Science Bulletin 63 (2018) 1096–1100 Contents lists available at ScienceDirect Science Bulletin journal homepage: www.elsevier.com/locate/scib 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 bÀ 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 bÀ 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 bÀ= Àx 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 À 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^ À 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 Àxc Àv 0 Àv a B ^ C B CB ^ C d B b C B Àv x Àv 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 Àxs b Thus, the coupled system can be described by the two hybrid eigen- modes with the eigenfrequencies satisfying rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ x2 þ ðx2 À x2Þ2 À v2x x c s c s 16 s c X Æ ¼ ; ð3Þ 1 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ x2 À ðx2 À x2Þ2 À v2x x c s c s 16 s c X Æ ¼ ; ð4Þ 2 2 where subscripts 1 and 2 stand for the two eigenmodes, and Xmþ (XmÀ) is the frequency of the creation (annihilation) operator of the eigenmodes. The normalized eigenvectors corresponding to X ¼ð 1 ; 2 ; 3 ; 4 ÞT mÆ are denoted as emÆ emÆ emÆ emÆ emÆ . ^ y ^y Each vector represents an operator in the basis ða^; b; a^ ; b Þ, i.e., ^ ^ ^ ^y ^y ^ ^ emÆ ¼ða; b; a ; b ÞemÆ, where the emþ and emÀ are the creation and annihilation operators of the m-th eigenmode satisfying ^ ^y emþ ¼ emÀ. 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 À : A ¼ e À 1 je Àj =je Àj and B ¼ e À 1 je Àj =je Àj m m m m m m m m m modes are mapped onto themselves under the time-reversal operation. The À4 (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 XmÆ 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 À x2Þ2 À 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 :  À4 cavity-atom detuning d xc À xs (Fig.