Reverse engineering of a Hamiltonian by designing the evolution operators

We propose an effective and flexible scheme for reverse engineering of a Hamiltonian by designing the evolution operators to eliminate the terms of Hamiltonian which are hard to be realized in practice. Different from transitionless quantum driving (TQD), the present scheme is focus on only one or parts of moving states in a D-dimension (D ≥ 3) system. The numerical simulation shows that the present scheme not only contains the results of TQD, but also has more free parameters, which make this scheme more flexible. An example is given by using this scheme to realize the population transfer for a Rydberg atom. The influences of various decoherence processes are discussed by numerical simulation and the result shows that the scheme is fast and robust against the decoherence and operational imperfection. Therefore, this scheme may be used to construct a Hamiltonian which can be realized in experiments.


Reverse engineering of a Hamiltonian
We begin to introduce the basic method of the scheme for reverse engineering of a Hamiltonian by designing the evolution operators. Firstly, we suppose that the system evolves along the state |φ 1 (t)〉 and the initial state of the system is |ψ(0)〉 . So, the condition |φ 1 (0)〉 = |ψ(0)〉 should be satisfied. We can obtain a complete orthogonal basis {|φ n (t)〉 } through a process of completion and orthogonalization. Therefore, the vectors in basis {|φ n (t)〉 } satisfy the orthogonality condition 〈 φ m (t)|φ n (t)〉 = δ mn and the completeness condition n n n . Since the system evolves along |φ 1 (t)〉 , the evolution operator can be designed as The Hamiltonian can be formally solved from Eq. (4), and be given as k k k the present scheme has more free parameters λ mn (t). Therefore, this scheme may construct some new and different Hamiltonians. Moreover, when parameters λ mn (m, n ≠ 1) are independent of time, Eq. (6) will degenerate into Eq. (7), which shows that the present scheme contains the results of TQD. On the other hand, once the unitary condition UU † = U † U = 1 for evolution operator is satisfied, the Hamiltonian given in Eq. (6) should be a Hermitian operator, because As an extension, for a N-dimension system (N ≥ 4), the evolution operator can be designed as Then, the initial state |ψ(0)〉 of the system can be expressed by the superposition of {|φ j (0)〉 } (j = 1, 2, … , s). Thus, the system can evolve along more than one moving states in this case. This might sometimes help us to simplify the design of the system's Hamiltonian.

The population transfer for a Rydberg atom
For the sake of clearness, we give an example to emphasize the advantages of the scheme. Here, we consider a Rydberg atom with the energy levels shown in Fig. 1. The transition between |1〉 and |3〉 is hard to realize. So, the Hamiltonian of the Rydberg atom is usually written as the following form i t 12 23 ( ) where, Ω 12 and Ω 23 are the Rabi frequencies of laser pulses, which drive the transitions |1〉 ↔ |2〉 and |2〉 ↔ |3〉 , respectively, and they are ϕ-dephased from each other. Suppose the initial state of the three-energy-level Rydberg atom is |1〉 , the target state is |Ψ tar 〉 = cos μ|1〉 + sin μ|3〉 . We choose a complete orthogonal basis as below With the unitary condition in Eq. (2), the evolution operator can take this form According to Eq. (6), the evolution operator in Eq. (12) gives the following Hamiltonian For simplicity, we set θ = 0 here, the Hamiltonian in Eq. (13) can be written by Here, the Hamiltonian in Eq. (14) is already a Hermitian operator. To eliminate the terms with |1〉 〈 3| and |3〉 〈 1|, which are difficult to realize for the three-energy-level Rydberg atom, we set λ β α For simplicity, we suppose the initial time is t i = 0 and the final time is t f = T, so T is the total interaction time. To satisfy the boundary conditions ( ) 0 and avoid the singularity of Hamiltonian, we choose the parameters as 4 where A is an arbitrary constant. Then, the Hamiltonian in Eq. (15) can be written by For the sake of obtaining a relatively high speed, the values of Ω 1 T and Ω 2 T in Eq. (17) should not be too large. Noticing that, with A increasing, πA increases while cot β decreases. Therefore, to obtain a relatively small |Ω 1 T| and |Ω 2 T|, A should be neither too large nor too small. Therefore, we choose A = 1 here. However, we can see from Eq. (17) that the functions of Rabi frequencies Ω 1 (t) and Ω 2 (t) are too complex for experimental realization. Fortunately, we can solve the problem by using simple functions to make a curve fitting for the Ω 1 (t) and Ω 2 (t). As an example, µ = π 4 is taken here. We use Ω′ t ( ) 1 and Ω′ t ( ) 2 in the following, which are linear superposition of the Gaussian or trigonometric functions, to make a curve fitting for the Ω 1 (t) and Ω 2 (t), In this case, we have Ω′ ≤ . To compare the values of Ω 1 (t) and Ω′ t ( ) 1 , Ω 2 (t) and Ω′ t ( ) 2 , we plot Ω 1 T and Ω′ t ( ) 1 versus t/T with μ = π/4 and A = 1 in Fig. 2(a) and plot Ω 2 T and Ω′ t ( ) 2 versus t/T with μ = π/4 and A = 1 in Fig. 2(b). From Fig. 2(a,b), one can find that the curves of Ω 1 (t) and Ω′ t ( ) 1 (Ω 2 (t) and Ω′ t ( ) 2 ) are well matched with each other. Therefore, we may use Ω′ t ( ) 1 Ω′ t ( ( )) 2 instead of Ω 1 (t) (Ω 2 (t)) to obtain the same effect. To test the effectiveness of the approximation by using Ω′ t ( ) 1 Ω′ t ( ( )) 2 instead of Ω 1 (t) (Ω 2 (t)), a simulation for the varies of populations of states |1〉 , |2〉 and |3〉 when the Rydberg atom is driven by laser pulses with Rabi frequencies Ω 1 (t) and Ω 2 (t) with parameters μ = π/4 and A = 1, is shown in Fig. 3(a). We can see from Fig. 3(a) that the evolution is consonant with the expectation coming from the evolution operator in Eq. (12). As a comparison, a simulation for the varies of populations of states |1〉 , |2〉 and |3〉 when the Rydberg atom is driven by laser pulses with Rabi frequencies Ω′ 1 (t) and Ω′ 2 (t) with parameters μ = π/4 and A = 1, is shown in Fig. 3(b). As shown in Fig. 3(a,b), we can conclude that the Scientific RepoRts | 6:30151 | DOI: 10.1038/srep30151 approximation by using Ω′ t ( ) 1 Ω′ t ( ( )) 2 instead of Ω 1 (t) (Ω 2 (t)) is effective here. In addition, seen from Fig. 3, the population of intermediate state |2〉 reaches a peak value about 0.72, because the system does not evolve along the dark state of the Hamiltonian of the system but a nonadiabatic shortcut, which greatly reduces the total evolution time.
Since most of the parameters are hard to faultlessly achieve in experiment, that require us to investigate the variations in the parameters caused by the experimental imperfection. We would like to discuss the fidelity F = |〈 Ψ tar |φ 1 (T)〉 | 2 with the deviations δT, δΩ′ 1 and δΩ′ 2 of total interaction time T, Rabi frequencies of laser pulses Ω′ 1 and Ω′ 2 being considered.
Firstly, we plot F versus δΩ′ Ω′ / 1 1 and δΩ′ Ω′ / 2 2 with parameters μ = π/4 and A = 1 in Fig. 4 (a). Moreover, we calculate the exact values of the fidelities F at some boundary points of Fig. 4 (a) and show the results in Table 1. According to Table 1 and Fig. 4 (a), we find that the final fidelity F is still higher than 0.9822 even when the deviation δ δ Ω′ Ω′ = Ω′ Ω′ = / / 10% 1 1 2 2 . Therefore, the realizing of the population transfer for a Rydberg atom given in this paper is robust against deviations δΩ′ 1 and δΩ′ 2 of Rabi frequencies Ω′ 1 and Ω′ 2 for laser pulses.
Thirdly, F versus δΩ′ Ω′ / 2 2 and δT/T with parameters μ = π/4 and A = 1 is plotted in Fig. 4 (c). And δΩ′ Ω′ / 2 2 and δT/T with corresponding fidelity F are given in Table 3. As indicated in Table 3 and Fig. 4 (c), the fidelity F is still high than 0.9588 even when the deviation δ δ Ω′ Ω′ = = T T / / 10% 2 2 . Moreover, when deviations of δΩ′ 2 and δT have the different signs (one negative and one positive), the fidelity F can still keep in a high level. Hence, we can say the scheme suffers little from deviations δΩ′ 2 and δT.
According to the analysis above, we summarize that, the scheme to realize the population transfer for a Rydberg atom is robust against operational imperfection.
To prove that the present scheme can be used to speed up the system's evolution and construct the shortcut to adiabatic passages, we make a comparison between the present scheme and the fractional stimulated Raman adiabatic passage (STIRAP) method via dark state  one can design the Rabi frequencies Ω 12 (t) and Ω 23 (t) as following fidelity F when the Rydberg atom is driven by laser pulses with Rabi frequencies Ω 12 (t) and Ω 23 (t) shown in Eq. (20) versus Ω 0 T. And a series of samples of Ω 0 T and corresponding fidelity F are shown in Table 5. From Fig. 5 and Table 5, we can see that, to meet the adiabatic condition and obtain a relatively high fidelity by using STIRAP method, one should take Ω 0 T about 30. Moreover, when Ω 0 T = 3.154, the adiabatic condition is badly violated and the fidelity is only 0.5538 for STIRAP method. But for the present scheme, we can obtain F = 1.000 while Ω′ ≤ . . Therefore, the evolution speed with the present scheme is faster a lot comparing with that using STIRAP method. It confirms that the present scheme can be used to speed up the system's evolution and construct the shortcut to adiabatic passages. Therefore, we conclude that the present scheme can construct a Hamiltonian with both fast evolution process and robustness against operational imperfection.   In the end, we discuss the fidelity F is robust to the decoherence mechanisms. In this scheme, the atomic spontaneous emission plays the major role. The evolution of the system can be described by a master equation in Lindblad form as following I l l l l l l l δΩ′ Ω′ /  where, L l is the Lindblad operator. There are two Lindblad operators here. They are = Γ L 1 2 1 1 and = Γ L 2 3 2 2 , in which, Γ 1 and Γ 2 are the atomic spontaneous emission coefficients for |2〉 → |1〉 and |3〉 → |2〉, respectively. Fidelity F versus Γ 1 T and Γ 2 T is plotted in Fig. 6. From Fig. 6, we can see that the fidelity F decreases when Γ 1 and Γ 2 increase. When in the case of strong coupling Ω Ω Γ Γ  , , 2 , the influence caused by atomic spontaneous emission is little. For example, if Γ 1 = Γ 2 = 0.01 × 3.154/T, the fidelity is 0.9901. Even when Γ 1 = Γ 2 = 0.1 × 3.154/T, the fidelity is 0.9101, still higher than 0.9. With current experimental technology, it is easy to obtain a laser pulse with Rabi frequency much larger than the atomic spontaneous emission coefficients. Therefore, the population transfer for a Rydberg atom with the reverse engineering scheme given here can be robustly realized.

Conclusion
In conclusion, we have proposed an effective and flexible scheme for reverse engineering of a Hamiltonian by designing the evolution operators. Different from TQD, the present scheme is focus on only one or parts of moving states in a D-dimension (D ≥ 3) system. The numerical simulation has indicated that the present scheme not only contains the results of TQD, but also has more free parameters, which make this scheme more flexible.  Table 5. Ω 0 T for STIRAP and corresponding fidelity F. Moreover, the new free parameters may help to eliminate the terms of Hamiltonian which are hard to be realized practically. Furthermore, owing to suitable choice of boundary conditions for parameters, by making a curve fitting, the complex Rabi frequencies Ω 1 and Ω 2 of laser pulses can be respective superseded by Rabi frequencies Ω′ 1 and Ω′ 2 expressed by the superpositions of the Gaussian or trigonometric functions, which can be realized with current experimental technology. The example given in Sec. III has shown that the present scheme can design a Hamiltonian to realize the population transfer for a Rydberg atom successfully and the numerical simulation has shown that the scheme is fast and robustness against the operational imperfection and the decoherence mechanisms. Therefore, the present scheme may be used to construct a Hamiltonian which can be realized in experiments.