囚禁在电介质球形微腔中类氢原子的内部无序性

刘雪; 王德华

doi:10.7498/aps.72.20222413

摘要

给出了一种研究囚禁在微腔中类氢原子的内部无序性的一种方法, 即利用体系的量子信息熵和形状复杂度对囚禁体系的无序性进行表征和研究. 计算和分析了囚禁在InN电介质球形微腔中类氢原子的位置和动量空间中香农信息熵和形状复杂度, 重点探究了量子囚禁效应对体系无序性的影响. 计算结果表明: 当微腔半径较小时, 量子囚禁现象显著, 形状复杂度曲线中有一系列极值点, 这是由信息熵和空间不均匀性的共同作用引起的. 随着微腔半径增大, 囚禁现象减弱, 囚禁类氢原子的香农信息熵和形状复杂度趋近于自由空间类氢原子的情形. 为囚禁量子体系内部无序性的研究提供了一种有效方法, 对囚禁量子体系的信息测量提供了一定的参考.

关键词:

Abstract

The research on the disorder of quantum system plays a very important role in the field of quantum information, and has received much attention from theoretical and experimental researchers. However, it is very difficult to study the disorder of atoms trapped in microcavity due to their complex nonlocal space-time evolution characteristics. To solve this problem, we present a method to study the internal disorder of hydrogenic atoms trapped in microcavity, that is, to characterize and investigate the disorder of the confined system by using the quantum information entropy and shape complexity of the system. The Shannon information entropy and shape complexity in position space and momentum space (S_r, S_p, C[r], C[p]) are calculated and analyzed for different quantum states of hydrogenic atom in InN dielectric spherical microcavity, and pay special attention to the exploration of the influence of quantum confinement effect on the disorder of the system. The results show that when the radius of the spherical microcavity is very small, the quantum confinement effect is more significant, and a series of extreme points appears in the shape complexity curve of the system, which is caused by the joint interaction of information entropy and spatial inhomogeneity. With the increase of the radius of the spherical cavity, the effect of quantum confinement is weakened, and the Shannon information entropy and shape complexity of the confined hydrogenic atom are similar to the counterparts of the hydrogenic atom in free space. Our work provides an effective method to study the internal disorder of a confined quantum. This work provides an effective method for studying the internal disorder of confined quantum systems and presents some references for the information measurement of confined quantum systems.

Keywords:

作者及机构信息

刘雪,
王德华

鲁东大学物理与光电工程学院, 烟台　264025

通信作者: 王德华, lduwdh@163.com

基金项目: 国家自然科学基金(批准号: 11374133)和山东省自然科学基金(批准号: ZR2019MA066)资助的课题.

Authors and contacts

School of Physics and Optoelectronic Engineering, Ludong University, Yantai 264025, China

Corresponding author: Wang De-Hua, lduwdh@163.com

Funds: Supported by the National Natural Science Foundation of China (Grant No. 11374133) and the Natural Science Foundation of Shandong Province, China (Grant No. ZR2019MA066).

文章全文 : translate this paragraph

参考文献

[1]	Connerade J P, Dolmatov V H, Lakshmi P A 2000 J. Phys. B: At. Mol. Opt. Phys. 33 251 Google Scholar
[2]	Sabin J R, Brändas E J, Cruz S A 2009 The Theory of Confined Quantum Systems, Parts I and II, Advances in Quantum Chemistry (Vols. 57, 58) (Amsterdam: Academic Press) pp2–19
[3]	Sen K D 2014 Electronic Structure of Quantum Confined Atoms and Molecules (Switzerland: Springer) pp1–253
[4]	Wang D H, Zhang J, Sun Z P, Zhang S F, Zhao G 2021 Chem. Phys. 551 111331 Google Scholar
[5]	Martínez-Sánchez M A, Vargas R, Garza J 2019 Quantum Rep. 1 208 Google Scholar
[6]	Guerra D, Vargas R, Fuentealba P, Garza J 2009 Adv. Quantum Chem. 58 1
[7]	Rodriguez-Bautista M, Díaz-García C, Navarrete-López A M, Vargas R, Garza J 2015 J. Chem. Phys. 143 034103 Google Scholar
[8]	Porras-Montenegro N, Pe´rez-Merchancano S T 1992 Phys. Rev. B 46 9780 Google Scholar
[9]	Şahin M 2008 Phys. Rev. B 77 045317 Google Scholar
[10]	Yuan J H, Zhang Y, Guo X X, Zhang J J, Mo H 2015 Physica E 68 232 Google Scholar
[11]	Wang D H, He X, Liu X, Chu B H, Liu W, Jiao M M 2022 Philos. Mag. 102 2302 Google Scholar
[12]	Longo G M, Longo S, Giordano D 2015 Phys. Scr. 90 085402 Google Scholar
[13]	Cottrell T L 1951 Trans. Faraday Soc. 47 337 Google Scholar
[14]	Zhu J L, Xiong J J, Gu B L 1990 Phys. Rev. B 41 6001 Google Scholar
[15]	Lopez-Rosa S, Manzano D, Dehesa J S 2009 Physica A 388 3273 Google Scholar
[16]	Liu S 2007 J. Chem. Phys. 126 191107 Google Scholar
[17]	Aquino N, Flores-Riveros A, Rivas-Silva J F 2013 Phys. Lett. A 377 2062 Google Scholar
[18]	Sun G H, Popov D, Camacho-Nieto O, Dong S H 2015 Chin. Phys. B 24 100303 Google Scholar
[19]	Najafizade S A, Hassanabadi H, Zarrinkamar S 2016 Chin. Phys. B 25 040301 Google Scholar
[20]	Bialynicki-Birula I, Mycielski J 1975 Comm. Math. Phys. 44 129 Google Scholar
[21]	Guevara N L, Sagar R P, Esquivel R O, 2003 J. Chem. Phys. 119 7030 Google Scholar
[22]	Fuentealba P, Melin J 2002 Int. J. Quantum Chem. 90 334 Google Scholar
[23]	López-Ruiz R, Mancini H L, Calbet X 1995 Phys. Lett. A 209 321 Google Scholar
[24]	Calbet X, López-Ruiz R 2001 Phys. Rev. E 63 066116 Google Scholar
[25]	Angulo J C, Antolín J 2008 J. Chem. Phys. 128 164109 Google Scholar
[26]	Majumdar S, Mukherjee N, Roy A K 2017 Chem. Phys. Lett. 687 322 Google Scholar
[27]	梁双, 吕燕伍 2007 56 1617 Google Scholar Liang S, Lv Y W 2007 Acta Phys. Sin. 56 1617 Google Scholar
[28]	Mata M, Zhou X, Furtmayr F, et al. 2013 J. Mater. Chem. C 1 4300 Google Scholar
[29]	Guo Y, Pan F, Ren Y J, et al. 2018 Phys. Chem. Chem. Phys. 20 24239 Google Scholar
[30]	Li P, David A, Li H, et al. 2021 Appl. Phys. Lett. 119 231101 Google Scholar
[31]	Binks D J, Dawson P, Oliver R A, Wallis D J 2022 Appl. Phys. Rev. 9 041309 Google Scholar
[32]	Chang C, Li X C 2022 Eur. Phys. J. D. 76 134 Google Scholar
[33]	González-Férez R, Dehesa J S, Patil S H, Sen K D 2009 Physica A 388 4919 Google Scholar

施引文献

图 1 囚禁在InN电介质球形微腔中氢原子的不同的量子态在位置空间和动量空间中的径向概率密度分布, 假设球形微腔的半径r_a = 10

Fig. 1. Position space and momentum space radial probability density for different quantum states of hydrogen atom in the InN dielectric spherical microcavity, suppose the radius of spherical microcavity r_a = 10

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图 2 InN电介质球形微腔内氢原子前几个态的本征能　(a) r_a = 10; (b) r_a = 50; (c) r_a = 200; (d) r_a = 500

Fig. 2. Intrinsic energy of the first few states of hydrogen atom in the InN dielectric spherical microcavity: (a) r_a = 10; (b) r_a = 50; (c) r_a = 200; (d) r_a = 500.

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图 3 囚禁在InN电介质球形微腔和真空球形微腔中氢原子的1s态的香农信息熵和形状复杂度随半径的变化　(a)位置空间和动量空间中的香农信息熵S_r和S_p; (b) 位置空间和动量空间的平均电子概率密度ln$\langle r \rangle$和ln$ \langle p \rangle $; (c) 位置空间的形状复杂度C [r]; (d)动量空间中的形状复杂度C [p]

Fig. 3. Shannon information entropy and shape complexity as a function of the radius of the microcavity for the 1 s state of the hydrogen atom confined in the InN dielectric spherical microvavtiy and vacuum spherical microcavity: (a) Shannon information entropy in the position space S_r and in the momentum space S_p; (b) averaging electron probability density in the position space ln$\langle r \rangle$ and in the momentum space ln$ \langle p \rangle $; (c) shape complexity in the position space C [r]; (d) shape complexity in the momentum space C [p].

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图 4 囚禁在InN电介质球形微腔中和真空球形微腔中氢原子的2s态的香农信息熵和形状复杂度随微腔半径的变化曲线 (a) S_r,S_p, ln$\langle r \rangle$和ln$\langle p \rangle $随r_a的变化曲线; (b)在位置空间和动量空间中形状复杂度C [r]和C [p]随r_a的变化曲线

Fig. 4. Shannon information entropy and shape complexity as a function of the radius of the microcavity for the 2s state of the hydrogen atom confined in the InN dielectric spherical microvavtiy and vacuum spherical microcavity: (a) Variation of S_r, S_p, ln$\langle r \rangle$ and ln$\langle p \rangle $ with r_a; (b) variation of shape complexity in the position space C [r] and in the momentum space C [p] with r_a.

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图 5 囚禁在InN电介质球形微腔中氢原子的3s态和4s态的香农信息熵和形状复杂度随微腔半径的变化曲线　(a) 3s态S_r, S_p, ln$\langle r \rangle$和ln$\langle p \rangle $随r_a的变化; (b) 3s态的C [r]和C [p]随r_a的变化; (c) 4s态S_r, S_p,ln$\langle r \rangle$和ln$ \langle p \rangle $随r_a的变化; (d) 4s态的C [r]和C [p]随r_a的变化

Fig. 5. Shannon information entropy and shape complexity as a function of the radius of the InN dielectric spherical microcavity for the 3s state and 4s state: (a) Variation of S_r, S_p, ln$\langle r \rangle$ and ln$\langle p \rangle $ with r_a for the 3s state; (b) variation of C [r] and C [p] with r_a for the 3s state; (c) variation of S_r, S_p, ln$\langle r \rangle$ and ln$\langle p \rangle $ with r_a for the 4s state; (d) variation of C [r] and C [p] with r_a for the 4s state.

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图 6 囚禁在InN电介质球形微腔中氢原子的2p, 3p和4p态的香农信息熵和形状复杂度随微腔半径的变化　(a) 2p态S_r, S_p, ln$\langle r \rangle$和ln$\langle p \rangle $随r_a的变化; (b) 2p态C [r]和C [p]随r_a的变化; (c) 3p态S_r, S_p, ln$\langle r \rangle$和ln$\langle p\rangle $随r_a的变化; (d) 3p态C [r]和C [p]随r_a的变化; (e) 4p态S_r, S_p, ln$ \langle r \rangle $和ln$\langle p\rangle $随r_a的变化; (f) 4p态C [r]和C [p]随r_a的变化

Fig. 6. Shannon information entropy and shape complexity as a function of the radius of the InN dielectric spherical microcavity for the 2p, 3p and 4p states: (a) Variation of S_r, S_p, ln$ \langle r \rangle $and ln$\langle p \rangle $ with r_a for the 2p state; (b) variation of C [r] and C [p] with r_a for the 2p state; (c) variation of S_r, S_p, ln$ \langle r \rangle $and ln$\langle p \rangle $ with r_a for the 3p state; (d) variation of C [r] and C [p] with r_a for the 3p state; (e) variation of S_r, S_p, ln$ \langle r \rangle $and ln$\langle p \rangle $ with r_a for the 4p state; (f) variation of C [r] and C [p] with r_a for the 4p state.

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图 7 囚禁在InN电介质球形微腔中氢原子的不同量子态的香农信息熵和S_t的比较

Fig. 7. Comparison of Shannon information entropy sum S_tfor different quantum states of hydrogenic atom confined in the InN dielectric spherical microcavity.

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图 8 囚禁在InN电介质球形微腔中氢原子的不同的s态和p态的动量空间的形状复杂度C [r]的比较

Fig. 8. Comparison of the momentum space shape complexity C [r] for different s and p states of hydrogenic atom confined in the InN dielectric spherical microcavity.

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图 9 囚禁在InN电介质球形微腔中氢原子的不同的s态和p态的动量空间的形状复杂度C [p]的比较

Fig. 9. Comparison of the momentum space shape complexity C [p] for different s and p states of hydrogenic atom confined in the InN dielectric spherical microcavity.

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表 1 InN电介质球形微腔中氢原子在不同囚禁半径下的前几个态的本征能

Table 1. Intrinsic energy of the first few states of hydrogen atoms at different confinement radii in InN dielectric spherical microcavity.

r_a/a.u.	0.5	10	30	50	100	200	300	500
1s	19.42009	0.03291	–0.00042	–0.00192	–0.00213	–0.00214	–0.00214	–0.00214
2s	78.54949	0.17680	0.01493	0.00365	–0.00011	–0.00053	–0.00053	–0.00053
2p	40.13943	0.08877	0.00709	0.00152	–0.00032	–0.00053	–0.00053	–0.00053
3s	177.19306	0.42100	0.04155	0.01305	0.00208	–0.00001	–0.00021	–0.00024
3p	119.05181	0.28297	0.02797	0.00880	0.00138	–0.00007	–0.00022	–0.00024
3d	66.22135	0.15538	0.01486	0.00447	0.00056	–0.00016	–0.00023	–0.00024
4s	315.33012	0.76461	0.07935	0.02653	0.00536	0.00074	0.00009	–0.00012
4p	237.44938	0.57693	0.06017	0.02023	0.00415	0.00057	0.00005	–0.00012
4d	165.17213	0.40025	0.04149	0.01385	0.00277	0.00034	–0.00001	–0.00013
4f	97.46420	0.23423	0.02381	0.00777	0.00143	0.00010	–0.00008	–0.00013

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表 2 囚禁在InN电介质球形微腔中氢原子1s, 2s, 3s和4s态在位置和动量空间中的香农信息熵和形状复杂度随半径的变化

Table 2. Variation of Shannon information entropy and shape complexity in the position and momentum spaces of 1s, 2s, 3s and 4s states for hydrogenic atom confined in the InN dielectric spherical microcavity as a function of the radius of the microcavity.

r_a/a.u.		0.5	2	4	6	10	40	100	400	1000	1500
1s	S_r	–1.4078	2.7388	4.8012	5.9998	6.8441	7.4938	11.2125	12.3083	12.3283	12.3283
	S_p	8.0224	3.8677	1.7946	0.5856	0.2690	–0.9287	–4.7199	–5.7473	–5.7617	–5.7617
	C [r]	1.3238	1.3331	1.3463	1.3604	1.3754	1.3915	1.7729	2.4742	2.5107	2.5107
	C [p]	1.5156	1.5110	1.5053	1.5002	1.4959	1.4924	1.6081	2.3007	2.3545	2.3545
2s	S_r	–1.6182	2.5393	4.6173	5.8324	6.6945	7.3633	11.5600	14.5011	16.2944	16.2945
	S_p	9.8106	5.6517	3.5722	2.3561	1.4934	0.8246	–3.3163	–6.3549	–8.9410	–8.9411
	C [r]	2.3122	2.3318	2.3584	2.3857	2.4135	2.4420	2.8872	2.6978	2.5882	2.5883
	C [p]	1.2987	1.2991	1.3004	1.3027	1.3059	1.3104	1.6255	3.5705	3.5862	3.5862
3s	S_r	–1.6941	2.4651	4.5451	5.7622	6.6260	7.2964	11.4868	14.3744	18.4189	18.6100
	S_p	10.7698	6.6122	4.5347	3.3205	2.4598	1.7931	–2.2946	–4.8614	10.0304	10.3718
	C [r]	3.2965	3.3183	3.3476	3.3772	3.4071	3.4372	3.8912	4.3150	2.4249	2.5897
	C [p]	1.2959	1.2934	1.2900	1.2866	1.2833	1.2802	1.2566	1.6164	4.9245	5.3478
4s	S_r	–1.7332	2.4264	4.5068	5.7242	6.5883	7.2589	11.4435	14.2816	18.7984	20.2300
	S_p	11.4216	7.2638	5.1857	3.9710	3.1098	2.4425	–1.6559	–4.1749	–8.9360	11.2692
	C [r]	4.2798	4.3022	4.3323	4.3624	4.3927	4.4230	4.8733	5.4856	3.3059	2.5188
	C [p]	1.3205	1.3193	1.3173	1.3153	1.3134	1.3115	1.2917	1.3738	3.2785	7.1624

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表 3 InN电介质球形微腔中氢原子2p, 3p和4p态位置空间和动量空间中的香农信息熵和形状复杂度

Table 3. Shannon information entropy and shape complexity of 2p, 3p and 4p states in the position space and momentum space for hydrogen atom confined in the InN dielectric spherical microcavity.

r_a/a.u.		0.5	2	4	6	10	40	100	400	1000	1500
2p	S_r	–1.1279	3.0287	5.1050	6.3181	7.8439	11.9414	14.4765	15.8804	15.8804	15.8804
	S_p	9.0214	4.8633	2.7851	1.5697	0.0398	–4.0817	–6.5889	–7.7092	–7.7092	–7.7092
	C [r]	1.1568	1.1577	1.1589	1.1602	1.1628	1.1887	1.2918	1.7107	1.7107	1.7107
	C [p]	1.3084	1.3068	1.3049	1.3024	1.2982	1.2749	1.3281	1.7639	1.7639	1.7639
3p	S_r	–1.3920	2.7660	4.8442	6.0595	7.5896	11.7333	14.4737	18.2711	18.4214	18.4214
	S_p	10.3320	6.1733	4.0942	2.8783	1.3468	–2.7953	–5.4546	–9.4770	–9.6406	–9.6406
	C [r]	1.6088	1.6113	1.6148	1.6183	1.6254	1.6833	1.8150	1.8267	1.9657	1.9657
	C [p]	1.2564	1.2547	1.2526	1.2506	1.2465	1.2240	1.2749	2.9511	2.9671	2.9671
4p	S_r	–1.5141	2.6444	4.7234	5.9394	7.4711	11.6261	14.3794	18.7629	20.1488	20.1494
	S_p	11.1181	6.9596	4.8808	3.6652	2.1344	–2.0027	–4.6540	–8.7725	10.8158	10.8150
	C [r]	2.0644	2.0676	2.0719	2.0762	2.0850	2.1533	2.2977	2.2520	2.0817	2.0829
	C [p]	1.2878	1.2862	1.2845	1.2827	1.2791	1.2542	1.2288	2.7123	3.8746	3.8744

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map

[1]	Connerade J P, Dolmatov V H, Lakshmi P A 2000 J. Phys. B: At. Mol. Opt. Phys. 33 251 Google Scholar
[2]	Sabin J R, Brändas E J, Cruz S A 2009 The Theory of Confined Quantum Systems, Parts I and II, Advances in Quantum Chemistry (Vols. 57, 58) (Amsterdam: Academic Press) pp2–19
[3]	Sen K D 2014 Electronic Structure of Quantum Confined Atoms and Molecules (Switzerland: Springer) pp1–253
[4]	Wang D H, Zhang J, Sun Z P, Zhang S F, Zhao G 2021 Chem. Phys. 551 111331 Google Scholar
[5]	Martínez-Sánchez M A, Vargas R, Garza J 2019 Quantum Rep. 1 208 Google Scholar
[6]	Guerra D, Vargas R, Fuentealba P, Garza J 2009 Adv. Quantum Chem. 58 1
[7]	Rodriguez-Bautista M, Díaz-García C, Navarrete-López A M, Vargas R, Garza J 2015 J. Chem. Phys. 143 034103 Google Scholar
[8]	Porras-Montenegro N, Pe´rez-Merchancano S T 1992 Phys. Rev. B 46 9780 Google Scholar
[9]	Şahin M 2008 Phys. Rev. B 77 045317 Google Scholar
[10]	Yuan J H, Zhang Y, Guo X X, Zhang J J, Mo H 2015 Physica E 68 232 Google Scholar
[11]	Wang D H, He X, Liu X, Chu B H, Liu W, Jiao M M 2022 Philos. Mag. 102 2302 Google Scholar
[12]	Longo G M, Longo S, Giordano D 2015 Phys. Scr. 90 085402 Google Scholar
[13]	Cottrell T L 1951 Trans. Faraday Soc. 47 337 Google Scholar
[14]	Zhu J L, Xiong J J, Gu B L 1990 Phys. Rev. B 41 6001 Google Scholar
[15]	Lopez-Rosa S, Manzano D, Dehesa J S 2009 Physica A 388 3273 Google Scholar
[16]	Liu S 2007 J. Chem. Phys. 126 191107 Google Scholar
[17]	Aquino N, Flores-Riveros A, Rivas-Silva J F 2013 Phys. Lett. A 377 2062 Google Scholar
[18]	Sun G H, Popov D, Camacho-Nieto O, Dong S H 2015 Chin. Phys. B 24 100303 Google Scholar
[19]	Najafizade S A, Hassanabadi H, Zarrinkamar S 2016 Chin. Phys. B 25 040301 Google Scholar
[20]	Bialynicki-Birula I, Mycielski J 1975 Comm. Math. Phys. 44 129 Google Scholar
[21]	Guevara N L, Sagar R P, Esquivel R O, 2003 J. Chem. Phys. 119 7030 Google Scholar
[22]	Fuentealba P, Melin J 2002 Int. J. Quantum Chem. 90 334 Google Scholar
[23]	López-Ruiz R, Mancini H L, Calbet X 1995 Phys. Lett. A 209 321 Google Scholar
[24]	Calbet X, López-Ruiz R 2001 Phys. Rev. E 63 066116 Google Scholar
[25]	Angulo J C, Antolín J 2008 J. Chem. Phys. 128 164109 Google Scholar
[26]	Majumdar S, Mukherjee N, Roy A K 2017 Chem. Phys. Lett. 687 322 Google Scholar
[27]	梁双, 吕燕伍 2007 56 1617 Google Scholar Liang S, Lv Y W 2007 Acta Phys. Sin. 56 1617 Google Scholar
[28]	Mata M, Zhou X, Furtmayr F, et al. 2013 J. Mater. Chem. C 1 4300 Google Scholar
[29]	Guo Y, Pan F, Ren Y J, et al. 2018 Phys. Chem. Chem. Phys. 20 24239 Google Scholar
[30]	Li P, David A, Li H, et al. 2021 Appl. Phys. Lett. 119 231101 Google Scholar
[31]	Binks D J, Dawson P, Oliver R A, Wallis D J 2022 Appl. Phys. Rev. 9 041309 Google Scholar
[32]	Chang C, Li X C 2022 Eur. Phys. J. D. 76 134 Google Scholar
[33]	González-Férez R, Dehesa J S, Patil S H, Sen K D 2009 Physica A 388 4919 Google Scholar

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