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We study fluctuations of survival probability in an open quantum system classically described by a map with a mixed phase space. Our results provide the first numerical support to theoretical predictions that such fluctuations have a fractal structure, quantitatively related to the algebraic decay of the classical survival probability.
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... that Q , Q max, t . It is easy to see from (15) that at 1 ø Q ø Q max, t the distribution F t ͑ Q ͒ displays the same decay exponent as P ͑ t ͒ and G ͑ Q f ͒ . It follows that the value of g to be used in (14) is g 1.323 ͑ 9 ͒ , as obtained from survival probability decays in Fig. 2 in agreement with known results [8,9]. So (14) predicts fractal dimension ഠ 1.33 . According to the above discussion, at a given time t this fractal structure should be observable on scales .df min ϳ 1 ͞ Q max, t ϳ 1 ͑͞ t ln r ͒ . Furthermore, as the validity of the theory is restricted to times less than the Heisenberg time t H ϳ r , at no time can formula (14) be expected to hold on scales smaller than ϳ 1 ͑͞ r ln r ͒ (which in our case is of order e 2 8 ). To check this prediction, we have computed fl uctuation patterns at four different times (two examples are given in Fig. 3), from which we have extracted fractal graph dimensions D and the exponents g C of correlation functions [formula (13)]. The inset of Fig. 4 shows such correlations. Fractal dimensions have been computed by means of the modi fi ed box counting algorithm used in [6]: results are shown in Fig. 4. We tested the algorithm, by computing fractal dimensions of curves generated ...
Context 2
... that Q , Q max, t . It is easy to see from (15) that at 1 ø Q ø Q max, t the distribution F t ͑ Q ͒ displays the same decay exponent as P ͑ t ͒ and G ͑ Q f ͒ . It follows that the value of g to be used in (14) is g 1.323 ͑ 9 ͒ , as obtained from survival probability decays in Fig. 2 in agreement with known results [8,9]. So (14) predicts fractal dimension ഠ 1.33 . According to the above discussion, at a given time t this fractal structure should be observable on scales .df min ϳ 1 ͞ Q max, t ϳ 1 ͑͞ t ln r ͒ . Furthermore, as the validity of the theory is restricted to times less than the Heisenberg time t H ϳ r , at no time can formula (14) be expected to hold on scales smaller than ϳ 1 ͑͞ r ln r ͒ (which in our case is of order e 2 8 ). To check this prediction, we have computed fl uctuation patterns at four different times (two examples are given in Fig. 3), from which we have extracted fractal graph dimensions D and the exponents g C of correlation functions [formula (13)]. The inset of Fig. 4 shows such correlations. Fractal dimensions have been computed by means of the modi fi ed box counting algorithm used in [6]: results are shown in Fig. 4. We tested the algorithm, by computing fractal dimensions of curves generated ...
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Citations
... Trajectories originating in the chaotic sea are trapped by the interface region for a long time before being released back into the chaotic sea. Quite remarkably, as shown in Ref. [34], this influence extends to include energy correlation functions in the quantum domain as well. ...
Kicked rotor is a paradigmatic model for classical and quantum chaos in time-dependent Hamiltonian systems. More than fifty years since the introduction of this model, there is an increase in the number of works that use kicked rotor model as a fundamental template to study a variety of questions in nonlinear dynamics, quantum chaos, condensed matter physics and quantum information. This is aided by the experimental approaches that have implemented many variants of the quantum kicked rotor model. The problems addressed using kicked rotor and its variants include the basic phenomenology of classical and quantum chaos, transport and localization in one- and higher dimensional kicked systems, effects of disorder and interactions, resonant dynamics and the relation between quantum correlations and chaos. This would also include a range of applications such as for constructing ratchet dynamics and for atom-optics based interferometry. This article reviews the current status of theoretical and experimental research devoted to exploring these ideas using the framework of kicked rotor model.
... As the Landau-level filling factor ν approaches its critical value, the localization length ξ diverges algebraically as ξ ∝ |ν − ν C | −γ . Theoretical studies have shown that observables like the distribution of the local density |ψ(r)| 2 [7,11] or the equilibrium current density |j(r)| 2 [12] display multifractality of the density fluctuations that leads to anomalous diffusion [10] and, consequently, a power-law decay of the density correlations, a slow decay of temporal wave-packet auto-correlations [13] and, most significantly for our purpose, multifractal conductance fluctuations [14][15][16][17][18]. ...
... Multifractality was initially introduced to characterize the statistical properties of fluid turbulence [19,20] and thereafter studied, not only in turbulent flows [21][22][23], but also in a variety of fields like the analysis of DNA sequences [24], atmospheric science [25,26], econophysics [27], heartbeat dynamics [28,29], and cloud structure [30], and many other parts of physics. In condensed-matter science, most investigations of multifractality, which manifests itself at some phase transitions, employ a combination of theoretical and numerical techniques [15][16][17][31][32][33][34][35]. The experimental characterizations of multifractality in such condensed-matter settings require high-precision experiments in good-quality samples, often at low temperatures and at high magnetic fields. ...
We present the first experimental evidence for the multifractality of a transport property at a topological phase transition. In particular, we show that conductance fluctuations display multifractality at the integer-quantum-Hall plateau-to-plateau transition in a high-mobility mesoscopic graphene device. We establish that to observe this multifractality, it is crucial to work with very high-mobility devices with a well-defined critical point. This multifractality gets rapidly suppressed as the chemical potential moves away from these critical points. Our combination of multifractal analysis with state-of-the-art transport measurements at a topological phase transition provides a novel method for probing such phase transitions in mesoscopic devices.
... If Fo 5 , a stable (automodel) oscillation process may be observed, with the profile shape repeating itself in an undamped oscillation process. The shape of the oscillations in diagrams is identical to the shape of quantum fractals [32][33][34][35][36][37][38][39][40]. Analysis of the wave function distribution shows that the plasma acts like a dispersion medium for electromagnetic wave propagation, i.e., the phase velocity depends on the wave length (oscillation frequency). ...
Electromagnetic oscillations in plasma contained in a rectangular channel were studied using the exact analytical solution of the Klein–Gordon relativistic equation obtained during the research. It has been shown that the oscillations occur at the same frequency at various points in the plasma. The coalescence of frequencies at these points in the plasma is evidence that the plasma oscillations are self-consistent. Research shows that the phase velocity depends on the wavelength (frequency). Therefore, plasma is a dispersion medium for electromagnetic waves, which is explained by its own internal and external scales of space and time. The obtained solution may be used to research the electron density in plasma by analyzing the conditions of electromagnetic wave propagation in plasma, e.g., when the wave is not fully reflected. Using experimentally obtained conditions of full reflection, it is possible to find the plasma frequency and the electron density in the plasma.
... 261,262 Several studies have also established the sensitivity of the averaged survival probability to the nature of the classical dynamics. 244,263,264 For example, Wilkie and Brumer 265 have shown that in the irregular case the average survival probability will fall below its long time average. More recently, for Hamiltonians whose underlying classical dynamics is chaotic but with local interactions, a thorough investigation of the various timescales associated with the average survival probability has been made. ...
Intramolecular vibrational energy redistribution (IVR) impacts the dynamics of reactions in a profound way. Theoretical and experimental studies are increasingly indicating that accounting for the finite rate of energy flow is critical for uncovering the correct reaction mechanisms and calculating accurate rates. This requires an explicit understanding of the influence and interplay of the various anharmonic (Fermi) resonances that lead to the coupling of the vibrational modes. In this regard, the local random matrix theory (LRMT) and the related Bose-statistics triangle rule (BSTR) model have emerged as a powerful and predictive quantum theories for IVR. In this Perspective we highlight the close correspondence between LRMT and the classical phase space perspective on IVR, primarily using model Hamiltonians with three degrees of freedom. Our purpose for this is threefold. First, this clearly brings out the extent to which IVR pathways are essentially classical, and hence crucial towards attempts to control IVR. Second, given that LRMT and BSTR are designed to be applicable for large molecules, the exquisite correspondence observed even for small molecules allows for insights into the quantum ergodicity transition. Third, we showcase the power of modern nonlinear dynamics methods in analysing high dimensional phase spaces, thereby extending the deep insights into IVR that were earlier gained for systems with effectively two degrees of freedom. We begin with a brief overview of recent examples where IVR plays an important role and conclude by mentioning the outstanding problems and the potential connections to issues of interest in other fields.
... There are many other important topics involving quantum billiards, such as nodal line structures and wavefunction statistics [148][149][150][151][152][153][154][155][156][157], quantum chaotic scattering [158][159][160][161][162][163][164][165][166] with experimental demonstrations on two dimensional electron gas (2DEG) [50, [167][168][169][170] where the electrons are described by the two dimensional Schrödinger equation, quantum pointer (preferred) states and decoherence [52, [171][172][173][174][175], universal conductance fluctuations [176][177][178][179][180], chaos-assisted quantum tunneling [181][182][183][184][185][186][187][188][189][190][191][192][193][194][195], effects of electron-electron interaction [30,[195][196][197][198][199] Other prototypical models that have been investigated extensively in the development of the field of quantum chaos include kicked rotors in terms of diffusion [274][275][276][277][278][279][280][281][282][283][284][285][286][287][288], entanglement as signatures of referring classical chaos [289][290][291], experimental realizations [292][293][294][295], and other related topics [285,[296][297][298][299], and the Dicke model [300][301][302][303][304][305][306] and the Lipkin-Meshkov-Glick model [307][308][309][310] to account for the many-body effects. In particular, in the phenomena of dynamical localization of kicked rotors with parameters in the classically chaotic region, the momentum localization length has an integer scaling property versus the reduced Planck constant ; while in the vicinity of the golden cantori, a fractional ÿ scaling is observed, which was argued as the quantum signature of the golden cantori [311][312][313][314][315][316]. ...
... For open quantum systems, an important topic of quantum chaos is quantum chaotic scattering [50, [158][159][160][161][162][163][164][165][166][167][168][169][170]. In nonrelativistic systems, a general observation is that, for classically mixed systems, the transmission (or conductance) of the corresponding quantum system exhibit many sharp resonances caused by the strongly localized states around the classically stable periodic orbits, while for classically chaotic system, the peaks are either broadened or removed. ...
... For example, in closed quantum systems, the basic phenomena that have been and continue to be studied include energy level-spacing statistics [4,5,[9][10][11]35,[41][42][43][44][45][46][47][48][49][50][51] and quantum scarring [1,[26][27][28][29]. In open Hamiltonian systems, quantum chaotic scattering has been investigated extensively [36][37][38][39][40][80][81][82][83][84][85][86][87][88][89][90]. In addition, the phenomena of quantum diffusion and wavefunction localization in classically driven chaotic systems have been studied [8,[12][13][14][15][16][17][18][19][20][21][22][23]31,91,92]. ...
... For the simplified equations (81)- (86), one can get the eigenvalues µ lm (α), where the magnetic flux α can be regarded as a control parameter, the quantities l and ν are related to each other by ν = l − α, and m represents the m'th solution for a given l. ...
... The seminal work of Jalabert, Baranger, and Stone [80] suggested that conductance fluctuations in the ballistic regime can be a probe of quantum chaos, establishing for the first time a connection between quantum transport in solid-state devices and classical chaos. Subsequent works [81][82][83][84][85][86][87][88] revealed that UCFs are intimately related to the study of quantum chaotic scattering [36,173]. A result in nonrelativistic quantum chaotic scattering is that, for those with integrable or mixed (nonhyperbolic) classical dynamics, sharp conductance fluctuations can occur. ...
Quantum chaos is generally referred to as the study of quantum manifestations or fingerprints of nonlinear dynamical and chaotic behaviors in the corresponding classical system, an interdisciplinary field that has been active for about four decades. In closed quantum systems, for example, the basic phenomena studied include energy level-spacing statistics and quantum scarring. In open Hamiltonian systems, quantum chaotic scattering has been investigated extensively. Previous works were almost exclusively for nonrelativistic quantum systems described by the Schrödinger equation. Recent years have witnessed a rapid growth of interest in Dirac materials such as graphene, topological insulators, molybdenum disulfide and topological Dirac semimetals. A common feature of these materials is that their physics is described by the Dirac equation in relativistic quantum mechanics, generating phenomena that do not usually emerge in conventional semiconductor materials. This has important consequences. In particular, at the level of basic science, a new field has emerged: Relativistic Quantum Chaos (RQC), which aims to uncover, understand, and exploit relativistic quantum manifestations of classical nonlinear dynamical behaviors including chaos. Practically, Dirac materials have the potential to revolutionize solid-state electronic and spintronic devices, and have led to novel device concepts such as valleytronics. Exploiting manifestations of nonlinear dynamics and chaos in the relativistic quantum regime can have significant applications. The aim of this article is to give a comprehensive review of the basic results obtained so far in the emergent field of RQC. Phenomena to be discussed in depth include energy level-spacing statistics in graphene or Dirac fermion systems that exhibit various nonlinear dynamical behaviors in the classical limit, relativistic quantum scars (unusually high concentrations of relativistic quantum spinors about classical periodic orbits), peculiar features of relativistic quantum chaotic scattering and quantum transport, manifestations of the Klein paradox and its effects on graphene or 2D Dirac material based devices, regularization of relativistic quantum tunneling by chaos, superpersistent currents in chaotic Dirac rings subject to a magnetic flux, and exploitation of relativistic quantum whispering gallery modes for applications in quantum information science. Computational methods for solving the Dirac equation in various situations will be introduced and physical theories developed so far in RQC will be described. Potential device applications will be discussed.
... In open Hamiltonian systems, quantum chaotic scattering has been investigated extensively. [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68] In addition, the phenomena of quantum diffusion and wavefunction localization in classically driven chaotic systems have been studied. 69-84 a) Electronic mail: Ying-Cheng.Lai@asu.edu ...
... 132,137,138,[175][176][177][178][179][180][181] The seminal work of Jalabert, Baranger, and Stone 55 suggested that conductance fluctuations in the ballistic regime can be a probe of quantum chaos, establishing for the first time a connection between quantum transport in solid-state devices and classical chaos. Subsequent studies [56][57][58][59][62][63][64][65] revealed that UCFs are intimately related to quantum chaotic scattering. 53,54,57 An intriguing phenomenon is that classical chaos can suppress the conductance (or transmission) fluctuations. ...
Quantum chaos is referred to as the study of quantum manifestations or fingerprints of classical chaos. A vast majority of the studies were for nonrelativistic quantum systems described by the Schrödinger equation. Recent years have witnessed a rapid development of Dirac materials such as graphene and topological insulators, which are described by the Dirac equation in relativistic quantum mechanics. A new field has thus emerged: relativistic quantum chaos. This Tutorial aims to introduce this field to the scientific community. Topics covered include scarring, chaotic scattering and transport, chaos regularized resonant tunneling, superpersistent currents, and energy level statistics—all in the relativistic quantum regime. As Dirac materials have the potential to revolutionize solid-state electronic and spintronic devices, a good understanding of the interplay between chaos and relativistic quantum mechanics may lead to novel design principles and methodologies to enhance device performance.
... Conductance fluctuations, as a function of the magnetic field, have been shown to have a fractional fractal dimensions in some simple, one-dimensional, quantum systems, e.g., the kicked rotor [64,68]. In these systems, such a fractional fractal dimension arises if one of the following conditions holds: (1) The PDF P (t), of the charge-carrier survival time t, has a power-law form P (t) ∝ t −γ at large t (as opposed to an exponential decay); (2) the energy correlation C(∆E) of elements of the S-matrix exhibit power laws (i.e., C(∆E) ∝ (∆E) −γ ) [31,34,68,69]. ...
... Conductance fluctuations, as a function of the magnetic field, have been shown to have a fractional fractal dimensions in some simple, one-dimensional, quantum systems, e.g., the kicked rotor [64,68]. In these systems, such a fractional fractal dimension arises if one of the following conditions holds: (1) The PDF P (t), of the charge-carrier survival time t, has a power-law form P (t) ∝ t −γ at large t (as opposed to an exponential decay); (2) the energy correlation C(∆E) of elements of the S-matrix exhibit power laws (i.e., C(∆E) ∝ (∆E) −γ ) [31,34,68,69]. Such survival probabilities are related to the conductance [64,68]; and their multifractal behaviour has been explored [25]. ...
... In these systems, such a fractional fractal dimension arises if one of the following conditions holds: (1) The PDF P (t), of the charge-carrier survival time t, has a power-law form P (t) ∝ t −γ at large t (as opposed to an exponential decay); (2) the energy correlation C(∆E) of elements of the S-matrix exhibit power laws (i.e., C(∆E) ∝ (∆E) −γ ) [31,34,68,69]. Such survival probabilities are related to the conductance [64,68]; and their multifractal behaviour has been explored [25]. ...
In quantum systems, signatures of multifractality are rare. They have been found only in the multiscaling of eigenfunctions at critical points. Here we demonstrate multifractality in the magnetic-field-induced universal conductance fluctuations of the conductance in a quantum condensed-matter system, namely, high-mobility single-layer graphene field-effect transistors. This multifractality decreases as the temperature increases or as doping moves the system away from the Dirac point. Our measurements and analysis present evidence for an incipient Anderson-localization near the Dirac point as the most plausible cause for this multifractality. Our experiments suggest that multifractality in the scaling behaviour of local eigenfunctions are reflected in macroscopic transport coefficients. We conjecture that an incipient Anderson-localization transition may be the origin of this multifractality. It is possible that multifractality is ubiquitous in transport properties of low-dimensional systems. Indeed, our work suggests that we should look for multifractality in transport in other low-dimensional quantum condensed-matter systems.
... Conductance fluctuations, as a function of the magnetic field, have been shown to have a fractional fractal dimensions in some simple, one-dimensional, quantum systems, e.g., the kicked rotor 64,68 . In these systems, such a fractional fractal dimension arises if one of the following conditions holds: (1) The PDF P(t), of the charge carrier survival time t, has a power-law form PðtÞ / t Àγ at large t (as opposed to an exponential decay); (2) the energy correlation C(ΔE) of elements of the S-matrix exhibit power laws (i.e., C(ΔE) / (ΔE) −γ ) 31,34,68,69 . ...
... Conductance fluctuations, as a function of the magnetic field, have been shown to have a fractional fractal dimensions in some simple, one-dimensional, quantum systems, e.g., the kicked rotor 64,68 . In these systems, such a fractional fractal dimension arises if one of the following conditions holds: (1) The PDF P(t), of the charge carrier survival time t, has a power-law form PðtÞ / t Àγ at large t (as opposed to an exponential decay); (2) the energy correlation C(ΔE) of elements of the S-matrix exhibit power laws (i.e., C(ΔE) / (ΔE) −γ ) 31,34,68,69 . Such survival probabilities are related to the conductance 64,68 ; and their multifractal behavior has been explored 25 . ...
... In these systems, such a fractional fractal dimension arises if one of the following conditions holds: (1) The PDF P(t), of the charge carrier survival time t, has a power-law form PðtÞ / t Àγ at large t (as opposed to an exponential decay); (2) the energy correlation C(ΔE) of elements of the S-matrix exhibit power laws (i.e., C(ΔE) / (ΔE) −γ ) 31,34,68,69 . Such survival probabilities are related to the conductance 64,68 ; and their multifractal behavior has been explored 25 . ...
In quantum systems, signatures of multifractality are rare. They have been found only in the multiscaling of eigenfunctions at critical points. Here we demonstrate multifractality in the magnetic-field-induced universal conductance fluctuations of the conductance in a quantum condensed-matter system, namely, high-mobility single-layer graphene field-effect transistors. This multifractality decreases as the temperature increases or as doping moves the system away from the Dirac point. Our measurements and analysis present evidence for an incipient Anderson-localization near the Dirac point as the most plausible cause for this multifractality. Our experiments suggest that multifractality in the scaling behaviour of local eigenfunctions are reflected in macroscopic transport coefficients. We conjecture that an incipient Anderson-localization transition may be the origin of this multifractality. It is possible that multifractality is ubiquitous in transport properties of low-dimensional systems. Indeed, our work suggests that we should look for multifractality in transport in other low-dimensional quantum condensed-matter systems.
... Physically, a sharper resonance corresponds to a longer trapping lifetime and scattering time delay. Previous works on wave or quantum chaotic scattering [66][67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85] established that classical chaos can smooth out (broaden) the sharp resonances and reduce the time delay markedly while integrable dynamics can lead to stable, longlived bound states (or trapping modes). We present concrete evidence for Dirac quantum chimera. ...
We uncover a remarkable quantum scattering phenomenon in two-dimensional Dirac material systems where the manifestations of both classically integrable and chaotic dynamics emerge simultaneously and are electrically controllable. The distinct relativistic quantum fingerprints associated with different electron spin states are due to a physical mechanism analogous to chiroptical effect in the presence of degeneracy breaking. The phenomenon mimics a chimera state in classical complex dynamical systems but here in a relativistic quantum setting - henceforth the term "Dirac quantum chimera," associated with which are physical phenomena with potentially significant applications such as enhancement of spin polarization, unusual coexisting quasibound states for distinct spin configurations, and spin selective caustics. Experimental observations of these phenomena are possible through, e.g., optical realizations of ballistic Dirac fermion systems.
