| THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS |
|
|
|
|
$\mathcal{PT}$ Symmetry of a Square-Wave Modulated Two-Level System |
| Liwei Duan1, Yan-Zhi Wang1, and Qing-Hu Chen1,2* |
1Department of Physics and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China
2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
|
|
| Cite this article: |
|
Liwei Duan, Yan-Zhi Wang, and Qing-Hu Chen 2020 Chin. Phys. Lett. 37 081101 |
|
|
|
|
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 $\gamma_{\rm eff}=\pm\varDelta_{\rm eff}$.
|
|
|
Received: 11 April 2020
Published: 28 July 2020
|
|
| PACS: |
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)
|
|
|
| Fund: Supported by the National Natural Science Foundation of China (Grant Nos. 11674285 and 11834005) and the National Key Research and Development Program of China (Grant No. 2017YFA0303002). |
|
|
|
| [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 |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
