# Connection between Inverse Engineering and Optimal Control in Shortcuts to Adiabaticity

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## Abstract

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## 1. Introduction

## 2. Fast Cooling in Time-Varying Harmonic Traps

#### 2.1. Model, Hamiltonian, and the Inverse Engineering Approach

#### 2.2. Optimal Control Theory

#### 2.2.1. Time-Optimal Solution

#### 2.2.2. Time-Averaged Energy Minimization

#### 2.3. Comparison between IE and OCT

## 3. Fast Transport of Atoms in Moving Harmonic Traps

#### 3.1. Classical and Quantum Inverse-Engineered Solutions

#### 3.2. Optimal Control Theory

#### 3.2.1. Time Minimization

#### 3.2.2. Mean Potential Energy Minimization

#### 3.3. Comparison between IE and OCT

#### 3.3.1. IE with Polynomial Ansatzs

#### 3.3.2. IE with Hyperbolic Ansatz

## 4. Spin Dynamics in the Presence of Dissipation

#### 4.1. Energy Minimization by OCT

#### 4.2. Case I: Reaching the Horizontal Plane of the Bloch Sphere

#### 4.3. Case II: Spin Flip

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

STA | shortcuts to adiabaticity |

IE | inverse engineering |

OCT | optimal control theory |

## References

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**Figure 1.**Fast cooling in time-varying harmonic traps: The 3-jump “bang-bang” control function, $u\left(t\right)={\omega}^{2}\left(t\right)/{\omega}_{0}^{2}$.

**Figure 2.**Fast cooling in time-varying harmonic traps: 2D color plot of the final normalized time ${t}_{f}{\omega}_{0}$ for a 3-jump “bang-bang” control as a function of the first pulse amplitude ${\omega}_{1}/{\omega}_{0}$, and the second pulse amplitude ${\omega}_{2}/{\omega}_{0}$.

**Figure 3.**Fast cooling in time-varying harmonic traps: (

**a**) Example of time-optimal trajectory of $b\left(t\right)$ from Equation (14). Parameters: ${\omega}_{f}^{2}={\omega}_{0}^{2}/5$. (

**b**) The time-averaged energy as a function of the normalized final time ${s}_{f}=\pi \gamma /2$, obtained from the time-optimal control solution.

**Figure 4.**Fast cooling in time-varying harmonic traps: Comparison of time-averaged potential energy for different optimal protocols: (1) energy-minimization (blue dashed line), (2) time-optimal protocol (with ${\omega}_{f}$ fixed, $\overline{{E}_{p}}$ constant) (red point); and inversed-engineered protocols: (1) 0-freedom polynomial in Equation (7) (black dotted line), (2) polynomial IE solution with two free parameters optimized for a given normalized final time (stars) (see Table 1): ${s}_{f}=1.1$ (green line), ${s}_{f}=\pi \gamma /2$ (red solid line), and ${s}_{f}=4$ (orange solid line). Parameters: ${\omega}_{f}^{2}={\omega}_{0}^{2}/5$.

**Figure 5.**Fast cooling in time-varying harmonic traps: Comparison of time-dependent normalized variable $b\left(\tau \right)$ obtained from optimal control theory (averaged energy optimization) (blue dashed line) and from inverse engineered solutions optimized to minimize the time-averaged energy (red solid line) for different final times (

**a**) ${s}_{f}=1.1$, (

**b**) ${s}_{f}=\pi \gamma /2$, and (

**c**) ${s}_{f}=4$. The corresponding optimal values of polynomial functions are detailed in Table 1. Parameters: ${\omega}_{f}^{2}={\omega}_{0}^{2}/5$.

**Figure 6.**Fast transport of atoms in a moving harmonic trap: Comparison of the trajectories of (

**a**) the center of mass $x(t/{t}_{f})/d$ and (

**b**) the trap center ${x}_{0}(t/{t}_{f})/d$, obtained from the OCT formalism by minimizing the time-averaged potential energy (blue dashed line) and using the IE approach (red solid line) based on a fifth-order polynomial ansatz. Parameters: ${\omega}_{0}=2\pi \times 50$ Hz and ${t}_{f}=22$ ms.

**Figure 7.**Fast transport of atoms in a moving harmonic trap: Time-averaged potential energy $\overline{{E}_{p}}/\epsilon $ (normalized to $\epsilon =m{\omega}_{0}^{2}{d}^{2}/2$) as a function of final time ${t}_{f}$ by using different protocols: time-optimal (orange dash-dotted line), energy-minimization with unbounded constraint (blue dashed line), and IE approaches with a fifth-order polynomial (black upper solid line), a seventh-order polynomial (purple solid line), and nineteenth-order polynomial (red lower solid line). Same parameters as Figure 6.

**Figure 8.**Fast transport of atoms in a moving harmonic trap: Comparison of trajectories of mass of center (

**a**) and trap center (

**b**), calculated from the OCT formalism (blue dashed line) and the IE approach (red solid line) with a 19th order polynomial ansatz. Same parameters as Figure 6.

**Figure 9.**Fast transport of atoms in a moving harmonic trap: comparison of trajectories of mass of center (

**a**) and trap center (

**b**), calculated from the OCT formalism (blue dashed line) and the IE approach with the optimized hyperbolic-function protocol in Equation (41) (red solid line). The “magic” values are ${a}_{1}=1.2$, ${a}_{2}=1.25$, and the other parameters are the same as those in Figure 6.

**Figure 10.**Fast transport of atoms in a moving harmonic trap: Time-averaged potential energy $\overline{{E}_{p}}\left({t}_{f}\right)/\epsilon $ (normalized to $\epsilon =m{\omega}_{0}^{2}{d}^{2}/2$) calculated from different protocols: time-optimal (orange dash-dotted line), energy-minimization with unbounded constraint (blue dashed line), and IE approach based on a hyperbolic-function-ansatz by choosing the “magic” values ${a}_{1}=1.2$ and ${a}_{2}=1.25$ (marked red point). Same parameters as Figure 7.

**Figure 11.**Spin dynamics in the presence of dissipation: Equivalent magnetic field ($B,{B}_{c}$) of transverse magnetic field ${B}_{\perp}$.

**Figure 12.**Spin dynamics in the presence of dissipation: Energy as a function of ${a}_{f}=ln{r}_{f}$ for the same target state $({\theta}_{f},{r}_{f},{t}_{f})=(\pi /2,0.6,3.6357955)$. We compare the results obtained from the optimal control theory (red star) with the inverse engineering results involving two different polynomial ansatz fulfilling the boundary conditions. The energy curve are plotted for different values of the polynomial coefficient ${a}_{1}$: (

**a**) for a second order polynomial ansatz and (

**b**) for a third order polynomial ansatz with ${a}_{3}=0.1$. The inset in (

**b**) shows the proximity of the inverse engineering result with that of the optimal control theory.

**Figure 13.**Spin dynamics in the presence of dissipation: For a $\pi /2$ rotation, we plot (

**a**) the magnetic field $B\left(t\right)$ and (

**b**) the corresponding variable $\theta \left(t\right)$ obtained from an inverse engineering technique based on an optimized third-order polynomial (red solid line) and from the optimal control theory formalism associated to a mean energy minimization (blue dashed line).

**Figure 14.**Spin dynamics in the presence of dissipation: Time evolution of the spin components ${S}_{z}\left(t\right)$ (red solid line), ${S}_{x}\left(t\right)$ (blue dashed line), and ${S}_{y}\left(t\right)$ (black dotted line) under the magnetic field obtained from the inverse engineering method. Same parameters as Figure 13. The inset depicts the corresponding spin trajectory on the Bloch sphere.

**Figure 15.**Spin dynamics in the presence of dissipation: (

**a**) The magnetic field $B\left(t\right)$ and (

**b**) the variable $\theta \left(t\right)$ as a function of time for a minimal-energy spin flip. An optimal ninth-order polynomial has been used for $\theta \left(t\right)$ to apply the inverse engineering method (red solid line). The optimal solution is plotted as a blue dashed line.

**Figure 16.**Spin dynamics in the presence of dissipation: In the case of spin flip (

**b**) obtained with magnetic field (

**a**), compared with OCT (blue dashed line), an tanh ansatz (instead of a polynomial) used in IE approach (red solid line) is chosen to reduce energy to $E=4.028$, with “magic” values ${a}_{5}=1.1$ and ${a}_{1}=3.104678$.

**Table 1.**Optimal values of the free parameters ${a}_{2}$ and ${a}_{3}$ in the three-order polynomial ansatz for the IE protocol that minimize the time-averaged energy. Parameter ${\omega}_{f}^{2}={\omega}_{0}^{2}/5$.

${\mathit{s}}_{\mathit{f}}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ |
---|---|---|

1.1 | −0.44893 | 0.10996 |

$\pi \gamma /2$ | −1.47741 | 0.34535 |

4 | −2.86194 | 0.62841 |

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Zhang, Q.; Chen, X.; Guéry-Odelin, D.
Connection between Inverse Engineering and Optimal Control in Shortcuts to Adiabaticity. *Entropy* **2021**, *23*, 84.
https://doi.org/10.3390/e23010084

**AMA Style**

Zhang Q, Chen X, Guéry-Odelin D.
Connection between Inverse Engineering and Optimal Control in Shortcuts to Adiabaticity. *Entropy*. 2021; 23(1):84.
https://doi.org/10.3390/e23010084

**Chicago/Turabian Style**

Zhang, Qi, Xi Chen, and David Guéry-Odelin.
2021. "Connection between Inverse Engineering and Optimal Control in Shortcuts to Adiabaticity" *Entropy* 23, no. 1: 84.
https://doi.org/10.3390/e23010084