1. Introduction

The development of metamaterial has drawn considerable attention and research effort in recent years, for the capability of bringing unusual and remarkable electromagnetic properties unseen in the natural world. Among the works, the transport properties of Hermitian lattice systems have always been a subject of intense investigation, while much less attention has been paid to non-Hermitian systems, until 1998 when Bender and Boettcher proposed that a non-Hermitian Hamiltonian exhibiting parity-time (𝒫𝒯) symmetry can also have entirely real energy eigen-spectrum in certain regions of parameter space [1,2]. Since then the concept of 𝒫𝒯-symmetry has been widely considered in fields like quantum field theories, non-Hermitian Anderson models, complex Lie algebras, lattice QCD theories, etc., within the theoretical study domain. On the other hand, due to the isomorphism between time-dependent Schrödinger equations and optical paraxial wave equations, optical systems provide the ideal test bed where the features of 𝒫𝒯-symmetry can be experimentally explored [3], and many intriguing properties, like double refraction and band merging [4, 5], power oscillations [6–8 ], nonreciprocal light propagation [9], coherent perfect absorbers [10–13 ], unidirectional invisibility [5,14–17 ], and so on, have been studied since then.

The study of 𝒫𝒯-symmetry on optical systems, either theoretically or experimentally, till this time has been carried out on coupled waveguides [6,18,19], photonic structures [5,20,21], transmission lines [13], whispering-gallery microcavities [22, 23], and optomechanical systems [24, 25]. However, there hasn’t been much attempt to utilize atomic clouds or vapors. Atomic media have their big advantages in all-optical tunable and reconfigurable features in real-time control as compared to solid-state systems. It is therefore of valuable importance to extend the study of 𝒫𝒯-symmetry to this area. Recently several works have come into sight [26–30 ] on this aspect. In mapping Schrödinger equations to paraxial wave propagation equations, the role of time variable t is cast to the propagation direction z, indicating that the 𝒫𝒯-symmetric potential V(x) manifests in the transversely modulated refractive index n(x). Many works have been done under this consideration, e.g., to study a pair of coupled waveguides with balanced gain and loss in the transverse plane of wave propagation [6,8,22,31–33 ]. Recently, the question has been asked about what if we implant the 𝒫𝒯-symmetric modulation of complex refractive index to the longitudinal direction z instead of the transverse plane x [34, 35]. Considering $n=\sqrt{1+{\chi }_p}\approx 1+{\chi }_p/2$, 1D atomic lattices have been designed with the N level configuration to realize the 𝒫𝒯-antisymmetric modulation of probe susceptibility ${\chi }_p(z)=-{\chi }_p^{*}(-z)$ in the absence of optical gain [29]. This scheme has the obvious flexibility in real-time control, all-optical tuning and reconfiguration and thus is more desired in the generation, manipulation, and application of optical 𝒫𝒯-antisymmetry.

In this work, we propose a new scheme of 1D atomic lattices for realizing both 𝒫𝒯-symmetry ${\chi }_p(z)={\chi }_p^{*}(-z)$ and 𝒫𝒯-antisymmetry ${\chi }_p(z)={\chi }_p^{*}(-z)$ along the lattice direction z by exploiting or avoiding the Raman gain. Here 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry are achieved via a careful sinusoidal modulation of one driving field in terms of intensity (Rabi frequency) or frequency (atom-field detuning) in the N level configuration. But it is also essential to have a Gaussian distribution of atomic density in each period of dipole traps forming the 1D optical lattices. A 𝒫𝒯-symmetric or 𝒫𝒯-antisymmetric metamaterial can be regarded as a complex photonic crystal because both real (χ′ _p) and imaginary (χ″ _p) susceptibilities are modulated with a common period. The critical point is that such twofold spatial modulations exhibit a fixed π/2 phase shift [36] so that χ′ _p and χ″ _p are even and odd functions or vice versa of lattice position z. These results may be extended to develop functional devices like photonic diodes and transistors, an impossible task utilizing real photonic crystals with only χ′ _p being nonvanishing and periodically modulated in space. The realization of such devices needs to completely break the optical reciprocity of light transport in a controlled way. Though 𝒫𝒯-symmetric or 𝒫𝒯-antisymmetric metamaterials support unidirectional light reflection [16, 29], it is rather involved to attain nonreciprocal light transmission so that temporal modulations may be further required in addition to spatial modulations.

2. Model and equations

We consider here a coupled system of 1D optical lattices composed of Gaussian-distributed bunches of cold ⁸⁷Rb atoms, with each bunch seated at the bottom of a dipole trap along the z direction. All atoms are coherently driven into the N configuration by three coherent laser fields at frequencies (amplitudes) ω_p (E _p), ω_c (E _c) and ω_d (E _d), as shown in Fig. 1. The N-configuration consists of two ground levels |1〉 and |2〉 and two excited levels |3〉 and |4〉, where the weak probe field ω_p interacts with transition |1〉 ↔ |4〉 while the strong pump fields ω_c and ω_d act upon transitions |1〉 ↔ |3〉 and |2〉 ↔ |4〉, respectively. The corresponding frequency detunings (Rabi frequencies) are defined as Δ_p = ω_p − ω ₄₁ (Ω_p = E _p · d ₁₄/h̄), Δ_c = ω_c − ω ₃₁ (Ω_c = E _c · d ₁₃/h̄), and Δ_d = ω_d − ω ₄₂ (Ω_d = E _d · d ₂₄/h̄) with ω_ij being resonant transition frequencies and d _ij relevant dipole moments.

 

Fig. 1 (a) Diagram of a four-level N configuration of ultracold atoms, which are driven by a weak probe field Ω_p, a strong coupling field Ω_c, and a strong driving field Ω_d. (b) An ensemble of N-type ultracold atoms trapped at the bottoms of dipole traps forming 1D optical lattices in the z direction. The atomic density in each dipole trap of spatial period a is assumed to exhibit a Gaussian distribution.

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With the rotating-wave and electric-dipole approximations, we can write down the interaction Hamiltonian and then obtain the density matrix equations

In the weak probe limit (Ω_p ≪ γ), it is viable to obtain first the zero-order solutions and then the first-order solutions of Eqs. (1) in the steady state in terms of Ω_p. We are only interested in the optical properties experienced by the probe field, so we obtain here

Then we consider the following two cases where either Ω_c (i) or Δ_d (ii) is periodically modulated in a standing-wave (SW) fashion along the z direction. In case (i), the coupling field is modulated in amplitude as

3. Results and discussion

Based on Eq. (2)–(7), we can implement numerical calculations to examine whether χ′ _p(z) and χ″ _p(z) can exhibit a fixed π/2 phase shift in spatial modulation for the two specific cases mentioned above.

We start by plotting in Figs. 2(a) and 2(b) the probe absorption and dispersion properties for a homogeneous atomic sample without suffering any spatial modulations (Ω_c(z) ≡ Ω_c0 and Δ_d(z) ≡ Δ_d0). It is obvious from the black curve in Fig. 2(a) that we observe a typical EIT spectrum with χ″ _p being negative (indicating loss) around the EIT window center when both Ω_c and Ω_d are not too large. With the increasing of Ω_c, the blue and red curves show that χ″ _p becomes positive (indicating gain) at both sides of the EIT window center to yield the spectra of coherent Raman gain without population inversion. Accordingly, we find from Fig. 2(b) that χ′ _p changes from positive dispersion (indicating slow light) to negative dispersion (indicating fast light) near the EIT window center as Ω_c is increased. Then we expect optical 𝒫𝒯-symmetry (requiring both gain and loss) and 𝒫𝒯-antisymmetry (requiring only absorption) may be attained by choosing relevant parameters with suitable modulations, e.g., as in Eq. (6) or Eq. (7).

 

Fig. 2 Imaginary (a) and real (b) parts of probe susceptibility χ_p vs. probe detuning Δ_p for a homogeneous sample with atomic density N ₀ = 5.0 × 10¹² cm⁻³ and dipole moment d ₁₄ = 2.0 × 10⁻²⁹ C·m. Other parameters are Γ₃₁ = Γ₃₂ = Γ₄₁ = Γ₄₂ = 3.0 MHz, γ ₁₂ = 2.0 kHz, Δ_c = Δ_d = 0, Ω_d = 2.0 MHz, and Ω_c = 2.0 (black); 2.5 (blue); 3.0 (red).

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3.1. 𝒫𝒯-symmetric susceptiblities

In Figs. 3(a)–3(d) we plot imaginary (χ″ _p) and real (χ′ _p) parts of the probe susceptibility in two periods of optical lattices with the trapped atoms exhibiting distributions as in Eq. (5) and the coupling field periodically modulated as in Eq. (6). The two upper panels of 2D plots clearly show that χ″ _p and χ′ _p are modulated along the z direction in different ways and χ″ _p seems more sensitive than χ′ _p to the change of probe detuning Δ_p. We further find from the two lower panels of 1D plots that χ″ _p(z) is an odd function while χ′ _p(z) is an even function, for Δ_p = 3.284 MHz, in reference to the ith lattice center z = z_i. This is a direct evidence for the realization of optical 𝒫𝒯-symmetry in terms of the probe susceptibility. The alternation of χ″ _p > 0 in one half period (e.g., 0.0 < (z − z_i)/a < +0.5) and χ″ _p < 0 in another half period (e.g., −0.5 < (z − z_i)/a < 0.0) indicates a balanced gain and loss on the whole. To understand why Δ_p = 3.284 MHz is chosen to demonstrate optical 𝒫𝒯-symmetry, we need to run back over Fig. 2(a) where χ″ _p → 0.0 at this point for Ω_c = 3.0 MHz and Ω_d = 2.0 MHz. It is thus easy to imagine that we can realize χ″ _p > 0.0 for Ω_c > 3.0 MHz and χ″ _p < 0.0 for Ω_c < 3.0 MHz at Δ_p = 3.284 MHz. Accordingly, optical 𝒫𝒯-symmetry is attained at this point with a balanced gain and loss when the coupling field is modulated as in Eq. (6) with Ω_c0 = 3.0 MHz and δΩ_c = 0.3 MHz.

 

Fig. 3 Imaginary (a) and real (b) parts of probe susceptibility χ_p vs. lattice position (z − z_i)/a and probe detuning Δ_p; imaginary (c) and real (d) parts of probe susceptibility χ_p vs. lattice position (z − z_i)/a with probe detuning Δ_p = 3.284 MHz. The coupling field is modulated as in Eq. (5) with Ω_c0 = 3.0 MHz and δΩ_c = 0.3 MHz. Other parameters used in the calculations are the same as in Fig. 2 except N ₀ = 5.0 × 10¹² cm⁻³ and σ = 0.2a for the Gaussian density distribution in Eq. (5).

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In Figs. 4(a) and 4(b) we further check what will happen when the coupling field shows a larger modulation amplitude δΩ_c. It is clear that χ′ _p is not sensitive to δΩ_c while χ″ _p changes remarkably with the increasing of δΩ_c. The black curve in Fig. 4(a) shows, in particular, that the maximal gain in one half period is obviously lower than the maximal loss in another half period for a sufficiently large δΩ_c so that optical 𝒫𝒯-symmetry is destroyed somewhat in the presence of an unbalanced gain and loss. This fact indicates that the response of χ″ _p to δΩ_c enters an unexpected nonlinear regime, e.g., with δΩ_c > 0.6 MHz, for the realization of optical 𝒫𝒯-symmetry.

 

Fig. 4 Imaginary (a) and real (b) parts of probe susceptibility χ_p vs. lattice position (z − z_i)/a for Δ_p = 3.284 MHz and δΩ_c = 0.3 MHz (red); δΩ_c = 0.6 MHz (blue); δΩ_c = 0.9 MHz (black) with other parameters being the same as in Fig. 3.

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3.2. 𝒫𝒯-antisymmetric susceptibilities

In Figs. 5(a)–5(d) we make similar plots as in Figs. 3(a)–3(d) with the Gaussian distribution of atomic density as in Eq. (5) and the driving field periodically modulated instead as in Eq. (7). The two upper panels of 2D plots once again show that χ″ _p and χ′ _p are modulated along the z direction in different ways and χ″ _p seems more sensitive than χ′ _p to the change of probe detuning Δ_p. The two lower panels of 1D plots further show that χ″ _p(z) is an even function while χ′ _p(z) is an odd function, for Δ_p = 0.0 MHz, in reference to the ith lattice center z = z_i. This is a direct evidence for the realization of optical 𝒫𝒯-antisymmetry in terms of the probe susceptibility. It is clear that optical 𝒫𝒯-antisymmetry requires the alternation of χ′ _p > 0 in one half period and χ′ _p < 0 in another half period with varying absorptive loss in each period of the considered atomic lattices. To understand why Δ_p = 0.0 MHz is chosen to demonstrate optical 𝒫𝒯-antisymmetry, we once again run back over Fig. 2(a) where χ″ _p → 0.0 at this point for Ω_c = 2.0 MHz and Ω_d = 2.0 MHz. An introduction of nonzero detuning Δ_d0 thus will result in the space-dependent probe absorption characterized with χ″ _p(z) < 0 at Δ_p = 0.0 MHz everywhere in each period. Accordingly, optical 𝒫𝒯-antisymmetry is attained at this point in the absence of gain when the driving field is modulated as in Eq. (7) with Δ_d0 = 1.0 MHz.

 

Fig. 5 Imaginary (a) and real (b) parts of probe susceptibility χ_p vs. lattice position (z − z_i)/a and probe detuning Δ_p; imaginary (c) and real (d) parts of probe susceptibility χ_p vs. lattice position (z − z_i)/a with probe detuning Δ_p = 0.0 MHz. The driving field is modulated as in Eq. (7) with Δ_d0 = 1.0 MHz. Other parameters used in the calculations are the same as in Fig. 2 except Ω_c = 2.0 MHz as well as N ₀ = 5.0 × 10¹² cm⁻³ and σ = 0.2a for the Gaussian density distribution in Eq. (5).

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In Figs. 6(a) and 6(b) we make similar plots as in Figs. 4(a) and 4(b) for different values of modulation detuning Δ_d0 of the driving field. It is clear that both χ′ _p and χ″ _p change remarkably, but to different extent, with the increasing of Δ_d0. It is more important that the maximal amplitude of χ′ _p > 0 in one half period is always the same as the maximal amplitude of χ′ _p < 0 in another half period. This means that optical 𝒫𝒯-antisymmetry is well reserved no matter Δ_d0 is small or large. The reason is simply that the increasing of Δ_d0 will weaken the optical pumping and therefore not result in a nonlinear response of χ′ _p to Δ_d0 for the realization of optical 𝒫𝒯-antisymmetry. Finally we note that the approach shown here for achieving optical 𝒫𝒯-antisymmetry is similar to that in Ref. [29] with also regard to four-level N-type atomic lattices. The main difference rests with two facts: (i) the probe, coupling, and driving fields are applied on three transitions in a way different from that in Ref. [29] where no population pumping occur and probe gain is always absent; (ii) the driving field is near resonant and modulated in frequency here but far detuned and modulated in amplitude in Ref. [29].

 

Fig. 6 Imaginary (a) and real (b) parts of probe susceptibility χ_p vs. lattice position (z − z_i)/a for Δ_p = 0.0 MHz and Δ_d0 = 1.0 MHz (red); Δ_d0 = 1.5 MHz (blue); Δ_d0 = 2.0 MHz (black) with other parameters being the same as in Fig. 5.

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4. Conclusions

In summary, we have studied the realization of optical 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in 1D atomic lattices as far as the probe susceptibility χ_p is concerned. Given that cold atoms in each dipole trap are driven into the N level configuration and have a Gaussian density distribution, we find real (χ′ _p) and imaginary (χ″ _p) parts of the probe susceptibility can be, respectively, an even function and an odd function or vice versa of the lattice position when the coupling or driving field is modulated along the z direction in a suitable way. This indicates optical 𝒫𝒯-symmetry requires as usual the alternation of balanced gain and loss in each period with positive χ′ _p everywhere, whereas optical 𝒫𝒯-antisymmetry requires the alternation of balanced χ′ _p > 0 and χ′ _p < 0 in each period with absorptive loss everywhere. Such complex photonic crystals of driven atomic lattices have the obvious advantages of real-time, all-optical tunable and reconfigurable features and may be used to develop functional devices inaccessible for real photonic crystals.