# Driving Interactions Efficiently in a Composite Few-Body System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Shortcut to Adiabaticity

## 4. Three Identical Particles

## 5. Driving in the Presence of Weak Fixed Interactions

## 6. Driving in the Presence of Strong Fixed Interactions

## 7. Conclusions & Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Accuracy of the Ansatz

**Figure A1.**Fidelity between instantaneous ground state obtained with exact diagonalization of the Hamiltonian and the interpolatory ansatz as a function of $g/{g}_{f}$. Panel (

**a**) is the fidelity for three identical particles, while panel (

**b**) shows the fidelity for system and impurity driving. The final interactions chosen are ${g}_{f}=1$ (blue lines), ${g}_{f}=5$ (red lines), and ${g}_{f}=40$ (black lines).

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**Figure 1.**(

**a**) Interaction potentials stemming from ${g}^{A}$ (dashed line) and ${g}^{AB}$ (solid lines) in the Jacobi coordinate plane. (

**b**) Examples of different interaction ramps when keeping ${g}^{A}\left(t\right)={g}^{AB}\left(t\right)$, with the reference ramp (blue dotted line) and the shortcut to adiabaticity (STA) ramp at ${t}_{f}=1.5$ (black solid line) and ${t}_{f}=10$ (red solid line).

**Figure 2.**(

**a**) Initial state in the relative $\{X,Y\}$ coordinate plane. Target states at (

**b**) ${g}_{f}=1$, (

**c**) ${g}_{f}=5$, and (

**d**) ${g}_{f}=40$. (

**e**) $\langle {W}_{irr}\rangle $ for three indistinguishable particles as a function of the ramp time ${t}_{f}$ for the STA (solid lines) and reference ramp (dotted lines). The final interactions are ${g}_{f}=1$ (blue lines), ${g}_{f}=5$ (red lines), and ${g}_{f}=40$ (black lines). Inset shows $\langle {W}_{irr}\rangle $ versus final interaction strength ${g}_{f}$ at ${t}_{f}=10$.

**Figure 3.**Target one-body density matrices (OBDMs, (white contour lines) on top of final OBDMs for three identical particles at ${t}_{f}=10$. Panels (

**a**–

**c**) correspond to the STA, while panels (

**d**–

**f**) correspond to the reference pulse. Panels (

**a**) and (

**d**) are for ${g}_{f}=1$, (

**b**) and (

**e**) for ${g}_{f}=5$, and (

**c**) and (

**f**) for ${g}_{f}=40$.

**Figure 4.**(

**a**) Initial state with ${g}^{AB}=1$ and ${g}_{i}^{A}=0$ and (

**b**–

**d**) target states for ${g}_{f}^{A}=\{1,5,40\}$. This case is referred to as system driving. (

**e**) Initial state with ${g}^{A}=1$ and ${g}_{i}^{AB}=0$ and (

**f**–

**h**) target states for ${g}_{f}^{AB}=\{1,5,40\}$. This case is referred to as impurity driving.

**Figure 5.**(

**a**) $\langle {W}_{irr}\rangle $ after driving the system interactions in the presence of a weak fixed impurity interaction ${g}^{AB}=1$. System interactions are driven to ${g}_{f}^{A}=1$ (blue lines), ${g}_{f}^{A}=5$ (red lines), and ${g}_{f}^{A}=40$ (black lines), with the solid lines showing the result of the STA and the dotted lines showing the result of the reference ramp. (

**b**) Impurity driving in the presence of weak fixed system interactions ${g}_{A}=1$, with final impurity interactions ${g}_{f}^{AB}=1$ (blue lines), ${g}_{f}^{AB}=5$ (red lines), and ${g}_{f}^{AB}=40$ (black lines). Insets show $\langle {W}_{irr}\rangle $ as a function of ${g}_{f}^{A}$ and ${g}_{f}^{AB}$, respectively, at ${t}_{f}=10$.

**Figure 6.**Panels (

**a**–

**f**): Target states (white contour lines) on top of final states (at ${t}_{f}=10$) for driving system interactions between A atoms in the presence of a fixed impurity interaction ${g}^{AB}=1$. Panels with index ($-1$) correspond to ${\rho}^{A}({x}_{1},{x}_{1}^{\prime})={\rho}^{A}({x}_{2},{x}_{2}^{\prime})$ while panels with index ($-2$) show ${\rho}^{B}({x}_{3},{x}_{3}^{\prime})$. Panels (

**a**–

**c**) show the STA final states while panels (

**d**–

**f**) show the final states for the reference pulse, where panels (

**a**) and (

**d**) are for ${g}_{f}^{A}=1$, (

**b**) and (

**e**) are for ${g}_{f}^{A}=5$, and (

**c**) and (

**f**) are for ${g}_{f}^{A}=40$. Panels (

**g**–

**l**): Target states (white contour lines) on top of final states (at ${t}_{f}=10$) for driving impurity interactions in the presence of a fixed interactions between the A atoms ${g}^{A}=1$. Panels (

**g**) and (

**j**) are for ${g}_{f}^{AB}=1$, (

**h**) and (

**k**) are for ${g}_{f}^{AB}=5$, and (

**i**) and (

**l**) are for ${g}_{f}^{AB}=40$.

**Figure 7.**Left panel: driving the interactions of the system when the impurity interaction is fixed at ${g}^{AB}=20$. (

**a**) Initial state with ${g}^{A}=0$, (

**b**) target state at ${g}_{f}^{A}=1$, (

**c**) ${g}_{f}^{A}=5$, and (

**d**) ${g}_{f}^{A}=40$. Right panel: driving the interaction with the impurity when the system interactions are fixed at ${g}^{A}=20$. (

**e**) Initial state with ${g}^{AB}=0$, (

**f**) target state at ${g}_{f}^{AB}=1$, (

**g**) ${g}_{f}^{AB}=5$, and (

**h**) ${g}_{f}^{AB}=40$.

**Figure 8.**(

**a**) $\langle {W}_{irr}\rangle $ after driving the system interactions in the presence of a strong fixed impurity interaction ${g}^{AB}=20$. System interactions are driven to ${g}_{f}^{A}=1$ (blue lines), ${g}_{f}^{A}=5$ (red lines), and ${g}_{f}^{A}=40$ (black lines), with the solid lines showing the result of the STA and the dotted lines showing the result of the reference ramp. (

**b**) Impurity driving in the presence of strong fixed system interactions ${g}^{A}=20$, with final impurity interactions ${g}_{f}^{AB}=1$ (blue lines), ${g}_{f}^{AB}=5$ (red lines), and ${g}_{f}^{AB}=40$ (black lines). Insets show $\langle {W}_{irr}\rangle $ as a function of ${g}_{f}^{A}$ and ${g}_{f}^{AB}$, respectively, at ${t}_{f}=10$.

**Figure 9.**Panels (

**a**–

**f**): Target states (white contour lines) on top of final states (at ${t}_{f}=10$) for driving system interactions between A atoms in the presence of a fixed impurity interaction ${g}^{AB}=20$. Panels with index ($-1$) correspond to ${\rho}^{A}({x}_{1},{x}_{1}^{\prime})={\rho}^{A}({x}_{2},{x}_{2}^{\prime})$, while panels with index ($-2$) show ${\rho}^{B}({x}_{3},{x}_{3}^{\prime})$. Panels (

**a**–

**c**) show the STA final states, while panels (

**d**–

**f**) show the final states for the reference pulse, where panels (

**a**) and (

**d**) are for ${g}_{f}^{A}=1$, (

**b**) and (

**e**) are for ${g}_{f}^{A}=5$, and (

**c**) and (

**f**) are for ${g}_{f}^{A}=40$. Panels (

**g**–

**l**): Target states (white contour lines) on top of final states (at ${t}_{f}=10$) for driving impurity interactions in the presence of a fixed interactions between the A atoms ${g}^{A}=20$. Panels (

**g**) and (

**j**) are for ${g}_{f}^{AB}=1$, (

**h**) and (

**k**) are for ${g}_{f}^{AB}=5$, and (

**i**) and (

**l**) are for ${g}_{f}^{AB}=40$.

**Figure 10.**(

**a,d**) Kinetic energy, (

**b,e**) potential trap energy, and (

**c,f**) interaction energy as a function of ${t}_{f}$ after using the STA (black solid line) and reference (orange solid line), with the adiabatic energies shown as the thin dotted line. (

**a–c**) show the result of driving the system to ${g}_{f}^{A}=40$, while (

**d–f**) show the result of driving the impurity to ${g}_{f}^{AB}=40$. Note the different scales on the subplots.

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## Share and Cite

**MDPI and ACS Style**

Kahan, A.; Fogarty, T.; Li, J.; Busch, T.
Driving Interactions Efficiently in a Composite Few-Body System. *Universe* **2019**, *5*, 207.
https://doi.org/10.3390/universe5100207

**AMA Style**

Kahan A, Fogarty T, Li J, Busch T.
Driving Interactions Efficiently in a Composite Few-Body System. *Universe*. 2019; 5(10):207.
https://doi.org/10.3390/universe5100207

**Chicago/Turabian Style**

Kahan, Alan, Thomás Fogarty, Jing Li, and Thomas Busch.
2019. "Driving Interactions Efficiently in a Composite Few-Body System" *Universe* 5, no. 10: 207.
https://doi.org/10.3390/universe5100207