Negative refraction without absorption via both coherent and incoherent fields in a four-level left-handed atomic systemSupported by National Natural Science Foundation of China (Grant No.60768001 and No.10464002).

Abstract

This paper attempts a probe into negative refraction without absorption by means of an incoherent pump field and a strong coherent field coupling the dense four-level atomic system.With the application of the incoherent pump field to manipulate the populations in atomic levels and the variable strong coherent field to create quantum coherence, the constraint condition of two equal transition frequencies responding to the probe field in the atomic system isn’t required.And these lead to the propagation transparency and strong magnetic response of the probe field,left-handedness with vanishing absorption in the atomic system.However,an excessive coherent field intensity would increase the absorption.

PACs

42.50.Gy

Keywords

negative refraction; without absorption; left-handed; coherent field; incoherent field.

I Introduction

Negative refraction of electromagnetic radiation[1]has recently attracted considerable attention because of its surprising and counterintuitive electromagnetical and optical effects,such as the reversals of both Doppler shift and Cherenkov effect,negative refraction[2],amplification of evanescent waves[3]and subwavelength focusing [3-5], negative Goos-H$\ddot{a}$nchen shift [6] and quenching spontaneous emission[7,8]and so on[9].And material with negative refraction index intrigues the researchers because of its many significant potential applications.The“ perfect lens ”is one of them.Since a slab of such materials has an ability to focus all frequency components of a two-dimensional image, it may become possible to make a “ perfect lens ”in which imaging resolution does not limited by the diffraction limit[3].Up to now, there have been several approaches to the realization of negative refractive index materials, including artificial composite metamaterials [10-11], photonic crystal structures [12],transmission line simulation[13] and chiral media[14-15] as well as photonic resonant materials(coherent atomic vapour)[16-19].In such a type of negative refractive materials, negative refraction is usually accompanied by a strong absorption especially towards higher frequencies. Thus,the realization of negative refraction material without absorption is of great significance.And some effort[20-23]has been made to realize negative refraction without absorption.K$\ddot{a}$stel et al[20] realized negative refraction with minimal absorption in a dense atomic gas via electromagnetically induced chirality. The key ingredient of the scheme is the electromagnetic chirality that results from coherently coupling a magnetic dipole transition with an electric dipole transition via atomic coherence induced by the two-photon resonant Raman transitions. In Ref.[21],K$\ddot{a}$stel et al also discussed negative refraction with reduced absorption due to destructive quantum interference in coherently driven atomic media.The negative refraction with deeply depressing absorption and without simultaneously requiring both negative electric permittivity and magnetic permeability (left-handedness)[2]was obtained by F.L. Li[23].Ref.[23] shows this at the ideal situation that the two chirality coefficients have the same amplitude but the opposite phase.

In this paper we propose an indirect coupling way of the atoms responding to the probe field via an incoherent pump field and a coherent coupling field.With the application of the incoherent pump field to manipulate the population distribution of each level and the variable coherent field to create quantum coherence,the magnetic response is amplified and the probe field propagates transparently. The atomic system displays negative refraction without absorption and left-handedness.The constraint condition of two equal transition frequencies responding to the probe field does not require,which is different from Ref.[18,24].

II Theoretical model

The level configuration of atoms under consideration is shown in Figure 1. The properties of the four atomic states are as following: levels $|1\rangle$,$|3\rangle$, and $|4\rangle$ have same parity, and the parity of level$|2\rangle$ is opposite with theirs. The two lower levels $|1\rangle$ and $|2\rangle$ have opposite parity and so $\langle 2|$$\hat{\vec{d}}$$|1\rangle$$\neq 0$ where $\hat{\vec{d}}$ is the electric dipole operator. The two upper levels,$|3\rangle$ and $|4\rangle$ have the same parity with $\langle 4|$$\hat{\vec{\mu}}$$|3\rangle$$\neq 0$ where $\hat{\vec{\mu}}$ is the magnetic-dipole operator. As observed in Fig.1,three electromagnetic fields are introduced to couple the four states:The electric and magnetic components of the probe light (corresponding Rabi frequency $\Omega_{p}$$=\frac{\vec{E_{P}}\vec{d_{21}}}{\hbar}$ ,$\Omega_{B}$$=\frac{\vec{B_{P}}\vec{\mu_{43}}}{\hbar}$) interact with the transitions $|2\rangle$ and$|1\rangle$ as well as$|4\rangle$ and$|3\rangle$, respectively.The incoherent pump field with pumping rate denoted by $\Gamma$ pumps atoms in level $|1\rangle$ into upper level $|3\rangle$, and then the atoms decay into metastable level $|2\rangle$ via rapid nonradiative transitions, whose decay rate is denoted as $\Gamma_{32}$.The strong coherent field $\Omega_{c}$ takes two effects on the system:(i) populations in level $|2\rangle$ are being pumped into level $|4\rangle$ and causes population reversion gain for magnetic-dipole transition $|3\rangle$$\leftrightarrow$$|4\rangle$. (ii)quantum coherence is induced by it and causes energy levels $|2\rangle$ and $|4\rangle$ split into two dressed sublevels and results in enough response of the electric part of probe field at certain frequency extent.

Figure 1: Four-level atomic medium interacting with a coherent $\Omega_{c}$ , an incoherent pumps $\Gamma$ and a probe field $\Omega_{p}$.

Using the density-matrix approach, the time-evolution of the system is described as

$$\frac{d\rho}{dt}=-\frac{i}{\hbar}[H,\rho]+\Lambda\rho,$$ (1)

Where $\Lambda\rho$ represents the irreversible decay part in the system.Under the dipole approximation and the rotating wave approximation the density matrix equations described the system are written as follows:

$$\dot{\rho_{11}}=\Gamma(\rho_{33}-\rho_{11})+\Gamma_{21}\rho_{22}+\Gamma_{31}\rho_{33}+\Gamma_{41}\rho_{44}+i\Omega_{p}(\rho_{21}-\rho_{12}),$$ (2)

$$\dot{\rho_{21}}=-(\gamma_{21}+i\Delta_{p})\rho_{21}-i\Omega_{p}(\rho_{22}-\rho_{11})+i\Omega_{c}\rho_{41},$$ (3)

$$\dot{\rho_{22}}=-\Gamma_{21}\rho_{22}+\Gamma_{32}\rho_{32}+\Gamma_{42}\rho_{44}+i\Omega_{p}(\rho_{12}-\rho_{21})+i\Omega_{c}(\rho_{42}-\rho_{24}),$$ (4)

$$\dot{\rho_{31}}=-[\gamma_{31}+i(\Delta_{c}+\delta)]\rho_{31}+i\Omega_{p}\rho_{23}+i\Omega_{B}\rho_{41},$$ (5)

$$\dot{\rho_{32}}=-[\gamma_{32}+i(\Delta_{c}-\Delta_{p}+\delta)]\rho_{32}-i\Omega_{c}\rho_{34}-i\Omega_{p}\rho_{31}+i\Omega_{B}\rho_{42},$$ (6)

$$\dot{\rho_{33}}=-\Gamma(\rho_{33}-\rho_{11})-\Gamma_{31}\rho_{33}-\Gamma_{32}\rho_{33}+\Gamma_{43}\rho_{44}+i\Omega_{B}(\rho_{43}-\rho_{34}),$$ (7)

$$\dot{\rho_{41}}=-[\gamma_{41}+i(\Delta_{p}+\Delta_{c})]\rho_{41}+i\Omega_{c}\rho_{21}-i\Omega_{p}\rho_{42}+i\Omega_{B}\rho_{31},$$ (8)

$$\dot{\rho_{42}}=-(\gamma_{42}+i\Delta_{c})\rho_{42}+i\Omega_{c}(\rho_{22}-\rho_{44})-i\Omega_{p}\rho_{41}+i\Omega_{B}\rho_{32},$$ (9)

$$\dot{\rho_{43}}=-[\gamma_{43}+i(\Delta_{p}-\delta)]\rho_{43}+i\Omega_{c}\rho_{23}+i\Omega_{B}(\rho_{33}-\rho_{44}),$$ (10)

where the above density matrix elements obey the conditions:$\rho_{11}+\rho_{22}+\rho_{33}+\rho_{44}$=1 and $\rho_{ij}$=$\rho_{ji}^{\ast}$.And $\Gamma_{ij}$(i,j=1,2,3,4)is the spontaneous emission decay rate from level $|i\rangle$ to level$|j\rangle$,ignoring the collision broaden effect.$\gamma_{21}$=$\Gamma_{21}$/2,$\gamma_{31}$=($\Gamma_{31}+\Gamma_{32}$)/2, $\gamma_{41}$=($\Gamma_{43}+\Gamma_{42}$)/2, $\gamma_{42}$=($\Gamma_{43}+\Gamma_{31}+\Gamma_{21}$)/2, $\gamma_{43}$=($\Gamma_{43}+\Gamma_{42}+\Gamma_{31}+\Gamma_{32}$)/2,$\gamma_{32}$= ($\Gamma_{32}+\Gamma_{31}+\Gamma_{21}$)/2 are the decay rates to the corresponding transitions.The detuning of the fields defined as $\Delta_{p}$=$\omega_{21}$-$\omega_{p}$,$\Delta_{c}$=$\omega_{42}$-$\omega_{c}$, $\Delta_{B}$=$\omega_{43}$-$\omega_{P}$,respectively.$\omega_{ij}$=$\omega_{i}$-$\omega_{j}$ is the transition frequency of level$|i\rangle$ and $|j\rangle$ (i, j=1,2,3,4) and we have $\delta$ =$\Delta_{p}$-$\Delta_{B}$.

According to the classical electromagnetic theory,the electric polarizability is a rank 2 tensor defined by its Fourier transform $\vec{P}_{e}(\omega_{P})$ $=\epsilon_{0}$$\alpha_{e}(\omega_{P})$$\vec{E}(\omega_{P})$,which is calculated as the mean value of the atomic electric-dipole moment operators by the definition $\vec{P}_{e}$ =Tr$\{$${\hat{\rho}\vec{d}}$$\}$=$\rho_{12}d_{21}$+c.c.where Tr stands for trace.In the following, we only consider the polarizability at the frequency $\omega_{P}$ of the incoming field $\vec{E}_{p}$.Therefore we drop the explicit $\omega_{P}$ dependence $\alpha_{e}(\omega_{P})\equiv\alpha_{e}$.Moreover,we choose $\vec{E}_{p}$ parallel to the atomic dipole $\vec{d}_{21}$ so that $\alpha_{e}$is a scalar,and its expression is as follows:

$$\alpha_{e}=\frac{\vec{d}_{21}\rho_{12}}{\epsilon_{0}\vec{E}_{p}}=\frac{\mid{d_{21}}\mid^{2}\rho_{12}}{\epsilon_{0}\hbar\Omega_{p}},$$ (11)

In the same way, the classical magnetic polarizations of the medium $\vec{P}_{m}(\omega_{P})$=$\mu_{0}\alpha_{m}\vec{E}(\omega_{P})$,which is related to the mean value of the atomic dipole moment operator through $\vec{P}_{m}$ =Tr$\{$${\hat{\rho}\vec{\mu}}$$\}$=$\rho_{34}$$\mu_{43}+c.c$.According to the classical Maxwell’s electromagnetic wavevector relation,we choose magnetic dipole is perpendicular to the induced electric dipole so that the magnetizability $\alpha_{m}$ is scalar, and its expression is as follows:

$$\alpha_{m}=\frac{\mu_{0}\vec{\mu}_{43}\rho_{34}}{\vec{B}_{p}}=\frac{\mu_{0}\mid\mu_{43}\mid^{2}\rho_{34}}{\hbar\Omega_{B}}.$$ (12)

According to the Clausius-Mossotti relations considering the local effect in dense medium[25], the relative permittivity and relative permeability are expressed as [18,26]

$$\epsilon_{r}=\frac{1+\frac{2}{3}N\alpha_{e}}{1-\frac{1}{3}N\alpha_{e}},$$ (13)

$$\mu_{r}=\frac{1+\frac{2}{3}N\gamma_{m}}{1-\frac{1}{3}N\gamma_{m}}.$$ (14)

In the above, we obtained the expressions for the electric permittivity and magnetic permeability of the atomic media.In the section that follows, we will demonstrate that the simultaneously negative both permittivity and permeability,negative refraction without absorption can be observed in the four-level atomic system.

III Results and discussion

In the following,with the stationary solutions to the density-matrix equations(2)-(10), we explore the property of simultaneously negative both electric permittivity and magnetic permeability through the numerical calculations.And several typical parameters should be selected before the calculation.The parameters for the electric and magnetic polarizabilities of atoms can be chosen as: electric and magnetic transition dipole moments $d_{21}$=2.5$\times$$10^{-29}$C$\cdot$m and $\mu_{34}$=7.0 $\times$$10^{-23}$C$\cdot m^{2}s^{-1}$[16], respectively.The density of atoms N was choosed to be $6.5\times 10^{25}m^{-3}$[27,28].And the other parameters are scaled by $\gamma=10^{6}s^{-1}$: $\Gamma_{31}$=$\Gamma_{41}$=$\Gamma_{42}$=$\Gamma_{43}$= 0.01$\gamma$,$\Gamma_{21}$=0.5$\gamma$, $\Gamma_{32}$=1$\gamma$. The Rabi frequency of the probe field is $\Omega_{p}=0.5\gamma$ ,and $\delta$ = - 20$\gamma$ without the condition of the two transition frequencies $\omega_{43}$=$\omega_{21}$. The incoherent pumping rate is $\Gamma=1.0\gamma$. The strong coherent optical field couples the atomic system with variational Rabi frequencies $\Omega_{c}=15\gamma$, 18$\gamma$,21$\gamma$, 24$\gamma$ and fixed frequency detuning $\Delta_{c}$=20$\gamma$.

Figure 2: The relative electric permittivity $\epsilon_{r}$ and magnetic permeability $\mu_{r}$ of the system as a function of probe frequency detuning $\Delta_{p}/\gamma$ for different values of $\Omega_{C}$.

Figure 2 shows the calculated electric permittivity $\epsilon_{r}$ and magnetic permeability $\mu_{r}$ as a function of probe field detuning.Under the cooperation of the strong coherent field and incoherent pumping,the dense four-level atomic system exhibits left-handedness with simultaneously negative permittivity and permeability at some frequency extents of the probe field.From the profiles of the real part of the relative electric permittivity $\epsilon_{r}$,we observe their values being proximately equal -2 in the probe frequency range[-26$\gamma$,26$\gamma$],although the phase of the figures are proximately opposite at $\Delta_{p}$=12.5$\gamma$ when the Rabi frequency $\Omega_{C}$ varies by 15$\gamma$, 18$\gamma$, 21$\gamma$ and 24$\gamma$.The influence of the strong coherent field on the real part of the relative magnetic permeability $\mu_{r}$ is different.When $\Omega_{C}$=15$\gamma$,the value is all negative in the probe frequency extent.When $\Omega_{C}$=18$\gamma$,21$\gamma$, 24$\gamma$,the positive values emerge in the ranges of [-3$\gamma$, -1.1$\gamma$], [-3.4$\gamma$, -0.4$\gamma$], [-3.85$\gamma$,0.36$\gamma$],respectively.And their amplitude values are increasing with the variation of $\Omega_{C}$.However,the ranges for simultaneously negative electric permittivity $\epsilon_{r}$ and magnetic permeability $\mu_{r}$ still exist.In Figure 2,the transparency is shown by the imaginary part of both the relative electric permittivity $\epsilon_{r}$ and the relative magnetic permeability $\mu_{r}$in some frequency extents.This is a very significant result for us.As is well known, the photon absorption of atom can be greatly depressed via electromagnetically induced transparency (EIT)[29].And the zero absorption phenomena may occur in the EIT extents.With observation of the profile of the imaginary part of $\epsilon_{r}$, the excessive strong coherent field intensity causes the shrinking gain and increasing absorption at $\Delta_{p}$=12.5$\gamma$.However,the imaginary part of $\mu_{r}$ displays increasing gain,and its amplitude is gradually amplified and gradually approaching to the resonant point when the coherent field is varied the Rabi frequencies by 15$\gamma$, 18$\gamma$,21$\gamma$ and 24$\gamma$. Comparing the images of the relative electric permittivity $\epsilon_{r}$ with the relative magnetic permeability $\mu_{r}$ in Figure 2,we notice that the magnetic response of the probe field is stronger than the electric response.The reason may come from that: the magnetic component of the probe field couples levels $|4\rangle$ and $|3\rangle$,and the strong coherent field drives levels $|2\rangle$ and $|4\rangle$.The effect of quantum coherence caused by the strong coherent field on the magnetic component is stronger than on the electric component which couples levels $|2\rangle$ and $|1\rangle$.

Figure 3: The refractive index as a function of the probe field detuning $\Delta_{p}/\gamma$for different values of $\Omega_{C}$.

Figure 4: The figure of merit (FOM: $|real(n)/imag(n)|$) as a function of the detuning parameter detuning $\Delta_{p}/\gamma$for different values of $\Omega_{C}$.

In Figure 3,the refraction index according to definition of the left-handed material ( $n(\omega)=-\sqrt{\epsilon_{r}(\omega)\mu_{r}(\omega)}$ ) [2] is plotted for different values of parameter $\Omega_{C}$.As observed in Figure 3,the real part of the refractive index shows negative values and their amplitudes enlarge gradually with the increasing of $\Omega_{C}$. The imaginary part of the refractive index displays absorption enhancing near the resonant point and absorption depression on the both sides,when the coherent field is varied the Rabi frequencies by 15$\gamma$, 18$\gamma$,21$\gamma$ and 24$\gamma$.The figure of merit(FOM)$|Re(n)/Im(n)|$ shows how much the absorption is suppressed .When the FOM is much larger than unity, it means that there is almost no absorption in this area. As shown in Figure 4, the FOM is far less than unity in the area near the resonant point.This illustrates that the increasing absorption occurs in this area ,which is consistent with the peaks of Im[n] near the resonance in Figure 3. And the FOM has the largest value at the identical re-scaled detuning parameter $\Delta_{p}/\gamma$ when $\Omega_{C}$= 15$\gamma$. It means that the absorption is depressed deeply when the coherent field varies its Rabi frequency to $\Omega_{C}$= 15$\gamma$,and the excessive coherent field intensity wouldn’t help to realize zero absorption.The markedly feature of present scheme is the absorption depression almost to zero in the left-handed atomic system. One may get the reason for this from the results shown in Figure 2:Because of the EIT effect, the electric and magnetic non-absorption extents appear in the re-scaled detuning parameter $\Delta_{p}/\gamma$ extent,which brings about the zero value of the imaginary part of refraction index. As a result, the negative refraction without absorption occurs in the left-handed atomic system .

In experimental investigation, the solid sample of $E_{r}^{3+}$ doped into calcium fluorophosphate at room temperature might be a good candidate because it has abundant energy levels, various electric magnetic transitions and high density in the order $10^{25}m^{-3}$[27,28].And the four levels $|1\rangle$,$|2\rangle$, $|3\rangle$,and $|4\rangle$ in Figure 1 correspond to the energy level configuration $I_{15/2}^{4}$,$I_{13/2}^{4}$, $I_{9/2}^{4}$ and $I_{11/2}^{4}$ of the rare earth ion $E_{r}^{3+}$ doped in calcium fluorophosphate, respectively.

IV Conclusion

In conclusion,we have demonstrated a scheme for realizing negative refraction without absorption via an incoherent pump field and a strong coherent field in the dense four-level atomic system.With the application of the incoherent pump field to manipulate the population distribution of each level and the variable strong coherent field to create quantum coherence,the magnetic response is amplified and the probe field propagates transparently in some frequency extents.Without the constraint condition of two equal transition frequencies responding to the probe field,the atomic system displays negative refraction with vanishing absorption and left-handedness. Therefore,our aim for searching the low-loss negative refraction is possible by choosing appropriate parameters in the scheme,in respect that the main applied limitation of the negative refractive materials is the large amount of dissipation and absorption.However,an excess coherent field intensity would increase the absorption near the resonance.This can also be obviously observed here.