$\mathcal{PT}$ Symmetry of a Square-Wave Modulated Two-Level System
Liwei Duan1 , Yan-Zhi Wang1 , and Qing-Hu Chen1,2*
1 Department of Physics and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China2 Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
Abstract ：We study a non-Hermitian two-level system with square-wave modulated dissipation and coupling. Based on the Floquet theory, we achieve an effective Hamiltonian from which the boundaries of the $\mathcal{PT}$ phase diagram are captured exactly. Two kinds of $\mathcal{PT}$ symmetry broken phases are found, whose effective Hamiltonians differ by a constant $\omega / 2$. For the time-periodic dissipation, a vanishingly small dissipation strength can lead to the $\mathcal{PT}$ symmetry breaking in the $(2k-1)$-photon resonance ($\varDelta = (2k-1) \omega$), with $k=1,2,3\dots$ It is worth noting that such a phenomenon can also happen in $2k$-photon resonance ($\varDelta = 2k \omega$), as long as the dissipation strengths or the driving times are imbalanced, namely $\gamma_0 \ne - \gamma_1$ or $T_0 \ne T_1$. For the time-periodic coupling, the weak dissipation induced $\mathcal{PT}$ symmetry breaking occurs at $\varDelta_{\rm eff}=k\omega$, where $\varDelta_{\rm eff}=(\varDelta_0 T_0 + \varDelta_1 T_1)/T$. In the high frequency limit, the phase boundary is given by a simple relation $\gamma_{\rm eff}=\pm\varDelta_{\rm eff}$.
收稿日期: 2020-04-11
出版日期: 2020-07-28
:
11.30.Er
(Charge conjugation, parity, time reversal, and other discrete symmetries)
42.82.Et
(Waveguides, couplers, and arrays)
03.65.Yz
(Decoherence; open systems; quantum statistical methods)
42.50.-p
(Quantum optics)
[1] Bender C M and Boettcher S 1998 Phys. Rev. Lett. 80 5243 [2] Bender C M, Brody D C and Jones H F 2002 Phys. Rev. Lett. 89 270401 [3] Bender C M 2005 Contemp. Phys. 46 277 [4] Bender C M 2007 Rep. Prog. Phys. 70 947 [5] Konotop V V, Yang J and Zezyulin D A 2016 Rev. Mod. Phys. 88 035002 [6] Ruschhaupt A, Delgado F and Muga J G 2005 J. Phys. A 38 L171 [7] El-Ganainy R, Makris K G, Christodoulides D N and Musslimani Z H 2007 Opt. Lett. 32 2632 [8] Makris K G, El-Ganainy R, Christodoulides D N and Musslimani Z H 2008 Phys. Rev. Lett. 100 103904 [9] Musslimani Z H, Makris K G, El-Ganainy R and Christodoulides D N 2008 Phys. Rev. Lett. 100 030402 [10] Guo A, Salamo G J, Duchesne D, Morandotti R, Volatier-Ravat M, Aimez V, Siviloglou G A and Christodoulides D N 2009 Phys. Rev. Lett. 103 093902 [11] Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M and Kip D 2010 Nat. Phys. 6 192 [12] Feng L, Ayache M, Huang J, Xu Y L, Lu M H, Chen Y F, Fainman Y and Scherer A 2011 Science 333 729 [13] Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N and Peschel U 2012 Nature 488 167 [14] Hodaei H, Miri M A, Heinrich M, Christodoulides D N and Khajavikhan M 2014 Science 346 975 [15] Luo X, Huang J, Zhong H, Qin X, Xie Q, Kivshar Y S and Lee C 2013 Phys. Rev. Lett. 110 243902 [16] Weimann S, Kremer M, Plotnik Y, Lumer Y, Nolte S, Makris K G, Segev M, Rechtsman M C and Szameit A 2017 Nat. Mater. 16 433 [17] Xiao L, Wang K, Zhan X, Bian Z, Kawabata K, Ueda M, Yi W and Xue P 2019 Phys. Rev. Lett. 123 230401 [18] Zhang D, Luo X Q, Wang Y P, Li T F and You J 2017 Nat. Commun. 8 1368 [19] Zhang G Q and You J Q 2019 Phys. Rev. B 99 054404 [20] Shen R C, Zhang G Q, Wang Y P and You J Q 2019 Acta Phys. Sin. 68 230305 [21] Lee T E 2016 Phys. Rev. Lett. 116 133903 [22] Shen H, Zhen B and Fu L 2018 Phys. Rev. Lett. 120 146402 [23] Kunst F K, Edvardsson E, Budich J C and Bergholtz E J 2018 Phys. Rev. Lett. 121 026808 [24] Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S and Ueda M 2018 Phys. Rev. X 8 031079 [25] Kawabata K, Higashikawa S, Gong Z, Ashida Y and Ueda M 2019 Nat. Commun. 10 297 [26] Kawabata K, Shiozaki K, Ueda M and Sato M 2019 Phys. Rev. X 9 041015 [27] Leykam D, Bliokh K Y, Huang C, Chong Y D and Nori F 2017 Phys. Rev. Lett. 118 040401 [28] Liu T, Zhang Y R, Ai Q, Gong Z, Kawabata K, Ueda M and Nori F 2019 Phys. Rev. Lett. 122 076801 [29] Yao S and Wang Z 2018 Phys. Rev. Lett. 121 086803 [30] Song F, Yao S and Wang Z 2019 Phys. Rev. Lett. 123 170401 [31] Xiao L, Zhan X, Bian Z H, Wang K K, Zhang X, Wang X P, Li J, Mochizuki K, Kim D, Kawakami N et al. 2017 Nat. Phys. 13 1117 [32] Yao S, Song F and Wang Z 2018 Phys. Rev. Lett. 121 136802 [33] Yokomizo K and Murakami S 2019 Phys. Rev. Lett. 123 066404 [34] Liu J S, Han Y Z and Liu C S 2020 Chin. Phys. B 29 010302 [35] Liu J S, Han Y Z and Liu C S 2019 Chin. Phys. B 28 100304 [36] Joglekar Y N, Marathe R, Durganandini P and Pathak R K 2014 Phys. Rev. A 90 040101 [37] Lee T E and Joglekar Y N 2015 Phys. Rev. A 92 042103 [38] Bagarello F, Lattuca M, Passante R, Rizzuto L and Spagnolo S 2015 Phys. Rev. A 91 042134 [39] Moiseyev N 2011 Phys. Rev. A 83 052125 [40] Gong J and Wang Q H 2015 Phys. Rev. A 91 042135 [41] Gong J and Wang Q H 2019 Phys. Rev. A 99 012107 [42] Xie Q, Rong S and Liu X 2018 Phys. Rev. A 98 052122 [43] Luo X, Yang B, Zhang X, Li L and Yu X 2017 Phys. Rev. A 95 052128 [44] Chitsazi M, Li H, Ellis F M and Kottos T 2017 Phys. Rev. Lett. 119 093901 [45] Li J, Harter A K, Liu J, de Melo L, Joglekar Y N and Luo L 2019 Nat. Commun. 10 855 [46] Jia C Y and Liang Z X 2020 Chin. Phys. Lett. 37 040502 [47] Lee T E, Reiter F and Moiseyev N 2014 Phys. Rev. Lett. 113 250401 [48] Shirley J H 1965 Phys. Rev. 138 B979 [49] Sambe H 1973 Phys. Rev. A 7 2203 [50] Chen C, An J H, Luo H G, Sun C P and Oh C H 2015 Phys. Rev. A 91 052122 [51] Tong Q J, An J H, Gong J, Luo H G and Oh C H 2013 Phys. Rev. B 87 201109 [52] Xiong T S, Gong J and An J H 2016 Phys. Rev. B 93 184306
[1]
.
[2]
.
[3]
.
[4]
.
[5]
ZHANG Jiao, YUE Chong-Xing. Radiative Dileptonic Decays Bs →l+ l− γ in the TC2 Model
[6]
ZHANG Feng;GAO Yuan-Ning;HUO Lei. Prospects on Determining Electric Dipole Moments of Σand Ξ Hyperons at BESIII
[7]
NI Wei-Tou. Probing the Microscopic Origin of Gravity via Precision Polarization and Spin Experiments
[8]
PING Rong-Gang;. Searching for Electrical Dipole and Chromodipole Moments of a Charm Quark in J/ψ → γФФ Decay
[9]
SHI Zhi-Qiang;NI Guang-Jiong;. Lifetime of Polarized Fermions in Flight
[10]
GUO Wei;HAN Ru-Shan. Time-Reversal Symmetry Breaking: an Evidence for Spin Pairing
in High-Tc Superconductors
[11]
WU Yue-liang. Probing New Physics from CP Violation in Radiative B Decays
[12]
GUO Han-ying;ZHAO Wan-yun. Induced Chern-Simons Term at High Temperature in Intrinsic Regularization
[13]
XING Zhizhong;DU Dongsheng. Constraints on Violation in the Fourth Generation Effect on CP
Penguin-Dominated b → s Decays
2020, Vol. 37 
