Parametrical amplification induced nonreciprocity in photonic band gaps

Abstract

For the first time, we experimentally investigate the parametrically amplified nonreciprocity of the transmission of the probe field, the reflection of the suppressed four-wave mixing (FWM), the enhanced six-wave mixing (SWM) and the radiation trap nonreciprocity of the fluorescence signals in a five level atomic system. The transition from the dressed suppressed dip to the enhanced peak is due to the nonreciprocity induced by parametrical amplification which is observed in FWM (SWM) signals noticeably. The SWM signal exhibits more obvious nonreciprocity than the FWM signal because the former suffers significant dressing enhancement. Moreover, the frequency offset of the signals on the two arm ramps of one round trip results from the optical bistability (nonreciprocity) which can be controlled by the frequency detunings, powers and angles of the dressing fields. Such a scheme could have potential applications in amplification processing of transistors and quantum information processing.

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I. Introduction

Electromagnetically induced transparency (EIT)^1–3 can effectively decrease the absorption of incident beams and has potential applications in nonlinear optics and wave mixing processes.⁴ Coherent population trapping (CPT) is one manifestation of EIT which was first observed in sodium atoms.⁵ For optically thick media, radiation trapping has been studied extensively in astrophysics, plasma physics, and atomic spectroscopy^6,7 and has been demonstrated to have a destructive effect on the orientation produced by optical pumping.^8,9 The increase of the effective decay rate of Zeeman coherence due to radiation trapping was reported, and the relaxation time of the coherent state is determined not only by the time of flight of the atom through the laser beam.⁹ Clear experimental evidence of random lasing action was observed, and the multi-stability is very sensitive to the induced atomic coherence in the system and can evolve from a normal bistable behavior with the change of two coupling fields.¹⁰ A spontaneous parametric FWM (SP-FWM) process generates two weak fields (Stokes field E_St and anti-Stokes field E_ASt) on a forward cone which can be injected into the input Stokes port of the SP-FWM process and lead to optical parametrical amplification (OPA). Finally, the EIT window will pick up the corresponding multi-wave mixing (MWM) signals with narrow line widths.^11,12 Optical bistability (OB) behavior based on atomic coherence and quantum interference are available here because the FWM process is capable of gain and oscillation.^13–15 The EIT-based nonlinear schemes can be driven by a standing wave due to two counterpropagating coupling fields^16–18 which possess a photonic band gap (PBG) structure.

In our paper, we report the observation and analysis of the parametrical amplification induced nonreciprocity process of the probe transmission signal (PTS), the reflection of the suppressed FWM, the enhanced SWM and the radiation trap of the fluorescence (FL) signals in the PBG structure. The nonreciprocity induced frequency offset (NFO) of the two signals on the two side ramps of the round trip caused by the feedback dressing effect are induced by the parametrical amplification. Moreover, the calculation of the nonlinear refractive index and the transitions of the signals are clearly observed which agree well with the theory. The organization of the paper is as follows: in Section 2, we briefly introduce the experimental scheme and theoretical model; in Section 3, we discuss the experimental results in detail; in Section 4, we conclude this paper.

II. Experimental setup and basic theory

A. Experimental setup

The experiment is performed in an atomic system containing a five level ⁸⁵Rb atomic vapor cell composed of 5S_1/2(F = 3) (|0〉), 5S_1/2(F = 2) (|3〉), 5P_3/2 (|1〉), 5D_3/2 (|4〉) and 5D_5/2 (|2〉) as shown in Fig. 1(a). As shown in Fig. 1(c1), except for the coupling counterpropagating laser beams E₃ (frequency ω₃, wave vector k₃ and Rabi frequency G₁) and E′₃ (ω′₃, k′₃ and G′₃) connecting the transition |3〉 → |1〉, the experimental setup is similar to the previous work.¹⁹ The dressing laser beams E₂ (ω₂, k₂ and G₂) and E₄ (ω₄, k₄ and G₄) connect |1〉 → |2〉 and |1〉 → |4〉, respectively. The probe laser beam E₁ (ω₁, k₁ and G₁) connects |0〉 → |1〉 (G_i = μ_iE_i/ћ is the Rabi frequency with a transition dipole moment μ_i). The coupling periodic dressing fields E₃ and E′₃ propagating through the ⁸⁵Rb vapor in opposite directions can be written as E₃₁ = y[E₃ cos(ω₃t − k₃z) + E′₃ cos(ω′₃t − k′₃z)], which will form a standing wave. It can be dressed by the dressing fields E₂ and E₄ periodically. The probe field E₁ propagates along the same direction as E′₃ with a small angle between them and the dressing fields E₂ and E₄ propagate in the opposite direction to E′₃ with a small angle as shown in Fig. 1(c1). The probe laser beam E₁ is from an external cavity diode laser (ECDL) with a wavelength of 780.245 nm and a horizontal polarization. The two coupling laser beams E₃ and E′₃ with a wavelength of about 780.238 nm and a vertical polarization are split from another ECDL. The dressing laser beam E₂ (E₄) with a wavelength of 775.978 nm (776.157 nm) and a vertical polarization is from two other ECDLs. The powers of E₁, E₂ (E₄), E₃ and E′₃ (beside the power dependence) are 2.1 mW, 21 mW (17 mW), 13.2 mW and 8.4 mW, respectively. The generated FWM signal (E_F) (satisfying the phase-matching condition k_F = k₃ + k₁ − k′₃), the SWM signal (E_S) (satisfying the phase-matching condition k_S = k₁ + k₂ − k₂ + k₃ − k′₃) and the PTS (E_P) are detected by a photodiode and avalanche photodiode detectors, respectively. Fig. 1(b1) shows the split energy level diagram with all five beams on. The level |1〉 will be split by |G₃₁|² into two dressing states expressed as |G₃₁±〉. For the double-dressed case, E₂ acts as one dressing field which will induce the first level dressing states |G₃₁±〉 to split into second level dressing states |G₃₁ − G₂ ±〉 and |G₃₁ + G₂ ±〉. When E₄ is turned on, the second level dressing states |G₃₁ ± G₂ ±〉 are further split into third level dressing states |G₃₁ ± G₂ ± G₄ ±〉. |G₃₁±〉, |G₃₁ ± G₂ ±〉 and |G₃₁ ± G₂ ± G₄ ±〉 are all periodic along the z direction, which induces periodic susceptibility (χ⁽¹⁾, χ⁽³⁾ and χ⁽⁵⁾) and furthermore a spatial periodic nonlinear refractive index, i.e., this generates the electromagnetically induced gain (EIG) which has a PBG. The probe beam with a frequency falling within this PBG will be reflected strongly to generate the FWM (SWM) signal. The two arm ramps of one round trip are shown in Fig. 1(b2), the frequency ranges from −10 GHz to 10 GHz, and the intensity of the signals changes from −500 nW to 500 nW. The annotation Δ_D indicates the detuning of the dressing field E₂ (E₄). In addition, if the E₃ beam has a sufficiently high power, the so-called conical emission²⁰ will be observed. E₃ can induce a SP-FWM process which generates two weak fields (Stokes field E_St and anti-Stokes field E_ASt) satisfying 2k₃ = k_St + k_ASt (Fig. 1(c2)). The generated FWM signal (E_F1) or SWM signal (E_S) beam is naturally injected into the input Stokes port of the SP-FWM process, and is parametrically amplified which enhances the efficiency of the FWM process.²¹ This amplifier involves a coupled Stokes channel and anti-Stokes channel and produces twin photons. One can express this coupling with the Hamiltonian H = g/v(â⁺b̂⁺âb̂), where â⁺(â) is the Boson-creation (annihilation) operator that acts on the electromagnetic excitation of the Stokes channel and b̂⁺(b̂) acts on the anti-Stokes channel. The E′₃ beam can induce conical emission and a spontaneous parametric PTS process will occur which generates two weak fields (Stokes field E′_St and anti-Stokes field E′_ASt) satisfying 2k′₃ = k′_St + k′_ASt. The phase-matching geometrical diagram of the SP-FWM process is shown in Fig. 1(c2). The probe beam (E_P) is naturally injected into the input Stokes port of the SP-FWM process.^11,12 It can be observed clearly that the generated probe and FWM (SWM) OPA mode have feedback dressing effects as shown in Fig. 2–5.

Fig. 1 (a) Five level energy system. (b1) Dressed energy level schematic diagrams. (b2) Two arm ramps of one round trip. (c1) Setup of our experiment: PBS, polarizing beam splitters; LD₁–LD₄, lasers; D₁–D₃, detectors. (c2) Phase-matching geometrical diagram of SP-FWM.

The FL signal caused by spontaneous decay is detected by another photodiode. Reabsorption of spontaneously emitted photons (or radiation trapping) will occur when light interacts with high density media (5.603 × 10¹¹ m⁻³ in our experiments) which will cause the trapping of FL signals by atoms.

B. PTS, FWM (SWM) and FL with feedback dressing

Parametrically amplified nonreciprocity induced by feedback dressing is similar with OB. According to the energy level system, Liouville pathways and considering the feedback dressing of the PTS and FWM (SWM) which are written as |G_1T|² and |G_FR|² (|G_SR|²), the first-order, third-order and fifth-order density-matrix elements with all five beams on are: and where d₁₀ = Γ₁₀ + iΔ₁, d₃₁ = Γ₃₀ + iΔ₁ − iΔ₃, d₂₁ = Γ₂₀ + iΔ₁ + iΔ₂, d₄₁ = Γ₄₀ + iΔ₁ + iΔ₄, Δ₁ = Ω₁₀ − ω₁, Δ₂ = Ω₂₁ − ω₂, |G′₃₁|² = |G₃|² + |G′₃|² + 2G₃G′ cos(2k₃x), frequency detuning Δ_i = Ω_i − ω_i (Ω_i is the resonance frequency of the transition driven by E_i), and Γ_ij is the transverse relaxation rate between |i〉 and |j〉. The intensity of the PTS is I_T = I₀ − I₀(2πχ′′l/λ₃) = I₀[1 − 2πNμ₃²Γ₂₀/ℏε₀Gλ₃(Γ₂₀² − Δ₁²)] (N is the atom density). The intensity of the polarizations are P⁽¹⁾₁ = iNμ₁₀G₁/(d₁₀ + |G′₃₁|²/d₃₁ + |G₂|²/d₂₁ + |G₄|²/d₄₁ + |G_1T|²/Γ₀₀), P⁽³⁾₁ = iNμ₁₀G₁|G₃|²/[(d₁₀ + |G′₃₁|²/d₃₁ + |G₂|²/d₂₁ + |G₄|²/d₄₁ + |G_FR|²/Γ₀₀)²/d₂] and P⁽⁵⁾₁ = iNμ₁₀G₁|G₂|²|G₃|²/[(d₁₀ + |G′₃₁|²/d₃₁ + |G₂|²/d₂₁ + |G₄|²/d₄₁ + |G_SR|²/Γ₀₀)²/d₂d₃]. The Kerr nonlinear coefficient can be defined by a general expression as n₂ ≈ Reρ^(1,3,5)₁₀/(ε₀cn₁)(Δn = n₂|u|²). From P = ε₀χE = Nμ₁₀ρ, the related first-order, third-order and fifth-order (nonlinear) susceptibilities χ⁽¹⁾, χ⁽³⁾ and χ⁽⁵⁾ can be obtained.

The nonlinear coupled wave equations,²² ∂E_p(z)/∂z = −αE_p(z) + ke^−iΔk_zzE_r(z) and −∂E_r(z)/∂z = −αE_r(z) + ke^iΔk_zzE_p(z), are given to estimate the reflection efficiency, where E_p(z) and E_r(z) stand for the probe and FWM (SWM) fields, respectively. By taking the reflected signals as reference, the photon numbers of the output Stokes (E^A_F or E^A_S) and anti-Stokes fields of the OPA are a_out = 〈â⁺_outâ_out〉 = G〈â⁺_inâ_in〉 + (G − 1) and b_out = 〈b̂⁺_outb̂_out〉 = (G − 1)〈â⁺_inâ_in〉 + (G − 1), where â (b̂) is the annihilation operator of E_St (E_ASt) (G = {cos[2t(AB)^1/2 sin(φ₁ + φ₂)/2] + cosh[2t(AB)^1/2 cos(φ₁ + φ₂)/2]}/2 is the gain of the process with the modules A and B (phases φ₁ and φ₂) defined in ρ^(1,3,5)_10(st) = Ae^iφ₁, and ρ^(1,3,5)_10(Ast) = Be^iφ₂ for E_St and E_ASt, respectively¹¹). The transmission signal can be obtained by following the same procedure.

As shown in Fig. 1(c2), we can get the amplified MWM â_out (probe â′′_out) with injections â_in(χ^(3,5)) and â′′_in(χ⁽¹⁾), respectively. The OPA of α and k can be written as α^A = (ω₃/c)Im χ′⁽¹⁾/2 which is the attenuation of the field due to the absorption of the medium and k^A = i(ω_p/c)χ′⁽³⁾/2(k = i(ω_p/c)χ′⁽⁵⁾/2) which is the gain due to the nonlinear susceptibility. χ′⁽¹⁾ and χ′⁽³⁾ (χ′⁽⁵⁾) are the zero-order coefficients of the Fourier expansion of â′′_out and â_out, respectively. The nonlinear coupled wave equations are amplified and finally the amplified reflectivity R^A and transmission T^A are: and , where d_z is length of the sample in the z direction. λ^±₁ = −iΔk_z/2 ± [(α − iΔk_z/2)² − k²]^1/2 and λ^±₂ = λ^±₁ + iΔk_z, where Δk_z is the phase mismatch magnitude.

Second, for the FL signal, the NFO due to the feedback dressing term |G_FL|²/Γ₀₀ induced by radiation trapping is investigated. By solving the coupled density-matrix equations, the density-matrix element of the second-order FL_R1 signal ρ⁽²⁾₁₁ can be written as ρ⁽²⁾₁₁ = −|G₁|²⁻/(d₁ + |G_FL1|²/Γ₀₀)Γ₁₁, the square of which is proportional to the intensity of the FL_R1 signal. With E₂ and E₄ on, and considering the feedback the dressing expression can be modified as:

Considering the feedback dressing term |G_FL2|²/Γ₀₀ and via the Liouville pathway , the density-matrix element of the fourth-order FL_R2 signal can be modified as:

ρ⁽⁴⁾₂₂ = |G₁|²|G₂|²/Γ₂₂d₁₀d₂₁d₅,

where d₅ = Γ₂₁ + iΔ₂. Considering the G₂ dressing and feedback dressing, the doubly dressed FL_R2 signal is given as:

Furthermore, the fourth-order FL_R4 signal (ρ⁽⁴⁾₄₄) is obtained by considering the dressing effect of E₂ which forms the triple-dressed FL_R4 signal which can be given as:

where d₆ = Γ₄₁ + i(Δ₁ + Δ₄). The intensity of the FL signals are I_FL1 = ρ′⁽²⁾₁₁ = Nρ⁽²⁾₁₁, I_FL2 = ρ′⁽⁴⁾₂₂ = Nρ⁽⁴⁾₂₂ and I_FL4 = ρ′⁽⁴⁾₄₄ = Nρ⁽⁴⁾₄₄.

C. Frequency offset, dressing suppression and enhancement conditions of nonreciprocity

We investigate the feedback nonreciprocity based on the influence of the feedback dressing term G′ (G_1T, G_FR, G_SR and G_FL) of the generated PTS, FWM (SWM) and FL signal on the intensity R^A, T^A and I_FL as discussed above.

Parametrical amplification is induced by feedback dressing on the signals, which will cause the nonlinear refractive index and finally induce the NFO. Because of the NFO caused by the feedback nonreciprocity (for the PTS, FWM and SWM) or radiation trap (for the FL signal), the signals on the rising edge and the falling edge of the frequency ramp are different. The nonreciprocity (or radiation trap) reflected from the change of the nonlinear refractive index change Δn′ is

Δn′ω_pl/c = N(n_2upI_up − n_2downI_down)ω_pl/c = Δσ. (7)

The nonreciprocity phase delay (Δσ = Δυn₁l/c), and the nonlinear refractive index (n₂) of the cell are related to the density of the ⁸⁵Rb vapor N (determined by the cell temperature) and Δυ is the NFO which can reflect the optical nonreciprocity (OB) directly. I_up and I_down (I_1T, I_FR, I_SR and I_FL) are the feedback intensities of the signals for the PTS, FWM (SWM) and FL signal on the two side ramps of one round trip. The relative transmission, relative reflection intensity and the FL signal intensities are approximated by I_P = T^A, I_F = R^A, and I_FL1 = ρ′⁽²⁾₁₁ = Nρ⁽²⁾₁₁ (I_FL2 = ρ′⁽⁴⁾₂₂ = Nρ⁽⁴⁾₂₂ and I_FL4 = ρ′⁽⁴⁾₄₄ = Nρ⁽⁴⁾₄₄), respectively. The change of the nonlinear refractive index change Δn′ and the intensity of the beam I is a function of the scanning field and the changing dressing field. The intensity of the three feedback signals can be approximately obtained by considering the PTS. The induced probe field will lead to |E| = |ω_nP⁽¹⁾l/ε₀cn| as the intensity of polarization is P⁽¹⁾₁ = iNμ₁₀G₁/(d₁₀ + |G′₃₁|²/d₃₁ + |G₂|²/d₂₁ + |G₄|²/d₄₁ + |G_1T|²/Γ₀₀). Then the intensity of the PTS can be obtained as I = 1/2ε₀c|E|², where the length of the ⁸⁵Rb cell is l = 7 cm. Therefore, we can get n₂|ΔG′|² = (n_2upI_up − n_2downI_down) = Δn′, where ΔG′ = G′_up − G′_down is the relative feedback dressing term between the two side ramps of one round trip.

The nonreciprocity suppression and enhancement on the rising edge and the falling edge are different. When we scan Δ₂, the primary A–T splitting^23,24 is caused by the external dressing field E₂ and the corresponding eigenvalues are λ_± = [Δ₂ ± (Δ₂² + 4|G₂|²)^1/2]/2, and the secondary A–T splitting is caused by the feedback dressing term G′, with the corresponding eigenvalues λ_+± = [Δ′₂ ± (Δ′₂ + 4|G′|²)^1/2]/2 (Δ′₂ = −Δ₁ − λ₊). Treating Δ′₂ = 0 to satisfy the resonance condition, we can get λ_± = G′, so the split energy levels are λ₊ ± G′ and λ₋. Therefore, the suppression and enhancement conditions of G₂ are Δ₁ + Δ₂ = 0 and Δ₁ + Δ₂ − λ_∓ = 0 (Δ₁ + λ_± = 0), respectively. With the double photonic feedback dressing, the suppression and enhancement conditions are Δ₁ + Δ′₂ = 0 and Δ₁ + λ₊ + λ_+± = 0 (Δ₁ + λ₋ + λ_−± = 0), respectively. Similarly, when we scan Δ₄, the split energy levels are λ′₊ ± G′ and λ′₋, where λ′_± = [Δ₄ ± (Δ₄² + 4|G₄|²)^1/2]/2.

III. Experimental results and discussion

Following the experiment results (Fig. 2–5), we compare the signals on the rising edge and the falling edge (Fig. 1(b2)) by folding them from the turning point of the round trip. The right lines show the signals on the rising edge of the ramp and the left ones stand for the signals on the falling edge.

A. PBG FWM nonreciprocity

First, we analyze the nonreciprocity of the PTS, FWM band gap signal (FWM BGS) and FL signal versus Δ₂ at Δ₁ = Δ₃ = 0 with different beams blocked as shown in Fig. 2. In Fig. 2(a), the background is the dressed PBG signal due to the term |G′₃₁|²/d₃₁ in eqn (1). The peaks on the baseline are the EIT satisfying Δ₁ + Δ₂ = 0 due to the term |G₂|/d₂₁ in eqn (1). A small dip induced by E₃ and E′₃ which splits the EIT satisfying the resonant condition Δ₁ − Δ₃ = 0 is due to the dressing term |G′₃₁|²/d₃₁ in eqn (1). There is offset of the frequency resulting from the different feedback term |G_1T|²/Γ₀₀ on the two arm ramps of one round trip due to the nonreciprocity induced by the OPA.^6,11 The feedback dressing term |G_1T|²/Γ₀₀ has a similar influence with the single dressing of |G₂|² as depicted in eqn (1). G₂ can dress the signals while ΔG′ can result in the non-overlap of the right signals and the left ones which we call NFO. The NFO is caused by ΔG′ which can be controlled by the nonlinear refractive index n_up (n_down) and the intensity of the beam I_up (I_down). It shows that the double-dressed PTS and the NFO both reach the biggest value as shown in Fig. 2(a1) because of the biggest ΔG′ caused by the difference of I_up and I_down due to the term |G_1T|² in eqn (1). When we block E₃ in Fig. 2(a2), the PTS decreases due to the term |G′₃₁|²/d₃₁ in eqn (1) which will induce the decrease of n_up (n_down).²⁵ So, the intensity of the feedback dressing also decreases, and the NFO becomes smaller due to the term |G_1T|². Similarly, with E′₃ blocked, the NFO decreases. Comparing the signals from Fig. 2(a1)–(a3), the two peaks of each split EIT signal are similar in Fig. 2(a1) but the left peak of each split EIT signal is bigger than the right one in Fig. 2(a2) and smaller than the right one (Fig. 2(a3)) because of the absence of the nonreciprocity of the dressed field E₃ (E′₃) due to the absence of the optic pumping of the fields. The intensity of the PTS and the NFO get the smallest values with both E₃ and E′₃ blocked in Fig. 2(a4) as depicted by the dashed lines. For the FWM BGS in Fig. 2(b), the suppression dip is the dressed field of the FWM BGS which is reflected by the PBG structure. The suppression dip becomes the deepest in Fig. 2(b1) with all four beams on due to the term |G₂|²/d₂₁ in ρ⁽³⁾₁₀. The FWM BGS disappeared with any beams blocked as shown in Fig. 2(b2)–(b4). The nonreciprocity of the FWM BGS can be observed in Fig. 2(b1) which has NFO due to the feedback dressing term |G_FR|²/Γ₀₀ in eqn (2). For the FL signal, Fig. 2(c) represents the fourth-order FL_R2 signal (ρ⁽⁴⁾₂₂) where the straight background is the second-order FL_R1 signal (ρ⁽²⁾₁₁) induced by the dressing field E₃ (E′₃) at the resonance point Δ₁ + Δ₃ = 0 according to the term |G′₃₁|/d₃₁ in eqn (4). The suppression dips decrease from Fig. 2(c1)–(c3) because of the absence of |G₃|² (or |G′₃|²) in |G′₃₁|²/d₃₁ (ρ⁽²⁾₁₁). The suppression dip in Fig. 2(c1) is deeper than that in Fig. 2(c4) because the power of E′₃ is bigger than that of E₃. The suppression dip reaches a minimum with both E₃ and E′₃ blocked due to the term |G′₃₁|²/Γ₀₀ in eqn (4). The fourth-order FL_R2 signal peaks decrease from Fig. 2(c1)–(c4) due to the dressing term |G′₃₁|²/Γ₀₀ in eqn (5). The nonreciprocity of both the FL_R1 signal and the FL_R2 signal decreases from top to bottom due to the feedback dressing term |G_FL|²/Γ₀₀ in eqn (4) and (5).

Fig. 2 Measured (a) PTS, (b) FWM BGS and (c) FL signal versus Δ₂ with Δ₁ = Δ₃ = 0 when we block (1) with no block, (2) E₃, (3) E′₃, and (4) E₃ and E′₃ from bottom to top, respectively.

Further, Fig. 3(a)–(c) shows the three signals which display the nonreciprocity behavior of the PTS, FWM BGS and FL signal with the relative phase (Δφ) by changing the incident angle of E₂.²⁶ The experimental results can be obtained by scanning Δ₂ with Δ₁ = 0 as shown in Fig. 3(a)–(c). By considering Δφ, eqn (1)–(4) can be modified as:

and

Fig. 3 Measured (a) PTS, (b) FWM BGS and (c) FL signal versus Δ₂ with Δ₁ = Δ₃ = −90 MHz, and Δ₄ = 90 MHz when we change the relative phase of E₂ as (1) π, (2) 2π/3, (3) π/3, (4) 0, and (5) −π/6, respectively. Measured (d) PTS, (e) FWM BGS and (f) FL signal versus Δ₄ with Δ₁ = Δ₃ = −90 MHz, and Δ₄ = 90 MHz when we change the power of E₂ as (1) 1.2 mW, (2) 7.5 mW, and (3) 13.8 mW, respectively. (g)–(i) Calculation results corresponding to (d)–(f), respectively.

The peaks of the PTS stand for the transmission enhancement of the probe signal. With the relative phase Δφ changing from π to −π/6, the highest enhancement peak stands at Δφ = π/3 and Δn′(Δ₂, Δφ) ≈ Δυ/ω ≈ 0.9 × 10⁻⁷ is the biggest. When Δφ is altered from π to −π/6, the EIT of the PTS changes from shallow to deep then to shallow due to the regulation of the relative phase Δφ. The variation of G₂ is caused by Δφ in Fig. 3(b) and (c), and the deepest suppression dips in the FWM BGS and the emission peak in the FL_R1 signal appear at Δφ = π/3 which corresponds to the strongest PTS peak according to the term |G₂|²e^iΔφ/d₂₁ in eqn (9) and (10). Under the abnormal configuration with Δφ = π and Δφ = −π/3, the enhancement peaks of the FWM BGS and FL_R1 signal become shallower because the classical effect plays the dominant role. The nonlinear refractive index change for the FWM BGS and FL signal is Δn′(Δ₂, Δφ) = Δυ/ω ≈ 0.85 × 10⁻⁷. The NFO of all these three signals is proportional to the suppression dips or the enhancement peaks due to the term σ = nI as shown in Fig. 3(a)–(c).

Fig. 3(d)–(f) investigate the nonreciprocity of the three signals versus Δ₄ with increasing P₂. When n₂ = 0, the signals will show a symmetric Lorenzian profile, while an asymmetric transmission profile is generated when n₂ ≠ 0. For example, both the left lines and the right lines are not in a Gaussian shape in Fig. 3(d)–(f), but they have the same tendency which was well investigated previously.²⁵ In Fig. 3(d1), the relative intensity of the PTS is about I(Δ₄, P₂) ≈ 30 nW and the NFO of the two signals is Δυ ≈ 150 MHz. We can get the nonlinear refractive index change Δn′(Δ₄, P₂) = Δυ/ω = 1.29 × 10⁻⁷. The intensity of the dressed PTS decreases from bottom to top due to the increasing cascade interaction between the dressing fields E₂ and E₄ according to the cascade dressing term |G₂|²/d₂₁ + |G₄|²/d₄₁ in ρ⁽¹⁾₁₀ of eqn (1). It can weaken the dressing of E₄ and finally make the PTS peak smaller which will decrease the nonlinear refractive index n_up(n_down). The NFO of the PTS decreases from Fig. 3(d1)–(d3) because of the decreasing ΔG′ caused by the decreasing n_up(n_down) due to the term |G_1T|²/Γ₀₀ in ρ⁽¹⁾₁₀. Correspondingly, the suppression dip of the FWM BGS (Fig. 3(e)) becomes smaller with the increasing of P₂ because the cascade dressing field E₂ has an inverse effect with E₄. The NFO of the FWM BGS decreases with the increasing P₂ due to the term |G_FR|²/Γ₀₀ in eqn (2). The nonlinear refractive index change is Δn′(Δ₄, P₂) = Δυ/ω = 1.2 × 10⁻⁷. The emission peak of the FL_R4 signal related to ρ⁽⁴⁾₄₄ becomes lower with the increasing G₂ as shown in Fig. 3(f1)–(f3). It results from the suppression caused by the dressing of E₂ according to the term d₁ + |G₂|²/d₂₁ in ρ⁽⁴⁾₄₄ of eqn (5). Similarly, the NFO decreases with the increasing G₂ because of the increasing ΔG′ induced by the feedback dressing term |G_FL|/Γ₀₀ in eqn (4). The nonlinear refractive index change of the FL_R4 signal is about Δn′(Δ₄, P₂) = Δυ/ω = 0.87 × 10⁻⁷. The corresponding calculations are presented in Fig. 3(g)–(i) which agree with the experimental results. At the resonance point Δ₁ = 0, the NFO of the signals is observed while the suppression and enhancement nonreciprocity is not obvious because of the significant atomic resonance effect.

B. PBG SWM nonreciprocity

In the above, the EIG (or electromagnetically induced absorption (EIA)) can be ignored at Δ₁ = 0 because the strong dressing suppression dip (Δ₁ + Δ₂ = 0) of the FWM BGS caused by the biggest suppression condition is big enough to fill up the other signals. Also, the dressing enhancement peak of the SWM BGS is much smaller than the dressing suppression dip (FWM BGS) which cannot be observed in Fig. 2 and 3. With Δ₁ set far away from the resonance point, the SWM BGS caused by the biggest enhancement condition Δ₁ + Δ₂ − λ_∓ = 0 can be obtained in Fig. 4(b), and Fig. 5(b) and (e). Thus, the nonreciprocity of the SWM is easier to obtain because the dressing enhancement is bigger than the dressing suppression.

Fig. 4 Measured (a) PTS, (b) FWM BGS and (c) FL signal versus Δ₂ with Δ₁ = Δ₃ = 700 MHz when we change the power of E₁ as (1) 4.9 mW, (2) 3.7 mW, (3) 2.5 mW, and (4) 1.2 mW, respectively. (d) Theoretical results corresponding to (a). (1)–(4) Four column signals on each line.

In Fig. 4, two column signals are observed. The EIG (EIA) signals caused by E₂ when E′₃ works as a probe field in the PTS and the EIG (EIA) signals caused by E′₃ when E₂ works as a probe field in the FWM BGS when Δ₁ is far away from the resonance point as columns (1) and (2) are depicted in Fig. 4(a) and (b), respectively. Considering the E′₃ field as a probe field, via the Liouville pathway , the density-matrix element ρ⁽¹⁾₁₃ can be written as:

ρ⁽¹⁾₁₃ = iG′₃(d₁₃ + |G₁|²/d₀₃ + |G₂|²/d₂₃ + |G₃₁|²/Γ₃₃ + |G′|²/Γ₃₃), (12)

and via the Liouville pathway , we can obtain the second-order density-matrix element ρ⁽²⁾₂₀ as:

ρ⁽²⁾₂₀ = −G₁G₂/[d₁₃ + |G₃₁|²/d³¹ + |G₂|²/d₂₃ + |G′|²/Γ₃₃]d₂₁, (13)

where d₁₃ = iΔ₃ + Γ₁₃, d₀₃ = i(Δ₃ − Δ₁) + Γ₀₃, and d₂₃ = i(Δ₂ + Δ₃)+Γ₂₃. Via the Liouville pathway , the fourth-order FL signal can be written as:

The PTS, the transform process from the FWM BGS to SWM BGS and the FL signal when we scan Δ₂ and change the power of E₁(P₁) from big to small are depicted in columns (3) and (4). In Fig. 4(a), there is a small dip located at Δ₂ + Δ₃ = 0 representing the suppression of the PTS caused by the EIA of E′₃ when E′₃ works as a probe field and E₂ as a dressing field in columns (1) and (2). When we change P₁ from large to small values in Fig. 4(a1)–(a4), the dips decrease because of the single photon effect of E₁ due to the term |G₁|²/d₀₃ in eqn (12). The peaks on the PTS stand for the EIT caused by the term |G₁|²/Γ₀₀ in ρ⁽¹⁾₁₀ located at the position of Δ₁ + Δ₂ = 0 as depicted in columns (3) and (4) which become higher due to the increasing dressing term G₁ = μE₁/ћ = μ(2P₁/ε₀cA)^1/2/ћ in ρ⁽¹⁾₁₀. The NFO of the peaks decreases due to the feedback term |G_1T|²/Γ₀₀ in ρ⁽¹⁾₁₀. The nonlinear refractive index change Δn′(Δ₄, P₂) is given as Δυ/ω = 2.2 × 10⁻⁷ in Fig. 4(a1). The peaks of the right signals (column (4)) are sharper than those of the left ones (column (3)) due to the different n₂ on the rising edge and the falling edge.²³ When Δ₁ is far away from the resonance point Δ₁ = 0, the FWM BGS will be weakened, and the enhancement in the reflected signals of the SWM BGS can be seen. Fig. 4(b) verifies the existence of the nonreciprocity of the SWM BGS. The left dips of each of the column signals (columns (1) and (2)) are the EIA of the E₂ field (acting as a probe field) and the E′₃ field (acting as a dressing field) at Δ₂ + Δ₃ = 0 which is depicted in ρ⁽²⁾₂₀ in eqn (13). It decreases because of the two-photon dressing of E₂ and E′₃ due to the decreasing G₁ in ρ⁽²⁾₂₀ as shown in Fig. 4(b1)–(b4). We mainly illustrate the right ones at the position of Δ₁ + Δ₂ = 0 (columns (3) and (4)) which are caused by the combination of suppression of the FWM BGS and the enhancement of the the SWM BGS. In Fig. 4(b1), P₁ is big enough to cause the suppression dip which can fill up the enhancement peak. With P₁ decreasing, the suppression dip decreases due to the term G₁ in ρ⁽³⁾₁₀ and the enhancement peak appears as shown in Fig. 4(b2)–(b4). The shapes of the two lines are not absolutely the same induced by the feedback nonreciprocity. The shapes switch between the FWM and SWM signals in Fig. 4(b3) and are more obvious than in Fig. 4(b1) due to the term G₁ in ρ⁽³⁾₁₀. Such as in Fig. 4(b3), the number (3) signals of each line represent an enhancement peak and the number (4) ones show a half enhancement peak and half suppression dip due to the different feedback dressing term |G_1T|²/Γ₀₀ (concluded in G′) in Δ₁ + Δ₂ − λ_∓ = 0, but they are both suppression dips in Fig. 4(b1). This can well explain that the delay is not only the phase delay, but also the changing of the shape caused by the feedback dressing term |G_FR|²/Γ₀₀ (|G_SR|²/Γ₀₀). For the FL signal in Fig. 4(c), the left column small peaks (columns (1) and (2)) on each FL signal are the emission peaks of E′₃ as depicted in eqn (14). The intensity and the NFO both decrease from top to bottom due to the term |G₁|²/d₀₃ in ρ′′⁽⁴⁾₂₂. The right one stands for the fourth-order FL_R2 signal related to ρ⁽⁴⁾₂₂. The emission peaks decrease from top to bottom due to the term G₁ in ρ⁽⁴⁾₂₂. The NFO also decreases due to the term |G_FL|²/Γ₀₀ in eqn (5). All the signals on the falling edge are bigger than that on the rising edge because the feedback dressing G′ is different. All the NFO of the left of each column signal is also wider than that of the right ones because of the different feedback dressing of the PTS, FWM BGS and FL signal. Fig. 4(d) is the theoretical model of Fig. 4(a), which is consistent with the experimental results.

Fig. 5 illustrates the three signals versus Δ₂ when we change Δ₁ from 300 MHz to 500 MHz. According to the analysis, the right peaks (dips) on columns (3) and (4) at Δ₂ + Δ₃ = 0 on each signal in Fig. 5(a) are the EIG of E′₃ when E₂ acts as a probe field and the left peaks (dips) of each signal at Δ₁ + Δ₂ = 0 on columns (1) and (2) are the PTS induced by E₁, E₂, E₃ and E′₃. Here we only research the transmission signals at the right side of each signal which are induced by the PBG structure due to the term |G′₃|²/d₃₁ in ρ⁽¹⁾₁₀. The NFO of the PTS in Fig. 5(a1) is almost 0 at Δ₁ = 300 MHz and reaches about 300 MHz at Δ₁ = 600 MHz in Fig. 5(a4) due to the term |G_T|/Γ₀₀ in eqn (1). The nonlinear refractive index change is Δn′(Δ₂, Δ₁) = 4 × 10⁻⁷ in Fig. 5(a4). From Fig. 5(a3)–(a4), the signals on the rising edge are weaker than those of the falling edge due to the difference feedback dressing term |G_T|/Γ₀₀ in eqn (1). Fig. 5(b) shows the SWM BGS for which the right peak (dip) of each column signal (columns (3) and (4)) are the EIG (EIA) of E′₃ when E₂ acts as a probe field and the left ones which are the signals at Δ₁ + Δ₂ = 0 when E₁ works as a probe field. In Fig. 5(b1), an impure SWM BGS both on the left and right sides of each column signal concludes the MWM signals and the EIG of E′₃. The SWM BGS has a blurry small peak in comparison with the FWM BGS as depicted by the dashed lines in (1) and (2). With Δ₁ moving far away from the resonance point, a moving half suppression and half enhancement signal appears at the left side of each column signal which stands for the mixing of the FWM BGS and SWM BGS. With Δ₁ set from 300 MHz to 500 MHz, the suppression dip (left signals of each column signal) becomes deeper and the enhancement peak becomes shallower as shown in Fig. 5(b). The NFO increases from Fig. 5(b1)–(b4) due to the term |G_SR|²/Γ₀₀ in ρ⁽⁵⁾₁₀. The EIA signal suppression dip becomes shallower than even disappears as shown in Fig. 5(b3)–(b4). The left peaks on each signal is higher than that of the right one due to the feedback dressing term |G′|²/Γ₃₃ in eqn (13). The location and the intensity of the EIG signals are constant both on the rising edge and falling edge due to the term |G₂|²/d₂₃ in eqn (13) as shown in Fig. 5(b). For the FL signal, two columns of signals are shown in Fig. 5(c) where the right peaks on the baseline (columns (3) and (4)) are the emission peak of E′₃ related to ρ′′⁽⁴⁾₂₂ in eqn (14) and it decreases from Fig. 5(c1)–(c4) due to the term |G₁|²/d₀₃ in eqn (14). The left dips of the column (1) and (2) signals are the FL_R1 signal related to ρ⁽²⁾₁₁ and the peaks on the dips stand for the fourth-order FL_R2 signal related to ρ⁽⁴⁾₂₂. The NFO of the FL_R1 signal increases from Fig. 5(b1)–(b5) as the dashed lines show due to the term |G_RF|/Γ₀₀ in eqn (2).

Fig. 5 Measured (a) PTS, (b) FWM BGS and (c) FL signal versus Δ₂ with the detuning of Δ₁ as (1) 300 MHz, (2) 350 MHz, (3) 400 MHz, (4) 450 MHz, respectively. (1)–(4) Four column signals on each line. Measured (d) PTS, (e) FWM BGS and (f) FL signal versus Δ₂ with Δ₁ = Δ₃ = −90 MHz, and Δ₄ = 90 MHz when we set N as (1) 2.498 × 10¹¹ m⁻³, (2) 5.603 × 10¹¹ m⁻³, and (3) 1.197 × 10¹² m⁻³, respectively.

The temperature was the most sensitive condition to observe the nonreciprocity and radiation trip as reported in ref. 7 which will change the atomic density N. Finally, we investigated the three signals’ nonlinear refractive index change Δn′ = N(n_2upI_up − n_2downI_down) by scanning Δ₂ with decreasing the density of the ⁸⁵Rb vapor at the resonance point Δ₁ = Δ₃ = −Δ₂. Symmetric signals are shown in Fig. 5 clearly, due to the term n₂ ≠ 0. Fig. 5(d1) is the PTS when N is 2.498 × 10¹¹ m⁻³. The nonlinear refractive index change is Δn′(Δ₂, T) = 2 × 10⁻⁷, where the NFO is about 200 MHz and the intensity of the PTS is about 15 nW. With the temperature rising, the intensity of the PTS increases from Fig. 5(d1)–(d3) due to the increasing atomic density N in eqn (7). The NFO of the signals increases with the rising N due to the stronger feedback dressing term |G_1T|² in eqn (1). Similarly, for the SWM BGS, with Δ₁ ≈ 1000 MHz far away from the single photon resonance point, a pure SWM BGS can be obtained as in Fig. 5(e). The NFO in Fig. 5(e1) is 160 MHz and the nonlinear refractive index change is Δn′(Δ₂, T) = 1.8 × 10⁻⁷ which increases with N. The nonlinear refractive index is not the same in the PTS signals and the SWM BGS due to |G_1T|² and |G_SR|² in eqn (1) and (3). Comparing with Fig. 5(b), the efficiency of the SWM BGS is obviously higher because of the strong atomic coherence at the resonance point Δ₁ = Δ₃ = −Δ₂. Both the intensities of the FL_R1 signal and the FL_R2 signal and the NFO of the FL_R2 signal increase from Fig. 5(e1)–(e3). And the NFO increases with increasing N in eqn (7). It indicates that nonreciprocity only sensitive to the density of the atomic cell is not general.

IV. Conclusion

In summary, generalized nonreciprocity of the PTS, FWM BGS (SWM BGS) and FL signal caused by the feedback dressing in a triple-dressed PBG structure are experimentally observed and theoretically verified. The nonreciprocities of SWM and FWM are compared which shows that the former case is more obvious. Parametrical amplification will cause the atomic trip and finally cause the nonreciprocity. All conditions in our experiments can induce nonreciprocity and not only the temperature as previously reported. Also, the nonlinear refractive index induced by nonreciprocity or the radiation trap is calculated.

Acknowledgements

This work was supported by the 973 Program (2012CB921804), NSFC (11474228, 61308015, 11104214, 61108017, 11104216, 61205112) and KSTIT of Shaanxi Province (2014KCT-10).

References

1. S. E. Harris, Phys. Today, 1997, 50(7), 36–42.
2. J. G. Banacloche, Y. Li and M. Xiao, Phys. Rev. A, 1995, 51, 576.
3. M. Fleischhauer, A. Imamoglu and J. P. Marangos, Rev. Mod. Phys., 2005, 77, 633.
4. M. D. Lukin, A. B. Matsko, M. Fleischhauer and M. O. Scully, Phys. Rev. Lett., 1999, 82, 1847.
5. E. Arimondo and G. Orriols, Lett. Nuovo Cimento Soc. Ital. Fis., 1976, 17, 333.
6. A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapour, Clarendon Press, Oxford, 1998.
7. A. B. Matsko, I. Novikova, M. O. Scully and G. R. Welch, Phys. Rev. Lett., 2001, 87, 13.
8. W. Happer, Rev. Mod. Phys., 1972, 44, 169.
9. G. Ankerhold, M. Schiffer, D. Mutschall, T. Scholz and W. Lange, Phys. Rev. A, 1993, 48, R4031.
10. J. F. Xu and M. Xiao, Appl. Phys. Lett., 2005, 87, 173117.
11. H. X. Chen, Y. Q. Zhang, X. Yao, Z. K. Wu, X. Zhang, Y. P. Zhang and M. Xiao, Sci. Rep., 2014, 4, 03619.
12. H. B. Zheng, X. Zhang, Z. Y. Zhang, Y. L. Tian, H. X. Chen, C. B. Li and Y. P. Zhang, Sci. Rep., 2013, 3, 01885.
13. A. Yariv and D. M. Pepper, Opt. Lett., 1977, 1, 16–18.
14. G. W. Herbert and J. H. Marburger, Appl. Phys. Lett., 1980, 36, 613.
15. Y. Q. Zhang, Z. K. Wu, H. B. Zheng, Z. G. Wang, Y. Z. Zhang, H. Tian and Y. P. Zhang, Laser Phys., 2014, 24, 045402.
16. B. J. Feldman and M. S. Feld, Phys. Rev. A, 1972, 5, 899.
17. J. X. Zhang, H. T. Zhou, D. W. Wang and S. Y. Zhu, Phys. Rev. A, 2011, 83, 053841.
18. H. T. Zhou, D. W. Wang, J. X. Zhang and S. Y. Zhu, Phys. Rev. A, 2011, 84, 053835.
19. Z. Y. Zhao, Z. G. Wang, P. Y. Li, G. P. Huang, N. Li, Y. Q. Zhang, Y. Q. Yan and Y. P. Zhang, Laser Phys. Lett., 2012, 9, 802.
20. V. Boyer, A. M. Marino, R. C. Pooser and P. D. Lett, Science, 2008, 321, 544–547.
21. M. D. Lukin, A. B. Matsko, M. Fleischhauer and M. O. Scully, Phys. Rev. Lett., 1999, 82, 1847–1850.
22. R. L. Abrams and R. C. Lind, Opt. Lett., 1978, 2, 94–96.
23. S. H. Autler and C. H. Townes, Phys. Rev., 1955, 100, 703–722.
24. W. Chalupczak, W. Gawlik and J. Zachorowski, Phys. Rev. A, 1994, 6, 4895–4901.
25. H. Wang, D. Goorskey and M. Xiao, Phys. Rev. Lett., 2001, 87, 7.
26. Z. G. Wang, P. Ying, P. Y. Li, D. Zhang, H. Q. Huang, H. Tian and Y. P. Zhang, Sci. Rep., 2013, 3, 3417.

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Footnote

† PACS numbers: 42.65.Tg, 42.50.Gy, 42.65.Jx, 42.65.Sf.

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