Ebook: Quantum Phenomena in Mesoscopic Systems

Within the last decade there has been considerable progress in the nanolithography of electronic structures, in atomic-scale imaging, as well as in self-assembling of materials and manipulation at the nanoscale. These new realities are presently triggering an outburst of interest in the enormous variety of applications of nanotechnology. With increasing capabilities, powerful new concepts in the physics of quantum devices, expertise on atomic and molecular interactions and knowledge of biological structure-function relationships are rapidly converging from different research fields.

One of the most socially pervasive and revolutionary applications of these achievements is in the field of information processing, storage and transmission.

Progress in computer large-scale integration will soon require devices beating the heat dissipation problem and pushing down the linear size below the one of the presently available microchips. This is becoming a reality at unprecedented pace.

On this scale, organic molecules, clusters and carbon nanotubes can be produced and self-assembled. They have very flexible and tunable conduction properties. In the next forthcoming computer generation, infoscience and nanoscience are intimately connected.

The collection of lectures included in this book explores the grey zone of mesoscopic condensed-matter physics: quantum phase coherence can be preserved in electronic systems at the nanometer scale, if temperature is low enough and dissipative effects due to the environment are controlled. Transport properties can no longer be described by the semiclassical drift diffusion models, because the size of the device becomes comparable with the Fermi wavelength of the electron wave function, leading to geometry- and size-dependent conduction properties. Electron-electron correlations and disorder of the medium give rise to unexpected features of macroscopic quantization and to possibilities of fine tuning with applied electric and magnetic fields.

It has been shown that the spin of molecular clusters and quantum dots can be manipulated and used to build entangled quantum states. Recent improvements in our understanding of spin relaxation processes open up the possibility of controlling the spin transport in nanodevices, such as spin valves or current-controlled magnetization switches. The new field of “spintronics” is taking off.

Quantum coherence is the basis for quantum algorithms of computing which have been shown to speed up some of the most time-consuming information processing (factorization algorithms, inverse telephone book algorithm and others).

The use of nanostructures for the implementation of quantum logical gates can become a reachable task, provided various fundamental problems are solved. Among these:

i) How to protect the superposition of quantum states and of entangled states from decoherence.

ii) How to control the device by operating from the outside, without affecting quantum coherence. The proposals that can be found in the literature cover the whole range from adiabatic (slow) control via external applied fields to ultrafast applied pulses (e.g., femtosecond laser pulses).

iii) How to optimize the mutual influence of different devices for scale integration, preparation of the quantum state and read-out purposes and develop efficient algorithms for quantum computing.

At present, the challenge is the fabrication of the building blocks of solid-state information devices based on quantum coherence: the “qubits”.

The material presented here by leading experts in the field shows to which extent the available semiconducting nanostructures and superconducting devices of highest performance can be used as tools to reach this task. Gate voltages or tunnel junctions confine electrons in semiconducting heterostructures, including quantum dots (QD), quantum wires (QW) or two-dimensional layers (2d electron gas). These electron systems can have striking quantization phenomena of charge and conductance as it occurs in quantum point contacts (QPC) and quantum Hall systems. Similar effects including coherent superposition of states involving one single Cooper pair, or quantized flux states are obtained in small superconducting systems involving Josephson junctions.

Our present mastering of nanotechnology allows for theoretical proposals to become a reality by tailoring appropriate quantum systems. It is amazing how devices which only one decade ago were still considered as being out of reach by the skeptics are presently used in the pioneer laboratories as probes of local quantum properties and as tools to check theoretical hypotheses. This is the case for the single-electron transistor or the single Cooper pair box.

In connection with these topics, fundamental questions were addressed about the metal insulator transition in disordered two-dimensional systems, quantum Hall conductance in heterostructures, and Kondo conductance in quantum dots, as well as electron decoherence.

These and other novel developments in mesoscopic physics were presented and discussed at the CLI Course of the International School of Physics “E. Fermi” held in Varenna (Italy), July 9-19, 2002.

We have organized the material as follows:

Lecture 1 by Aleiner and coworkers deals with the corrections to the conductivity in two-dimensional disordered systems due to the electron-electron interaction. The presentation sets the stage in a simple and physical way to discuss the temperature dependence of the conductivity on impurity and e-e scattering processes.

Inelastic scattering processes limit the phase coherence time in a way that is still strongly debated as concernes the expected and measured low-temperature limit. Lecture 2 by Birge and coworker gives an overall review of the dispute which started in 1998 when Mohanty, Jariwala and Webb published measurements on mesoscopic Au wires in which the phase-coherent time seemed to saturate at low temperature. Were this the case, many theoretical expectations would be shaken from their very foundations. The approach is pedagogical and stresses the possible role of other coherent scattering mechanisms as Kondo effect due to very low unavoidable concentration of magnetic impurities (as low as 10 ppm!). These could be responsible for the anomalous inelasticity and electron decoherence.

Kondo effect has been unambiguously proven to affect conductance in quantum dots at the Coulomb blockade regime in beauteful experiments by Kastner and coworkers and by many others since then. In lecture 3, Kastner reports on these and more recent findings. Some speculations on the mesoscopic conductance fluctuations in an open quantum dot under ac pumping appear in the contribution by Kravtsov. Computational approaches for electronic transport in molecular devices are presented in lecture 5 by Car and coworker, with special consideration of non-equilibrium conditions.

The total electron spin in a quantum dot can be thought of as a collective variable and qubits can be envisaged in which entanglement of various dot spins is produced. Loss and coworker review the physics of gate operations based on quantum dot spin-qubits in lecture 6 and discuss decoherence in these systems.

Josephson-junction–based qubits are presented by Tsai and coworkers in lecture 7. They report how they defeated decoherence in a Cooper pair box for the first time. Major source of decoherence can be attributed to the background charge fluctuation in the device. This is discussed theoretically by Falci and coworkers in the next lecture.

The search for a Hilbert subspace spanned by a quantum device during its dynamics, decoupled from the environment, using quantum-dynamical algebras is outlined by Rasetti in lectures 9 and 10. Operating in this space would provide noiseless and error-avoiding encoding of the quantum information.

In lecture 11 a recent work on tunneling spectroscopy in cleaved edge wires by Yacoby and coworkers appears, reprinted from Science. There are various hints that the collective excitation spectrum shows features which can only be interpreted within the Luttinger-liquid paradigm for electrons interacting in one-dimensional systems. This experimental paper is complemented by the contribution of Sassetti and coworkers on the theory of transport in quantum wires using bosonization technique, which separates the electronic charge and spin degrees of freedom.

Lecture 13, presented by Lerner, deals with one of the main analytical approaches to quantum transport and thermodynamics in disordered electronic systems, that is the non-linear σ model. The ensemble average over the disorder configurations follows the classical approach by Efetov and others.

The strong e-e correlations in low-dimensional system could be responsible for the large variety of transitions to different states of matter. The combined effects of e-e interations and disorder in electronic solids is discussed in lecture 14 by Giamarchi. In the following lectures we have collected the contributions given to the School on non-Fermi-liquid model approaches to superconductivity, one-dimensional systems, fractional quantum Hall effect and frustrated spin systems by Fabrizio, Levitov, Giuliano, Kramer, Capriotti and Ercolessi, respectively.

The book benefits from the fact that the lecturers have rewritten and rearranged the material discussed at the School, together with their coworkers. This enhances its readability and effectiveness for a reader that is cut out of the pedagogical environment where these lectures were first presented.

Regrettably, a few of the key contributions given to the School are not included in this book. Also, a wealth of posters that were presented by the participants are not included here.

Unfortunately, there is no effective way to record and convey the benefit of the numerous discussions between participants that are a significant integral feature of the School.

In spite of being a deforming mirror of what was debated at the School, still we believe that this book can be useful both to the students that attended to the school and to the interested reader.

We are most grateful to the staff of the Italian Physical Society who proposed and supported, with the help of the European Community, this School and helped us to organize it. In particular we would like to acknowledge the excellent and warm hospitality by Barbara Alzani and her team at Villa Monastero and their perfect organization and charming favour, even in front of the most unusual queries of the participants. Thanks to them the Villa became an exceptionally lively place, elected by the speakers and the students for exchanging views and opinions about physics and surroundings, round the clock. Gratitude goes to Carmen Vasini for her invaluable but unpleasant task: without her firm and obstinate determination this book would have never come to print.

B. Altshuler, A. Tagliacozzo and V. Tognetti

1. Introduction

2. Most important results

3. Friedel oscillations as the source of the interaction corrections to transport coefficients

4. Longitudinal conductivity: Perturbation theory

5. Kinetic-equation approach

1. Introduction

2. Background

3. Two surprises of 1997

4. A correlation between energy and phase relaxation

5. Magnetic impurities—the signs

6. Magnetic impurities—the tests

7. Conclusions and outlook

1. Introduction

2. Weak-coupling regime

3. Kondo regime

4. Fano regime

1. Introduction

2. The Landauer conductance in the time domain

3. The limit of high frequencies

4. Conductance fluctuations for the noise and harmonic ac pumping

5. Conductance fluctuations for an almost periodic ac field and statistics of seros of the dephasing functions

6. Effect of commensurability in the frequency domain

1. Introduction

2. Landauer formalism

3. Master equation for the electronic density matrix

4. Effective single-particle approximation

5. Application to a 1D tunneling system: the current continuity problem

6. Master equation and charge continuity

7. 1D tunneling revisited, conclusions

1. Introduction

2. Quantum entanglement between distinguishable parties

3. Quantum entanglement with electron spins in quantum-dots

4. Quantum correlations between indistinguishable particles

5. Dynamics of entanglement in quantum gate operations

6. Conclusions

1. Superconducting qubits

2. Charge qubit

3. Device

4. Qubit control

5. Multi-pulse experiment

6. Coexistence of charge and phase oscillation

7. Summary

1. Introduction

2. Simple estimates of decoherence

3. Model for 1/f noise

4. Qubit at the optimal point

5. Pure dephasing

6. 1/f noise in single-shot measurements

7. Comparison with the oscillator environment

8. Repeated measurements and inhomogeneuos broadening

9. Charge echo

10. Conclusions

1. Quantum computation, decoherence

2. Error-avoiding codes, universality

3. Symmetrization and computation within error-avoiding codes

1. Josephson junctions: Quantum dynamical algebra

2. Josephson junctions: The phase-number problem

1. Introduction

2. The Tomonaga-Luttinger model

3. The effective action

4. Current transport

5. Results

6. Conclusion

1. Introduction

2. From the TOE model to a functional integral

3. Averaging over disorder

4. The NLsigmaM functional

5. Mesoscopic fluctuations within the NLsigmaM

6. Coulomb and pairing interactions in the sigma model

1. Introduction

2. Basic description

3. Basic description and preconceived ideas

4. Methods

5. Bragg glass and disordered Wigner crystal

6. Further steps and unsolved questions

1. Introduction

2. A model for alkali-doped fullerenes

3. Dynamical mean-field results

4. Conclusions

1. Introduction

2. Dirac fermions in 2D carbon

3. Electrons in a nanotube

4. Field effect

5. Screening and chiral anomaly

6. Energy anomaly for 1D dirac fermions

7. Conclusions

1. Introduction

2. The Haldane-Shastry model of a one-dimensional antiferromagnet and its elementary excitations

3. Spinon dynamics in the Haldane-Shastry model

4. The Kuramoto-Yokoyama model of a one-dimensional insulator and its elementary excitations

5. One-spinon one-holon dynamics in the Kuramoto-Yokoyama model

6. Physical consequences of the interaction between spinons and holons

7. Conclusion

1. Introduction

2. Introduction to Composite Fermion theory

3. The Chern-Simons transformation with spin

4. The propagator of the gauge field fluctuations

5. The propagator of the Composite Fermions

6. Solution of the Dyson equation

7. The energy gap

8. Conclusion