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Abstract
Here, we focus on using composite pulses to realize high-robustness and high-fidelity coherent control in three-level quantum systems. We design the dynamic parameters (Rabi frequency and detuning) for three-level Hamiltonians for high-fidelity quantum state control using five well-known coherent control techniques including a composite adiabatic passage (CAP). Furthermore, we compare their performance against the Rabi frequency and systematic errors, and accordingly show that the CAP is the most robust against them. It features a broad range of high efficiencies above 99.9%. Thus, it provides an accurate approach for manipulating the evolution of quantum states in three-level quantum systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Accurate initialization and manipulation of quantum states have generated extensive interest in the fields of quantum optics, quantum control, and quantum simulations [1,2]. In addition, they have widespread successful applications in atomic, molecular, and optical physics, chemistry, and many other aspects [3–5]. To achieve these targets, there are numerous control methods that can be employed, such as resonance excitation (RE) [6], adiabatic passages (APs), shortcuts to adiabaticity (STAs) [7–11], dynamical correction technique [12,13], reinforcement learning [14–17], frequency tuning method [18], and composite schemes [19–21]. RE uses $\pi$ or $\pi /2$ resonant pulses to realize rapid quantum state control. APs, including rapid adiabatic passages [22], piecewise adiabatic passages [23], and stimulated Raman adiabatic passages [24,25], can cause quantum systems to evolve adiabatically along an eigenstate of the Hamiltonian, leading to efficient and robust population transfer. STAs, including counter-diabatic (CD) shortcuts [26–31] and Lewis-Riesenfeld invariants (LRIs) [32–36], can accelerate a quantum adiabatic process to reproduce the same final population.
Composite pulses (CPs) are well-studied protocols for realizing specific coherent quantum control using a sequence of pulses. Their advantage is the control of the phases of pulses to produce desired results, instead of the control of the amplitudes of pulses. Two typical types of CPs are composite adiabatic passage (CAP) and universal composite pulses (UCPs). A CAP is a combination of an adiabatic scheme and a CP, which can eliminate experimental errors by appropriately selecting the phases of the pulse sequences; a UCP is a more general composite technology, without specific requirements for pulses. Both are regarded as powerful tools for precise quantum state manipulations in various quantum systems [37–40]. For example, in 2011, Torosov et al. [37] proposed a CAP to optimize the defect that an AP cannot achieve complete population inversion in two-level systems. In 2014, Genov et al. [38] used a UCP to realize robust high-fidelity population inversion in two-state quantum systems, and experimentally demonstrated its effectiveness and universality in a $\rm {\Pr ^{3 +}}$:$\rm {Y_2}Si{O_5}$ crystal. In 2018, Torosov et al. [39] presented three classes of symmetric broadband CP sequences, which compensate the imperfections in the pulse area to an arbitrary high order. In 2021, Shi et al. [40] proposed a method to realize robust general single-qubit gates using CPs in a three-level system.
Imperfect experimental implementations and conditions generate experimental control errors and parameter fluctuations of the Hamiltonian. For example, atoms adopting different Rabi frequencies induced by different positions and fields [41] cause the original Hamiltonian elements to present small global shifts. In addition, atoms or ions in the nonuniform spatial distribution of an external laser, a microwave, or the radiofrequency field produce fluctuations of the Rabi frequency. These errors affect the accuracy and efficiency of quantum state control. A CAP is a useful method for suppressing these experimental errors, because it is insensitive to the pulse shape or pulse area. In 2013, Schraft et al. [42] experimentally demonstrated the robustness and efficiency advantages of a CAP even under weak adiabatic conditions. In 2018, Bruns et al. [43] experimentally demonstrated the improvement in the transfer efficiency and robustness of composite stimulated Raman adiabatic passages compared to those of conventional and repeated stimulated Raman adiabatic passages. In 2021, Torosov et al. [44] compared six well-known techniques for coherent control of two-state quantum systems with respect to various sources of errors, and showed that a CAP is more resilient to experimental errors than the other methods.
Here, we establish a method to use CPs to realize complete and robust coherent control in three-level quantum systems. The CPs change a quantum state using CP sequences, with the relative phases serving as the control parameters, without dependence on the specific shapes of the pulses. More importantly, we compare the performance of a CAP with RE, an AP, a CD, and an LRI with respect to common experimental errors, and find that the CAP is the most robust against these errors among all considered methods. Moreover, it can maintain the transition probability at ultrahigh efficiencies above $99.9\%$ over broad ranges of the error parameters. This offers an effective route for achieving high-fidelity coherent control of three-level quantum systems using CAPs.
2. Preliminaries
We consider here arrays of optical wells, where the dipole force of a red detuned laser field is used to store neutral atoms in each of the foci of a set of microlenses [45]. Three in-line dipole wells are modeled as three harmonic potentials, and we assume here that we can initially store no or one neutral atom per well. The Hamiltonian, in the basis of $\{\left |L\right \rangle,\left |C\right \rangle, \left |R\right \rangle \}$, is expressed as
Here, our objective is to manipulate the evolution of an atom wavefunction from the central well to the external wells with the same probability amplitude, i.e., $\left |C\right \rangle \rightarrow 1/\sqrt {2}\left (\left |L\right \rangle +\left |R\right \rangle \right )$. This is similar to a crucial device of many optical experimental and measurement systems–a beam splitter–which can split a beam of light in two (see Fig. 1). For this purpose, we can rewrite the Hamiltonian, ${H_0}(t)$, in the basis of $\{\left |C\right \rangle,\left |\Phi _+\right \rangle, \left |\Phi _-\right \rangle \}$, as
3. Quantum state control using five coherent techniques
3.1 RE
First, we show how to achieve the above-mentioned aim using RE. RE requires that the frequency of the external drive field should be the same as the Bohr transition frequency, i.e., $\Delta = 0$. We can write analytically the time-dependent evolution of the transition probability as
where $A(t) = \int _{{t_0}}^{t} {\Omega (t')dt'}$ is the pulse area. Understandably, when $A(t)$ is selected as an odd multiple of $\pi$ and $P=1$, complete population inversion can be achieved, regardless of the pulse shape. To explain, we take a Gaussian pulse [51] as an example, which is expressed as where $\sqrt \pi /T$ is the peak Rabi frequency and $T$ is the pulse width, which are plotted in Fig. 2(a). It satisfies $A=\int _{ - \infty }^{ + \infty } {\Omega (t')dt'} = \pi$. In practice, the pulse duration cannot be infinite; it is typically sufficient to choose the condition, $t \in [ -3T,3T]$, to realize population inversion, as shown in Fig. 2(b).
Fig. 2. Time dependence of parameters $\Omega$ and population $P(t)$ of state $\left | {{\Phi _ + }} \right \rangle$ of RE method: (a) $\Omega$ (blue, solid line), (b) Population $P(t)$ (red, solid line).
3.2 AP
AP is an important method to realize population inversion of two-state systems. It requires that the parameters of Hamiltonian change slowly enough. If the system starts in an eigenstate of Hamiltonian, then the system will evolve adiabatically along this eigenstate. Specifically, if we take a linearly chirped Gaussian pulse,
the adiabatic condition is [44] where ${\Omega _0}$ is the peak Rabi frequency and $\alpha$ is the chirp rate. Here we adoptFig. 3. Time dependence of parameters $\Omega$ and population $P(t)$ of state $\left | {{\Phi _ + }} \right \rangle$ of AP method: (a) $\Delta$ (red, dashed line) and $\Omega$ (blue, solid line), (b) Population $P(t)$ (red, solid line).
3.3 CD control
CD control is a method of adding a CD Hamiltonian $H_c$ to original Hamiltonian $H(t)$ for adiabatic evolution, allowing to control the system by perfectly following the instantaneous eigenstate of the original Hamiltonian. Based on the theory of Berry [27], the CD Hamiltonian is written as
The time evolutions of the Rabi frequency, detuning, and CD term are plotted in Fig. 4(a). The time evolutions governed by Hamiltonian $H' = H + {H_c}$ can achieve accurate population transfer for this system, as shown in Fig. 4(b).
Fig. 4. Time dependence of parameters $\Delta$, $\Omega$, and population $P(t)$ of state $\left | {{\Phi _ + }} \right \rangle$ of CD method: (a) $\Delta$ (red, dashed line), $\Omega$ (blue, solid line), and CD term ${\Omega _c}$ (green, dashed-dotted line), (b) Population $P(t)$ (red, solid line).
3.4 LRIs
We consider an LRI for comparison. Using the concept of invariants [32], first, the dynamics of the system can be designed, by choosing a time-dependent invariant, and subsequently the Hamiltonian elements are inversely obtained by inverting the time-dependent Schrödinger equation. Specifically, for Hamiltonian $H(t)$, there exists an invariant
By solving this equation, the following constraint conditions are obtained:
Using eigenstates ${\left | {{\varphi _n}(t)} \right \rangle }$ of this invariant, the solution of Schrödinger equation $i\hbar \partial \left | {\psi (t)} \right \rangle /\partial t=H(t)\left | {\psi (t)} \right \rangle$ and the propagator [32] can be written as
This yields
To realize population inversion along eigenstate $\left | {{\varphi _ + }(t)} \right \rangle$, $\theta (t)$ can be chosen as
Once $\theta (t)$ and $\beta (t)$ are determined, Rabi frequency $\Omega$ and detuning $\Delta$ leading to rapid population inversion are found using Eq. (16). When an error representing a weak disturbance to the system occurs, the Hamiltonian becomes $H_\delta =H+\delta V$, where $\delta$ is an unknown time-independent parameter representing the error in the description of the model and $V$ is the error Hamiltonian. For example, when $V = \Omega {\sigma _x}/2$, it implies that the system deviates only at Rabi frequency $\Omega$, which is called the Rabi frequency error. In contrast, when $V = H$, it implies that the system deviates at detuning $\Delta$ and Rabi frequency $\Omega$ of the Hamiltonian simultaneously, which is called the systematic error. These errors decrease the accuracy and efficiency of the control of quantum states.
Therefore, we should design different robust schemes against various errors. To this end, we define the control error sensitivity as
where $P({t_f})$ is the transition probability at the final moment. When ${q_s} = 0$ is satisfied, the error sensitivity is zero, suggesting that the optimal LRI shortcuts are robust against such types of errors. The Rabi frequencies and detunings for the optimal LRI against the Rabi frequency error and the systematic error are plotted in Fig. 5(a), and the corresponding population of state, $\left |\Phi _+\right \rangle$, is shown in Fig. 5(b).Fig. 5. Time dependence of parameters $\Delta$, $\Omega$, and population $P(t)$ of state $\left | {{\Phi _ + }} \right \rangle$ of LRI method. (a) n = $-$0.5, $\Delta$ (red, dashed line) and $\Omega$ (blue, solid line); n = 0.125, $\Delta$ (yellow, dashed line) and $\Omega$ (purple, solid line), (b) Population $P(t)$ (red, solid line). It is worth noting that for all n, time-dependent functions of populations are same, because n affects only phases.
3.5 CAP
Finally, we explicitly explain how use a CAP to produce a specific coherent change in the quantum system. For this purpose, we take pulses with the same shapes, widths, and detunings of each pulses, but different phases. For Hamiltonian $H(t)$, its propagator $U$ can be expressed by Cayley–Klein parameters $a$ and $b$,
Thus, the propagator $U$ is
For a composite sequence of $N$ identical pulses, the total propagator is expressed as
where phases ${\phi _k}$ are arbitrary control parameters. For simplicity, the first phase is set as zero because the global phase can be removed. To further reduce the constraint equations, phase sequences having symmetric distributions are applied, i.e., ${\phi _{N + 1 - k}} = {\phi _k}$, where $N$ is the total number of pulses. If the following constraints act the time-dependent variation in Hamiltonian $H$: parameter $a$ of propagator $U$ is real, i.e., $a \in R$. For a three-pulse sequence, only the phase of the second pulse needs to be adjusted. Here, we choose the sequence of phases as $(0,{\phi _2},0)$. From Eqs. (27) and (28), we obtainTo realize a high-fidelity population transfer, an appropriate condition should be set to minimize $U_{11}^{(3)}$. To this end, we should only choose the condition that ${\phi _2} = 2\pi /3$, to ensure that the second term on the right-hand side of Eq. (30) is zero, and ${a^{3}}$ is a three-order small quantity of $a$. In addition, even if the population transfer of a single pulse is not completely reached, i.e., ${U_{11}} \ne 0$, this deviation can be exponentially reduced by a composite technique to achieve perfect transfer. For a five-pulse sequence with phases $(0,{\phi _2},{\phi _3},{\phi _2},0)$, we can similarly obtain
Again, we choose ${\phi _2} = 4\pi /5,{\phi _3} = 2\pi /5$; therefore, $U_{11}^{(5)}$ is only left with the highest-order term of $a$, i.e., $U_{11}^{(5)} = {a^{5}}$. Similarly, we can obtain the phases corresponding to a seven-pulse sequence: ${\phi _2} = 6\pi /7,{\phi _3} = 4\pi /7,$ and ${\phi _4} = 8\pi /7$. In fact, for any CP sequence, there is a general formula for the selection of phases [37]. It is worth noting that the choices of phases are nonunique because the constrained equations are nonlinear functions of phases, from Eqs. (30) and (31). For any set of pulses, if $\{ {\phi _k}\} _2^{N - 1}$ is a solution, then $\{ 2\pi - {\phi _k}\} _2^{N - 1}$ is also a solution. In addition, for a CAP with numerous pulses, there are frequently more than two independent solutions $\{ {\phi _k}\} _2^{N - 1}$ and $\{ 2\pi - {\phi _k}\} _2^{N - 1}$, e.g., $(0,2\pi /5,6\pi /5,2\pi /5,0)$ is also a solution of a five-pulse sequence. For an $N$-pulse sequence, the transition probability can be written as
If $N$ is sufficiently large, the fidelity of the prepared quantum state is infinitely close to $1$. In fact, $N=5$ or $7$ may be sufficient for the requirements of fidelity in quantum information processing.
Here, to realize population inversion, an odd number of pulses must be satisfied. For simplicity, we take a five-pulse sequence as an example. For single pulse, the Rabi frequency and the detuning are a Gaussian pulse and a linear chirp, respectively, i.e.,
Fig. 6. Time dependence of parameters $\Delta$, $\Omega$, and population $P(t)$ of state $\left | {{\Phi _ + }} \right \rangle$ of CAP method. In protocol, we employ five-pulse sequence to construct CAP, in which all pulses have the same Rabi frequencies and detunings. $\Delta$ (red, dashed line) and $\Omega$ (blue, solid line) of single pulse are plotted in (a). (b) Population $P(t)$ of CAP with five-pulse sequence (red, solid line), where duration of each pulse is 6T.
4. Robustness against different experimental errors
Here, we discuss how the effectiveness of population inversion is affected by the Rabi frequency error and the systematic error for the quantum control protocols discussed above.
First, the error originating from the Rabi frequency is considered. For RE, we can write the relationship between the transition probability and the error coefficient analytically as $P({t_f}) = \frac {1}{2}\left [ {1 + \cos (\delta \pi )} \right ]$. For a CD, the errors of both original Hamiltonian $H$ and additional Hamiltonian ${H_c}$ should be considered, i.e., $(\Omega + i{\Omega _c} ) \to (1 + \delta )(\Omega + i{\Omega _c})$. For an LRI, the choice of parameters can be further optimized. When $n=-0.5$, ${q_s} = 0$ is satisfied; thus, the Rabi frequency and detuning for the optimal LRI scheme with respect to the Rabi frequency error are obtained from Eq. (16) (see Fig. 5). In Fig. 7, we compare the accuracies of the five methods by plotting the transition probability as a function of error parameter $\delta$. We find that in the vivinity of $\delta =0$, the RE, CD, LRI and CAP techniques behave very well, with the transition probability being 1. However, as $|\delta |$ increases, their accuracies are affected. The CAP method outperforms the other techniques, followed by the LRI and then the CD method. The RE method is sensitive to the variations in the Rabi frequency. The AP method cannot achieve complete population transfer when $\delta =0$, but it improves as $\delta$ increases. The CAP method maintains the probability at ultrahigh efficiencies above $99.9\%$ over broad ranges of $\delta$, showing its ultrahigh robustness against the Rabi frequency error.
Fig. 7. Transition probability $P({t_f})$ versus Rabi frequency error parameter $\delta$ (dimensionless parameter): RE (black, dashed line), CD (yellow, solid line), LRI (blue, dashed line), AP (green, dashed-dotted line), and CAP (red, solid line).
In the following, the effect of the systematic error on quantum state control is considered. The systematic error of RE is also the Rabi frequency error. The CD method considers both original Hamiltonian $H$ and the error of additional Hamiltonian ${H_c}$, i.e., $(H + {H_c}) \to (1 + \delta )(H + {H_c})$. The optimal LRI scheme with respect to the systematic error is obtained when $n=0.125$, and the Rabi frequency and detuning are also determined using Eq. (16) (see Fig. 5). In Fig. 8, we observe that the effect of the variation in the systematic error is similar to that of the Rabi frequency variation. The efficiencies of the LRI and CD methods show robustness against the systematic error. As prior, the CAP technique outperforms its competitors and features a broad range of high efficiencies.
Fig. 8. Transition probability $P({t_f})$ versus systematic error parameter $\delta$ (dimensionless parameter): RE (black, dashed line), CD (yellow, solid line), LRI (blue, dashed line), AP (green, dashed-dotted line), and CAP (red, solid line).
5. Summary
In summary, we show that CPs can be used to achieve complete and robust quantum state engineering in three-level quantum systems. A CAP combines the robustness of AP with the high fidelity of RE using a sequence of pulses to cause a change in the quantum state. For coherent control, RE and CAP do not depend on specific shapes of pulses; however, RE is sensitive to the pulse area and detuning. Different from the CD and LRI methods, a CAP is constructed by controlling the phases of the pulse sequence; thus, it has more freedom in designing the Hamiltonian parameters. By comparing the sensitivities of the CAP, RE, AP, CD, and LRI techniques to the Rabi frequency and systematic errors, we find that the LRI method needs to be changed corresponding to the Rabi frequency and the detuning to achieve robustness against the different experimental errors. The CAP shows ultrahigh robustness against these errors with a broad range of high efficiencies over $99.9\%$, without the need for varying the Hamiltonian parameters. All these features make a CAP a promising alternative for coherent control of three-level quantum systems.
Funding
Natural Science Foundation of Anhui Province (2008085QA43); National Natural Science Foundation of China (12004006, 12075001, 12175001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data produced by numerical simulations in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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