## Abstract

Quantum coherence excitation onto spin ensembles by resonant Raman optical fields and coherence transfer back to an optical emission are discussed in a three-level optical system composed of inhomogeneously broadened spins, where the spin decay time is much slower than the optical decay time. Dynamic quantum coherent control of the spin excitations and coherence conversion are also discussed at a strong coupling field limit for practical applications of optical information processing.

©2008 Optical Society of America

## 1. Introduction

Coherence excitation in an optical medium by resonant optical fields occurs whenever an atomic population transfer is involved: absorption. The medium’s optical parameters such as optical susceptibility are modified during the action of optical excitations, so that a dynamic behavior of the refractive index of the medium can be utilized. When two orthogonal photons interact with an optically dense medium, the absorption spectrum can be modified to be transparent. This nonabsorption resonance phenomenon is called coherent population trapping [1] or electromagnetically induced transparency (EIT) [2]. Thus, two-photon excitations have been studied for the refractive index modification toward a group velocity control of traveling light [3–6]. The group velocity modification gives a great benefit to optical information processing such as an optical buffer memory [7] and quantum information [8] owing to lengthened interaction time based on enhanced nonlinear optics.

In an optical medium experiencing refractive index modification, photon density per unit volume is inversely proportional to the group velocity of traveling light. With a dramatic decrease of the group velocity of a traveling light pulse, highly efficient nonlinearity (giant Kerr effect) has also been studied [9–15] and presented for potential applications of quantum information [13–18]. In this case, the resulting strong modification of the dispersion spectrum can be applied for π-phase shift on a weak signal field, while keeping nearly zero absorption under EIT. Thus, the EIT-based giant Kerr effect can be a good candidate for Schrödinger’s cat [13], quantum entanglement [14], quantum switching [16, 17], and quantum wavelength conversion [10–12,18].

Researches on quantum coherent control in an optical system composed of spin inhomogeneous broadening have been carried out in a rare-earth doped solid medium [4,17,19–22]. The spin inhomogeneous broadening is a common phenomenon in solid media especially rare-earth doped crystals, whose spin decay rate is much slower than the optical counterpart. The slow spin decay rate is an origin of the (optical) spectral hole-burning phenomenon [24]. The hole-burning phenomenon gives a great benefit using a rare-earth doped crystal to optical information processing owing to spectral modifications [4, 17, 19–22]. Thus, the rare-earth doped crystals have been intensively studied for quantum optical memories utilizing photon echoes [20–23] and EIT [4, 19]. Theoretical studies of quantum optical memories using EIT in rare-earth doped solids, however, have not been performed well yet. Here, we discuss the role of the spin inhomogeneous broadening to the quantum coherent control such as the light-matter coherence conversion process [4, 17–19,22]. Understanding the coherent transfer mechanism between photons and spin ensembles (inhomogeneously broadened spins) is important for the intensive studies of slow-light based optical routing [25] as well as quantum optical data storage time versus spin decay rate. Coherence excitations by optical Raman fields onto the spin ensemble and the spin coherence transfer back into optical emission will be discussed below.

## 2. Theory

The density matrix approach is an excellent tool to denote ensemble behavior of
light-matter interactions in an optical system. In a lambda-type three-level optical
system interacting with two optical Raman fields Ω_{P} and
Ω_{C} (see Fig. 1(a)), the time-dependent density matrix equation is
denoted by [26]:

where {Γ,ρ}=Γρ+ρΓ. Interaction Hamiltonian H is

where δ_{1}=ω_{31}-ω_{P},
δ_{2}=ω_{32}-ω_{C},
ω_{ij}=ω_{i}-ω_{j}, and
ω_{P} and ω_{C} are the frequencies of the
probe (Ω_{P}) and coupling (Ω_{C}) fields,
respectively. In Fig. 1(a), the thick line on level |2>
stands for spin inhomogeneous broadening between |1> and
|2>.

## 3. Results and discussions

Figure 1(b) shows the pulse sequence of the applied Raman
optical fields for Fig. 1(a). The coupling field Ω_{C} and
the probe field Ω_{P} temporally overlap each other. Initially
equal population distribution between two low lying levels |1>
and |2> are assumed: ρ_{11}=1/2;
ρ_{22}=ρ_{33}=1/2. Even if optical
transitions Ω_{P} for
(|1>-|3>) and Ω_{C}
for (|2>-|3>) satisfy the resonant
two-photon condition, the spin inhomogeneous broadening between
|1>-|2> makes an intrinsic detuning. We
choose only δ_{2} as an intrinsic detuning for the
inhomogeneously broadened spins to the two-photon optical fields (in this case only
to the coupling field Ω_{C}) for the following numerical
calculations.

In Fig. 2, a Gaussian shaped spin inhomogeneous broadening is
assumed. The optical system, however, is homogeneous. For the calculations using the
density matrix equations given in Eq. (1), the spin inhomogeneous broadening, whose full width at half
maximum is 50 kHz, is divided into 180 groups from δ_{2}=-90 kHz
to δ_{2}=+90 kHz at a step of 2 kHz.

For the numerical calculations, experimental parameters in Pr^{3+}
doped Y_{2}SiO_{5} are used [19, 24, 27]. Each applied laser Rabi frequency of
Ω_{P} and Ω_{C} is 20 kHz for a weak field
limit, where the optical phase decay rate is
γ_{31}=γ_{32}=50 kHz. The atom-field
interaction time of 60 µs) is identified as much longer than the optical
phase decay time T_{2}
(T_{2}=1/(πγ_{32}=6.4 µs).

In Fig. 3, numerical calculations for optical
(*i*(ρ_{13})) and spin coherence
(*r*(ρ_{12})) are presented. For the
calculations all groups of atoms (spins) with different detuning
δ_{2} are considered. As seen in Figs. 3(a) and 3(b), the probe field absorption
(*i*(ρ_{13})) and spin coherence
(*r*(ρ_{12})) are affected by both population
differences between levels |3> and |1>
and between |2> and |1>. According to
the theory of coherent population trapping (in a weak field limit) the maximum spin
coherence (*r*(ρ_{12})) is -0.5, where the current
coherence magnitude in Fig. 3 should be reduced by a higher optical decay rate
γ and the spin inhomogeneous broadening.

In Fig. 4, however, we demonstrate
population-difference-independent spin coherence excitation for a detuning zero atom
group (center in Fig. 1) satisfying two-photon resonance for
Ω_{P} and Ω_{C}. Unlike Fig. 3 discussed for population-dependent coherence
excitations, here the spin coherence is independent of the population difference
between levels |2> and |1>. This is
because the optical system in Fig. 1(a) is population shelved, in which the optical
population decay rate is much weaker than the applied optical Rabi frequencies.
Moreover, both optical Rabi frequencies are exactly the same in magnitude and phase.
Thus, the two optical fields incur population difference only between levels either
|1> and |3>, or
|2> and |3>, under perfect resonance
condition (δ_{2}=0).

According to the density matrix equations, the time derivative of the spin coherence
is composed of three parts (spin coherence ρ_{12}, optical
coherence ρ_{13}, and optical coherence
ρ_{23}), where the population difference should not be a factor.
However for the optical coherence the population difference must be considered as
shown in Fig. 3. This means that the population difference between two
low-lying ground levels is not a necessary condition to induce the spin coherence
excitation by resonant two-optical fields.

Figure 5 shows all groups of atoms for spin coherence
excitation versus ground levels population difference. As seen in Fig. 5, at line center of the spin inhomogeneous broadening,
the spin coherence is excited even without population difference (see the red-dotted
lines). This fact is important for coherence retrieval processes such as photon
echoes using reversible optical inhomogeneous broadening for the population reverse
in a two-level system, because no population difference is created before and after
the retrieval action (This is beyond the scope of this paper and will be discussed
elsewhere). For all other atoms which two-photon resonance is not satisfied, the
spin coherence excitation depends on the population difference
(ρ_{22}-ρ_{11}) via those between optical
transitions as discussed in Fig. 4.

For detuned atoms (spins) in Fig. 5, it is clear that the excited coherence oscillates
with δ_{2} as a function of interaction time (see Fig. 6): The more detuning it has, the faster oscillation it
experiences. Of course its amplitude becomes weaker as the detuning gets wider. As
is well known in the area of optical transient phenomena such as free induction
decay as shown in Fig. 3, the overall coherence decay time becomes shorter than
the spin phase decay time T_{2}
^{Spin} as the detuning becomes
wider: Here T_{2}
^{Spin} is
1/(πγ_{21})=1,590 µs. Thus, Figs. 3 and 6 prove a spin coherence transient effect induced by two
optical fields under the spin broadened system: nutation. This optical field-induced
spin coherence transient effect is important in analyzing the EIT-based nonlinear
Kerr effect such as nondegenerate four-wave mixing processes as experimentally
demonstrated already [4,17,19,28]. Unlike spin homogeneously broadened atomic systems [3,5,6,10–12], the read-out pulse must come in a shorter
time compared with the spin T_{2} [28].

In general, however, the population-independent spin-coherence excitation is not
satisfied if either balanced optical fields
(Ω_{P}=Ω_{C}) or equal population
distributions on the low lying two ground levels
(ρ_{11}=ρ_{22}) are not provided. Figure 7 shows a general case of coherence excitation induced
by two optical fields whose Rabi frequency is the same, where initial population
distribution is unbalanced: ρ_{11}=1;
ρ_{22}=ρ_{33}=0. As seen in Fig. 7, the spin coherence excitation strongly depends on the
population difference (see also Eqs. 3(b) and 3(c)).

Now we increase each optical Rabi frequency up to 50 kHz which is equal to the
optical phase decay rate. In this case, one can expect a damped oscillation
phenomenon in coherence excitations. Figure 8 shows coherence excitations versus population
differences for the initial condition of equal population
(ρ_{11}=ρ_{22}). All other parameters are
the same as above. As seen in Figs. 8, spin coherence excitation
(*r*(ρ_{12})) results from optical absorption
(*i*(ρ_{13})) with a phase delay. The phase
delay is a common characteristic of the response function in a physical system. For
example, the absorption and dispersion relation is characterized as the
Kramers-Kronig relation, where the absorption and dispersion are coupled to each
other resulting in a response function. Here the relation between linear absorption
(*i*(ρ_{13})) and two-photon dispersion
(*r*(ρ_{12})) mimics the Kramers-Kronig
relation as a response function. Surprisingly time evolution of the spin coherence
(*r*(ρ_{12})) matches that of the population
difference between the excited and ground levels
(ρ_{33}-ρ_{11}). This is because the atom
population on both levels must be Rabi flopping, and then the coherence excitation
directly results from the population difference: See Eqs. (3). Thus, spin coherence excitation can be controlled only by
adjusting one of the optical fields’ Rabi frequencies. Figure 8(a) depicts overall atoms, and 8b only two-photon
resonant atoms (δ_{2}=0). As seen in Fig. 8(b), the ground levels population difference is
negligible for the spin coherence excitation.

So far we have discussed resonant Raman field induced spin coherence excitations at a
moderate or strong probe limit. However, optical signals in information processing,
in general, are expected to be weak. Owing to the discussion in Fig. 8 that one optical laser field (coupling
Ω_{C}) can be used to control the spin coherence excitation,
the following is considered: Ω_{P}≪γ;
Ω_{P}≪Ω_{C};
γ≪Ω_{C}.

Figure 9 shows a quantum coherent control for spin coherence
excitation by using a strong coupling field with a weak probe field. As we have
discussed above, due to a very weak optical decay rate (population shelved model)
and weak probe field Ω_{P}, each atomic population on levels
|2> and |3> strongly depends on the
coupling field Ω_{C}: Rabi flopping. The Rabi flopping-induced
coherence oscillation at a frequency of 200 kHz is the same as the coupling Rabi
frequency in Fig. 9. So is the population difference between levels
|3> and |1> as wells as between
|2> and |1>: See the blue dotted and red
curves, respectively. Unlike the weak coupling limit
(Ω_{C}≪γ) discussed Figs. 1–8, spin coherence excitation now strongly
depends on the population on level |2> via the coupling field
interaction. Region A in Fig. 9 shows spin free-induction decay, where the decay time
is 1/Δ_{spin}: Δ_{spin}=50 kHz.

Figure 10 represents coherence damping, whose decay time
relies on the optical phase decay rate γ_{32} (=50 kHz), where
the optical phase decay time is
T_{2}=1/(πγ_{32})=6.4 µs. In Fig. 10(b) and 10(c) the dotted exponential curve is for the best fit to the
coherence envelope: Fig. 10(b) shows overall inhomogeneous spins, while Fig. 10(c) shows only two-photon resonance spins at line
center in Fig. 2. As seen in Fig. 10, the spin coherence
(*r*(ρ_{12})) evolution nearly perfectly
follows the population change on level |2>
(ρ_{22}). For the calculations in Fig. 10(c), both optical and spin population decay rates are
set to zero for analysis purposes. From these results, we conclude that the spin
coherence excitation depends on both optical phase decay rate and Rabi flopping by
the coupling field Ω_{C}.

So far we have discussed resonant Raman optical field excited spin coherence behavior for both a weak field limit and a strong field limit of the coupling field. In the strong field limit, we have demonstrated that the magnitude of the spin coherence excitation can be controlled by adjusting only the coupling field interaction time. Thus, one may seek relationship between the spin coherence and the Kerr effect in nonlinear optics. To answer to this question, the coherence transfer phenomenon will be discussed below.

Figure 11 repeats the resonant Raman field excited spin
coherence shown in Figs. 1–3, but for a reference of coherence
transfer from the spin coherence to the optical coherence. Here, we discuss the
relation between the spin coherence and the optical coherence (emission) caused by a
third optical pulse of Ω_{A} immediately following the excitation
pulse composed of Ω_{P} and Ω_{C}, in which
Ω_{A} has the same transition as Ω_{C}.

In Fig. 12, three different cases of coherence transfer are
demonstrated. From Fig. 12(a) to 12(c) the spin coherence magnitude created by the Raman
excitation pulse is chosen in the order of degradation. The Raman excitation pulse
length for Figs. 12(a), 12(b), and 12(c) is 25, 30, 40 µs, respectively, as shown in Fig. 12(d), 12(e), and 12(f). The retrieval pulse length of Ω_{A}
is 10 µs for all cases. From Figs. 12, we conclude that the spin coherence is transferred
to optical coherence resulting in optical emission by the action of the retrieval
pulse Ω_{A}, and the strength of the optical coherence (magnitude
of emission) is proportional to that of the spin coherence (denoted by dotted
arrows). This phenomenon is actually EIT-based nondegenerate four-wave mixing
experimentally demonstrated in Pr^{3+} doped
Y_{2}SiO_{5} [28].

For a detailed investigation of the coherence transfer process, we apply a time
derivative to the spin coherence and plot the results in Fig. 13. The Eq. (4) is obtained intuitively from the delay relation in Eq. (3). For the numerical calculations in Fig. 13, Fig. 12(b) is chosen as a model. The result of Eq. (14) for
the region (dash-dot, 30<t<40 µs) of retrieval
pulse Ω_{A} is denoted by blue-colored “o,”
which is a perfect match to the optical coherence (red curve) obtained in Fig. 12(b) for the zero detuned spins
(δ_{2}=0, green curve). For overall spins spanned over 50kHz, the
coherence transfer induced optical emission lags a little bit behind (see the black
curve).

Figure 14 shows three-dimensional simulations of the
one-photon absorption (*i*(ρ_{13})) and the spin
coherence (*r*(ρ_{12})) for all inhomogeneous
spins. As seen in Fig. 14, the overall one-photon optical coherence (emission;
red area in Fig. 14(a)) is from the retrieval of the overall two-photon
spin coherence (blue area in Fig. 14(b)) as discussed in Fig. 13. This phenomenon has already been studied
experimentally in Pr^{3+} doped Y_{2}SiO_{5} [28].

So far we have discussed resonant Raman optical field excited spin coherence and its coherence transfer back to an optical emission at a weak control limit. To comply with dynamic quantum coherent control at higher speed, we need to discuss it at a strong coupling limit. As we mentioned above in Figs. 9 and 10 at a strong coupling field limit with weak probe field, both optical and spin coherence excitations depend on the control field directly affecting the population on level |2>, while remaining nearly unchanged on level |1>.

Figures 15 and 16 demonstrate dynamic coherence retrieval processes with a
retrieval pulse Ω_{A}. Figure 15 shows coherence oscillation at a strong coupling
limit. Surprisingly, without population inversion between the ground level
|1> and the excited level |3>, Figs. 15(b) and 15(e) show photon emission:
ι(ρ_{13})>0;
(ρ_{33}-ρ_{11})<0. This is due
to the coupling induced Rabi flopping on level |3>, so that
the slopes of population differences for both transitions are negative (see Fig. 15(c)), which means photon emission. As discussed in Fig. 12, the retrieval pulse whose Rabi frequency is 100 kHz
follows the two-photon excitation pulse composed of Ω_{P} and
Ω_{C} with no time delay (see Fig. 16). To demonstrate a dynamic coherence transfer of the
excited spin coherence to optical one-photon coherence, we choose three different
moments at t=12, 15, and 20 µs (see the green dashed line in Fig. 15(b)). For the coherence transfer at t=15 µs
and t=20 µs (see respectively Figs. 16(b)/(e)/(h) and (c)/(f)/(i)), the photon emission at t=20 µs is bigger
because of a stronger spin coherence excited. Moreover, there are extra photon
emission peaks as a result of Fig. 15(c): see pink shaded area in Fig. 15(c). This kind of dynamic coherence control is
important in a population shelved system to avoid unwanted signals.

## 4. Conclusion

We have presented a quantum coherent control and discussed resonant Raman optical field excited spin coherence and the spin coherence transfer back to the optical field by using a third retrieval pulse for both a weak and a strong coupling limit. In the weak coupling limit, we numerically demonstrated that the spin coherence excitation does not have to rely on a population difference between the ground levels, which is important for photon echo like quantum memories [4,19]. In a strong coupling limit, however, the spin coherence excitation nearly perfectly depends on population change induced by the coupling field only. This one-field quantum coherent control for spin excitation is important in practical applications using an optically shelved medium such as a rare-earth doped crystal. For the spin coherence transfer into optical coherence (emission), we numerically demonstrated that the magnitude of the one-photon coherence nearly perfectly depends on the magnitude of the spin coherence. For the strong coupling limit in an optically shelved model, we also theoretically proved that an emitted photon pulse train is possible without population inversion.

## Acknowledgment

This work was supported by Creative Research Initiative Program (Center for Photon Information Processing) of MOST and KOSEF, S. Korea.

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