## Abstract

We propose two schemes of holographic imaging with an object that has no any macro structure itself. The tunable electromagnetically induced grating (EIG) is such a kind of object. We obtain an EIG based on the periodically modulated strong susceptibility in a three-level ladder-type hybrid artificial molecule, which is comprised of a semiconductor quantum dot and a metal nanoparticle coupled via the Coulomb interaction. The holographic interference pattern is detected either directly in the way of classical holographic imaging with a coherent field being the imaging light, or indirectly and nonlocally in the way of two-photon coincidence measurement with a pair of entangled photons playing the role of imaging light. This work provides a practical prototype of electromagnetically induced transparency-based holographic solid-state devices for all-optical classical and quantum information processing.

© 2015 Optical Society of America

## 1. Introduction

Quantum coherence and interference effects, the most familiar one of which is electromagnetically induced transparency (EIT) [1], have been paid much attention due to its significant action on modifying and controlling the optical properties of a medium. It is generally studied in the multilevel atomic system. While the much more promising practical applications require that the effects are implemented in solid-state medium, where metal nanoparticle, quantum well and quantum dot have been deeply investigated due to their significant advantages [2,3]. In addition, the modern nanotechnology also opens a possibility to build nano-superstructures with various combinations for exploring new physical effects and embraces their advantages in one system. Hybrid structure composed of semiconductor quantum dots (SQDs) and metal nanoparticles (MNPs) is a case in point [4–8], and brings out some new physical effects, such as Fano-type asymmetric features in absorption spectra [4,5], exciton/plasmonic induced transparency [6,7], enhancement of Rabi flopping [8]. For achieving the above effects, the coherence is essential, and is usually realized by a traveling wave coupling field.

Recently, the atomic and solid-state systems drove by the standing wave have been well adopted to modulate periodically coherence in space, which is advantageous for both the generation of photonic band gaps [9, 10] and the creation of electromagnetically induced grating (EIG) [11–13]. Since the EIG is presented, its diffraction behaviors have been deeply investigated by making Fourier transform of the optical transfer function either in the far-field limit or in the near-field situation. In the former case, high first-order diffraction efficiency close to that of an ideal sinusoidal phase grating (approximately 34%) has been realized in the both atomic and solid-state systems [14,15]. In the later one, based on the Talbot effect [16], the EIG is employed for the imaging of ultracold atoms or molecules, which is so called electromagnetically induced Talbot effect [17, 18]. While based on the principle only the amplitude information of the EIG can be imaged, and they are only applied for the imaging of periodic objects enslaved to the basic principle of Talbot effect.

Holography, firstly proposed by Gabor in 1948 [19], is also a lensless imaging technique and is capable of recording both the amplitude and phase information of an arbitrary shaped object. Due to the significant advantages, it has important applications in many fields, such as optical storage, reconstruction and information processing [20–22]. Traditionally, the laser is a crucial prerequisite for the holographic imaging because it requires a coherent source with both better temporal and spatial coherence to perform spatial interference between object field and reference field. Recently, researchers find that the entanglement light, even the thermal light is capable of performing interference and diffraction by correlation measurement [23–25]. In the paper, we propose two types of holographic imaging schemes for an EIG in the SQD-MNP hybrid system based on EIT, i.e., the electromagnetically induced classical holographic imaging (EICHI) and the electromagnetically induced quantum holographic imaging (EIQHI). Based on the principle of holographic imaging, both the amplitude and phase information of the EIG can be imaged, and the schemes can be applied for the object without periodic structure. Compared to the EICHI case, it should be noted that, in the EIQHI case the EIG can be imaged nonlocally, and the imaging obtained can be magnified on demand simply by adjusting the system settings. This all-optical and solid-state device might broaden variety important applications in self-imaging techniques, quantum information processing and quantum networking.

## 2. Electromagnetically induced grating

The hybrid nanostructure depicted in Fig. 1(a) composes of a spherical MNP with radius *r _{m}* and a spherical SQD with radius

*r*. They are separated by a distance

_{s}*R*and coupled by the Coulomb interaction. The energy scheme of the system is shown in Fig. 1(b). The plasmonic excitations of the MNP are a continuous spectrum, the excitations of the SQD are excitons with discrete energy levels. The interband transition |

*g*〉 ↔ |

*s*〉 is excited by a weak object field with Rabi frequency Ω

*. A strong standing wave controlling field drives the interband transition |*

_{o}*s*〉

*↔ |e*〉, which is formed by two strong coupling fields displaced symmetrically with respect to the object field path (see Fig. 1(c)). For simplicity and without loss of generality, here we consider the simplest case, i.e., one-dimensional standing wave. Then the Rabi frequency of the controlling field can be written as Ω

*cos(*

_{c}*πx/*Λ), where Λ is the spatial period of the standing wave, and can be made arbitrarily larger than the wavelength of the object field in principle by varying the angle between the two wave vectors of two coupling fields.

Under the dipole and rotating-wave approximation, the density-matrix equations for the three-level SQD are given as

*γ*and

_{sg}*γ*are the dephasing rates of SQDs, Δ

_{eg}*and Δ*

_{o}*denote the detunings of the object and controlling fields with respect to the corresponding transitions, $\mathrm{\Omega}={\mathrm{\Omega}}_{p}(1+\frac{6{\epsilon}_{b}({\epsilon}_{m}-{\epsilon}_{b}){r}_{m}^{3}}{{\left(2{\epsilon}_{b}+{\epsilon}_{m}\right)}^{2}{R}^{3}})+\frac{{\mu}_{gs}^{2}}{\pi \overline{h}}\frac{9{\epsilon}_{b}({\epsilon}_{m}-{\epsilon}_{b}){r}_{m}^{3}}{{\left(2{\epsilon}_{b}+{\epsilon}_{m}\right)}^{2}(2{\epsilon}_{b}+{\epsilon}_{s}){R}^{6}}({\rho}_{gs}+{\rho}_{sg})$ is the Rabi frequency of the total electric field felt by the SQD which consists of the external applied object field and the induced internal field from the MNP [5, 6], where*

_{c}*μ*is the dipole moments of the transition|

_{gs}*g*〉 ↔ |

*s*〉,

*ε*and

_{s}*ε*are the dielectric constants of the SQD and MNP,

_{m}*ε*is the background dielectric constant, and the external applied fields are assumed to be parallel to the major axis of the hybrid system. The periodical manipulation of the strong standing wave about the response of the medium to the object field realizes when the object photon goes through the medium.

_{b}In the weak probe field limit and the steady state, we can solve the density-matrix equations under the initial conditions *ρ _{gg}* = 1,

*ρ*=

_{ss}*ρ*= 0,

_{ee}*ρ*= 0 (

_{ij}*i*≠

*j*), and obtain the susceptibility with respect to the object field frequency

*N*being the SQD density. The transmission profile of the object field at the output surface of the medium can be obtained by solving Maxwell’s equation of the object field and reads where

*χ*=

*χ*′ +

*iχ*″,

*k*is the wave number of the object field,

_{o}*L*is the length of the medium, and

*E*(

_{o}*x*,0) is the object photons profile before it enters the atomic medium. From the Eq. (2), we can see that the absorption of the hybrid system vanishes accompanied with eliminated refraction at near resonant frequency (Δ

*= Δ*

_{o}*= 0.0) rang as a result of exciton-plasmon interaction. Under the standing wave intensity pattern of the controlling field, the absorption and refraction for the object field will experience a periodic variation and the EIG can be obtained in the standing wave direction. Figure 2 shows the amplitude of the transmission function with different inter-particle distance*

_{c}*R*, where the profile of the amplitude EIG can be seen clearly. With the increasing of

*R*, the intensity of the transmission function decreases. This can be understood as that the dipole-dipole interaction between the SQD and the MNP becomes weak when the value of

*R*increases. The

*R*-dependent destructive or constructive interference causes the change of the object field absorption.

As shown in Fig. 2, because of the transmission profile, i.e., the EIG, periodicity, the Eq. (3) can be recast into Fourier series

*c*is the coefficient of the

_{n}*n*-th harmonic, and its detailed format can be calculated easily.

## 3. Electromagnetically induced holographic imaging

In this section, based on the one-dimensional EIG obtained in the above, the schemes of EICHI and the EIQHI are given, respectively. It should be noted that, as a proof-of-principle experiment, the one-dimensional EIG to be holographically imaged is an amplitude grating.

**EICHI** The scheme of the EICHI under consideration is sketched in Fig. 3. A coherent field is split by a beam splitter (BS) into an object field traveling along the object path and a reference field freely along the reference path. These two fields are then interfered and the intensity is recorded by the detector *D*. The medium, i.e., the hybrid artificial molecules, modulated by the standing wave, is placed on the object path. The distance from the source of the coherent field to the atomic medium is *z _{o}*

_{1}, and to the detector

*D*is

*z*; the distance between the medium and detector

_{r}*D*is

*z*

_{o}_{2}. The fields at the detection planes are related to the fields at the source plane by the Fresnel diffraction integral.

In holography, the object field is usually much weaker than the reference field. When the temporal coherence condition is satisfied, the holographic pattern is dominated by the interference term

*E*

_{0}(

*x*

_{0}) is the light field distribution in the source plane.

*E*(

_{o}*x*) and

*E*(

_{r}*x*) are the fields in the recording planes for the object field and reference field, respectively.

*x*

_{0},

*x*′

_{0}and

*x*are the transverse positions across the fields. Under the paraxial approximation, the impulse response functions of the object field and reference field are respectively written as

Under the condition of coherent object field, i.e., $\u3008{E}_{0}^{*}({{x}^{\prime}}_{0}){E}_{0}({x}_{0})\u3009={\alpha}^{*}\alpha $, the interference term (5) can be factorized to be

*z*=

_{r}*z*

_{o}_{1}+

*z*

_{o}_{2}) in the interferometry does not need to meet due to the object field and reference field split from a coherent source with better temporal and spatial coherence to perform spatial interference.

Substituting Eq. (4) into Eq. (7) and completing the integration on *x*′, we can recover the expression

Calling to mind the Talbot effect, the above Eq. (8) has the same form with the result of traditional Talbot effect. According to the Fresnel diffraction integral, *z _{o}*

_{2}can be regards as their effective diffraction length, and some conclusions are immediately in order from the Eqs. (3 and 8). The first exponential term of the Eq. (8), which describes the phase changes of the diffraction orders, tells ones whether self-imaging occurs or not. The imaging of the EIG occurs at the planes

*z*

_{o}_{2}=

*m*Λ

^{2}

*/λ*, where

*m*denotes a positive integer referred to as the self-imaging number, then the imaging when

*m*is an odd integer shifts by a half-period with respect to that when

*m*is an even integer. The second exponential term of Eq. (8) releases the information of the self-imaging, its period, for example. Apparently, the period of the obtained imaging in the classical scheme equals that of the EIG itself.

**EIQHI** The scheme of the EIQHI under consideration is sketched in Fig. 4. A pair of entangled photons emerges from a nonlinear crystal via a spontaneous parametric down-conversion (SPDC), and then is split by a BS into a signal field
${E}_{s}^{(+)}(x)$ (with the wave number *k _{s}*) traveling along the signal path and an idler field
${E}_{i}^{(+)}(x)$ (with the wave number

*k*) freely along the idler path. The signal path contains two arms and is an interferometer. The signal field is divided into two parts, ${E}_{so}^{(+)}(x)$ and ${E}_{sr}^{(+)}(x)$, serving as the object and reference fields of the interferometer, respectively. Interference intensity of the interferometer is recorded by the detector

_{i}*D*, while the idler field propagates directly to the detector

_{s}*D*. The medium modulated by the standing wave is placed on the object field path. The distance from the output surface of the crystal to the atomic medium is

_{i}*z*

_{so}_{1}, to the detector

*D*(

_{s}*D*) through reference (idler) path is

_{i}*z*(

_{sr}*z*); the distance between the medium and detector

_{i}*D*is

_{s}*z*

_{so}_{2}.

For obtaining the holographic imaging in the quantum scheme, correlation measurement between idler and signal fields should be implemented. The two-photon coincidence counting rate, *R*(*x _{s},x_{i}*), according to Glauber’s quantum measurement theory [26] is

*x*(

_{s}*x*) is the transverse coordinates of the position on the detected plane of

_{i}*D*(

_{s}*D*). Because of ${E}_{s}^{(+)}={E}_{so}^{(+)}+{E}_{sr}^{(+)}$, the two-photon coincidence counting rate consists of four parts: two parts are the two-photon intensities and the other two parts are the two-photon interference terms given by

_{i}These two terms define the spatial interference of two two-photon wavepackets. It does not exist in the other quantum imaging schemes, and may include the holographic information.

For simplicity, we consider an ideal two-photon entangled state at the source, satisfying $C({{x}^{\prime}}_{0},{{x}^{\u2033}}_{0})=\delta ({{x}^{\prime}}_{0}-{{x}^{\u2033}}_{0})$. The two-photon wavepacket can be calculated by

*h*and

_{i}*h*(

_{j}*j*=

*so, sr*) are the impulse response functions of idler and signal fields showing as follows

Substituting Eqs. (4, 11 and 12) into Eq. (10) and completing the integration on *x*_{0} and *x*′, we recover the interference amplitude

*β*=

*k*. By comparison, we can see that the above result has the same form with that in the EICHI case, while the effective diffraction length in the present case is $\frac{{z}_{so2}({z}_{so1}+\beta {z}_{i})}{{z}_{so1}+{z}_{so2}+\beta {z}_{i}}(\equiv {Z}_{q})$. It is apparent that, according to the Talbot effect, the self-imaging occurs at ${Z}_{q}=\frac{m}{2}{z}_{T}$ (here

_{s}/k_{i}*z*= 2Λ

_{T}^{2}

*/λ*and is called as the Talbot length), where

_{s}*m*is a positive integer. The Eq. (13) also releases that the period of the imaging is not any more always the same with that of the EIG itself, it is decided by the way in which the two detectors scan across the signal and idler fields. Three special scanning ways can be employed. The first is that the both detectors scan across the fields synchronously, i.e.,

*x*=

_{s}*x*. The interference amplitude

_{i}*I*(

*x*) at the

_{s},x_{i}*m*th Talbot plane reduces to $I({x}_{i},{x}_{i})\propto {\displaystyle \sum _{n=-\infty}^{+\infty}{c}_{n}\mathrm{exp}[-im{n}^{2}\pi ]\mathrm{exp}[-i\frac{2n\pi {x}_{i}}{\mathrm{\Lambda}}]}$ and the imaging is completely the same as that obtained in the EICHI case. The second is that one of the two detectors (

*D*and

_{s}*D*) is fixed at its origin, the detector

_{i}*D*is located at

_{s}*x*= 0 but

_{s}*D*moves, for example. One has $I(0,{x}_{i})\propto {\displaystyle \sum _{n=-\infty}^{+\infty}{c}_{n}\mathrm{exp}[-im{n}^{2}\pi ]\mathrm{exp}[-i\frac{2n\pi {x}_{i}}{\mathrm{\Lambda}}/\left(1+\frac{{z}_{so1}+\beta {z}_{i}}{{z}_{so2}}\right)]}$ and, compared to the original EIG, the imaging is magnified by the factor of $1+\frac{{z}_{so1}+\beta {z}_{i}}{{z}_{so2}}$. The third is that the two detectors scan across the fields along the opposite directions, i.e.,

_{i}*x*=

_{s}*−x*, and the obtained imaging in size is $1+\frac{2{z}_{so2}}{{z}_{so1}+\beta {z}_{i}-{z}_{so2}}$ times larger than the original EIG. In the later two cases, by modulating the parameters (

_{i}*z*

_{so}_{1},

*z*

_{so}_{2},

*z*) properly, an arbitrary larger holographic imaging can be achieved in theory.

_{i}In the above analysis, we discuss the point detection case for the both two detectors. Now we consider the case of bucket detection of the signal photons, the interference term, *I*(*x _{s},x_{i}*), can be rewritten as

From Eqs. (8 and 13) we have predicted analytically the features of the EICHI and EIQHI of the one-dimensional EIG. Now, we testify the prediction by numerical simulation. Figure 5(a) displays the imaging of the EIG as *R* = 10*nm* in the EICHI case, the solid (black) and the dashed (red) curves are related to the self-imaging number *m* = 1 and *m* = 2, respectively. The imaging does be the same with the EIG itself (see the Fig. 2), and the imaging when *m* is an odd integer does shift by a half-period with respect to that when *m* is an even one. Figures 5(b) and 5(c) show the imaging of the same EIG in the EIQHI case with *m* being an even. In Fig. 5(b), detector *D _{i}* scans across the idler field but

*D*is fixed at

_{s}*x*= 0.0. The black solid, the red dashed and green dash-dotted curves are related to the effective diffraction lengths

_{s}*Z*= 2

_{q}*z*(

_{T}*m*= 4,

*z*

_{so}_{1}= 4

*cm*,

*z*

_{so}_{2}= 10

*cm*,

*z*= 6

_{i}*cm*),

*Z*= 2

_{q}*z*(

_{T}*m*= 4,

*z*

_{so}_{1}= 4

*cm*,

*z*

_{so}_{2}= 7.5

*cm*,

*z*= 11

_{i}*cm*) and

*Z*= 3

_{q}*z*(

_{T}*m*= 6,

*z*

_{so}_{1}= 4

*cm*,

*z*

_{so}_{2}= 10

*cm*,

*z*= 26

_{i}*cm*), respectively, and the corresponding magnification factors are 2, 3 and 4. In Fig. 5(c), the two detectors scan two fields in the three different ways on a given Talbot plane (

*m*= 6,

*z*

_{so}_{1}= 4

*cm*,

*z*

_{so}_{2}= 10

*cm*,

*z*= 26

_{i}*cm*), where the black solid (

*x*=

_{s}*x*), red dashed (

_{i}*x*= 0) and green dash-dotted (

_{s}*x*= −

_{s}*x*) curves correspond to these three scanning ways and have magnifications 1, 4 and 2, respectively. It is apparent that no matter what kind of circumstances, in such an amplitude EIG case the visibility of the obtained imaging at the Talbot planes approaches almost unity. We note here that in the case with other parameters, the imaging process is exactly the same as that in the cases analyzed above, except for the variation of the original EIG as shown in Fig. 2, which will influence the maximum amplitude contrast of the imaging.

_{i}## 4. Conclusion

In conclusion, we have theoretically proposed two schemes of holographic imaging, i.e., the EICHI and the EIQHI, for an one-dimensional EIG in a SQD-MNP hybrid nanostructure based on the EIT. The effect is capable of lensless imaging and may reduce the influence from vibrations in the experiment. The results show that the parameters of the obtained EIG can be easily modulated due to the existence of the MNP, which can effectively change the absorption and the refraction of the hybrid artificial molecule. The period of the obtained imaging in the classical scheme is completely the same with that of the EIG itself, is about several tens of millimeters for the visible light, therefore a second imaging process would be necessary to magnify the self-imaging as implemented in paper [27], while the imaging obtained in the quantum one, similar to the results in our previous work [18], has the characteristic of nonlocality and of arbitrarily controllable imaging variation in size. In contrast to the previous atomic imaging method [17, 18], the present one can extend the rang of the objects to be imaged to an arbitrary shaped object, and the obtained imaging can include both the amplitude and phase information. All the advantages make the schemes have important potential applications in the imaging and classical or quantum information processing. The present schemes can also be easily extend to the imaging of two- or three-dimensional EIG, which can be realized by applying two or three mutually perpendicular standing waves in a multilevel atomic system. The related works can be seen elsewhere.

## Acknowledgments

This research was supported by National Natural Science Foundation of China, Project Nos. 11447156, and by Natural Science Foundation of Shandong Province, Project No. ZR2014AP006.

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