Large and tunable negative refractive index via electromagnetically induced chirality in a semiconductor quantum well nanostructure
aa r X i v : . [ phy s i c s . op ti c s ] D ec Large and tunable negative refractive index via electromagnetically induced chiralityin a semiconductor quantum well nanostructure
Shun-Cai Zhao, ∗ Shuang-Ying Zhang, and You-Yang Xu Physics department, Kunming University of Science and Technology, Kunming, 650500, PR China (Dated: December 20, 2018)Large and tunable negative refractive index (NRI) via electromagnetically induced chirality isdemonstrated in a semiconductor quantum wells (SQWs) nanostructure by using the reported ex-perimental parameters in Ref.[19]. It is found: the large and controllable NRI with alterable fre-quency regions is obtained when the coupling laser field and the relative phase are modulated, whichwill increase the flexibility and possibility of implementing NRI in the SQWs nanostructure. Thescheme rooted in the experimental results may lead a new avenue to NRI material in solid-statenanostructure.
I. INTRODUCTION
In recent years there has been a huge interest in semi-conductor quantum wells(SQWs), on account of the in-tersubband transitions (ISBTs) in SQWs are believedto have great potential applications in solid-state op-toelectronics and quantum information science[1–4]. Inmany case the characteristics of ISBT dephasing mecha-nisms in SQWs behave essentially as “artificial atoms”,which allowed us to regard the SQW as a single quan-tum object for many kinds of quantum optical phe-nomena, in which the nonlinear quantum optical phe-nomena have been extensively discussed, such as Kerrnonlinearity[5], ultrafast all optical switching[6], coher-ent population trapping[7], electromagnetically inducedtransparency[8–10], gain without inversion[11], enhanc-ing index of refraction[12] and other novel phenomena[13–18]. But not only that, the devices based on ISBTsin SQWs also have many inherent advantages that theatomic systems don’t have, such as the high nonlinearoptical coefficients, the large transition energies and elec-tric dipole moments for the small effective electron mass,the great flexibilities in symmetries design as well as indevices design by choosing the materials and structuredimensions.
II. THEORETICAL MODEL
In this work, we investigate the refractive indexin a bulk SQW nanostructure via electromagneticallyinduced chirality. And the results demonstrate thelarge and tunable NRI can be realized by the coupling ∗ [email protected] laser field and the relative phase. The SQW sam-ples we simulate here are reported to be grown by themolecular beam epitaxy (MBE) method and each con-tains a 4.8nm In . Ga . As / 0.2nm Al . In . As /4.8nm In . Ga . As coupled quantum well, separatedby a 36nm Al . In . As modulation-doped barriers[19,20]. And the transition energies of the ISBTs were mea-sured as ω =124 meV, ω =185 meV, implying ω =309meV[19]. The corresponding transition dipoles were cal-culated as d =2.335 nm, d =2.341 nm, and d =0.120nm[21]. So the synthesized 3-level cascade electronic sys-tem of ISBTs in such SQW forms a familiar ladder con-figuration, as shown in Fig.1. FIG. 1.
Fig . Schematic diagram of the SQW structure. Theelectric-dipole transition | i - | i is driven by the coupling laserfield, and the level pairs | i - | i and | i - | i are coupled to theelectric and magnetic fields of the weak probe light, respec-tively. The equivalent 3-level loop “ atom” system simulatedin our work is mainly based on the above reported SQWsample. In such a 3-level system, the parity of level | i is set to be opposite to those of the levels | i and | i ,which have the same parity. The possible optical tran-sition | i - | i is mediated by a coupling laser field withcentral frequency ω c and Rabi frequency Ω c . A weakprobe laser field with central frequency ω p and Rabi fre-quency Ω pe is applied to the ISBT | i - | i . Because ofthe parity selection rules, the two levels | i and | i withelectric dipole element d = h | ˆ ~d | i6 = 0 are coupled bythe electric component of the weak probe field, where ˆ ~d is the electric dipole operator. The two levels | i and | i with the magnetic dipole element µ = h | ˆ ~µ | i6 = 0 arecoupled by the magnetic component of the probe fieldwith Larmor frequency Ω pb = ~B p µ / ~ , where ˆ ~µ is themagnetic-dipole operator. The coherent cross-couplingbetween electric and magnetic dipole transitions drivenby the electric and magnetic components of the probefield may lead to chirality[25, 26]. Because the transition | i - | i is mediated by a coupling laser field, the transi-tions | i - | i and | i - | i are coupled respectively by theelectric and magnetic components of the probe field; itthen forms a closed-loop system. It is well known thatin a loop configuration the dynamics behaviour becomesquite sensitive to phases[22].Under the rotating-wave approximations in the inter-action representation, with the assumption of ~ =1 andemploying the quantum regression theorem and the es-tablished treatment for 3-level atoms[23], the resultingHamiltonian of the three-level configuration can be writ-ten as H int = ∆ pe | ih | + (∆ c + ∆ pb ) | ih | − (Ω pe exp ( − iϕ pe ) | ih | + Ω pb exp ( − iϕ pb ) | ih | + Ω c (1) exp ( − iϕ c ) | ih | + H.c ) , where ∆ pe = ω - ω p , ∆ pb = ω - ω p and ∆ c = ω - ω c are theISBTs detunings of the corresponding fields, and ∆ c depicts the two photon detuning process. Accordingto the measured transition energies in the above SQWsample[19], we can get the relationship ∆ c =∆ pe + ∆ pb ,which can avert the major obstacle mentioned in Ref.[24]in realizing the predicted effects at a realistic experimen-tal setting. The ϕ pb , ϕ pe and ϕ c are denoted the rel-evant phases of the three coherent fields. The symbolH.c. means the Hermitian conjugate. Then the equationof the time-evolution for the system can be described as dρdt = − i ~ [ H, ρ ] + Λ ρ , where Λ ρ represents the irreversibledecay part in the system. Under the dipole approxima-tion the density matrix equations described the system are written as follows:˙ ρ = i Ω pb ( ρ − ρ ) + i Ω pe ( ρ − ρ ) − γ ρ + γ ρ , ˙ ρ = i Ω pb ( ρ − ρ ) + i Ω c exp ( − iφ ) ρ − i Ω c exp ( iφ ) ρ − ( γ + γ ) ρ , ˙ ρ = i Ω pe ( ρ − ρ ) + i Ω c exp ( iφ ) ρ − i Ω pb ( ρ + ( i ∆ pe − Γ ρ , (2)˙ ρ = i Ω c exp ( iφ )( ρ − ρ ) + i Ω pe ( ρ − i Ω pb ρ + [ i (∆ pe + ∆ pb ) − Γ ρ , ˙ ρ = i Ω pb ( ρ − ρ ) + i Ω pe ρ − i Ω c ( ρ +( i ∆ pb − Γ ρ , where φ = ϕ c − ϕ pe − ϕ pb is the relative phase of thethree corresponding optical fields. And the above den-sity matrix elements comply with the conditions: ρ + ρ + ρ =1 and ρ ij = ρ ∗ ji . The total decay rates Γ ij areadded phenomenologically[19, 25] in above density ma-trix equations, which include the population decay ratesand dephasing decay rates. Among them, the popula-tion decay rates from ISBTs, denoted by γ ij , are pri-marily due to longitudinal optical(LO) phonon emissionevents at low temperature. And the dephasing decayrates originate from electron-electron, interface rough-ness and phonon scattering processes. Thus the totaldecay rates Γ ij can been written by Γ = γ + γ dph ,Γ = γ + γ + γ + γ dph , Γ = γ + γ + γ dph . Whenthe probe field is weak, i.e. Ω pe , Ω pb ≪ Ω c , Γ ij , almost allthe atoms can be assumed to be in the ground state | i ,the steady-state values of the density matrix elements ρ and ρ can be written in a linear approximation ρ = 2 A d iA A E + 4 e − iφ Ω c (Γ + 2 i ∆ pb ) u A A B , (3) ρ = 4 e iφ A Ω c d A A ∗ E + 8 u e iφ Ω c iA ∗ A ∗ B (4)where A ∗ i ( i =1 , , ,
4) is the conjugate complex of A i , andwith A =Γ (Γ + 2 i ∆ pb ) − i Γ ∆ pe + 4(∆ pe ∆ pb + Ω c ) ,A =Γ (Γ + 2 i ∆ pb ) + Γ (Γ + 2 i ∆ pb ) − A − pb ,A =Γ + 2 i (∆ pe + ∆ pb ), A =Γ + Γ − i ∆ pb . Theensemble electric polarization and magnetization of thebulk SQWs to the probe field are given by ~P = N ~d ρ and ~M = N ~µ ρ , respectively, where N is the density ofSQWs. Then the coherent cross-coupling between elec-tric and magnetic dipole transitions driven by the elec-tric and magnetic components of the probe field maylead to chirality[26, 27]. Substituting equations (3) and(4) into the formula for the ensemble electric polar-ization ( ~P = N ~d ρ ) and magnetization ( ~M = N ~µ ρ ),we have the relations ~P = α EE ~E + α EB ~B , ~M = α BE ~E + α BB ~B . Considering both electric and magnetic local fieldeffects[28], ~E and ~B in ~P and ~M must be replaced by thelocal fields: ~E L = ~E + ~P ε , ~B L = µ ( ~H + ~M ). Then we ob-tain ~P = 3 ε ( µ α BB α EE − µ α BE α EB − α EE ) B ~E + − µ ε α EB B ~H, (5) ~M = 3( µ α BE α EB − µ α BB α EE + 3 ε µ α BB ) B ~H + 9 ε α BE B ~E where B = µ α BE α EB + 3 α EE − µ α BB α EE − ε +3 µ ε α BB , B = µ α BB α EE + 9 ε − µ α BE α EB − α EE − ε µ α BB with α EE = − (Γ + 2 i ∆ pb ) +2 i Γ (∆ pb + ∆ pe ) − pb + ∆ pb ∆ pe + Ω c )] d / [2(∆ pe +∆ pb ) − i Γ ][Γ (Γ + 2 i ∆ pb ) − i Γ ∆ pe + 4(∆ pe ∆ pb +Ω c )] ~ , α EB =4 e − iφ (Γ + 2 i ∆ pb )Ω c µ / [Γ + 2 i (∆ pe +∆ pb )][Γ (Γ + 2 i ∆ pb ) − i Γ ∆ pe + 4(∆ pe ∆ pb + Ω c )] ~ , α BE =4 e iφ (Γ + Γ − i ∆ pb )Ω c d / [Γ − i (∆ pe +∆ pb )][Γ (Γ − i ∆ pb ) + 2 i Γ ∆ pe + 4(∆ pe ∆ pb +Ω c )] ~ , α BB =8 e iφ Ω c µ / [Γ − i (∆ pe +∆ pb )][Γ ( i Γ +2∆ pb ) − ∆ pe + 4 i (∆ pe ∆ pb + Ω c )] ~ . The key idea ofelectromagnetic induced chirality is to use the magneto-electric cross-coupling in which the electric polarization ~P is coupled to the magnetic field ~H of an electromag-netic wave and the magnetization ~M is coupled to theelectric field ~E [27]: ~P = ε χ e ~E + ξ EH c ~H, ~M = ξ HE cµ ~E + χ m ~H (6)Here χ e and χ m , ξ EH and ξ HE are the electricand magnetic susceptibilities, and the complex chiral-ity coefficients, respectively. They lead to additionalcontributions to the refractive index for one circularpolarization[27, 29]: n = r εµ − ( ξ EH + ξ HE ) i ξ EH − ξ HE ) (7)By comparison with equations (5) and (6), we obtainthe permittivity by ε =1 + χ e and the permeability by µ =1 + χ m , and the complex chirality coefficients ξ EH and ξ HE . In the above, we obtained the expressions forthe electric permittivity and magnetic permeability ofthe bulk SQWs. Substituting equations from (3) to (6) into (7), the expression for refractive index can also bepresented. In the section that follows, we will discussthe refractive index of the SQWs with the experimentalparameters from Ref.[19]. III. RESULTS AND DISCUSSION
In this work, some key parameters are selected fromthe Ref.[19], which has the advantage than the photonic-resonant materials because of the parameters used inthe numerical simulation coming from the reported ex-perimental results. The key parameters of the abovementioned SQWs sample are the measured transitionenergies[19], according to which the detunings of electric-dipole and magnetic-dipole transitions are defined by therelation ∆ pb =∆ pe +61 mev, so we depict the two tran-sitions to be different by setting ∆ pb =∆ pe (i.e. twotransition frequencies are not near the same frequencyin this SQWs system)[24]. In order to ensure linearresponse, the probe laser field is kept more than 100times weaker to the coupling laser field. As far as thementioned bulk SQWs sample be concerned, we esti-mate its average density in the cubic volume element asN ≈ . × m − from the mentioned experimental pa-rameters. The parameter for the electric transition dipolemoment from | i ↔ | i is chosen from the measured pa-rameter: d = 2 . × . × − Cm , and the mag-netic transition dipole moment is chosen from the typicalparameter µ = 7 . × − Cm s − [30]. Thus the totaldecay rates Γ ij are set Γ ij =5 mev for all three ISBTstransitions from the measured parameters[19]. Anotherparameter is the coupling laser intensity, which shouldbe below the damage threshold of SQWs. For its Rabifrequency, in this paper we choose the ranges of Ω c ≤ c = 20 meV to make a calculation, the electricfield amplitude is obtained E ≈ V /cm according tothe relationship ~ Ω = µE . Using of the connection be-tween the electric field amplitude E and the intensity ofthe radiation I : E=27.4682 ×√ I , we can get the intensityof the radiation I=13 MW/ cm , which shows the above-mentioned laser intensities may be satisfied below thedamage threshold of quantum object[31]. For the sake ofsimplification, we take the unit with ε = µ = 1.Firstly, we discuss the refractive index dependence onthe coupling laser field’s Rabi frequences. We concen-trate on the situation when the negative refraction ismost prominent, i.e., under the condition of φ = 0. InFig. 2, the plots of Re[n] is shown as a function of ∆ pe with different Ω c = 9 mev, 11 mev, 13 mev and 15 mev.And the dotted line, dashed dotted line, dashed line andsolid line correspond to the four different values of Ω c ,respectively. The dotted line has the most narrow inter-val of [-62.7mev, 58.3mev], in which the refraction indexis negative and its maxima is -0.7. The dashed dottedline shows its maxima of -1.32 in the interval [-64.5 mev,-56.65 mev] for NRI. The maxima of NRI arrives to -2.3in the interval [-64.0 mev, -54.5 mev] when Ω c was tunedto 13 mev. The interval expands to [-64.9 mev, -51.8mev] in which the maxima of NRI is -3.7 when Ω c =15mev. The increasing maxima of NRI in these expand-ing intervals is obtained by the gradually increasing Ω c ,which can be explained via the quantum interference andcoherence. The increasing coupling laser field drives thetransition | i - | i in the SQWs, which enhances interfer-ence between the electric dipole element and magneticdipole element from the probe laser in ISBTs, then mod-ifies the refractive index properties of the SQWs sample.The adjustable bandwidth for NRI in different frequencyregions can also be drawn from equations (3) and (4).In equations (3), the coherent term ρ is composed oftwo items: the former item is independent Ω c , while thelatter cross-coupling item is proportional to Ω c then besensitive to Ω c . In the coherent term ρ , the two partsare proportional to Ω c and Ω c , respectively. Then the in-duced chirality depends strongly on Ω c , which causes therefractive index being negative and the variable band-widths. FIG. 2.
Fig . The real parts of the refractive index n as afunction of the probe detuning ∆ pe with different Rabi fre-quencies Ω c of coupling laser field: 9 mev(dotted line), 11mev(dashed dotted line), 13 mev(dashed line), 15 mev(solidline). The parameters are: φ =0, Γ =Γ =Γ =5emv,Ω pe =0.02emv, ∆ pb =∆ pe +61 mev. After studying the refractive index dependence on thecoupling laser field’s Rabi frequencies, we next study how the relative phase φ bring changes in the refractive indexof the SQWs sample. As mentioned before, the dynam-ics behaviour is quite sensitive to phases[22] in a loopconfiguration, as can be see from equations (3) and (4).Equation (3) shows that the former coefficient in thecoherent term ρ is independent of the relative phase φ , while the latter coefficient is sensitive to the relativephase φ through e − iφ . Similarly, equation (4) has thephase-dependent chirality expression before coefficient ~E through e iφ and its second chirality part is dependent ofthe relative phase φ with e iφ . Thus, due to the presenceof the induced chirality depends strongly on the relativephase φ , which causes the refractive index to be negativevia the adjusting relative phase. FIG. 3.
Fig . The real parts of the refractive index n asa function of the probe detuning ∆ pe with different valuesof relative phase φ : 1.92 π (dotted line), 1.88 π (dashed dottedline), 1.86 π (dashed line) and 1.84 π (solid line), Ω c =20mev.The other parameters are the same as Fig.2. In Fig. 3, the refraction index Re[n] is plot-ted as a function of ∆ pe with the relative phase φ being 1.92 π (dotted line), 1.88 π (dashed dotted line),1.86 π (dashed line) and 1.84 π (solid line) and Ω c =20mev.The sensitive adjustability via the relative phase φ isshown by the curves in Fig. 3. Interestingly, the Re[n]varies significantly with the different values of φ i.e., φ =1.92 π , 1.88 π , 1.86 π to 1.84 π when Ω c =20mev. Theamplitudes of variation in the relative phase φ are 0.04 π ,0.02 π and 0.02 π between the dotted line, dashed dottedline, dashed line and solid line, which causes the maxi-mum of NRI changing from -0.65, -1.78, -2.50 to -3.45,and the corresponding intervals enlarge from [4.6 mev,55.68 mev], [0.48 mev, 55.68 mev], [-0.78 mev, 55.68 mev]to [-1.70 mev, 55.68 mev]. Hence, the bandwidths forNRI can also be adjusted by the relative phase. Com-paring Fig.2 with Fig.3, the relative phase is much moresensitive than the coupling laser field in controlling thebandwidths for NRI. IV. CONCLUSION
In summary, the large and tunable NRI is theoreticallydemonstrated in a SQWs nanostructure by using the re-ported experimental parameters. In our scheme, the co-herent cross-coupling between electric and magnetic tran-sitions leads to chirality and the chirality coefficients aremodulated by the coupling laser field and relative phase,so the SQWs nanostructure can become NRI material.When the coupling laser field and relative phase are prop-erly modulated, the large and tunable NRI with alterablefrequency regions can be obtained. The flexibly modulat-ing parameters and using reported experimental param- eters increase the possibility of implementing NRI in theSQWs nanostructure, which may give us a newt way toNRI material in solid state nanotructure. And we hopethe coming experiment will achieve this in the future.
V. ACKNOWLEDGMENT
The work is supported by the National Natural Sci-ence Foundation of China (Grant No. 61205205)and theFoundation for Personnel training projects of YunnanProvince, China (Grant No. KKSY201207068). [1] R. Tsu, L. L. Chang, G. A. Sai-Halasz, and L. Esaki,
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