×

This course (not surprisingly) focuses on the mechanics of very small objects. In particular, we will discuss the regime of nanometer-scale objects where classical theory begins to break down and quantum mechanical behavior emerges. In order to do so, we will touch on continuum mechanics, statistical mechanics, opto-mechanics, and quantum mechanics. After covering the fundamentals of the field, we will study its most important results up to and including contemporary work. We will discuss efforts to bring nanomechanical resonators into the quantum mechanical regime. We will also emphasize the most practical and successful applications of these devices, i.e. as sensors of force, mass, and displacement.

The main topics to be covered include: mechanical sensors, cantilever mechanics (statics and dynamics), dissipation and noise in mechanical systems, nanomechanical transducers, cooling mechanical resonators, the standard quantum limit on displacement measurement, nanomechanical mass and force sensing, and current trends and applications.

The course consists of one 2-hour lecture per week and one 1-hour exercise session per week. Exercise sessions will be a forum to discuss and resolve assigned exercises. Exercises will not be graded, but their content will form the basis for the final exam and their completion is the best way to prepare for it. The final exam will be oral and will cover the major topics of the course. Grades will be on a scale of 1 to 6 based on this exam. The course will be conducted in English.

This course will be aimed at 3rd-year bachelor and master students in physics and nanoscience. Physics III is a prerequisite. Previous course-work in solid-state physics and statistical mechanics is expected.

Most of the source material and reading in this class will be drawn from original papers in scientific journals and will be provided in class. Some reading will be based on short sections of *Foundations of Nanomechanics*, A. N. Cleland (Springer, 2003), *Fundamentals of Nanomechanical Resonators*, S. Schmid, L. G. Villanueva, and M. L. Roukes (Springer, 2016), and *Fundamentals of Statistical and Thermal Physics*, F. Reif (McGraw-Hill, 1965). Copies of these readings will be distributed in class.

Lectures: Wednesdays, 10.00-12.00, Alter Hörsaal 2, 1.22

Exercise Session: TBA

Course outline and expectations; What is nanomechanics? Why study nanomechanics? Cantilever basics (static case). Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session: **None.

**Reading: ***Foundations of Nanomechanics*: p. 191-211.

**Downloads:** Lecture notes, Lecture slides.

Cantilever basics (dynamic case). Zener’s model for an anelastic solid. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session: **Problem Set 1, harmonic oscillator (Giulio Romagnoli)

**Reading: ***Foundations of Nanomechanics*: p. 233-237; *Fundamentals of Nanomechanical Resonators: p. 1-54.*

**Downloads:** Lecture notes.

Driven cantilevers; cantilevers as harmonic oscillators. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session: **Problem Set 2, bending of beams (Giulio Romagnoli)

**Reading: ***Foundations of Nanomechanics*: p. 277-301.

**Downloads:** Lecture notes.

Power spectral density; Fluctuation-dissipation theorem; thermal displacement. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session: **None

**Reading: ** *Fundamentals of Statistical and Thermal Physics: p. 560-573; Fundamentals of Nanomechanical Resonators: p.151-154.*

**Downloads:** Lecture notes, excerpt from thesis of Dr. Nicola Rossi about flexural motion of a beam.

Thermal displacement, phase, and frequency noise; thermally limited mechanical transducer; thermal limits on force, mass, and force gradient sensitivity. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session: **Paper Discussion 1, mass sensing (Lukas Schneider)

**Reading:** *Fundamentals of Statistical and Thermal Physics: p. 560-573; Fundamentals of Nanomechanical Resonators: p.151-154.*

**Downloads:** Lecture notes.

Measurement of displacement, frequency, and dissipation; types of transducers for nanomechanics; focus on optical interferometry. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session: **Paper Discussion 2, cantilever magnetometry (Giulio Romagnoli)

**Reading:** *Fundamentals of Nanomechanical Resonators: p.115-147.*

**Downloads:** Lecture notes.

Freezing out thermal motion; motivations for ground-state cooling; quantum harmonic oscillator; cryogenic ‘brute-force’ cooling; feedback cooling. Lecture videos (Part 1, Part 2).

**Exercise Session:**

**Reading:**M. Poggio et al., *Phys. Rev. Lett.* **99**, 017201 (2007).

**Downloads:** Lecture notes.

Feedback cooling: practical considerations, measurement bandwidth, dynamic range, sensitivity. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session:** Paper Discussion 3, brute-force ground-state cooling (Lukas Schneider)

**Reading:** F. Marquardt et al., *J. Mod. Opt.* **55**, 3329 (2008).

**Downloads:** Lecture notes.

Cavity or optical cooling: semi-classical picture. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session:** Problem Set 3, quantum harmonic oscillator (Lukas Schneider)

**Reading: **

**Downloads:** Lecture notes.

Cavity or optical cooling: quantum picture; quantum fluctuation-dissipation theorem. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session:** Paper Discussion 4, cavity-assisted ground-state cooling (Giulio Romagnoli)

**Reading:** A. A. Clerk et al., *Rev. Mod. Phys.* **82**, 1155 (2010) and Appendix.

**Downloads:** Lecture notes.

Back-action; standard quantum limit (SQL); approaching SQL. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session:** None

**Reading:**

**Downloads:** Lecture notes.

We will discuss nanowires (NWs) as nanomechanics resonators, including applications as mechanical sensors. Mechanical modes, non-linearity, mode coupling will be discussed. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session:** Paper discussion 5 (Dr. Estefani Marchiori)

**Reading:** N. Rossi, Ph.D. Thesis in Physics (2019)*.*

**Downloads:** Lecture slides; N. Rossi, *Ph.D. Thesis in Physics *(2019).

We will discuss the origin of mechanical dissipation in nanomechanical elements and explain how this dissipation can be minimized. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session: **Problem Set 4 (Dr. Estefani Marchiori)

**Reading: ***Fundamentals of Nanomechanical Resonators*: p.57-90.

**Downloads:** Lecture notes.

We will discuss applications of nanomechanical resonators. In particular, we will cover devices used for doing torque magnetometry and torque-detected magnetic resonance. We will review the major concepts covered in the course in view of the final exam. Lecture video (Part 1).

**Exercise Session: **None

**Reading:** None

**Downloads:** Lecture notes.