1. Introduction

Since proposed originally by Feynman [1] in 1982, the quantum computer has attracted much interest due to its powerful computation ability. In theory, the quantum computation can be able to provide speedups for some certain tasks at an exponential rate, which may revolutionize the computer science in the future [2–4]. Until now, various physical systems have been presented to realize the quantum computer, such as trapped ions [5–7], nuclear magnetic resonance [8, 9], quantum dots [10, 11], superconducting circuit [12–14], etc. In 2002, DeMille [15] suggested encoding the qubits in pendular states of the ultracold polar molecules which are trapped in optical lattice, oriented along or against the external electric field and correlated by the dipole-dipole interaction. On the other hand, methods and techniques for cooling, trapping and controlling polar molecules have been developing rapidly for several decades [16–23]. Nowadays, the polar molecules can be considered as a hopeful candidate for quantum computation due to their long coherence time and strong dipole-dipole interactions [24–28].

Quantum entanglement and coherence play essential roles in quantum computation, both of which arise from the superposition principle of the quantum mechanics [29–31]. It is well accepted that the multitude of superpositions of the molecular qubits are of significance for quantum parallelism. Thus, a number of researches on the quantum correlations and coherence of the polar molecules have been carried out theoretically and experimentally in recent years [32–41]. For example, Mishima et al. [32] investigated the generation of the entanglement between molecular ro-vibrational states using chirped pulses. Wei et al. [33] examined the influences of the Starkeffect and dipole interaction on the linear dipoles. Liao [35] proved that the entanglement of the adsorbed polar molecules is important with the assistance of anticrossing between the energy levels. Moreover, See ßelberg et al. [41] achieved the enhancement of the rotational coherence time for a dilute gas of polar molecules in a spin-decoupled magic trap. Yu et al. [42] performed the optimal control simulations for the coupled field-free quantum planar rotors in rotational states, and realized the maximal orientation and entanglement. To our knowledge, the polar molecules in the static electric fields with gradient may be more suitable for quantum computing, since the pendular qubit states of the dipoles with different Stark effect can be individually addressable [15, 33]. The aim of the present paper is to design optimal laser pulses by the optimal control theory (OCT) for generating the intermolecular entanglement and coherence in the presence of external electric fields. In general, the quantum OCT is an effective approach to find the optimal solution of a physical problem under constrain conditions, which has been applied to the orientation and alignment of molecules [43, 44], the implementation of basic quantum logic gates and quantum algorithm [45–49], the discrimination of isotopologues mixture [50, 51], etc. Here, the BaI molecules [52] are taken as a demonstration, and the OCT is adopted to achieve the transition from the molecular ground states to the Bell states and maximally coherent states. Our results indicate that the entanglement and the coherence of the two coupled polar molecules, which are measured by the concurrence and the relative entropy respectively, can be markedly enhanced by utilizing the optimized laser pulses. In the meantime, the dynamics of the molecular populations are given to exhibit the generation processes of the target states in numerical simulations. Moreover, we also realize the conversion of the entangled state to the coherent state. Furthermore, by using the OCT we create a two-qubit state lying between the separable state and the maximally entangled state, called Hardy state, which can be able to demonstrate the differences of the quantum theory and the classical physics. Our results could shed some new light on quantum information processing based on pendular qubit states of molecular systems.

This paper is organized as follows. In Section II, we briefly review the model of coupled polar molecules in pendular states, the definitions of concurrence and relative entropy of coherence (REC), as well as the rapidly convergent iteration methods for OCT. In Section III, we present the numerical results of the optimal control simulations, including the optimal control fields, the evolutions of concurrence and REC, and the population transfer from the initial states to the target states. We summarize our conclusions in Section IV.

2. Theory

2.1. Polar molecules in pendular states

For a trapped ultracold polar diatomic molecule in a static electric field, the Hamiltonian is given by [33]

Here, B is the rotational constant, J is the angular momentum operator, θ is the angle between the molecular permanent dipole moment $\mathit{\mu }$ and the external electric field ϵ. The molecular dipole is induced to oscillate about the field direction in an angular range, and then the low rotational states are superposed as the pendular states $|\tilde{J},M〉$. Note that $\tilde{J}$ is not a good quantum number any more because of the Stark effect, whereas M (the projection of J on the field axis) remains good. According to DeMille’s proposal, the qubits $|0〉$ and $|1〉$ are encoded by the two lowest pendular states $|\tilde{J}=0,M=0〉$ and $|\tilde{J}=1,M=0〉$ respectively, which can be written as

For the case of two identical polar diatomic molecules at a distance of r₁₂, the dipole-dipole interaction for M = 0 can be simplified by averaging the azimuthal angle and takes the following form [33, 34]

Here, $\mathrm{\Omega }={\mu }^2/r_{12}^3$, α is the angle between r₁₂ and the field direction, and ${\theta }_i\left(i=1,2\right)$ is the angle between the dipole moment and the electric field. Therefore, the total Hamiltonian of two coupled polar molecules in external field is given by

Under the basis of the pendular qubits $\left\{\left|00〉,\right|01〉,\left|10〉,\right|11〉\right\}$, the items of the above Hamiltonian can be expanded as

Here, ${\lambda }_0^i$ and ${\lambda }_1^i$ are the eigenenergies corresponding to the pendular states $|0〉$ and $|1〉$ respectively, I is a two-dimensional identity matrix, $C_0^i$, $C_1^i$ and $C_t^i$ are molecular orientations and transition dipole moment mentioned above.

2.2. Quantifications of entanglement and coherence

To evaluate the amount of the entanglement among the pendular states of the two coupled molecules, here we employ the concurrence which is defined as [53]

Here, ρ is the density matrix of an arbitrary two-qubit state, ${\rho }^{*}$ denotes the complex conjugate of ρ, and σ_y is the Pauli operator in y direction. It is well accepted that for the maximally entangled states C = 1 and for the completely separable states C = 0, the value of C is between 0 and 1 while the bipartite system is partially entangled.

Recently, Baumgratz et al. [54] demonstrated that the coherence for a physical system can be quantified by the minimal distance between the arbitrary quantum state ρ and the incoherent state δ, which can be given in the form of relative entropy:

2.3. Methods for quantum optimal control

The basic principle of the quantum OCT is to maximize the transition probability and to simultaneously minimize the control laser energy. To this end, the objective function for optimal control can be constructed as follows [55]

Here, $\mathit{E}\left(t\right)$ is the laser field which can drive the wave function ${\psi }_i\left(t\right)$ from the initial state ${\psi }_i\left(0\right)$ to the target state ${\varphi }_f\left(T\right)$, α₀ is the penalty factor employed to control the laser intensity, $S\left(t\right)={\text{sin}}^2\left(\pi t/T\right)$ is the laser shape function with duration time T, ${\psi }_f\left(t\right)$ is the Lagrange multiplier which forms the Schrödinger equation with the wave function ${\psi }_i\left(t\right)$ under boundary conditions. Note that the first term on the right-hand side can be defined as the fidelity between the final state of laser-driven wave function and the target state, i.e., $Fidelity\equiv \left|〈{\psi }_i\left(T\right)\right|{\varphi }_f\left(T\right)〉{|}^2$. To obtain the optimal control laser, the maximum value of the objective function needs to be calculated. Requiring $\delta {\mathit{J}}_{fi}=0$, we have [47, 55]

Table 1. Energy gaps and transition frequencies ω_j between different energy levels Λ_j of two coupled BaI molecules.

View Table

 

Fig. 1 Optimization results for the transition of $|00〉\to 1/\sqrt{2}\left(\left|00〉+\right|11〉\right)$. (a) Concurrence, REC, and fidelity as a function of iteration numbers, (b) converged laser pulse versus the last iteration, (c) fourier transform of the optimized laser pulse, (d) population evolution driven by the optimized laser pulse.

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3. Results

The numerical results of the optimal control for intermolecular entanglement and coherence are shown in this section. For the BaI molecule, the permanent dipole moment is $\mu =6.63D$ and the rotational constant is $B=0.0267c{m}^{-1}$ [52]. To achieve the addressability of the pendular qubits, the two polar molecules are assumed to be located in different electric fields with the intensities being $0.4795kV/cm$ and $0.7192kV/cm$, respectively. For simplicity, the electric fields are supposed to be perpendicular to the direction of r₁₂, i.e., $\alpha =\pi /2$. Here, the intermolecular distance r₁₂ is set as $85nm$. Based on the above physical parameters, we can obtain the transition frequencies ${\omega }_j\left(j=1,2,3,4\right)$ between the energy levels of the two coupled BaI molecules in the presence of dipole-dipole interaction, as shown in Table I. It is found that the frequency shift $△\omega ={\omega }_2-{\omega }_3={\omega }_4-{\omega }_1\approx 5.6MHz$, which is in good agreement with the results in [33]. Accordingly, the duration of the laser pulse is set to be $T=10\hslash /△\omega$ [56]. Generally speaking, the polar molecules mostly populate in ground state at ultracold temperatures, which means that the ground state can be regarded as a reasonable initial state ${\psi }_i\left(0\right)$ of the laser-driven function ${\psi }_i\left(t\right)$. Because of the dipole-dipole interaction, there might exist entanglement between the pendular qubits even in the absence of control laser fields. However, the entanglement of the coupled polar molecules in ground state is so weak, about $6.5\times {10}^{-4}$ (characterized by the concurrence), that almost it can be negligible. Thereby, we consider the ground product state to be the initial state in Eq. (10), i.e., ${\psi }_i\left(0\right)=\left|0〉\otimes \right|0〉$. Moreover, in the following we adopt the fourth-order Runge-Kutta algorithm with steps of $0.25ps$ to solve the time-dependent Schrödinger equations, and take the penalty factor ${\alpha }_0=1.3\times {10}^5$ to limit the laser energies.

In order to generate the strong entanglement of the pendular qubits, we first select the Bell state $1/\sqrt{2}\left(\left|00〉+\right|11〉\right)$ as the target state. The convergence criterion in iterative process is set to be $\delta C={10}^{-5}$, where $\delta C$ is the difference of the concurrence between the adjacent iteration numbers. Fig. 1(a) exhibits the variation trends of the concurrence, REC, and fidelity with respect to the number of iteration steps. It indicates that the concurrence and the fidelity have the similar convergent behaviors, both of which first increase rapidly and then tend to steadily with the growth of iteration numbers. After 2561 iterations, the value of concurrence increases to $0.9828$ from 0 while the one of fidelity increases to $0.9906$ from $0.5$. Compared with the concurrence, the value of REC eventually tends to 1 rather than the ideal value of 2 under the control field, which is because the target state $1/\sqrt{2}\left(\left|00〉+\right|11〉\right)$ is not maximally coherent. We present the converged laser pulse versus the last iteration in Fig. 1(b). The amplitude of the optimized field is about $0.7519kV/cm$, which is smaller than the maximum value of the initial field $E\left(t\right)={\text{sin}}^2\left(\pi t/T\right)\times \cos ({\omega }_{12}t)$ with ${\omega }_{12}={\omega }_1+{\omega }_2$. Driven by the optimized laser, the concurrence approaches its maximum value through the evolution over $2.868\times {10}^{5}ps$. It implies that the entanglement between the two coupled dipoles can be significantly enhanced by means of the optimal control. Moreover, the power spectrum of the optimal field is obtained by Fourier transform. From Fig. 1(c), we can see that the spectrum consists of one main peak and a lot of small peaks. The frequency of the main peak approximately coincides with the transition frequency between the ground state $|00〉$ and the excited state $|11〉$. To illustrate the generation process of the Bell state more intuitively, in Fig. 1(d) we plot the populations as a function of evolution time. In what follows, the populations in different energy levels are defined as $P_k\equiv \left|C_k{(t)}^2\right|\left(k=1,2,3,4\right)$, where $C_k\left(t\right)$ are the coefficients of wave function ${\psi }_i\left(t\right)=C_1\left(t\right)\left|00〉+C_2\left(t\right)\right|01〉+C_3\left(t\right)\left|10〉+C_4\left(t\right)\right|11〉$. As we expect, the population P₁ for ground state $|00〉$ reduces from 1 to $0.4974$ whereas the P₄ for state $|11〉$ increases from 0 to $0.4932$. At the same time, the populations P₂ andP₃ in the two intermediate levels come close to 0 in the end, nearly returning to their initial values. Additionally, the fast beating of the populations can be observed during the evolution, which reflects the sharp and repetitive transitions among different energy levels.

 

Fig. 2 Optimization results for the transition of $|00〉\to 1/\sqrt{2}\left(\left|01〉+\right|10〉\right)$. (a) Concurrence, REC, and fidelity as a function of iteration numbers, (b) converged laser pulse versus the last iteration, (c) fourier transform of the optimized laser pulse, (d) population evolution driven by the optimized laser pulse.

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In previous works, we have investigated the effects of intrinsic decoherence on the entanglement of the coupled dipoles in different initial states [38, 39]. The Bell state $1/\sqrt{2}\left(\left|01〉+\right|10〉\right)$ is proven to be more robust against the intrinsic noises from the external environment than the Bell state $1/\sqrt{2}\left(\left|00〉+\right|11〉\right)$. Thus, we turn to choose $1/\sqrt{2}\left(\left|01〉+\right|10〉\right)$ as the target state. Here, the initial laser is set to be $E\left(t\right)={\text{sin}}^2\left(\pi t/T\right)\times 0.5\left[\cos ({\omega }_{1}t\right)+\cos ({\omega }_{31}t)]$ with ${\omega }_{31}={\omega }_3-{\omega }_1$. As shown in Fig. 2(a), the behaviors of the concurrence, REC, and fidelity versus iteration numbers are similar to the case of Fig. 1(a), except that the initial value of the fidelity is 0 instead of $0.5$. Under the same convergence criterion, after 300 iterations the values of the concurrence and the fidelity can reach $0.9647$ and $0.9813$, respectively. As before, the REC of the molecular system approximates to 1 at the end of iteration. The converged control field after the optimization is given in Fig. 2(b) and its power spectrum is plotted in Fig. 2(c). It is found that the optimized laser has two high and narrow peaks with similar intensities, corresponding to the transition frequencies of the two coupled BaI molecules from $\left|00〉\to \right|10〉$ and $\left|10〉\to \right|01〉$. The dynamics of populations during the evolution is depicted in Fig. 2(d), which can clearly indicate that the population of ground state $|00〉$ is transferred to the intermediate states $|01〉$ and $|10〉$ after plenty of repetitive oscillations.

 

Fig. 3 Optimization results for the transition of $|00〉\to 1/2\left(\left|00〉+\right|01〉-\left|10〉+\right|11〉\right)$. (a) REC, concurrence, and fidelity as a function of iteration numbers, (b) converged laser pulse versus the last iteration, (c) fourier transform of the optimized laser pulse, (d) population evolution driven by the optimized laser pulse.

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As we know, the computational basis states are required to be largely superposed for quantum computation, so it is necessary to design the control laser to achieve the enhancement of the coherence between the polar molecules. According to the definition for REC, we can choose $1/2\left(\left|00〉+\right|01〉+\left|10〉+\right|11〉\right)$ as the maximally coherent state. However, it is not hard to find that this state actually is a product state with the form of $\left|+〉\otimes \right|+〉$, where $|+〉=1/\sqrt{2}\left(\left|0〉+\right|1〉\right)$. Therefore, we turn to consider another maximally coherent state $1/2\left(\left|00〉+\right|01〉-\left|10〉+\right|11〉\right)$ to be the target state instead. This state is maximally entangled as well, since it can be rewritten as the form of Bell state $1/\sqrt{2}(\left|0〉\right|+〉-\left|1〉\right|-〉)$ with $|-〉=1/\sqrt{2}\left(\left|0〉-\right|1〉\right)$. To generate the intermolecular coherence as much as possible, here we reset the convergence criterionas $\delta R={10}^{-5}$, where $\delta R$ is the difference of the REC between the adjacent iteration numbers. The initial laser control field is chosen as $E\left(t\right)={\text{sin}}^2\left(\pi t/T\right)\times 0.25{\displaystyle \sum }_{j=1}^4\cos ({\omega }_{j}t)$ to attain good convergence in numerical calculations. From Fig. 3(a), we find that the REC and concurrence of the final state can be very close to their respective maximum values just through 1 iteration, where after they hardly increase as the iteration number grows. Additionally, the fidelity of the desired state can reach $0.9943$ after the final iteration. This maybe lies in that the initial control field contains four transition frequencies, which can contribute to the generation of the target state consisting of four eigenvectors. However, due to the fact that the transition frequencies ${\omega }_1\approx {\omega }_4$ and ${\omega }_2\approx {\omega }_3$ (see Table I), the power spectrum of the converged laser pulse only exhibits two obvious main peaks rather than four ones, as plotted in Fig. 3(c). Again, applying the optimized laser pulse (shown in Fig. 3(b)) to the two coupled molecules, we obtain the time evolutions of the populations (see Fig. 3(d)). One can find that the final populations are $P_1=0.2510$, $P_2=0.2418$, $P_3=0.2580$, and $P_4=0.2492$. This means that the population of the ground state almost evenly distributes to the four energy levels at the end of the evolution. Moreover, we observe that the optimized laser is composed of two sub-pulses with different amplitudes. By contrasting Figs. 3(b) and 3(d), it is found that the oscillation frequencies of the populations seem to have a positive correlation with the maximum strength of the sub-pulses, i.e., the optimized control field with the larger amplitude will lead to more rapid oscillations during the evolution.

 

Fig. 4 Optimization results for the transition of $1/\sqrt{2}\left(\left|00〉+\right|11〉\right)\to 1/2\left(\left|00〉+\right|01〉+\left|10〉+\right|11〉\right)$. (a) REC, concurrence, and fidelity as a function of iteration numbers, (b) converged laser pulse after the optimization, (c) fourier transform of the optimized laser pulse, (d) population evolution driven by the optimized laser pulse.

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Fig. 5 Optimization results for the transition of $|00〉\to (\sqrt{\sqrt{5}-2}\left|00〉+\sqrt{\left(3-\sqrt{5}\right)/2}\right|10〉+\sqrt{\left(3-\sqrt{5}\right)/2}|01〉)$. (a) Fidelity and concurrence as a function of iteration numbers, (b) converged laser pulse after the optimization, (c) fourier transform of the optimized laser pulse, (d) population evolution driven by the optimized laser pulse.

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We have performed the optimal control for the molecular transitions from the ground state to the entangled or coherent states in the above studies. At present, an interesting question is whether the transition can be realized between the maximally entangled state and the maximally coherent state. To check this, we set the Bell state $1/\sqrt{2}\left(\left|00〉+\right|11〉\right)$ as the initial state, the non-entangled coherent state $1/2\left(\left|00〉+\right|01〉+\left|10〉+\right|11〉\right)$ as the target state, and the laser pulse $E\left(t\right)={\text{sin}}^2\left(\pi t/T\right)\times 1/3\left[\cos ({\omega }_{1}t\right)+\cos ({\omega }_{12}t)+\cos ({\omega }_{31}t)]$ as the initial control field. In this case, the value of the REC increases from 1 to $1.9999$ after 124 iterations, whereas the concurrence monotonously decays with the iteration numbers growing and evolves to 0 asymptotically (see Fig. 4(a)). The converged laser pulse is depicted in Fig. 4(b), whose power spectrum has three frequencies versus the level transitions $\left|00〉\to \right|11〉$, $\left|00〉\to \right|10〉$, and $\left|10〉\to \right|01〉$, as shown in Fig. 4(c). Moreover, the results of the population transfer shown in Fig. 4(d) are in line with our expectations as well. Furthermore, it should be mentioned that the opposite transfer $1/2\left(\left|00〉+\right|01〉+\left|10〉+\right|11〉\right)\to 1/\sqrt{2}\left(\left|00〉+\right|11〉\right)$

can also be successfully achieved through the optimization, here we do not go into details any more.

Up to now, we only choose several special quantum states as the target states for the generation of intermolecular entanglement and coherence. In fact, arbitrary superposition states can be created by using the method of OCT. Hardy state is one of the non-maximally entangled states for a two-particle system, which can be used to reveal the conflict between quantum physics and local realism [57]. On the basis of Hardy’s theorem, the Hardy state can be abbreviated as $\mathrm{\Phi }=\alpha \left|00〉+\beta \right|10〉+\gamma |01〉$, where $\alpha \cdot \gamma \cdot \beta \ne 0$ [58]. The probability of the Hardy state violating the local realism is

When $|\alpha {|}^2=\sqrt{5}-2$, $\left|\beta {|}^2=\right|\gamma {|}^2=\left(3-\sqrt{5}\right)/2$, the probability has the maximum value. Now, the Hardy state with the maximum violation probability is taken as the target state in the optimal control simulation. Here, we set the convergence criterion as the difference of the fidelity $\delta F={10}^{-5}$ and suppose that the initial control field is same as the one of Fig. 2. From Fig. 5(a), we find that the fidelity of the final state can reach $0.9979$ after 257 iterations. However, the concurrence versus the last iteration is far from its maximum value. The converged laser pulse and its power spectrum are plotted in Figs. 5(b) and 5(c), respectively. Finally, in Fig. 5(d) we provide the evolution behaviors of the populations driven by the optimized laser pulse. We can see that the terminal values of the populations are $P_1=0.2587$, $P_2=0.3674$, $P_3=0.3725$, and $P_4=0.0014$, nearly matching with their respective theoretical values. Thus, it maybe turns out that the OCT is a valid method to create arbitrary molecular states that we need.

4. Conclusions

In summary, by taking the BaI molecule as a test case, we have theoretically performed the optimal control on the two coupled dipoles in static fields with appreciable intensity gradient for generating the entanglement and coherence between the pendular states. Our results show that the fidelity, concurrence, and REC of the final states monotonously increase with the iteration numbers, and eventually reach the convergence limit in an asymptotical way. The desired control fields are obtained after multiple iterations, which can drive the molecular system from the ground state to the maximally entangled or coherent states. Accordingly, the concurrence and the REC of the two coupled dipoles are approximately enhanced to their respective maximum values after optimizing. During the evolutions, the populations in different energy levels oscillate many times, possibly resulting from the short evolution period of the molecular system and the strong couplings between different states. Moreover, the transition dynamic between the Bell state and the coherent state is achieved as well. Finally, we have created the Hardy state by designing a suitable laser pulse, which indicates that the OCT is a general approach to prepare arbitrary quantum states for dipole-dipole system.

It is worth mentioning that the coherence time on the scale of millisecond was demonstrated experimentally in the system of polar molecules. For example, the rotational coherence time of the ultracold polar ${ }^{40}{\mathrm{K}}^{87}\text{Rb}$ molecules trapped in optical lattice is able to reach a few milliseconds at a magic angle [59], and can also be extended to tens of milliseconds via spin-echo techniques [60]. For a dilute gas of ultracold ${ }^{23}]{\text{Na}}^{40}\mathrm{K}$ molecules, the rotational coherence time of almost 10 ms was realized in a spin-decoupled magic lattice [41]. Furthermore, for the polar YbF molecules which are cooled below $100\mu \mathrm{K}$, its coherence time can even exceed 150 ms [61]. In our work, the duration of the designed optimal laser pulses is about $2.868\times {10}^5$ ps, which is much less than the millisecond-scale coherence time. In such a short duration, the decoherence effect of pendular qubits is almost negligible, i.e., before decoherence the population transfer between the pendular states of the polar molecules is completed very quickly under the optimal control fields. Additionally, it should be noted that the current work only involves the case of two polar molecules in pendular states. In general, the generation of the multipartite entanglement or coherence may be more meaningful for quantum information processing. We will explore the applications of the OCT to quantum tasks [62] based on the pendular states of multiple polar molecule arrays in the future.