Photonic bandgap via nonlinear modulation assisted by spontaneously generated coherence

Physics Letters A 377 (2013) 1416–1420

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Physics Letters A www.elsevier.com/locate/pla

Photonic bandgap via nonlinear modulation assisted by spontaneously generated coherence Yin-Ping Yao a , Tong-Yi Zhang a , Jun Kou b , Ren-Gang Wan a,∗ a b

State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, PR China Quantum Engineering Center, Beijing Institute of Control Device, Beijing 100854, PR China

a r t i c l e

i n f o

Article history: Received 22 January 2013 Received in revised form 26 March 2013 Accepted 7 April 2013 Available online 9 April 2013 Communicated by R. Wu Keywords: Spontaneously generated coherence Photonic bandgap Nonlinear modulation

a b s t r a c t A four-level double-ladder atomic system with two upper states coupled to the excited state by a standing-wave trigger field is explored to generate photonic bandgap (PBG) structure. With the assistance of spontaneously generated coherence (SGC) from the two decay pathways, we can obtain single or double fully developed PBG when the trigger field is far away from resonance or resonant. While in the absence of SGC, the atomic medium becomes strong absorptive to the probe field, and therefore the resulting PBGs are severely malformed or even cannot be opened up. Numerical results show that the PBG structure is originated from the third-order cross Kerr nonlinear modulation between the probe and trigger fields. This mechanism differs from the recent schemes based on linear modulation. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Techniques to manipulate the susceptibility of medium and the propagation of light have been attracted considerable interest in the past few decades due to their importance in both fundamental science and practical applications. Atomic coherence effect is an effectively avenue to control the interaction between light and matter. Based on atomic coherence, many fascinating phenomena have been revealed and investigated, such as laser without inversion (LWI) [1,2], electromagnetically induced transparency (EIT) [3,4]. Using EIT technique, one can well manipulate the absorption, dispersion and nonlinearity of the matter as well as the propagation dynamics of light. Generally, a traveling-wave (TW) coupling field is adopted in the schemes of slow light [5], optical storage [6], enhancement of nonlinearity [7], etc. Yet, when the same three-level atom is driven by a standing-wave (SW), the optical response of the probe is modulated periodically in the space. This one-dimensional periodic structure works as photonic crystal, and then a fully developed photonic bandgap can be opened up for the probe with frequency matching the Bragg condition [8–11]. The resulting PBG has been explored to generate stationary light pulse [12–15], to devise a dynamic controlled cavity [16], to implement optical routing [17,18], etc. For more practical applications, double or triple PBGs have been studied most recently [19–22]. Most recently, one-dimensionally ordered atomic structures by using cold

*

Corresponding author. E-mail address: [email protected] (R.-G. Wan).

0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.04.013

atoms trapped in SW has been experimentally investigated to obtain PBG [23]. Moreover, in SW driven media, electromagnetically induced gratings can also be formed to diffract light into highorder direction [24,25] or to generate dipole soliton [26]. In recent years, enhancing third-order Kerr nonlinearity with low absorption via quantum interference in the phase-coherence medium has been studied extensively both theoretically and experimentally since it has many applications in polarization phase gates [27–29], optical switch [30,31], the generation of optical solitons [32,33], etc. The laser induced quantum coherence in multilevel atoms lies at the heart of these fascinating phenomena and it is crucial to have at least one coupling laser to create the necessary coherence. Nevertheless, the atomic coherence can also be generated from the process of spontaneous emission, in which the atom decays from closely placed upper levels to a single ground level. This spontaneously generated coherence (SGC) can lead to many phenomena, such as amplification without inversion [34,35], narrowing and quenching of spontaneous emission [36,37], decay induced transparency [38], and enhanced Kerr nonlinearity [39,40]. With the assistance of spontaneously generated coherence (SGC), one of the authors (with collaborators) recently proposed a cross Kerr nonlinearity enhancement scheme in four-level double-ladder atomic system [41]. The enhanced nonlinear phase modulation is then utilized to achieve phase grating with high diffractive efficiency. In this Letter, further investigations on the optical response of the system are carried out in detail. Differing from the earlier paper, we concentrate on the PBG structure induced by an SW trigger field. Numerical results show that the emergence of PBG structure is attributed to the nonlinear refractive modulation with

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where N = N 0 (λ p /2π )3 with N 0 being the atomic density, γ21 = Γ2 /2, γ2 =  p − δ + i Γ2 /2, γ3 =  p + δ + i Γ3 /2, γ4 =  p +  T + √ i Γ4 /2 and γ23 = η Γ2 Γ3 /2. Γ2 , Γ3 and Γ4 represent the decay rates from the corresponding states.  p = ω p − (ω21 + ω31 )/2 and  T = ω T − (ω42 + ω43 )/2 are the detunings of the probe and trigger fields, respectively. The frequency difference of the two closely lying states |2 and |3 is 2δ . γ23 denotes the cross coupling term between the two spontaneous emission pathways, which refers to 







the quantum interference or SGC effect. η = d12 · d13 /|d12 ||d13 | = cos α depends on the alignment of the two dipole matrix elements. 



In the case of d12  d13 (α = 0), the system exhibits maximum quantum interference generated from the two decay channels. 

Fig. 1. (Color online.) (a) The considered four-level double ladder-type system with two closely doublets |2 and |3. A trigger field E T couples levels |2 and |3 to an exited level |4, while a weak probe field E p drives the |1 ↔ |2 and |1 ↔ |3 



transitions. (b) Arrangement of the probe (trigger) electric field E p ( E T ) and the 









While d12 ⊥d13 (α = π /2), SGC effect vanishes. In order to investigate the role of nonlinear modulation between the trigger and probe fields, we expand the probe susceptibility χ ( p ) into the second-order of Ω T in the weak field approximation of Ω T Γ4 which can be written as

relevant dipole moments d12 and d13 (d24 and d34 ).



negligible absorption assisted by the SGC effect. This mechanism is different from the recent schemes based on linear modulation [8–11,16–22]. To our knowledge, SGC effect can only exist in the atoms that have closely spaced upper levels with dipole matrix elements to the lower level being nonorthogonal [42]. But these rigorous conditions are rarely met in real atoms, so there has been no experiment carried out in atomic system to observe the SGC effect directly. However, such type quantum interference results from the incoherent decay processes can be realized in many other systems. Examples include spontaneous emission in molecules [43], autoionizing resonances [44], tunneling effect in quantum wells [45,46]. Also, modified vacuum, such as cavity field or anisotropy vacuum, can result in quantum interference among the decay channels even with orthogonal dipole moments [47,48]. Moreover, SGC can be simulated with atoms in the dressed-state picture [49,50]. Therefore, our proposed scheme can be equally applied to the above systems. 2. Atomic model and equations The considered four-level double-ladder atomic system is shown in Fig. 1(a), where a weak probe field couples the ground state |1 to the closely lying doublets |2 and |3 with Rabi frequen







cies Ω p1 = E p · d12 /2h¯ and Ω p2 = E p · d13 /2h¯ , and the transitions |2 ↔ |3 and |2 ↔ |4 are simultaneously driven by a trigger field 







with Rabi frequencies Ω T 1 = E T · d24 /2h¯ and Ω T 2 = E T · d34 /2h¯ . E p ( E T ) represents the amplitude of the probe (trigger) field and



di j denotes the electric-dipole moment of transition |i  ↔ | j . For simplicity, we assume in the following that the misalignment angle









α between d13 and d12 is equally partitioned by E p while d34 

is antiparallel to d24 , as shown in Fig. 1(b). We further set that d31 /d21 = p and d43 /d42 = q, and then Ω p1 = Ω p2 / p = Ω p and ΩT 1 = ΩT 2 /q = ΩT . Under the electric-dipole and rotating-wave approximation, utilizing the Weisskopf–Wigner theory of spontaneous emission, we can obtain the probe susceptibility by solving the equations of motion for the probability amplitudes in steady-state (the detailed equations can be seen in our earlier paper, i.e., Ref. [41]), which is given as

χ ( p ) = 3π Nγ21 ·

−γ4 ( p 2 γ2 + γ3 − 2ip γ23 ) + ( p − q)2 ΩT2

2 γ4 (γ2 γ3 + γ23 ) − ΩT2 (q2 γ2 + γ3 − 2iqγ23 )

,

(1)



χ ( p ) = 3π Nγ21 χ (1) + χ (3) ΩT2 ,

(2a)

with χ (1) and χ (3) corresponding to the first-order linear and third-order cross Kerr nonlinear parts of the probe susceptibility,

χ (1 ) = − χ (3 ) =

p 2 γ2 + γ3 − 2ip γ23 2 γ2 γ3 + γ23

(2b)

,

( p − q)2 . 2 2 3 + γ23 )

(2c)

γ  (γ  γ  4

When the trigger field is in SW pattern, we further obtain its squared Rabi frequency,

  ΩT (x)2 = 4Ω 2 cos2 (π x/Λ), T0

(3)

where Ω T 0 represents the Rabi frequency of the forward and backward trigger fields and Λ = λ T /[2 cos(θ/2)] is the spatial periodicity in the x direction, which can be adjusted by changing the intersection angle θ between the two components of the SW. To obtain the steady optical spectra of a periodically  modulated medium with complex refractive index n( p , x) = 1 + χ ( p , x), we use the transfer-matrix method where light propagation through a single period of length a is governed by a 2 × 2 unimodular transfer matrix M ( p ). The translational invariance of the periodic medium is fulfilled by imposing the Bloch condition on the photonic eigenstates, i.e.



E + (x + a) E − (x + a)



 = M ( p )

E + (x) E − (x)

 =

 iκ a +  e E (x) , e i κ a E − (x)

(4)

where E + and E − denote the amplitudes of the forward and backward electric field of the probe, respectively. κ = κ  + i κ  is the complex Bloch wave vector, which represents the PBG structure, and can be derived from the solutions of equation e 2i κ a − Tr[ M ( p )]e i κ a + 1 = 0 with det M = 1. The Bloch wave vector describes the photonic band structure for a probe in infinite periods. For a sample of length l = Na with N being the number of the SW periods, the reflection and transmission spectra for the probe are given by

   M N (12) ( P ) 2   R ( P ) =  M N (22) ( P )  2    M 12 sin( N κ a)  , =  M 22 sin( N κ a) − sin[( N − 1)κ a] 

(5a)

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Fig. 2. (Color online.) Periodic modulation of complex refractivity n( p , x) in the direction of the standing-wave trigger field. The real part (a) and imaginary part (b) correspond respectively to the absorption and refraction for the probe field. The black solid and blue dashed lines represent the case with (η = 1) and without SGC (η = 0) when considering all-order susceptibilities. The red circle line displays the case of only taking into account the first- and third-order susceptibilities. Parameters are as follows: Γ2 = Γ3 = 2γ , Γ4 = γ , δ = 3, Ω T 0 = 0.8γ ,  T = 80γ .

2    1   T ( P ) =  M N (22) ( P )  2    sin(κ a)  , =  M 22 sin( N κ a) − sin[( N − 1)κ a] 

(5b)

with M N (i j ) being the matrix elements of M N = M N .

Fig. 3. (Color online.) Photonic band gap structure near the first Brillouin zone boundary in terms of Bloch wave vector, the probe reflectivity and transmissivity spectra for a 1.0 cm long sample (atomic density is N 0 = 1.0 × 1012 cm−3 ) with λ p = 795 nm and Λ = 397.53 nm. Other parameters are the same as in Fig. 2. The black solid and blue dashed–dotted lines represent the case with and without SGC when considering all-order susceptibilities. The red dashed line displays the case of only taking into account the first- and third-order susceptibilities.

3. Numerical results Due to the trigger field has an intensity-dependent SW pattern, the probe field experiences periodic modulation of the refractive index. Therefore, the probe with proper frequency, when incident upon such a periodic structure, will be reflected via photonic Bragg scattering. In this section, we show the numerical results about the coherently induced PBG and the corresponding reflectivity (transmissivity) spectra. As demonstrated in our earlier paper [41], pq < 0 can lead to constructive quantum interference in the cross Kerr nonlinearity. Then in the following we assume that p = 1 and q = −1 to attain large refractive modulation. Moreover, we set Γ2 = Γ3 = 2γ and all the other parameters are scaled by γ . 3.1. Single PBG via nonlinear modulation assisted by SGC We first consider the case that the trigger field is far away from resonance, i.e.  T Γ4 . In the absence of trigger field, SGC effect (with η = 1) as well as destructive quantum interference can make the light-atom system evolve into dark sate with  all the population trapping in |ψDark  = [δ|1 + Ω p (|2 − |3)]/ δ 2 + 2Ω p2 , which

corresponds to a transparency window and a steep normal dispersion at  p = 0. Under the condition of Ω T2 / T Γ4 , the trigger field can cause a small ac-Stark shift to the states |2 and |3. As a result, the refractive index is dramatically changed while the absorption can be neglected. This change of refractive index is owing to the giant third-order cross Kerr nonlinearity. However, if there is no SGC effect (with η = 0), although the trigger field can also lead to sufficient phase modulation to the probe, it is accompanied by strong absorption. In Fig. 2, we plot the modulation of the absorptive part [Im(n)] and the refractive part [Re(n)] of the complex refractivity along the direction of SW trigger field with SGC (black solid line) and without SGC (blue dashed line). As we can see, SGC cannot only enhance the modulation depth of refractivity, but also strongly inhibit the absorption, which is in accordance with the above discussions. To give insight into the inherent mechanism, we find that the enhancement of refractive modulation is almost originated from the cross Kerr nonlinearity between the probe and the trigger fields [see the red circle line in Fig. 2(b)]. This dynamically induced

periodic structure then can exhibit PBG for the probe with frequency matching the Bragg condition. The resulting PBG structure is shown in Fig. 3 in terms of the Bloch wave vector and the reflectivity and transmissivity spectra. From Figs. 3(a) and 3(b) we can see that, in the presence of SGC, a well developed PBG is opened up in the frequency ranges where κ  = π /a and κ  = 0. Within the gap, κ  = 0 corresponds to reflection rather than absorption. The lines of Bloch wave vector calculated from the all-order susceptibility (black solid line) and the third-order Kerr nonlinearity (red dashed line) coincide with each other, which indicates that the PBG structure is attributed to the nonlinear modulation. Nevertheless, Eq. (2) is derived in the assumption of Ω T Γ4 , due to the high-order absorption in the case of Ω T ∼ Γ4 , the reflectivity and the transmissivity with the all-order susceptibility (black solid line) are a little lower than that with the third-order Kerr nonlinearity (red dashed line) [see Figs. 3(c) and 3(d)]. While in the case with no SGC, the medium is completely opaque to the probe field. Hence, both the reflectivity and the transmissivity is zero [blue dashed–dotted line in Figs. 3(c) and 3(d)], and PBG cannot be opened up. 3.2. Double PBGs via nonlinear modulation assisted by SGC We then consider the case that the trigger field is resonant, i.e.

 T = 0. As pointed above, in the presence of SGC (with η = 1), the atomic system is finally in the dark state |ψDark  at  p = 0 when trigger field is absent. As this single dark state is coherently coupled to another level |4 by a resonant trigger √ field, double dark resonances arise at the frequency of  p = ± 2Ω T (x), which indicate two distinct transparency window for the probe field. The corresponding dark states are |ψDark±  = [δ|1 + Ω p (|2 − |3 ∓

√



2|4)]/ δ 2 + 4Ω p2 .

When the trigger field has SW pattern, √ the resulting transparency windows, which locate at  p = ± 2Ω T (x), are positiondependent with the same periodicity as the SW. This modulation of the transparency windows results in spatial modified absorption for the probe field. Therefore, the dispersion or refractivity is also modulated with the same period due to Kramer–Kronig relations. In Fig. 4, we plot the absorptive [Im(n)] and the refractive [Re(n)] modulation along the direction of SW trigger field with SGC (black

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between the two reflectivity curves in Fig. 5(c) is due to the highorder nonlinear absorption which is ignored when only take into account the first- and third-order susceptibilities as in Eq. (2). 4. Conclusion

Fig. 4. (Color online.) Periodic modulation of complex refractivity n( p , x) in the direction of the standing-wave trigger field. The real part (a) and imaginary part (b) correspond respectively to the absorption and refraction for the probe field. The black solid and blue circle lines represent the case with (η = 1) and without SGC (η = 0) when considering all-order susceptibilities. The red dashed line displays the case of only taking into account the first- and third-order susceptibilities. Parameters are as follows: Γ2 = Γ3 = 2γ , Γ4 = 0.2γ , δ = 15γ , Ω T 0 = 2γ ,  T = 0.

In summary, we have investigated the steady optical response of a double-ladder atomic system with two closely spaced doublets that coupled to the excited by SW trigger field. Consequently, the probe refractive index is periodically modulated in the direction of the SW and the atoms may work as one-dimensional photonic crystal. With an off-resonant trigger, single PBG with high reflectivity can be opened up in the transparency window via enhanced nonlinear phase modulation assisted by SGC. While in the case of no SGC effect, the medium is strongly absorptive to the probe. Under condition of a resonant trigger field, the spatial modulated double dark resonances can lead to two well developed PBGs in the presence of SGC. However, the double PBGs become severely malformed and blurred due to the unavoidable absorption when SGC vanishes. Such induced PBG structure can be used to devise novel photonic devices, e.g. all-optical routing and switching, in optical network and information processing. It also has applications in the deterministic quantum logic [51] and the enhancement of nonlinear interactions between light pulses [52]. Acknowledgements This work is supported by the NSFC (Grant Nos. 11204367, 61176084, 11174282 and 11204011) and the China Postdoctoral Science Foundation funded project (Grant No. 2012M512040). R.G. Wan gratefully acknowledges the support of K.C. Wong Education Foundation, Hong Kong. References

Fig. 5. (Color online.) Photonic band gap structure near the first Brillouin zone boundary in terms of Bloch wave vector, the probe reflectivity and transmissivity spectra for a 1.0 cm long sample (atomic density is N 0 = 1.0 × 1012 cm−3 ) with λ p = 795 nm and Λ = λ p /2. Other parameters are the same as in Fig. 4. The black solid and blue dashed–dotted lines represent the case with and without SGC when considering all-order susceptibilities. The red dashed line displays the case of only taking into account the first- and third-order susceptibilities.

solid line) and without SGC (blue circle line). As we can see, although SGC cannot enhance the modulation depth of refraction, it significantly reduces the absorption via destructive quantum interference. We also plot the third-order nonlinear absorptive and refractive modulation (red dashed line). It is clear that the phase modulation is owing to the cross Kerr nonlinearity. Due to the symmetry of the transparency windows, we expect double PBGs can be opened up at both sides of the resonant frequency. The corresponding double PBGs structure is shown in Fig. 5. From the Bloch wave vector in Fig. 5(a) we can see that, in the absence of SGC, double PBGs are poorly developed and hardly defined (blue dashed–dotted line), and the reflectivity of both PBGs is relatively low due to the unavoidable probe absorption [blue dashed–dotted line in Fig. 5(c)]. However, when SGC effect is present, the Bloch wave vector remains κ  = π /a and κ  = 0 in two frequency ranges [black solid line in Fig. 5(a)]. Hence, double PBGs structure is well generated with reflectivity about 85% [black solid line in Fig. 5(c)]. In Figs. 5(a) and 5(c), comparing the curves calculated from the all-order susceptibility (black solid line) with that from the thirdorder Kerr nonlinearity (red dashed line), it is obvious that they are almost coincident. Hence, we can conclude that the double PBGs structure also results from the nonlinear modulation. Similarly, as a result of the nonnegligible intensity of Ω T , the difference

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