Fast generations of tree-type three-dimensional entanglement via Lewis-Riesenfeld invariants and transitionless quantum driving

Recently, a novel three-dimensional entangled state called tree-type entanglement, which is likely to have applications for improving quantum communication security, was prepared via adiabatic passage by Song et al. Here we propose two schemes for fast generating tree-type three-dimensional entanglement among three spatially separated atoms via shortcuts to adiabatic passage. With the help of quantum Zeno dynamics, two kinds of different but equivalent methods, Lewis-Riesenfeld invariants and transitionless quantum driving, are applied to construct shortcuts to adiabatic passage. The comparisons between the two methods are discussed. The strict numerical simulations show that the tree-type three-dimensional entangled states can be fast prepared with quite high fidelities and the two schemes are both robust against the variations in the parameters, atomic spontaneous emissions and the cavity-fiber photon leakages.


Preliminary Theory
Lewis-Riesenfeld invariants. Here we give a brief description about Lewis-Riesenfeld invariants theory 52 .
A quantum system is governed by a time-dependent Hamiltonian H(t), and the corresponding time-dependent Hermitian invariant I(t) satisfies The solution of the time-dependent Schrödinger equation can be expressed by a superposition of invariant I(t) dynamical modes |Φ n (t)〉 n n i n n where C n is the time-independent amplitude, α n is the time-dependent Lewis-Riesenfeld phase, and |Φ n (t)〉 is one of the orthogonal eigenvectors of the invariant I(t) with a real eigenvalue λ n . The Lewis-Riesenfeld phases are defined as n t n n 0 Transitionless quantum driving. Suppose a system is dominated by a time-dependent Hamiltonian H 0 (t) with instantaneous eigenstates |φ n (t)〉 and eigenvalues E n (t), n n n 0 When a slow change satisfying the adiabatic condition happens, the state of the system governed by H 0 (t) can be expressed as 26,53  Because the instantaneous eigenstate |φ n (t)〉 do not meet the Schrödinger equation, there may be transitions between the eigenstates of H 0 (t) with a finite probability during the whole evolution process even under the adiabatic condition. In order to construct the Hamiltonian H(t) that exactly drives the instantaneous eigenstates |φ n (t)〉 s, i.e., there are no transitions between different eigenstates during the whole evolution process, the simplest choice of the Hamiltonian H(t) can be written as n t n n  Therefore, as long as H(t) is constructed, the system will evolves with no transitions between different eigenstates.
Quantum Zeno dynamics. Assume that a quantum system's dynamics evolution is governed by the K obs m eas where H obs can be viewed as the Hamiltonian of the quantum system investigated and H meas as an additional interaction Hamiltonian performing the measurement. K is a coupling constant, and in the strong coupling limit K → ∞ , the whole system is governed by the Hamiltonian 54 K n n n n n obs where P n is one of the eigenprojections of H meas with eigenvalues λ n ( λ = ∑ H P n n n meas ). Interestingly, it is easy to deduce that the system state will remain in the same Zeno subspace as that of its initial state. In particular, if the system is initially in the dark state |Ψ d 〉 of H meas , i.e., H meas |Ψ d 〉 = 0, the the Hamiltonian (8) reduces to n n n obs

Description of the physical model
The schematic setup for generating the tree-type three-dimensional entanglement is shown in Fig. 1(a). Three atoms are trapped respectively in three spatially separated optical cavities which are connected by two fibers. Under the short fiber limit (lv)/(2πc) ≤ 1, only the resonant modes of the fibers interact with the cavity modes 55 , where l is the length of the fiber and v is the decay rate of the cavity field into a continuum of fiber modes. The atomic level configurations and relevant transitions are shown in Fig. 1(b). As shown in Fig. 1(b), the five-level atom1 and atom3 are both M-type with two excited states |e L 〉 and |e R 〉 and three ground states |g L 〉 , |g 0 〉 and |g R 〉 . The four-level atom2 is tripod-type with one excited state |e 0 〉 and three ground states |g L 〉 , |g 0 〉 and |g R 〉 . The atomic transition ↔ e g L R j j ( ) 0 (j = 1, 3) is resonantly coupled to the left-circularly (right-circularly) polarized mode of jth cavity with corresponding coupling constant g j,L(R) , and | 〉 ↔ | 〉 e g L R 0 2 ( ) 2 is resonantly coupled to the left-circularly (right-circularly) polarized mode of cavity2 with corresponding coupling constant g 2,L(R) ).
al acf where H total is the total Hamiltonian of the whole system, H al (H acf ) is the interaction between the atoms and the classical laser fields (the cavity-fiber system), v 1(2),L(R) is the coupling strength between the modes of the cavity1,2 (cavity2,3) and the modes of the fiber1(2), a k,L(R) (k = 1, 2, 3) is the annihilation operator of left-circularly (right-circularly) polarized mode of kth cavity, and † b L R 1(2), ( ) is the creation operator of fiber1(2) left-circularly (right-circularly) polarized mode. For simplicity, we assume g k,L(R) and v 1(2),L(R) are real, g k,L(R) = g, and v 1(2),L(R) = v.
Suppose that the total system is initially in the state φ = g g g 00000 c f c f c g g g L g g g R g g g L g g g L g g g R g g g R g g g L g g g L 00000 , 00000 , 00 00 , 0 0 00 , 0 000 , 000 0 , 0 000 , 000 0 , 0000 , 0000 , Obviously, the system is initially in the dark state of H acf , i.e., H acf |φ 1 〉 = 0. Therefore, under the Zeno limit condition , v(k = 1, 2, 3), the whole system can approximatively evolve in an invariant Zeno subspace consisting of dark states corresponding to the zero eigenvalue of H acf : corresponding to the projections Here setting v = g and Ω 3 (t) = Ω 1 (t), we can obtain an effective Hamiltonian of the system Scientific RepoRts | 6:33669 | DOI: 10.1038/srep33669 18 19 20 . The instantaneous eigenstates of H 0 (t) corresponding respectively to the eigenvalues λ 0 = 0 and λ = ±Ω Two methods used to generate tree-type three-dimensional entanglement The method of Lewis-Riesenfeld invariants. In order to construct a shortcut by the LRI method for fast generating of tree-type three-dimensional entanglement, we are supposed to chase down the Hermitian invariant operator (16), I(t) can be easily given by 27,56 c os sin s in cos sin 0 c os cos sin cos cos 0 , 1 3 where χ is an arbitrary constant in the unit of frequency keeping I(t) in the unit of energy, and ν and β are time-dependent auxiliary parameters satisfying the equations So Ω 1 (t) and Ω 2 (t) can be easily deduced as follows: The solution of Shrödinger with respect to the instantaneous eigenstates of I(t) can be written as Then we consider a series of boundary conditions satisfying 2 . Therefore, to avoid infinite Rabi frequencies, we can choose the boundary conditions for ν and β as follows: where ε is a time-independent small value. Then the parameters can be easily set as with the Lewis-Riesenfeld phases 2 0 is the tree-type three-dimensional entanglement generated by the LRI method. Therefore, for the appropriate Rabi frequencies and the fidelity F = 1, we can choose . arctan 2 sin 2 , i e arcsin arctan 2 2 0 177 (28) Thus, the transformation Ψ → Ψ LRI 1 is achieved and we have constructed a shortcut by the LRI method to speed up the generation of the tree-type three-dimensional entanglement.
The method of transitionless quantum driving. Because the instantaneous eigenstates of H 0 (t) do not meet the Schrödinger equation, there may be transitions between the eigenstates in Eq. (17). Therefore, we need to construct the TQD Hamiltonian H(t) that exactly drives the instantaneous eigenstates with no transitions between different eigenstates. Based on Eq. (6), we learn the simplest Hamiltonian H(t) is derived in the form (30) is constructed by a direct coupling of |Ψ 1 〉 and |Ψ 2 〉 , and it is too hard to be implemented in experiment for such a complex system we use 24,35 . However, fortunately inspired by the references 44,46 , we find an alternative physically feasible (APF) Hamiltonian whose effect is equivalent to H(t). The APF design is shown in Fig. 1(c). Comparing Fig. 1(b,c), we change all of the resonant atomic transitions into non-resonant atomic transitions with detuning Δ .
The interaction Hamiltonian of the non-resonant system reads Then similar to the approximation for the Hamiltonian from Eqs (10-16), an effective Hamiltonian for the present non-resonant system can be obtained , i.e., 2 0 is the tree-type three-dimensional entanglement generated by the TQD method.

Numerical simulations and comparisons between LRI and TQD
In the following, we will give the numerical simulations in three subsections to discuss respectively the selections of parameters of the two methods, the feasibility of generating tree-type three-dimensional entanglement and the robustness of our schemes. Also the comparisons between the LRI method and the TQD method will be included in every subsection.

Selections of parameters.
Firstly, to determine the parameters of the LRI method, we plot the fidelity 2 versus the operation time t f and ε in Fig. 2, where |Φ (t f )〉 is the state at the time t = t f of the whole system governed by the total Hamiltonian H total in Eq. (10). In Fig. 2(a), we plot the relation between the fidelity and the operation time t f with ε = 0.177 which is determined in Eq. (28). And we can see that in a very short operation time t f = 80/g the fidelity is already almost unity: F(t f = 80/g) = 0.996. From Fig. 2(b), we can find that under t f = 80/g when ε = 0.177 the fidelity is highest. Thus we can choose t f = 80/g and ε = 0.177 as the parameters of the LRI method in the following discussion. Furthermore, in order to consider the joint effects of t f and ε on the fidelity we plot the three dimensional image of the fidelity versus t f /g −1 and ε in Fig. 2(c). From the three dimensional image, it is clear that the effects of t f and ε on the fidelity are not dependent on each other.
Next we determine the parameters of the TQD method. In order to satisfy the boundary conditions in Eq. (22), the Rabi frequencies Ω 1 (t) and Ω 2 (t) in the original Hamiltonian H total are chosen as 46 where Ω 0 is the pulses' amplitude, t f is the operation time, and τ and T are related parameters. The time-dependent Ω 1 (t) and Ω 2 (t) are shown in Fig. 3. Based on the correlation in Eq. (35), the Rabi frequencies of the TQD method can be figured out. As an illustration, we plot the fidelity = |〈Ψ |Φ 〉| F t ( ) TQD f 2 versus the detuning Δ and t f in Fig. 4, where |Φ (t f )〉 is the state at the time t = t f of the whole system governed by the total Hamiltonian ′ H total in Eq. (31). To compare with each other effectively, we choose the same operation time t f = 80/g in the TQD method as that in the LRI method. From Fig. 4(a), we can find that under t f = 80/g when Δ = 6 g the fidelity is highest. Besides, we can see that the fidelity is almost unity: F(t f = 80/g) = 0.996 at the point t f = 80/g from Fig. 4(b). Thus we choose t f = 80/g and Δ = 6 g as the parameters of the TQD method in the following discussion. Similar to the LRI method, in order to consider the joint effects of t f and Δ on the fidelity we plot the three dimensional image of the fidelity versus t f /g −1 and Δ /g in Fig. 4(c). However, from Fig. 4(c), we are not able to judge whether the effects of t f and Δ on the fidelity are dependent or not dependent on each other. We will make a detailed discussion about the joint effects of t f and Δ on the fidelity later in the last subsection.
Next, a brief discussion is very necessary to show our schemes are fast. Firstly, it is quite apparent that the operation time t f = 80/g is far shorter than t = 3000/g which is the generation time of the tree-type three-dimensional entanglement generated in the ref. 23. Further more, for a more effective discussion, we consider the adiabatic condition θ ∆ t E ( ) 57 , where ∆ = Ω E t ( )/ 3 is the minimum separation between the eigenvalues of the Hamiltonian (16). For convenience, we just consider the LRI method for the shortcut to adiabatic passage. With a simple calculation, for the shortcut to adiabatic passage, we can obtain the constants θ = .  0 014 LRI and Δ E LRI = 0.077 by using Eq. (25). However, for ref. 23, we obtain time-dependent but single-peak θ  t ( ) and Δ E, and the amplitudes are θ = .  0 003 max and Δ E max = 0.067, respectively. Obviously, according to the information above, the adiabatic condition θ ∆ t E ( ) is met in the ref. 23 but needs not to be met in our schemes, which means that the slow process of the generation of the tree-type three-dimensional entanglement is sped up.

Discussion of feasibility.
In this subsection, we will give the numerical simulations for discussing the feasibility of our two schemes. For the LRI method, we plot the time-dependent Rabi frequencies Ω 1 (t) and Ω 2 (t) which are described by Eq. (25) in Fig. 5(a). The evolutions of populations ∼ P LRI 1 (17 20) of states |φ 1(17~20) 〉 governed by H total are shown in Fig. 5(b). For the TQD method, we plot the time-dependent Rabi frequency Ω′ t ( ) 2 Ω′ t ( ( ) ) 1 which is described by Eq. (35) in Fig. 5(c). The evolutions of populations ∼ P TQD 1 (17 20) of states |φ 1(17~20) 〉 governed by ′ H total are shown in Fig. 5(d). In addition, in Fig. 6(a)  (blue dashed line) corresponding to the cases of the LRI method and the TQD method respectively. In Fig. 6(b), we plot the fidelities of the tree-type three-dimensional entanglement generated by the LRI method (red dashed line) and the TQD method (blue solid line), respectively.
Here we first consider Fig. 6(b). From Fig. 6(b), we know that both of the two lines of fidelity illustrated based on the two methods reach a very high point close to unity at the time t = 80/g, and thus both of our schemes are feasible. Figure 6(b) also shows that the fidelity of the TQD method can reach a very high value within a shorter time than that of the LRI method. Next we consider the populations of the target states in Fig. 5(b,d). We can see that a near perfect result we expect is obtained in Fig. 5(b), but a little bit imperfect result appears in Fig. 5(d) in which there exists a small gap between two lines of the populations of |φ 1 〉 and |φ 17~20 〉 . Therefore, for the transformation of populations, the LRI method is a bit better than the TQD method. However, when compare the pulse types in Fig. 5(a,c), we find that the TQD method is more feasible than the LRI method. Because the pulses in LRI method are short-time truncations of two harmonic pulses and the truncations of the two harmonic pulses in a short time are too hard to be achieved. But the pulses in TQD method are almost complete Gaussian pulses which are relatively easy to be achieved. Moreover, the populations of atomic and cavity-fiber excited states for two methods are shown in Fig. 6(a), and all of the populations of excited states are near zero at the time t = 80/g.  (c) the three dimensional image of the fidelity for the TQD method versus Δ /g and t f /g −1 .
Scientific RepoRts | 6:33669 | DOI: 10.1038/srep33669 So we can deduce that whichever method is employed, the state of the whole system almost populates in tree-type three-dimensional entanglement.
It is worth explaining the gap between the two lines of the populations of |φ 1 〉 and |φ 17~20 〉 in Fig. 5(d). For the TQD method, there are two limit conditions , v(k = 1, 2, 3) and ∆  g, v applied to prepare tree-type three-dimensional entanglement. However, as shown in Fig. 5(c), the amplitude of Ω′ t ( ) 2 Ω′ t ( ( ) ) 1 is 0.8 g which does not strictly meet the limit condition , v. And also the detuning Δ = 6 g does not strictly meet the limit condition ∆  g, v. In fact, these two limit conditions are difficult to be coordinated. With no assignments of t f and Δ , we calculate the amplitude of Ω′ t ( ) 2 with the parameters τ = 0.14t f and T = 0.19t f to obtain which is not dependent on the amplitude Ω 0 of Ω 1,2 (t) in Eq. (36) but only proportional to ∆ t / f . Thus, Ω′ 0 roughly equals to ∆ /3 if the operation time is chosen as t f = 80/g. Nevertheless, the ratio 1/3 is not small enough to satisfy both two limit conditions , v will not be satisfied if the limit condition ∆  g, v is satisfied and vice versa. Therefore, there exists a gap between the two lines of the populations of |φ 1 〉 and |φ 17~20 〉 in Fig. 5(d). In addition, Eq. (37) reveals that Ω′ t ( ) 2 's amplitude Ω′ ∝ ∆ t / f 0 . It is known that the fidelity of the TQD method is strongly dependent on Ω′ 0 , so we can deduce that the fidelity of the TQD method is strongly dependent on the value of Δ /t f . As a result, in Fig. 4(c), the effects of t f and Δ on the fidelity are dependent on each other.
Based on the discussion above, for fast generating tree-type three-dimensional entanglement, both the LRI method and the TQD method are feasible. Besides, the two methods have their own advantages and disadvantages and we can choose a certain method depending on the conditions in experiment.  Discussion of robustness. In the above discussion, the whole system are perfect and considered as absolutely isolated from the environment. Therefore, it is necessary to give the discussions of robustness of our schemes against the variations in the parameters and decoherence induced by the atomic spontaneous emissions and photon leakages of the cavity-fiber system. For discussing the effects of the variations in the parameters, we plot the fidelity of the LRI method versus the variations in t f and ε in Fig. 7(a) and the fidelity of the TQD method versus the variations in t f and Δ in Fig. 7(b). Here we define δx = x′ − x as the deviation of x, in which x denotes the ideal value and x′ denotes the actual value. In Fig. 7(a), the fidelity decreases with the increase of |δε| as described in Fig. 2(b). From Eq. (25), we know that the Rabi frequencies decrease with the increase of the operation time t f . According to the limit condition , v we use, the values of the Rabi frequencies are the smaller the better, so the operation time t f is the longer the better as described in Fig. 2(a). Therefore, the fidelity of the LRI method increases with the increase of δt f in Fig. 7(a). In Fig. 7(b), we can clearly see that the effects of t f and Δ on the fidelity of the TQD method are dependent on each other and even the fidelity of the TQD method is apparently dependent on the value of Δ /t f as mentioned in the last subsection. Significantly, we notice that the fidelities of the two methods are both over 0.98 even when δ = .
. Therefore, both of our schemes are robust against the variations in the parameters.
We can also see that the smallest fidelity of the TQD method in Fig. 7(b) is slightly higher than the smallest fidelity of the LRI method in Fig. 7(a). This fact can easily be found by comparing Figs 2(c) and 4(c), in which there is a greater advisable range of the parameters in the TQD method than in the LRI method for preparing tree-type three-dimensional entanglement with a high fidelity.
Next taking the decoherence induced by the atomic spontaneous emissions and photon leakages of the cavity-fiber system into account, the whole system is dominated by the master equation   . Based on the master equation, we plot the fidelities of the LRI method and the TQD method versus κ g / and γ/g in Fig. 8. As we can see from the decrease of the fidelity of the LRI method with the increases of κ g / and γ/g in Fig. 8(a), we learn that the influence of atomic spontaneous emissions on the fidelity is greater than that of photon leakages of the cavity-fiber system. However, in Fig. 8(a) the influence of cavity-fiber photon leakages on the fidelity of the TQD method plays a full role, but that of atomic spontaneous emissions is little. As a cross reference, we can get some inspiration from Fig. 6(a). In Fig. 6(a), the highest value of the cavity-fiber excited populations (blue dashed line) of the TQD method is over 0.2 during the evolution process but that of the atomic excited populations (blue solid line) of the TQD method is near zero which caused by the detuning Δ . The highest values of the atomic excited populations (red solid line) and the cavity-fiber excited populations (red dashed line) of the LRI method are slightly higher and slightly lower than 0.05, respectively. Therefore, it is no doubt that the results of Figs 6 and 8 are corresponding to each other. Finally, it is necessary to emphasize that the fidelities of the LRI method and the TQD method are near 0.94 and over 0.955 respectively, even when κ γ = = . g 0 02 . Therefore, our two schemes of the LRI method and the TQD method both are robust against the decoherence induced by the atomic spontaneous emissions and photon leakages of the cavity-fiber system.

Experimental feasibility and conclusion
Now we show the experimental feasibility of our schemes. As mentioned in the ref. 23 of 5P 3/2 can be used as the excited states |e L 〉 , |e 0 〉 and |e R 〉 respectively. Based on a set of cavity QED parameters g = 2π × 750 MHz, γ = 2π × 3.5 MHz and κ = 2π × 2.62 MHz, which are achieved in recent experiments 58-60 , we can obtain the very high fidelities F LRI = 0.984 and F TQD = 0.990 corresponding to the LRI method and the TQD method respectively, which show our schemes for generating tree-type three-dimensional entangled states both are feasible in the experiment.
In conclusion, we have proposed two schemes to speed up the generations of the tree-type three-dimensional entanglement via Lewis-Riesenfeld invariants and transitionless quantum driving. The two different tree-type three-dimensional entangled states are prepared among three atoms trapped respectively in three spatially separated optical cavities connected by two fibers. The strict numerical simulations show that the LRI method and the TQD method both are feasible and robust against the variations in the parameters, atomic spontaneous emissions and photon leakages of the cavity-fiber system. Besides, comparing the two methods, we know they both have their own advantages and disadvantages. So we can choose different methods depending on different conditions in experiment. In short, both of our schemes are fast, robust and feasible. We hope that the tree-type three-dimensional entanglement would contribute to the improvement of quantum communication security and our work would be useful for the experimental realization of quantum information processing in the near future.