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Temperature effect of a quantum pseudodot qubit
Chinese Journal of Physics 55 (2017) 2336–2340
Contents lists available at ScienceDirect
Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph
Temperature effect of a quantum pseudodot qubit
MARK
Sun Yong, Ding Zhao-Hua, Xiao Jing-Lin* Institute of Condensed Matter Physics, Inner Mongolia National University, Tongliao 028043, China
AR TI CLE I NF O
AB S T R A CT
Keywords: Quantum pseudodot Qubit Probability density Oscillatory frequency Temperature
Considering a variational method of the Pekar type and the quantum statistics theory, the temperature effects on the probability density and the oscillating frequency of an electron-LOphonon strongly coupling in a RbCl quantum pseudodot (QPD) are investigated. It is discovered that a qubit may be made up of a QPD two-level-system. Furthermore, when an electron locates in the superposition state of the ground and first excited states, the electron's probability density varies periodically with time and oscillates in the RbCl QPD with an independent period. The electron's probability density increases (decreases) with increasing temperature at low (high) temperature. At low (high) temperature, the oscillating frequency decreases (increases) with an increase of the temperature, the chemical potential of the two-dimensional electron gas and the zero point of the pseudoharmonic potential. The oscillating frequency is a decreasing function of the polaron radius.
1. Introduction In the last few years quantum computing and information science has rapidly developed into an information science field involving the contributions of physicists and computer engineers. The discoveries have not only solved quantum algorithm problems, but also developed quantum error correction and fault-tolerance, and have shown the ultimate feasibility of the experimental realization of quantum computing and information science in quantum systems (especially quantum dot (QD) systems). In the experiments, such as Bianucci et al. [1], an experimental method was employed to obtain the Deutsch–Jozsa algorithm of one qubit in a QD. Shi et al. [2] realized the qubit of a fast hybrid silicon double QD. Kim et al. [3] achieved quantum control and the quantum process tomography of a hybrid qubit in a semiconductor QD. In these reports, the qubit which incorporated two-levels ground and first excited state (GFES) in the QD is the fundamental component in quantum computing and information science. The QD qubit [4–8] is easier to implement than many other qubit designs, and has lots of effects from external environmental factors (temperature, noise, magnetic field, electric field, and so on). However, changing the external environment factors to adjust the qubit in the QD is crucial for achieving quantum computation. Numerous groups throughout the world currently build QD qubits and are considering temperature factors for QD qubits that are much less than those expected based on the theoretical work. For example, Xiao [9] investigated the temperature and electric field effects on the qubit of an asymmetric RbCl QD. Chen and Xiao [10] studied the temperature and magnetic field influences on the qubit of a parabolic QD. Khordad and Ghanbari [11] researched the phonon effect on the optical properties of RbCl QPD qubits. In these researches, the QD qubit is not only at different temperatures and in variant type fields but also is in different potentials. So studying the QD qubit at different temperatures and in variant potentials has been a hot topic in quantum computing and information science. Recently many researchers have studied the variant properties of QPD qubits. A work including one of the authors of this article, Sun et al. [12] has investigated the properties of a QPD qubit. However, the studies of temperature effects on a QPD qubit are very few. *
Corresponding author. E-mail address: [email protected] (J.-L. Xiao).
http://dx.doi.org/10.1016/j.cjph.2017.09.017 Received 22 June 2017; Received in revised form 11 September 2017; Accepted 25 September 2017 0577-9073/ © 2017 Published by Elsevier B.V. on behalf of The Physical Society of the Republic of China (Taiwan).
Chinese Journal of Physics 55 (2017) 2336–2340
Y. Sun et al.
In this article, the ground and first excited states’ eigenenergies and eigenfunctions of the strong electron-LO-phonon coupling in this RbCl QPD are employed. The qubit is built through a two-level in the QPD system. Considering the temperature factor, the electron's probability density is obtained in the QPD oscillating with temperature and time (a few complete periods) when electrons are in the ground and first excited states’ superposition state. The effect of the chemical potential of the two-dimensional electron gas, the zero point of the pseudoharmonic potential, the polaron radius and the temperature on the oscillating frequency are discussed. The important findings are useful for qubit building in quantum computing and information science. 2. Theoretical model 2.1. Qubit The electron moving in a RbCl crystal QPD interacts with bulk longitudinal optical (LO) phonons. The system Hamiltonian of the electron-phonon interaction can be expressed as
H=
p2 + V (r ) + 2m
∑ ℏωLO aq† aq + ∑ [Vq aq exp(iq·r) + h. c], q
(1)
q
aq†
and aq denote the creation and annihilation operators of the bulk LO phonon, q is the where m is the band mass of the electron, wave vector, p and r are the momentum and the electron position vector. The pseudoharmonic potential of the above Eq. (1) is 2
r r V (r ) = V0 ⎛ − 0 ⎞ , r⎠ ⎝ r0 ⎜
⎟
(2)
where V0 is the chemical potential of the two-dimensional electron gas and r0 is the zero point of the pseudoharmonic potential. Vq and α in Eq. (1) are 1
1
1
ℏω ℏ ⎞ 4 4πα 2 e 2 ⎞ 2mωLO 2 ⎛ 1 1 ⎛ ⎞ ,α=⎛ ⎛ ⎞ Vq = i ⎜⎛ LO ⎟⎞ ⎛ − ⎞. ɛ0 ⎠ ⎝ 2ℏωLO ⎠ ⎝ ℏ ⎠ ⎝ ɛ∞ ⎝ q ⎠ ⎝ 2mωLO ⎠ ⎝ V ⎠ ⎜
⎟
⎜
⎟
⎜
⎟
(3)
Following a variational method of the Pekar type [13, 14], the trial wavefunction of the strongly-coupled polaron, separated into two parts individually describing the electron and the phonon, is written as
ψ = φ U 0ph ,
(4)
where |0ph〉 is the phonon's vacuum state with aq 0ph = 0 , and U|0ph〉 represents the coherent state of the phonon. The state |φ〉 depends only on the electron coordinate,
⎡ ⎤ U = exp ⎢∑ aq† fq − aq f q* ⎥, ⎣ q ⎦
(5)
where fq ⎛⎜f q* ⎞⎟ is the variational function, the trial ground and first excited state wavefunctions [16] of the electron-phonon system ⎝ ⎠ take the following forms:
φ0 (λ 0) = 0 0ph ,
(6)
φ1 (λ1) = 1 0ph ,
(7)
where λ0 and λ1 are the variational parameters. By minimizing the expectation value of the Hamiltonian, we then obtain the polaron ground and first excited state energies
E0 = φ0 (λ 0) U−1HU φ0 (λ 0) ,
(8)
E1 = φ1 (λ1) U−1HU φ1 (λ1) .
(9)
The electron ground and first excited state energies in the QPD can be calculated as
E0 (λ 0) =
3ℏ2 2 3V0 λ0 + + 2V0 λ 02 r02 − 2V0 − 4m 2λ 02 r02
E1 (λ1) =
5ℏ2 2 5V0 2 3 2 λ1 + + V0 λ12 r02 − 2V0 − α ℏωLO λ1 R 0 , 4m 3 4 π 2λ12 r02
2 α ℏωLO λ 0 R 0 , π
1 (ℏ/2mωLO ) 2
(10)
(11)
is the polaron radius. It is known that one can obtain λ0 and λ1 with the variational method to get the ground where R 0 = and first excited states energies and wave-functions. Therefore, the two-level system of the single qubit is built up. The superposition state can be written as 2337
Chinese Journal of Physics 55 (2017) 2336–2340
Y. Sun et al.
ψ01 =
1 ( 0 + 1 ). 2
(12)
The time evolution of the electron quantum state is described as
ψ01 (r , t ) =
1 iE t 1 iE t ψ (r )exp ⎛− 0 ⎞ + ψ (r )exp ⎛− 1 ⎞. 2 0 2 1 ⎝ ℏ ⎠ ⎝ ℏ ⎠
(13)
The electron's probability density in the QPD has the following form:
Q (r , t ) = ψ01 (r , t ) 2 1 [ ψ (r ) 2 + ψ1 (r ) 2 + ψ0* (r ) ψ1 (r )exp(iω01 t ) + ψ0 (r ) ψ1* (r )exp(−iω01 t )], = 2 0
(14)
where ω01 = (E1 − E0)/ℏ is the transition frequency between the ground and first excited states. The oscillation frequency of the electron probability density is obtained as
f=
E1 − E0 . h
(15)
2.2. Temperature effect The polaron properties depend on the statistical average values of the different states at finite temperature. According to quantum statistics theory, the statistical average number of the bulk LO phonons is obtained as −1
ℏω N = ⎡exp ⎛ LO ⎞ − 1⎤ ⎢ ⎥ ⎝ KB T ⎠ ⎣ ⎦ ⎜
⎟
, (16)
where KB and T are the Boltzmann constant and the temperature, respectively. The mean optical phonon number in the superposition state around the electron is as follows:
N=
1 3 αr0 λ 0 + αr0 λ1. 2π 4 2π
(17)
Through the self-consistently calculation of Eqs. (16) and (17) one obtains the relationship between λ0, λ1 and T. It is seen that the quantities (probability density, oscillation frequency and mean phonon number) depend on the variational parameters (λ0 and λ1) from Eqs. (13), (14) and (17), and so are connected with the temperature T. 3. Numerical simulation and discussion In the numerical simulation and discussion section, the numerical calculation for the RbCl crystal QPD has been chosen. The adopted experiment parameters are α = 3.81, m = 0.432m 0 , ωLO = 21.45 meV [15]. The numerical calculated results of the electron's probability density Q(r, t) varying with the temperature T and the time t, and oscillating frequency f versus the temperature T, the chemical potential of two-dimensional electron gas V0, the zero point of the pseudoharmonic potential r0 and the polaron
0.06
Q(r,t)
0.05 0.04 0.03 0.02 0.01 0.00 100
200
2.0 1.6
300
T/K
400
500
0.4 600
0.8
1.2 t/T0
0.0
Fig. 1. The probability density Q(x,t) versus the temperature T and the time t.
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Chinese Journal of Physics 55 (2017) 2336–2340
Y. Sun et al.
1.0
V0=1.0meV
f/(1015Hz)
V0=3.0meV
0.8
V0=5.0meV
0.6 0.4 0.2 0.0 200
300
400
500
600
700
800
900
T(K) Fig. 2. The oscillating frequency f versus the temperature T and the chemical potential of the two-dimensional electron gas V0.
radius R0 are indicated in Figs. 1–4. In Fig. 1, it is shown that the electron's probability density Q(r, t) changes with the temperature T and the time t when the electrons are in the superposition state in the RbCl crystal QPD, setting V0 = 10.0 meV , R 0 = 0.5 nm , r0 = 1.3 nm , x = 1.0 nm , y = 1.0 nm , z = 1.0 nm and cos θ = 1. It turns out that the electron oscillates with a certain oscillatory period T0 = h/(E1 − E0) in the RbCl crystal QPD. Simultaneously, with increasing the temperature, the electron's probability density increases at the low temperature region, but decreases at the high temperature region. Owing to the increasing of the temperature, the electron velocity increases, and this makes the electron's probability density located in the superposition state increase. Furthermore, the electron interacts with more phonons, thus leading to the destruction of the superposition state. The contribution from the former is greater than the latter in the low temperature region, resulting in the electron's probability density increasing with temperature. However, the contribution from the former is smaller than the latter in the high temperature region, resulting in the electron's probability density decreasing with temperature. This result is in agreement with the results of the asymmetric QD [9], the quantum rod [16] and the parabolic QD [17]. As the above suggests, we can change the electron's probability density by tuning the temperature of the QPD system. In the low (high) temperature regime, the QPD system works at a higher (lower) temperature to amplify the probability density. Fig. 2 illustrates the oscillating frequency f as a function of the temperature T and the chemical potential of a two-dimensional electron gas V0 when the electrons are in the superposition state for the RbCl crystal QPD, R 0 = 0.2 nm , r0 = 0.15 nm . Fig. 3 shows the oscillating frequency f as a function of the temperature T and the zero point of the pseudoharmonic potential r0 when the physical parameters are R 0 = 0.2 nm , V0 = 5.0 meV . From the two figures, we discover that the oscillating frequency is decreasing (increasing) with temperature at low (high) temperature. By virtue of the temperature increasing, the speed of the electron and the phonon increases, and finally, the electron interacts with more phonons. When the temperature is lower, the contribution from the electron interacting with more phonons to destroy the superposition state is not as strong as that from the increment of the electron speed. Therefore, the electron's lifetime in the superposition state is prolonged and the oscillating frequency decreases with temperature. When the temperature is higher, the two different contributions are reversed. And the oscillating frequency increases with temperature. We also find that the oscillating frequency is a decreasing function of the chemical potential of the two-dimensional electron
1.0
r0=0.15nm r0=0.18nm
f/(1015Hz)
r0=0.20nm
0.8 0.6 0.4 0.2 200
300
400
500
600
700
800
900
T(K) Fig. 3. The oscillating frequency f versus the temperature T and the zero point of the pseudoharmonic potential r0.
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Chinese Journal of Physics 55 (2017) 2336–2340
Y. Sun et al.
1.8
f/(1015Hz)
1.6
R0=0.20nm
1.4
R0=0.22nm R0=0.24nm
1.2 1.0 0.8 0.6 0.4 0.2 200
300
400
500
600
700
800
900
T(K) Fig. 4. The oscillating frequency f versus the temperature T and the polaron radius R0.
gas and the zero point of the pseudoharmonic potential at lower temperatures and an increasing function of them at higher temperatures. Fig. 4 shows the oscillating frequency f as a function of the temperature T and the polaron radius R0 when the electrons are in the superposition state for the RbCl crystal QPD r0 = 0.15 nm , V0 = 5.0 meV . From Fig. 4, we can see the oscillating frequency is decreasing (increasing) with the temperature at low (high) temperatures. The result is in agreement with the result of Figs. 2 and 3. We also discover that the oscillating frequency is a decreasing function of the polaron radius. Consequently, changing the physical quantities (such as the temperature, the chemical potential of a two-dimensional electron gas, the zero point of the pseudoharmonic potential and the polaron radius) to adjust the oscillating frequency is of great importance. Finally, concerning prolonging the life time of the qubit. Here is a suggestion as a new method for decoherence suppression. This is to prolong the qubit life by decreasing the oscillating frequency. 4. Conclusions On the bases of a variational method of the Pekar type and quantum statistics theory, the electron's probability density versus temperature and time, and the oscillatory frequency versus the temperature, the chemical potential of a two-dimensional electron gas, the zero point of the pseudoharmonic potential and the polaron radius are studied. The numerically calculated results have indicated that: firstly, the electron's probability density versus time is periodic and oscillates with a certain period in the RbCl crystal QPD system; secondly, the oscillating frequency increases (decreases) with the chemical potential of the two-dimensional electron gas and the zero point of the pseudoharmonic potential at the low (high) temperature regime. Finally, the oscillating frequency is a decreasing function of the polaron radius. Acknowledgement This project was supported by Natural Science Foundation of Inner Mongolia Autonomous Region of China under Grant No. 2017MS (LH) 0107 and the National Science Foundation of China under Grant No. 10964005. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
P. Bianucci, A. Muller, C.K. Shih, et al., Phys. Rev. B 69 (2004) 161303. Z. Shi, C.B. Simmons, J.R. Prance, et al., Phys. Rev. Lett. 108 (2012) 140503. D. Kim, Z. Shi, C.B. Simmons, et al., Nature 511 (2014) 70–74. J.K. Sun, H.J. Li, J.L. Xiao, Superlatt. Microstruct. 46 (2009) 476–482. A.V. Kuhlmann, J. Houel, A. Ludwig, et al., Nature Phys. 9 (2013) 570–575. Y. Sun, Z.H. Ding, J.L. Xiao, J. Low Temp. Phys. 177 (2014) 151–156. S.S. Li, J.B. Xia, J.L. Liu, et al., J. Appl. Phys. 90 (2001) 6151–6155. K. Eng, T.D. Ladd, A. Smith, et al., Sci. Adv. 1 (2015) e1500214. J.L. Xiao, J. Phys. Soc. Jpn. 83 (2014) 160034004. Y.J. Chen, J.L. Xiao, J. Low Temp. Phys. 186 (2017) 241–249. R. Khordad, A. Ghanbari, Opt. Quantum Electron. 49 (2017) 76. Y. Sun, Z.H. Ding, J.L. Xiao, J. Electron. Mater. 45 (2016) 3576–3580. L.D. Landau, S.I. Pekar, Zh. Eksp. Teor. Fiz. 16 (1946) 341. S.I. Pekar, Untersuchungen über Die Elektronen-Theorie Der Kristalle, Akademie Verlag, Berlin, 1954. J.T. Devreese, Polarons in Ionic Crystals and Polar Semiconductors, North-Holland Publ. Co, Amsterdam, 1972. J.L. Xiao, Superlatt. Microstruct. 60 (2013) 248. Y.J. Chen, J.L. Xiao, Acta. Phys. Sin. 57 (2008) 6758.
2340
Contents lists available at ScienceDirect
Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph
Temperature effect of a quantum pseudodot qubit
MARK
Sun Yong, Ding Zhao-Hua, Xiao Jing-Lin* Institute of Condensed Matter Physics, Inner Mongolia National University, Tongliao 028043, China
AR TI CLE I NF O
AB S T R A CT
Keywords: Quantum pseudodot Qubit Probability density Oscillatory frequency Temperature
Considering a variational method of the Pekar type and the quantum statistics theory, the temperature effects on the probability density and the oscillating frequency of an electron-LOphonon strongly coupling in a RbCl quantum pseudodot (QPD) are investigated. It is discovered that a qubit may be made up of a QPD two-level-system. Furthermore, when an electron locates in the superposition state of the ground and first excited states, the electron's probability density varies periodically with time and oscillates in the RbCl QPD with an independent period. The electron's probability density increases (decreases) with increasing temperature at low (high) temperature. At low (high) temperature, the oscillating frequency decreases (increases) with an increase of the temperature, the chemical potential of the two-dimensional electron gas and the zero point of the pseudoharmonic potential. The oscillating frequency is a decreasing function of the polaron radius.
1. Introduction In the last few years quantum computing and information science has rapidly developed into an information science field involving the contributions of physicists and computer engineers. The discoveries have not only solved quantum algorithm problems, but also developed quantum error correction and fault-tolerance, and have shown the ultimate feasibility of the experimental realization of quantum computing and information science in quantum systems (especially quantum dot (QD) systems). In the experiments, such as Bianucci et al. [1], an experimental method was employed to obtain the Deutsch–Jozsa algorithm of one qubit in a QD. Shi et al. [2] realized the qubit of a fast hybrid silicon double QD. Kim et al. [3] achieved quantum control and the quantum process tomography of a hybrid qubit in a semiconductor QD. In these reports, the qubit which incorporated two-levels ground and first excited state (GFES) in the QD is the fundamental component in quantum computing and information science. The QD qubit [4–8] is easier to implement than many other qubit designs, and has lots of effects from external environmental factors (temperature, noise, magnetic field, electric field, and so on). However, changing the external environment factors to adjust the qubit in the QD is crucial for achieving quantum computation. Numerous groups throughout the world currently build QD qubits and are considering temperature factors for QD qubits that are much less than those expected based on the theoretical work. For example, Xiao [9] investigated the temperature and electric field effects on the qubit of an asymmetric RbCl QD. Chen and Xiao [10] studied the temperature and magnetic field influences on the qubit of a parabolic QD. Khordad and Ghanbari [11] researched the phonon effect on the optical properties of RbCl QPD qubits. In these researches, the QD qubit is not only at different temperatures and in variant type fields but also is in different potentials. So studying the QD qubit at different temperatures and in variant potentials has been a hot topic in quantum computing and information science. Recently many researchers have studied the variant properties of QPD qubits. A work including one of the authors of this article, Sun et al. [12] has investigated the properties of a QPD qubit. However, the studies of temperature effects on a QPD qubit are very few. *
Corresponding author. E-mail address: [email protected] (J.-L. Xiao).
http://dx.doi.org/10.1016/j.cjph.2017.09.017 Received 22 June 2017; Received in revised form 11 September 2017; Accepted 25 September 2017 0577-9073/ © 2017 Published by Elsevier B.V. on behalf of The Physical Society of the Republic of China (Taiwan).
Chinese Journal of Physics 55 (2017) 2336–2340
Y. Sun et al.
In this article, the ground and first excited states’ eigenenergies and eigenfunctions of the strong electron-LO-phonon coupling in this RbCl QPD are employed. The qubit is built through a two-level in the QPD system. Considering the temperature factor, the electron's probability density is obtained in the QPD oscillating with temperature and time (a few complete periods) when electrons are in the ground and first excited states’ superposition state. The effect of the chemical potential of the two-dimensional electron gas, the zero point of the pseudoharmonic potential, the polaron radius and the temperature on the oscillating frequency are discussed. The important findings are useful for qubit building in quantum computing and information science. 2. Theoretical model 2.1. Qubit The electron moving in a RbCl crystal QPD interacts with bulk longitudinal optical (LO) phonons. The system Hamiltonian of the electron-phonon interaction can be expressed as
H=
p2 + V (r ) + 2m
∑ ℏωLO aq† aq + ∑ [Vq aq exp(iq·r) + h. c], q
(1)
q
aq†
and aq denote the creation and annihilation operators of the bulk LO phonon, q is the where m is the band mass of the electron, wave vector, p and r are the momentum and the electron position vector. The pseudoharmonic potential of the above Eq. (1) is 2
r r V (r ) = V0 ⎛ − 0 ⎞ , r⎠ ⎝ r0 ⎜
⎟
(2)
where V0 is the chemical potential of the two-dimensional electron gas and r0 is the zero point of the pseudoharmonic potential. Vq and α in Eq. (1) are 1
1
1
ℏω ℏ ⎞ 4 4πα 2 e 2 ⎞ 2mωLO 2 ⎛ 1 1 ⎛ ⎞ ,α=⎛ ⎛ ⎞ Vq = i ⎜⎛ LO ⎟⎞ ⎛ − ⎞. ɛ0 ⎠ ⎝ 2ℏωLO ⎠ ⎝ ℏ ⎠ ⎝ ɛ∞ ⎝ q ⎠ ⎝ 2mωLO ⎠ ⎝ V ⎠ ⎜
⎟
⎜
⎟
⎜
⎟
(3)
Following a variational method of the Pekar type [13, 14], the trial wavefunction of the strongly-coupled polaron, separated into two parts individually describing the electron and the phonon, is written as
ψ = φ U 0ph ,
(4)
where |0ph〉 is the phonon's vacuum state with aq 0ph = 0 , and U|0ph〉 represents the coherent state of the phonon. The state |φ〉 depends only on the electron coordinate,
⎡ ⎤ U = exp ⎢∑ aq† fq − aq f q* ⎥, ⎣ q ⎦
(5)
where fq ⎛⎜f q* ⎞⎟ is the variational function, the trial ground and first excited state wavefunctions [16] of the electron-phonon system ⎝ ⎠ take the following forms:
φ0 (λ 0) = 0 0ph ,
(6)
φ1 (λ1) = 1 0ph ,
(7)
where λ0 and λ1 are the variational parameters. By minimizing the expectation value of the Hamiltonian, we then obtain the polaron ground and first excited state energies
E0 = φ0 (λ 0) U−1HU φ0 (λ 0) ,
(8)
E1 = φ1 (λ1) U−1HU φ1 (λ1) .
(9)
The electron ground and first excited state energies in the QPD can be calculated as
E0 (λ 0) =
3ℏ2 2 3V0 λ0 + + 2V0 λ 02 r02 − 2V0 − 4m 2λ 02 r02
E1 (λ1) =
5ℏ2 2 5V0 2 3 2 λ1 + + V0 λ12 r02 − 2V0 − α ℏωLO λ1 R 0 , 4m 3 4 π 2λ12 r02
2 α ℏωLO λ 0 R 0 , π
1 (ℏ/2mωLO ) 2
(10)
(11)
is the polaron radius. It is known that one can obtain λ0 and λ1 with the variational method to get the ground where R 0 = and first excited states energies and wave-functions. Therefore, the two-level system of the single qubit is built up. The superposition state can be written as 2337
Chinese Journal of Physics 55 (2017) 2336–2340
Y. Sun et al.
ψ01 =
1 ( 0 + 1 ). 2
(12)
The time evolution of the electron quantum state is described as
ψ01 (r , t ) =
1 iE t 1 iE t ψ (r )exp ⎛− 0 ⎞ + ψ (r )exp ⎛− 1 ⎞. 2 0 2 1 ⎝ ℏ ⎠ ⎝ ℏ ⎠
(13)
The electron's probability density in the QPD has the following form:
Q (r , t ) = ψ01 (r , t ) 2 1 [ ψ (r ) 2 + ψ1 (r ) 2 + ψ0* (r ) ψ1 (r )exp(iω01 t ) + ψ0 (r ) ψ1* (r )exp(−iω01 t )], = 2 0
(14)
where ω01 = (E1 − E0)/ℏ is the transition frequency between the ground and first excited states. The oscillation frequency of the electron probability density is obtained as
f=
E1 − E0 . h
(15)
2.2. Temperature effect The polaron properties depend on the statistical average values of the different states at finite temperature. According to quantum statistics theory, the statistical average number of the bulk LO phonons is obtained as −1
ℏω N = ⎡exp ⎛ LO ⎞ − 1⎤ ⎢ ⎥ ⎝ KB T ⎠ ⎣ ⎦ ⎜
⎟
, (16)
where KB and T are the Boltzmann constant and the temperature, respectively. The mean optical phonon number in the superposition state around the electron is as follows:
N=
1 3 αr0 λ 0 + αr0 λ1. 2π 4 2π
(17)
Through the self-consistently calculation of Eqs. (16) and (17) one obtains the relationship between λ0, λ1 and T. It is seen that the quantities (probability density, oscillation frequency and mean phonon number) depend on the variational parameters (λ0 and λ1) from Eqs. (13), (14) and (17), and so are connected with the temperature T. 3. Numerical simulation and discussion In the numerical simulation and discussion section, the numerical calculation for the RbCl crystal QPD has been chosen. The adopted experiment parameters are α = 3.81, m = 0.432m 0 , ωLO = 21.45 meV [15]. The numerical calculated results of the electron's probability density Q(r, t) varying with the temperature T and the time t, and oscillating frequency f versus the temperature T, the chemical potential of two-dimensional electron gas V0, the zero point of the pseudoharmonic potential r0 and the polaron
0.06
Q(r,t)
0.05 0.04 0.03 0.02 0.01 0.00 100
200
2.0 1.6
300
T/K
400
500
0.4 600
0.8
1.2 t/T0
0.0
Fig. 1. The probability density Q(x,t) versus the temperature T and the time t.
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Chinese Journal of Physics 55 (2017) 2336–2340
Y. Sun et al.
1.0
V0=1.0meV
f/(1015Hz)
V0=3.0meV
0.8
V0=5.0meV
0.6 0.4 0.2 0.0 200
300
400
500
600
700
800
900
T(K) Fig. 2. The oscillating frequency f versus the temperature T and the chemical potential of the two-dimensional electron gas V0.
radius R0 are indicated in Figs. 1–4. In Fig. 1, it is shown that the electron's probability density Q(r, t) changes with the temperature T and the time t when the electrons are in the superposition state in the RbCl crystal QPD, setting V0 = 10.0 meV , R 0 = 0.5 nm , r0 = 1.3 nm , x = 1.0 nm , y = 1.0 nm , z = 1.0 nm and cos θ = 1. It turns out that the electron oscillates with a certain oscillatory period T0 = h/(E1 − E0) in the RbCl crystal QPD. Simultaneously, with increasing the temperature, the electron's probability density increases at the low temperature region, but decreases at the high temperature region. Owing to the increasing of the temperature, the electron velocity increases, and this makes the electron's probability density located in the superposition state increase. Furthermore, the electron interacts with more phonons, thus leading to the destruction of the superposition state. The contribution from the former is greater than the latter in the low temperature region, resulting in the electron's probability density increasing with temperature. However, the contribution from the former is smaller than the latter in the high temperature region, resulting in the electron's probability density decreasing with temperature. This result is in agreement with the results of the asymmetric QD [9], the quantum rod [16] and the parabolic QD [17]. As the above suggests, we can change the electron's probability density by tuning the temperature of the QPD system. In the low (high) temperature regime, the QPD system works at a higher (lower) temperature to amplify the probability density. Fig. 2 illustrates the oscillating frequency f as a function of the temperature T and the chemical potential of a two-dimensional electron gas V0 when the electrons are in the superposition state for the RbCl crystal QPD, R 0 = 0.2 nm , r0 = 0.15 nm . Fig. 3 shows the oscillating frequency f as a function of the temperature T and the zero point of the pseudoharmonic potential r0 when the physical parameters are R 0 = 0.2 nm , V0 = 5.0 meV . From the two figures, we discover that the oscillating frequency is decreasing (increasing) with temperature at low (high) temperature. By virtue of the temperature increasing, the speed of the electron and the phonon increases, and finally, the electron interacts with more phonons. When the temperature is lower, the contribution from the electron interacting with more phonons to destroy the superposition state is not as strong as that from the increment of the electron speed. Therefore, the electron's lifetime in the superposition state is prolonged and the oscillating frequency decreases with temperature. When the temperature is higher, the two different contributions are reversed. And the oscillating frequency increases with temperature. We also find that the oscillating frequency is a decreasing function of the chemical potential of the two-dimensional electron
1.0
r0=0.15nm r0=0.18nm
f/(1015Hz)
r0=0.20nm
0.8 0.6 0.4 0.2 200
300
400
500
600
700
800
900
T(K) Fig. 3. The oscillating frequency f versus the temperature T and the zero point of the pseudoharmonic potential r0.
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1.8
f/(1015Hz)
1.6
R0=0.20nm
1.4
R0=0.22nm R0=0.24nm
1.2 1.0 0.8 0.6 0.4 0.2 200
300
400
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600
700
800
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T(K) Fig. 4. The oscillating frequency f versus the temperature T and the polaron radius R0.
gas and the zero point of the pseudoharmonic potential at lower temperatures and an increasing function of them at higher temperatures. Fig. 4 shows the oscillating frequency f as a function of the temperature T and the polaron radius R0 when the electrons are in the superposition state for the RbCl crystal QPD r0 = 0.15 nm , V0 = 5.0 meV . From Fig. 4, we can see the oscillating frequency is decreasing (increasing) with the temperature at low (high) temperatures. The result is in agreement with the result of Figs. 2 and 3. We also discover that the oscillating frequency is a decreasing function of the polaron radius. Consequently, changing the physical quantities (such as the temperature, the chemical potential of a two-dimensional electron gas, the zero point of the pseudoharmonic potential and the polaron radius) to adjust the oscillating frequency is of great importance. Finally, concerning prolonging the life time of the qubit. Here is a suggestion as a new method for decoherence suppression. This is to prolong the qubit life by decreasing the oscillating frequency. 4. Conclusions On the bases of a variational method of the Pekar type and quantum statistics theory, the electron's probability density versus temperature and time, and the oscillatory frequency versus the temperature, the chemical potential of a two-dimensional electron gas, the zero point of the pseudoharmonic potential and the polaron radius are studied. The numerically calculated results have indicated that: firstly, the electron's probability density versus time is periodic and oscillates with a certain period in the RbCl crystal QPD system; secondly, the oscillating frequency increases (decreases) with the chemical potential of the two-dimensional electron gas and the zero point of the pseudoharmonic potential at the low (high) temperature regime. Finally, the oscillating frequency is a decreasing function of the polaron radius. Acknowledgement This project was supported by Natural Science Foundation of Inner Mongolia Autonomous Region of China under Grant No. 2017MS (LH) 0107 and the National Science Foundation of China under Grant No. 10964005. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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