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Physica E: Low-dimensional Systems and Nanostructures journal homepage: http://www.elsevier.com/locate/physe

Quantum coherence in quantum dot systems K. Berrada a, b, * a b

Imam Mohammad Ibn Saud Islamic University (IMSIU), College of Science, Department of Physics, Riyadh, Saudi Arabia The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Miramare-Trieste, Italy

A R T I C L E I N F O

A B S T R A C T

Keywords: Quantum dot system Dot total spin Spin quantum number Quantum coherence Geometric measures Relative entropy Quantum correlation

We investigate the quantum coherence in a vertical quantum dot system (QDS) with the magnetic field considering electron-electron interaction and Coulomb-blocked systems. We show how the important physical parameters can affect the variation of the quantum coherence. Moreover, we demonstrate that there is a tem perature optimal value such that the quantum coherence is maximal and that the retardation of the coherence loss may take place by adjusting the physical parameters. Finally, we display the monotonic dependence of the coherence measures and total quantum correlation on the physical parameters according to the projective measurements.

1. Introduction Considerable physical phenomena in quantum optics and informa tion have been theoretically developed and enjoy a diversity of utilizes as resources in different tasks of quantum information communication and processing (QICP). During the progress of the resource theory of quantum entanglement, diverse measures have been considered. Nevertheless, the quantum entanglement can not be considered as the lone measure of the quantum correlation since separable states can contain nonclassical correlations. Recently, the notion of coherence has attracted an intense attention since it renders as a resource in QICP tasks [1–5] and in developing fields such as nanoscale thermodynamics [6,7], quantum metrology [8–12] and quantum biology [13–15]. With regard to the essential significance of coherence, the detection and measure of coherence become an enthusiastic subject in recent years. A accurate structure to measure the coherence of quantum states has been intro duced. Baumgratz et al. [16] have demonstrated a set of conditions to measure the coherence. Various appropriate coherence measures focused on numerous physical situations have been introduced and discussed, such as the l1 norm of coherence and relative entropy coherence. On the other hand, the nonlocal correlation or convex-roof construction can be used to quantify the coherence [17,18]. According to the coherence measures, Winter et al. [19] introduced an operational theory of coherence in physical quantum systems. Currently, an substantial aim in solid materials is to understand the behavior of the coherence and correlation for such kind of physical

system. The reason behind this consideration due to both from the fact that coherence for electrons in the structure of a solid state has not yet well been demonstrated and from the recent empirical development in the quantum information theory for these quantum systems, which has led to empirical realization of coherent control of spins in diamond [20] and manipulations electron spin qubits in QDSs [21–23]. Actually, several effective active topics of research considering the comportment of these quantum systems, such as hyperfine coupling to the nuclear spins [24,25] the spin blockade [26,27], application of the singlet-triplet qubit located below the surface of a GaAs-AlGaAs heterostructure in a 2-dimensional electron gas (2DEG) [28], and the influences of executing a slanting magnetic field [29]. In the present paper, we examine the quantum coherence in an isolated vertical QDS (Fig. 1), electron-electron interaction at the mean field level, considering the influence of the important physical parameter in the QDS for various ranges of the temperatures. By modifying the values of the physical parameters, we show the relationship between the quantum coherence measures and the total quantum correlation in the QDS according to projective measurements. This manuscript is organized as follows. In Sec. 2, we give a brief review on the l1 norm of coherence and relative entropy coherence that are used for studying the quantum coherence in the QDS. In Sec. 3, we describe the physical model associated to the QDS and present the re sults with discussion. Sec. 4 is devoted to explore the monotonic dependence of the coherence and quantum correlation on the physical parameters involved in the QDS state. Finally, we summarize our work

* Imam Mohammad Ibn Saud Islamic University (IMSIU), College of Science, Department of Physics, Riyadh, Saudi Arabia. E-mail address: [email protected] https://doi.org/10.1016/j.physe.2019.113784 Received 17 July 2019; Received in revised form 20 September 2019; Accepted 11 October 2019 Available online 21 October 2019 1386-9477/© 2019 Elsevier B.V. All rights reserved.

K. Berrada

Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113784

P where N ¼ ns dyns dns is the electrons number in the QDS and N0 can be controlled with a nearby gate voltage [32]. In the following, we suppose that the QDS is tuned into a Coulomb-blockade valley with an even integer number of the electron (N0 ¼ 2). Hence, we label the two active orbital levels with n ¼ 1 and n ¼ þ 1. For ES ≪ δE, the dot total spin, S¼

1X y d σ 0 dy 0 2 0 ns ss ns

(4)

nss

is 0 for N even and 1/2 for N odd. When ES of the order of δ and δ ¼ εþ1 ε 1 > 0 can be random, the singlet-triplet transition will be occurred. Thus, the lowest energy states can be determined as a function of the total spin quantum number S and the z-projection Sz (∣S, Sz〉):

Fig. 1. (a) Schematic diagram of a vertical double QD structure [33]. (b) Schematic of the orbital occupation for the triplet and singlet state. Only in the singlet case can both electrons occupy the ground state ∣gg〉 whereas in the triplet case they have to be in an asymmetric combination of ground and pffiffiffi excited states e.g. 1= 2ð∣ge〉 ∣eg〉Þ.

dyþ1↓ dy 1↓ ∣0〉;

∣1; 1〉 ¼

in Sec. 5. 2. Quantum coherence

∣1; 1〉

¼

dyþ1↑ dy 1↑ ∣0〉;

∣0; 0〉

¼

dy 1↑ dy 1↓ ∣0〉;

∣1; 0〉

� 1 ¼ pffiffi dyþ1↑ dy 1↓ þ dyþ1↓ dy 1↑ ∣0〉; 2

where ∣0〉 presents the ground state of the dot with N0-2 electrons. The 2electron quantum dot acting actually when a 2-level system close to the singlet-triplet transition. Then, we can consider two fictitious 1/2-spins, S1 and S2, and determine the correspondence

Preferably, the coherence of a quantum state is quantified as its distance to the closest incoherent state. Usually, the properties of the coherence for a quantum state are referred to the diagonal elements of its density operator with regard to a selected reference basis. The l1 norm of coherence measures the coherence by considering the absolute value of the quantum state non-diagonal elements. Whereas, the relative en tropy coherence is defined in terms of the distance between the quantum state and its closest incoherent state. The l1 norm of coherence is defined by X Cl 1 ¼ jjρij ; (1)

∣↓1 ↓2 〉⟺∣1; 1〉 ∣↑1 ↑2 〉⟺∣1; 1〉 1 pffiffi ð∣↑1 ↓2 〉 ∣↓1 ↑2 〉Þ⟺∣0; 0〉 2 1 pffiffi ð∣↑1 ↓2 〉 þ ∣↓1 ↑2 〉Þ⟺∣1; 0〉: 2

i6¼j

Evidently, with these spins the model of the isolated QDS given by Eq. (3) can be converted into the low-energy Hamiltonian [32].

which can be considered as a formal distance measure (geometric measure). Here i and j are the row and column index, respectively, and ij denotes the density matrix elements. On the other hand, the relative entropy has been determined as a useful measure of the coherence for a given basis: � � � (2) Cr ¼ S ρ�ρdiag ¼ S ρdiag SðρÞ

H¼

k0 S1 ⋅S2 4

γB0 Sz :

(5)

Here, the Zeeman effect is incorporated via the applied magnetic field ! B ¼ B0 b z and the parameter γ presents the dot gyromagnetic ratio. The exchange coupling is represented by the first term in Eq. (5) with k0 ¼ δ 2ES is the difference of energy between the single and triplet states. In the limit k0 > 0 and B0 ¼ 0, the ground state is a singlet with energy E0,0 and the triplet states become three-fold degenerate. With positive magnitudes B0 > 0, the magnetic field inserts a splitting of the triplet states. Therefore, at the transition point B0 ¼ k0/4γ, the energy levels E0,0 and E1,1 become degenerate. The eigenvectors and eigenvalues of the Hamiltonian in Eq. (5) are given by � � k0 H∣ψ 1 〉 ¼ E1 ∣ψ 1 〉 ¼ þ γB0 ∣↓1 ↓2 〉 16 � � k0 H∣ψ 2 〉 ¼ E2 ∣ψ 2 〉 ¼ γB0 ∣↑1 ↑2 〉 16

where SðρÞ ¼ Trðρlog2 ρÞ defines the von Neumann entropy and ρdiag denotes the incoherent state getting from the density matrix ρ by elim inating all the off-diagonal entries. Cl1 and Cr are known to follow strong monotonicity for all quantum states. It was proven that Cl1 presents the upper bound for the quantity Cr for qubit states pure states [30]. 3. Quantum dot system model We introduce a QDS for which the charging energy EC to be the largest energy scale, that is empirically convenient situation. For a lateral quantum dot (i.e., dot with completely broken spatial symme tries), the energy levels of the single-particle εn are distributed with respect to the Wigner-Dyson ensemble and the random matrix theory satisfies. On the average, the charging energy EC is much larger than the mean level spacing δE. Moreover, when the dot electron number is considered even, we assume intradot exchange interactions that support ferromagnetic configurations, which is similar to the case of the Hund’s rule. The interaction strength, ES > 0, verifies the inequality ES ≪ δE ≪ EC. Consequently, the Hamiltonian of the quantum-dot uni versal reads [31]. X Hdot ¼ εn dþns dns ES S2tot Ez Sz þ EC ðN N0 Þ (3)

H∣ψ 3 〉 H∣ψ 4 〉

k0 1 pffiffi ð∣↑ ↓ 〉 þ ∣↓1 ↑2 〉Þ 16 2 1 2 3k0 1 pffiffi ð∣↑1 ↓2 〉 ∣↓1 ↑2 〉Þ : ¼ E4 ∣ψ 4 〉 ¼ 16 2 ¼

E3 ∣ψ 3 〉 ¼

∣ψ 3〉 and ∣ψ 4〉 are two of maximally entangled states, while ∣ψ 1〉 and ∣ψ 2〉 are factorizable states with zero value of entanglement. Next, we will consider ℏ ¼ 1 and the Boltzmann constant K ¼ 1.

ns

2

K. Berrada

Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113784

Fig. 2. (Color online) Quantum coherence of the QDS versus the parameters k0 and r for two different values of the temperature T. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

4. Coherence in quantum dot systems At thermal equilibrium, the quantum dot density operator is described as the probabilistic combination of the eigenvectors of the Hamiltonian, X ρT ¼ pi ∣ψ i 〉〈ψ i ∣; (6) i

where pi ¼ expð Ei =KTÞ=Z defines the probability distribution with Z ¼ Trðexpð H=TÞ Þ is the partition function. In the standard orthonormal basis, the density operator ρT of the dot system takes the following form 0 1 a 0 0 0 1B0 b c 0C C; (7) ρT ¼ B z @0 c b 0A 0 0 0 d where the diagonal and anti-diagonal elements are � � k0 16γB0 a ¼ exp 16T b

� � 1 ¼ exp 2

k0 16T

� � 1 exp 2

k0 16T

c ¼

d

¼

� exp

�

�

3k0 þ exp 16T

�

k0 þ 16γB0 16T

� exp

3k0 16T

Fig. 3. Coherence Cl1 for the state of QDS versus the temperature T for different values of r in the case of k0 ¼ 10. r ¼ 1 is for dashed blue line, r ¼ 2 is for dashdotted red line, r ¼ 3 is for solid green line, and r ¼ 5 is for dotted black line. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

(8)

c Cl1 ¼ 2jj z

�� (9)

(12)

The analytical formula for the relative entropy coherence is obtained as � � � � � � � � � � � � b c b c bþc bþc 2b b Cr ¼ log2 þ log2 log2 z z z z z z (13)

�� (10)

� (11)

The coherence measures of the QDS are ploted in Fig. 2 as functions of the dimensionless quantities r and k0, for two values of the temperature T ¼ 0.2 and T ¼ 1. It can be seen evident differences of the coherence for the two cases of different temperature. It is the most obvious that the

with r ¼ γB0. The l1 norm of coherence is found to be 3

K. Berrada

Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113784

examine the variation of the coherence for the QDS. The coherence Cl1 and Cr are displayed in Fig. 3–6 versus the temperature T for various values of the dimensionless quantities r and k0. Generally, we find that the rise of the temperature will damage the amount of the coherence in the QDS. On other hand, we can see that the coherence becomes larger (smaller) with the rise (drop) of k0 (r). This shows that r may restrain the amount of the coherence, nevertheless, k0 may enhance the coherence. Interestingly, the coherence is a monotonically decreasing function to the temperature T with k0 � kcritical , and the coherence curves eventually 0 approach the zero value when the temperature significantly large. When

k0 < kcritical , the coherence will firstly increase from zero value to a 0 maximum, then monotonically decrease to approach the zero value. The

result may be understood as follows. More precisely, when k0 > kcritical , 0 the ground state will be ∣ψ 4〉, that is the Bell state, so Cl1 ¼ Cr ¼ 1 in the T → 0 limit. With the increase of the parameter T, ∣ψ 4〉 will mix with the higher energy levels ∣ψ 2〉, ∣ψ 3〉, and ∣ψ 1〉, respectively, so the coherence monotonically decreases with the temperature. When the ground state

Fig. 4. Coherence Cl1 for the state of the QDS versus the temperature T for different k0 in the case of r ¼ 1. k0 ¼ 10 is for dashed blue line, k0 ¼ 5 is for dashdotted red line, k0 ¼ 4 is for solid green line, and k0 ¼ 3 is for dotted black line. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

gets ∣ψ 2〉, factorizable state, in the case of k0 < kcritical , the quantum 0 coherence reaches the zero value at T ¼ 0 and it will mix with ∣ψ 4〉, ∣ψ 3〉, and ∣ψ 1〉 and the quantum coherence will firstly increase and then

decrease. For k0 ¼ kcritical , the ground state is the generate state of ∣ψ 4〉 0 and ∣ψ 1〉 with probabilities 50% and values of coherence Cl1 ¼ Cr ¼ 0:5 in the T → 0 limit. In addition, there is a critical value of the temperature Tcritical that leads to disappear the coherence and that the relation of Tcritical and k0 is monotonic increasing. These result indicate that k0 can be utilized as a converter for Tcritical, be used to adjust the value of Tcritical, namely, change the temperature of turning on or off the coherence. So k0 can be a switch to coherence, and is tunable, e.g., by means of an external magnetic field B. With these properties, more possible applications can be expected in the future. 5. Coherence versus quantum discord Fig. 5. Coherence Cr for the state of the QDS versus the temperature T for different values of r in the case of k0 ¼ 10. r ¼ 1 is for dashed blue line, r ¼ 2 is for dash-dotted red line, r ¼ 3 is for solid green line, and r ¼ 5 is for dotted black line. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

In the present section, we shall compare the quantum correlation to the variation of the coherence Cl1 and Cr for the QDS with respect the values of the physical parameters. The quantum discord appears as a substantial quantity for new quantum technilogies and considers other types of quantum correlations than entanglement [34–41]. The quantum discord is introduced as the difference between the quantum mutual information and the classical correlation [4]: δ← AB ¼ I ðρAB Þ

maxfΠk g IðρA ∣fΠk gÞ

(14)

where I ðρAB Þ presents the total amount of correlation (quantum mutual information), which includes both classical and quantum character. When a quantum measurement is executed on subsystem B, the amount maxfΠk g IðρA ∣fΠk gÞ is the maximal classical mutual information. If ρA (ρB) is the reduced matrix operator of subsystem A (B), therefore the mutual information reads as I ðρAB Þ ¼

SðρAB Þ þ SðρA Þ þ SðρB Þ;

(15)

where S(⋅) denotes the von Neumann entropy of the quantum system. The classical correlation is defined by:

Fig. 6. Coherence Cr for the quantum dot state versus the temperature T for different k0 in the case of r ¼ 1. k0 ¼ 10 is for dashed blue line, k0 ¼ 5 is for dashdotted red line, k0 ¼ 4 is for solid green line, and k0 ¼ 3 is for dotted black line. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

C← AB

¼ ¼ SðρA Þ

maxfΠk g I ðρA ∣fΠk gÞ X minfΠk g pk SðρA ∣fΠk gÞ;

(16)

k

where, ρA∣{Πk} ¼ TrB(ΠkρABΠk)/TrAB(ΠkρABΠk) is the state of A after getting the outcome k in B and Πk is a set of positive operator valued measures that results in the outcome k with probability pk ¼ TrAB(ΠkρABΠk). Finally, the quantum discord is then given by

region of coherence becomes smaller with the rise of temperature. The region of entanglement locates at k0 bigger and r smaller. Moreover, the maximal value of the coherence for the QDS becomes smaller at T ¼ 0.2 than T ¼ 1. That is to say, the increasing the value of the temperature will restrain the value of the coherence in the QDS. According to the analytical formulae of the coherence, we are able to

δ← AB ¼ I ðρAB Þ

C← AB ;

(17)

which is always a positive quantity. The quantum discord has zero value 4

Physica E: Low-dimensional Systems and Nanostructures 116 (2020) 113784

K. Berrada

quantum coherence is maximal and that the retardation of the coherence loss may take place by adjusting the important physical quantities. It is found that the dependence of the coherence in the QDS on the magnetic field, r, and bare value, k0, shows that k0 can enhance the amount of the coherence, whereas the decrease of r results the destroy of the coher ence. Finally, we have compared the behavior of the coherence mea sures with the quantum discord in the QDS state according to the projective measurements. Acknowledgments K. Berrada pleased to express his sincere and profound gratitude for the hospitality at ICTP where part of this work is done. We acknowledge the Reviewer comments and suggestions very much, which are valuable in improving the quality of our manuscript.

Fig. 7. Quantum correlation and coherence for the state of the QDS versus temperature T for different values of r in the case of k0 ¼ 10. Dashed blue (r ¼ 1) and dash-dotted red lines (r ¼ 3) are for Cl1 , solid green (r ¼ 1) and dotted black lines (r ¼ 3) are for Cr, and solid red (r ¼ 1) and solid black lines (r ¼ 3) are for ! δ . (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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Fig. 8. Quantum correlation and coherence for the state of the QDS versus the temperature T for different k0 in the case of r ¼ 1. Dashed blue (k0 ¼ 10) and dash-dotted red lines (k0 ¼ 3) are for Cl1 , solid green (k0 ¼ 10) and dotted black lines (k0 ¼ 3) are for Cr, and solid red (k0 ¼ 10) and solid black lines (k0 ¼ 3) are ! for δ . (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

P for a quantum state of the form ρ ¼ lpl∣l〉〈l∣ � ρB, where ∣l〉 is an orthonormal basis for the subsystem A, and ρB is the density operator for the subsystem B, and pl is the probability distribution. Figs. 7 and 8 depict the dependence of the quantum coherence and quantum discord on k0 and r in terms of the temperature. From the figures, the behavior and variation of the relative entropy Cr and quantum discord are found to be similar with respect to the values of the parameters k0 and r and the quantities coincide as the temperatures become significantly large. On the other hand, we can see that there exist a slight change in the behavior of the coherence Cl1 and quantum discord with different values during the change of the temperature. These results assure that the coherence is a beneficial by extracting the information from the QDS, and execute it in various domains of QICP. 6. Conclusion In conclusion, by using geometric measures, we have investigated the quantum coherence in a vertical QDS with the magnetic field considering weak electron-electron interaction and Coulomb-blocked systems. We have showed how the important physical quantities can affect the variation of the quantum coherence. Moreover, we have demonstrated that there is a temperature optimal value for which the

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