Reverse engineering of a Hamiltonian by designing the evolution operators

Abstract

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.

Introduction

Executing computation and communication tasks^1,2,3,4 with time-dependent interactions in quantum information processing (QIP)^{5,6,7,8,9,10,11,12} have attracted more and more interests in recent years. It has been shown that, the adiabatic passage, resonant pulses and some other methods can be used to realize the evolution process. Among of them, the adiabatic passage techniques are known for their robustness against variations of experimental parameters. Therefore, many schemes have been proposed with adiabatic passage techniques in quantum information processing field. For example, rapid adiabatic passage, stimulated Raman adiabatic passage and their variants^{13,14,15,16,17,18,19,20,21,22} have been widely used to perform population transfers in two- or three-level systems. The system keeps in the instantaneous ground state of its time-dependent Hamiltonian during the entire evolution process under an adiabatic control of a quantum system. To ensure that the adiabatic condition is always satisfied, the control parameters in the Hamiltonian should be well designed, which usually issue in relatively long execution time. Although little heating or friction will be created when the system remains in the instantaneous ground state, the long time required may make the operation useless or even impossible to implement because decoherence would spoil the intended dynamics. On the other hand, using resonant pulses, the scheme may has a relatively high speed, but it requires exact pulse areas and resonances. Therefore, accelerating the adiabatic passage towards the perfect final outcome is a good idea and perhaps the most reasonable way to actually fight against the decoherence that is accumulated during a long operation time. Consequently, some alternative approaches have been put forward by combining the virtues of adiabatic techniques and resonant pulses together for achieving controlled quantum state evolutions with both high speed and fidelity, such as optimal control theory^23,24,25 and composite pulses^26,27. Recently, by designing nonadiabatic shortcuts to speed up quantum adiabatic process, a new technique named “shortcuts to adiabaticity” (STA)^{28,29,30,31,32,33,34,35,36,37,38,39} opens a new chapter in the fast and robust quantum state control. As two famous methods of STA, “Transitionless quantum driving” (TQD)^31,32,33,34 and inverse engineering^{34,35,36,37,38} based on Lewis-Riesenfeld invariants⁴⁰ have been intensively focused, They have been applied in different kinds of fields including “fast quantum information processing”, “fast cold-atom”, “fast ion transport”, “fast wave-packet splitting”, “fast expansion”, etc.^{41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60}. For example, with invariant-based inverse engineering, a fast population transfer in a three-level system has been achieved by Chen and Muga⁵⁶. Chen et al.⁵⁷ have proposed a scheme for fast generation of three-atom singlet states by TQD. These schemes have shown the powerful application for invariant-based inverse engineering and TQD in QIP.

It has been pointed out in ref. 34 that, invariant-based inverse engineering and TQD are strongly related and potentially equivalent to each other. Invariant-based method is convenience and effective with a Hamiltonian which admits known structures for the invariants. But for most systems, the invariants are unknown or hard to be solved. As for TQD, it will not meet this difficult point. However, some terms of Hamiltonian constructed by TQD, which are difficult to be realized in experiments, may appear when we accelerate adiabatic schemes. Therefore, how to avoid these problematic terms is a notable problem. Till now, some schemes^{61,62,63,64,65,66,67,68,69,70} have been proposed to solve the problem of the TQD method recently. For example, Ibáñez et al.⁶⁴ have produced a sequence of STA by examining the limitations and capabilities of superadiabatic iterations. Ibáñez et al.⁶⁵ have also studied the STA for a two-level system with multiple Schrödinger pictures and subsequently, Song et al.⁶⁶ have expanded the method in a three-level system based on two nitrogen-vacancy-center ensembles coupled to a transmission line resonator. Moreover, without directly using the counterdiabatic Hamiltonian, Torrontegui et al.⁶⁷ have used the dynamical symmetry of the Hamiltonian to find alternative Hamiltonians that achieved the same goals as speed-up schemes with Lie transforms. Chen et al.⁷⁰ have proposed a method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms.

In this paper, inspired by TQD and the previous schemes^{61,62,63,64,65,66,67,68,69,70}, a new scheme for reverse engineering of a Hamiltonian by designing the evolution operators is proposed for eliminating the terms of Hamiltonian which are hard to be realized in practice. The present scheme is focus on only one or parts of moving states in a D-dimension (D ≥ 3) system, that is different from TQD with which all instantaneous eigenstates evolve parallel. According to the numerical simulation, the present scheme not only contains the results of TQD, but also has more free parameters, which make this scheme more flexible. Moreover, the problematic terms of Hamiltonian may be eliminated by suitably choosing these new free parameters. For the sake of clearness, an example is given to realize the population transfer for a Rydberg atom, where numerical simulation shows the scheme is effective. Therefore, this scheme may be used to construct a Hamiltonian which can be realized in experiments.

The article is organized as follows. In the section of “Reverse engineering of a Hamiltonian”, we will introduce the basic principle of the scheme for reverse engineering of a Hamiltonian by designing the evolution operators. In the section of “The population transfer for a Rydberg atom”, we will show an example using the present scheme to realize the population transfer for a Rydberg atom. Finally, conclusions will be given in the section of “Conclusion”.

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 |ϕ₁(t)〉 and the initial state of the system is |ψ(0)〉. So, the condition |ϕ₁(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 . Since the system evolves along |ϕ₁(t)〉, the evolution operator can be designed as

where parameters λ_mn(t) (m, n ≠ 1) are chosen to satisfy the unitary condition UU^† = U^†U = 1. Submitting the unitary condition into Eq. (1), we obtain

Secondly, according to Schrödinger equation (ħ = 1), we have

On account of the arbitrariness of |ψ(0)〉, Eq. (3) can be written by

The Hamiltonian can be formally solved from Eq. (4) and be given as

By submitting Eq. (2) into Eq. (5), the Hamiltonian in Eq. (5) can be described as

Different from TQD, which gives Hamiltonian in the following from

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

Figure 1

Energy levels of the three-energy-level Rydberg atom.

where, Ω₁₂ and Ω₂₃ 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 . Eq. (14) will be changed into

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 α(0) = 0, α(T) = μ, , β(0) = β(T) = 0, and avoid the singularity of Hamiltonian, we choose the parameters as

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 Ω₁T and Ω₂T 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 |Ω₁T| and |Ω₂T|, 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 Ω₁(t) and Ω₂(t) are too complex for experimental realization. Fortunately, we can solve the problem by using simple functions to make a curve fitting for the Ω₁(t) and Ω₂(t). As an example, is taken here. We use and in the following, which are linear superposition of the Gaussian or trigonometric functions, to make a curve fitting for the Ω₁(t) and Ω₂(t),

In this case, we have and .

To compare the values of Ω₁(t) and , Ω₂(t) and , we plot Ω₁T and versus t/T with μ = π/4 and A = 1 in Fig. 2(a) and plot Ω₂T and 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 Ω₁(t) and (Ω₂(t) and ) are well matched with each other. Therefore, we may use instead of Ω₁(t) (Ω₂(t)) to obtain the same effect. To test the effectiveness of the approximation by using instead of Ω₁(t) (Ω₂(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 Ω₁(t) and Ω₂(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 approximation by using instead of Ω₁(t) (Ω₂(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.

Figure 2

(a) Ω1T and versus t/T with μ = π/4. (b) Ω₂T and Ω_2′T versus t/T with μ = π/4 and A = 1.

Figure 3

(a) Populations of states |1〉, |2〉 and |3〉 versus t/T when the Rydberg atom is driven by laser pulses with Rabi frequencies Ω₁ and Ω₂. (b) Populations of states |1〉, |2〉 and |3〉 versus t/T when the Rydberg atom is driven by laser pulses with Rabi frequencies and . Here we set the parameters μ = π/4 and A = 1.

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|ϕ₁(T)〉|² with the deviations δT, and of total interaction time T, Rabi frequencies of laser pulses and being considered.

Firstly, we plot F versus and 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 . Therefore, the realizing of the population transfer for a Rydberg atom given in this paper is robust against deviations and of Rabi frequencies and for laser pulses.

Table 1 δΩ′₁/Ω′₁ and δΩ′₂/Ω′₂ with corresponding fidelity F.

Figure 4

(a) Fidelity F of the target state versus and . (b) Fidelity F of the target state versus and δT/T. (c) Fidelity F of the target state versus and δT/T. Here we set the parameters μ = π/4 and A = 1.

Secondly, we plot F versus and δT/T with parameters μ = π/4 and A = 1 in Fig. 4 (b). Moreover, and δT/T with corresponding fidelity F are shown in Table 2. Seen from Table 2 and Fig. 4 (b), we obtain that the fidelity F is still high than 0.9729 even when the deviation . So, the scheme is insensitive to deviations and δT.

Table 2 δΩ′₁/Ω′₁ and δT/T with corresponding fidelity F.

Thirdly, F versus and δT/T with parameters μ = π/4 and A = 1 is plotted in Fig. 4 (c). And 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 . Moreover, when deviations of 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 and δT.

Table 3 δΩ′₂/Ω′₂ and δT/T with corresponding fidelity F.

Fourthly, we discuss the fidelity F when , and δT are all considered. Some samples are given in Table 4. Table 4 shows that the fidelity F is still with a high level when the three deviations , and δT are all considered. Moreover, in the worst case, when , the fidelity F is still higher than 0.9469.

Table 4 δΩ′₁/Ω′₁, δΩ′₂/Ω′₂ and δT/T with corresponding fidelity F.

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 of Hamiltonian shown in Eq. (10). According to STIRAP method, by setting boundary condition

one can design the Rabi frequencies Ω₁₂(t) and Ω₂₃(t) as following

where Ω₀ denotes the pulse amplitude, t_c and t₀ are some related parameters. Setting t_c = 0.19t_f and t₀ = 0.14t_f, Rabi frequencies Ω₁₂(t) and Ω₂₃(t) can well satisfy the boundary condition in Eq. (19). We plot Fig. 5 to show the fidelity F when the Rydberg atom is driven by laser pulses with Rabi frequencies Ω₁₂(t) and Ω₂₃(t) shown in Eq. (20) versus Ω₀T. And a series of samples of Ω₀T 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 Ω₀T about 30. Moreover, when Ω₀T = 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 and . 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.

Table 5 Ω₀T for STIRAP and corresponding fidelity F.

Figure 5

Fidelity F of the target state versus Ω₀T with the STIRAP method.

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

where, L_l is the Lindblad operator. There are two Lindblad operators here. They are and , in which, Γ₁ and Γ₂ are the atomic spontaneous emission coefficients for |2〉 → |1〉 and |3〉 → |2〉, respectively. Fidelity F versus Γ₁T and Γ₂T is plotted in Fig. 6. From Fig. 6, we can see that the fidelity F decreases when Γ₁ and Γ₂ increase. When in the case of strong coupling , the influence caused by atomic spontaneous emission is little. For example, if Γ₁ = Γ₂ = 0.01 × 3.154/T, the fidelity is 0.9901. Even when Γ₁ = Γ₂ = 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.

Figure 6

Fidelity F of the target state versus Γ₁/Ω₀ and Γ₂/Ω0.

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. 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 Ω₁ and Ω₂ of laser pulses can be respective superseded by Rabi frequencies and 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.