## Abstract

We demonstrate the feasibility of a laser induced complete population transfer to
and from a continuum of states. We study the two-photon dissociation of
*υ* = 28, *J* = 1,…, 10
sodium dimers. We demonstrate that using just a pair of “counter
intuitively” ordered pulses we can dissociate 100% of the molecules
in an ensemble. The scheme is shown to be stable with respect to the initial
choice of rotational level and to fluctuations in the laser frequency and
intensity. We also study the reverse phenomenon of complete population transfer
*from* the continuum. We perform calculations on the
radiative association of Na atoms to form the Na_{2} molecule in
specific vib-rotational states. It is shown that two pulses of 20 nsec duration
and as little as 6 MW/cm^{2} peak power can photoassociate more than 98%
of the atoms within a (pulse and velocity determined) relative effective
distance, to yield Na_{2} molecules in the chosen
*υ* = 28, *J* = 10 vib-rotational
state. This means that given a density of 10^{16} atoms/cm^{3}
and a temperature of 7*K*, a 10Hz pulsed laser source of the
above parameters can convert *half* of all the Na atoms in the
ensemble to *υ* = 28, *J* = 10
Na_{2} molecules within 15 seconds of operation.

© 1999 Optical Society of America

## 1. Introduction

In recent years, a number of schemes for the production of ultracold molecules have been proposed [1–12]. These schemes include buffer-gas loading into magnetic traps [1], sequential cooling of rotation, translation and vibration [2], and far off-resonance trapping (FORT) [4] of molecules. One of the most promising technique is the photoassociation (PA) of cold atoms [5–12], which was recently demonstrated experimentally [10].

In a typical PA scheme, pairs of colliding atoms are radiatively excited to form bound molecules in an excited electronic state. These molecules are either allowed to decay spontaneously to the the ground molecular state [5–10], resulting in a vibrational population distribution, or subjected to a second appropriately tuned laser, which stimulates a bound-bound transition to a single target vibrational level in the ground electronic state [11,12].

In previous work we have developed an exact time-dependent formalism for treating
photodissociation (PD) [13–15] and PA [11] processes using strong pulses. This formalism aims at
extending the stimulated Raman adiabatic passage (STIRAP) technique of Bergmann et
al. [16–22s] to the case of an initial or final continua.
It was shown that enhanced two-photon association of ultracold sodium atoms can be
attained in an efficient way, thereby producing *υ* = 0,
*J* = 0 translationally cold Na_{2} molecules.

In this article we show, by applying the same formalism, that adiabatic passage via
two-photon dissociation of sodium dimers and the reverse two-photon association
process is possible in a molecular beam environment. It is shown that complete
dissociation of specific quantum states, such as the *υ* =
28, *J* = 0 to 10 Na_{2} molecules is achievable using
nanosecond laser pulses no stronger than a few MW/cm^{2}, and that
stimulated Raman photoassociation of sodium atoms in a beam (translational
temperature of 5–10 K) can be utilized for the production of
translationally cold molecules in specific vib-rotational levels. These findings
prove that coherent population transfer is possible, even for a final (or an
initial) continuum state.

The organization of this article is as follows: In section **2** we review
the theory of two-photon dissociation and association by strong laser pulses and in
section **3** we apply this formalism to the simulation of resonantly
enhanced two-photon dissociation of sodium dimers and stimulated two-photon
association of Na atoms.

## 2. Theory of Two Photon Dissociation and Association

#### 2.1 The Slowly Varying Continuum Approximation

We consider a system with two bound molecular states ∣ 1⟩
and ∣ 2⟩, and a continuum of scattering states
∣ *E*, **n**
^{±}) (it is convenient
to use “incoming” scattering states ∣
*E*, **n**
^{-}) for the dissociation
problem and “outgoing” scattering states ∣
*E*, **n**
^{+}) for the
association process), subjected to the combined action of two laser pulses of
central frequencies *ω*
_{1} and
*ω*
_{2}. We assume that
*ω*
_{1} is in near resonance with the
bound-bound transition, ∣ 1⟩ ↔ ∣
2⟩, and that *ω*
_{2} is in near
resonance with the bound-free transition, ∣ 2⟩
↔ ∣*E*,
**n**
^{±}). Depending on the initial state of the
system and on the pulse configuration, molecules in the bound manifold can
dissociate to the continuum, or colliding atoms initialy in the continuum, may
associate to form a bound molecular states. The situation is depicted in Fig. 1 for a Λ-type configuration. Other
configurations such as the ladder system, may be equally treated.

The total Hamiltonian of the system is written as

where *H* is the radiation-free Hamiltonian,
*ϵ*
_{1}(*t*) and
*ϵ*
_{2}(*t*) are
“slowly varying” electric field amplitudes and
${\overrightarrow{\mu}}_{1}$ and
${\overrightarrow{\mu}}_{2}$ are (electronic)
transition dipole operators. The material wave function of the system may be
expanded as,

$$+\sum _{n}\int \mathit{dE}{b}_{E,n}\left(t\right)\mid E,{\mathbf{n}}^{\pm}\u3009\mathrm{exp}\left(\frac{-\mathit{iEt}}{\mathit{\u0127}}\right),$$

where

Substitution of the expansion of Eq. (2) into the time-dependent Schrodinger equation,
*iħ∂*Ψ/*∂t*
= *H*_{tot}
Ψ, and use of the orthogonality of
the ∣1⟩, ∣2⟩ and ∣
*E*, **n**
^{±}) basis states,
results in an (indenumerable) set of first-order differential equations for the
expansion coefficients. In the rotating wave approximation, and neglecting the
low amplitude inter-continuum transitions, this set of equations is of the form,

where *N* is the number of (asymptotically) open channels,

and

and we have assumed for simplicity that
⟨2∣*μ*
_{1}∣1⟩
*ϵ*
_{1}(*t*) and
*ϵ*
_{2}(*t*) are real. In
the above we ignored spontaneous emission from ∣ 2⟩,
assuming that the pulse intensities are such that the stimulated emission rates
are much faster than the spontaneous emission rates.

In order to obtain a unique solution for Eqs. (4)–(6), we need to specify initial conditions, denoted here as,

Substituting the formal solution of Eq. (6)

into Eq. (5), we obtain,

$$-\sum _{\mathbf{n}}\int \mathit{dE}{\int}_{0}^{t}\mathit{dt}\prime {\Omega}_{2,E.\mathbf{n}}\left(t\right){\Omega}_{2,E,\mathbf{n}}^{*}\left(t\prime \right)\mathrm{exp}\left[-i{\Delta}_{E}\left(t-t\prime \right)\right]{b}_{2}\left(t\prime \right),$$

If the molecular continuum is unstructured, as in the present Na-Na system at
threshold energies, where the bound-free dipole matrix-elements vary with energy
by less than 1% over the (nsec) pulse bandwidth, we can invoke the slowly
varying continuum approximation (SVCA) [13–15] and replace the energy-dependent
bound-free dipole-matrix elements at energies spanning the laser profile by
their value at the pulse center, given (in the Λ configuration of Fig. 1) as *E*_{L}
=
*E*
_{2} -
*ħ*
*ω*
_{2},

The use of the SVCA (whose range of validity has been thoroughly researched [14]) greatly simplifies the equations because upon
substitution of Eq. (7) and Eq. (12) into Eq. (11) we can perform the integration over *E*
and *t*′ analytically. We obtain that,

where

The source term *F*
_{2}(*t*) is given as,

where

#### 2.2 The Adiabatic Approximation

Equation (13) and Eq. (4), can be expressed in matrix notation as,

where

and

and the initial conditions are obtained from Eq. (9) and Eq. (17) as,

As a first step towards solving Eq. (16) we can diagonalize the 𝖧 matrix,

thereby defining an *adiabatic* basis set. In the above, the
eigenvalue matrix, *ε̂*, is given as,

The complex-orthogonal eigenvector matrix 𝖴, satisfying the equation,

can be parameterized in the 2 × 2 case in terms of a
*complex* “mixing angle”
*θ* [15],

where

Operating with 𝖴(*t*) on Eq. (16), and defining,

we obtain that,

where

is the “non-adiabatic” coupling matrix. The source-vector 𝗀 is given as,

The adiabatic approximation amounts to ignoring 𝖠. This can be done whenever the rate of change of 𝖴 with time is slow. Equation (27) then becomes,

with the initial condition that

The adiabatic solution for Eq. (30) with the initial conditions given in Eq. (31) is of the form

where

and

#### 2.3 Adiabatic Two-Photon Dissociation

We first consider the case of PD from the bound manifold into the continuum. In
PD the entire population is taken to belong initialy to the bound manifold, i.e.
${b}_{E,\mathbf{m}}^{0}$ =
0 for all *E* and **m**. As a result
${\stackrel{\u0305}{\mu}}_{2}$(*t*)
of Eq. (15) is zero and g(*t*) vanishes
for all times. The adiabatic solution of Eq. (32) now becomes

Using Eq. (26) and Eq. (17), we obtain for the
*b*
_{1}(*t*) and
*b*
_{2}(*t*) coefficients:

$$-\mathrm{sin}\theta \left(t\right)\mathrm{exp}\{i{\int}_{0}^{t}{\epsilon}_{2}\left(t\prime \right)\mathit{dt}\prime \}\left(-\mathrm{sin}\theta \left(0\right){b}_{1}^{0}+\mathrm{cos}\theta \left(0\right){b}_{2}^{0}\right)\}$$

$$+\mathrm{cos}\theta \left(t\right)\mathrm{exp}\{i{\int}_{0}^{t}{\epsilon}_{2}\left(t\prime \right)\mathit{dt}\prime \}\left(-\mathrm{sin}\theta \left(0\right){b}_{1}^{0}+\mathrm{cos}\theta \left(\theta \right){b}_{2}^{0}\right).$$

If only state ∣1⟩ is initialy populated we have that
𝖻_{0} = (1,0) and

$$+\mathrm{sin}\theta \left(t\right)\mathrm{exp}\{i{\int}_{0}^{t}{\epsilon}_{2}\left(t\prime \right)\mathit{dt}\prime \}\mathrm{sin}\theta \left(0\right)\}$$

$$-\mathrm{cos}\theta \left(t\right)\mathrm{exp}\{i{\int}_{0}^{t}{\epsilon}_{2}\left(t\prime \right)\mathit{dt}\prime \}\mathrm{sin}\theta \left(0\right).$$

#### 2.4 Adiabatic Two-Photon Association

In the PA process, the initial conditions are such that 𝖺_{0}
= 0 (the entire population is initialy in the continuum). Hence the adiabatic
solutions of Eq. (32) are of the form,

or, using the definitions of Eq. (26) and Eq. (17),

$$-\mathrm{sin}\theta \left(t\right){\int}_{0}^{t}\mathrm{exp}\{i{\int}_{t\prime}^{t}{\epsilon}_{2}\left(t\prime \prime \right)\mathit{dt}\prime \prime \}{F}_{2}\left(t\prime \right)\mathrm{cos}\theta \left(t\prime \right)\mathit{dt}\prime \},$$

$$+\mathrm{cos}\theta \left(t\right){\int}_{0}^{t}\mathrm{exp}\{i\underset{t\prime}{\overset{t}{\int}}{\epsilon}_{2}\left(t\prime \prime \right)\mathit{dt}\prime \prime \}{F}_{2}\left(t\prime \right)\mathrm{cos}\theta \left(t\prime \right)\mathit{dt}\prime \}.$$

Given *b*
_{2}(*t*), the (channel specific)
continuum coefficients
*b*
_{E,n}(*t*)
are obtained directly via Eq. (10).

## 3. Numerical Results

#### 3.1 Photodissociation of Na_{2} Molecules

The formalism of section **2** enables an easy computation of PD and PA
processes. In this section, we study the pulsed two-photon dissociation of
Na_{2} molecules in characteristic molecular-beam conditions. In
order to perform the calculation, the transition-dipole matrix elements of Eq. (7), obtained by solving the radial Schrödinger
equation with known [23] Na_{2} potential curves, need be computed only
once for all pulse configurations.

We consider the pulsed PD of molecules in the
*X*
^{1}${\sum}_{g}^{+}$ (*υ* = 28,*J*) state with
*J* in the range of 0 to 10, to the (*E*, 3** s** +3

**) continuum, with the bound**

*s**A*

^{1}${\sum}_{u}^{+}$ (

*υ*′ = 37,

*J*+1) state acting as an intermediate resonance. Given the

*ab-initio*[23] electronic dipole-moments and potential curves of Fig. 2, the bound eigenfunctions and eigenenergies are obtained using the renormalized Numerov method [24]. The continuum wavefunctions are expressed, to excellent accuracy, in terms of the Uniform Airy functions [25–29]. The overlap integrals between the bound states are calculated using Simpson quadrature. The calculation of the bound-continuum matrix elements is performed with a high-order Gauss-Legendre quadrature.

Bound-bound and bound-continuum transition-dipole matrix elements for various
choices of *υ*′ are plotted in Fig. 3. Choosing
*υ*′ = 37 for the intermediate state
clearly maximizes the bound-free transition probability without compromising the
bound-bound transitions.

Since the pulses used in our simulations typically last 5–10 nsec,
their small bandwidth allows for the resolution of individual rotational levels.
Transition-dipole matrix elements for *J* in the range of 0 to 10
are plotted in Fig. 4. Both vibrational states, which lie well below
their respective dissociation thresholds, are hardly affected by the centrifugal
barrier. As a result, the variation in the bound-bound transition matrix
elements with *J* is less than 1%.

In contrast, the radial continuum wavefunctions are more sensitive to the
rotational quantum number, resulting in the variation of the bound-free dipole
matrix elements with *J* by as much as 5%. These small variations
are found, however, to have only a marginal effect on the overall population
transfer probabilities, which are very insensitive to changes in the Rabi
frequencies Ω_{1} and Ω_{2}.

Having computed all the input matrix elements, the dynamical equations are solved
using either the Runge Kutta Merson (RKM) algorithm for direct integration of
the full non-adiabatic equation (Eq. (16)) or the adiabatic solutions of Eqs. (38) and (39). For pulse parameters of relevance to this work, the
adiabatic solutions are found to be practically
*indistinguishable* from the numerically-exact RKM solutions.

The results of a PD process conducted in a
“counter-intuitive” fashion, in which the
“dump”
*ϵ*
_{1}(*t*) pulse is applied
before the “pump”
*ϵ*
_{2}(*t*) pulse, are
shown in Fig. 5. As shown by Bergmann et al. [16–22] for bound-bound Λ-type
systems, this configuration enables complete population transfer from the
initial state to the final state without ever populating the
intermediate-resonance. In Fig. 5 we show, in agreement with our previous work on
two-photon dissociation [15], that, with judicious choice of pulse parameters, a
“counter-intuitive” pulse sequence is capable of
dissociating every molecule in our ensemble, while keeping at all times the
intermediate state population low. In this way, losses due to spontaneous
emission from the intermediate state are avoided. Thus, although not following a
perfect adiabatic *passage* scenario, population transfer to the
continuum is nevertheless adiabatic. [As pointed out above, the adiabatic
solutions of Eq. (16) (Eq. (38) and Eq. (39)) are in perfect agreement with the RKM solutions].

In Fig. 6 the calculated dissociation probability is plotted
as a function of the Δ_{1} detuning, at three pulse
intensities. The resulting symmetric lineshapes have widths that increase with
pulse intensity. This power-broadening is related to the saturation of the
bound-continuum transition at high intensities. The lineshapes of Fig. 6 are smooth functions of the detuning and the pulse
intensity. We see that at large enough intensities, the two-photon dissociation
yield of the present scheme is stable with respect to fluctuations in laser
frequency and power.

#### 3.2 Photoassociation of a Coherent Na+Na Wavepacket

We now turn to the reverse process: the pulsed photoassociation (PA) of a coherent wavepacket of cold Na atoms. In this process the initial wavefunction is described by a moving Gaussian wave packet composed of of radial waves:

where

and *t*
_{0} is the peak time of the Na+Na wave
packet (i.e. the time of maximum overlap with the ∣ 2⟩
state). In our simulations we have chosen the mean initial collision energies to
be *E*
_{0} = 1 - 10*K* and wave packet
widths *δ*_{E}
= 10^{-4} -
10^{-3}cm^{-1}. Radial waves with *J* in the
range of 0 to 10 were considered, keeping in mind that individual rotational
transitions could be resolved due to the energetic-narrowness of the wavepacket
and laser profiles.

As depicted in Fig. 1, the combined effect of the two laser pulses of
central frequencies *ω*
_{2} and
*ω*
_{1} (taken to be in resonance with the
*X*
^{1}${\sum}_{g}^{+}$ (*υ* = 28,*J*) to
*A*
^{1}${\sum}_{u}^{+}$ (*υ*′ = 37,
*J*+1) transition), is the transfer of population from
the continuum to a single vib-rotational state
*X*
^{1}${\sum}_{g}^{+}$ (*υ* = 28, *J*), with the bound *A*
^{1}${\mathrm{\Sigma}}_{u}^{+}$
(*υ*′ = 37, *J*+1) state
acting as an intermediate resonance.

It was shown in subsection 3.2 (Fig. 4) that the material transition-matrix elements of
Eq. (7) are only slightly affected by the choice of initial
radial wavefunctions. Therefore, the final PD population transfers are also
insensitive to the initial state. We now examine the stability of the PA
probabilities with respect to the initial state, i.e., the translational
temperature of the atomic ensemble. Free-bound transition-dipole matrix elements
for translational temperatures typical to the relative lateral motion within a
Na beam of 1–10K, are plotted in Fig. 7. We find, as in the bound-bound case, that the
variation of the free-bound dipole-matrix elements with collision energy, which
over the 1–10K range of temperatures is of the order of 5%, has
almost no effect on the population transfer efficiencies. In addition, as
demonstrated in Fig. 7, the
*μ*
_{2,E} bound-continuum
matrix elements are practically constant over the pulse spectral bandwidth
(typically in the range of 5–10 mK), thus justifying the use of the
SVCA of Eq. (12).

Making use of the SVCA, we rewrite Eq. (15) as

and obtain from Eq. (44) and Eq. (45) that,

Choosing a pair of Gaussian pulses of the form

we can write Eq. (46) as,

The solutions of Eq. (16) are obtained using either the RKM algorithm or the PA
adiabatic expressions of Eqs. (41) and (42). As in the PD calculations, adiabaticity is found to be
maintained for the ${\u03f5}_{1,2}^{0}$ and
Δ*t*
_{1,2} pulse parameters used in the
calculations. In all the results presented here the adiabatic solutions are
found to be virtually identical to the RKM numerical-solutions.

The photoassociative production of Na_{2} molecules in the
*υ* = 28, *J* = 10 level of the
ground electronic state is shown in Fig. 8. Contrary to the PD process, in the PA process
“counter-intuitive” pulse ordering means that the
*ϵ*
_{1}(*t*) pulse,
coupling the two bound states, is made to precede
*ϵ*
_{2}(*t*) pulse, which
couples the bound to the continuum states.

We find that population is transferred monotonically from the continuum into the
“target” bound-state, almost without ever populating the
intermediate (*A*
^{1}${\sum}_{u}^{+}$, *υ*′ = 37,
*J*′ = 11) level. As in the adiabatic PD process, in
this way the spontaneous emission from the intermediate state is eliminated,
thus preventing the formation of molecules in vib-rotational states other than
the the *X*
^{1}${\sum}_{g}^{+}$ (*υ* = 28, *J* = 10) level of
choice. We observe that 98% of all *J* = 10 atom-pairs that
collide during the pulse form *υ* = 28,
*J* = 10 Na_{2} molecules.

Based on the above near unity association probability per pulse, we can estimate
the PA yield in a thermal ensemble. In order to estimate the probability for a
collision with a given *J*, we use the semiclassical relation
between *J* and the impact parameter ${b}_{J},{b}_{J}=\frac{\u0127\left(J+\frac{1}{2}\right)}{\mathit{m\upsilon}}$, where *m* is the reduced mass of the collision
pair and *υ* is their relative velocity. Due to the
rotational selection rules for optical transitions, by tuning the laser central
frequencies to a specific *J* → *J*
± 1 → *J* sequence, only those colliding
pairs whose impact parameter lies between
*b*
_{J-1} and
*b*_{J}
are affected by the laser. Hence all atoms
contained in a cylinder (see Fig. 9) whose height is
*υ*Δ*t*
_{2}
(where Δ*t*
_{2} is the duration of the pump
pulse), and whose area is *π*(${b}_{J}^{2}$ - ${b}_{J-1}^{2}$) will be
associated with a given atom.
*η*_{J}
(*T*) - the fraction
of recombining atoms per pulse at temperature *T* is therefore
given as,

where *n* is the number-density of Na atoms in the beam and
*δ*_{i,j}
is the Kronecker delta
function. Taking the atom density and average lateral velocity in a typical Na
atomic beam to be *n* = 10^{16} cm^{-3} and
*υ* = 1 × 10^{4} cm/sec
(corresponding to a translational temperature of ~
7*K*), and using a 20 nsec pump pulse, we find that the
association yield to form *J* = 10 molecules is
*η*
_{10}(7*K*) = 4
× 10^{-3} per pulse. This means that with a 10Hz pulsed laser
source we can recombine half of all the ensemble of Na atoms in about 15
seconds.

The above PA yields can be further increased using longer pulses. This is possible due to the existence of an exact scaling relations in Eq. (16). The initial wave packet energetic-width is inversely proportional to the size of the wavepacket, i.e., the effective Na-Na distance for which the association is complete. By scaling down the wavepacket’s energetic-width together with the pulse intensities as,

it is possible to scale up the duration of both pulses as,

since it follows from Eq. (48) and Eq. (14) that under these transformations,

and Eq. (16) becomes,

where 𝖻̄ denotes the vector of solutions of the scaled
equations. We see that the scaled coefficients at time *t* are
identical to the unscaled coefficients at times *t*/* s*. Thus, pulses’ durations can be made longer and their
intensities concomitantly scaled down, without changing the final
population-transfer yields. This behavior is demonstrated in Fig. 10 where pulse widths and intensities are scaled as
above, with

*= 10. It is evident that the resulting time evolution of the system is scaled up by a factor of 10, with the same final populations. As mentioned above, longer pulses increase the Na-Na distances for which collisions are effective in bringing about radiative recombination. Since the intermediate state population is low at all times, one needn’t worry about spontaneous emission losses when pulse durations are taken beyond the radiative lifetime of that state.*

**s**## 4. Conclusions

We have established that complete population transfer to and from a continuum is achievable in cold thermal ensembles, using pulses of realistic durations and intensities. We have shown that, contrary to results based on a discretized quasi-continuum approach [30], the 3-level STIRAP technique may be extended to consider adiabatic passage into a final continuum, and that stimulated two photon association is an efficient mechanism for producing translationally cold molecules in specific vib-rotational states.

While an alternative “intuitive” approach may be taken in both processes [11], the “counter-intuitive” pulse scheme has the advantage of maintaining a low intermediate state population, thus minimizing spontaneous emission losses. Since the final population distribution is unaltered when longer pulses (of lower intensities) are used, this means that higher ensemble yields may be obtained from the “counter-intuitive” scheme, by increasing the pulse duration [11].

We have performed detailed calculations for the PA of cold Na atoms to produce
*X*
^{1}${\sum}_{g}^{+}$(*υ* = 28, *J* = 10) molecules. Due
to more favorable transition dipole matrix elements, the required intensities for a
given pulse duration are almost two orders of magnitude lower than those required to
produce *X*
^{1}${\sum}_{g}^{+}$ (*υ* = 0, *J* = 0) ultracold
molecules, calculated in our previous work [11]. In addition, the molecular beam number-densities used in
this work are four to five orders of magnitude higher than the those available in a
magneto-optical trap (MOT). The resulting 4 × 10^{-3} PA yield
per 20 nsec pulse is three orders of magnitude higher than the efficiency we
obtained for PA in a MOT.

Given the low intensities required in this work, a two-step scheme may be devised for
the production of cold Na_{2} molecules in the ground
*X*
^{1}${\sum}_{g}^{+}$ (*υ* = 0, *J* = 0) state. Starting
from a translationally cold ensemble of Na atoms, the first stage is the two-photon
association process outlined in this article. Once enough
*X*
^{1}${\sum}_{g}^{+}$ (*υ* = 28, *J* = 0) molecules are
formed, a second 3-level STIRAP stage may be employed to transfer molecules from the
*X*
^{1}${\sum}_{g}^{+}$ (*υ* = 28, *J* = 0) level into the
ground*X*
^{1}${\sum}_{g}^{+}$(v = 0,*J* = 0) state. This four-photon, two-step approach is
admittedly more complex from an experimental point of view than the two-photon
one-step scheme [11] of our previous work. Nevertheless, the gained efficiency
may compensate for the increase in experimental complexity.

## Acknowledgments

This work was supported by the German-Israeli foundation for scientific research and development (GIF) and by the Israel Science foundation (ISF).

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