Abstract

We propose an all-optical-control scheme to simultaneously realize parity-time (𝒫𝒯)-symmetric and 𝒫𝒯-antisymmetric susceptibilities along the propagation direction of light by applying an external magnetic field. Through the light-atom interaction within a double-Λ configuration, the resulting position-dependent susceptibilities for the interacting fields can be manipulated through the relative phase between them. In particular, for the probe field, one can switch its refractive index from the 𝒫𝒯-symmetry to 𝒫𝒯-antisymmetry by just varying the phase. Based on the quantum interference among transition channels in a closed loop, analytical formulas are also derived to illustrate the conditions for 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Even though parity-time (𝒫𝒯) symmetry was initially introduced to generalize quantum mechanics with Hermitian Hamiltonians to non-Hermitian ones [1], quantum entanglement gives negative results for the no-signalling principle when applying the local 𝒫𝒯-symmetric operation on one of the entangled particles [2]. Nevertheless, 𝒫𝒯-symmetry could still be used as an interesting model for open systems in the classical limit [3].

In classical optics, with the correspondence between paraxial wave equation and Schrodinger equation, one can realize optical 𝒫𝒯-symmetric systems by asking the refractive index in the medium to satisfy some specific symmetries. For non-magnetic materials, 𝒫𝒯-symmetric condition is said to be satisfied when the optical susceptibility χ has the form: χ(η) = χ* (−η). Here, the spatial coordinate η can be a transverse or longitudinal one, with respect to the propagation direction. One can see that such a sufficient condition for 𝒫𝒯-symmetry requires an even function in the real part of susceptibility, along with an odd function in the imaginary part. The latter one corresponds to a perfect balance between gain and loss [4,5]. Moreover, in stead of embedding gain or loss mechanics to implement 𝒫𝒯-symmetry [6], one can also have 𝒫𝒯-antisymmetry by asking the susceptibility in the form, χ(η) = −χ*(−η) [7,8].

With the addition degree of freedom from a non-conservative Hamiltonian, as well as the existence of exceptional points to induce phase transition due to the broken 𝒫𝒯-symmetry [9], optical 𝒫𝒯-symmetric devices are studied theoretically and experimentally for directional couplers [10, 11], optical lattices [12–14], soliton dynamics [15], wave localization [16, 17], Bloch oscillations [18], and light diffraction [19]. In addition to play with the optical refractive index directly, 𝒫𝒯-symmetric conditions can also be realized in different physical systems, such as whispering-gallery microcavities [20], moving media [21], RLC circuits [22], and optomechanically-induced transparency systems [23].

In terms of optical refractive index, it is well-known that one can manipulate optical properties of a probe field through the light-atom interaction [24]. As a consequence, a variety of configurations for atom-photon interactions have been proposed and demonstrated to realize 𝒫𝒯-symmetric systems in different experimental settings [25–29]. Even though there already exist many schemes to have 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry, a single scheme to simultaneously realize both of them is still missing. Here, we propose a scheme to realize 𝒫𝒯-symmetric and 𝒫𝒯-antisymmetric susceptibilities for probe and signal fields passing through an atomic system in a double-Λ configuration [30]. By using an external magnetic field with its magnitude linearly increasing along the propagation direction, the position-dependent Zeeman effect can map the intrinsic symmetry or antisymmetry in the optical susceptibilities with respect to the frequency detuning into the direction of propagation. Then, 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in the susceptibilities can be controlled by all optical approaches. Simultaneous realizations of 𝒫𝒯-symmetry-antisymmetry or 𝒫𝒯-antisymmetry-antisymmetry in the probe-signal fields will be illustrated by varying the relative phase between these two fields. The potential applications of 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in the longitudinal direction include coherent perfect absorbers [31–34], unidirectional invisibility [35–38], and unidirectional light reflection [39]. Further, the 𝒫𝒯-symmetry in the longitudinal direction has the flexibility for real time control, all optical tuning, and re-configuration.

This paper is organized as following. In Sec. 2, we will give the light-atom interaction Hamiltonian for a double-Λ configuration, with the corresponding optical susceptibilities for probe and signal fields, respectively. Then, in Sec. 3, illustrations on the simultaneous realization of 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in the susceptibilities of probe and signal fields are shown along the propagation distance. The underline physical picture and discussions will be given with the analytical formula in Sec. 4. Finally, we summary this work in Sec. 5.

2. Field susceptibilities in a double-Λ atomic system

We consider a photon-atom interaction system with allowable transitions in a four-level double-Λ configuration, as shown in Fig. 1. Here, we have four interacting fields, including two weak (denoted as probe and signal) fields and two strong (denoted as coupling and driving) fields. The corresponding transitions are characterized by its Rabi frequencies, denoted as Ωp, Ωs, Ωc, and Ωd, respectively. Then, the one-photon detunings of these four fields are defined as Δp = ωpω31, Δs = ωsω41, Δc = ωcω32, and Δd = ωdω42, where the notations ωμ (μp, s, c, d) and ωµν (EμEν) / ħ represent the field frequency and the corresponding energy difference between energy states |μ〉 and |ν〉, respectively.

 

Fig. 1 Our four-level atomic system in a double-Λ configuration. Here, the four fields are denoted as Ωs, Ωp, Ωc, and Ωd, with the corresponding Rabi frequencies for signal, probe, coupling, and driving fields, respectively. The one-photon detunings of these fields are characterized by Δs, Δp, Δc, and Δd, respectively.

Download Full Size | PPT Slide | PDF

With the rotating wave approximation, one can write down the interaction Hamiltonian for this four-level atomic system in a double-Λ configuration:

H^=[Δp|33|+(ΔpΔc)|22|+Δs|44|]2(Ωp|31|+Ωc|32|+Ωs|41|+Ωd|42|+H.C.).
Here, we have asked the detunings to satisfy the condition: Δp + Δd −Δs − Δc = 0, corresponding to a zero net detuning. Moreover, as the intensities of probe and signal fields are much weaker than those of coupling and driving fields, i.e., Ωp, Ωs ≪ Ωc, Ωd, one can safely derive the equations of motion for density matrix elements in the lowest order, which have the form:
tρ21=γ˜21ρ21+i2Ωc*ρ31+i2Ωd*ρ41,
tρ31=γ˜31ρ31+i2Ωcρ21+i2Ωp,
tρ41=γ˜41ρ41+i2Ωdρ21+i2Ωs.
Here, decays in the upper atomic levels (|e〉, e = 2, 3, 4) to the ground state (|1〉) are introduced phenomenologically through γ̃21γ21ip − Δc), γ̃31γ31iΔp, and γ̃41γ41iΔs.

Macroscopically, the susceptibility for probe field can be found by collecting the atomic matrix component ρ31, i.e., χp=(n312/ε0Ωp) ρ31 with the number density of atomic medium, n = NV, and the atomic dipole transition from |1〉 to |3〉, 31. For steady states, one can ignore the time derivative terms and obtain the susceptibility of probe field by solving Eqs. (2)(4), i.e.,

χp=χp0[i(4γ˜21γ˜41Ωp+|Ωd|2ΩpΩcΩd*Ωs)4γ˜21γ˜31γ˜41+γ˜41|Ωc|2+γ˜31|Ωd|2],
where we have defined χp0n312/2ε0Ωp=(α/8)(λp/L)(Γ/Ωp), with the optical density of atomic ensemble denoted by α = nσL, the wavelength of probe field λp, the length L, the absorption cross section of probe field σ=3λp2/2π2, and the excited state spontaneous decay rate Γ (associated with γ31 = γ41 = 1.25Γ). It is noted that in general the Rabi frequency of each field is a complex number. As shown in Fig. 1, due to such a closed-loop configuration for the interaction channels, the relative phase in the form: ϕr = ϕp + ϕdϕsϕc, plays a crucial parameter in our system [30].

For the signal field, we can apply the same analysis to have its susceptibility by calculating the density element ρ41. The corresponding susceptibility for signal field can be found as χs=(n412/ε0Ωs) ρ41, or explicitly in the form:

χs=χs0[i(4γ˜21γ˜31Ωs+|Ωc|2ΩsΩdΩc*Ωp)4γ˜21γ˜31γ˜41+γ˜41|Ωc|2+γ˜31|Ωd|2].
Here, we have defined χs0 = (α/8) (λs/L) (Γ/Ωs), with the wavelength of signal field λs.

3. 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in the longitudinal direction

With Eqs. (5)(6), one can see that, in terms of the frequency detuning of signal field, Δs, the corresponding susceptibilities for probe and signal fields are manifested with respect to frequency detuning, but not to spatial coordinate. As a simple way to realize optical 𝒫𝒯-symmetric condition, we propose to apply a magnetic field B(z) along the propagation direction, z, see Fig. 2. If this magnetic field has a linearly increasing function along the z-direction, the induced Zeeman effect will also be position dependent. Then, as the Zeeman effect splits the energy level, we can map the detuning information Δs to the longitudinal position, i.e.,

Δs(z)=Δs(0)gLmjμBB(z)/.
Here, μB = /2me = 9.27 × 10−24 (in the unit of Joule/Tesla) is Bohr magneton, and gL = 1 + [J(J + 1) + S(S + 1) − L(L + 1)] /2J (J + 1) is the Landé g-factor, with S, L, and J = L + S denoting the spin, orbital, and total angular momentums, respectively. Moreover, we have assumed that B(z) = 0 = 0 and B(z > 0) > 0.

Now, as the detuning Δs becomes a linear function along the z-direction, depending on the sign of the quantum number mj, the corresponding Δs (0) can be positive or negative. By adjusting B(z) and Δs(0), we can implement the symmetry condition of Δs(L) = −Δs (0), with magnetic field in the form:

B(z)=[2Δs(0)gLmjμBL]z.
By taking Rubidium atoms 87Rb as our atomic ensemble, from Eq. (8), one can estimate the gradient of a magnetic field along z-direction as 4.27 (Tesla/m), for |Δs(0)| = 10Γ, Γ = 2π × 6 MHz, gL = 4/3, mj = 3/2, and L ≃ 1 mm.

 

Fig. 2 Setup of our photon-light interaction scheme, where four fields propagate along the z-direction and the atomic ensemble having photon-atom interaction in a double-Λ configuration, as shown in Fig. 1. With a magnetic field B(z), which has its magnitude linearly increasing along the propagation z-direction, the resulting susceptibilities for probe and signal fields can support a 𝒫𝒯-symmetry or 𝒫𝒯-antisymmetry in the longitudinal z-direction.

Download Full Size | PPT Slide | PDF

Based on this scheme, we can transfer the optical susceptibility from the (detuning) spectrum to a position-dependent distribution. In particular, as shown in Fig. 3, we reveal the susceptibilities of probe and signal fields, i.e., χp(z) and χs(z), in blue- and red-curves, respectively, for different relative phases ϕr = π/2, π, 3π/2, and 2π.

 

Fig. 3 Real (Left-column) and imaginary (Right-column) parts of the susceptibilities for probe and signal fields, depicted in Blue- and Red-curves, respectively. The relative phase difference ϕr is: (a–b) π/2, (c–d) π, (e–f) 3π/2, and (g–h) 2π. One can see that χp satisfies the 𝒫𝒯-symmetric condition when ϕr = π/2, 3π/2; and satisfies the 𝒫𝒯-antisymmetric condition when ϕr = π, 2π. However, χs only gives the 𝒫𝒯-antisymmetric condition. Here, the parameters used are |Ωc| = |Ωd| = 20Γ, Ωp = 0.01Γ, and Ωs = 1Γ, respectively.

Download Full Size | PPT Slide | PDF

For probe field, χp(z) in Blue-curves, as one can see, when the relative phase is ϕr = (2n + 1) π/2, n ∈ integers, as shown in Figs. 3(a)–3(b) and 3(e)–3(f), the real part of probe susceptibility is an even function; while the imaginary part is an odd function with respect to the center of length. This is the optical 𝒫𝒯-symmetric condition. Moreover, one can see from the real part of susceptibility in Figs. 3(a) and 3(c), we have a negative refractive index for the probe field when ϕr = π/2, along with a dip in the central position; while a positive refractive index happens when ϕr = 3π/2, along with a peak in the central position.

Nevertheless, when the relative phase is ϕr = , with n ∈ integers, as shown in Figs. 3(c)–3(d) and 3(g)–3(h), the real and imaginary parts of χp become odd and even functions, respectively. Now, we have the 𝒫𝒯-antisymmetric susceptibility. Moreover, the imaginary part of susceptibility shows that we have a gain peak for ϕr = π, but a absorption dip for ϕr = 2π. Most importantly, only a change in the relative phase can give us such an all-optical-control to switch from a 𝒫𝒯-symmetry to 𝒫𝒯-antisymmetry or vice versa.

In addition to the probe field, the susceptibility for signal field is also depicted in Fig. 3, but in Red-color. Nevertheless, no matter what the relative phase is, we always have an odd function in the real part of susceptibility for signal field, and an even function in its imaginary part of susceptibility. Here, for signal field, we have the 𝒫𝒯-antisymmetric susceptibility. Moreover, the imaginary part of susceptibility gives a gain peak at the central position. Simultaneously, when ϕr = (2n + 1) π/2, n ∈ integers, we can realize 𝒫𝒯-symmetric and 𝒫𝒯-antisymmetric conditions in the susceptibilities of probe and signal fields, respectively. When ϕ = , both the susceptibilities of probe and signal fields satisfy the 𝒫𝒯-antisymmetric condition at the same time.

To check the symmetry on the probe susceptibility, we calculate the deviation in the susceptibility by calculating [ξ (z) − ξ(Lz) ] / Θ for an even function and [ξ (z) + ξ(Lz)] / Θ+ for an odd function, respectively. Here, ξ can be the real or imaginary part of the susceptibility in the probe or signal field, i.e., Re (χp), Im (χp), Re(χs), or Im (χs). A normalized factor Θ± is also introduced, which is defined by the maximal value of |ξ|. As an illustration, we fix Ωp = 0.01 in Fig. 4 for the condition of a small probe intensity. One can see that only a negligible deviation, i.e., less than 5%, is found in Fig. 4(a) for the relative phase ϕr = (2n) × π/2 and in Fig. 4(b) for the relative phase ϕr = (2n + 1) × π/2. These results reveal that the symmetry in our probe susceptibility is almost perfect along the longitudinal direction.

 

Fig. 4 The deviation of symmetry on the probe susceptibility. For an even function, we calculate ξ(z) − ξ(Lz)/Θ; while for an odd function, we calculate [ξ (z) + ξ(Lz)] / Θ+. Here, ξ can be the real (in Blue-color) or imaginary (in Red-color) part of the susceptibility in the probe field, and Θ± is the normalized factor. All the parameters used are the same as those shown in Fig. 3, but the relative phases are choose for (a): ϕr = (2n) × π/2; and (b): ϕr = (2n + 1) × π/2.

Download Full Size | PPT Slide | PDF

4. Discussions

In order to illustrate the physical picture behind the results of these 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry, we can further simplify the susceptibility for probe and signal fields given in Eqs. (5)(6). First of all, let us consider the condition with the relative phase ϕr = π/2 and |Ωc| = |Ωd| ≫ Ωp, Ωs. Then, the corresponding susceptibility of probe field can be approximated as:

Reχpχp0(ΔsΩp+2γ31ΩsΔs2+4γ312),
Imχpχp0(2γ31ΩpΔsΩsΔs2+4γ312).
From Eqs. (9)(10), one can see that the real part of probe susceptibility is almost an even function with respect to the frequency detuning Δs when Ωsp ≫ |Δs|/2γ31. At the same time, the imaginary part of χp becomes an odd function. Then, the linear transformation between the detuning Δs and propagation distance z maps these even/odd functions into the longitudinal position.

Instead, when the relative phase is set as ϕr = 3π2, the corresponding probe susceptibility χp becomes

Reχpχp0(ΔsΩp2γ31ΩsΔs2+4γ312),
Imχpχp0(2γ31Ωp+ΔsΩsΔs2+4γ312).
By comparing Eqs. (9)(10) and Eqs. (11)(12), the change of the sign in front of 2γ31 of the numerator gives us an odd function for the real part of probe susceptibility; along with an even function for the imaginary part. According to the discussions above, now we have the 𝒫𝒯-symmetric condition.

Next, when the relative phase is ϕr = π or ϕr = 2π, the probe susceptibility can be approximated as:

Reχpχp0(Δs(Ωp±Ωs)Δs2+4γ312),
Imχpχp0(2γ31(Ωp±Ωs)Δs2+4γ312).
Here, the + sign applies for ϕr = π, while the − sign applies for ϕr = 2π, respectively. As one can see, we have an odd function for real part of probe susceptibility, and an even function for the imaginary part, resulting in the 𝒫𝒯-antisymmetric condition.

As for the signal field, when the relative phase is ϕr = π/2 n ∈ integers, Eq. (6) can be approximated as:

Reχsχs0(ΔsΩs±2γ31ΩpΔs2+4γ312),
Imχsχs0(2γ31Ωs±ΔsΩp)Δs2+4γ312).
Here, the sign, + or −, applies for even or odd number of n, respectively. Similarly, the signal susceptibilities for the relative phases ϕr = (2n + 1) × π, n ∈ integers and ϕr = 2n × π, n ∈ integers have the forms:
Reχsχs0(Δs(Ωs±Ωp)Δs2±4γ312),
Imχsχs0(2γ31(Ωs±Ωp)Δs2±4γ312).
Again, the sign, + or −, applies for even or odd number of n, respectively. Nevertheless, with the comparison to the probe susceptibility, when Ωsp ≫ 2γ31/|Δs|, the real part of signal susceptibility is always an odd function, while the imaginary is an even one. It means that the 𝒫𝒯-antisymmetric condition for signal field is fulfilled.

Before the conclusion, let us remark the roles played by the probe and signal fields. In our 4-level double-Λ configuration, the role of Ωp and Ωs can be exchanged. Nevertheless, such an even or odd function in the susceptibility comes from the detuning of signal field, Δs. That is, if we perform the detuning of probe field, Δp, 𝒫𝒯-symmetry can be manifested in the signal field. Moreover, to simultaneously realize 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry for probe and signal fields, their Rabi frequencies are either Ωp ≪ Ωs or Ωp ≫ Ωs. Nevertheless, by only manipulating the relative phase, ϕr, one can also change the energy flow between probe and signal fields.

To illustrate the propagation effect on the optical susceptibilities under 𝒫𝒯-symmetric and 𝒫𝒯-antisymmetric conditions, we apply the Maxwell-Schrodinger equations for probe Ωp and signal Ωs, i.e.,

Ωpz=iγ31α2ρ31,
Ωsz=iγ41α2ρ41.
By solving Eqs. (2)(4) as well as Eqs. (19)(20) numerically and taking the spatial dependent detuning given in Eq. (7), we can obtain the field propagation behaviors.

In Fig. 5, we reveal the relationships between probe and signal field intensities and propagation distances under different phases ϕr. For the probe field with the 𝒫𝒯-symmetry, as shown in Fig. 5(a), it suffers loss in the first half part of medium, and then gets gain in the rest part. It can be understood by Fig. 3(b), in which the imaginary part of probe field susceptibility is positive (loss) from 0 < z < L/2, and negative (gain) when L/2 < z < L. Similarly, when ϕr = 3π/2, Fig. 5(e) shows that the probe field gets gain first and loss afterward. On the other hand, the signal field has 𝒫𝒯-antisymmetry condition, which means that it does not have gain and loss simultaneously in the longitudinal direction. As a result, the signal field intensity always decays because the corresponding imaginary part of susceptibilities is always positive. The field intensity propagation also gives agreement to the field susceptibilities shown in Fig. 3.

 

Fig. 5 Probe and signal field intensity verse the propagation distance with different phases: (a,b) ϕr = π/2, (c,d) ϕr = 2 × π/2, (e,f) ϕr = 3 × π/2, and (g,h) ϕr = 4 × π/2

Download Full Size | PPT Slide | PDF

5. Conclusion

In this work, through the interaction with an ensemble of 4-level atoms in a double-Λ configuration, we reveal a scheme to simultaneously realize 𝒫𝒯-symmetric and 𝒫𝒯-antisymmetric conditions. It is the linearly increasing magnetic field that maps the spectrum information into the longitudinal direction of wave propagation. Compared to previous proposals [8], in which different parameter conditions are needed for achieving 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry, our system provides a simple and all-optical-controllable way to realize 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry, as well as the switching process between them. With these phase-dependent susceptibilities, such a double-Λ system can provide a platform to study the field dynamics in 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry conditions.

Funding

This work was supported by the Ministry of Science and Technology of Taiwan under Grant No. 105-2119-M-007-004.

References

1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998). [CrossRef]  

2. Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014). [CrossRef]  

3. S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017). [CrossRef]  

4. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010). [CrossRef]  

5. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014). [CrossRef]  

6. Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015). [CrossRef]  

7. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015). [CrossRef]  

8. X. Wang and J.-H.. Wu, “Optical 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in coherently driven atomic lattices,” Opt. Express 24, 4289–4298 (2016). [CrossRef]   [PubMed]  

9. L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016). [CrossRef]  

10. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007). [CrossRef]   [PubMed]  

11. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008). [CrossRef]  

12. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008). [CrossRef]  

13. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef]  

14. A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012). [CrossRef]   [PubMed]  

15. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008). [CrossRef]  

16. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009). [CrossRef]  

17. C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015). [CrossRef]   [PubMed]  

18. S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009). [CrossRef]  

19. Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017). [CrossRef]   [PubMed]  

20. B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014). [CrossRef]  

21. M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 90, 013842 (2014). [CrossRef]  

22. J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011). [CrossRef]  

23. W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

24. H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003). [CrossRef]  

25. C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013). [CrossRef]  

26. H.-J. Li, J.-P. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency,” Opt. Express 21, 32053–32062 (2013). [CrossRef]  

27. J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013). [CrossRef]  

28. Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015). [CrossRef]  

29. Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016). [CrossRef]  

30. Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016). [CrossRef]   [PubMed]  

31. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008). [CrossRef]   [PubMed]  

32. Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010). [CrossRef]   [PubMed]  

33. S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010). [CrossRef]  

34. Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014). [CrossRef]  

35. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13, 3068–3078 (2005). [CrossRef]   [PubMed]  

36. S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011). [CrossRef]  

37. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011). [CrossRef]  

38. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013). [CrossRef]  

39. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  2. Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
    [Crossref]
  3. S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
    [Crossref]
  4. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
    [Crossref]
  5. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
    [Crossref]
  6. Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
    [Crossref]
  7. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
    [Crossref]
  8. X. Wang and J.-H.. Wu, “Optical 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in coherently driven atomic lattices,” Opt. Express 24, 4289–4298 (2016).
    [Crossref] [PubMed]
  9. L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
    [Crossref]
  10. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [Crossref] [PubMed]
  11. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
    [Crossref]
  12. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref]
  13. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
    [Crossref]
  14. A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
    [Crossref] [PubMed]
  15. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [Crossref]
  16. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
    [Crossref]
  17. C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
    [Crossref] [PubMed]
  18. S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
    [Crossref]
  19. Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
    [Crossref] [PubMed]
  20. B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
    [Crossref]
  21. M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 90, 013842 (2014).
    [Crossref]
  22. J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
    [Crossref]
  23. W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).
  24. H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
    [Crossref]
  25. C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
    [Crossref]
  26. H.-J. Li, J.-P. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency,” Opt. Express 21, 32053–32062 (2013).
    [Crossref]
  27. J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
    [Crossref]
  28. Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
    [Crossref]
  29. Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
    [Crossref]
  30. Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
    [Crossref] [PubMed]
  31. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
    [Crossref] [PubMed]
  32. Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
    [Crossref] [PubMed]
  33. S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
    [Crossref]
  34. Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
    [Crossref]
  35. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13, 3068–3078 (2005).
    [Crossref] [PubMed]
  36. S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
    [Crossref]
  37. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  38. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
    [Crossref]
  39. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
    [Crossref] [PubMed]

2017 (2)

S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
[Crossref]

Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
[Crossref] [PubMed]

2016 (5)

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

X. Wang and J.-H.. Wu, “Optical 𝒫𝒯-symmetry and 𝒫𝒯-antisymmetry in coherently driven atomic lattices,” Opt. Express 24, 4289–4298 (2016).
[Crossref] [PubMed]

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

2015 (4)

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
[Crossref] [PubMed]

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

2014 (6)

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 90, 013842 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

2013 (4)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

H.-J. Li, J.-P. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency,” Opt. Express 21, 32053–32062 (2013).
[Crossref]

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

2012 (1)

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

2011 (3)

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

2010 (3)

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

2009 (3)

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

2008 (4)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

2007 (1)

2005 (1)

2003 (1)

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Artoni, M.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Azaña, J.

Bélanger, N.

Bender, C. M.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Chaung, Y.-L.

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

Chen, H.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Chen, Y.-C.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Chen, Y.-F.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Chen, Y.-H.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Chong, Y.-D.

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Christodoulides, D. N.

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Chuang, Y.-L.

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

Dou, J.-P.

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Ellis, F. M.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

Fan, C.-H.

Fan, S.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Feng, L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Flammia, S. T.

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

Gao, F.

Ge, L.

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Gianfreda, M.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Gunther, U.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Hang, C.

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
[Crossref] [PubMed]

C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Hsieh, M.-H.

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

Hua, S.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Huang, G.

Jiang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Jiang, X.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Jiang, Y.

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Kang, H.

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

Kartashov, Y. V.

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

Klaiman, S.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Konotop, V. V.

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40, 2758–2761 (2015).
[Crossref] [PubMed]

C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Kottos, T.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

Kulishov, M.

La Rocca, G. C.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Landy, N. I.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Laniel, J. M.

Lee, R.-K.

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

Lee, Y.-C.

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

Lei, F.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Li, A.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

Li, C.

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Li, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Li, H. Q.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Li, H.-J.

Li, J.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Li, W.

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Liu, J.

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

Liu, Y.-M.

Liu, Z.-Y.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Lo, H.-Y.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Long, G. L.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Longhi, S.

S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
[Crossref]

S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Miri, M. A

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Miri, M. A.

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

Mock, J. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Moiseyev, N.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Monifi, F.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Nori, F.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Oliveira, J. E. B.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Onishchukov, G.

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Ozdemir, S. K.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Padilla, W. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Peng, B.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Peschel, U.

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Plant, D. V.

Praxmeyer, L.

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Regensburger, A.

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

Sajuyigbe, S.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Schindler, J.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

Sheng, J.

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 90, 013842 (2014).
[Crossref]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Smith, D. R.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Song, H.

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Stone, A. D.

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Sun, Y.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Tan, W.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Tsai, P.-J.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Wang, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Wang, X.

Wen, J.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Wen, L.

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

Wu, J.-H.

Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
[Crossref] [PubMed]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Wu, J.-H..

Xiao, M.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

Xu, Y.-L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Yang, C.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Yang, L.

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Yang, P.

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

Yu, I. A.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Zheng, M. C.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

Zhu, Y.

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

Ziauddin,

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

Europhys. Lett. (2)

S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
[Crossref]

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “𝒫𝒯-symmetry in Rydberg atoms,” Europhys. Lett. 115, 14005 (2016).
[Crossref]

J. Phys. A (1)

S. Longhi, “Invisibility in 𝒫𝒯-symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

Nat. Photonics (1)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Nature (1)

A. Regensburger, C. Bersch, M. A Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Nature Mater. (1)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mater. 12, 108–113 (2013).
[Crossref]

Nature Phys. (2)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

B. Peng, Ş. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (9)

S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “𝒫𝒯-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (2013).
[Crossref]

Ziauddin, Y.-L. Chuang, and R.-K. Lee, “Giant Goos-Hanchen shift using 𝒫𝒯 symmetry,” Phys. Rev. A 92, 013815 (2015).
[Crossref]

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[Crossref]

M. G. Silveirinha, “Spontaneous parity-time-symmetry breaking in moving media,” Phys. Rev. A 90, 013842 (2014).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A 84, 040101(R) (2011).
[Crossref]

L. Praxmeyer, P. Yang, and R.-K. Lee, “Phase-space representation of a non-Hermitian system with 𝒫𝒯 symmetry,” Phys. Rev. A 93, 042122 (2016).
[Crossref]

Y.-C. Lee, J. Liu, Y.-L. Chuang, M.-H. Hsieh, and R.-K. Lee, “Passive PT-symmetric couplers without complex optical potentials,” Phys. Rev. A 92, 053815 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91, 033811 (2015).
[Crossref]

Phys. Rev. Lett. (15)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Y.-C. Lee, M.-H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local 𝒫𝒯 symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112, 130404 (2014).
[Crossref]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile 𝒫𝒯 symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[Crossref]

C. Hang, G. Huang, and V. V. Konotop, “𝒫𝒯 symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with 𝒫𝒯 phase transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref]

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[Crossref] [PubMed]

Y.-D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: Time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by 𝒫𝒯-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Sci. Rep. (1)

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 Our four-level atomic system in a double-Λ configuration. Here, the four fields are denoted as Ωs, Ωp, Ωc, and Ωd, with the corresponding Rabi frequencies for signal, probe, coupling, and driving fields, respectively. The one-photon detunings of these fields are characterized by Δs, Δp, Δc, and Δd, respectively.
Fig. 2
Fig. 2 Setup of our photon-light interaction scheme, where four fields propagate along the z-direction and the atomic ensemble having photon-atom interaction in a double-Λ configuration, as shown in Fig. 1. With a magnetic field B(z), which has its magnitude linearly increasing along the propagation z-direction, the resulting susceptibilities for probe and signal fields can support a ����-symmetry or ����-antisymmetry in the longitudinal z-direction.
Fig. 3
Fig. 3 Real (Left-column) and imaginary (Right-column) parts of the susceptibilities for probe and signal fields, depicted in Blue- and Red-curves, respectively. The relative phase difference ϕr is: (a–b) π/2, (c–d) π, (e–f) 3π/2, and (g–h) 2π. One can see that χp satisfies the ����-symmetric condition when ϕr = π/2, 3π/2; and satisfies the ����-antisymmetric condition when ϕr = π, 2π. However, χs only gives the ����-antisymmetric condition. Here, the parameters used are |Ωc| = |Ωd| = 20Γ, Ωp = 0.01Γ, and Ωs = 1Γ, respectively.
Fig. 4
Fig. 4 The deviation of symmetry on the probe susceptibility. For an even function, we calculate ξ(z) − ξ(Lz)/Θ; while for an odd function, we calculate [ξ (z) + ξ(Lz)] / Θ+. Here, ξ can be the real (in Blue-color) or imaginary (in Red-color) part of the susceptibility in the probe field, and Θ± is the normalized factor. All the parameters used are the same as those shown in Fig. 3, but the relative phases are choose for (a): ϕr = (2n) × π/2; and (b): ϕr = (2n + 1) × π/2.
Fig. 5
Fig. 5 Probe and signal field intensity verse the propagation distance with different phases: (a,b) ϕr = π/2, (c,d) ϕr = 2 × π/2, (e,f) ϕr = 3 × π/2, and (g,h) ϕr = 4 × π/2

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

H ^ = [ Δ p | 3 3 | + ( Δ p Δ c ) | 2 2 | + Δ s | 4 4 | ] 2 ( Ω p | 3 1 | + Ω c | 3 2 | + Ω s | 4 1 | + Ω d | 4 2 | + H . C . ) .
t ρ 21 = γ ˜ 21 ρ 21 + i 2 Ω c * ρ 31 + i 2 Ω d * ρ 41 ,
t ρ 31 = γ ˜ 31 ρ 31 + i 2 Ω c ρ 21 + i 2 Ω p ,
t ρ 41 = γ ˜ 41 ρ 41 + i 2 Ω d ρ 21 + i 2 Ω s .
χ p = χ p 0 [ i ( 4 γ ˜ 21 γ ˜ 41 Ω p + | Ω d | 2 Ω p Ω c Ω d * Ω s ) 4 γ ˜ 21 γ ˜ 31 γ ˜ 41 + γ ˜ 41 | Ω c | 2 + γ ˜ 31 | Ω d | 2 ] ,
χ s = χ s 0 [ i ( 4 γ ˜ 21 γ ˜ 31 Ω s + | Ω c | 2 Ω s Ω d Ω c * Ω p ) 4 γ ˜ 21 γ ˜ 31 γ ˜ 41 + γ ˜ 41 | Ω c | 2 + γ ˜ 31 | Ω d | 2 ] .
Δ s ( z ) = Δ s ( 0 ) g L m j μ B B ( z ) / .
B ( z ) = [ 2 Δ s ( 0 ) g L m j μ B L ] z .
Re χ p χ p 0 ( Δ s Ω p + 2 γ 31 Ω s Δ s 2 + 4 γ 31 2 ) ,
Im χ p χ p 0 ( 2 γ 31 Ω p Δ s Ω s Δ s 2 + 4 γ 31 2 ) .
Re χ p χ p 0 ( Δ s Ω p 2 γ 31 Ω s Δ s 2 + 4 γ 31 2 ) ,
Im χ p χ p 0 ( 2 γ 31 Ω p + Δ s Ω s Δ s 2 + 4 γ 31 2 ) .
Re χ p χ p 0 ( Δ s ( Ω p ± Ω s ) Δ s 2 + 4 γ 31 2 ) ,
Im χ p χ p 0 ( 2 γ 31 ( Ω p ± Ω s ) Δ s 2 + 4 γ 31 2 ) .
Re χ s χ s 0 ( Δ s Ω s ± 2 γ 31 Ω p Δ s 2 + 4 γ 31 2 ) ,
Im χ s χ s 0 ( 2 γ 31 Ω s ± Δ s Ω p ) Δ s 2 + 4 γ 31 2 ) .
Re χ s χ s 0 ( Δ s ( Ω s ± Ω p ) Δ s 2 ± 4 γ 31 2 ) ,
Im χ s χ s 0 ( 2 γ 31 ( Ω s ± Ω p ) Δ s 2 ± 4 γ 31 2 ) .
Ω p z = i γ 31 α 2 ρ 31 ,
Ω s z = i γ 41 α 2 ρ 41 .

Metrics