This textbook is drawn from notes for a two-semester graduate course in quantum mechanics. It begins with the most constrained quantum system, and recovers the rest of the subject by relaxing those constraints one at a time. The starting point is a single qubit, the smallest nontrivial Hilbert space with the strongest possible restriction on its dynamics, made concrete by a Bloch cube whose six faces are the cardinal states of a spin-1/2 system. Tensor products admit many qubits; lattices give them a place to live; time evolution sets them in motion; the continuum limit produces wavefunctions; three-dimensional angular momentum, the hydrogen atom, and perturbation theory follow; Lorentz invariance promotes the lattice of spinors to the Dirac equation; and the renormalization group asks how theories at different scales relate. Each chapter loosens one feature of the qubit while keeping the others fixed, so that the standard apparatus of graduate quantum mechanics arrives as a sequence of controlled generalizations rather than as separate topics. Discrete-to-continuous transitions recur at four scales: in Hilbert-space dimension, in real space, in time, and in coupling. The book closes by reimposing one of the original constraints, returning to a two-level system that is now a logical qubit protected by quantum error correction, with the fault-tolerance threshold appearing as an unstable RG fixed point and supplying the reason a logical qubit, independent of its underlying hardware, can exist at all.

Conventional physics publications systematically omit the failed experiments and informal collaborations that contribute to physicists’ knowledge. We propose an inverse seminar format to narrow the gap between published and actual physics.

The LaAlO3/SrTiO3 (LAO/STO) interface hosts a gate-tunable superconducting two-dimensional electron gas (2DEG), which can be programmed to create quantum devices, such as ballistic electron waveguides and quantum dots. To fully exploit this platform for quantum transport, a key requirement is the ability to shuttle single electrons, electron pairs, and other exotic states between spatially separated devices with precision. Surface acoustic waves (SAWs), which travel along the surface of a solid, offer a powerful route to achieve this through their moving electrical potential that captures and transfers electrons. In particular, SAWs in the GHz regime enable fast, controlled transport of individual quantum particles. Although this approach is well-explored in GaAs-based 2DEG, SAW generation in STO remains largely unexplored due to the lack of intrinsic piezoelectricity at room temperature. Here, we investigate room-temperature SAWs in LAO/STO and observe SAW modes up to 2.2 GHz with very low propagation loss of the order 10(-3) dB per wavelength. To directly visualize these modes, we employ atomic acoustic force microscopy, achieving sub-micron resolution imaging of the SAW wave forms, providing insight into the electrostriction-induced SAW generation mechanism. Our measurements indicate a shear horizontal-type mode, which provides the ability to couple to in-plane degrees of freedom for future acoustoelectric and quantum device applications. This work studies the fundamentals of SAW excitation and propagation on STO, a widely used and commercially available substrate, enabling straightforward coupling of SAWs to a broad range of materials that can be grown or transferred onto STO.

Superconductivity in strontium titanate has remained enigmatic for over 50 years. The LaAlO3/SrTiO3 (LAO/STO) heterointerface enables systematic dimensional confinement, from two-dimensional electron gas to quasi-one-dimensional nanostructures, providing unprecedented access to this quantum state. While transport measurements in patterned devices reveal puzzling phenomena, including width-independent critical currents and anomalous pairing, suggesting one-dimensional behavior, direct local probes for the patterned interface and its superconducting response have remained lacking. Here, we use ultra-low temperature non-contact atomic force microscopy with dissipation spectroscopy and Kelvin probe force microscopy to locally probe signatures of superconductivity in patterned LAO/STO devices. Spatially resolved energy dissipation measurements reveal signatures of superconductivity with some features confined to edge channels of order ≈200 nm in width. Dissipation spectra exhibit a characteristic nonlinear bias dependence that provides a local diagnostic of superconductivity consistent with the intermediate carrier density near the superconducting dome, which persists up to the critical field. These results demonstrate the ability of atomic force microscopy to probe superconductivity in patterned LAO/STO structures, potentially addressing fundamental, longstanding questions about quantum confinement and transport anomalies in these correlated nanostructures. Superconductivity at the oxide interface of lanthanum aluminate and strontium titanate can be patterned down to the nanometer scale, but its properties have remained a mystery for many decades. The authors present measurements obtained using atomic force microscopy and energy dissipation, which enable imaging and characterization of the superconducting state at the nanoscale under ultra-low-temperature conditions.

We report the details of construction and testing of a Quantum Twisting Microscope, a recently developed scanning probe instrument that enables twist angle dependent electronic measurements on layered materials. Our implementation is based on a commercial atomic force microscope whose open geometry beneath the scan head allows integration of the rotation and translation stages required for QTM operation. We describe the complete fabrication process including tip preparation by focused ion beam deposition and graphite transfer, custom stage assembly with integrated rotation capability, and multistep alignment procedures. To validate the instrument, we perform conductance measurements between graphite layers as a function of twist angle, observing clear 60 degree periodicity consistent with the hexagonal lattice symmetry and conductance enhancements near the commensurate twist angles of 21.8 and 38.2 degrees. These results confirm the instruments ability to resolve crystallographic twist angle dependent transport features. By providing detailed construction and operational guidelines, we aim to make QTM technology accessible to research groups with standard AFM infrastructure, enabling investigations of twist angle dependent phenomena in van der Waals materials, complex oxide heterostructures and chiral systems.

The supercurrent diode effect (SDE), characterized by nonreciprocal critical currents, represents a promising building block for future dissipationless electronics and quantum circuits. Realizing SDE requires breaking both time-reversal and inversion symmetry in the device. Here we use conductive atomic force microscope (c-AFM) lithography to pattern reconfigurable superconducting weak links (WLs) at the LaAlO3/KTaO3 (LAO/KTO) interface. By deliberately engineering the WL geometry at the nanoscale, we realize SDE in these devices in the presence of modest out-of-plane magnetic fields. The SDE polarity can be reversed by simply changing the WL position, and the rectification efficiency reaches up to 13% under optimal magnetic field conditions. Time-dependent Ginzburg-Landau simulations reveal that the observed SDE originates from asymmetric vortex motion in the inversion-symmetry-breaking device geometry. This demonstration of SDE in the LAO/KTO system establishes a versatile platform for investigating and engineering vortex dynamics, forming the basis for engineered quantum circuit elements.

Electrically tunable nonlinear optical responses at the nanoscale remain challenging to achieve because conventional nonlinear materials lack the combination of large susceptibility, nanoscale confinement, and in situ reconfigurability. Here we report electric-field-induced second harmonic (EFISH) generation from a nanoscale tunnel junction defined by conductive atomic force microscope lithography at the LaAlO_3/SrTiO_3 interface. A conducting channel written at the interface is interrupted by a nanoscale insulating gap, across which applied DC bias produces local electric fields exceeding 10^7 V/m. The SHG signal is spatially localized at the junction, exhibits a quadratic bias dependence described by I(2ω) ∝ |χ^(2)_0 + χ^(3) E_DC|^2 with no hysteresis, a modulation depth exceeding 380

Freestanding complex oxide membranes enable the release and transfer of epitaxial films, offering new design freedoms for next-generation electronics. While the LaAlO3/SrTiO3 (LAO/STO) heterostructure exhibits remarkable tunable conductivity at its interface, the active interface remains buried beneath the substrate, limiting access to this functionality. Here, we demonstrate how the LAO/STO heterostructure, in membrane form, can be flipped and precisely positioned on silicon and other platforms using polymer-free micromanipulation. The transferred membranes preserve atomically smooth surfaces, high crystallinity, and key electronic properties. Through the 44-nm insulating STO layer, ultra-low-voltage electron-beam lithography (ULV-EBL) writes conductive nanostructures at the now-accessible STO/LAO interface, offering the potential to function as programmable local gates that modulate charge carriers in the underlying silicon. The platform establishes a general strategy for integrating complex oxide heterostructures with semiconductors, quantum materials, and flexible substrates, enabling new architectures for reprogrammable nanoelectronic devices.

The LaAlO_3/SrTiO_3 (LAO/STO) interface hosts a gate-tunable superconducting two-dimensional electron gas (2DEG) which can be programmed to create quantum devices such as ballistic electron waveguides and quantum dots. To fully exploit this platform for quantum transport, a key requirement is the ability to shuttle single electrons, electron pairs, and other exotic states between spatially separated devices with precision. Surface acoustic waves (SAWs), which travel along the surface of a solid, offer a powerful route to achieve this through their moving electrical potential that captures and transfers electrons.

Superconductivity at oxide interfaces has intrigued researchers for decades, yet the underlying pairing mechanism remains elusive. Here we demonstrate that proximity to a ferroelectric quantum critical point dramatically enhances interfacial superconductivity in KTaO3. By precisely tuning KTaO3 to its quantum critical composition through 0.8

Traditional approaches to undergraduate-level quantum mechanics require extensive mathematical preparation, preventing most students from enrolling in a quantum mechanics course until the third year of a physics major. Here we describe an approach to teaching quantum formalism and postulates that can be used with first-year undergraduate students and even high school students. The only pre-requisite is a familiarity with vector dot products. This approach enables students to learn Dirac notation and core postulates of quantum mechanics at a much earlier stage in their academic career, which can help students prepare for careers in quantum science and engineering and advance the Second Quantum Revolution.

Conjugational defects, also known as solitons, play an important role in the electronic, magnetic, and optical properties of materials. Understanding solitons can uncover intriguing physics and provide insights for designing quantum materials with tailored band structures and electronic properties. Here, we propose a framework to create and control solitons via topological phase transitions in a class of graphene nanoribbons (GNRs) called square-root GNRs, using a transverse electric field. To demonstrate the experimental feasibility, we design and synthesize a representative GNR with a bottom-up approach, with first-principles calculations revealing topological soliton states at the domain wall induced by the electric field. The framework introduced in this Letter can potentially enable direct manipulation of solitons and provide a platform to study them systematically.

Conductive atomic force microscope (c-AFM) lithography can be utilized to create a wide range of LaAlO3 /SrTiO3 (LAO/STO)-based nanoelectronic devices in a reconfigurable manner. Experiments were generally performed with intrinsically insulating LAO/STO heterostructures, with LAO thickness less than the critical value at which a polar catastrophe takes place [< 4 unit cell (u.c.)]. Here, we use inductively coupled plasma reactive ion etching (ICPRIE) to fabricate c-AFM "canvases" on intrinsically conducting LAO/STO samples with >= 4 u.c. LAO. We observe that its interfacial two-dimensional electron gas (2DEG) can be pinched off and then switched back on by c-AFM lithography. Nanowires created with initially conductive LAO/STO interfaces have an order-of-magnitude longer lifetime in ambient conditions, when compared to an identically created 3.4 u.c. LAO/STO nanowire. We also demonstrate key nanoscale properties such as ballistic transport in a quasi-one-dimensional electron waveguide at a 5 u.c. LAO/STO interface. This approach frees c-AFM-written nanodevice designs from time constraints in air associated with <4 u.c. LAO/STO heterostructures. (c) 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial 4.0International (CC BY-NC) license (https://creativecommons.org/licenses/by-nc/4.0/).

In this article, we reflect upon our positive experiences incorporating or working on extra credit projects in algebra-based introductory physics that asked students to connect physics with humanities, social sciences, and everyday life. We give an example of a student project that reflects their creativity and ingenuity and encourages other instructors to offer similar projects.

Quantum mechanics is a notoriously abstract subject, and therefore challenging to teach at pre-college and introductory college levels. Here we introduce the Bloch Cube, a hands-on educational tool which can illustrate key quantum concepts without equations. A series of videos have been created showing how Bloch Cubes can be used to teach concepts such as quantum measurement, quantum dynamics, pure states versus mixed states, and quantum decoherence. Bloch Cube states can assist in the development of more sophisticated concepts such as the Bloch Sphere, which plays a central role in the quantum mechanics of two-state systems and quantum information science.

The origin and function of chirality in DNA, proteins, and other building blocks of life represent a central question in biology. Observations of spin polarization and magnetization associated with electron transport through chiral molecules, known collectively as the chiral induced spin selectivity effect, suggest that chirality improves electron transfer. Using reconfigurable nanoscale control over conductivity at the LaAlO3/SrTiO3 interface, we create chiral electron potentials that explicitly lack mirror symmetry. Quantum transport measurements on these chiral nanowires reveal enhanced electron pairing persisting to high magnetic fields (up to 18 tesla) and oscillatory transmission resonances as functions of both magnetic field and chemical potential. We interpret these resonances as arising from an engineered axial spin-orbit interaction within the chiral region. The ability to create one-dimensional electron waveguides with this specificity creates opportunities to test, via analog quantum simulation, theories about chirality and spin-polarized electron transport in one-dimensional geometries.

The discovery of two-dimensional superconductivity in LaAlO3/KTaO3 (111) and (110) interfaces has raised significant interest in this system. In this manuscript we report the first successful fabrication of a superconducting quantum interference device (DC-SQUID) in the KTO system. The key device elements, superconducting weak links, are created by conductive atomic force microscope (c-AFM) lithography which can reversibly control the conductivity at the LAO/KTO(110) interface with nanoscale resolution. The periodic modulation of the SQUID critical current, Ic(B), with magnetic field corresponds well with our theoretical modeling, which reveals a large kinetic inductance of the superconducting two-dimensional electron gas in KTO. The kinetic inductance of the SQUID is tunable by electrical gating from the back, due to the large dielectric constant of KTO. The demonstration of weak links and SQUIDs in KTO broadens the scope for exploring the underlying physics of KTO superconductivity, including the role of spin-orbit-coupling, pairing symmetry, and inhomogeneity. It also promotes KTO as a versatile platform for a growing family of quantum devices, which could be applicable in the realm of quantum computing and information.

Interface engineering at complex oxide heterostructures enables a wide range of electronic functionalities critical for next-generation devices. Here it is demonstrated that ultra-low-voltage electron beam lithography (ULV-EBL) creates high-quality mesoscale structures at LaAlO3/SrTiO3 (LAO/STO) interfaces with greater efficiency than conventional methods. Nanowires, tunnel barriers, and electron waveguides are successfully patterned that exhibit distinctive transport characteristics including 1D superconductivity, nonlinear current-voltage behavior, and ballistic electron flow. While conductive atomic force microscopy (c-AFM) previously enabled similar interface modifications, ULV-EBL provides significantly faster patterning speeds (10 mm s(-)(1) vs 1 mu m s(-)(1)), wafer-scale capability (>(10 cm)(2) vs <(90 m)(2)), and maintenance of pattern quality under vacuum conditions. Additionally, an efficient oxygen plasma treatment method is developed for pattern erasure and surface cleaning, which reveals novel surface reaction dynamics at oxide interfaces. These capabilities establish ULV-EBL as a versatile approach for scalable interface engineering in complex oxide heterostructures, with potential applications in reconfigurable electronics, sensors, and oxide-based devices.

Advancements in materials synthesis have been key to unveil the quantum nature of electronic properties in solids by providing experimental reference points for a correct theoretical description. Here, we report hidden transport phenomena emerging in the ultraclean limit of the archetypical correlated electron system SrVO3. The low temperature, low magnetic field transport was found to be dominated by anisotropic scattering, whereas, at high temperature, we find a yet undiscovered phase that exhibits clear deviations from the expected Landau Fermi liquid, which is reminiscent of strange-metal physics in materials on the verge of a Mott transition. Further, the high sample purity enabled accessing the high magnetic field transport regime at low temperature, which revealed an anomalously high Hall coefficient. Taken with the strong anisotropic scattering, this presents a more complex picture of SrVO3 that deviates from a simple Landau Fermi liquid. These hidden transport anomalies observed in the ultraclean limit prompt a theoretical reexamination of this canonical correlated electron system beyond the Landau Fermi liquid paradigm, and more generally serves as an experimental basis to refine theoretical methods to capture such nontrivial experimental consequences emerging in correlated electron systems.