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
Even though parity-time (𝒫𝒯) symmetry was initially introduced to generalize quantum mechanics with Hermitian Hamiltonians to non-Hermitian ones , quantum entanglement gives negative results for the no-signalling principle when applying the local 𝒫𝒯-symmetric operation on one of the entangled particles . Nevertheless, 𝒫𝒯-symmetry could still be used as an interesting model for open systems in the classical limit .
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 , 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 , optical 𝒫𝒯-symmetric devices are studied theoretically and experimentally for directional couplers [10, 11], optical lattices [12–14], soliton dynamics , wave localization [16, 17], Bloch oscillations , and light diffraction . 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 , moving media , RLC circuits , and optomechanically-induced transparency systems .
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 . 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 . 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 . 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 = ωp − ω31, Δs = ωs − ω41, Δc = ωc − ω32, and Δd = ωd − ω42, where the notations ωμ (μ ∈ p, s, c, d) and ωµν ≡ (Eμ − Eν) / ħ represent the field frequency and the corresponding energy difference between energy states |μ〉 and |ν〉, respectively.
With the rotating wave approximation, one can write down the interaction Hamiltonian for this four-level atomic system in a double-Λ configuration:
Macroscopically, the susceptibility for probe field can be found by collecting the atomic matrix component ρ31, i.e., 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.,Fig. 1, due to such a closed-loop configuration for the interaction channels, the relative phase in the form: ϕr = ϕp + ϕd − ϕs − ϕc, plays a crucial parameter in our system .
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 ρ41, or explicitly in the form:
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.,
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: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.
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.
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 = nπ, 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 = nπ, 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(L − z) ] / Θ− for an even function and [ξ (z) + ξ(L − z)] / 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.
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: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 Ωs/Ωp ≫ |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 becomesEqs. (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:
As for the signal field, when the relative phase is ϕr = π/2 n ∈ integers, Eq. (6) can be approximated as:
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.,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.
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 , 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.
This work was supported by the Ministry of Science and Technology of Taiwan under Grant No. 105-2119-M-007-004.
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]
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]
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]
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]
18. S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009). [CrossRef]
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]
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]
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]