
For large ␶ 共␶ / T ⬎ 1兲, the global condition (A4) is less
relevant because then the splitting ⍀共t兲 is a two-peaked function, which has a minimum at t = 0, where the nonadiabatic
˙ 共t兲 has its maximum. Then the adiabatic condition
coupling ␽
can be obtained by integrating the local condition (A2)
within the interval 关−T␽˙ / 2 , T␽˙ / 2兴, which gives ␲ / 2
ⱗ ⍀共0兲T␽˙ , i.e.,
The global condition (A4) does not involve the pulse delay ␶. Upper and lower bounds on ␶ can be obtained by using
the local adiabaticity condition (A2).
The left-hand side of Eq. (A2) is a bell-shaped function
centered at t = 0, with a characteristic width T␽˙ = T2 / ␶ and
area ␲ / 2. The right-hand side has different behavior for
small and large ␶ / T.
For small ␶ 共␶ / T ⬍ 1兲, the global condition (A4) provides
˙ 共t兲
a good criterion for adiabaticity since both the coupling ␽
and the splitting ⍀共t兲 are bell-shaped functions centered at
the same time t = 0. The splitting, however, has a different
characteristic width, T⍀ = 2T + ␶. The adiabaticity can be broken only at large times because the eigenvalue splitting ⍀共t兲
decreases faster (in a Gaussian manner) than the nonadia-
For example, for ⍀0T = 10, 30, 100, 300, 1000, as in Fig. 4,
Eq. (A7) gives respectively ␶ / T ⱗ 1.84, 2.32, 3.00, 3.58, 4.12
(shown by arrows in Fig. 4).
[1] U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, J.
Chem. Phys. 92, 5363 (1990).
[2] K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys.
70, 1003 (1998).
[3] N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann,
Annu. Rev. Phys. Chem. 52, 763 (2001).
[4] N. V. Vitanov, M. Fleischhauer, B. W. Shore, and K. Bergmann, Adv. At., Mol., Opt. Phys. 46, 55 (2001).
[5] B. W. Shore, The Theory of Coherent Atomic Excitation
(Wiley, New York, 1990).
[6] L. P. Yatsenko, V. I. Romanenko, B. W. Shore, and K. Bergmann, Phys. Rev. A 65, 043409 (2002).
[7] M. Demirplak and S. A. Rice, J. Chem. Phys. 116, 8028
(2002).
[8] Q. Shi and E. Geva, J. Chem. Phys. 119, 11773 (2003).
[9] M. P. Fewell, B. W. Shore, and K. Bergmann, Aust. J. Phys.
50, 281 (1997).
[10] N. V. Vitanov and S. Stenholm, Opt. Commun. 135, 394
(1997).
[11] N. V. Vitanov and S. Stenholm, Phys. Rev. A 56, 1463 (1997).
[12] It can be shown from the Liouville equation (15) that the function ␦␳共t兲 = ␳+0共t兲 − ␳0−共t兲 satisfies a homogeneous differential
equation with null initial condition; hence ␦␳共t兲 = 0.
␶ 2 2
⍀0T ⲏ ␲ e␶ /4T ,
T
共A6兲
which defines a lower bound for the peak Rabi frequency ⍀0.
We can solve this inequality for ␶ / T by taking into account
2
2
that as ␶ / T increases the relation e␶ /4T Ⰷ ␶ / T applies. A
simple calculation gives
␶
ⱗ2
T
063409-8
冑
ln
冉
冊
⍀ 0T 1
⍀ 0T
− ln 4 ln
− ¯.
2
␲
2␲
共A7兲