## Abstract

We study both steady and dynamic optical responses of three samples with the same amounts of cold atoms but very different density functions. These samples are driven into the regime of electromagnetically induced transparency by a probe and a coupling in the Lambda configuration. When the coupling is in the traveling-wave pattern, all samples have the same transmission spectra and therefore identical transmitted pulses at the sample exits. In the case of a standing-wave coupling, however, very different reflection and transmission spectra are found for the three samples. Accordingly, reflected pulses at the sample entrances and transmitted pulses at the sample exits are quite sensitive to the spatial inhomogeneity of cold atoms. These interesting phenomena are qualitatively analyzed in terms of constructive and destructive interference between forward and backward probe photons scattered by a standing-wave atomic grating.

© 2011 Optical Society of America

## 1. Introduction

Under the condition of electromagnetically induced transparency (EIT) [1, 2], laser induced quantum coherence in atomic samples has been extensively studied both experimentally and theoretically, e.g., to enhance resonant optical nonlinearities with little absorption [3, 4, 5, 6], to manipulate the propagation dynamics of weak light pulses [7, 8, 9], to achieve quantum storage and retrieval of photonic states [10, 11, 12, 13, 14], etc. These applications of laser induced quantum coherence are essential and stimulating for the further developments of quantum nonlinear optics and quantum information science as well as other related fields in physics. Originally, laser fields dressing atomic samples are always set in the traveling-wave (TW) pattern to establish spatially homogeneous quantum coherence. Recently, the standing-wave (SW) driving configuration has been well adopted to induce spatially periodic quantum coherence, which is advantageous for both the coherent generation of photonic bandgaps (PBGs) [15, 16, 17, 18, 19] and the dynamic creation of stationary light pulses (SLPs) [20, 21, 22, 23, 24]. All-optically controlled PBGs and SLPs in atomic samples represent, in fact, an important progress for achieving the flexible manipulation of photonic flow and interaction [25, 26] as required in classical and quantum information processing.

So far atomic samples dressed by TW or SW laser fields are always assumed to have spatially homogeneous densities for the simplicity in theoretical treatments, which is not true at least for cold atoms confined in magneto-optical traps (MOT). It is well known that cold atomic clouds in MOT usually have the Gaussian [27] or cigar-shaped [28] profiles in spatial density. In this paper, we investigate by numerical calculations how the spatial inhomogeneity affects both steady and dynamic optical properties of cold atoms driven (by a weak probe and a strong coupling) into the EIT regime. Steady spectra of probe reflectivity and transmissivity are examined by the transfer-matrix method with the probe susceptibility analytically solved from density matrix equations while the propagation dynamics of an incident pulse is explored via the Fourier-transform method starting from the probe reflectivity and transmissivity.

Our numerical results show that, in the TW driving configuration, the probe transmissivity characterized by an EIT window is irrelevant to the spatial inhomogeneity of cold atoms. Therefore the transmitted pulses are not distinguishable for three typical samples (one homogeneous and two inhomogeneous) with the same amounts of cold atoms but very different density functions. In the SW driving configuration, however, both spectra of probe reflectivity and transmissivity are very sensitive to the spatial inhomogeneity of cold atoms. For instance, coherently induced PBGs denoted by a platform of reflectivity near 100% become narrower and narrower when the inhomogeneity of cold atoms is gradually increased. Consequently, an incident pulse perfectly reflected by the homogeneous sample will be split into a partially reflected one and a partially transmitted one in the two inhomogeneous samples with the same driving parameters. The underlying physics is that constructive and destructive interferences between forward (FW) and backward (BW) probe photons (scattered by an atomic grating generated by the SW coupling) depend critically on the spatial inhomogeneity of cold atoms. These results are expected to be instructive for the design of relevant experiments for light information manipulation in cold atomic samples driven by at least one SW field.

## 2. Model and Equations

We consider here a simplified three-level Λ system as shown in Fig. 1. The transition from level |1〉 to level |3〉 is probed by a weak light beam of frequency *ω _{p}* and amplitude

*E*while the transition from level |2〉 to level |3〉 is coupled by a strong light beam of frequency

_{p}*ω*and amplitude

_{c}*E*, which may be either in the TW pattern or in the SW pattern. Under the electric-dipole and rotating-wave approximations, we can write the interaction Hamiltonian of this coherently driven Λ system as

_{c}*ω*−

_{p}*ω*

_{31}and

*δ*=

*ω*−

_{p}*ω*−

_{c}*ω*

_{21}are, respectively, single-photon and two-photon detunings relevant to the probe and coupling fields. Ω

*=*

_{p}*E*

_{p}d_{13}/2

*h̄*(Ω

*=*

_{c}*E*

_{c}d_{23}/2

*h̄*) is the probe (coupling) Rabi frequency with

*d*

_{13}(

*d*

_{23}) denoting the dipole matrix element on transition |1〉 ↔ |3〉 (|2〉 ↔ |3〉). In particular, when the coupling field is set in the SW pattern, its Rabi frequency takes the form which varies along the

*z*direction with a spatial periodicity of

*λ*/2 and depends on the space-independent Rabi frequencies Ω

_{c}_{c+}and Ω

_{c−}of the FW and BW coupling beams.

Starting from Eq. (1) and Eq. (2), it is straightforward to attain the following (expanded) *Liouville* equations for density matrix elements in the interaction picture

*ρ*

_{11}+

*ρ*

_{22}+

*ρ*

_{33}= 1. In Eqs. (3), Γ

*describes the population decay rate from level |*

_{ij}*i*〉 to level |

*j*〉 whereas

*γ*denotes the dephasing rate of the coherence term

_{ij}*ρ*.

_{ij}In the limit of a weak probe, we are allowed to assume *ρ*_{11} = 1 and *ρ*_{22} = *ρ*_{33} = *ρ*_{23} = 0 so that the steady-state solution of *ρ*_{31} can be found from Eqs. (3) to the first order in Ω* _{p}*, from which the probe susceptibility may be written as

*N*(

_{a}*z*) has been taken to be space-dependent. With Eq. (4) in hand, we can further attain the complex refractive index which governs absorptive and dispersive properties on the probe transition.

Due to joint modulations of the spatially periodic coupling field and the spatially inhomogeneous atomic density, the complex refractive index *n _{p}* (Δ,

*z*) in Eq. (5) becomes very complicated so that the derivation of relevant transmission and reflection spectra are rather intractable. That is, we have to first partition the atomic sample into a large number of laminas, then derive the individual transfer matrix of each lamina with the complex refractive index, and finally multiply the individual transfer matrices of all laminas in succession to attain the total transfer matrix of the whole sample [29]. To be specific, the propagation of a probe field through the

*nth*atomic lamina (extending from

*z*=

*nd*−

*d*to

*z*=

*nd*) is described by a 2 × 2 unimodular transfer matrix

*M*(Δ,

_{n}*d*) via

*E*

_{p+}and

*E*

_{p−}denote, respectively, the FW and BW probe fields. With

*M*(Δ,

_{n}*d*) determined by Eq. (6), the total transfer matrix finally turns out to be

*M*(Δ,

*L*) =

*M*

_{1}(Δ,

*d*) ⋯

*M*(Δ,

_{n}*d*) ⋯

*M*(Δ,

_{N}*d*) for a sample of length

*L*=

*Nd*. Then we can write the probe reflectivity and transmissivity in terms of matrix elements

*M*

_{(ij)}(Δ,

*L*) as

_{c−}= 0 or Ω

_{c+}= 0).

Now we briefly introduce the Fourier transform method [19] for studying the propagation dynamics of an incident light pulse with *r*(Δ, *L*) and *t*(Δ, *L*) given in Eqs. (7). In the weak probe limit, the linear response functions *r*(Δ, *L*) and *t*(Δ, *L*) contain all relevant information on the optical responses of the atomic system under consideration to a monochromatic probe field. The basic procedure for the Fourier transform method is: first to write the incident pulse in the time domain *E _{It}* (

*t*) and decompose it into the Fourier components in the frequency domain

*E*(Δ); then to multiply the incident Fourier components

_{If}*E*(Δ) with

_{If}*r*(Δ,

*L*) and

*t*(Δ,

*L*) to attain the reflected and transmitted Fourier components

*E*(Δ) =

_{Rf}*E*(Δ) ·

_{If}*r*(Δ,

*L*) and

*E*(Δ) =

_{Tf}*E*(Δ) ·

_{If}*t*(Δ,

*L*); finally to perform the inverse Fourier transform so that the reflected pulse

*E*(

_{Rt}*t*) at the sample entrance (

*z*= 0) and the transmitted pulse

*E*(

_{Tt}*t*) at the sample exit (

*z*=

*L*) can be reconstructed as

*t*

_{0}and

*δt*(Δ

_{0}and

*δ*) are, respectively, the center and the width of the incident probe pulse in the time (frequency) domain. Eqs. (7) and Eqs. (8) together with Eqs. (9) are the main results that we have derived to examine various effects of the atomic spatial inhomogeneity on both steady and dynamic probe responses.

_{p}## 3. Results and Discussions

With the formulas developed in the last section, we now perform numerical calculations to investigate the probe spectra of reflectivity and transmissivity as well as the propagation dynamics of an incident pulse in one homogeneous sample and two inhomogeneous samples of cold atoms. The three samples under consideration have the same medium length *L* = 1.5 mm, the same average atomic density *N*_{0} = 2.0 × 10^{10} mm^{−3}, but very different density functions *N _{ai}*(

*z*), which is constrained by ${\int}_{0}^{L}{N}_{ai}\left(z\right)dz={N}_{0}L$ (see Fig. 2). In particular, the two inhomogeneous samples have either the sinusoid profile or the Gaussian profile as denoted by the caption of Fig. 2 and the latter is quite typical for cold atomic clouds in MOT [27].

First we consider the simple case where a TW coupling field is switched on, which can be examined by setting Ω_{c}_{−} = 0 in Eq. (2). We see from Fig. 3(a) that the transmission spectra with a characteristic EIT window are clearly independent of atomic density functions as long as all three samples contain the same amounts of cold atoms. Fig. 3(b) further shows that the three transmitted pulses with a remarkable time delay (relative to the incident pulse) are almost indistinguishable for different atomic density functions constrained by
${\int}_{0}^{L}{N}_{ai}\left(z\right)dz={N}_{0}L$. Thus we may conclude that, in the TW driving configuration, it is a good choice to simplify relevant calculations and analyses by ignoring the spatial inhomogeneity of real atomic samples. Note, however, that the pulse distribution in the homogeneous sample is distinct from those in the inhomogeneous samples because the pulse intensity and the group velocity at a given spatial position are inversely proportional to the local atomic density.

Then we consider the complicated case where a SW coupling field is switched on, i.e. both Ω_{c}_{+} and Ω_{c}_{−} are nonzero in Eq. (2). We see from Fig. 4(a) that three PBGs, characterized by a platform of reflectivity over 95%, are well developed around the probe resonance with suitably chosen parameters. But these dynamically induced PBGs have quite different widths indicating that they are very sensitive to atomic density functions. Fig. 4(b) further shows that the three transmission spectra depend critically on atomic density functions due to their correlation with the three reflection spectra in Fig. 4(a). In Fig. 4(c) and 4(d), we examine instead the propagation dynamics of a probe pulse with its most carrier frequencies fallen into the widest PBG in Fig. 4(a). It is clear that the incident pulse is perfectly reflected with a short time delay and experiences little energy loss and profile deformation in the homogeneous sample. For the inhomogeneous samples, however, the reflected pulses become depleted and distorted more or less due to the loss of some carrier frequencies. Among the lost carrier frequencies, most make up the transmitted pulses with a long time delay at the sample exits while little are indeed absorbed because we always work in the regime of EIT. It is clear that, in the SW driving configuration, the spatial inhomogeneity of real atomic samples have to be duly addressed, otherwise theoretical predictions may deviate largely from relevant experimental results.

It is straightforward to understand why the inhomogeneity of cold atomic samples is trivial in the TW driving configuration: the probe susceptibility *χ _{p}* depends on the spatial variable

*z*only through the atomic density

*N*(

_{a}*z*) so that the probe transmissivity is proportional to ∫

*N*(

_{a}*z*)

*dz*(i.e. the total amounts of cold atoms) while the probe reflectivity is always negligible. In the SW driving configuration, however, it is not so simple to explain the importance of the inhomogeneity of cold atomic samples. In this case, the probe susceptibility

*χ*depends on

_{p}*z*not only through

*N*(

_{a}*z*) but also through Ω

*(*

_{c}*z*), which makes both transmissivity and reflectivity significant around the probe resonance. In particular, dynamically induced PBGs may be observed as a platform of high reflectivity when perfect constructive (destructive) interference occurs between the BW (FW) probe photons scattered by an atomic grating generated by the SW coupling. In a homogeneous sample, different lattices of the atomic grating are exactly the same in optical responses and thus have identical scattering abilities, which is favorable for achieving perfect interference between the probe photons in a large spectral region. When the sample becomes inhomogeneous, however, each lattice of the atomic grating may be very different from others in optical responses, especially in scattering abilities, so that perfect interference can only be achieved between the probe photons in a small spectral region. This is why distinct steady optical spectra and light propagation dynamics have been observed in Fig. 4 for three samples with the same amounts of cold atoms but very different density functions.

One may wonder whether can we compensate the undesirable effect of the spatial inhomogeneity of cold atomic samples to attain wide enough PBGs? Fig. 5 shows that, for the two inhomogeneous samples with fixed medium lengths and density functions, it is viable to widen the dynamically induced PBGs while maintaining their high reflectivities by suitably increasing Rabi frequencies Ω_{c+} and Ω_{c−} of the FW and BW coupling beams. It is also clear that the amended PBGs (red-dashed) in Fig. 5(a) and Fig. 5(b) well match the widest PBG (black-solid) in Fig. 4(a) as far as widths and heights are concerned.

## 4. Conclusions

In summary, we have calculated steady optical spectra and light propagation dynamics in three samples of cold atoms with either homogeneous or inhomogeneous density functions. These samples are assumed to be driven into the EIT regime by a weak probe and a strong coupling in the Λ configuration. When the coupling is a TW field, the transmissivity spectra characterized by typical EIT windows are exactly identical for the three samples with the same amounts of cold atoms but very different spatial inhomogeneity while the reflectivity spectra are not necessary to be taken into account. Consequently, an incident pulse with its most carrier frequencies fallen into the overlapped EIT windows will go through the three samples with the same time delay while suffering little energy loss and profile deformation. The spatial inhomogeneity of cold atomic samples becomes rather important when the coupling is a SW field instead. That is, both transmissivity and reflectivity spectra may change dramatically if the spatial inhomogeneity of a sample is increased or decreased. In particular, the dynamically induced PBG for the homogeneous sample looks much wider than those for the two inhomogeneous samples. As a result, an incident pulse perfectly reflected by the homogeneous sample is seen to split into a partially reflected one and a partially transmitted one when impinging upon the two inhomogeneous samples. Qualitative analyses have been given in terms of constructive (destructive) interference between the BW (FW) probe photons scattered by an atomic grating generated by the SW coupling. These findings should be helpful to amend relevant theoretical results attained without considering the spatial inhomogeneity of real atomic samples coherently driven by one or more SW fields.

## Acknowledgments

The authors would like to thank the financial supports from NSFC under grants 10874057 and 10904047, from NBRP under grant 2011CB921603, from the Basic Scientific Research Foundation of Jilin University under grant 200905019, and from the Graduate Innovation Foundation of Jilin University under grant 20101051.

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