In this paper, we briefly introduce the differences between classical phase transitions and quantum phase transitions, as well as the characteristics of quantum phase transitions, and give examples of typical quantum phase transitions and related model applications, such as topological fermion condensation and superfluid-insulator phase transitions. Among them, the quantum Rabi model has played an important role in the fields of quantum information and quantum optics in recent years.
Written by | Yue Chen (2023 Ph.D. graduate, Institute of Theoretical Physics; Supervisor: Prof. Xiaosong Chen; Research Interests: Statistical Physics and Complex Systems)
Phase transitions in quantum many-body systems have attracted a great deal of theoretical and experimental research in naturally occurring systems (e.g., condensed matter) and in man-made systems produced by cold atomic gases [1-14]. A phase transition is a fundamental change in the state of a system when a parameter of the system (the order parameter) passes through a special value (phase transition point). The states on both sides of the phase transition point are characterized by different types of order, usually from symmetrical or disordered states (which contain some symmetry of the Hamiltonian) to symmetry breaking or ordered states (which do not have that symmetry), although the Hamiltonian still possesses this symmetry.
Classical (thermodynamic) phase transitions are phase transitions at finite temperatures that are the result of the thermal motion and interaction of particles competing with each other. For the stable phase of the system, the free energy is taken as a minimum, and the free energy is determined by the result of the internal energy of the system competing with the entropy representing the disordered degree at a given temperature. For example, the free energy curve of water intersects with the free energy curve of ice at 0°C (at a standard atmospheric pressure), 0°C is the freezing point of water, the free energy of water is low at high temperatures, and the free energy of ice is low at low temperatures, so water is liquid at high temperatures and solid at low temperatures. So, at absolute zero, will the system still have a phase transition? According to classical physics, there is no entropy at zero temperature, so there should be no phase transitions, for example, the molecules in water are all motionless at the lowest energy position, forming a kind of ice with only one phase. However, this is not the case. The motion of microscopic particles does not essentially follow classical Newtonian mechanics, but should be described by quantum mechanics. Classical phase transitions can be described entirely by thermodynamics without the use of quantum mechanics, but phase transitions that occur at zero temperatures need to be described by quantum mechanics. According to the uncertainty principle in quantum mechanics, the position and momentum of microscopic particles cannot be determined at the same time. Therefore, at absolute zero, the particles still do not stop moving, and there is the so-called "zero-point energy", which causes quantum fluctuations, just as thermal motion causes thermal fluctuations. Therefore, the competition of kinetic and potential energy of particles at absolute zero leads to the existence of different phases and the phase transitions between them. Unlike the classical thermodynamic phase transitions at finite temperatures, it is not the thermal fluctuations that come into play here, but the quantum fluctuations. Quantum fluctuations will cause the transformation of the system from order to disorder.
One might think that absolute zero is impossible to achieve according to the third law of thermodynamics, and therefore it is not practical to discuss such phase transitions at absolute zero, which is not. Similar to classical critical phenomena, quantum critical phenomena are also described by some general laws, and in many cases, the properties of a system at non-zero temperatures are determined by quantum phase transition points.
Unlike classical phase transitions, quantum phase transitions can only be achieved by changing physical parameters at zero temperature, such as magnetic fields or pressures. At zero temperature, the equilibrium state system is always in its lowest energy state (or in a superposition state of equal weight in the degenerate state if the lowest energy is degenerate). A quantum phase transition describes a sudden change in the ground state of a many-body system due to quantum fluctuations, which can be a second-order phase transition. Quantum phase transitions occur at quantum critical points, when quantum fluctuations drive correlation length divergence.
Topological fermion condensed quantum phase transitions are an example. In the case of a three-dimensional Fermi liquid, this phase transition transforms the Fermi surface into a Fermi body. Such a phase transition can be a first-order phase transition, as it transforms the two-dimensional structure of the Fermi surface into a three-dimensional one. As a result, the topological charge of the Fermi liquid is abrupt, as it can only take discrete values. Another example is the "superfluid-insulator" phase transition of rarefied atoms in an optical trap. In 1995, the Bose-Einstein condensation of dilute rubidium atomic gas was achieved by laser cooling, magnetic trapping and evaporative cooling. The earliest experiments were to trap gas molecules in a potential well and achieve condensation. Some people use the method of laser standing waves to form a lattice of potential wells and barriers. When the temperature drops to several tens of nanokelvins (10^(-9)K), a Bose-Einstein condensation is produced, and a sharp peak can be observed from the velocity distribution of the escaping gas molecules near the origin, as well as some satellite peaks, which react to periodic barriers. This indicates that molecules can move through the barrier (tunneling effect) because these satellite peaks are caused by diffraction. If the barrier is raised, when a certain threshold is exceeded, the central peaks and satellite peaks disappear and become a blur. This means that the particles cannot move within the barrier and are "localized". The former state is called the "superfluid" state, while the latter state is called the "insulator" state. The "superfluid-insulator" phase transition here is a quantum phase transition because it is caused by quantum fluctuations, not thermal fluctuations.
The band theory of solids, developed after the establishment of quantum mechanics, explains why some metals are conductors and others are insulators. Later research found that some materials should be metals in terms of bands, but they are actually insulators. Further studies have shown that this is related to the Coulomb repulsion between electrons, which cannot be described by a simple image of the motion of a single particle, and this type of material is called "Mott insulator". The study of the phase transition of such insulators into conductors under doping and pressurization conditions is a hot topic, which is related to the study of high-temperature superconductors with wide application prospects. It has been argued that the study of Bose-Einstein condensation of cold atoms provides a new way to solve this problem: the direct regulation of this phase transition by experimental means.
The quantum Rabi model, which describes the interaction between a photon field and a two-level atomic system, is one of the simplest models for studying light-matter interactions, having its origins in a semi-classical model more than 80 years ago. At that time, Rabi introduced a model to discuss the effect of a rapidly changing weak magnetic field on directional atoms with nuclear spin [19, 20]. The simplest case corresponds to a two-quantum state system. The motion of atoms is described in quantum mechanics, and the field is regarded as a classical rotational field. Bloch and Siegert later discussed the effects of non-rotating alternating fields [21], and they discovered an offset in the resonance position—now known as the Bloch-Siegert shift. This shift has been observed in experiments driving superconducting qubits [22].
A similar quantum model was introduced in 1963 by Jaynes and Cummings, describing a two-level atom that interacts with the quantization mode of the optical cavity [23]. Their initial goal was to study the relationship between the quantum theory of radiation and the corresponding semi-classical theory. Despite its simplicity, the quantum Rabi model was not considered exact and solvable at the time. To solve this model, a rotational wave approximation is used. In this approximation, known as the Jaynes–Cummings (JC) model, the inverse rotation term is ignored, which turns out to be a valid approximation of the region of near-resonance and weakly coupled parameters associated with many experiments. The JC model is easy to solve and has been very successfully applied to understand a range of experimental phenomena, such as vacuum Rabi mode splitting [24] and quantum Rabi oscillation [25].
In important experimental developments for engineered quantum systems, all relevant system parameters are table, making it possible to reach new regions of quantum coupling. These systems include superconducting qubits coupled to microwave waveguide resonators [26-29], LC resonators [30-32], and mechanical resonators [33-35]. In particular, it is possible to achieve a super-strong coupling region. In addition, a classical simulator of the quantum Rabi model in the deeply strongly coupled state has been implemented in the waveguide superlattice written by femtosecond lasers [36]. In addition, results have been reported for superconducting qubit resonator circuits in both the super-coupled state and the stronger coupled state [37, 38]. In this coupling region, the usual rotational wave approximation is no longer valid, and the counterrotation term cannot be ignored. Direct evidence of JC model failure has been reported [26]. Various methods have been proposed to solve the problem of strongly coupled regions, including what came to be known as generalized rotational wave approximation [39-45], which was used to obtain continuous approximations of the eigenspectra of quantum Rabi models. Based on this, some interesting phenomena caused by the reverse rotation term have been predicted [46-52].
In another aspect of development, Braak discovered in 2011 that the quantum Rabi model is precisely solvable. Braak gives the exact solution of the quantum Rabi model in the Bargmann–Fock space of the analytic function, and derives the conditions for determining the energy spectrum [53, 54]. Subsequently, other researchers reproduced this condition using the Bogoliubov transform [55]. It is further found that the analytic solution of the quantum Rabi model can be given by the confluent Heun function [56, 57], in which the famous Judd isolated exact solution [58] appears as a truncation of the infinite series defining the confluence Heun function. Braak's analytical solution of the quantum Rabi model has led to a wave of solutions to the complete eigenspectra of various known generalized models of the quantum Rabi model.
Now, with the rapid experimental progress in detecting the coupling regions of strength [59, 60], superstrength [26, 31, 61], and deep strength [27,28], the quantum Rabi model has attracted much attention [40, 42, 46, 53, 55, 56, 62-78], and it plays an important role in quantum optics [23], condensed matter physics [79], and quantum information [80]. One needs to acquire the properties of its quantum phase transition. Although the analytical solution of the quantum Rabi model has been obtained, it is not in closed form and cannot be directly applied to the study of quantum phase transitions, so approximate analytical (such as perturbation theory) and numerical methods are usually used to study quantum phase transitions. For the quantum Rabi model of single-mode cavity field, Hwang et al. (2015) discovered that there is a second-order phase transition from the normal phase to the super-radioactive phase, and studied its critical behavior and dynamics [16]. In 2021, Shen et al. proposed a dual-mode quantum Rabi model Hamiltonian that only involves the interaction between the quantized electric field and one component of the atomic spin, and its ground-state phase diagram is consistent with the single-mode quantum Rabi model, but the critical point is different [82]. In 2023, Chen et al. studied the dual-mode quantum Rabi Hamiltonian [95], which contains both the interaction of the quantized electric field with one component of the atomic spin and the interaction of the quantized magnetic field with the other spin component [95], and found that the ground-state phase diagram contains four phases: the normal phase, the electrical superradiating phase, the magnetic superradiating phase, and the electromagnetic superradiating phase, and the intersection point of the four phases is a quadruple critical point. The phase transition from the normal phase to the electrical superradiating phase and the magnetic superradiating phase is second-order, the order parameter is the average number of photons, and this phase transition breaks the discrete spatial reflection invariance. The phase transition from the electrical superradiating phase to the magnetic superradioactive phase is of the first order. If the atom-photon coupling strength parameters of the collective are equal, there is a continuous unitary transformation invariance. This continuous unitary transformation invariance fails in the electromagnetic superradiating phase because the phase is infinitely degenerate. In the excitation energy spectrum, there are three critical lines in which the excitation energy becomes zero and the Nambu-Goldstone mode appears. The critical exponents of the excitation energy and photon number are the same as those of the single-mode quantum Rabi model [95].
In addition to the two-level model, the interaction between the two-mode cavity field and the three-level system leads to a number of important phenomena, such as electromagnetically induced transparency [83] and dark state [84], which are advantageous in the precise control of the capture and transfer of coherent quantum states [85]. Three-level systems, often referred to as "qutrit", are also important in quantum information. Compared with two-level schemes, quantum key distribution based on qutrits is more resistant to attacks [86, 87], and quantum computation using qutrits is faster and has a lower error rate [88, 89]. Qutrit quantum computers with trapped ions have been proposed [90]. In addition, three-level systems are used to construct quantum thermal engines [91, 92]. Identifying the quantum phases and quantum phase transitions that may be involved in the dual-mode model can help to further understand these light-matter interaction models and expand their applications. The two-mode and three-level interaction model at the thermodynamic limit has attracted much attention. Hayn et al. studied the quantum phase transition using the generalized Holstein-Primakoff transform and revealed that it exhibits two superradiation quantum phase transitions, which can be first- or second-order [93]. Cordero et al. found that multicolor ground state phase diagrams can be divided into monochromatic regions by variational analysis [94]. Zhang et al. reported the analytical calculations of the ground state phase diagram, scaling function, and critical exponent of the two-mode three-level quantum Rabi model [8]. After the discovery of quantum phase transitions and critical phenomena in the quantum Rabi model, further research on the scaling behavior of the Rabi and Dicke models showed that the two models belong to the same universal class. These advances have led to new insights into quantum phase transitions that are not at thermodynamic limits.
Whether it is a condensed matter system or a quantum optical system, the study of quantum phase transitions has led to an in-depth understanding of the abrupt phenomena in nature. With the progress of theory and experiments, people will surely discover more essential laws about phase transitions in quantum many-body systems.
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