# Nanoparticle Interferometer by Throw and Catch

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{2}particles and that this design can be extended to even ${10}^{8}$ amu particles by using flight times below the typical Talbot time of the system.

## 1. Introduction

_{60}) were shown [16] in the far-field and shortly later in the molecular TLI scheme as well [17].

_{2}nanoparticles using a ‘throw and catch’ design. Silica nanoparticles have large polarisabilities, which opens the opportunity for manipulation and, indeed, cooling to the motional ground state by optical techniques beside others in the emerging field of levitated mechanics [20]. The interferometer approach will be heavily based on the proposal suggested by Bateman et al. in 2014 [21], with the throw and catch scheme intended to alleviate the practical issues of the original proposal, such as inefficient reloading. We will show the expected interference patterns produced whilst using realistic experimental parameters for both ${10}^{6}$ amu and ${10}^{8}$ amu particles including all major sources of decoherence.

## 2. Experimental Setup

^{−1}and a positional uncertainty of less than 1 nm for a ${10}^{8}$ amu particle. The particle in our trap, hence, acts as a coherent source of matter waves1.

## 3. Theoretical Model

#### 3.1. Background

#### 3.2. Accounting for Decoherence and Particle Size

## 4. Practical Considerations

^{−1}, then we can expect the same speed as it returns. The challenge now is to restart the laser at the right time in order to catch and re-cool the particle. The trapping of a particle can be understood as the balancing of two forces, a gradient force and a scattering force. The laser’s electric field polarises the particle, and the gradient force pulls the dipole towards the region where the electric field is highest4, which is the centre of the focal spot. The scattering force arises due to the presence of the particle in the light field, which modifies the latter’s energy flow [30]. This force pushes the particle away from the centre of the trap. In Figure 3, we show that a good time to re-start the trapping laser is when the particle is in region III. There, the gradient and scattering forces act in the direction opposite the direction of travel of the particle, and their contributions add to the greatest achievable deceleration. This region gives us approximately 2 µm to stop the particle. Region II can be thought of as the acceptable uncertainty in the arrival of the pulse that triggers the laser to turn back on. The particle falling at around 1 ms

^{−1}travels the 145 nm of region II in 145 ns. If the laser is turned on at ${t}_{0}\pm 145$ ns, where ${t}_{0}$ is the time at which the particle passes trough the focal plane ($z=0$), then the particle will be decelerated in the fastest time. Region I should be avoided since, in this region, the gradient force further accelerates the particle in the direction of gravity. We see that we now also have a relatively strict timing requirement to recapture the particle, as well as a strict cooling requirement. If the trap is turned on too soon, the gradient force will accelerate the particle too much for recapture to be possible, and if the trap is turned on too late, the gradient force will be too weak to stop the particle.

^{−1}particle, we would need a laser power of $1.5$ W, whilst our trap typically operates at 100 mW. We must note that only the scattering force is dissipative and actually takes away energy from the falling particle. The gradient force will spring the particle back upwards after the latter was brought to a halt. So, in order to properly brake the particle, one would need to modulate the laser power, in synchronisation with the motion of the particle, starting from $1.5$ W down to the 100 mW stationary trapping power, until the latter is brought to the stationary regime, where the regular feedback cooling protocol can be applied.

## 5. Results

#### 5.1. Expected Interference Patterns

#### 5.2. Quantum/Classical Distinctions

#### 5.3. Throw- and Catch-Specific Decoherence

#### 5.4. Experimental Progress

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Incoherent Sources

_{2}particle for every run of our interferometer, we performed simulations that allowed the mass of the particle used for each run to vary. This would roughly simulate the situation where a new particle would be reloaded for each run. Figure A1 shows the results of these simulations when particles of a mean mass of ${10}^{6}$ amu were used, with other parameters matching those listed in Table 1. Table A1 shows that the visibility of the fringes, that we expect to see in the quantum interference pattern, decreased as the spread of masses that the particles can take increased. In Figure A1, we also see that, if the spread in the mass of the particles exceeded $30\%$, then the maximal intensity regions of the quantum and classically predicted interference patterns would overlap, leading to additional difficulty in distinguishing between the two cases. Therefore, we see that re-using the same particle can lead to drastic increases in visibility depending on the variation in mass of the different particles that would be used otherwise. It is also important to note that these simulations only took into account a change in mass of the particle. In reality, different particles could have different shapes or densities depending on the purity of the batch that is being used. This would lead to even further washing out of the fringes. We also note that these visibility drops could be partially avoided, when not re-using the same particle, by post-selecting data where only similar particles were used. However, this would lead to more runs being necessary, where, again, the problem of inefficient re-loading arises.

**Figure A1.**Expected interference patterns for the interferometry of a SiO

_{2}particle, with a mean mass of ${10}^{6}$ amu, when accounting for a spread of different particle masses being used for each run of the experiment. Errors in mass range from 0 to 50%. (

**a**) shows the expected patterns for the classical case, whilst (

**b**) shows the quantum case.

**Table A1.**Table of visibility values for the plots presented in Figure A1.

Mass Error | Classical Fringe Visibility | Quantum Fringe Visibility |
---|---|---|

0% | 77.3% | 98.2% |

10% | 75.6% | 95.1% |

20% | 72.4% | 88.3% |

30% | 66.0% | 81.1% |

40% | 67.1% | 74.5% |

50% | 68.7% | 71.4% |

## Appendix B. Mie Scattering Correction

## Notes

1 | Interference effects between the coherent part of the wave function can only occur if the size of the original source is smaller than the grating period ${\sigma}_{x}/d<1$. One furthermore needs ${\sigma}_{p}d/h\gg 1$, to ensure that the initial trapped state extends over many grating momenta, a necessary condition to guarantee the validity of the theoretical model used to describe the interferometric setup [21,23]. Both of these conditions are fulfilled for the case of study presented here. |

2 | |

3 | For the case of finite-size particles, we refer the reader to the derivation in [25]. |

4 | This is true for a particle optically levitated in vacuum. If the refractive index of the medium is greater than that of the particle, then the particle is pushed away from the maximum field strength region. |

5 | Despite the cooling requirements detailed in the previous section, we chose a 1 mK temperature here to demonstrate the relatively low cooling requirements needed to see visible fringes. In theory, should a better solution for recapture be found, this would be the new cooling requirement. Using the temperatures from the previous section would lead to slightly higher visibility fringes. |

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**Figure 1.**Schematic of the proposed experimental setup, with different coloured beams representing different wavelength lasers. The particle is shown as being kicked when offset from the trapping centre to demonstrate the generally parabolic trajectory. Please note that, in the following text, the parabolic mirror will be assumed to be placed such that the longitudinal axis of the trapping light is along the z direction. The kicking pulse will then be focused by the parabola onto the particle. This figure aims to give a rough concept of the design.

**Figure 2.**Matching point of optical potential energy barrier and particle kinetic energy. (

**a**) The particle’s transverse displacement must be less than 2.4 µm for it to be recaptured. (

**b**) To achieve that transverse displacement during the total flight time of $285.6$ ms, to and from a height of 10 cm, the transverse speed must be less than 8.5 µm/s. (

**c**) That speed corresponds to a temperature of the oscillation along x of less than 5 µK.

**Figure 3.**Gradient and scattering forces’ equilibrium on the z-axis. The particle is falling from the left, in the direction of the gravitational acceleration,

**g**, indicated by the blue arrow. Its diameter is 100 nm, and the laser parameters are: 100 mW, 1550 nm. The focusing element is an NA = 1 parabola of $3.6$ mm diameter. In the first region,

**I**, where ${F}_{grad}>{F}_{scat}$, the particle receives an additional acceleration, ${a}_{I}$, from the gradient force. The second region,

**II**, begins at 145 nm in front of the focal plane, where ${F}_{grad}={F}_{scat}$ for the particle considered, and ends at $z=0$, where ${F}_{grad}=0$. Here, the deceleration begins, because the scattering force is greater than, and of opposite sign to, the gradient force. In the third region,

**III**, both forces act in the direction opposite that of the particle. The deceleration, ${a}_{III}$, is highest here, with a peak at 0.58 µm. The red, horizontal line, marks the average force of this region, denoted as ${F}_{avg}$. Its value is $-0.84$ pN. In the fourth region,

**IV**, after 1.9 µm, the deceleration drops because both forces fade away as we depart from the focal plane.

**Figure 4.**Particle return speed against trap power needed to stop it. To stop a particle returning at $1.4$ m/s, we need a laser power of $1.5$ W.

**Figure 5.**Expected quantum interference patterns (blue) and Moiré shadow patterns (orange) for a (

**a**) ${10}^{6}$ amu nanoparticle and (

**b**) a ${10}^{8}$ amu nanoparticle. (

**c**) shows a zoomed in section of (

**b**) for clarity. These patterns assume parameters that were found by optimising for the visibility of the quantum fringes in each case and are detailed in Table 1. These patterns include all interference effects mentioned in the previous section, aside from the decoherence from imperfect kicks, which is shown later.

**Figure 6.**Expected variation of quantum (blue) and classically (orange) predicted visibilities of fringes when varying the phase modulation parameter ${\varphi}_{0}$, controlled by the pulse energy of the grating laser. (

**a**) shows the expected visibility variation for a SiO

_{2}particle of mass ${10}^{6}$ amu, whilst (

**b**) demonstrates the same plot for a particle of mass ${10}^{8}$ amu.

**Figure 7.**Expected variation of quantum (blue) and classically (orange) predicted visibilities of fringes with varying flight times. (

**a**) shows visibility variation for a ${10}^{6}$ amu SiO

_{2}particle, whilst (

**b**) shows the same plot for a ${10}^{8}$ amu particle.

**Figure 8.**Expected interference patterns for the interferometry of a ${10}^{6}$ amu SiO

_{2}particle when accounting for the error in initial kick velocity, with errors ranging from 0 to 30%. (

**a**) shows the expected patterns for the classical case, whilst (

**b**) shows the quantum case.

**Figure 9.**Expected interference patterns for the interferometry of a ${10}^{8}$ amu SiO

_{2}particle when accounting for the error in the initial kick velocity, with errors ranging from 0 to 25%. (

**a**) shows the expected patterns for the classical case, whilst (

**b**) shows the quantum case. (

**c**,

**d**) give a zoomed in picture of (

**a**,

**b**), respectively.

**Figure 10.**Example of a measurement of a particle’s re-entry position in the vertical (z) direction. The trap was turned off at 0 ms and turned back on at $0.05$ ms. Position is measured from the equilibrium position of the particle, which is slightly offset from the centre of the trap.

**Table 1.**Table of relevant parameters used for the generation of quantum and classical patterns throughout this section.

Parameter | ${10}^{6}$ amu Particle | ${10}^{8}$ amu Particle |
---|---|---|

Pressure | ${10}^{-10}$ mbar | ${10}^{-10}$ mbar |

Initial temperature | 1 mK | 1 mK |

Flight time | 58 ms | 142 ms |

Phase modulation (${\varphi}_{0}$) | $\pi /2$ | $8\pi $ |

**Table 2.**Table of visibility values for the plots presented in Figure 8 for a ${10}^{6}$ amu particle.

Velocity Error | Classical Fringe Visibility | Quantum Fringe Visibility |
---|---|---|

0% | 77.3% | 98.2% |

10% | 75.0% | 94.9% |

20% | 73.5% | 88.3% |

30% | 74.5% | 81.1% |

**Table 3.**Table of visibility values for the plots presented in Figure 9 for a ${10}^{8}$ amu particle.

Velocity Error | Classical Fringe Visibility | Quantum Fringe Visibility |
---|---|---|

0% | 92.2% | 79.6% |

5% | 93.0% | 79.6% |

10% | 92.5% | 79.1% |

15% | 91.5% | 78.4% |

20% | 90.3% | 77.2% |

25% | 89.0% | 75.5% |

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**MDPI and ACS Style**

Wardak, J.; Georgescu, T.; Gasbarri, G.; Belenchia, A.; Ulbricht, H.
Nanoparticle Interferometer by Throw and Catch. *Atoms* **2024**, *12*, 7.
https://doi.org/10.3390/atoms12020007

**AMA Style**

Wardak J, Georgescu T, Gasbarri G, Belenchia A, Ulbricht H.
Nanoparticle Interferometer by Throw and Catch. *Atoms*. 2024; 12(2):7.
https://doi.org/10.3390/atoms12020007

**Chicago/Turabian Style**

Wardak, Jakub, Tiberius Georgescu, Giulio Gasbarri, Alessio Belenchia, and Hendrik Ulbricht.
2024. "Nanoparticle Interferometer by Throw and Catch" *Atoms* 12, no. 2: 7.
https://doi.org/10.3390/atoms12020007