Study of quantum speed limit of  of transport process of single quantum dot system in dissipative environment

Liu Tian; Li Zong-Liang; Zhang Yan-Hui; Lan Kang

doi:10.7498/aps.72.20222159

Abstract

We theoretically study the quantum speed limit (QSL) of the single dot system in dissipative environment based on quantum dot transport theory and Bures angle metric method. The theoretical results show that in the dissipative environment, different tunneling probabilities have different effects. The increase of left tunneling probability has a weak effect on the accelerating capability of the system, due to the Coulomb blocking effect and quantum coherence. On the other hand, the right tunneling probability has a significant influence on the accelerating capability of the system, the accelerating capability is promoted with the increase of right tunneling probability because of the effect of channel blocking and co-tunneling. The increase of energy displacement promotes the accelerating capability of the system and changes the oscillation frequency of the system, owing to its taking longer time for the system to evolve to a target state. The effect of the relaxation rate for the system's accelerating capability is not monotonic, there is an interesting turning point due to the change of electron layout number. When the relaxation rate is less than this point, the accelerating capability of the system will oscillate. When the relaxation rate is higher than this point, the change of accelerating capability is monotonically suppressed by the relaxation rate. In general, the increase of the relaxation rate weakens the acceleration capability of the system. Our results provide theoretical reference for studying the QSL time in a semiconductor device affected by numerous factors.

Keywords:

Authors and contacts

1.
School of Physics and Electronics, Shandong Normal University, Jinan 250014, China
2.
School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China

Corresponding author: Zhang Yan-Hui, yhzhang@sdnu.edu.cn

Funds: Project supported by the Natural Science Foundation of Shandong Province(Grant No. 11974217).

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References

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图 1 单量子点系统输运过程示意图, $ {E_{\text{e}}} $和$ {E_g} $表示激发态与基态能级, $ {E_{\text{l}}} $和$ {E_{\text{r}}} $为左右两侧化学势, $ {\varGamma _{{\text{e}}\left( {\text{g}} \right)}} $为左侧电子隧穿进入激发态(基态)能级的隧穿概率, $ {\varGamma _{{\text{E}}\left( {\text{G}} \right)}} $为激发态(基态)电子隧穿进入右侧电子库的隧穿概率

Figure 1. The diagram of the single quantum dot system in transport process , $ {E_{\text{e}}} $and $ {E_g} $denotes two energy levels in quantum dot, $ {E_{\text{l}}} $ and$ {E_{\text{r}}} $denotes the left and right electrode potentials $ {\varGamma _{{\text{e}}\left( {\text{g}} \right)}} $ is the tunneling probability of the left electron tunneling into the excited(ground) state energy level, $ {\varGamma _{{\text{E}}\left( {\text{G}} \right)}} $ is the tunneling probability of the excited (ground) state electron tunneling into the right electron library.

DownLoad: Full-Size Img PowerPoint

图 2 初态为g态与相干态下量子速度极限时间比值${{{\tau _{{\text{QSL}}}}} /\tau }$随驱动时间变化图　(a)初态为g态时$ {{{\tau _{{\text{QSL}}}}}/ \tau } $随驱动时间的变化; (b)初态为相干态时$ {{{\tau _{{\text{QSL}}}}} \mathord{\left/ {\vphantom {{{\tau _{{\text{QSL}}}}} \tau }} \right. } \tau } $随驱动时间的变化, 左侧量子点与电子库耦合强度${\varOmega _{\text{r}}} = 1$, 电子的隧穿概率$ {\varGamma _{\text{E}}} = 0.5 $

Figure 2. The ratio of the quantum speed limit time $ {{{\tau _{{\text{QSL}}}}} / \tau } $ varies with the driving time in the initial g state and the superposition state: (a) The change of the$ {{{\tau _{{\text{QSL}}}}}/\tau } $with driving time when the initial state is coherent state; (b)the change of the $ {{{\tau _{{\text{QSL}}}}} /\tau } $with driving time when the initial state is superposition state ; the left coupling strength parameter ${\varOmega _{\text{r}}} = 1$, tunneling probability of electrons${\varGamma _{\text{E}}} = 0.5$.

DownLoad: Full-Size Img PowerPoint

图 3 单量子点系统的速度极限时间比值$ {{{\tau _{{\text{QSL}}}}} / \tau } $随左侧隧穿概率与驱动时间的变化图　(a)不同左侧激发态隧穿概影响下$ {{{\tau _{{\text{QSL}}}}} / \tau } $随驱动时间变化; (b)不同左侧基态隧穿概率影响下$ {{{\tau _{{\text{QSL}}}}} / \tau } $随驱动时间变化, 量子点与左侧电子库耦合强度${\varOmega _{\text{r}}} = 1$, 能级差$\varepsilon = 5{\varOmega _{\text{r}}}$

Figure 3. The diagram of the quantum speed limit time ratio $ {{{\tau _{{\text{QSL}}}}}/ \tau } $as a function of left tunneling probability and driving time for a single quantum dot system: (a) Variation of $ {{{\tau _{{\text{QSL}}}}}/\tau } $with driving time under the influence of left different tunneling probabilities into excited states; (b) variation of $ {{{\tau _{{\text{QSL}}}}} / \tau } $with driving time under the influence of left different tunneling probabilities into ground states, the left coupling strength parameter ${\varOmega _{\text{r}}} = 1$, the energy displacement $\varepsilon = 5{\varOmega _{\text{r}}}$.

DownLoad: Full-Size Img PowerPoint

图 4 单量子点系统的速度极限时间比值$ {{{\tau _{{\text{QSL}}}}} / \tau } $随右侧隧穿概率与驱动时间的变化图　(a)不同右侧激发态隧穿概率影响下$ {{{\tau _{{\text{QSL}}}}} / \tau } $随驱动时间的变化; (b)不同右侧基态隧穿概率影响下$ {{{\tau _{{\text{QSL}}}}} / \tau } $随驱动时间的变化. 量子点与左侧电子库耦合强度${\varOmega _{\text{r}}} = 1$, 能极差$\varepsilon = 5{\varOmega _{\text{r}}}$

Figure 4. The diagram of the quantum speed limit time ratio $ {{{\tau _{{\text{QSL}}}}} / \tau } $as a function of right tunneling probability and driving time for a single quantum dot system: (a) Variation of $ {{{\tau _{{\text{QSL}}}}} / \tau } $with driving time under the influence of right different tunneling probabilities from excited states ; (b) variation of $ {{{\tau _{{\text{QSL}}}}} / \tau } $with driving time under the influence of right different tunneling probabilities from ground states.The left coupling strength parameter ${\varOmega _{\text{r}}} = 1$, the energy displacement $\varepsilon = 5{\varOmega _{\text{r}}}$.

DownLoad: Full-Size Img PowerPoint

图 5 单量子点系统的速度极限时间比值$ {{{\tau _{{\text{QSL}}}}} / \tau } $随弛豫速率与驱动时间的变化图　(a) 不同弛豫速率影响下$ {{{\tau _{{\text{QSL}}}}} / \tau } $随驱动时间的变化; (b) $ {{{\tau _{{\text{QSL}}}}} / \tau } $随弛豫速率$ {\gamma _r} $与驱动时间的三维变化图. 量子点与左侧电子库耦合强度${\varOmega _{\text{r}}} = 1$, 能极差$ \varepsilon = 5{\varOmega _{\text{r}}} $

Figure 5. The diagram of the quantum speed limit time ratio $ {{{\tau _{{\text{QSL}}}}} / \tau } $as a function of relaxation rates and driving time for a single quantum dot system:(a) Variation of $ {{{\tau _{{\text{QSL}}}}} / \tau } $with driving time under different relaxation rates ; (b) three-dimensional diagram of $ {{{\tau _{{\text{QSL}}}}} / \tau } $ as a function of relaxation rate and driving time. The left coupling strength parameter ${\varOmega _{\text{r}}} = 1$, the energy displacement$\varepsilon = 5{\varOmega _{\text{r}}}$.

DownLoad: Full-Size Img PowerPoint

图 6 单量子点系统的速度极限时间比值$ {{{\tau _{{\text{QSL}}}}} / \tau } $随能级差$ \varepsilon $与驱动时间的变化图　(a)不同能级差$ \varepsilon $影响下的$ {{{\tau _{{\text{QSL}}}}} / \tau } $随驱动时间变化; (b)能级差产生跃迁原理图. 量子点与左侧电子库耦合强度${\varOmega _{\text{r}}} = 1$, $ {\varGamma _{\text{E}}} = 0.5 $

Figure 6. The diagram of the quantum speed limit time ratio $ {{{\tau _{{\text{QSL}}}}} / \tau } $ as a function of energy displacement $ \varepsilon $ and driving time for a single quantum dot system: (a) Variation of $ {{{\tau _{{\text{QSL}}}}} / \tau } $ with driving time under different energy displacement $ \varepsilon $; (b) the schematic diagram of transition generation by difference energy level. The left coupling strength parameter ${\varOmega _{\text{r}}} = 1$, Tunneling probability of electrons $ {\varGamma _{\text{E}}} = 0.5 $.

DownLoad: Full-Size Img PowerPoint

[1]	Poggi P M 2019 Phys. Rev. A 99 042116 Google Scholar
[2]	Marvian I, Spekkens R, Zanardi P 2016 Phys. Rev. A 93 052331 Google Scholar
[3]	Pires D P, Modi K, Celeri L C 2021 Phys. Rev. E 103 032105 Google Scholar
[4]	Negrneac V, Ali H, Muthusubramanian N, Battistel F, Sagastizabal R, Moreira M S, Marques J F, Vlothuizen W J, Beekman M, Zachariadis C, Haider N, Bruno A, DiCarlo L 2021 Phys. Rev. Lett. 126 220502 Google Scholar
[5]	Villamizar D V, Duzzioni E I 2015 Phys. Rev. A 92 042106 Google Scholar
[6]	Farhi E, Goldstone J, Gutmann S, Sipser M 1998 Phys. Rev. Lett. 81 5442 Google Scholar
[7]	Russell B, Stepney S 2014 Phys. Rev. A 90 012303 Google Scholar
[8]	Becker S, Datta N, Lami L, Rouze C 2021 Phys. Rev. Lett. 126 190504 Google Scholar
[9]	Xu Z Y, Luo S L, Yang W L, Liu C, Zhu S Q 2014 Phys. Rev. A 89 012307 Google Scholar
[10]	Liu H B, Yang W L, An J H, Xu Z Y 2016 Phys. Rev. A 93 020105 Google Scholar
[11]	Rossatto D Z, Pires D P, de Paula F M, de Saneto O P 2020 Phys. Rev. A 102 053716 Google Scholar
[12]	Hegerfeldt G C 2013 Phys. Rev. Lett. 111 260501 Google Scholar
[13]	Kobe D H, Aguilera-Navarro V C 1994 Phys. Rev. A 50 933 Google Scholar
[14]	Mandelstam L, Tamm I 1945 J. Phys. USSR 9 249
[15]	Margolus N, Levitin L B 1998 Phys. D 120 188 Google Scholar
[16]	Deffner S, Lutz E 2013 Phys. Rev. Lett. 111 010402 Google Scholar
[17]	Bagheri Harouni M 2021 Chin. Phys. B 30 090301 Google Scholar
[18]	冯海冉, 李鹏, 岳现房 2019 物理学报 68 050201 Google Scholar Feng H R, Li P, Yue X F 2019 Acta Phys. Sin. 68 050201 Google Scholar
[19]	Du K Y, Ma Y J, Wu S X, Yu C S 2021 Chin. Phys. B 30 090308 Google Scholar
[20]	Huang J H, Qin L G, Chen G L, Hu L Y, Liu F 2022 Chin. Phys. B 31 110307 Google Scholar
[21]	孙舒宁, 郑雨军 2021 科学通报 66 1946 Google Scholar Sun S N, Zheng Y J 2021 Chin. Sci. Bull. 66 1946 Google Scholar
[22]	Cai X J, Zheng Y J 2016 Phys. Rev. A 94 042110
[23]	Sun S N, Zheng Y J 2019 Phys. Rev. Lett. 123 180403 Google Scholar
[24]	Sun S, Peng Y, Hu X, Zheng Y 2021 Phys. Rev. Lett. 127 100404 Google Scholar
[25]	Gurvitz S A 1997 Phys. Rev. B 56 15215 Google Scholar
[26]	Gurvitz S A, Fedichkin L, Mozyrsky D and Berman G P 2003 Phys. Rev. Lett. 91 066801 Google Scholar
[27]	Ouyang S H, Lam C H, You J Q 2010 Phys. Rev. B 81 075301 Google Scholar
[28]	Lan K, Du Q, Kang L S, Tang X, Jiang L J, Zhang Y H, Cai X J 2020 Phys. Rev. B 101 174302 Google Scholar
[29]	Jiang L J, Lan K, Lin Z Y, Zhang Y H 2021 Chin. Phys. B 30 040307 Google Scholar
[30]	Kang L S, Zhang Y H, Xu X L, Tang X 2017 Phys. Rev. B 96 235417 Google Scholar
[31]	Du Q, Lan K, Zhang Y H, Jiang L J 2020 Chin. Phys. B 29 030302 Google Scholar
[32]	Lin Z Y, Zhang Y H, L T, Li Z L 2022 Chin. Phys. B 31 070307 Google Scholar
[33]	Wu S X, Yu C S 2018 Phys. Rev. A 98 042132 Google Scholar
[34]	Hu X H, Sun S N, Zheng Y J 2020 Phys. Rev. A 101 042107 Google Scholar
[35]	蓝康, 杜倩, 康丽莎, 姜露静, 林振宇, 张延惠 2020 物理学报 69 040504 Google Scholar Lan K, Du Q, Kang L S, Jiang L J, Lin Z Y, Zhang Y H 2020 Acta Phys. Sin. 69 040504 Google Scholar
[36]	栗军, 刘玉, 平婧, 叶银, 李新奇 2012 物理学报 61 137202 Google Scholar Li J, Liu Y, Ping J, Ye Y, Li X Q 2012 Acta Phys. Sin. 61 137202 Google Scholar
[37]	Ektesabi A, Behzadi N and Faizi E 2017 Phys. Rev. A 95 022115 Google Scholar
[38]	Deffner S 2017 New J. Phys. 19 103018 Google Scholar
[39]	Deffner S and Campbell S 2017 J. Phys. A Math. Theor. 50 453001 Google Scholar
[40]	Cai X J and Zheng Y J 2017 Phys. Rev. A 95 052104 Google Scholar
[41]	Lu X, Zhang Y J and Xia Y J 2021 Chin. Phys. B 30 020301 Google Scholar
[42]	Lan K, Xie S J and Cai X J 2022 New J. Phys. 24 055003 Google Scholar

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