Reverse-Engineered Exact Control of Population Transfer in Lossy Nonlinear Three-State Systems

Abstract

We introduce a reverse-engineered scheme for achieving the precise control of population transfer in nonlinear quantum systems characterized by a 1:2 resonance. This scheme involves the use of two resonant laser pulses that transition from initial and final states to an intermediate level exhibiting irreversible losses. In comparison to alternative techniques, our approach offers computational efficiency advantages. Notably, the analytically defined form of the pump pulse enables tailored control strategies, enhancing robustness against decoherence and imperfections. This flexibility extends to choosing dump pulses and designing time evolution scenarios. These features open doors for practical implementation and scalability in quantum technologies, with potential applications in quantum information processing, quantum computing, and quantum communication.

1. Introduction

Effectively manipulating the distribution of particles among their quantum states is a cornerstone of current quantum science and technology research, with continued exploration in this area holding the promise of breakthroughs in various fields [1,2]. This makes it a hot topic with significant ongoing efforts (see, e.g., [3,4,5,6,7,8,9,10] and references therein).

In this paper, we present a novel exact scheme for controlling population transfer in a lossy three-state nonlinear quantum system characterized by a 1:2 resonance through reverse engineering. By reverse engineering, we mean that we start from the desired population dynamics and work backward to determine the appropriate form of the pump pulse, rather than prescribing both the pump and dump pulses from the outset. The technique offers a computationally efficient approach for manipulating quantum systems, allowing tailored control strategies without requiring the same level of computational resources as some alternative methods.

Schematically, achieving population transfer from the initial state to the desired final state involves leveraging an intermediate state, typically characterized by irreversible losses. In this context, “lossy” refers to the assumption that the population in the intermediate state is subject to decay, which can significantly degrade the fidelity of the transfer. By minimizing the occupation of the intermediate state, our scheme enhances overall efficiency and accuracy. Two laser fields are used, each of which resonates with one of the transitions to the intermediate state. Remarkably, the time dynamics of the probability amplitude of the final state can be arbitrarily prescribed. This process exhibits robustness, as the form of the dump pulse can also be freely chosen.

The key control parameter lies in the analytically definable form of the pump pulse. Additionally, it is noteworthy that the freedom in choosing the dump pulse and time evolution scenario for the final state allows for suppressing, via appropriate choice of these functions, the effects of decoherence, experimental imperfections, and non-adiabatic effects, ultimately leading to a higher fidelity process. While minimizing losses in the cintermediate state is a key challenge, the system’s nonlinear nature introduces additional complexity that must also be addressed. Since the system is nonlinear, being characterized by a 1:2 resonance, interactions between states are not proportional to the applied fields. As a result, small changes in control parameters can significantly impact population dynamics. Our reverse-engineered approach handles this nonlinearity effectively, allowing precise control over the population transfer.

Notably, we achieve near-complete population transfer to the target state while minimizing population in the lossy intermediate level through a counterintuitive pulse sequence similar to the case of the STIRAP process. In our approach, the dump pulse is applied before the pump pulse, as in STIRAP, which allows us to effectively suppress the population in the lossy intermediate state. This counterintuitive sequencing is crucial for enhancing fidelity, especially in systems with significant decay.

Beyond the immediate applications in quantum control, this proposed scheme holds the potential for advancements in quantum information processing, quantum computing, and quantum communication. The ability to tailor control strategies with reduced computational demands opens avenues for practical implementation and scalability in quantum technologies. Moreover, the robustness in the face of various challenges positions this scheme as a promising candidate for real-world quantum applications.

2. Derivations

Consider the radiation–matter interaction scheme depicted in Figure 1 (a lambda scheme, though the scheme can also be of a ladder type). Such a scheme is encountered, for example, in the photo- or magneto-association of cold atoms in degenerate quantum gases into diatomic molecules [11,12,13]. Other examples include photonic systems, quantum dots, and superconducting qubits utilized in diverse fields such as quantum optics (“chi-two” process), quantum information processing, and quantum technology [14,15,16].

Thus, we examine a three-state nonlinear quantum system subject to excitation by two monochromatic laser fields: one resonant, with the transition |1⟩→|2⟩, and the other with the transition |2⟩→|3⟩. The direct transition from the first level to the final third one is forbidden, and the intermediate level 2 decays with a rate of $\Gamma$. The nonlinearity in the system resembles that of the 1:2 resonance between the quantum states. As quantum mechanics is generally linear, there are several mechanisms through which nonlinearity can arise in quantum systems [17,18,19,20]. One well-known example is the Gross–Pitaevskii equation, where nonlinearity emerges from the reduction of the linear many-body problem into a mean-field theory, as seen in the theoretical description of Bose–Einstein condensates.

The 1:2 resonance creates a nonlinear relationship between the population dynamics and the applied fields, even though the two laser fields pump different transitions in the Lambda scheme. The interaction with the intermediate state and the decay further contribute to the nonlinear behavior of the system.

In the rotating-wave approximation, the time-dependent Schrödinger equations that describe the temporal dynamics of such an interaction in the case of one- and two-photon resonances are written as follows:

$$iħ{\displaystyle \frac{d{c}_1}{dt}}=P{c}_2{c}_1^{*},$$

(1)

$$iħ{\displaystyle \frac{d{c}_2}{dt}}=-i\Gamma c_2+P{c}_1^2+S{c}_3,$$

(2)

$$iħ{\displaystyle \frac{d{c}_3}{dt}}=S{c}_2,$$

(3)

where $c_{1,2,3}$ are the probability amplitudes of the levels indicated by the corresponding indices, $c_1^{*}$ denotes the complex conjugate of c₁, and $ħ$ is the reduced Planck constant. The functions P = P (t) (pump) and S = S (t) (dump) are the time-dependent Rabi frequencies of the exciting optical fields. Specifically, P acts on the transition from state |1⟩ to state |2⟩, while S drives the transition from state |2⟩ to state |3⟩. In this analysis, we use dimensionless time for all time-dependent variables. This approach allows us to present the population transfer dynamics in a generalized form, independent of specific system parameters, such as frequency or time units.

The system comprising (1)–(3), which in the lossless case with $\Gamma$ = 0 assumes normalization ${\left|c_1\right|}^2+{\left|c_2\right|}^2+{\left|c_3\right|}^2=1$ can be generated by the following Hamiltonian:

$$H={\displaystyle \frac{P}{2}}\left({c_1}^2{c_2}^{†}+c_2{(c_1^{†})}^2\right)+S\left(c_2{c_3}^{†}+c_3{c_2}^{†}\right)-iħ\Gamma c_2^{†}{c}_2=H_0-iħ\Gamma c_2^{†}{c}_2$$

(4)

Here, $H_0$ is the Hermitian Hamiltonian describing the 1:2 resonance from state 1 to state 2, and then the transition to state 3. The term $-iħ\Gamma c_2^{†}{c}_2$ is introduced phenomenologically to simulate the decay of excited state 2 (see, e.g., [17,18,19,20]). This Hamiltonian produces operator analogues of Equations (1)–(3) if the Heisenberg equations $iħd{c}_i/dt=[c_i,H]$ are applied. After deriving the operator equations in the Heisenberg picture, we proceed by treating the operators as c-numbers, as performed, e.g., in Ref. [18] for the case of mean-field equations for the collective two-color photoassociation of a freely interacting gas. This transition allows us to describe the system using probability amplitudes $c_{1,2,3}$ in the Schrödinger picture, where these amplitudes represent the evolution of state probabilities. Note that to meet the applied normalization, one should change $c_1\to \sqrt{2}{c}_1$ in the derived equations.

Since amplitude $a_1$ can be chosen to be real (so that $c_3$ is also real, and $c_2$ is imaginary), it is convenient to rewrite system (1) as follows:

$${\displaystyle \frac{iħ}{2}}{\displaystyle \frac{d{c}_1^2}{dt}}=P{c}_1^2{c}_2$$

(5)

$$c_1^2={\displaystyle \frac{1}{P}}\left(iħ{\displaystyle \frac{d{c}_2}{dt}}+i\Gamma c_2-S{c}_3\right),$$

(6)

$$c_2={\displaystyle \frac{iħ}{S}}{\displaystyle \frac{d{c}_3}{dt}}$$

(7)

Substituting Equations (6) and (7) into Equation (5), we obtain a real equation for $c_3$ [21]:

$${\displaystyle \frac{d}{dt}}\ln \left({ħ}^2{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{c_3^{\prime }}{S}}\right)+ħ\Gamma {\displaystyle \frac{c_3^{\prime }}{S}}+S{c}_3\right)={\displaystyle \frac{P^{\prime }}{P}}+{\displaystyle \frac{2c_3^{\prime }}{S}}P,$$

(8)

where the prime denotes differentiation with respect to time.

In the context of our analysis, an important point to note here is that when we consider an equation with respect to $P(t)$, this presents a first-order ordinary differential equation, which is known as the Bernoulli equation [22,23]:

$$P^{\prime }+AP+B{P}^2=0$$

(9)

where

$$A\left(t\right)={\displaystyle \frac{1}{S}}\left({ħ}^2{c}_3^{″}+\left(ħ\Gamma -S^{\prime }/S\right)c_3^{\prime }+S^2{c}_3\right), B\left(t\right)={\displaystyle \frac{2a_3^{\prime }}{S}}$$

(10)

This point presents the main step of our approach as compared to most of the alternative approaches applied in the past. Indeed, in most research completed previously, when discussing Equation (8) (or, generally, the governing system (1)–(3)), researchers have typically defined particular forms for the functions $P(t)$ and $S\left(t\right)$, developed the solution for the probability amplitudes (either analytically or numerically), and further attempted to optimize the outcomes within the parameter space involved in $P(t)$ and $S\left(t\right)$, (see, for example, [24,25,26,27]). In contrast, we consider the pump $P(t)$ as an unknown function while assuming the desired form of $c_3\left(t\right)$ (as well as the dump $S\left(t\right)$) as given. We note that for a two-level case, an approach somewhat akin to ours in ideology has been applied in [28,29].

The integration of Equation (9) is straightforward, yielding an exact result [22,23]:

$$P\left(t\right)={\displaystyle \frac{A\left(t\right)}{C_0+\int A\left(\tau \right)B\left(\tau \right)d\tau }}.$$

(11)

Here, the integration constant $C_0$ should be determined from the initial conditions, using Equations (6) and (7). Thus, the pump pulse can be precisely defined for any combination of the dump pulse form $S\left(t\right)$ and the time evolution function $c_3\left(t\right)$ of the target state’s probability amplitude. Finally, we note that in a similar manner, one can track $c_1$ by adjusting the Rabi frequency $S\left(t\right)$ through the corresponding equation for $c_1$.

Example 1.

For

$$c_3={\displaystyle \frac{a}{2}}\left(1+\tanh \left(t\right)\right), S=S_0\mathrm{sech}\left(t\right),$$

(12)

where $a$ is an arbitrarily variable parameter, we obtain [21]

$$C_0=-{\displaystyle \frac{2S_0}{a}}+a\left(S_0+{\displaystyle \frac{\Gamma }{S_0}}\right),$$

(13)

$$P={\displaystyle \frac{2a{S}_0\left(S_0^2{\mathrm{e}}^t-\sinh \left(t\right)+\Gamma \cosh \left(t\right)\right)}{S_0^2\left(a^2\sinh \left(2t\right)+\left(a^2-2\right)\cosh \left(2t\right)-2\right)+a^2\left(1+\Gamma +\Gamma {\mathrm{e}}^{2t}\right)}}.$$

(14)

The pulse shapes and the occupation probabilities p_k = |c_k|² and $k=1, 2, 3$ for parameters $S_0=1.5$, $\Gamma =0.4$, $a=-0.85$, $ħ=1$ are shown in Figure 2.

We note that configuration (12)–(14) is capable of providing almost-complete population transfer to the third level, achieved by setting $a\approx -1$. For a given decay rate of $\Gamma$, one should then choose a sufficiently large value of the parameter $S_0$ to avoid the denominator of the function $P(t)$ from vanishing at a certain point in time (see Equation (14)). This regime is demonstrated in Figure 3 for $a=-0.99$ and $\Gamma =0.3$. We note that the population of the lossy intermediate level |2⟩ remains almost zero throughout the entire process, and the pulse sequence in which the dump pulse $S\left(t\right)$ precedes the pump pulse $P(t)$ is counterintuitive, as in the case of the STIRAP process [14,15,16].

Example 2.

In the case when (compare this with (12))

$$c_3={\displaystyle \frac{a}{2}}\left(1+\tanh \left(t\right)+{\mathrm{sech}}^2\left(t\right)\right)\mathrm{and} S=S_0\mathrm{sech}\left(t\right),$$

(15)

the integration constant is found to be [21]

$$C_0={\displaystyle \frac{a^2}{6}}\left(3+{\displaystyle \frac{7\Gamma }{S_0^2}}\right)-1,$$

(16)

and the resulting occupation probabilities are shown in Figure 4.

3. Conclusions

This paper introduces an exact scheme for controlling population transfer in a lossy three-state nonlinear quantum system through reverse engineering. The assumed nonlinearity is local and resembles a 1:2 resonance. The key development lies in the precise definition of the pump pulse, enabling tailored control strategies with reduced computational demands compared to alternative methods.

The derivations section details the radiation–matter interaction scheme, providing a clear foundation. The time-dependent Schrödinger equations derived in the rotating-wave approximation offer a comprehensive description of the system’s behavior under the influence of two monochromatic laser fields. The re-parametrization of the system in terms of real amplitudes simplifies the equations, facilitating a more manageable and insightful analysis. The integration of the first-order differential equation for the probability amplitude of the final state underscores the effectiveness of the proposed scheme. The obtained result provides a means to precisely define the pump pulse for arbitrary combinations of the dump pulse form and the time evolution function of the target state’s probability amplitude.

The robustness in the face of challenges, such as decoherence, experimental imperfections, and non-adiabatic effects, positions this scheme as a promising candidate for practical implementation in real-world quantum technologies, such as quantum information processing, quantum computing, and quantum communication. Two illustrative examples showcase the dynamics of the system under the proposed scheme, providing insight into the occupation probabilities for different configurations.

In conclusion, the paper contributes to the field of quantum control by presenting a novel reverse-engineered approach to population transfer in three-state nonlinear quantum systems. The presented exact scheme, along with its potential applications and robustness, opens new avenues for advancements in quantum science and technology. Notably, the achieved high-fidelity population transfer to the target state while minimizing the population in the lossy intermediate level through a counterintuitive pulse sequence, similar to the case of the STIRAP process, demonstrates the potential of this approach for practical applications.