Thermal entanglement and quantum coherence of a single electron in a double quantum dot with Rashba interaction

At thermal equilibrium, the double quantum dot density operator $\rho$ is described as

$$\rho_{AB}(T)=\left[\begin{array}[]{cccc}\rho_{11}&\rho_{12}&\rho_{13}&\rho_{14}\\ \rho_{12}^{*}&\rho_{22}&\rho_{14}&\rho_{24}\\ \rho_{13}&\rho_{14}^{*}&\rho_{11}&-\rho_{12}\\ \rho_{14}^{*}&\rho_{24}&-\rho_{12}^{*}&\rho_{22}\end{array}\right].$$

(5)

The elements of this density matrix, after a cumbersome algebraic manipulation, are given by

$$\begin{array}[]{ccl}\rho_{11}&=&\frac{A_{+}^{2}a_{+}^{2}e^{-\beta\varepsilon_{1}}+A_{-}^{2}a_{-}^{2}e^{-\beta\varepsilon_{2}}+B_{+}^{2}b_{+}^{2}e^{-\beta\varepsilon_{3}}+B_{-}^{2}b_{-}^{2}e^{-\beta\varepsilon_{4}}}{Z},\\ \rho_{12}&=&\frac{i[-A_{+}^{2}a_{+}e^{-\beta\varepsilon_{1}}-A_{-}^{2}a_{-}e^{-\beta\varepsilon_{2}}+B_{+}^{2}b_{+}e^{-\beta\varepsilon_{3}}+B_{-}^{2}b_{-}e^{-\beta\varepsilon_{4}}]}{Z},\\ \rho_{13}&=&\frac{A_{+}^{2}a_{+}^{2}e^{-\beta\varepsilon_{1}}+A_{-}^{2}a_{-}^{2}e^{-\beta\varepsilon_{2}}-B_{+}^{2}b_{+}^{2}e^{-\beta\varepsilon_{3}}-B_{-}^{2}b_{-}^{2}e^{-\beta\varepsilon_{4}}}{Z},\\ \rho_{14}&=&\frac{i[A_{+}^{2}a_{+}e^{-\beta\varepsilon_{1}}+A_{-}^{2}a_{-}e^{-\beta\varepsilon_{2}}+B_{+}^{2}b_{+}e^{-\beta\varepsilon_{3}}+B_{-}^{2}b_{-}e^{-\beta\varepsilon_{4}}]}{Z},\\ \rho_{22}&=&\frac{A_{+}^{2}e^{-\beta\varepsilon_{1}}+A_{-}^{2}e^{-\beta\varepsilon_{2}}+B_{+}^{2}e^{-\beta\varepsilon_{3}}+B_{-}^{2}e^{-\beta\varepsilon_{4}}}{Z},\\ \rho_{24}&=&\frac{-A_{+}^{2}e^{-\beta\varepsilon_{1}}-A_{-}^{2}e^{-\beta\varepsilon_{2}}+B_{+}^{2}e^{-\beta\varepsilon_{3}}+B_{-}^{2}e^{-\beta\varepsilon_{4}}}{Z},\end{array}$$

where $Z=\underset{i}{\sum}e^{-\beta\varepsilon_{i}}$.

Since $\rho_{AB}(T)$ represents a thermal state in equilibrium, the corresponding entanglement is then called thermal entanglement. In this paper, we consider a single electron spin in a double quantum dot with Rashba interaction. We found that, the charge qubit controlled by the interdot tunneling and the spin qubit driven by the Rashba interaction are responsible for the thermal entanglement of the model.

In order to quantify the amount of entanglement associated with a given two-qubit state $\rho$, we consider concurrence $\mathcal{C}$ defined by Wootters wootters ; woo

$\displaystyle\mathcal{C}={\rm{max}\left\{0,2max\left(\sqrt{\lambda_{i}}\right)-\sum_{i}\sqrt{\lambda_{i}}\right\},}$

(6)

here $\lambda_{i}\>(i=1,2,3,4)$ are the eigenvalues in descending order of the matrix

$\displaystyle R=\rho\left(\sigma^{y}\otimes\sigma^{y}\right)\rho^{\ast}\left(\sigma^{y}\otimes\sigma^{y}\right),$

(7)

with $\sigma^{y}$ being the Pauli matrix. After straightforward calculations, the eigenvalues of the matrix $R$ can be expressed as

	$\displaystyle\lambda_{1}$	$\displaystyle=$	$\displaystyle\Theta+G+\sqrt{\Xi_{+}\Sigma_{+}},$
	$\displaystyle\lambda_{2}$	$\displaystyle=$	$\displaystyle\Theta+G-\sqrt{\Xi_{+}\Sigma_{+}},$
	$\displaystyle\lambda_{3}$	$\displaystyle=$	$\displaystyle\Theta-G+\sqrt{\Xi_{-}\Sigma_{-}},$
	$\displaystyle\lambda_{4}$	$\displaystyle=$	$\displaystyle\Theta-G-\sqrt{\Xi_{-}\Sigma_{-}},$		(8)

where

$$\begin{array}[]{ccl}G&=&-2\rho_{14}\rho_{12}+\rho_{11}\rho_{24}-\rho_{13}\rho_{22},\\ \Theta&=&\rho_{11}\rho_{22}-\rho_{13}\rho_{24}+|\rho_{14}|^{2}+|\rho_{12}|^{2},\\ \Xi_{\pm}&=&2\left(\rho_{12}\pm\rho_{14}\right)\left(\rho_{22}\pm\rho_{24}\right),\\ \Sigma_{\pm}&=&2\left(\rho_{13}\mp\rho_{11}\right)\left(\rho_{14}\pm\rho_{12}\right).\end{array}$$

Thus, the concurrence of this system can be written as cao

$\displaystyle\mathcal{C}={\rm{max}\left\{0,\mid\sqrt{\lambda_{1}}-\sqrt{\lambda_{3}}\mid-\sqrt{\lambda_{2}}-\sqrt{\lambda_{4}}\right\},}$

(9)

In this case, the analytical expression for the thermal concurrence is too large to be explicitly provided here, but it easy to recover following the above steps.

Quantum coherence is an important feature in quantum physics and is of practical significance in quantum information processing task. Quantum coherence in a bipartite system can be contained both locally and in the correlations among the subsystems. The difference between the amount of coherence contained in the global state and the coherences that are purely local, is called correlated coherence, $\mathcal{C}_{cc}$ tan . For a bipartite quantum system, it becomes

$$\mathcal{C}_{cc}(\rho_{AB})=\mathcal{C}_{l_{1}}(\rho_{AB})-\mathcal{C}_{l_{1}}(\rho_{A})-\mathcal{C}_{l_{1}}(\rho_{B}),$$

(10)

where $\rho_{A}=Tr_{B}(\rho_{AB})$ and $\rho_{B}=Tr_{A}(\rho_{AB})$. Here, $A$ and $B$ stand for local subsystems.

In accordance with the set of properties that any appropriate measure of coherence should satisfy baum , a number of coherence measures have been put forward. Here we are concerned with the $l_{1}$-norm, it is a bona fide measure of coherence. The definition of the $l_{1}$-norm of coherence $\mathcal{C}_{l_{1}}$ is

$$\mathcal{C}_{l_{1}}(\rho)=\sum_{i\neq j}|\langle i|\rho|j\rangle|.$$

(11)

Quantum coherence is a basis-dependent concept, but we can choose an incoherent one for the local coherence, which will allow us to diagonalize $\rho_{A}$ and $\rho_{B}$. From Eq.(5), the reduced density matrix $\rho_{A}(T)$ will be given by

$$\rho_{A}(T)=\left(\begin{array}[]{cc}\rho_{11}+\rho_{22}&\rho_{13}+\rho_{24}\\ \rho_{13}+\rho_{24}&\rho_{11}+\rho_{22}\end{array}\right).$$

(12)

In a similar way, we obtain

$$\rho_{B}(T)=\left(\begin{array}[]{cc}2\rho_{11}&0\\ 0&2\rho_{22}\end{array}\right).$$

(13)

In order to analyze the correlated coherence, we perform a unitary transformation in the reduced density matrix $\rho_{A}(T)$. Thus, the unitary matrix results in

$$U=\left(\begin{array}[]{cc}\cos\theta&-e^{i\varphi}\sin\theta\\ e^{-i\varphi}\sin\theta&\cos\theta\end{array}\right).$$

(14)

So, let us have $\widetilde{\rho}_{A}(T)=U\,\rho_{A}(T)\,U^{\dagger}$. For $\rho_{B}(T)$ it is not necessary to perform any transformation, the operator $\rho_{B}(T)$ is already incoherent. On the other hand, the unitary transformation of the bipartite quantum state $\rho_{AB}(T)$ is given by $\widetilde{\rho}_{AB}(T)=\widetilde{U}\,\rho_{AB}(T)\,\widetilde{U}^{\dagger}$, where $\widetilde{U}=U\otimes\mathbb{I}$.

The unitary transformation will show the relationship between the global coherence and the local coherence for several choices of $\theta$ and $\varphi$ parameters. In particular, by setting $(\theta=\frac{\pi}{4},\varphi=0)$ in the Eq.(14), we obtain a matrix that diagonalize $\rho_{A}(T)$. This step provide us the basis set, where $A$ is locally incoherent. Thus, by inserting Eq.(14) into the Eq.(10), fixing $\theta=\frac{\pi}{4}$ and $\varphi=0$, we obtain an explicit expression for correlated coherence, that is,

$$\mathcal{C}_{cc}(\rho_{AB}(T))=|\rho_{14}+\rho_{12}|+|\rho_{14}+\rho_{12}^{*}|+|\rho_{12}-\rho_{14}|+|\rho_{12}^{*}-\rho_{14}|.$$

(15)

The mixed-state fidelity can be defined as jo ; zhou

$$F(\rho_{1},\rho_{2})=Tr\sqrt{\rho_{2}^{1/2}\rho_{1}\rho_{2}^{1/2}}.$$

(16)

This quantity measures the degree of distinguishability between the two quantum states $\rho_{1}$ and $\rho_{2}$. Conversely, the quantum fidelity between the input pure state and the output mixed state is defined by

$$F=\langle\psi|\rho\left|\psi\right>,$$

(17)

where $\left|\psi\right>$ is the pure state and $\rho$ is the density operator state. This measurement provides the information of the overlap between the pure state $|\psi\rangle$ and the mixed state $\rho$. In the our case, we will study the thermal fidelity between the ground state $\left|\varphi_{2}\right>$ and the state of the system at temperature $T$. After some algebra, one finds

$$F(T)=\frac{\left[a_{+}^{2}(\rho_{11}+\rho_{13})+(\rho_{22}-\rho_{24})+2ia_{+}(\rho_{12}-\rho_{14}\right]}{\left[a_{+}^{2}+1\right]}.$$

(18)

Although this work is theoretical, a possible implementation of the device of a single electron in a double quantum dot with Rashba interaction is to consider the introduction of micromagnets in the device for spin-orbit interaction (SOI), see bor ; holl .

Firstly, we investigate the influence of the tunneling coefficient $t$ and Rashba coupling $\alpha$ on the energy levels in zero temperature. The energy levels versus Zeeman splitting $\Delta$ is plotted in Fig. 2. Initially, we show in the same graph the two energies, each two-fold degenerate, for $t=0$ and $\alpha=0$ as indicated by dashed lines, red ($\varepsilon_{1}=\varepsilon_{3}$) and blue ($\varepsilon_{2}=\varepsilon_{4}$), respectively. On the other hand, for the solid curves, the tunneling between quantum dots ($t=2$) breaks the degeneracy at $\Delta=0$. Meanwhile, the Rashba coupling ($\alpha=0.1$) induces two anti-crossing points. In $\Delta=4$ for energy levels $\varepsilon_{3}$, and $\varepsilon_{4}$, and in $\Delta=-4$ for energy levels $\varepsilon_{1}$ and $\varepsilon_{2}$ From the above analysis, it is easy see that there is a strong correlation between interdot tunneling rates and degeneracy breaking of the eigenstates. As well as, one clear signature of the spin-orbit interaction is the formation of anti-crossing points in the electron energy spectrum.

In Fig. 3 we plot the concurrence $\mathcal{C}$ versus Rashba coupling $\alpha$, at zero temperature for fixed $t=0.1$ (solid curves), and $t=2$ (dashed curves), assuming several values of the $\Delta$. For tunneling parameter $t=0.1$, we observe a vigorous increase of the concurrence until reaching $\mathcal{C}\approx 0.9993$ for weak Zeeman splitting $\Delta=0.5$ and

Rashba coupling $\alpha=10$, in this case a single non-zero eigenvector that contributes to the entanglement is $\left|\varphi_{2}\right>\approx-0.491i\left(\left|L0\right>+\left|R0\right>\right)+0.508\left(-\left|L1\right>+\left|R1\right>\right)$, whereas when we consider $\alpha\rightarrow\infty$, the ground state reduces to $\left|\varphi_{2}\right>=-0.5i\left(\left|L0\right>+\left|R0\right>\right)+0.5\left(-\left|L1\right>+\left|R1\right>\right)$ and achieving maximum concurrence $(\mathcal{C}=1)$. Moreover, the curves show that the entanglement between the spin-charge qubits is smaller as the Zeeman splitting increases. From the same figure, we can see that as soon as the tunneling parameter increase say $t=2$, the concurrence is weaker than for weak tunneling regime (see dashed curves). Furthermore, still in same figure, it is observed that the concurrence is null at $\alpha=0$ for each parameter $t$ and $\Delta$ considered. Here the unentangled ground state is given by $\left|\varphi_{2}\right>=\frac{1}{\sqrt{2}}\left(-\left|L1\right>+\left|R1\right>\right)$.

Firstly, we study how the concurrence $\mathcal{C}$ is affected by temperature $T$. In Fig. 4 we depict the concurrence $\mathcal{C}$ as a function of the temperature $T$ in the logarithmic scale and for different values of the Rashba coupling $\alpha$, with $\Delta=2$ and $t=1$. It is clear to see that there are two different regimes: the first one corresponds to a strong Rashba coupling $\alpha=10$ (blue curve), where we can see the concurrence for $T=0$ becomes $\mathcal{C}\approx 0.98$. It is also observed that the concurrence monotonously leads to zero at the threshold temperature $T_{th}\approx 4.558$. For $\alpha=2$ (green curve), the concurrence $(\mathcal{C}\approx\frac{1}{\sqrt{2}})$ is smaller than to the previous case at low temperature. However, it decreases quickly as temperature raise and finally vanishes at threshold temperature $T_{th}\approx 1.728$. The second one corresponds to weak Rashba coupling strength, e.g., $\alpha=1$ (red curve), where we obtain a weak entanglement at zero temperature $\mathcal{C}\approx 0.447$, which remains almost constant at low temperature. Then, the concurrence monotonically decreases with increasing temperature until it completely vanishes at the threshold temperature $T_{th}\approx 1.224$. This result shows that $\alpha$ can be used for either tuning on or off the entanglement.

In Fig. 5(a), we illustrate the density plot of concurrence $\mathcal{C}$ as a function of $T$ and $t$, for fixed values of $\Delta=2$ and $\alpha=1$. The blue color corresponds to the entangled region, while the white color corresponds to the unentangled region. One interesting feature observed here is that the system is strongly entangled around $t=0$ and at low temperatures. There is a threshold temperature above which the entanglement becomes zero. We also observed that the concurrence gradually decreases with the increase of the tunnel effect parameter, which indicates that the tunnel effects weakens the quantum entanglement. Furthermore, a similar density plot for the concurrence is reported in Fig. 5(b) as a function of $T$ and $\Delta$ for fixed values of $t=0.5$ and $\alpha=1$. Still, in the same panel, we can notice that when the Zeeman splitting is null, the model is weakly entangled in a low temperature region. But quickly, the concurrence disappears due to the thermal fluctuations as the temperature increases. Additionally, the density plot also shows that the entanglement is strong for weak Zeeman splitting values at zero temperature, but the entanglement decreases as the Zeeman parameter increases. On the other hand, when $T$ increases, the concurrence $\mathcal{C}$ decreases rapidly until achieving the threshold temperature, above which the thermal entanglement becomes null.

In Fig. 6, the thermal effects on populations $\rho_{11}$ (red curve), $\rho_{22}$ (green curve) and concurrence (black curve), are reported for two values of the Rashba coupling. In this figure, the blue dashed line shows the threshold temperature going from the region of constant concurrence to the region where concurrence monotonously decreases as the temperature increases, this threshold temperature also describes the beginning of population change. In Fig. 6(a) for the Rashba coupling $\alpha=0.1$. We have observed that for low temperatures, the population and concurrence remain constant in a small range of temperature, in this region we find that the populations are $\rho_{11}\approx 0.003$ (red curve) and $\rho_{22}\approx 0.4996$ (green curve). These results suggest that the weakly entangled qubits are in the ground state $\left|\varphi_{2}\right>\approx-0.017i\left(\left|L0\right>+\left|R0\right>\right)-0.7068\left(-\left|L1\right>+\left|R1\right>\right)$ for low temperature regimes, so the concurrence is $\mathcal{C}\approx 0.0499$. In this figure, the blue dashed line shows the threshold temperature is $T_{th}\approx 0.1777$. Thus, we found that quantum entanglement is sensitive to population change as a consequence of increasing temperature. On the other hand, in Fig.6(b) for a strong Rashba coupling $\alpha=10$, we observe a sudden increase of $\rho_{11}$ which attains the value $\rho_{11}\approx 0.2$ (red curve), a decrease for $\rho_{22}\approx 0.3$ (green curve) and the concurrence reaches the value $\mathcal{C}\approx 0.9805$ at low temperatures, this concurrence is constant until the threshold temperature $T_{th}\approx 0.0191$, see the blue dashed line. Therefore, due to thermal fluctuations, the populations undergo a change and concurrence decreases until it disappears. In any case, with increasing temperature regardless of the value of the Rashba coupling, the population corresponding to the $\rho_{11}$ state increases, while the population $\rho_{22}$ decreases until at higher temperature the eigenstates are distributed equally, reaching the value $0.25$.

In Fig.7, we plot the fidelity $F$ between the ground state $\left|\varphi_{2}\right>$ and the thermal state $\rho_{AB}(T)$ as a function of temperature $T$ in the logarithmic scale. We can see that the mixed-state fidelity approaches ground-state fidelity i.e. $F=1$, when the temperature leads to zero. On the other hand, when the temperature increases the ground state mixes with the excited states, allowing the fidelity to decrease monotonically as the temperature increases. It is also observed that for $T=0$, the figure exhibits the change of the fidelity $F=0.5$ (red curve), since the ground states become the degenerate states $\left|\varphi_{2}\right>$ and $\left|\varphi_{4}\right>$ for fixed tunneling parameter $t=0$.

Finally, in Fig. 8, we give the plot of correlated coherence and the concurrence as a function of temperature at a fixed value of the tunneling parameter $t=1$, Rashba coupling $\alpha=10$, Zeeman parameter $\Delta=2,$ and for different values of the parameter $\theta$. Note that in these figures, we include the curves of total quantum coherence $\mathcal{C}_{l_{1}}(\rho_{AB})$ (black curve) and the local quantum coherence $\mathcal{C}_{l_{1}}(\rho_{A})+\mathcal{C}_{l_{1}}(\rho_{B})$ (black dashed curve) for a better understanding of these amounts. In Fig. 8(a), we plot the correlated coherence and the concurrence as a function of temperature $T$, in the basis of the eigenenergies which corresponds to the angle $\theta=0$ and to $\varphi=0$ in the transformation $U$(see Eq. 14). These curves show that, for $T\rightarrow 0$, the correlated coherence $\mathcal{C}_{cc}$ (solid blue curve) is higher than the thermal entanglement $\mathcal{C}$(solid red curve). The difference between them is the untangled quantum correlation (quantum discord). We can also notice the presence of a plateau in the correlated coherence in this low temperature regime, this is due to the fact that the correlated coherence of the ground state $(\left|\varphi_{2}\right>)$ is weak affected by thermal fluctuations in this regime. From this figure, it is also easy to see that, as the temperature increases, the entanglement (red curve) decays up to threshold temperature $T_{th}\approx 4.5$, while the total quantum coherence gradually decreases as the temperature increases. In Fig. 8(b), we repeat the analysis for a starting angle of $\theta=\frac{\pi}{8}$. Here, we observed a decrease in local quantum coherence that accompanies the lowering of total quantum coherence, which follows as a consequence of the reduction of correlated coherence. Interestingly, the behavior of correlated coherence, as well as total and local quantum coherence qualitatively follows the same pattern as in Fig. 8(a). In Fig. 8(c), we choose $\theta$ close to $\frac{\pi}{4}$, $\left(\theta=0.95\frac{\pi}{4}\right)$ and $\varphi=0$, for this choice of the $\theta$ parameters, we observed a dramatic decrease in correlated coherence. In addition, we can see that the local quantum coherence (dashed black curve) is almost null. Then, it can be seen that the correlated coherence almost entirely constitute the total quantum coherence (solid black curve) for this particular choice of $\theta$. On the other hand, for high temperatures and after the concurrence and the local coherence have disappeared, the total quantum coherence is composed solely of non-entangled quantum correlations.

To recover the independence of the correlated coherence basis, we choose the local natural basis of $\rho_{A}$, which is obtained by choosing $\theta=\frac{\pi}{4}$ and $\varphi=0$ (the reduced density matrix $\rho_{B}$ is already diagonal). Thus, in Fig. 8(d), the concurrence and quantum coherence are analyzed for the incoherent basis $\theta=\frac{\pi}{4}$ and $\varphi=0$. It is interesting to note that at low temperatures, the entangled quantum correlations of the system are stored entirely in the quantum coherence; this indicates that, in this case, the correlated coherence captures all the thermal entanglement information. As the temperature increases, the thermal fluctuations generate a slight increase in quantum coherence, while the entanglement decays and disappears at the threshold temperature, $T\approx 4.5$. Finally, the correlated coherence leads monotonically to zero.