LDRD Template

Thomas jefferson National Accelerator facility

Thomas jefferson National Accelerator facility

Proposal for FY2021 Laboratory Directed Research and Development Funds

Title: SRF Levitation and Trapping of Nanoparticles Experiment

Topic: Quantum information science; new research directions using existing JLab facilities

Lead Scientist or Engineer:

RONGLI GENG

Phone:

757-269-7649

Email:

geng@jlab.org

Date:

5/29/20

Department/Division:

SRF R&D/Accelerator Division

Electric Engineering/Engineering Division

Detector Group/Physics Division

Other Personnel:

John Musson (ED), Frank Marhauser, Haipeng Wang (AD)

Wenze Xi (PD)

Mentor (if needed)

Proposal Term:

From: 10/2020

Through: 10/2022

If continuation, indicate year (2nd/3rd):

Division Budget Analyst

Kelly Webster

Phone:

757-269-7575

Email:

hanifan@jlab.org

ii SLAC Annual Laboratory Plan

This document and the material and data contained herein were developed under the sponsorship of the United States Government. Neither the United States nor the Department of Energy, nor the Thomas Jefferson National Accelerator Facility, nor their employees, makes any warranty, express or implied, or assumes any liability or responsibility for accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use will not infringe privately owned rights. Mention of any product, its manufacturer, or suppliers shall not, nor it is intended to imply approval, disapproval, or fitness for any particular use. A royalty-free, non-exclusive right to use and disseminate same for any purpose whatsoever, is expressly reserved to the United States and the Thomas Jefferson National Accelerator Facility.

3 of 26JLab LDRD Proposal

Abstract

Levitate and trap mesoscopic particles by radio frequency fields in a superconducting RF (SRF) cavity, aimed at overcoming a major limit faced by the state-of-the-art laser trapping techniques by exploiting intrinsic characteristics of an SRF cavity. Establish a foundation to enable observation of quantum phenomena of an isolated mechanical oscillator interacting with microwave fields. Address R&D issues relevant for new research directions using existing SRF facilities at JLab, specifically its groundbreaking relevance to quantum information science and technology, targeting applications for precision force measurement sensors, quantum memories and alternative quantum computing implementations with promises for superior coherence characteristics and scalability well beyond the start-of-the-art.

Summary of ProposalDescription of Project

While levitation and trapping have always attracted attention and found applications such as for containerless processing and spectroscopy of microparticles [1], recent developments in the field of optomechanics have brought about levitation of nanoparticles enabling observation of quantum phenomena in the context of quantum science and technology.

Quantum properties occur when a macroscopic matter particle, being trapped in an optical field and behaving like a mechanical oscillator, is further cooled, either by the same trapping optical fiend or by an external laser, to its fundamental quantum ground state. Coherent control of a macroscopic quantum system has potentially game changing implications for fundamental physics as well as technology. It could allow the exploration of the classical-to-quantum boundary, the development of precision force measurement sensors, and could provide a fundamental building block for quantum information [2]. The motion of a “membrane” type mechanical oscillator, embodied by a compliant capacitor in an LC resonator, has been demonstrated to entangle with a propagating electrical signal, with one half of the entangled state being stored in the mechanical oscillator [3]. That result highlighted the potential for the mechanical oscillators to serve as quantum memories for microwave signals and to allow information retrieval from the memory on demand.

Levitated nanoparticles offer the unique and appealing feature of being highly decoupled from the environment allowing the observation of extremely high quality factors (theoretically expected up to 1012) in a mechanical oscillator. As the main source of heating is avoided by being free from mechanical attachment to other mechanical objects, reaching the quantum ground state of the mechanical motion of levitated particles is hence greatly facilitated. Recent experiments, in which optical trapping of a dielectric silica sphere of ~140 nm in diameter at ambient temperature in vacuum, have successfully demonstrated the realization of quantum ground state [4, 5].

Four current challenges are identified for nanoparticle trapping and cooling with the light fields in an optical cavity [6]: (1) stable trapping at high vacuum; (2) minimizing the mechanical occupation; (3) minimizing the photon shot noise; (4) maximizing the optomechnical coupling.

We propose an experiment on levitation and trapping mesoscopic particles by radio frequency fields in a superconducting RF cavity, SRF levitation and trapping of nanoparticles, aimed at enabling the observation of quantum phenomena in the context of quantum science and technology. Our approach will bring a new tool that is unfamiliar to the field of optomechanics. Most critically, by virtue of much longer wavelength (factor of 105), ultra-high vacuum (down to 10-10 - 10-11 mbar) and cryogenic temperatures (1 – 4 K), it addresses three of the four currently identified challenges faced by optomechnical nanoparticle levitation, therefore promises to advance nanoparticle levitation well beyond the state of the art demonstrated by optical cavities (see Appendix I for details).

· Photon scattering, scales as , reduced by 1020.

· Mechanical phonon occupation, scales as , reduced by 102.

· Cavity internal loss kcav reduced by a factor of 104.

This experiment, if successful, will establish a platform enabling exploration of an electromechanical system consisting of a trapped macroscopic mechanical oscillator and a superconducting RF resonator, aimed at ultimately generating entanglement between the motion of a trapped particle mechanical oscillator and a propagating microwave field [7]. Such a platform would offer an opportunity for developing qubits with superior scalability as compared to trapped ions due to the large trap volume and coherence time, , up to 10 sec, 3-4 order of magnitude longer than that in the state of art SQUID based quantum computer architecture [16]. In addition, it provides opportunities for developing precision force measurement sensors and testing quantum theory in macroscopic domain.

Expected Results

Our proposed approach is to levitate and trap tiny particles (size 100 nm to 10 µm) in a radio frequency field (RF electric field amplitude in the range of 1 to 20 MV/m) of a superconducting RF resonator cooled to cryogenic temperatures (1.4 – 4.2 K). Fig. 1 show the block diagram of the testing concept.

Particle deliver arm

& LED light source

Laser interferometer

RF amplifier

Particle

Directional coupler

VCO

Mixer

SRF cavity

High speed camera

Figure 1: SRF levitation and trapping experiment concept.

Initial tests will be carried out using existing single-cell SRF niobium cavities such as the 1.3 GHz niobium cavity shown in Fig. 2 [8]. That cavity was originally developed at JLAB for the International Linear Collider (ILC) high gradient R&D. It possesses the capability of attaining strong fields and a large intrinsic quality factor, Q0, and is therefore suitable for levitating nanoparticles.

Figure 2: An SRF cavity with measured high field capability suitable for levitation.

The particles will be initially selected to be microscopic dielectric particles, size being in the range of 1 - 100 µm like metallic particles such as those shown in Fig. 3 as observed on the surface of SRF cavities operated at Jefferson Lab for electron beam acceleration [9]. Those metallic particles were discovered in recent years within the vacuum space of the CEBAF electron accelerator. These particles are suspected to be a source of field emission. Prior work has established the possibility of field emitter particles being levitated (albeit not trapped) in SRF cavities [10].

Figure 3: Particles naturally observed in SRF cavities showing a variety of shapes with nominal dimensions of a few micrometers: Left to right: titanium/tantalum, niobium, and iron. They will guide the selection of target particles for initial SRF levitation test.

The particle levitation and trapping will be monitored by electronic and optical methods. An ordinary RF phase lock loop is used to control the RF field excited in the SRF cavity. The system may be further improved for the detection of detuning arising from the movement of a trapped particle, which is to be used for evaluation of a device parameter , needed for the determination of the single-photo coupling strength in future cooling experiments. A high speed camera is used to monitor and observe the particle position. A laser interferometry based on an existing lidar device is to be used for monitoring the vibration of a trapped particle.

The cavity is to be exited in the fundamental TM010 mode and higher order modes such as the TM110 mode or TE-like modes to provide a varying spatial distribution of the electromagnetic field hence manipulation of various particles including dielectric and superconducting ones (see Appendix I for justifications).

The goal of the project is to demonstrate levitation and trapping of a particle 100 nm in diameter for 10 seconds or longer in an SRF cavity.

Levitation and trapping of nanoparticles is the foundation to enable observation of quantum phenomena. The success of this project will set a stage for cooling nanoparticles into the quantum ground state in the context of quantum science and technology, via the side-band cooling of a mechanical motion arising from the strong coupling between photos and phonons [14, 15]. In summary, the expected results are highlighted in the following box guided by three key questions.

1) Is this intended to be purely speculative or is it answering a request/interest that’s based on some underlying theory?

The idea in our proposal is based on sound theory of interaction between matters and electromagnetic fields. The salient feature is a 3D restoring force arising from the field intensity gradient (hence the well-known "gradient force"). The famous "optical tweezers" are based on the gradient force experienced by a small dielectric particle in the electric field of a laser beam. Similarly SRF trapping is based on the same theory with the wavelength of the EM field much longer as compared to optical trapping. Moreover, the SRF cavity allows us to select regions of "magnetic gradient force", again there is published theoretical work, for levitating magnetic particles including superconducitng particles. It is estimated that for a simple pill box shape SRF cavity, microscopic particle levitation and trapping is possible with a RF field of ~ 10 MV/m, which is well within the capability of existing cavities at hand. We certainly need analytical and numerical calculations for optimizing the gradient force and we have tools needed to do just that.

There are of course some speculative features in our proposal, which makes an experiment necessary. For example, we anticipate the longer wavelength offers an advantage for reduced energy absorption by the trapped particles. We anticipate the cold environment would allow us to trap superconducting niobium particles. All these have to be addressed by an experimental study.

2) What is your description of the minimum achievement to consider this project successful?

Demonstration of levitated particles 1-10 micron in diameter and trapped for 10 seconds.

3) Will the project demonstrate relevance to QIS after the end of the project?

The "minimum" achievement goal is levitation and trapping. This is the necessary but not sufficient evidence to prove relevance to QIS.

To proof directly the relevance to QIS, additional figure of merit, namely phonon occupation number needs to be measured. This may require the development a new cavity with adequate single photo coupling strength, leading to cooling of trapped particles. Our pitch in the current proposal is to reuse existing cavities and test stand. We will strive to measure nonetheless a crucial device parameter, the single photo coupling strength, which maybe in our reach by improving the low level RF control system.

Success in demonstrating SRF trapping of nanoparticle will place ourselves in a strong position to launch into the next phase: SRF cooling of trapped particles. It offers a promising research path in raising the coherence time of a qubit by as much as a factor of 1000 compared to the state-of-the-art superconducting SQUID quibits and superior scalability as compared to trapped ion qubits due to large trapping volume.

Moreover, our SRF levitation and trapping of nanoparticles experiment promises to provide an immediate test bed for developing sensitive force, position, or electromagnetic field measurement down to quantum limit (see Appendix II for more details).

Proposal Narrative

Quantum properties occur when macroscopic particles are cooled to their fundamental quantum ground state, the lowest possible energy level. Levitated nanoparticles offer the unique and appealing feature of being highly decoupled from the environment allowing the observation of extremely high quality factors. Bringing a levitated oscillator into the quantum regime is the first necessary step toward the creation of the quantum superposition states. Recent experiments have indeed demonstrated the realization of such a ground state by optical trapping of a dielectric silica sphere of ~140 nm in diameter at ambient temperature in vacuum and an active feedback cooling to its center-of-mass motion with or without an optical cavity [4, 5].

RF levitation was demonstrated and used for vacuum evaporation of ultrapure silicon and germanium [11]. The restoring force arises from the gradient of the time averaged field square [12]. NASA’s Jet Propulsion Laboratory at Caltech reported a proposal for microwave levitation of small objects, calling out the use of an SRF cavity to provide the required large RF electric field of 10 MV/m [13]. Experimental evidence exists in showing microwave levitation, although apparently no trapping, of natural field emitter particles in SRF cavities [10].

Purpose/Goals

Our experiment is aimed at the demonstration of levitation and trapping mesoscopic particles by radio frequency fields in a superconducting RF cavity, a foundation to enable observation of quantum phenomena of an isolated mechanical oscillator interacting with microwave fields, in the context of quantum science and technology. Our approach promises to overcome a major limit faced by the state-of-the-art laser trapping techniques by exploiting intrinsic characteristics of an SRF cavity. Our proposal addresses R&D issues relevant for new research directions using existing SRF facilities at JLab, specifically its groundbreaking relevance to quantum information science and technology, targeting applications for precision force measurement sensors, quantum memories and alternative quantum computing implementations with superior coherence characteristics.

Approach/Methods

A mechanical oscillator made of levitated microscopic particle has a significant advantage as compared to other type of oscillators that are mechanically attached to other components. Very high quality factors (up to 1012) can be anticipated, hence it offers a promising approach to extend the coherence time, a critical need based on the current quantum computing implementations based on trapped ions or SQUID [16].

Compared to the optical levitation and trapping, SRF levitation and trapping has many advantages. The relative long wave length allows much reduced photo scattering hence levitating light-absorbing particles. It may be even possible to levitate a large number of isolated particles because of large field volume, therefore allowing interrogation of inter-particle interaction. By choosing different resonant modes, regions of high electric field or high magnetic field can be selected, allowing levitation of varying particles, including dielectric particles commonly used in laser trapping and cooling, and moreover paramagnetic or diamagnetic particles. In addition, the cryogenic temperature offers the possibility of levitating superconducting particles such as a niobium sphere. The excellent vacuum condition, on the order of 10-10 Torr with the possibility of further extension down to 10-11 Torr, provides an excellent stage for next particle cooling studies by significantly reducing the thermal bath effect from residual gas molecules. Moreover, an SRF cavity is able to generate strong fields with very small intensity fluctuation, provides a perfectly shielded space against external electrical or magnetic field (conducting cavity wall and Meissner effect), therefore one can expect improved fidelity and integrity of a quantum system based on the proposed technology. In summary, successful demonstration of the proposed experiment may ultimately lead to a surprising alternative path to a qubit architecture with 3-4 order of magnitude increase in the coherence time without even using a dilution refrigerator.

Our proposed approach is to reuse an existing SRF cavity to demonstrated levitation and trapping of tiny particles (size 100 nm to 10 µm) in a radio frequency field (RF electric field amplitude in the range of 1 to 20 MV/m, see Appendix II for further analysis). The test will be carried out in the existing VTA facility with the superconducting RF resonator cooled to cryogenic temperatures (1.4 – 4.2 K). We will reuse an existing laser interferometer device for monitoring the vibration of the trapped particles.

To support the realization of the experiment goals, the following tasks will be carried out leading to the experimental studies of SRF levitation and trapping of nanoparticles:

· A deterministic formulism of suitable RF field conditions for levitation and trapping of nanoparticles in a SRF cavity excited in TM010 mode and other modes.

· A precision mechanism and apparatus for particle delivery. Three options under consideration: (1) free fall delivery; (2) pneumatic capillary gun injection; (3) piezoelectric vibrator launching.

· A high-speed camera system for imaging and recording the levitated trapped particles.

· A system based on laser interferometry for monitoring the vibrations of trapped particles.

Goals for FY2021

· Quarter 1

· Identify test existing cavity and test stand.

· Design the precision mechanism and apparatus for particle delivery.

· Place order of high speed camera.

· Develop formulism of RF field conditions for levitation and trapping of nanoparticles in a SRF cavity excited in TM010 mode and other modes

· Identify suppliers of dielectric and niobium particles.

· Start evaluation on the suitability of invention disclosure.

· Quarter 2

· Modify test stand.

· Order particles.

· Prototype the precision mechanism and apparatus for particle delivery.

· Offline test of the high speed camera and laser interferometry.

· Numerical simulations of RF field conditions for levitation and trapping.

· Quarter 3

· Finalize the precision mechanism and apparatus for particle delivery.

· Finalize RF field parameters for levitation and trapping experiments.

· Novel structures exploration for optimized trapping force and mechanical oscillation frequency.

· Quarter 4

· Integrate the particle delivery system, high speed camera system and the laser interferometry system into the test stand.

· Integrated test of the whole test set up at room temperature.

· Novel structures exploration for optimized trapping force and mechanical oscillation frequency.

· File invention disclosure if deemed suitable.

Goals for FY2021

· Quarter 1

· Integrated test of the whole test set up at cryogenic temperatures.

· Trouble shoot the hardware and software.

· Establish procedures for particle delivery and monitoring at cryogenic temperatures

· Quarter 2

· Experimental studies with dielectric particles.

· Tuning cavity fields to measure trapping force and oscillation dependence and compare with theory.

· Evaluate the resolution of detection system toward the measurement of the device parameter G=dω_c/dx, needed for the determination of the single-photo coupling strength in future cooling experiments.

· Quarter 3

· Experimental studies with superconducting niobium particles.

· Quarter 4

· Write the project report and publish the results.

Required Resources

This work will be carried out at JLab. The experiments will be done in the VTA facility in Test lab. Stating of the test set up will be done in the VSA next to the VTA. Offline testing will be done in RF Structure Lab in TLA or in the ARC building. Cavity processing and preparation will be done in the SRF Chem Room and Clean Room in Test Lab. Required labor and new equipment are described in the budget explanation section.

Anticipated Outcomes/Results

The proposed work takes advantage of JLab’s strength in SRF science and technology and the latest advancement in the related field of optomechanics. Its aimed R&D direction, however, radically departs from the conventional portfolio, hence drives the exploitation of the high-value and hard-to-get-by SRF infrastructures to meet the next generation challenge in quantum information science and technology. It is worth noting that Fermilab has already started an initiative to extend their SRF science and technology into the regime of quantum computing. The approach there is to use the high Q0 nature of SRF cavities. Our JLab approach adopted in this proposal is based on SRF cavity levitation and trapping of nanoparticles, hence sharply distinct from Fermilab’s R&D. It is also worth noting that BNL, based on their strength and track record in formation and acceleration of ion beams, has started an initiative in exploring crystalline ion beams for quantum computing with their LDRD support. In view of these new developments in other DOE labs, this proposal, if successful, may provide JLab an opportunity to stay competitive with other sister labs in moving the lab’s traditional use of accelerator infrastructure and expertise into the regime of quantum information science. In addition, it provides a platform for engaging the lab’s talents in mastering quantum noise, measurement, and amplification, which naturally leads to attracting and training of next generation talents. The adaptive reuse of JLab’s SRF infrastructure, as embodied in this proposal, offers a path, rooted in credible past SRF achievements, towards a possible launching pad for the lab to enter into the realm of quantum information science for both fundamental research and practical applications.

Accomplishments in Previous Years

N/A

Budget Explanation

Personnel:

Rongli Geng is the PI of this proposed work. He has more than two decades for experience in SRF science and technology and is an expert on high-gradient SRF cavity technology. He will lead the design of the experiment including the source of particles and the delivery mechanism of particles, assist the test stand modification, and lead the experimental tests. The estimated level of effort is 8.8 person-weeks.

John Musson is an expert on electronic controls and instrumentations. He will lead the instrumentation of the experiment, including RF electronics and the high-speed imaging. The estimated level of effort is 8.8 person-weeks.

Frank Marhauser is an expert on electromagnetisms related to RF cavities. He will carry out EM simulations with codes such as Microwave Studio and COMSOL and select suitable RF field configurations for strong particle trapping. The estimated level of effort is 4.4 person-weeks.

Haipeng Wang is an expert on high power RF and microwave theories. He will carry out theoretical analysis of the interaction between a dielectric microsphere and an RF field and select particle attributes for optimal trapping potential. The estimated level of effort is 4.4 person-weeks.

Wenze Xi is an expert in optics and detectors. He will evaluate the existing partile-sensitive lidar-based detector system and adapt it into the instrumentation system of the proposed experiment for the diagnostic of the particle vibrations. The estimated level of effort is 4.4 person-weeks.

This proposal includes a request of technicians for test stand modification, cavity assembly, leak checking, test stand movement, liquid helium transfer etc. the estimated level of effort in 8.8 person-weeks.

Purchases/procurements:

The major purchases include $60,000 for a fast camera, $20,000 for a precision linear translation stage, and $10,000 for lenses and optical fibers.

In addition, $10,000 is requested for machine shop for test stand modification.

Needed hiring:

No hiring is planned. We will actively seek collaborations with local and national partners, especially university groups with strong interest in advancing the field and train future talents in Quantum Information Sciences by building and using the quantum test bed based on SRF trapping and cooling at Jefferson Lab.

Top level budget summary:

Item

amount(k$)

Labor

155

Procurement

100

Total direct

255

Total including overhead

289

References

[1] E. H. Brandt, Levitation in Physics, Science, 243, 349-355 (1989).

[2] G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, A. A. Geraci, Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum”, , Phys. Rev. A 91, 051805(R) (2015).

[3] T. A. Palomaki, J. D. Teufel, R. W. Simmonds, K. W. Lehnert, Entangling Mechanical Motion with Microwave Fields, Science, 342, 710-713 (2013).

[4] F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, Motional Sideband Asymmetry of a Nanoparticle Optically Levitated in Free Space, Phys. Rev. Lett. 124, 013603 (2020).

[5] U. Delić, M. Reisenbauer, K. Dare, D. Grass, V. Vuletić, N. Kiesel, M. Aspelmeyer, Cooling of a levitated nanoparticle to the motional quantum ground state, Science, 367, 892-895 (2020).

[6] J. Millen, T. S. Monteiro, R. Pettie, A. N. Vamivakas, Optomechanics with levitated particles, Rep. Prog. Phys. 83 026401 (2020).

[7] T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, K. W. Lehnert, Coherent state transfer between itinerant microwave fields and a mechanical oscillator Nature 495, 210–214 (2013).

[8] R. L. Geng, C. Adolphsen, Z. Li, J. K. Hao, K. X. Liu, and H. Y. Zhao, “New Results of Development on High Efficiency High Gradient Superconducting RF Cavities”, in Proc. 6th Int. Particle Accelerator Conf. (IPAC'15), Richmond, VA, USA, May 2015, pp. 3518-3520. doi:10.18429/JACoW-IPAC2015-WEPWI013.

[9] R. L. Geng, J. F. Fischer, E. A. McEwen, and O. Trofimova, “Nature and Implication of Found Actual Particulates on the Inner Surface of Cavities in a Full-Scale Cryomodule Previously Operated With Beams”, in Proc. 17th Int. Conf. RF Superconductivity (SRF'15), Whistler, Canada, Sep. 2015, paper MOPB035, pp. 164-168.

[10] P. L. Anthony, J. R. Delayen, D. Fryberger, W. S. Goree, J. Mammosser, Z. M. Szalata, J. G. Weisend II, Experimental studies of light emission phenomena in superconducting RF cavities, Nucl. Inst. and Meth. In Phys. Res. A612 (2009) 1–45.

[11] E. A. Roth, E. A. Magerum, and J. A. Amick, Evaporation of Sillicon and Germanium by rf Levitation, Rev. Sci. Inst. 33, 686-687 (1962).

[12] T. B. Jones, A necessary condition for magnetic levitation, J. Appl. Phys. 50, 5057-5058 (1979).

[13] J. L. Watkins, H. W. Jackson, Microwave Levitations of Small Objects, NASA Tech Briefs, 15(9), 109 (1991).

[14] J. D. Teufel, Dale Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker & R. W. Simmonds, Circuit cavity electromechanics in the strong-coupling regime, Nature 471, 204–208 (2011).

[15] J. D. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert & R. W. Simmonds, Sideband cooling of micromechanical motion to the quantum ground state, Nature, 475, 359–363 (2011).

[16] M. Kjaergaard, M. E. Schwartz, J. Braumuller, P. Krantz, J. I-J Wang, S. Gustavsson, and W. D. Oliver, Superconducting Qubits: Current State of Play, Annual Review of Condensed Matter Physics 11.1 (2020): 369–395. arXiv:1905.13641

Attachments

Appendix I Responses to Andrei Seryi’s Questions raised over the course of the formation of this proposal

1. Is the plan to trap metallic or dielectric particles?

Baseline particles: dielectric

a) Fused silica (SiO2)

b) Sapphire (single crystal Al2O3)

Alternative particles: metallic in superconducting state

a) Niobium (superconducting when cooled to cavity temperature of ~ 2K)

2. What is the mechanism of trapping, what force is acting on particles?

The gradient force is the major driving force behind the SRF levitation and trapping. Such a force arises in an EM field where a spatial variation in the field amplitude, electric or magnetic, exists.

For a dielectric particle in air or vacuum, in Rayleigh regime with the particle radius a being sufficiently smaller than the wavelength of the electromagnetic field λ, , the time averaged gradient force arises from the spatial variation of the electric field amplitude,

, (1)

where α is the polarizability of particle,

, (2)

here r is the radius of the particle, is the permittivity of vacuum, is the relative permittivity of the particle. The same force is behind “laser trapping” of dielectric particles (SiO2) with the particle radius in the range of 70 nm to 1.5 µm [1-3]. The same gradient force is behind “optical tweezers” invented by Arthur Ashkin who was awarded a Nobel Prize in 2018.

Other forces involved are the gravitational force and the photon scattering force,

, (3)

, (4)

where is the mass of the particle, the gravity constant, the density of the particle, the speed of light in vacuum, the time averaged Poynting’s vector. The scattering force is in the direction of the laser propagation in optical tweezers. is the scattering cross section,

, (5)

Where is the wave number in air or vacuum. The scattering force can be dominant in laser optical tweezers. It is insignificant in the context of particle trapping, but is important for cooling of the mechanical motion of a trapped particle (note: cooling is not an immediate goal of the proposed SRF levitation and trap experiment, but an ultimate goal toward the demonstration of its relevance to QIS)

For a superconducting niobium particle, the gradient force arises from the spatial variation of the magnetic field amplitude,

, (6)

where is the permeability of vacuum, a parameter dependent on the material property and the frequency of the EM field [4].

3. Do you assume that particles will be cryo-cold? What is the justification?

(One can imagine that particles, which do not have heat sink, but have some heat source (surface currents for metallic, some dielectric effects) will heat up and may actually be so hot that they will shine, as Bob mentioned earlier.)

The bulk temperature of a particle trapped in an SRF cavity is expected to be at cryogenic temperatures. For a dielectric particle, heating due to absorption of the EM energy is determined by the product of electric field squared and the photon absorption cross section

, (7)

where is the imaginary part of the polarizability [5]. can be found from [6], consequently we have

. (8)

The salient feature is the scaling. This leads to a decrease by a 20 orders of magnitude in heat production for the same particle size and the same electric field amplitude. As a result, the heating of the trapped particle is significantly reduced in case of SRF trapping, opening up stable trapping without damaging the trapped dielectric particle in ultra-high vacuum, which being one of the major challenges currently faced by the laser trapping community [5].

Present day laser optical cavity trapping experiments are typically done at a vacuum level 10-5 (with the best down to 10-8) mbar, with a quality factor of the mechanical oscillator being Q=107. In the Knudsen regime (the mean free path of the background gas is much larger than the radius of the trapped particle), the momentum damping rate of a levitated mechanical oscillator arising from collision with the background gas is linearly proportional to the gas pressure [5]. As a result, there is a theoretical expectation in reaching the mechanical oscillator Q to 1011 - 1012 by pushing the optical cavity vacuum to 10-9 - 10-10 [2,7]. It is yet to be shown if such a high vacuum levels will pose a limit, due to overheating damage, to the cooling of the particle mechanical oscillation in a room temperature laser optical cavity.

SRF levitation and trapping has an additional advantage for dielectric particle trapping – the cold cavity wall provides better cooling of the trapped particle via black body radiation.

As for SRF levitation and trapping of superconducting niobium particles, it is expected that the surface current heating is small, as a result of vanishing surface resistance.

The challenge to the SRF levitation and trapping is managing the gradient force. To the 0th order, the trapping force for a dielectric particle is proportional to . The present day laser optical cavity trapping typically involves a laser power at a wavelength of 1064 nm focused to a beam waist radius . The peak electric field amplitude can be found via

(9)

to be in the range of 15-20 MV/m. This corresponds to an accelerating gradient of 9-12 MV/m (assuming a transient time factor ) which is easily achievable by using today’s SRF niobium cavity technology. For an ordinary 1.5 GHz elliptical cavity with cut-off beam tubes, the half-cell length sets the length scale of 5 cm in determining , which is to be compared to 1 µm for the laser trapping. Consequently, the mechanical oscillation frequency decreases by 4-5 orders of magnitude. The leads to an expected mechanical oscillation of ~ 1 Hz for a dielectric particle trapped in a 1.5 GHz cavity excited at ~10 MV/m. For our proposed SRF levitation and trapping, this is an adequate parameter regime. However, in order to demonstrate cooling of the mechanical oscillation, for example via side-band resolved cooling [5], the mechanical oscillation frequency needs to be increased much larger than the resonance width of the Nb cavity, which is about 1 Hz for a Q=109 of a 1.5 GHz cavity. This rough estimates leads to the need of engineering the cavity mode or cavity geometry for maximizing . An alternative avenue is to explore engineered thin- walled dielectric sphere.

The SRF trapping of a superconducting Nb sphere has never been explored before. Its inquiry remains a strong motivation in addition to the justification made above with regard to dielectric particle levitation.

4. Where the statement of 1E12 quality factor of mechanical oscillator coming from? Is it a guess, or an estimation? What is taking into account? E.g. what is effect of friction on residual gas?

The mechanical quality factor of 1012 is expected on a theoretical ground (Eq (8) in Ref. [5]). It is based on the fact that the gas collision heating rate is linearly proportional to the back ground gas pressure. For a laser trapped particle, a Q value of 1011 is expected at vacuum of 10-9 mbar as shown in the Fig. 1 (also taken from Ref. [5]). In another reference a mechanical Q = 1012 is explicitly stated [7].

Figure 1: Mechanical quality factors [5].

5. What is your assumption about the value of mechanical frequency of the trapped particle and why?

(In case if you assumed that this frequency is the same as RF frequency, I think this is not correct, and mechanical frequency is much lower. Similarly, frequency of a particle trapped in an optical tweezer is not equal to laser frequency, but defined by gradient of laser intensity, particle mass, etc.).

We expect the mechanical frequency of a trapped dielectric particle in an ordinary 1.5 GHz single-cell SRF Nb cavity excited at Eacc = 10 MV/m to be ~ 1Hz. This is based on the expected gradient force in our proposed experiment to be a factor of 10-5 that of the achieved value of 40 kHz for the laser trapping.

The mechanical oscillation frequency needs to be increased at least to 10 Hz in order to proceed to cooling experiment. We have several ideas to explore that: (1) Increase the spring constant of the mechanical oscillator potential well by RF mode superposition, field squeezing by 3D shaped cavity; (2) Decrease particle mass with thin-walled dielectric sphere.

6. What parameters exactly are assumed on page 3, in three lines where you estimate reduction factors of 1E20, 1E2 and 1E4 for photon scattering and so on? Taking into account the questions above, are these estimations well justified?

The factor of 1020 is based on Eq. (5) where the scattering cross section scales . The wavelength of 1.5 GHz microwave and 1064 nm laser wavelength ratio is ~ 105, hence scattering cross section scales 1020. Similar scaling holds for power absorption (see Eq. (8)), which justifies significantly reduced particle heating – potentially overcoming a major challenge faced by the laser trapping.

Factor of 102 in the mechanical phonon occupation, , is based on the temperature decrease from 300 K to 2K for the same mechanical frequency (~ 40 kHz). With trapping force engineering or thin-walled dielectric sphere engineering, we expect the mechanical frequency of SRF levitated oscillator will reach a range of 100 Hz to 1 KHz, which would be large enough for side-band cooling. Consider this effect, this gain factor may diminish to 10 or just break even with laser trapping.

Cavity internal loss kcav reduced by a factor of 104 is based on the fact that Fabry-Perot optical cavity Q is in the range of 105 -106 and SRF cavity cavity Q being in the range of 109 – 1010.

References:

[1] Tongcang Li, Simon Kheifets & Mark G. Raizen, Millikelvin cooling of an optically trapped microsphere in vacuum, Nature Physics, 7, 527–530 (2011).

[2] Jan Gieseler, Bradley Deutsch, Romain Quidant, and Lukas Novotny, Subkelvin Parametric Feedback Cooling of a Laser-Trapped Nanoparticle, Phys. Rev. Lett. 109, 103603 (2012).

[3] U. Delić, M. Reisenbauer, K. Dare, D. Grass, V. Vuletić, N. Kiesel, M. Aspelmeyer, Cooling of a levitated nanoparticle to the motional quantum ground state, Science, 367, 892-895 (2020).

[4] T. B. Jones, A necessary condition for magnetic levitation, J. Appl. Phys., 50, 5057-8, (1979).

[5] James Millen et al, Optomechanics with levitated particles, Rep. Prog. Phys. 83, 026401 (2020).

[6] Jan Gieseler and James Millen, Levitated Nanoparticles for Microscopic Thermodynamics – A Review, Entropy, 20, 326 (2018).

[7] Z. Q. Yin, A. A. Geraci, T. Li, Optomechanics of levitated dielectric particles, International Journal of Modern Physics B, 27, 1330018 (2013)

Appendix II Trapping force and oscillation frequency estimation in a single-cell 1.3 GHz TESLA shape cavity excited in TM010 mode

Ftrap,z

Ftrap,r

Fg

r

z

Figure 1 illustrates a single-cell 1.3 GHz

TESLA shape single-cell cavity with a

dielectric particle levitated and trapped in

the electric field of TM010 mode. The cavity

vacuum space is shown as shaded area

with overlapping lines representing electric

field lines. The trapping force is directed

toward the center of the cavity where

has a local maximum. The gravity force Fg is

balanced by the axial trapping force Ftrap, z,

together with radial trapping force Ftrap, r,

the dielectric sphere is levitated and trapped.

(1)

(2)

r is the radius of the particle,

is the permittivity of vacuum,

is the relative permittivity of the particle.

Figure 1: Sketch of levitated dielectric particle. Inset: contour map of TM010 mode, trapping center at cavity center.

N.B. As will be shown by this analysis, a 1.3 GHz TESLA single-cell cavity is adequate for the proposed nanoparticle levitation and trapping experiment. While this choice provides the benefit of cost saving in the current stage of the proposed innovation, optimized SRF structures will have to be the scope of a follow up project. Of course, the explorative nature of LDRD program is likely to inspire us to advance our concept without any delay, if the proposed work is funded. This approach remains highly consistent with our intention for adaptive reuse of expensive SRF infrastructures at JLab to address compelling science and technology questions of national importance.

Next, we demonstrate levitation is achievable with modest cavity RF fields and derive the mechanical oscillation frequency of the trapped particle as a harmonic oscillator. We will focus on the axial trapping force along the cavity axis. SUPERFISH calculation output for along the axis (normalized to , where ω is cavity resonance frequency and the stored energy) is shown in Figure 2. The term for the normalized axial trapping force is also shown. It’s apparent linear dependence near is consistent with the fact that the trapping force is a restoring force as required for stable trapping.

r

z

Figure 2: Axial gradient of (black dashed line) from SUPERFISH calculation output for (red solid line). Normalized to .

We use a fused silica sphere as an example (the popular particle used in laser trapping). All quantities are in SI units. The particle parameters: relative permittivity ; particle mass density ; vacuum permittivity ; standard gravitational acceleration . We define a constant

. (3)

From Figure 2, the peak value of is found at , we define

, (4)

we shall define the peak axial trapping force, directed toward the center of the cavity,

. (5)

The gravitational force is

, (6)

we shall define another constant

, (7)

Which leads to

. (8)

A critical condition is such when the gravitational force acted on the sphere, is balanced by the trapping forward, directed toward the center of the cavity, or

. (9)

Eq. (9) is the necessary condition for dielectric particle levitation and trapping. As will be shown later, this condition determines a critical RF electric field. For convenience, we choose to express it in terms of the familiar “acceleration gradient”, . Successful levitation and trapping requires the RF electric field be greater than . Substituting the trapping force and gravitational force in Eq. (9) by Eq. (5) and (8), we have

. (10)

Note the particle radius dependence cancels out, leading to the successful levitation and trapping is independent on the particle size. Combing Eq. (9) and (10), a critical value for is found, .

Recall the familiar cavity parameter R/Q,

, (11)

where is the power dissipation on cavity surface, the acceleration voltage (for relativistic electron), the shun impedance, the acceleration gradient, the cell length (typically taken to be , being the wavelength of the RF field), and the unloaded quality factor of the cavity which is defined as

. (12)

Combining Eq. (11) and (12), we have

. (13)

Here we introduce the customary symbol , a cavity device parameter. For the TESLA shaped single-cell cavity, . Now we may define the critical acceleration gradient,

. (14)

This is a field level easily attainable in today’s SRF niobium cavities. For example, the cavity G2, a single-cell TESLA end cell shape cavity (see Figure 2 of this proposal) reached a maximum Eacc = 40 MV/m. At Eacc = 16 MV/m, the cavity , which means only 1 W of RF power dissipation. For the TESLA end cell shape single-cell cavity, the general relationship can be derived by combining Eq. (10) and (13),

. (15)

In summary, our analysis established the access to SRF levitation and trapping with comfortable margins by using modern SRF cavities. Moreover, we have shown that

· SRF levitation and trapping of dielectric particles is independent of the particle radius, or uniform trapping insensitive to particle size variation.

· Tuning the RF field can efficiently adjust the ratio of the trapping force to the gravitational force.

· Particle size is an effective knob for trapping force, allowing wide range of 60 zN - 60 µN for particle radius 10 nm – 1 mm, hence SRF levitation and trapping offers excellent opportunities of sensitive force measurements for quantum limited sensor development or containerless material purification.

We now move on to find the mechanical oscillation frequency of the trapped particle as a harmonic oscillator. Again, for purpose of illustration, only the oscillation in the axial z direction is discussed. From the SUPERFISH calculation output (see Figure 2), by limiting the z range from -3 cm to 3 cm, a good linear fit of with z is found,

, (16)

with .

Similar to Eq. (5), the general expression of the axial trapping force can be written as

, (17)

by defining

, (18)

we finally arrive at .

The angular frequency of the axial oscillation is

, (19)

recalling and substituting by Eq. (18), we have

. (20)

Notice the term cancels out, leading to the oscillation frequency being independent of the particle size. Finally, for the TESLA end cell shape single-cell cavity,

. (21)

At the critical gradient identified previously, , the corresponding angular frequency is , or the linear frequency .

In summary, our analysis established the mechanical oscillation frequency for SRF levitation and trapping using an existing TESLA single-cell cavity to be . Moreover, we have shown that

· The oscillation frequency of an SRF trapped dielectric particle is independent of the particle radius, or uniform frequency insensitive to particle size variation.

· The mechanical oscillation frequency tuning knobs are identified to be the RF electric field energy density, particle mass density and particle dielectric constant.

· A preliminary estimate suggests that the mechanical oscillation frequency on the order of 1 kHz is attainable by proper choice of particle and novel SRF structures optimized for high gradient and high energy density. This promises bright opportunities for exploring SRF trapped and cooled nanoparticles as qubits with superior coherence time and scalability far beyond the current state-of-art in qubits based on superconducting SQUIDs or trapped ions.

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