Open Access
Add to BibSonomy
Abstract
The non-Hermitian models, which are symmetric under parity (P) and time-reversal (T) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually demands precise control of nonlinear processes, limiting their application. In this paper, to bypass this obstacle, we introduce a class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet) that can describe a two-level system with temporally modulated on-site potential and couplings. We show that implementing an appropriate non-Unitary gauge transformation converts the original system to an effective one with a balanced gain and loss. This will allow us to derive the evolution of states analytically. Our proposed class of Hamiltonians can be employed in different platforms such as electronic circuits, acoustics, and photonics to design structures with hidden PT-symmetry potentially without imaginary onsite amplification and absorption mechanism to obtain an exceptional point.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The non-Hermitian models in physics have become a vast research area in recent years, including condensed matter physics [1], photonics [2,3], biophysics [4], and acoustic [5]. Among them, the non-Hermitian generalization of topological tight-binding systems introduces new phenomena absent in Hermitian ones, such as non-Hermitian skin effect [6], distinct transport effects [7], relocation of topological edge states [8], and noise-resilient [9], to name a few.
A well-known approach to making a tight-binding model non-Hermitian is introducing complex onsite potential, playing the role of gain (loss) in photonic models [10]. One can also stagger inter (intra) cell hopping amplitudes and develop non-reciprocity through the system [6,11,12].
Among non-Hermitian hamiltonians with gain and loss, the ones with balanced amplification and absorption are known as parity time-reversal ($PT$) symmetric hamiltonians and attracted more intention. $PT$-symmetric systems have been studied in both continuous and discrete models [13,14] and extensively investigated theoretically and experimentally in photonic lattices [15–18], microring resonator [19–21], and electrical circuit [22–24].
In $PT$-symmetric non-Hermitian systems, specific singular points are known as exceptional points (EPs) where at least two or more eigenvalues and corresponding eigenvectors of the hamiltonians coalesce. In recent years, the EPs have led to many interesting phenomena observed in different physical systems [25–30].
Indeed, EPs separate the broken $PT$-symmetric regions, in which the eigenvalues of the non-Hermitian Hamiltonian become complex, from the unbroken $PT$-symmetric phase with real-valued eigenvalues. Many nontrivial features such as unidirectional invisibility [31], invisible sensor [32], single-mode lasing [33], sensitive readout [34] and robust wireless power transfer [35] can be investigated by creating and developing $PT$-symmetric structure. However, from a practical point of view, constructing delicate balanced gain and loss is a rigorous requirement that hinders the possibility of implementing $PT$-symmetry and studying its interesting features in specifically quantum systems.
In addition, fundamental obstacles such as gain-induced noise and instabilities arise motivation to look for a different route of achieving $PT$-symmetry without gain. The gain-free models with a hidden $PT$-symmetry capture various research interests [19,36–41].
The key approaches to design a system benefit hidden $PT$-symmetry, are based on nonlinear phenomena [42] or by making use of a temporally modulated coupling. In the latter method, a unitary gauge transformation leads to an effective system whose Hamiltonian is invariant under parity and time reversal. A well-known platform to explore such a gauge transformation is an electronic system. For example, a transient $PT$-symmetry can be triggered by the switching on and off of electronic devices [43].
A larger class of Hamiltonian with hidden $PT$-symmetry can be developed and studied by releasing the unitarity condition on the transformation to reach the effective Hamiltonian. This approach shapes the main idea of this paper.
Indeed, in this study, we introduce a general class of time-dependent two-level non-Hermitian Hamiltonians with a hidden $PT$-symmetry which can be revealed by implementing a non-unitary gauge transformation.
By implementing a specific non-Unitary transformation introduced in this study, one can reduce a time-dependent system to the $PT$-symmetric one. The main aspect of this approach is its generality. Indeed, our proposed Hamiltonian is not necessarily floquet which makes it applicable to the vast range of time-dependent two-level systems. In addition, we formulated our two-level Hamiltonian with an arbitrary onsite potential. In the case of onsite potential with zero imaginary part, the system can be transformed into the effective one with balanced gain and loss. Therefore our proposed method can bypass the usual difficulties in realizing amplification and absorption mechanism. The two-level Hamiltonian investigated in this letter can model acoustics, electronics, and photonics systems whose couplings are time or coordinate dependant.
Besides, we show that the analysis of state evolution described by this class of Hamiltonian can be analytically obtained regardless of adiabatic condition. The exact solutions are helpful to predict and manipulate the system’s behavior precisely when it encircles EPs [44].
We organize the paper as follows. In section 2, we introduce the Hamiltonian, non-unitary time dependant transformation and manifest the $PT$-symmetry in the model. In section 3, we apply our method to an electronic system to illustrate different $PT$-symmetric phases. The system’s time evolution is studied analytically in Supplement 1 (SM).
2. Time-dependant model
Let us consider a class of two-level systems described by the following time-dependent Hamiltonian
We introduce time-dependent transformation $\Psi (t) =A(t)\chi (t)$, which converts the Hamiltonian (1) into the effective Hamiltonian
where $A$ is an invertible ($\det A\neq 0$) matrix. For a closed two-level system with a reciprocal coupling between their sites, the Hamiltonian is given by a Hermitian matrix (i.e., $H=H^{\dagger }$). For such a system, the transformation $A$ needs to be given by a unitary matrix to conserve the norm of state (i.e., $\|\Psi (t)\|= \|\chi (t)\|$), in this case, the two terms on the right side of (3) are also Hermitian. However, in our study, we do not restrict our approach to the similarity condition and consider a general time-dependent transformation. Specifically, by choosing a suitable form of time-dependent transformation $A(t)$, one can reduce the Hamiltonian $H(t)$ to a traceless matrix (i.e., Tr($H_{\text {eff}})=0$) which describes the effective system with balanced gain and loss. This can be achieved byBy choosing a specific form of time transformation $A$ given by (4), the Schrödinger Eq. (7) can be recasted into the following second-order differential equation for $\zeta _{\mp }$
In driving the above differential equations, we scaled variables such as
If we suppose $\omega _2(t)=0$, then, the vector state $\Psi (t)$ is given by
For the nonzero $\omega _2(t)\neq 0$, the evolution of $\psi _{1,2}$ are given by the above equation by replacing $\Gamma (\tau )\rightarrow \omega _2(\tau )-\Gamma (\tau )$.
In SM, by considering three different forms for the effective onsite potential $\Gamma (t)$, we show that the evolution of the system governed by (7) can be obtained analytically. From there, implementation of an inverse time transformation leads to the dynamics of the original system given by (1).
To conclude the section, we are looking through the eigenbasis and eigenvalues of the original Hamiltonian (1) and its effective counterpart (5). For the effective system, the time-dependent eigenvalues are given by
In the case of static $PT$-symmetric where $\Gamma$ is time independent and $\nu =\nu '$, the system is in $PT$ unbroken (broken) phase for $\vert \nu \vert >\Gamma$ ($\vert \nu \vert <\Gamma$), where $\vert \nu \vert =\Gamma$ is an exceptional point. In the case of the periodic Hamiltonian, then the $PT$ (un)broken phase can be found by looking at the exceptional points of the Floquet Hamiltonian.
For the general time-dependant $\Gamma (t)$, as a consequence of non-Hermicity, the eigenbasis of the system is not orthogonal, but one can define biorthogonal eigenbasis by using its right and left eigenvectors which are defined via
For the original system, if $\Omega _-=\pm i\nu$, it reaches exceptional points (EPs). By following the same calculations for the effective system given by the Hamiltonian $H_{\text {eff}}$, the EPs exist if
To study the dynamic of EPs under time-dependent transformation, we furthermore define the following complex parameter
In regarding the original Hamiltonian (1), the complex parameter $B$ takes a complex-valued number in the parameter space (${\rm Re}(B)$-${\rm Im}(B)$), which are
Under the transformation $A(t)$, the loci of EPs for the effective system in its parameter space (${\rm Re}(\bar {B})$-${\rm Im}(\bar {B})$) are given by $\bar {B}^{P}=\pm i$ where
Tracking the loci of EPs for the effective system in the $B$-parameter space shows that they are no longer constant and given in time as
where $b(t):=\frac {\partial _t\ln \frac {f_1}{f_2}}{2\nu }=b_r(t)+ib_i(t)$. To derive (22), we put $\cot g=\cot \bar {g}=i$ in Eq. (16). It shows that, in the $B$-parameter space, the real (imaginary) coordinate at $t=t_1$ of EPs shift by the factor of $-b_i(t_1)$($b_r(t_1)$).In the following section, by making use of an electronic platform, we illustrate the hidden $PT$-symmetric in the system whose dynamics are given by the Hamiltonian (1).
3. Application: $PT$ symmetry in an electronic system
We implement our approach to design a $PT$-symmetric model using an electronic platform consisting of an RLC circuit. By following the Kirchhoff’s laws, one can find the coupled equation for the voltage in capacitor $V(t)$ and the current across the inductor $I(t)$:
By taking $\Psi (t)=(V(t),I(t))^{T}$, the dynamic of $RLC$ circuit can be given with $H(t)$
For a time-dependent resistance $R(t)$, capacitance $C(t)$ and inductance $L(t)$, the above Hamiltonian-like matrix can be written in the form of the Hamiltonian (1) if we set
Then in terms of our denotations, we have
If we modulate the resistance such that $g(t)=L_0\partial _t f$, then the effective Hamiltonian (5) reduces to
where $\omega _0=1/\sqrt {L_0C_0}$. To get the above relation, we modified the time transformation (4) such that $A(t)\rightarrow \text {diag}(C_0^{-1/2},L_0^{-1/2})A(t)$ [46] We first study the time-independent, effective Hamiltonian corresponding to case (a) in the SM. This can be achieved by setting $f(t)=e^{-\gamma t}$. Then, the electronic $RLC$ circuit stimulate a static $PT$-symmetric system whose Hamiltonian is given by $H_{\text {eff}}^{PT}=i\gamma \sigma _z+\omega _0\sigma _y$. The eigenvalues of this Hamiltonian are given by $\lambda _{\pm }=\pm \sqrt {\omega _0^2-\gamma ^2}$. From these eigenvalues, we can find the $PT$-symmetry breaking threshold where the system is referred to as being in a $PT$-broken phase for $\gamma >\omega _0$. Figure 1 demonstrates the time evolution of the circuit’s energy $U_L+U_C$ for both LC ($R(t)=0$) and RLC circuit where
Fig. 1. Time evolution of the circuit’s energy for LC circuit (a,b) and RLC circuit (c,d). In $\gamma <\omega _0$ ($\gamma <2\omega _0$) regime, the system is in the unbroken $PT$-symmetric phase, and the energy oscillates. This oscillation is depicted in plots (a) and (c), where, in the latter, the resistance causes the damping oscillation. In the broken phase, the energy amplifies for both electronic systems.
In the regime $\gamma <\omega _0$ ($\gamma <2\omega _0$), which defines unbroken $PT$-symmetric phase, one can see from the diagram that the energy of the RLC (LC) system oscillates while in the broken phase, the energy of both systems amplifies. Note that in the RLC system, the oscillation is damping by the factor of $e^{-R/2L}$ because of the resistance.
In the second step, we consider a periodic time-dependent $f(t)=\epsilon _1\cos (\Omega _0 t)+\epsilon _2$ with the period $T=2\pi /\Omega _0$. In this case, the effective Hamiltonian becomes periodic, and therefore its dynamics are determined by the one-period, non-unitary time evolution operator obtained from the following time-ordered product [47],
where the time-independent operator $H_F$ is the so-called Floquet Hamiltonian in which their eigenvalues and eigenstates are called quasi-energies, and quasi-states [48]. The corresponding Floquet Hamiltonian $H_F$ can determine the $PT$ broken and unbroken phases of the system. To show this, we consider the eigenvalues of Floquet Hamiltonian $\varepsilon _{1,2}$ and the time evolution $\lambda _{1,2}$ that related to each other through $\lambda _{1,2}=e^{-iT\varepsilon _{1,2}}$. The quasi-energies $\varepsilon _{1,2}$ take real value in the phase where the system is invariant under parity and time-reversal transformation. In this phase, the eigenvalues of the evolution operator $U_F$ satisfies $\vert \lambda _1\vert$=$\vert \lambda _2\vert =1$ (see Fig. 2). The energy of the time-dependent RLC circuit oscillates in this phase while it is damping when the Floquet Hamiltonian enters the broken $PT$ phase.Fig. 2. The magnitude of the eigenvalues $\lambda _k$ belong to the time evolution operator $U_F$ versus $\Omega _0$. Orange and blue regions represent the $PT$ unbroken and broken phases of the system, respectively.
We note that for time-dependent coupling in the form of the step function, the $PT$ (un)broken phases have been studied in [49] for the LC circuit. However, our approach can encompass a general class of time-dependent systems with hidden $PT$-symmetric phase.
4. Conclusion
In summary, in this paper, we propose a class of time-dependent Hamiltonian describing a two-level system with the hidden $PT$-symmetry. The hidden symmetric can be revealed by implementing a specific non-Unitary gauge transformation which transforms the original Hamiltonian with temporally modulated couplings and onsite potentials to an effective one with balanced gain and loss. The dynamics of the effective system can be analytically extracted. Time-dependent systems are not usually analytically solvable. Thus, using our method we can propose a class of systems that one can analytically drive the condition for the existence of exceptional points and thus use it for determining other applications of exceptional points including sensing. The approach of this paper can provide a protocol to design and develop a different type of system with a $PT$-symmetric characteristic without encountering the obstacles of injecting gain. We demonstrated how our method can be applied to electronic devices. However, photonics models may have a potential application. The main experimental challenge in developing a photonics application is designing a system with time-varying couplings and onsite potentials. Microwave Smoothly deformed metallic waveguides in the microwave and helical waveguides in optics have the potential to be used in the design of driven photonics systems. In such a system, the deformed waveguide is extended along the $x$-axis, allowing us to achieve the spatially modulated couplings [50,51], where $x$ now represents time in our approach.
Funding
National Science Foundation (PHY-2012172, OMA-2231387); Army Research Office (W911NF-20-1-0276).
Acknowledgments
H. R. acknowledge the support by the Army Research Office Grant No. W911NF-20-1-0276 and NSF Grant No. PHY-2012172 and OMA-2231387.
Disclosures
“The authors declare no conflicts of interest.”
Data availability
The data that support the findings of this study are available from the corresponding author upon 168 reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. W. B. Rui, M. M. Hirschmann, and A. P. Schnyder, “$\mathcal {PT}$-symmetric non-hermitian dirac semimetals,” Phys. Rev. B 100(24), 245116 (2019). [CrossRef]
2. S. Longhi, “Spectral singularities and bragg scattering in complex crystals,” Phys. Rev. A 81(2), 022102 (2010). [CrossRef]
3. K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional $\mathcal {P}\mathcal {T}$-symmetric photonic crystals,” Phys. Rev. B 92(23), 235310 (2015). [CrossRef]
4. T. Chou, K. Mallick, and R. Zia, “Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport,” Rep. Prog. Phys. 74(11), 116601 (2011). [CrossRef]
5. X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “$\mathcal {P}\mathcal {T}$-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014). [CrossRef]
6. S. Yao and Z. Wang, “Edge states and topological invariants of non-hermitian systems,” Phys. Rev. Lett. 121(8), 086803 (2018). [CrossRef]
7. H. Ghaemi-Dizicheh and H. Schomerus, “Compatibility of transport effects in non-hermitian nonreciprocal systems,” Phys. Rev. A 104(2), 023515 (2021). [CrossRef]
8. H. Ramezani, “Anomalous relocation of topological states,” arXiv, arXiv:2207.12193 (2022). [CrossRef]
9. W. Tuxbury, R. Kononchuk, and T. Kottos, “Non-resonant exceptional points as enablers of noise-resilient sensors,” Commun. Phys. 5(1), 210 (2022). [CrossRef]
10. H. Schomerus, “Topologically protected midgap states in complex photonic lattices,” Opt. Lett. 38(11), 1912–1914 (2013). [CrossRef]
11. T. E. Lee, “Anomalous edge state in a non-hermitian lattice,” Phys. Rev. Lett. 116(13), 133903 (2016). [CrossRef]
12. S. Lieu, “Topological phases in the non-hermitian su-schrieffer-heeger model,” Phys. Rev. B 97(4), 045106 (2018). [CrossRef]
13. L. Feng, R. El-Ganainy, and L. Ge, “Non-hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017). [CrossRef]
14. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-hermitian physics and pt symmetry,” Nat. Phys. 14(1), 11–19 (2018). [CrossRef]
15. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical pt-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007). [CrossRef]
16. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in $\mathcal {P}\mathcal {T}$ symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef]
17. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]
18. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef]
19. B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014). [CrossRef]
20. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346(6212), 975–978 (2014). [CrossRef]
21. B. Peng, Ş. K. Özdemir, 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,” Nat. Phys. 10(5), 394–398 (2014). [CrossRef]
22. J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active lrc circuits with $\mathcal {PT}$ symmetries,” Phys. Rev. A 84(4), 040101 (2011). [CrossRef]
23. M. Chitsazi, H. Li, F. M. Ellis, and T. Kottos, “Experimental realization of floquet $\mathcal {P}\mathcal {T}$-symmetric systems,” Phys. Rev. Lett. 119(9), 093901 (2017). [CrossRef]
24. R. d. J. León-Montiel, M. A. Quiroz-Juárez, J. L. Domínguez-Juárez, R. Quintero-Torres, J. L. Aragón, A. K. Harter, and Y. N. Joglekar, “Observation of slowly decaying eigenmodes without exceptional points in floquet dissipative synthetic circuits,” Commun. Phys. 1(1), 1–11 (2018). [CrossRef]
25. M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54(10), 1039–1047 (2004). [CrossRef]
26. C. M. Bender, “Making sense of non-hermitian hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]
27. I. Rotter, “A non-hermitian hamilton operator and the physics of open quantum systems,” J. Phys. A: Math. Theor. 42(15), 153001 (2009). [CrossRef]
28. N. Moiseyev, Non-Hermitian quantum mechanics (Cambridge University Press, 2011).
29. W. Heiss, “The physics of exceptional points,” J. Phys. A: Math. Theor. 45(44), 444016 (2012). [CrossRef]
30. H. Cao and J. Wiersig, “Dielectric microcavities: Model systems for wave chaos and non-hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015). [CrossRef]
31. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by $\mathcal {P}\mathcal {T}$-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011). [CrossRef]
32. R. Fleury, D. Sounas, and A. Alu, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015). [CrossRef]
33. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014). [CrossRef]
34. Z. Dong, Z. Li, F. Yang, C.-W. Qiu, and J. S. Ho, “Sensitive readout of implantable microsensors using a wireless system locked to an exceptional point,” Nat. Electron. 2(8), 335–342 (2019). [CrossRef]
35. S. Assawaworrarit and S. Fan, “Robust and efficient wireless power transfer using a switch-mode implementation of a nonlinear parity–time symmetric circuit,” Nat. Electron. 3(5), 273–279 (2020). [CrossRef]
36. H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. 124(19), 193901 (2020). [CrossRef]
37. H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time symmetry based on time modulation,” Phys. Rev. Appl. 14(3), 031002 (2020). [CrossRef]
38. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of $\mathcal {P}\mathcal {T}$-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). [CrossRef]
39. M. Ornigotti and A. Szameit, “Quasi-symmetry in passive photonic lattices,” J. Opt. 16(6), 065501 (2014). [CrossRef]
40. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef]
41. H. Liu, D. Sun, C. Zhang, M. Groesbeck, R. Mclaughlin, and Z. V. Vardeny, “Observation of exceptional points in magnonic parity-time symmetry devices,” Sci. Adv. 5(11), eaax9144 (2019). [CrossRef]
42. Y. Jiang, Y. Mei, Y. Zuo, Y. Zhai, J. Li, J. Wen, and S. Du, “Anti-parity-time symmetric optical four-wave mixing in cold atoms,” Phys. Rev. Lett. 123(19), 193604 (2019). [CrossRef]
43. X. Yang, J. Li, Y. Ding, M. Xu, X.-F. Zhu, and J. Zhu, “Observation of transient parity-time symmetry in electronic systems,” Phys. Rev. Lett. 128(6), 065701 (2022). [CrossRef]
44. A. U. Hassan, B. Zhen, M. Soljačić, M. Khajavikhan, and D. N. Christodoulides, “Dynamically encircling exceptional points: Exact evolution and polarization state conversion,” Phys. Rev. Lett. 118(9), 093002 (2017). [CrossRef]
45. T. J. Milburn, J. Doppler, C. A. Holmes, S. Portolan, S. Rotter, and P. Rabl, “General description of quasiadiabatic dynamical phenomena near exceptional points,” Phys. Rev. A 92(5), 052124 (2015). [CrossRef]
46. One can consider a rigorous version of kirchhoff’s laws where the time derivative of capacitance (c(t)) is taken into account. then there is an extra onsite potential (i.e., ω1 is no longer zero). however, in our approach to transforming the original hamiltonian into an effective one with balanced gain and loss, we consider an arbitrary onsite potentials. as a result, hidden pt symmetry can still be revealed in such a system with different broken and unbroken phases, but the location of exceptional point shifts.
47. J. H. Shirley, “Solution of the schrödinger equation with a hamiltonian periodic in time,” Phys. Rev. 138(4B), B979–B987 (1965). [CrossRef]
48. S. Barone, M. Narcowich, and F. Narcowich, “Floquet theory and applications,” Phys. Rev. A 15(3), 1109–1125 (1977). [CrossRef]
49. M. A. Quiroz-Juárez, Z. A. Cochran, J. L. Aragón, Y. N. Joglekar, and R. d. J. León-Montiel, “Parity-time symmetry via time-dependent non-unitary gauge fields,” arXiv, arXiv:2109.03846 (2021).
50. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496(7444), 196–200 (2013). [CrossRef]
51. J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016). [CrossRef]
