Ebook: Quantum Coherence in Solid State Systems

This Volume originates from the lectures delivered at the CLXXI Course of the “Enrico Fermi” School, which was held in Varenna from July 1 to July 11, 2008. The purpose of the Course was to give an overview of the manifestations of quantum coherence in different solid-state systems including semiconductor confined systems, magnetic systems, crystals and superconductors. Besides being of paramount importance in fundamental physics, the study of quantum coherence furnishes the starting point for important applications like quantum computing or secure data transmission. Quantum coherence has been studied in photon systems in quantum optics as well as in atomic systems after the demonstration of Bose-Einstein condensation. Effects analogous to the ones predicted and observed in atoms and photons have been predicted and observed in different solid-state systems during the last few years. Some examples are polariton condensation and superfluidity in semiconductor quantum wells, magnon condensation in magnetic systems, entanglement in confined semiconductor systems, and in ionic crystals. We notice that the coherent effects discussed during the Course mainly involve elementary excitations in solids like polaritons, excitons, magnons, macroscopic quantities like superconductor currents, and electron spins. Therefore, several new aspects of the physics of quasi-particles are understood and discussed in this context. Finally, due to the variety of systems in which quantum coherence may be observed, solid-state systems are the natural candidates for applications that rely on coherence like e.g. quantum computers.

The basic theoretical tools relevant to the field of quantum coherence have been introduced by Serge Haroche who lectured on quantum coherence in quantum optical system and by Sandro Stringari in his lecture on Bose-Einstein condensation in atomic systems. The theory of polariton condensation was the subject of the lectures by Vincenzo Savona, whereas experiments on polariton condensation were presented and discussed in the lectures by Yoshisha Yamamoto and Benoît Deveaud-Plédran (unfortunately, the latter encurred in an accident and the lecture was delivered by Maxime Richard). The lecture by Sergej Demokritov delt with condensation of magnons. The lecture by Cristiano Ciuti was devoted to parametric effects and superfluidity in low-dimensional polariton systems (polaritonics), whereas Wolfgang Langbein discussed experiments of coherent effects in quantum dot excitons. A link between quantum optics and polariton physics was presented in the lecture by Raffaello Girlanda, whereas Atac Imamoglu lectured on quantum optics in mesoscopic systems, and Keiichi Edamatsu discussed the emission of entangled photons from quantum dots. The topic of manipulation of excitons into a cooperative state was developed in the lecture by Matoko Kuwata-Gonokami, whereas Giuseppe La Rocca devoted his contribution to the self-transparency in semiconductors. The lecture of Daniel Loss was devoted to the physics of spin q-bits in quantum dots, and Michael Mehring discussed spin entanglement in fullerenes. Finally, Michel Devoret showed the insurgence of quantum coherence in Josephson circuits, and Marcus Arndt lectured on experimens on coherence and decoherence with clusters and molecules.

The Course was attended by a large number of students from all over the world, who followed the lectures with great interest and assiduity. They had the opportunity to present their research work in two poster sessions, which were very successful. Thanks to the enthusiasm of the lecturers and of the participants, we found again the friendly and exciting atmosphere we enjoyed years ago as students in the Villa Monastero. We take the occasion to thank warmly everybody. However, the Course could have never been successful without the precious help of the team of the Società Italiana di Fisica including Barbara Alzani, Ramona Brigatti, Marta Pigazzini, Laura Strolin, Marcella Missiroli and Angela Di Giuseppe. We also mention in particular the support of the President of the Società Italiana di Fisica Luisa Cifarelli. Finally, we kindly acknowledge the financial support of the Società Italiana di Fisica and the Ecole Polytechnique Fédérale de Lausanne.

This volume is dedicated to the memory of Franco Bassani, the former President of the Società Italiana di Fisica and our long-time friend. With his great enthusiasm, he gave us an important support during the preparation of this Course. He was not able to join us in Varenna and passed away in the fall of 2008.

A. Quattropani, B. Deveaud-Plédran and P. Schwendimann

These lecture notes review microwave cavity experiments in which Rydberg atoms interact one by one with superconducting cavities. These experiments, performed at Ecole Normale Supérieure (ENS), illustrate the concepts of complementarity, entanglement and decoherence and demonstrate basic steps of quantum logic operations. The recent development of super high-Q cavities storing quantum fields for times larger than a tenth of a second has made possible the Quantum Non-Demolition (QND) counting of photons and the first observation of quantum jumps of light. By combining QND counting procedures with homodyne methods, non-classical states of trapped fields have been reconstructed, among which Schrödinger cat states of light. A time-resolved procedure yields snapshots of the evolving decoherence of these states. In future developments, we plan to implement feedback methods to maintain quantum coherence over long time periods and to study the non-local properties of mesoscopic fields stored in two cavities.

We review the recent progress in the generation of entangled photons in semiconductors. Of particular interest is the entangled photon generation via biexciton-resonant hyperparametric scattering.

A quantum dot with an excess conduction band electron constitutes a new paradigm for solid-state quantum optics. When an external magnetic field is applied along the growth direction, the lowest-lying optical excitations could be modeled as a pair of weakly coupled two-level systems; this system could be used for spin-photon entanglement and optical read-out of the electron spin. If the external field is perpendicular to the growth direction, the two low-energy spin states could be coupled via Raman transitions, allowing for efficient spin cooling and coherent optical spin rotation. The optical physics of a singly charged quantum dot is enriched by the presence of coupling to nuclear and electron spin reservoirs.

Several topics on the implementation of spin qubits in quantum dots are reviewed. We first provide an introduction to the standard model of quantum computing and the basic criteria for its realization. Other alternative formulations such as measurement-based and adiabatic quantum computing are briefly discussed. We then focus on spin qubits in single and double GaAs electron quantum dots and review recent experimental achievements with respect to initialization, coherent manipulation and readout of the spin states. We extensively discuss the problem of decoherence in this system, with particular emphasis on its theoretical treatment and possible ways to overcome it.

This paper reviews recent experiments on matter wave interferometry with large molecules. Starting from an elementary introduction to matter wave physics we discuss far-field diffraction and near-field interferometry with thermally excited many-body systems. We describe the constraints imposed by decoherence and dephasing effects, and present an outlook to the future challenges in macromolecule and cluster interferometry.

This contribution discusses the basic aspects of entanglement and decoherence of spin 1/2 qubits and provides different examples with electron and nuclear spins.

Amplifiers are crucial in every experiment carrying out a very sensitive measurement. However, they always degrade the information by adding noise. Quantum mechanics puts a limit on how small this degradation can be. Theoretically, the minimum noise energy added by a phase-preserving amplifier to the signal it processes amounts at least to half a photon at the signal frequency. In this article, we show that we can build a practical microwave amplifying circuit that fulfills the minimal requirements to reach this quantum limit. The readout of solid-state qubits, and more generally, the measurement of very weak signals in various areas of science, can benefit from this new superconducting device. We also discuss how our circuit can be the basic buiding block for a variety of practical applications such as frequency conversion with and without photon number gain, dynamic cooling and production of entangled signal pairs.

Controlling quantum coherence in light-matter interactions is a key step for advanced ultra-fast and quantum information technology. An ever increasing effort has then been devoted over the years to the development of techniques for manipulating quantum coherence in advanced materials optical devices. We here focus on one of the most attractive recent techniques, electromagnetically induced transparency, which has spawned a flurry of interesting effects in ultra-cold atoms in typical three-level lambda or ladder configurations that are now being extended to solid materials. We review our recent results on electromagnetically induced transparency based on intrinsic free exciton and biexciton states in bulk semiconductors and microcavities. We specifically examine copper oxide crystals Cu₂O) where, akin to the atomic case, transparency may be induced through a lambda configuration obtained from the yellow series of exciton states, and copper chloride crystals (CuCl) where a tunable transparency effect may be induced through a ladder configuration of exciton-biexciton transitions via a wave-vector–selective polaritonic mechanism.

Coherent optical spectroscopy of semiconductor nanostructures is a well-established field with several decades of history. This contribution discusses the selected topics of imaging spectroscopy, speckle analysis, and four-wave mixing. Imaging spectroscopy is getting increasingly popular due to the availability of suited detectors, and for the intuition for the, physics behind the data provoked by a two- or three-dimensional view on the measured quantities. A general discussion on the optical imaging and detector requirements is presented, followed by examples concerning microcavity polaritons in spectral and time domain as well as in real space and reciprocal space. Speckle analysis is a linear optical technique conceived a decade ago capable of extracting microscopic properties such as dephasing from ensembles of localized optical excitations even in the presence of inhomogeneous broadening. Four-wave mixing is a well-known technique of coherent spectroscopy, which in the last decade has been improved by the introduction of heterodyne detection and spectral interferometry, enabling to investigate quantum wires and quantum dots, both in ensembles and also for individual localized transitions.

In these lecture notes of the Summer School “Quantum coherence in solid-state systems”, I will present in a pedagogical way an overview of recent advances on the physics of polariton excitations in semiconductor microcavities in the strong light-matter coupling regime and I will discuss how the control of their dynamics can provide interesting quantum phenomena and devices with peculiar properties. In particular, I will present two different types of cavity polariton excitations, originating from two different optically active electronic transitions: i) exciton transitions in undoped quantum wells; ii) intersubband transitions in doped quantum wells containing a dense two-dimensional electron gas. I will draw the fundamental analogies and differences between these two types of polaritons. Concerning exciton-polaritons, I will explain how polariton-polariton phase-coherent interactions have been exploited for the experimental realization of micro-optical parametric amplifiers and for the generation of quantum correlated twin photon beams. The same kind of interactions is shown to lead to quantum fluid propagation properties. Concerning intersubband cavity polaritons, I will discuss: a) the essential features of the peculiar ultrastrong coupling regime and quantum vacuum radiation phenomena; b) very recent advances on electrically driven intersubband polariton devices emitting in the mid and far infrared; c) the possibility of lasing without population inversion based on stimulated scattering. In these lecture notes, accompanied by a comprehensive bibliography, the degree of technicality is reduced as much as possible in order to focus on the key physical issues, whose qualitative and intuitive aspects are discussed in detail.

Here we present a microscopic quantum theory able to describe the quantum optical effects originating from nonlinear optical interactions involving excitons. This theory has been exploited to predict the generation of entangled photon pairs in semiconductors. We also discuss the influence of many-body and correlation effects on the transient optical response of quantum well excitons embedded in semiconductor microcavities.

We present a short introduction to a series of review papers and books published in the last 10 years, relative to the theoretical and experimental developments in the field of ultracold atomic gases.

Magnons are Bose particles, therefore under particular conditions they should demonstrate Bose-Einstein condensation. However, to reach BEC at room temperature one needs to increase the chemical potential of the magnon gas above the zero value characterizing the state of true thermal equilibrium. In this lecture I present our recent results on BEC in magnons gas, driven by a microwave pumping. The room temperature kinetics and thermodynamics of the magnons gas was investigated by means of the Brillouin light scattering technique. We show that for high enough pumping powers the relaxation of the driven gas results in a quasi-equilibrium state described by the Bose-Einstein statistics with a non-zero chemical potential. Further increase of the pumping power causes BEC in the magnon gas documented by an observation of the magnon accumulation at the lowest energy level. Using the sensitivity of the Brillouin light scattering to the coherence degree of the scattering magnons we confirm spontaneous emergence of coherence of the magnons accumulated at the bottom of the spectrum, if their density exceeds a critical value. Interference of two magnon condensates is observed as well.

Collective quantum mechanical phenomena such as superfluidity and superconductivity are observed for particles at a high density and low temperature. These collective effects can also take place in optically manipulated materials through a precise control of light-matter interactions. The most successful example is the creation of ultra cold atomic gas prepared by laser cooling and evaporation cooling. Various quantum mechanical phenomena including Bose-Einstein condensation and paring of Fermionic atoms have been extensively studied. The rapid development of ultra cold atom quantum physics stimulates other fields including the study of excitons in semiconductors. Several new experimental results triggered a growing interest to revisit many body phenomena of excitonic particles, BEC of excitons in two-dimensional coupled quantum wells, condensation of cavity polaritons in semiconductor microcavities and exciton BEC in bulk semiconductors. Other new aspects of such quantum degenerate excitonic particles are their potential for manipulating and controlling the quantum coherence of photons, which is a key technique for quantum information technology. Phase-preserving control of photons is possible by the formation of “dark states”, i.e. optically inactive states with a long-lived phase coherence of excitonic particles. Highly quantum degenerate excitonic particles, such as ensemble of ultracold biexcitons and spin-flipped excitons, could be potential candidate for the media for coherence storage. In these lecture notes, we present recent experiments on the creation and detection of high-density and cold excitonic particles into a highly quantum degenerate regime with precise control of optical excitation. We discuss new experimental aspects regarding coherence storage and observation of collective quantum phenomena in semiconductors. We also discuss the prospects of spontaneous Bose Einstein condensation of excitons in a bulk crystal of Cu₂O.

We report in this review the experimental observations that led us to claim the occurrence of Bose-Einstein condensation for polaritons in microcavities. First, we will recall the properties of microcavity polaritons that are of interest for Bose condensation. Then, we will list the observed properties in the highest quality sample that we have been able to study a CdTe monolithic microcavity. In this sample, that shows a very large Rabi splitting of 26 meV, we first observe a thermalized distribution of polaritons below threshold. At threshold, observed to occur for ground-state occupation factors of the order of 1, we observe a strong increase of the ground-state emission. At the same density, a strong reduction of the linewidth and a build-up of the first-order coherence are observed. Also, the distribution of the polaritons appears to change from Boltzmann-like to a Bose-like distribution. Polarization of the emission is observed that is not due to spontaneous symmetry breaking but to a slight birefringence of the material. Second-order coherence shows that the polariton condensate still contains a significant excitonic fraction. The main observation in favor of BEC is the clear evidence for long-range spatial coherence of the whole polariton gas. This measurement is obtained through the use of a specially designed Michelson interferometer. Importantly enough, we discuss the possible similarities and differences between a polariton condensate and a Vertical Cavity Surface Emitting Laser (VCSEL). Although many of the properties that we have observed might show similar behavior for a VCSEL-like laser, some of the differences clearly allow to differentiate a polariton condensate from a standard laser. According to the accepted terminology, as we are observing a thermalized distribution below threshold, what we observe is not a polariton laser but a Bose-Einstein condensate indeed. Eventually, we study some of the consequences of the inherent disorder in semiconductor microstructures. This brings along some important observations. The first one is mode synchronization that allows the condensate to overcome residual disorder. The second one is the observation of pinned vortices, an indication of possible superfluid-like effects in the presence of disorder.

In the last decade, a two-dimensional bosonic system called microcavity exciton-polariton has emerged as a new, promising candidate of Bose Einstein Condensation (BEC) in solids. Many pieces of important evidence of polariton BEC have been established very recently in GaAs and CdTe microcavities at liquid-helium temperature, opening a door to rich many-body physics inaccessible in experiments before. In parallel with experimental progresses, theory and numerical simulations are developed, and our understanding of the system has greatly advanced. In this article, we review the recent experimental and theoretical results obtained at Stanford University.

We derive a theory of polariton Bose-Einstein condensation based on the many-body quantum field theory of interacting Bose particles. In particular, we describe self-consistently the linear exciton-photon coupling and the exciton-non-linearities, by generalizing the Hartree-Fock-Popov description of BEC to the case of two coupled Bose fields at thermal equilibrium. In this way, we compute the density-dependent one-particle spectrum, the energy occupations and the phase diagram. The results quantitatively agree with the existing experimental findings. We discuss the conditions under which polaritons can be considered as a dilute Bose gas at thermal equilibrium. While the diluteness is enforced by the very peculiar energy-momentum dispersion, thermal equilibrium is only partially achieved in common experiments. A basic tool to model the kinetics of polariton BEC is then derived in terms of Boltzmann equations, including polariton-phonon and polariton-polariton scattering. We discuss under which conditions thermal equilibrium condensation can be reached and how to design future microcavity samples in order to reach this goal.

Coupled electron-nuclear spins are promising physical systems for quantum information processing: By combining the long coherence times of the nuclear spins with the ability to initialize, control, and measure the electron spin state, the favorable properties of each spin species are utilized. We present a vision of how these systems could be used as the fundamental processor of a quantum computer, in which the nuclear spins are analogous to local memory units, whereas the electron spins act as buses. In particular, we focus on control of a system in which a single electron spin is coupled to N nuclear spins via resolvable anisotropic hyperfine interactions. High-fidelity universal control of this 1-electron–N-nuclei system is possible exclusively using excitations on a single electron spin transition. This electron spin actuator control is implemented by using optimal control theory to find the modulation sequences that generate the desired unitary operations. A model to fully characterize decoherence of the nuclear qubits in this context is currently under investigation. Up to now, coherent control using an electron spin actuator in an ensemble of anisotropic hyperfine-coupled 1-electron–1-nucleus systems has been achieved, providing evidence that we can generate any unitary operation on such systems while sitting on a single transmitter frequency (Hodges J. et al., Phys. Rev. A, 78 (2008) 010303). Data was acquired using a custom-built pulsed electron spin resonance spectrometer. Complex modulation sequences found by the GRadient Ascent Pulse Engineering (GRAPE) algorithm (Khaneja N. et al., J. Magn. Res., 172 (2005) 296) were used to perform simple-preparation–quantum-operation–readout experiments. The next generation of the experimental setup will include an improved spectrometer, bandwidth-constrained GRAPE (Aiello C. and Hodges J., in preparation), and samples with larger Hilbert spaces.

We have introduced a quasi–one-dimensional (1D) bosonic gas of indirect excitons in semiconductor nanotubes of type-II quantum wells and studied theoretically the coherent properties of this interacting gas. We firstly obtain the spectrum of a single exciton in such cylindrical geometry. Thus, we calculate the exciton-exciton interaction at the ground state and show that the excitons form a quasi-1D bosonic gas with repulsion interaction. Finally, by mapping the exciton gas onto an exactly solvable quasi-1D bosonic gas at the low temperature and low density limit where the bosonic features of the gas are dominant, we show that the gas is in the strong-coupling regime, and the excitons become fermions.

We have theoretically studied the radiative decay of excitons in a film with nano-to-macro crossover thickness. Under the phase matching condition between exciton center-of-mass motion and the radiation field, the exciton radiative decay rate gets larger with increasing thickness because of the increment of the interaction volume. This is called exciton superradiance. However, it is suppressed at thickness over a particular length because the Fermi's golden rule is broken in exciton-photon interaction, i.e., the emitted photon begins to be reabsorbed and then the exciton behaves as a polariton. Its suppression condition is the main result of our study. We define an apparent propagation speed of the superradiant exciton as an effective thickness, which renormalizes the multiple reflection effect in the film, divided by the radiative decay time. This propagation speed also increases together with the thickness, and the exciton superradiance is suppressed when it reaches the group velocity of exciton polaritons. Afterward, the exciton radiative decay rate decreases inversely proportional to the thickness.

Since the birth of quantum mechanics, interference effects between different paths have represented one of the most fascinating signatures of this theory. Systems of three-level atoms show a remarkable example of this effect. If one couples a long-living metastable state to an excited state by means of a strong laser field and one then probes the ground-excited transition of the dressed atom by means of another weak laser field, one will see no absorption and all the radiation will be transmitted through the atomic medium. This phenomenon is called Electromagnetically Induced Transparency (EIT) and it can be explained in terms of quantum interference between different atomic excitation schemes. An atomic medium under EIT conditions shows a polaritonic dispersion, with three branches near the Raman two-photon transition between the ground and metastable states. The central polariton is characterized by a huge reduction of the group velocity of light which can be controlled by the intensity of the dressing field. The energy of the incoming radiation then coherently oscillates between electromagnetic field and atomic excitations: this gives rise to a trapping effect which eventually slows down the radiation, as can be seen in a full quantum treatment. In the limit of a vanishing coupling field, all the radiation energy is stored as a coherent collective atomic excitation. We have first studied the dispersion of light for a lattice of three-level atoms in the linear regime by means of the semi-classical Transfer Matrix technique: we have obtained the different polariton bands and the reflection spectrum at the interface of the atomic medium. The latter shows a dip in correspondence of the Raman resonance which is associated to the refractive index going to 1. We have verified that this feature still holds in the case of a hole (lack of one atom) in the lattice. Instead, in the case of an impurity consisting of an undressed two-level atom, a peak appears at Raman resonance and the radiation is fully scattered back. This case can be important for the behaviour of the system in the non-linear regime. Beyond the steady-state picture, the slow propagation of light opens the possibility of modulating the dressing laser field in order to manipulate in real time a travelling polariton. This is an example of a dynamical photonic structure; similar schemes have been suggested for photon lifter applications. We are presently investigating the effect of dynamical changes of the dressing field on the scattering amplitude from a defect: this problem is of interest in the perspective of probing the new quantum phases of ultracold gases by means of slow light.

The competition between the Zeeman energy and the Rashba and Dresselhaus spin-orbit couplings is studied for ferromagnetic states in the fractional quantum Hall regime. A transition of the spin-polarization direction, which acquires an in-plane component even if the magnetic field is perpendicular to the quantum well, is predicted to occur at small values of the Zeeman energy, as an effect of the spin-orbit interaction. For a given fractional state, we theoretically investigate this phenomenon in the perturbative limit of high magnetic fields. We consider the Laughlin wave functions and the Pfaffian state as specific examples of possible ground states, and show that the quantitative features of this transition provide valuable information about the nature of the correlated ground-state. In particular, a relation to the pair-correlation function is derived. We also discuss indications of non-analytic features around the fractional states and include effects of the nuclear bath polarization, which are significant in a relevant range of temperatures and magnetic fields.

Recent mid-infrared absorption (Anappara A. A. et al., Appl. Phys. Lett., 87 (2005) 051105) and electroluminescence (Sapienza L. et al., Phys. Rev. Lett., 100 (2008) 136806) experiments on microcavities embedding quantum wells have shown strong coupling between a cavity photon mode and the transition between two conduction subbands, being the lowest one filled with a dense two-dimensional electron gas. We theoretically investigated the effect of this strong light-matter coupling on intersubband electroluminescence. In a first work (De Liberato S. and Ciuti C., Phys. Rev. B, 77 (2008) 155321), using a cluster factorization method, we derived a closed set of dynamical equations for the quantum well carrier and cavity photon occupation numbers, the correlation between the cavity field and the intersubband polarization, as well as polarization-polarization contributions. Solving the resulting set of equations in the stationary regime, we were able to fully characterize the transport and electroluminescence properties as a function of the applied voltage. We discovered a strong-coupling equivalent of the Purcell effect, that can increase the efficiency of intersubband light emitting devices (normally of the order of 10⁻⁵) of various orders of magnitude. In a second work (De Liberato S. and Ciuti C., arxiv:0802.409) we calculated the spectral function of the electrons inside the microcavity. We were thus able to investigate how the electronic states are modified by the coupling to the microcavity vacuum field and showed that resonant electron tunneling from a narrow-band injector can selectively excite superradiant states and produce ultraefficient polariton electroluminescence.