Science Bulletin 63 (2018) 1096–1100
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Article Spontaneous T -symmetry breaking and exceptional points in cavity quantum electrodynamics systems ⇑ Yu-Kun Lu a,b,1, Pai Peng a,c,1, Qi-Tao Cao a,b,1,DaXua,b, Jan Wiersig d, Qihuang Gong a,b, Yun-Feng Xiao a,b, a State Key Laboratory for Mesoscopic Physics and Collaborative Innovation Center of Quantum Matter, School of Physics, Peking University, Beijing 100871, China b Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China c Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA d Institut für Physik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany article info abstract
Article history: Spontaneous symmetry breaking has revolutionized the understanding in numerous fields of modern Received 3 June 2018 physics. Here, we theoretically demonstrate the spontaneous time-reversal symmetry breaking in a cav- Received in revised form 22 July 2018 ity quantum electrodynamics system in which an atomic ensemble interacts coherently with a single res- Accepted 24 July 2018 onant cavity mode. The interacting system can be effectively described by two coupled oscillators with Available online 31 July 2018 positive and negative mass, when the two-level atoms are prepared in their excited states. The occur- rence of symmetry breaking is controlled by the atomic detuning and the coupling to the cavity mode, Keywords: which naturally divides the parameter space into the symmetry broken and symmetry unbroken phases. Exceptional point The two phases are separated by a spectral singularity, a so-called exceptional point, where the eigen- Spontaneous symmetry breaking Topological singularity states of the Hamiltonian coalesce. When encircling the singularity in the parameter space, the quasi- Cavity QED adiabatic dynamics shows chiral mode switching which enables topological manipulation of quantum states. Ó 2018 Science China Press. Published by Elsevier B.V. and Science China Press. All rights reserved.
1. Introduction [32–34], unconventional lasers [35–39], highly efficient phonon- lasing [40], slow light [41] and highly sensitive nanoparticle detec- Spontaneous symmetry breaking (SSB), a phenomenon where tion [42–45]. the symmetric system produces symmetry-violating states, exists While EPs in open systems are well understood, their existence ubiquitously in diverse fields of modern physics, such as particle in closed systems has been elusive. The reason is that for a closed physics [1–4], condensed matter physics [5], cosmology [6], and system with an n-dimensional Hilbert space, the Hamiltonian has n optics [7–10]. One of the great triumphs of SSB is to classify orthogonal eigenstates, which prohibit the occurrence of EPs (see different phases of matter. For instance, the paramagnetic- Ref. [46] and the Supplementary data). In this paper, we demon- ferromagnetic phase transition occurs by breaking the strate the spontaneous T -symmetry breaking and the resulting spin-rotation symmetry [11], the time-crystal phase is realized EPs in a cavity quantum electrodynamics (QED) system without by breaking the temporal translation symmetry [12–16], and the any gain or loss. The time-reversal operator T replaces i ! i superconducting phase transition emerges by breaking the more while the PT operator replaces i ! i as well as exchanging the subtle gauge symmetry [17]. Recently, in open (non-Hermitian) two modes, thus the spontaneous T -symmetry breaking serves systems, parity-time (PT ) symmetry breaking has also been pro- as the counterpart of PT -symmetry breaking in open systems. posed theoretically [18,19] and demonstrated experimentally in Analogically, EPs emerge at the edge of T -symmetry broken and optical, microwave and acoustic systems [20–26]. In particular, unbroken phases, which is verified by the coalescence of the eigen- PT symmetry breaking gives rise to exceptional points (EPs), frequencies and the eigenmodes. In the presence of dissipations, which are non-Hermitian degeneracies that are not only of sub- further study reveals that the final state depends only on the chi- stantial theoretical interest [27–31], but also lead to fascinating rality of the evolution trajectory encircling an EP, exhibiting the applications such as unidirectional-invisible optical devices topological mode switching [47,48]. Spontaneous T -symmetry breaking and EPs in quantum systems are of substantial interests not only for fundamental studies in physics, but also applications ⇑ Corresponding author. in various fields including quantum information processing and E-mail address: [email protected] (Y.-F. Xiao). 1 These authors contributed equally to this work. precise metrology. https://doi.org/10.1016/j.scib.2018.07.020 2095-9273/Ó 2018 Science China Press. Published by Elsevier B.V. and Science China Press. All rights reserved. Y.-K. Lu et al. / Science Bulletin 63 (2018) 1096–1100 1097
2. EPs and T -symmetry breaking
The system consists of N identical neutral two-level atoms interacting with a single-mode optical cavity (Fig. 1a), described ^y^ ^ ^y ^ bþ b by the Hamiltonian H ¼ xca a þ xsSz þ gða þ aÞðS þ S Þ. Here a^ (a^y) denotes the annihilation (creation) operator of the cavity P b ¼ = N rðjÞ mode, Sz 1 2 j¼1 z represents the collective operator of the ðjÞ two-level atoms with rz being the z-component spin of the j-th bþ b atom, and S ðS Þ is the collective raising (lowering) operator.
The real parameters xc, xs, and g represent the resonant frequency of the cavity mode, the transition frequency of the atoms, and the atom-photon coupling strength. The atoms are assumed to be approximately in excited states j1i for most of the time, and their collective spin can be approximated as a harmonic oscillator with a negative mass [49–51], described by the bosonic operator pffiffiffiffi T ^y ¼ b = x Fig. 1. Scheme of the system and the illustration of spontaneous symmetry b S N with a negative frequency s. For a sufficiently large breaking. (a) An ensemble of two-level atoms coupled to a single-mode cavity. The atom number N and a weak atom-photon coupling atoms are initialized at their excited states. (b) Blue (orange) region represents T b ^y^ symmetry unbroken (broken) phase with real (complex) eigenfrequencies in g; Sz N=2 b b [52,53]. The linearized Hamiltonian reads [50,54], parameter space spanned by the effective coupling strength v and the cavity-atom d ¼ x ^y^ x ^y^ þ vð^y þ ^Þð^y þ ^Þ; ð Þ detuning . (c) The system is described by two coupled oscillators with a positive H ca a sb b a a b b 1 mass (the cavity mode) and a negative mass (the collective spin of the atoms). The pffiffiffiffi upper (bottom) panel shows T symmetry broken (unbroken) phase where the pair- y ^y ^ where v ¼ g N describes the effective coupling strength. creation term a^ b and the pair-annihilation term a^b are on (off) resonance. The The Heisenberg equations of the system are given by solid arrows represent the materialized processes while the dotted arrows describe 0 1 0 10 1 virtual processes (quantum fluctuations). ^ ^ a xc v 0 v a B ^ C B CB ^ C d B b C B v x v 0 CB b C B C ¼ iB s CB C: ð2Þ @ ^y A @ A@ ^y A dt a 0 v xc v a ^y ^y b v 0 v xs b
Thus, the coupled system can be described by the two hybrid eigen- modes with the eigenfrequencies satisfying rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ x2 þ ðx2 x2Þ2 v2x x c s c s 16 s c X ¼ ; ð3Þ 1 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ x2 ðx2 x2Þ2 v2x x c s c s 16 s c X ¼ ; ð4Þ 2 2 where subscripts 1 and 2 stand for the two eigenmodes, and Xmþ
(Xm ) is the frequency of the creation (annihilation) operator of the eigenmodes. The normalized eigenvectors corresponding to X ¼ð 1 ; 2 ; 3 ; 4 ÞT m are denoted as em em em em em . ^ y ^y Each vector represents an operator in the basis ða^; b; a^ ; b Þ, i.e., ^ ^ ^ ^y ^y ^ ^ em ¼ða; b; a ; b Þ em , where the emþ and em are the creation and annihilation operators of the m-th eigenmode satisfying ^ ^y emþ ¼ em . Specifically, the m-th eigenmode is the superposition of the optical andqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oscillator mode, with coefficientsqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi derived from Fig. 2. Time evolution of the eigenmodes. (a), (b) Time evolution of the hybrid T 1 3 2 1 2 2 4 2 2 2 modes 1 and 2 in the -symmetry-unbroken regime, which shows that the e : A ¼ e 1 je j =je j and B ¼ e 1 je j =je j m m m m m m m m m modes are mapped onto themselves under the time-reversal operation. The 4 (see the Supplementary data for details). parameters are chosen as xc=xs ¼ 0:57 and v=xs ¼ 2:5 10 . (c), (d) Time T It is clear that both the eigenfrequencies Xm are real, when evolution of the eigenmodes in the -symmetry-broken regime, which shows that the modes are mapped onto each other under the time-reversal operation. Inset is ðx2 x2Þ2 v2x x > c s 16 s c 0 corresponding to the blue region in v the zoomed-in view, which shows the displacement of mode 1 and mode 2 the parameter space spanned by the coupling strength and the oscillates at the same frequency. The parameters are chosen as xc=xs ¼ 1 and v=x ¼ : 4 cavity-atom detuning d xc xs (Fig. 1b). In this case, the time s 2 5 10 . evolution of the two modes exhibits harmonic oscillations, as y^y shown in Fig. 2a and b. The two-mode squeezing terms va^ b and ^ va^b merely result in quantum fluctuations (virtual processes) and On the other hand, the eigenfrequencies become complex when ðx2 x2Þ2 v2x x < cancel with each other in the sense of average (Fig. 1c, bottom c s 16 s c 0 (for example when the cavity mode is panel). In this situation, the two eigenmodes can be mapped to on resonance with the atoms), resulting in the instability of the themselves under the time-reversal operation, which preserves system (Fig. 1c, upper panel). The instability originates from the T symmetry of the Hamiltonian (see the Supplementary data the spontaneous T symmetry breaking of the system. While the for details). Hamiltonian is invariant under the time-reversal operation, 1098 Y.-K. Lu et al. / Science Bulletin 63 (2018) 1096–1100 satisfying T HT 1 ¼ H (T is the time-reversal operator which the parameter space form an ‘‘exceptional curve” which is exactly replaces i ! i), the individual eigenmodes are not necessarily the critical curve in Fig. 1b. y^y T -invariant, and the two-mode squeezing interactions va^ b and ^ va^b play the key role in the spontaneous T -symmetry breaking. 3. Topological structure of EP Note that the spontaneous T symmetry breaking caused by the squeezing interaction differs from the parity symmetry breaking In a realistic system, the inescapable coupling to the environ- in previous works [55,56], which is also a consequence of squeez- ment leads to dissipation of the cavity mode and atoms with decay ing interaction. The key difference is that those previous models rate j and C, respectively. The difference between the decay rates have to work in the ultra-strong coupling regime, and most impor- c ¼ j C provides a new degree of freedom to study the dynamics tantly, the parity symmetry breaking does not lead to EPs. In this around EPs [47,48]. As a result, the evolution matrix in Eq. (2) is case, the energy of one mode grows exponentially while the other modified to decays at the same rate (Fig. 2c and d). Thus, the two eigenmodes 0 1 xc þ ij v 0 v are mapped onto each other by the time-reversal operation, and T B C B v x þ iC v 0 C symmetry is broken spontaneously. The above argument about T M ¼ B s C: ð5Þ @ v x þ j v A symmetry breaking is in analogy with PT symmetry breaking in 0 c i
Ref. [21]. Note that here the Hilbert space is infinite dimensional, v 0 v xs þ iC and for unbounded operators in it, it is self-adjointness rather than The real parts of the eigenfrequencies exhibit a square-root Rie- Hermicity that guarantees the spectrum to be real [57–60]. Thus it mann surface structure (Fig. 4) in the parameter space spanned by is reasonable for eigenfrequencies to acquire imaginary parts when d and c. The evolution trajectory in the parameter space is set as a the Hamiltonian, though remaining Hermitian, fails to be self- circle, i.e., dðtÞ¼d þ q cosð2pt=TÞ and cðtÞ¼q sinð2pt=TÞ, where adjoint. In the parameter space in Fig. 1b, the EP separates the 0 q is the radius of the circle and T denotes the period of the evolu- T -symmetric and the T -symmetry-broken regions (phases), mark- tion. The point (d , 0) is the center of the circle, which is set to the ing the onset of spontaneous symmetry breaking. 0 EP unless specifically mentioned. When the system starts with the The exceptional curve corresponds to a critical detuning d sat- c upper (lower) branch evolving along the clockwise (counterclock- isfying ðx2 x2Þ2 16v2x x ¼ 0, where two eigenfrequencies c s psffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic wise) direction, the state remains on the Riemann surface for a suf- X ¼ X ¼ x2 þ x2= d x coalesce, i.e., 1 2þ c s 2. Since sðcÞ in the opti- ficiently large T, in accordance with the adiabatic theorem (Fig. 4a cal domain, dc 2v (Fig. 1b). When jdj > jdcj, the real parts of the and c). On the other hand, when the system starts with the upper eigenfrequencies of the two modes show an attraction behavior (lower) branch evolving along the trajectory counterclockwise instead of an anti-crossing, with the imaginary parts being zero (clockwise), the adiabatic theorem breaks down, causing the (Fig. 3a and b). When jdj < jdcj, the gap between the real parts of detachment from the Riemann surface even for a large T (Fig. 4b the two eigenfrequencies closes while the imaginary parts bifur- and d). As a result, the system always evolves to the lower (upper) cate into a complex-conjugate pair. This kind of coalescence of branch when going clockwise (counterclockwise). This behavior eigenfrequencies is the typical characteristics of an EP. can be explained by the significant amplification of one mode rel- To further confirm the occurrence of the EP, the dependence of ative to the other mode [63]. For the amplified mode, its dynamical the two eigenmodes on d is presented on a Bloch sphere [61,62] phase has a positive imaginary part which leads to the dominance (also see the Supplementary data)inFig. 3c. Each point ðh; /Þ on in the final state. the Bloch sphere represents a unique mode, where the polar angle Based on the evolution in Fig. 4, the intensities of two instanta- h and the azimuthal angle / are obtained by the complex ampli- neous eigenmodes (see the Supplementary data for details) are tude Am and Bm with h ¼ 2 arctanðjBm=AmjÞ and / ¼ argðAm=BmÞ.In quantitatively studied by normalizing with respect to the total the largely detuned limit (d v), all energy of the mode 1 (mode intensity at each moment, as shown in Fig. 5. For loops centered 2) resides in the cavity mode (atoms), and the system is located at the EP, one mode (blue solid curve) dominates the output when at the north pole (south pole) of the Bloch sphere. When jdj going counterclockwise, while the other mode (red dashed curve) decreases from infinity, the eigenmodes evolve toward the equa- dominates the output when going clockwise (Fig. 5a and b). When tor, and merge to a single mode at the critical detuning dc where the loop encloses the EP eccentrically, the above phenomenon two EPs appear. As jdj decreases further, they part into two modes remains the same as the centered case (Fig. 5a and c). However, again (Fig. 3c). The coalescence of the eigenfrequencies and modes if the loop excludes the EP, the final state is no longer dominated directly verifies that EPs do exist in this closed system. The EPs in by one state, and mode switching does not occur (Fig. 5d). Thus
Fig. 3. (Color online) Verification of EPs. (a), (b) Dependence of real and imaginary parts of the eigenfrequencies Xm on the cavity-atom detuning d. Red dashed and blue solid 4 curves stand for hybrid modes 1 and 2, respectively. The parameters are chosen as v=xs ¼ 10 . (c) Dependence of eigenmodes on d on a Bloch sphere, where the azimuthal angle / denotes the relative phase, and the polar angle h represents the relative intensity of the uncoupled cavity mode and collective spin. The black dots mark the onset of the EPs. Y.-K. Lu et al. / Science Bulletin 63 (2018) 1096–1100 1099
mode in the unstable phase can be used as a probe: if the system is prepared in the stable phase near the exceptional curve, a weak perturbation such as the attachment of a nanoparticle can drive the system across the boundary, resulting in fast amplification of both the optical field and the oscillator motion, which has been observed experimentally [50,64]. Note here the significant amplifi- cation originates from spontaneous T -symmetry breaking instead of artificial gain media, so that our scheme is particularly beneficial for systems where gain is not available. Additionally, by coupling more than two bosonic modes together, higher-order EPs can be realized, which further boosts the sensitivity [44] (see the Supple- mentary data for details).
4. Summary
In summary, we have demonstrated the existence of sponta- neous T -symmetry breaking in closed systems without construct- ing the balance of gain or loss, an analogy of PT -symmetry breaking in open systems. By showing the coalescence of eigenfre- quencies as well as eigenmodes in closed cavity QED systems, it has been proved that EP emerges as a consequence of spontaneous T -symmetry breaking. Furthermore, the topological nature of EP is Fig. 4. (Color online) Riemann sheet structure of the eigenfrequencies in the explored, and robust mode switching is achieved by encircling the parameter space. Real parts of eigenfrequencies Xm in the parameter space spanned by the cavity-atom detuning d and the decay rate c, which exhibit a EP. Similar Hamiltonians can also be realized beyond cavity QED Riemann surface structure. Arrows represent the evolution trajectories of the states. systems [65], such as optomechanical systems [66–68], spin sys- (a) The state starts with the upper branch, clockwise. (b) Starts with the upper tems [69], atomic systems [50], Josephson junctions [70], etc. branch, counterclockwise. (c) Starts with the lower branch, counterclockwise. (d) Spontaneous T -symmetry breaking in closed systems not only Starts with the lower branch, clockwise. Through (a) to (d), the parameters are 4 1 broadens the understanding of SSB and singularities in quantum chosen as v=xs ¼ 2 10 ; jTj¼10v ; q ¼ 1:5v and d0 ¼ 2v. physics, but also reveals the rich physics in infinite dimensional systems. Apart from its fundamental interest, spontaneous T -symmetry breaking in closed systems without gain or loss also provides a new platform for various applications, such as sensing and quantum information processing.
Conflict of interest
The authors declare that they have no conflict of interest.
Acknowledgments
The authors would like to thank Y.-C. Liu, H. Jing, Y.-X. Wang, L. Yang, S. Rotter, K. An, and H. Schomerus for fruitful discussions. This work was supported by the National Key R&D Program of China (2016YFA0301302), the National Natural Science Foundation of China (61435001, 11654003, 11474011), and High-performance Computing Platform of Peking University.
Appendix A. Supplementary material
Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.scib.2018.07.020. Fig. 5. (Color online) Time evolution of relative intensities of the two modes. The intensities are normalized with respect to the total intensity in two modes at each moment. The black point represents the EP, the arrow denotes the evolution References trajectory and T is the period of the evolution. The inset shows the evolution trajectory in the parameter space spanned by detuning d and dissipation rate c. The [1] Nambu Y, Jona-Lasinio G. Dynamical model of elementary particles based on 4 1 parameters are set as v=xs ¼ 2 10 ; q ¼ 1:5v and jTj¼10v . (a) A loop an analogy with superconductivity. i. Phys Rev 1961;122:345. 1 centered at EP (d0 ¼ 2v), counterclockwise (T ¼ 10v ). (b) A loop centered at EP [2] Nambu Y, Jona-Lasinio G. Dynamical model of elementary particles based on 1 an analogy with superconductivity. ii. Phys Rev 1961;124:246. (d0 ¼ 2v), clockwise (T ¼ 10v ). (c) An excentric loop enclosing EP (d0 ¼ 3:25v), 1 [3] Englert F, Brout R. Broken symmetry and the mass of gauge vector mesons. counterclockwise (T ¼ 10v ). (d) A loop excluding EP (d0 ¼ 6v), counterclockwise (T ¼ 10v 1). Phys Rev Lett 1964;13:321. [4] Higgs PW. Broken symmetries and the masses of gauge bosons. Phys Rev Lett 1964;13:508. [5] Altland A, Simons BD. Condensed matter field theory. Cambridge University the asymmetric state transfer is protected by the topology of the Press; 2010. EP, immune to the smooth deformation of the evolution trajectory. [6] Albrecht A, Steinhardt PJ. Cosmology for grand unified theories with T radiatively induced symmetry breaking. Phys Rev Lett 1982;48:1220. The existence of spontaneous -symmetry breaking holds [7] Cao Q-T, Wang H, Dong C-H, et al. Experimental demonstration of spontaneous potential for high-precision sensing. The exponentially growing chirality in a nonlinear microresonator. Phys Rev Lett 2017;118:033901. 1100 Y.-K. Lu et al. / Science Bulletin 63 (2018) 1096–1100
[8] Del Bino L, Silver JM, Stebbings SL, et al. Symmetry breaking of counter- [49] Møller CB, Thomas RA, Vasilakis G, et al. Quantum back-action-evading propagating light in a nonlinear resonator. Sci Rep 2017;7:43142. measurement of motion in a negative mass reference frame. Nature [9] Hamel P, Haddadi S, Raineri F, et al. Spontaneous mirror-symmetry breaking in 2017;547:191. coupled photonic-crystal nanolasers. Nat Photonics 2015;9:311. [50] Kohler J, Gerber JA, Dowd E, et al. Negative-mass instability of the spin and [10] Rodríguez-Lara B, El-Ganainy R, Guerrero J. Symmetry in optics and photonics: motion of an atomic gas driven by optical cavity backaction. Phys Rev Lett a group theory approach. Sci Bull 2018;63:244. 2018;120:013601. [11] Schumann F, Buckley M, Bland J. Paramagnetic-ferromagnetic phase transition [51] Khalili FY, Polzik ES. Overcoming the standard quantum limit in gravitational during growth of ultrathin Co/Cu (001) films. Phys Rev B 1994;50:16424. wave detectors using spin systems with a negative effective mass. Phys Rev [12] Wilczek F. Quantum time crystals. Phys Rev Lett 2012;109:160401. Lett 2018;121:031101. [13] Wilczek F. Superfluidity and space-time translation symmetry breaking. Phys [52] Holstein T, Primakoff H. Field dependence of the intrinsic domain Rev Lett 2013;111:250402. magnetization of a ferromagnet. Phys Rev 1940;58:1098. [14] Sacha K. Modeling spontaneous breaking of time-translation symmetry. Phys [53] Klein A, Marshalek ER. Boson realizations of Lie algebras with applications to Rev A 2015;91:033617. nuclear physics. Rev Mod Phys 1991;63:375. [15] Else DV, Bauer B, Nayak C. Floquet time crystals. Phys Rev Lett [54] Walls DF, Milburn GJ. Quantum Optics. Springer Science & Business Media; 2016;117:090402. 2007. [16] Yao NY, Potter AC, Potirniche I-D, et al. Discrete time crystals: rigidity, [55] Emary C, Brandes T. Quantum chaos triggered by precursors of a quantum criticality, and realizations. Phys Rev Lett 2017;118:030401. phase transition: the Dicke model. Phys Rev Lett 2003;90:044101. [17] Greiter M. Is electromagnetic gauge invariance spontaneously violated in [56] Garziano L, Stassi R, Ridolfo A, et al. Vacuum-induced symmetry breaking in a superconductors? Ann Phys 2005;319:217. superconducting quantum circuit. Phys Rev A 2014;90:043817. [18] Bender CM, Boettcher S. Real spectra in non-hermitian hamiltonians having [57] Hall BC. Quantum theory for mathematicians, vol. 267. Springer; 2013. PT symmetry. Phys Rev Lett 1998;80:5243. [58] Hassani S. Mathematical physics: a modern introduction to its [19] Mostafazadeh A. Pseudo-hermiticity versus PT symmetry: the necessary foundations. Springer Science & Business Media; 2013. condition for the reality of the spectrum of a non-hermitian hamiltonian. J [59] Simon B. Quantum mechanics for Hamiltonians defined as quadratic Math Phys 2002;43:205. forms. Princeton University Press; 2015. [20] Guo A, Salamo G, Duchesne D, et al. Observation of PT -symmetry breaking in [60] Gieres F. Mathematical surprises and Dirac’s formalism in quantum complex optical potentials. Phys Rev Lett 2009;103:093902. mechanics. Rep Prog Phys 2000;63:1893. [21] Rüter CE, Makris KG, El-Ganainy R, et al. Observation of parity–time symmetry [61] Okamoto H, Gourgout A, Chang C-Y, et al. Coherent phonon manipulation in in optics. Nat Phys 2010;6:192. coupled mechanical resonators. Nat Phys 2013;9:480. [22] Bittner S, Dietz B, Günther U, et al. PT symmetry and spontaneous symmetry [62] Faust T, Rieger J, Seitner MJ, et al. Coherent control of a classical breaking in a microwave billiard. Phys Rev Lett 2012;108:024101. nanomechanical two-level system. Nat Phys 2013;9:485. [23] Chang L, Jiang X, Hua S, et al. Parity–time symmetry and variable optical [63] Hassan AU, Zhen B, Soljai M, et al. Dynamically encircling exceptional points: isolation in active–passive-coupled microresonators. Nat Photonics Exact evolution and polarization state conversion. Phys Rev Lett 2014;8:524. 2017;118:093002. [24] Peng B, Özdemir SßK, Lei F, et al. Parity-time-symmetric whispering-gallery [64] Kohler J, Spethmann N, Schreppler S, et al. Cavity-assisted measurement and microcavities. Nat Phys 2014;10:394. coherent control of collective atomic spin oscillators. Phys Rev Lett [25] Zhu X, Ramezani H, Shi C, et al. PT -symmetric acoustics. Phys Rev X 2017;118:063604. 2014;4:031042. [65] Xiao Y-F, Gong Q. Optical microcavity: from fundamental physics to functional [26] Shi C, Dubois M, Chen Y, et al. Accessing the exceptional points of parity-time photonics devices. Sci Bull 2016;61:185. symmetric acoustics. Nat Commun 2016;7:11110. [66] Aspelmeyer M, Kippenberg TJ, Marquardt F. Cavity optomechanics. Rev Mod [27] Yi C-H, Kullig J, Wiersig J. Pair of exceptional points in a microdisk cavity under Phys 2014;86:1391. an extremely weak deformation. Phys Rev Lett 2018;120:093902. [67] Bernier NR, Tóth LD, Feofanov AK, et al. Level attraction in a microwave [28] Makris KG, El-Ganainy R, Christodoulides D, et al. Beam dynamics in PT optomechanical circuit. arXiv:1709.02220. symmetric optical lattices. Phys Rev Lett 2008;100:103904. [68] Li T, Yin Z-Q. Quantum superposition, entanglement, and state teleportation of [29] Chong Y, Ge L, Stone AD. PT -symmetry breaking and laser-absorber modes in a microorganism on an electromechanical oscillator. Sci Bull 2016;61:163. 1 optical scattering systems. Phys Rev Lett 2011;106:093902. [69] Manousakis E. The spin-2 Heisenberg antiferromagnet on a square lattice and [30] Longhi S. Bloch oscillations in complex crystals with PT symmetry. Phys Rev its application to the cuprous oxides. Rev Mod Phys 1991;63:1. Lett 2009;103:123601. [70] Nataf P, Ciuti C. Vacuum degeneracy of a circuit QED system in the ultrastrong [31] Liu Z-P, Zhang J, Özdemir SßK, et al. Metrology with PT -symmetric cavities: coupling regime. Phys Rev Lett 2010;104:023601. Enhanced sensitivity near the PT -phase transition. Phys Rev Lett 2016;117:110802. [32] Lin Z, Ramezani H, Eichelkraut T, et al. Unidirectional invisibility induced by PT -symmetric periodic structures. Phys Rev Lett 2011;106:213901. Yu-Kun Lu is an undergraduate from the Department of [33] Feng L, Ayache M, Huang J, et al. Nonreciprocal light propagation in a silicon Physics, Peking University. He is doing research in photonic circuit. Science 2011;333:729. quantum optics under the supervision of Prof. Yun-Feng [34] Regensburger A, Bersch C, Miri M-A, et al. Parity-time synthetic photonic Xiao. He is interested broadly in AMO, quantum infor- lattices. Nature 2012;488:167. mation and condensed matter theory. [35] Peng B, Özdemir SßK, Rotter S, et al. Loss-induced suppression and revival of lasing. Science 2014;346:328. [36] Peng B, Özdemir SßK, Liertzer M, et al. Chiral modes and directional lasing at exceptional points. Proc Natl Acad Sci USA 2016;113:6845. [37] Brandstetter M, Liertzer M, Deutsch C, et al. Reversing the pump dependence of a laser at an exceptional point. Nat Commun 2014;5:4034. [38] Feng L, Wong ZJ, Ma R-M, et al. Single-mode laser by parity-time symmetry breaking. Science 2014;346:972. [39] Hodaei H, Miri M-A, Heinrich M, et al. Parity-time–symmetric microring lasers. Science 2014;346:975. [40] Jing H, Özdemir SK, Lü X-Y, et al. PT -symmetric phonon laser. Phys Rev Lett 2014;113:053604. Yun-Feng Xiao received his Ph.D. degree in Physics from [41] Jing H, Özdemir SßK, Geng Z, et al. Optomechanically-induced transparency in University of Science and Technology of China in 2007. parity-time-symmetric microresonators. Sci Rep 2015;5:9663. He joined the faculty of Peking University in 2009, and [42] Wiersig J. Enhancing the sensitivity of frequency and energy splitting was promoted to a tenured professor in 2014. His detection by using exceptional points: application to microcavity sensors for research interests lie in microcavity optics and pho- single-particle detection. Phys Rev Lett 2014;112:203901. tonics. [43] Wiersig J. Sensors operating at exceptional points: General theory. Phys Rev A 2016;93:033809. [44] Hodaei H, Hassan AU, Wittek S, et al. Enhanced sensitivity at higher-order exceptional points. Nature 2017;548:187. [45] Chen W, Özdemir SßK, Zhao G, et al. Exceptional points enhance sensing in an optical microcavity. Nature 2017;548:192. [46] Moiseyev N. Non-Hermitian quantum mechanics. Cambridge University Press; 2011. [47] Xu H, Mason D, Jiang L, et al. Topological energy transfer in an optomechanical system with exceptional points. Nature 2016;537:80. [48] Doppler J, Mailybaev AA, Böhm J, et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 2016;537:76.