An analytical solution for quantum scattering through a PT-symmetric delta potential

doi:10.1007/s11467-021-1061-1

Frontiers of Physics

2021, Vol. 16

Issue (4): 43503 https://doi.org/10.1007/s11467-021-1061-1

本期目录

An analytical solution for quantum scattering through a

P T

-symmetric delta potential

Ying-Tao Zhang¹, Shan Jiang¹, Qingming Li¹, Qing-Feng Sun^2,^3,⁴

¹. College of Physics, Hebei Normal University, Shijiazhuang 050024, China
². International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
³. Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
⁴. CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China

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Abstract：

We employ the Lippmann–Schwinger formalism to derive the analytical solutions of the transmission and reflection coefficients through a one-dimensional open quantum system, in which particle loss or gain on one lattice site located at x = 0, or particle loss and gain on the lattice sites located at $x = ± L 2$ are considered respectively. The gain and loss on the lattice site are modeled by the delta potential with positive and negative imaginary values. The analytical solution reveals the underlying physics that the sum of the transmission and reflection coefficients through an open quantum system (even a $P T$ -symmetric open system) may not be 1, i.e., qualitatively explains that the number of particles is not conserved in an open quantum system. Furthermore, we find that the resonance states can be formed in the $P T$ -symmetric delta potential, which is similar to the case of real delta potential. The results of our analysis can be treated as the starting point of studying quantum transport problems through a non-Hermitian system using Green’s function method, and more general cases for high-dimensional systems may be deduced by the same procedure.

Key words： transmission non-Hermitian PT-symmetry')" href="#">

P T

-symmetry Green function

收稿日期: 2021-01-18 出版日期: 2021-04-15

引用本文:

. [J]. Frontiers of Physics, 2021, 16(4): 43503.
Ying-Tao Zhang, Shan Jiang, Qingming Li, Qing-Feng Sun. An analytical solution for quantum scattering through a

P T

-symmetric delta potential. Front. Phys. , 2021, 16(4): 43503.

链接本文:

https://academic.hep.com.cn/fop/CN/10.1007/s11467-021-1061-1
https://academic.hep.com.cn/fop/CN/Y2021/V16/I4/43503

1	C. M. Bender and S. Boettcher, Real spectra in non- Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80(24), 5243 (1998) https://doi.org/10.1103/PhysRevLett.80.5243
2	C. M. Bender, D. C. Brody, and H. F. Jones, Complex extension of quantum mechanics, Phys. Rev. Lett. 89(27), 270401 (2002) https://doi.org/10.1103/PhysRevLett.89.270401
3	C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70(6), 947 (2007) https://doi.org/10.1088/0034-4885/70/6/R03
4	A. Mostafazadeh, Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians, J. Math. Phys. 43(12), 6343 (2002) https://doi.org/10.1063/1.1514834
5	A. Mostafazadeh, Pseudo-Hermiticity versus PTsymmetry (II): A complete characterization of non-Hermitian Hamiltonians with a real spectrum, J. Math. Phys. 43(5), 2814 (2002) https://doi.org/10.1063/1.1461427
6	A. Mostafazadeh, Pseudo-Hermiticity versus PTsymmetry (III): Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries, J. Math. Phys. 43(8), 3944 (2002) https://doi.org/10.1063/1.1489072
7	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 (2010) https://doi.org/10.1038/nphys1515
8	S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, PT symmetry and spontaneous symmetry breaking in a microwave billiard, Phys. Rev. Lett. 108(2), 024101 (2012) https://doi.org/10.1103/PhysRevLett.108.024101
9	L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators, Nat. Photonics 8(7), 524 (2014) https://doi.org/10.1038/nphoton.2014.133
10	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 (2014) https://doi.org/10.1126/science.1258479
11	H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Parity–time-symmetric microring lasers, Science 346(6212), 975 (2014) https://doi.org/10.1126/science.1258480
12	Y. Wu, B. Zhu, S. F. Hu, Z. Zhou, and H. H. Zhong, Floquet control of the gain and loss in a PT-symmetric optical coupler, Front. Phys. 12(1), 121102 (2017) https://doi.org/10.1007/s11467-016-0642-x
13	N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Observation of asymmetric transport in structures with active nonlinearities, Phys. Rev. Lett. 110(23), 234101 (2013) https://doi.org/10.1103/PhysRevLett.110.234101
14	S. Assawaworrarit, X. Yu, and S. Fan, Robust wireless power transfer using a nonlinear parity–time-symmetric circuit, Nature 546(7658), 387 (2017) https://doi.org/10.1038/nature22404
15	Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators, Nat. Commun. 9(1), 2182 (2018) https://doi.org/10.1038/s41467-018-04690-y
16	Y. Wu, W. Liu, J. Geng, X. Song, X. Ye, C. K. Duan, X. Rong, and J. Du, Observation of parity–time symmetry breaking in a single-spin system, Science 364(6443), 878 (2019) https://doi.org/10.1126/science.aaw8205
17	G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48(2), 119 (1976) https://doi.org/10.1007/BF01608499
18	V. Gorini, A. Kossakowski, and E. C. Sudarsahan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17(5), 821 (1976) https://doi.org/10.1063/1.522979
19	S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Topology by dissipation in atomic quantum wires, Nat. Phys. 7(12), 971 (2011) https://doi.org/10.1038/nphys2106
20	F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5(9), 633 (2009) https://doi.org/10.1038/nphys1342
21	J. Dalibard, Y. Castin, and K. Molmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68(5), 580 (1992) https://doi.org/10.1103/PhysRevLett.68.580
22	H. J. Carmichael, Quantum trajectory theory for cascaded open systems, Phys. Rev. Lett. 70(15), 2273 (1993) https://doi.org/10.1103/PhysRevLett.70.2273
23	L. Jin and Z. Song, Physics counterpart of the PT non- Hermitian tight-binding chain, Phys. Rev. A 81(3), 032109 (2010) https://doi.org/10.1103/PhysRevA.81.032109
24	A. E. Miroshnichenko, Nonlinear fano-Feshbach resonances, Phys. Rev. E 79(2), 026611 (2009) https://doi.org/10.1103/PhysRevE.79.026611
25	L. Jin, and Z. Song, Hermitian scattering behavior for a non-Hermitian scattering center, Phys. Rev. A 85(1), 012111 (2012) https://doi.org/10.1103/PhysRevA.85.012111
26	G. Zhang, X. Q. Li, X. Z. Zhang, and Z. Song, Transmission phase lapse in the non-Hermitian Aharonov–Bohm interferometer near the spectral singularity, Phys. Rev. A 91(1), 012116 (2015) https://doi.org/10.1103/PhysRevA.91.012116
27	B. G. Zhu, R. Lü, and S. Chen, PT symmetry in the non-Hermitian Su–Schrieffer–Heeger model with complex boundary potentials, Phys. Rev. A 89(6), 062102 (2014) https://doi.org/10.1103/PhysRevA.89.062102
28	L. L. Zhang, G. H. Zhan, Z. Z. Li, and W. J. Gong, Effect of PT symmetry in a parallel double-quantum-dot structure, Phys. Rev. A 96(6), 062133 (2017) https://doi.org/10.1103/PhysRevA.96.062133
29	L. L. Zhang, and W. J. Gong, Transport properties in a non-Hermitian triple-quantum-dot structure, Phys. Rev. A 95(6), 062123 (2017) https://doi.org/10.1103/PhysRevA.95.062123
30	L. L. Zhang, Z. Z. Li, G. H. Zhan, G. Y. Yi, and W. J. Gong, Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots, Phys. Rev. A 99(3), 032119 (2019) https://doi.org/10.1103/PhysRevA.99.032119
31	K. L. Zhang, X. M. Yang, and Z. Song, Quantum transport in non-Hermitian impurity arrays, Phys. Rev. B 100(2), 024305 (2019) https://doi.org/10.1103/PhysRevB.100.024305
32	P. O. Sukhachov and A. V. Balatsky, Non-Hermitian impurities in Dirac systems, Phys. Rev. Research 2(1), 013325 (2020) https://doi.org/10.1103/PhysRevResearch.2.013325
33	Y. Liu, X. P. Jiang, J. Cao, and S. Chen, Non-Hermitian mobility edges in one-dimensional quasicrystals with parity–time symmetry, Phys. Rev. B 101(17), 174205 (2020) https://doi.org/10.1103/PhysRevB.101.174205
34	C. Wang and X. R. Wang, Level statistics of extended states in random non-Hermitian Hamiltonians, Phys. Rev. B 101(16), 165114 (2020) https://doi.org/10.1103/PhysRevB.101.165114
35	S. Datta, Quantum Transport: From Atoms to Transistors, Cambridge, New York: Cambridge University Press, 2005
36	E. N. Economou, Greens Functions in Quantum Physics, 3rd Ed., Springer-Verlag, Germany, 2006 https://doi.org/10.1007/3-540-28841-4
37	Here we use the form of Green′s function G0(x, x′)=−imℏ2κeiκ\|x−x′\|. While the energy E expands into the complex energy E→E±i0, in fact there are two form solutions for the Green′s function G0±(x,x′)=−∓imℏ2κe±iκ\|x−x′\|. Here we only choose G0+(x,x′) as our solution because it can promise only the scattering waves traveling toward the positive direction exist in the limite x → ∞.
38	D. Boese, M. Lischka, and L. E. Reichl, Resonances in a two-dimensional electron waveguide with a single δ-function scatterer, Phys. Rev. B 61(8), 5632 (2000) https://doi.org/10.1103/PhysRevB.61.5632
39	F. Erman, M. Gadella, and H. Uncu, One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials, Phys. Rev. D 95(4), 045004 (2017) https://doi.org/10.1103/PhysRevD.95.045004
40	F. Erman, M. Gadella, and H. Uncu, On scattering from the one-dimensional multiple Dirac delta potentials, Eur. J. Phys. 39(3), 035403 (2018) https://doi.org/10.1088/1361-6404/aaa8a3
41	P. Molinàs-Mata and P. Molinàs-Mata, Electron absorption by complex potentials: One-dimensional case, Phys. Rev. A 54(3), 2060 (1996) https://doi.org/10.1103/PhysRevA.54.2060
42	H. F. Jones, Scattering from localized non-Hermitian potentials, Phys. Rev. D 76(12), 125003 (2007) https://doi.org/10.1103/PhysRevD.76.125003
43	J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Vol. 6, Part 2, Springer-Verlag, New York, 2001; G. Chew, The Analytic S-Matrix, W.A. Benjamin, New York, 1966
44	J. G. Muga, J. P. Palao, B. Navarro, and I. L. Egusquiza, Complex absorbing potentials, Phys. Rep. 395(6), 357 (2004) https://doi.org/10.1016/j.physrep.2004.03.002
45	R. Zavin and N. Moiseyev, One-dimensional symmetric rectangular well: From bound to resonance via selforthogonal virtual state, J. Phys. Math. Gen. 37(16), 4619 (2004) https://doi.org/10.1088/0305-4470/37/16/011
46	F. Erman, M. Gadella, S. Tunalı, and H. Uncu, A singular one-dimensional bound state problem and its degeneracies, Eur. Phys. J. Plus 132(8), 352 (2017) https://doi.org/10.1140/epjp/i2017-11613-7

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