We consider the Rabi Hamiltonian which exhibits a quantum phase transition (QPT) despite consisting only of a single-mode cavity field and a two-level atom. We prove QPT by deriving an exact solution in the limit where the atomic transition frequency in unit of the cavity frequency tends to infinity. The effect of a finite transition frequency is studied by analytically calculating finite-frequency scaling exponents as well as performing a numerically exact diagonalization. Going beyond this equilibrium QPT setting, we prove that the dynamics under slow quenches in the vicinity of the critical point is universal, that is, the dynamics is completely characterized by critical exponents. Our analysis demonstrates that the Kibble-Zurek mechanism can precisely predict the universal scaling of residual energy for a model without spatial degrees of freedom. Moreover, we find that the onset of the universal dynamics can be observed even with a finite transition frequency.Introduction.-Universality plays a key role for our understanding of quantum phase transitions (QPT) in interacting quantum systems [1]. While the concept of universality is well-established in equilibrium QPT, the question to what extent the concept of universality could be extended to non-equilibrium dynamics of QPT remains largely to be explored [2,3]. For a slow quench across a QPT, the closing spectral gap at a critical point leads to a breakdown of the adiabacity regardless of the quench rate. The scaling of defect formation has been shown to be entirely controlled by critical exponents and quench rate through a successful application of the Kibble-Zurek mechanism (KZM) [4][5][6][7], originally developed for classical phase transitions, to QPT in shortrange interaction models [8][9][10][11][12]. However, whether this scaling holds for fully-connected models [13], which lack spatial degrees of freedom, such as Dicke [14] or Lipkin-MeshkovGlick (LMG) model [15] remains an open problem [16][17][18].The Dicke model considers a system of a quantized singlemode cavity field uniformly coupled to N two-level atoms. It exhibits a superradiant QPT in the thermodynamic limit (N → ∞) [19][20][21][22]. While tremendous efforts have been devoted to understand the QPT of the Dicke model both in and out of equilibrium [19][20][21][22][23][24][25][26][27], a criticality of the Rabi model [27][28][29][30][31][32][33], the most simplified version of Dicke model with N = 1, has been hitherto largely overlooked. Having only two constituent particles, the Rabi model is far from being in the thermodynamic limit where a QPT typically occurs; however, a ratio of the atomic transition frequency Ω to the cavity field frequency ω 0 that approaches infinity, Ω/ω 0 → ∞, can play the role of a thermodynamic limit [27] that allows the spectral gap to be precisely closed at the critical point [1].In this letter, we firstly establish the theory of equilibrium QPT of the Rabi model. At the core of our analysis is a lowenergy effective Hamiltonian that is valid for Ω/ω 0 1 and becom...

We demonstrate that the quantum phase transition (QPT) of the Rabi model and critical dynamics near the QPT can be probed in the setup of a single trapped ion. We first show that there exists equilibrium and nonequilibrium universal functions of the Rabi model by finding a proper rescaling of the system parameters and observables. We then propose a scheme that can faithfully realize the Rabi model in the limit of a large ratio of the effective atomic transition frequency to the oscillator frequency using a single trapped-ion and therefore the QPT. It is demonstrated that the predicted universal functions can indeed be observed based on our scheme. Finally, the effects of realistic noise sources on probing the universal functions in experiments are examined.Introduction.-The experimental realization of quantum phase transition (QPT) in a well-controlled quantum system is of considerable interest [1][2][3][4][5][6][7][8][9]. This is particularly important for the study of the dynamics of QPT where a controlled change of the system parameters are necessary [10][11][12]. Understanding the dynamics of QPT is at the frontier of the study of critical phenomena; the full extent of the universality in non-equilibrium dynamics of a system that undergoes a QPT remains to be determined [13,14] and its theoretical underpinnings are being actively investigated [15][16][17][18][19][20].Trapped ions are a particularly promising platform for this purpose thanks to the possibility of precise coherent quantum controls and high-fidelity measurements [3][4][5] as exemplified by the recent observation of the dynamics of classical phase transitions [21,22]. A major challenge, however, lies in the fact that the QPT typically occurs in a thermodynamic limit where the number of system constituents diverges [23]. Although the universality manifests itself even for a system of finite size in the form of finite-size scaling relations [24,25], it emerges only when the system size is sufficiently large; moreover, a controlled change in the system size under otherwise unchanged conditions is necessary in order to observe the critical exponents. In this respect, and despite the advances in trapped-ion technologies, it is still a formidable challenge to scale up the system size sufficiently to enable the observation of critical phenomena while maintaining the controllability and the coherence of the system [5].Recently, it has been shown in Ref. [20,26] that even a single two-level atom coupled to a harmonic oscillator may undergo a second-order QPT. The experimental realization of such a finite-system QPT is highly desirable, as it opens a possibility to study the dynamics of QPT in a small, fully controlled quantum system with a high degree of coherence without the necessity of the scalability in the number of system components; however, the required parameter regime [20,26] that includes simultaneously extremely large detuning [27][28][29] and large coupling strength [30][31][32] has made it difficult to find a suitable experimental platform ...

We study the phase diagram of the Dicke model in terms of the excitation energy and the radiation-matter coupling constant λ. Below a certain critical value λ c , all the energy levels have a well-defined parity. For λ > λ c the energy spectrum exhibits two different phases separated by a critical energy E c that proves to be independent of λ. In the upper phase, the energy levels have also a well-defined parity, but below E c the energy levels are doubly degenerated. We show that the long-time behavior of appropriate parity-breaking observables distinguishes between these two different phases of the energy spectrum. Steady states reached from symmetry-breaking initial conditions restore the symmetry only if their expected energies are above the critical. This fact makes it possible to experimentally explore the complete phase diagram of the excitation spectrum of the Dicke model.

The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M. -J. Hwang et al, Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which shows a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states leads to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.Comment: 9 pages, 6 figure

We propose a robust realization of the two-photon quantum Rabi model in a trapped-ion setting based on a continuous dynamical decoupling scheme. In this manner the magnetic dephasing noise, which is identified as the main obstacle to achieve long time coherent dynamics in ion-trap simulators, can be safely eliminated. More specifically, we investigate the ultrastrong coupling regime of the two-photon quantum Rabi model whose realization in trapped ions involves second-order sideband processes. Hence, the resulting dynamics becomes unavoidably slow and more exposed to magnetic noise requiring an appropriate scheme for its elimination. Furthermore, we discuss how dynamical decoupling methods take a dual role in our protocol, namely they remove the main source of decoherence from the dynamics while actively define the parameter regime of the simulated model.