Charge qubits based on ultra-thin topological insulator films

doi:10.1007/s11467-023-1364-5

Front. Phys.

2024, Vol. 19

Issue (3) : 33208 https://doi.org/10.1007/s11467-023-1364-5

RESEARCH ARTICLE

Charge qubits based on ultra-thin topological insulator films

Kexin Zhang^1,²(

), Hugo V. Lepage², Ying Dong¹, Crispin H. W. Barnes²(

)

¹. Research Center for Quantum Sensing, Intelligent Perception Research Institute, Zhejiang Lab, Hangzhou 311121, China
². Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, United Kingdom

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Abstract

We study how to use the surface states in a Bi₂Se₃ topological insulator ultra-thin film that are affected by finite size effects for the purpose of quantum computing. We demonstrate that: (i) surface states under the finite size effect can effectively form a two-level system where their energy levels lie in between the bulk energy gap and a logic qubit can be constructed, (ii) the qubit can be initialized and manipulated using electric pulses of simple forms, (iii) two-qubit entanglement is achieved through a $SWAP$ operation when the two qubits are in a parallel setup, and (iv) alternatively, a Floquet state can be exploited to construct a qubit and two Floquet qubits can be entangled through a Controlled-NOT operation. The Floquet qubit offers robustness to background noise since there is always an oscillating electric field applied, and the single qubit operations are controlled by amplitude modulation of the oscillating field, which is convenient experimentally.

Keywords topological insulator quantum computing nanodevices

Corresponding Author(s): Kexin Zhang,Crispin H. W. Barnes

Issue Date: 20 December 2023

Cite this article:

Kexin Zhang,Hugo V. Lepage,Ying Dong, et al. Charge qubits based on ultra-thin topological insulator films[J]. Front. Phys. , 2024, 19(3): 33208.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1364-5
https://academic.hep.com.cn/fop/EN/Y2024/V19/I3/33208

Fig.1 Top: Graphical representation of five Bi

2

3

QLs. The wave densities of the hybridized surface states and the combined qubit state

| 1 ?

in a 5-QL TI slab (finite in the

z

direction). The red dashed curve is anti-bonding state

Φ U

located at the upper energy level of the gapped Dirac Cone, while the blue dotted curve is the bonding state

Φ L

located at the lower energy level of the gapped Dirac cone. The black solid curve is the combined qubit state.

Fig.2 The Bloch sphere representation of basic rotation. (a) The initialization from

Φ 0

| 0 ?

and

| 1 ?

. The initialization to

| 0 ?

is achieved with a static electric field applied in the

− z

direction and the initialization to

| 1 ?

is achieved with a static electric field applied in the

+ z

direction. (b) The relation between the field amplitude and the axes of rotations. The axis of rotation tilts from the

x

axis as the electric field increases; it always located in the

x z

plane.

Fig.3 The Bloch sphere representation of

R x

R y

and

R z

rotations of angle 95° from

O

A

and the corresponding pulses. (a) The path of the

R x

rotation, (b) the path of the

R y

rotation, and (c) the path of the

R z

rotation on the Bloch sphere. (d) The pulses used to generate each rotation. The pulse times are calculated in picoseconds. The

R z

rotation is longer to achieve than the

R x

and

R y

rotations in the case of 95°.

Fig.4 (a) The electron density of

| 1 (t) ?

(left) and

| 0 (t) ?

(right) of a Floquet qubit vs. time. (b) The trajectory of a

R z

rotation from

O z

A

in the Floquet frame. (c) The pulses used to control the single qubit: initialization (black dash-dotted line),

R x

(blue dashed line), and

R z

(red solid line). (d) The trajectory of a

R x

rotation from

O x

B

in the static frame.

Fig.5 Setup of the two TI thin films supporting qubits 1 and 2 aligned in parallel with a separation

l x

, which is finite in the

z

direction.

Fig.6 Bands of the Hamiltonian Eq. (28) along the line

? 1 = ? 2

with various

l x

. Band 1 is a black solid line, band 2 a red dashed line, band 3 a cyan dash-dotted line, and band 4 a green dotted line. (a)

l x =

2 nm,

Δ / J = 0.05

. (b)

l x =

10 nm,

Δ / J = 0.15

. (c)

l x =

100 nm,

Δ / J = 1.5

Fig.7 Time evolution of the state

| L R ?

at various

l x

. (a) Time evolution of the state

| L R ?

l x =

2 nm. A SWAP gate occurs with a period of 521 ps. (b) Time evolution of the state

| L R ?

l x =

20 nm. A SWAP gate occurs with a period of 6266 ps.

Fig.8 Bands of the Hamiltonian Eq. (40) at

l x =

0.5 nm vs.

? i

. The bands remain in the Floquet states over the range of

? i

. Band 1 is in black solid line, band 2 in red dashed line, band 3 in cyan dash-dotted line, and band 4 in green dotted line. (a) Along the line

? 1 = 0

. (b) Along the line

? 2 = 0.40 eV = ? A

. (c) Along the line

? 2 = 0.57 eV > ? A

. (d) Along the line

? 2 = 0.23 eV < ? A

Fig.9 Time evolution of the state

| R L ?

at various

? 2

. (a) Time evolution of the state

| R L ?

? 2 = ? A = 0.40 eV

. CROT operations have a period of 0.25 ps. A CNOT gate is observed at 0.12 ps. (b) Time evolution of the state

| R L ?

? 2 = 0.57 eV > ? A

. CROT operations have a period of 0.19 ps. (c) Time evolution of the state

| R L ?

? 2 = 0.23 eV < ? A

. CROT operations have a period of 0.19 ps.

1	Ando Y.. Topological insulator materials. J. Phys. Soc. Jpn., 2013, 82(10): 1 https://doi.org/10.7566/JPSJ.82.102001
2	Z. Hasan M., L. Kane C.. Colloquium: Topological insulators. Rev. Mod. Phys., 2010, 82(4): 3045 https://doi.org/10.1103/RevModPhys.82.3045
3	Z. Hasan M., E. Moore J.. Three-dimensional topological insulators. Annu. Rev. Condens. Matter Phys., 2011, 2(1): 55 https://doi.org/10.1146/annurev-conmatphys-062910-140432
4	Z. Lu H.Q. Shen S., Weak localization and weak anti-localization in topological insulators, in: Proc. SPIE, Vol. 9167 (2014)
5	Pesin D., H. MacDonald A.. Spintronics and pseudospintronics in graphene and topological insulators. Nat. Mater., 2012, 11(5): 409 https://doi.org/10.1038/nmat3305
6	He M., Sun H., H. Qing L.. Topological insulator: Spintronics and quantum computations. Front. Phys., 2019, 14(4): 43401 https://doi.org/10.1007/s11467-019-0893-4
7	Cho S., Kim D., Syers P., P. Butch N., Paglione J., S. Fuhrer M.. Topological insulator quantum dot with tunable barriers. Nano Lett., 2012, 12(1): 469 https://doi.org/10.1021/nl203851g
8	M. Herath T.Hewageegana P.Apalkov V., A quantum dot in topological insulator nanofilm, J. Phys.: Condens. Matter 26(11), 115302 (2014)
9	Kirczenow G.. Perfect and imperfect conductance quantization and transport resonances of two-dimensional topological-insulator quantum dots with normal conducting leads and contacts. Phys. Rev. B, 2018, 98(16): 165430 https://doi.org/10.1103/PhysRevB.98.165430
10	Li G., L. Zhu J., Yang N.. Magnetic quantum dot in two-dimensional topological insulators. J. Appl. Phys., 2017, 121(11): 114302 https://doi.org/10.1063/1.4978632
11	Steinberg H., B. Laloë J., Fatemi V., S. Moodera J., Jarillo-Herrero P.. Electrically tunable surface-to-bulk coherent coupling in topological insulator thin films. Phys. Rev. B, 2011, 84(23): 233101 https://doi.org/10.1103/PhysRevB.84.233101
12	J. Ferreira G., Loss D.. Magnetically defined qubits on 3D topological insulators. Phys. Rev. Lett., 2013, 111(10): 106802 https://doi.org/10.1103/PhysRevLett.111.106802
13	A. Castro-Enriquez L., F. Quezada L., Martín-Ruiz A.. Optical response of a topological-insulator–quantum-dot hybrid interacting with a probe electric field. Phys. Rev. A, 2020, 102: 013720 https://doi.org/10.1103/PhysRevA.102.013720
14	Islam S., Bhattacharyya S., Nhalil H., Banerjee M., Richardella A., Kandala A., Sen D., Samarth N., Elizabeth S., Ghosh A.. Low-temperature saturation of phase coherence length in topological insulators. Phys. Rev. B, 2019, 99(24): 245407 https://doi.org/10.1103/PhysRevB.99.245407
15	X. Xiu F., T. Zhao T.. Topological insulator nanostructures and devices. Chin. Phys. B, 2013, 22(9): 096104 https://doi.org/10.1088/1674-1056/22/9/096104
16	W. Liu C., Wang Z., L. J. Qiu R., P. A. Gao X.. Development of topological insulator and topological crystalline insulator nanostructures. Nanotechnology, 2020, 31(19): 192001 https://doi.org/10.1088/1361-6528/ab6dfc
17	Li H., Peng H., Dang W., Yu L., Liu Z., insulator nanostructures:Materials synthesis Topological. Raman spectroscopy, and transport properties. Front. Phys., 2012, 7(2): 208 https://doi.org/10.1007/s11467-011-0199-7
18	B. Hu Y.H. Zhao Y.F. Wang X., A computational investigation of topological insulator Bi2Se3 film, Front. Phys. 9(6), 760 (2014)
19	X. Liu C., J. Zhang H., Yan B., L. Qi X., Frauenheim T., Dai X., Fang Z., C. Zhang S.. Oscillatory crossover from two- dimensional to three-dimensional topological insulators. Phys. Rev. B, 2010, 81(4): 041307 https://doi.org/10.1103/PhysRevB.81.041307
20	Z. Lu H., Y. Shan W., Yao W., Niu Q., Q. Shen S.. Massive Dirac fermions and spin physics in an ultrathin film of topological insulator. Phys. Rev. B, 2010, 81(11): 115407 https://doi.org/10.1103/PhysRevB.81.115407
21	Oka T., Kitamura S.. Floquet engineering of quantum materials. Annu. Rev. Condens. Matter Phys., 2019, 10(1): 387 https://doi.org/10.1146/annurev-conmatphys-031218-013423
22	H. Kolodrubetz M., Nathan F., Gazit S., Morimoto T., E. Moore J.. Topological Floquet-Thouless energy pump. Phys. Rev. Lett., 2018, 120(15): 150601 https://doi.org/10.1103/PhysRevLett.120.150601
23	Oka T., Aoki H.. Photovoltaic Hall effect in graphene. Phys. Rev. B, 2009, 79(8): 81406 https://doi.org/10.1103/PhysRevB.79.081406
24	Bilitewski T., R. Cooper N.. Scattering theory for Floquet-Bloch states. Phys. Rev. A, 2015, 91(3): 033601 https://doi.org/10.1103/PhysRevA.91.033601
25	H. Wang Y., Steinberg H., Jarillo-Herrero P., Gedik N.. Observation of Floquet−Bloch states on the surface of a topological insulator. Science, 2013, 342(6157): 453 https://doi.org/10.1126/science.1239834
26	Boyers E., Pandey M., K. Campbell D., Polkovnikov A., Sels D., O. Sushkov A.. Floquet-engineered quantum state manipulation in a noisy qubit. Phys. Rev. A, 2019, 100(1): 012341 https://doi.org/10.1103/PhysRevA.100.012341
27	V. Lepage H., A. Lasek A., R. M. Arvidsson-Shukur D., H. W. Barnes C.. Entanglement generation via power-of-swap operations between dynamic electron-spin qubits. Phys. Rev. A, 2020, 101(2): 022329 https://doi.org/10.1103/PhysRevA.101.022329
28	X. Liu C., L. Qi X., J. Zhang H., Dai X., Fang Z., C. Zhang S.. Model Hamiltonian for topological insulators. Phys. Rev. B, 2010, 82(4): 045122 https://doi.org/10.1103/PhysRevB.82.045122
29	B. Visscher P., A fast explicit algorithm for the time-dependent Schrödinger equation, Comput. Phys. 5(6), 596 (1991)
30	R. M. Arvidsson-Shukur D., V. Lepage H., T. Owen E., Ferrus T., H. W. Barnes C.. Protocol for fermionic positive-operator-valued measures. Phys. Rev. A, 2017, 96(5): 052305 https://doi.org/10.1103/PhysRevA.96.052305
31	Lepage H., Fermionic quantum information in surface acoustic waves, PhD thesis, University of Cambridge, 2020
32	Takada S., Edlbauer H., V. Lepage H., Wang J., A. Mortemousque P., Georgiou G., H. W. Barnes C., J. B. Ford C., Yuan M., V. Santos P., Waintal X., Ludwig A., D. Wieck A., Urdampilleta M., Meunier T., Bäuerle C.. Sound-driven single-electron transfer in a circuit of coupled quantum rails. Nat. Commun., 2019, 10(1): 4557 https://doi.org/10.1038/s41467-019-12514-w
33	H. Shirley J.. Solution of the Schrödinger equation with a Hamiltonian periodic in time. Phys. Rev., 1965, 138(4B): B979 https://doi.org/10.1103/PhysRev.138.B979
34	Linder J., Yokoyama T., Sudbø A.. Anomalous finite size effects on surface states in the topological insulator Bi2Se3. Phys. Rev. B, 2009, 80(20): 205401 https://doi.org/10.1103/PhysRevB.80.205401
35	Lasek A.V. Lepage H.Zhang K.Ferrus T.H. W. Barnes C., Pulse-controlled qubit in semiconductor double quantum dots, arXiv: 2303.04823 (2023)
36	Gorman J., G. Hasko D., A. Williams D.. Charge-qubit operation of an isolated double quantum dot. Phys. Rev. Lett., 2005, 95(9): 090502 https://doi.org/10.1103/PhysRevLett.95.090502
37	Choi K.Liu H., Amplitude modulation, pp 90–100 (2016)
38	Fujisawa T., Shinkai G., Hayashi T., Ota T.. Multiple two-qubit operations for a coupled semiconductor charge qubit. Physica E, 2011, 43(3): 730 https://doi.org/10.1016/j.physe.2010.07.040
39	A. De Moura C.S. Kubrusly C., The Courant–Friedrichs–Lewy (CFL) Condition, Birkhäuser Boston, MA, 2012
40	Pandey L., Husain S., Chen X., Barwal V., Hait S., K. Gupta N., Mishra V., Kumar A., Sharma N., Kumar N., Saravanan L., Dixit D., Sanyal B., Chaudhary S.. Weak antilocalization and electron-electron interactions in topological insulator BixTey films deposited by sputtering on Si(100). Phys. Rev. Mater., 2022, 6(4): 044203 https://doi.org/10.1103/PhysRevMaterials.6.044203
41	Zhao C., Zheng Q., Zhao J.. Excited electron and spin dynamics in topological insulator: A perspective from ab initio non-adiabatic molecular dynamics. Fundamental Research, 2022, 2(4): 506 https://doi.org/10.1016/j.fmre.2022.03.006
42	Z. Lu H.Q. Shen S., Weak localization and weak anti-localization in topological insulators, in: Spintronics VII, Vol. 9167, pp 263–273, SPIE (2014)
43	Shiranzaei M., Parhizgar F., Fransson J., Cheraghchi H.. Impurity scattering on the surface of topological-insulator thin films. Phys. Rev. B, 2017, 95(23): 235429 https://doi.org/10.1103/PhysRevB.95.235429

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