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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: Mondays, 14:00-15:00, 4.1

Course outline and expectations; What is nanomechanics? Why study nanomechanics? Cantilever basics (static case).

**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.

**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; power spectral density; fluctuation-dissipation theorem.

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

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

**Downloads:** Lecture notes.

Fluctuation-dissipation theorem; thermal displacement, phase, and frequency noise; thermally limited mechanical transducer.

**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.

**Exercise Session: **Paper Discussion 1, mass sensing (Lorenzo Ceccarelli)

**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.

**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.

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

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

**Downloads:** Lecture notes.

Feedback cooling: practical considerations, measurement bandwidth, dynamic range, sensitivity.

**Exercise Session:** Problem Set 3, quantum harmonic oscillator (Lorenzo Ceccarelli)

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

**Downloads:** Lecture notes.

Cavity or optical cooling: semi-classical picture, quantum picture.Back-action; standard quantum limit (SQL); approaching the SQL.

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

**Reading: **

**Downloads:** Lecture notes.

Back-action; standard quantum limit (SQL); approaching SQL.

**Exercise Session:**

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

**Downloads:** Lecture notes.

We will discuss experiments carried out in the last decade, which have succeeded in demonstrating the preparation of a nano-mechanical oscillator in its ground state.

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

**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.

**Exercise Session:** Problem Set 4 (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.

**Exercise Session: **None

**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. We will also take a look at nanomechanics-related experiments being carried out here in our department.

**Exercise Session: **None

**Reading:**

**Downloads:** Lecture notes.