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We conjecture a chaos energy bound, an upper bound on the energy dependence of the Lyapunov exponent for any classical/quantum Hamiltonian mechanics and field theories. The conjecture states that the Lyapunov exponent $\lambda(E)$ grows no faster than linearly in the total energy $E$ in the high energy limit. In other words, the exponent $c$ in $\lambda(E) \propto E^c \,(E\to\infty)$ satisfies $c\leq 1$. This chaos energy bound stems from thermodynamic consistency of out-of-time-order correlators (OTOC's) and applies to any classical/quantum system with finite $N$ / large $N$ ($N$ is the number of degrees of freedom) under plausible physical conditions on the Hamiltonians. To the best of our knowledge the chaos energy bound is satisfied by any classically chaotic Hamiltonian system known, and is consistent with the cerebrated chaos bound by Maldacena, Shenker and Stanford which is for quantum cases at large $N$. We provide arguments supporting the conjecture for generic classically chaotic billiards and multi-particle systems. The existence of the chaos energy bound may put a fundamental constraint on physical systems and the universe., Comment: 3 pages, plus 7 pages of supplemental material; v2: minor revision with a reference and a footnote added

We propose a way to observe the photon ring of the asymptotically anti-de Sitter black hole dual to a superconductor on the two-dimensional sphere. We consider the electric current of the superconductor under the localized time-periodic external electromagnetic field. On the gravity side, the bulk Maxwell field is sent from the AdS boundary and then diffracted by the black hole. We construct the image of the black hole from the asymptotic data of the bulk Maxwell field that corresponds to the electric current on the field theory side. We decompose the electric current into the dissipative and non-dissipative parts and take the dissipative part for the imaging of the black hole. We investigate the effect of the charged scalar condensate on the image. We obtain the bulk images that indicate the discontinuous change of the size of the photon ring.

We study rotating global AdS solutions in five-dimensional Einstein gravity coupled to a multiplet complex scalar within a cohomogeneity-1 ansatz. The onset of the gravitational and scalar field superradiant instabilities of the Myers-Perry-AdS black hole mark bifurcation points to black resonators and hairy Myers-Perry-AdS black holes, respectively. These solutions are subject to the other (gravitational or scalar) instability, and result in hairy black resonators which contain both gravitational and scalar hair. The hairy black resonators have smooth zero-horizon limits that we call graviboson stars. In the hairy black resonator and graviboson solutions, multiple scalar components with different frequencies are excited, and hence these are multioscillating solutions. The phase structure of the solutions are examined in the microcanonical ensemble, i.e. at fixed energy and angular momenta. It is found that the entropy of the hairy black resonator is never the largest among them. We also find that hairy black holes with higher scalar wavenumbers are entropically dominant and occupy more of phase space than those of lower wavenumbers., 40 pages, 8 figures

The AdS soliton is a nonsingular spacetime that has a flat conformal boundary with a compact $S^1$ direction. We find a horizonless cohomogeneity-1 metric that describes nonlinear gravitational oscillations of the AdS soliton in five dimensions. We call this spacetime the resonating AdS soliton. This solution is obtained as the nonlinear extension of normal modes of the AdS soliton dual to spin-2 glueball excitations. The boundary energy momentum tensor of the resonating AdS soliton has time periodic components, and it is interpreted as a coherently excited state in the dual field theory. Physical quantities of the resonating AdS soliton are multivalued at a fixed energy, suggesting a transition between different frequency solutions. The energy of the resonating AdS soliton is higher than that of the undeformed AdS soliton, in accordance with the positive energy conjecture proposed by Horowitz and Myers., 20 pages, 3 figures, 1 table. v2: published version

Thermal states in some quantum field theories (QFTs) correspond to black holes in asymptotically AdS spacetime in the AdS/CFT correspondence. We propose a direct procedure to construct holographic images of the black hole in the bulk from a given response function of the QFT on the boundary. The response function with respect to an external source corresponds to the asymptotic data of the bulk field generated by the source on the AdS boundary. According to the wave optics, we can obtain the images from the bulk field propagating in the bulk spacetime. For a thermal state on two-dimensional sphere dual to Schwarzschild-AdS4 black hole, we demonstrate that the holographic images gravitationally lensed by the black hole can be constructed from the response function. In particular, the Einstein rings on the image can be clearly observed and their radius depends on the total energy of the QFT thermal state. These results are consistent with the size of the photon sphere of the black hole calculated in geometrical optics. This implies that, if there exists a dual gravitational picture for a given quantum system, we would be able to probe existence of the dual black hole by the Einstein rings constructed from observables of the quantum system.

Rapidly rotating Myers-Perry-AdS$_5$ (MPAdS$_5$) black holes are shown to be unstable against rotational superradiance of a Maxwell field. From the onset of the instability, time-periodic neutral black hole solutions equipped with a nontrivial electromagnetic wave are obtained, which we call {\it photonic black resonators}. In the horizonless limit, they reduce to geon solutions which may be called {\it photon stars}. Specifically, we introduce a cohomogeneity-1 ansatz for the metric and Maxwell field and construct such solutions with an $R\times SU(2)$ isometry group. We compute thermodynamic quantities and obtain phase diagrams. It turns out that a photonic black resonator has a higher entropy than a MPAdS$_5$ black hole, while it also has a smaller entropy than a black resonator without the Maxwell field. This suggests what is expected for nonlinear dynamics following the Maxwell superradiant instability with the $SU(2)$ isometry., 23 pages, 10 figures

It is challenging to quantify chaos of QCD, because non-perturbative QCD accompanies non-local observables. By using holography, we find that QCD strings at large $N_c$ and strong coupling limit exhibit chaos, and measure their Lyapunov exponent at zero temperature. A pair of a quark and an antiquark separated by $L_q$ in the large $N_c$ QCD is dual to a Nambu-Goto string hanging from the spatial boundary of the D4-soliton geometry. We numerically solve the motion of the string after putting a pulse force on its boundaries. The chaos is observed for the amplitude of the force larger than a certain lower bound. The bound increases as $L_q$ grows, and its dependence is well approximated by a hypothesis that the chaos originates in the endpoints of the QCD string., 12 pages, 13 figures

Clarifying conditions for the existence of a gravitational picture for a given quantum field theory (QFT) is one of the fundamental problems in the AdS/CFT correspondence. We propose a direct way to demonstrate the existence of the dual black holes: imaging an Einstein ring. We consider a response function of the thermal QFT on a two-dimensional sphere under a time-periodic localized source. The dual gravity picture, if it exists, is a black hole in an asymptotic global AdS$_4$ and a bulk probe field with a localized source on the AdS boundary. The response function corresponds to the asymptotic data of the bulk field propagating in the black hole spacetime. We find a formula that converts the response function to the image of the dual black hole: The view of the sky of the AdS bulk from a point on the boundary. Using the formula, we demonstrate that, for a thermal state dual to the Schwarzschild-AdS$_4$ spacetime, the Einstein ring is constructed from the response function. The evaluated Einstein radius is found to be determined by the total energy of the dual QFT. Our theoretical proposal opens a door to gravitational phenomena on strongly correlated materials., Comment: 5+2 pages, 5 figures, accepted for publication in Physical Review Letters. arXiv admin note: substantial text overlap with arXiv:1811.12617

With the aim to reveal universal features of hadronic matter and correlated Dirac insulators in strong AC-electric fields, we study the $\mathcal{N}=2$ supersymmetric QCD with a finite quark mass driven by a rotating electric field $\mathcal{E}_x+i\mathcal{E}_y= E e^{i\Omega t}$. The analysis is done in the holographically dual D3/D7 system in the co-rotating frame, effectively. The nonequilibrium phase diagram is determined from the threshold electric field at which the insulator phase breaks down to a conductive phase due to the AC version of the Schwinger mechanism. The external field induces a rotating current $\mathcal{J}_x + i \mathcal{J}_y = J e^{i\Omega t}$ originating from vacuum polarization and dissipative current in the insulating and conductive phases respectively. Intriguing features are observed as the frequency $\Omega$ approaches resonance with the meson excitation energy $\Omega_{\rm meson}$. There, the threshold minimizes and a condensate of vector mesons with oscillating current exists even in the zero driving field limit. This state, which we call Floquet condensate of vector mesons, is expected to be dynamically stable realizing a non-thermal fixed point that breaks time translational and reversal symmetries. Our finding has many similarities with exciton BEC discussed in solid state systems, where the semiconductor is to be replaced by materials hosting gapped Dirac electrons, e.g. 3D topological insulators or bismuth. Vector meson Floquet condensate may also have implications in the pre-thermalized dynamics in heavy ion collision experiments., Comment: 36 pages, 15 figures

Floquet states can be realized in quantum systems driven by continuous time-periodic perturbations. It is known that a state known as the Floquet Weyl semimetal can be realized when free Dirac fermions are placed in a rotating electric field. What will happen if strong interaction is introduced to this system? Will the interaction wash out the characteristic features of Weyl semimetals such as the Hall response? Is there a steady state and what is its thermodynamic behavior? We answer these questions using AdS/CFT correspondence in the $\mathcal{N}=2$ supersymmetric massless QCD in a rotating electric field in the large $N_c$ limit realizing the first example of a "holographic Floquet state". In this limit, gluons not only mediate interaction, but also act as an energy reservoir and stabilize the nonequilibrium steady state (NESS). We obtain the electric current induced by a rotating electric field: In the high frequency region, the Ohm's law is satisfied, while we recover the DC nonlinear conductivity at low frequency, which was obtained holographically in a previous work. The thermodynamic properties of the NESS, e.g., fluctuation-dissipation relation, is characterized by the effective Hawking temperature that is defined from the effective horizon giving a holographic meaning to the "periodic thermodynamic" concept. In addition to the strong (pump) rotating electric field, we apply an additional weak (probe) electric field in the spirit of the pump-probe experiments done in condensed matter experiments. Weak DC and AC probe analysis in the background rotating electric field shows Hall currents as a linear response, therefore the Hall response of Floquet Weyl semimetals survives at the strong coupling limit. We also find frequency mixed response currents, i.e., a heterodyning effect, characteristic to periodically driven Floquet systems., 30 pages, 12 figures

We study chaotic motion of classical closed strings in the five-dimensional Anti-de Sitter (AdS) soliton spacetime. We first revisit classical chaos using a cohomogeneity-1 string ansatz. We then consider turbulent behaviors of the classical strings when the spatial dependence of the string world-sheet is included. Sensitivity to initial conditions in chaotic systems suggests that the string under chaos tends to stretch in the AdS soliton spacetime in a Lyapunov timescale. In this process, the orbital angular momentum transfers to internal spin due to the turbulence on the string. It follows that the string stays around the tip of the AdS soliton with a jumbled profile. We evaluate the spectra of conserved quantities and discuss their universal power-law scalings in the turbulent behaviors., 23 pages, 10 figures

The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time., 30 pages, 13 figures

Assigning a chaos index for dynamics of generic quantum field theories is a challenging problem, because the notion of Lyapunov exponent, which is useful for singling out chaotic behaviors, works only in classical systems. We address the issue by using the AdS/CFT correspondence, as the large $N_c$ limit provides a classicalization (other than the standard $\hbar \to 0$) while keeping nontrivial quantum condensation. We demonstrate the chaos in the dynamics of quantum gauge theories: Time evolution of homogeneous quark condensates $\langle \bar{q}q\rangle$ and $\langle \bar{q} \gamma_5 q\rangle$ in an ${\cal N}=2$ supersymmetric QCD with the $SU(N_c)$ gauge group at large $N_c$ and at large 't Hooft coupling $\lambda \equiv N_c g_{\rm YM}^2$ exhibits a positive Lyapunov exponent. The chaos dominates the phase space for energy density $E \gtrsim (6\times 10^2)\times m_q^4(N_c/\lambda^2) $ where $m_q$ is the quark mass. We evaluate the largest Lyapunov exponent as a function of $(N_c,\lambda,E)$ and find that the ${\cal N}=2$ supersymmetric QCD is more chaotic for smaller $N_c$., Comment: 9 pages, 5 figures. v2: Analysis of Lyapunov exponent of linear sigma model added, references and a footnote added. v3: Published version. An appendix added for explaining relation between classical chaos and out-of-time-ordered correlator. Title modified, abstract refined v4: Typos and grammatical errors corrected

We examine the nonlinear classical dynamics of a fundamental string in anti-de Sitter spacetime. The string is dual to the flux tube between an external quark-antiquark pair in $\mathcal{N}=4$ super Yang-Mills theory. We perturb the string by shaking the endpoints and compute its time evolution numerically. We find that with sufficiently strong perturbations the string continues extending and plunges into the Poincar\'e horizon. In the evolution, effective horizons are also dynamically created on the string worldsheet. The quark and antiquark are thus causally disconnected, and the string transitions to two straight strings. The forces acting on the endpoints vanish with a power law whose slope depends on the perturbations. The condition for this transition to occur is that energy injection exceeds the static energy between the quark-antiquark pair., Comment: 29 pages, 15 figures

We review the stability analysis of Myers-Perry black holes. The D-dimensional MyersPerry spacetime has spherical horizon and is parametrized by its mass and n = (D − 1)/2 angular momenta. In general, the spacetime has Rt × U(1) isometry group, which corresponds to time translation and rotational symmetries. This symmetry is not enough to separate the gravitational perturbation equations and the perturbation equations are given by partial differential equations of (D − n − 1) coordinates. However, for special values of the angular momentum, the symmetry of the spacetime can be enhanced and it makes the stability analysis easy. In this paper, we focus on two kinds of Myers-Perry black holes. The first one is odd dimensional Myers-Perry black holes with equal angular momenta. This spacetime is cohomogeneity-1, namely, it depends on a single radial coordinate. In this spacetime, gravitational perturbation equations reduce to system of ordinary differential equations. In five dimensions, we find that there is no evidence of instability. On the other hand, for D = 7, 9, 11, 13, we see that the spacetime becomes unstable for large angular momenta. The second spacetime is singly rotating Myers-Perry black holes. This spacetime is cohomogeneity-2, namely, it depends on two coordinates. In this spacetime, the gravitational perturbation equation is given by 2-dimensional partial differential equations. We see the numerical evidence of the instability of this spacetime. At the onset of the instability of these spacetimes, there are statinary perturbations. Thus, these results indicate existence of new phases of rotating black holes in higher dimensions.

We consider a black hole solution whose spatial boundary is a squashed three sphere in Einstein gravity with negative cosmological constant. We solve the Einstein equations numerically and find a warped AdS black hole solution with arbitrary squashing parameter. This solution becomes the ordinary AdS-Schwarzschild solution when the squashing parameter is chosen appropriately. Motivated by this fact, we study N=4 super Yang-Mills theory with zero coupling constant on a squashed three sphere and show that the thermodynamical entropy of this theory roughly agrees with that of the warped AdS black hole up to a factor of 3/4. We also study the confinement/deconfinement transition of the gauge theory. We evaluate the approximate Hawking-Page transition temperature of the warped AdS black hole and find a qualitative agreement between the transition temperatures of the gravity and the gauge theory. These results suggest a duality between the warped AdS solution and N=4 super Yang-Mills theory on a squashed three sphere., 19 pages, 7 figures

We study gravitational perturbations of the Kerr-AdS_5 x S^5 spacetime with equal angular momenta. In this spacetime, we found the two kinds of classical instabilities, superradiant and Gregory-Laflamme instabilities. The superradiant instability is caused by the wave amplification via superradiance, and by wave reflection due to the potential barrier of the AdS spacetime. The Gregory-Laflamme instability appears in Kaluza-Klein modes of the internal space S^5 and breaks the symmetry SO(6). Taken into account these instabilities, the phase structure of Kerr-AdS_5 x S^5 spacetime is revealed. The implication for the AdS/CFT correspondence is also discussed., Comment: 30 pages, 7 figures

We study N =4 super Yang-Mills theories on a three sphere with two kinds of chemical potentials. One is associated with the R-symmetry and the other with the rotational symmetry of S^3 (SO(4) symmetry). These correspond to the charged Kerr-AdS black holes via AdS/CFT. The exact partition functions at zero coupling are computed and the thermodynamical properties are studied. We find a nontrivial gap between the confinement/deconfinement transition line and the boundary of the phase diagram when we include more than four chemical potentials. In the dual gravity, we find such a gap in the phase diagram to study the thermodynamics of the charged Kerr-AdS black hole. This shows that the qualitative phase structures agree between the both sides. We also find that the ratio of the thermodynamical quantities is almost well-known factor 3/4 even at the low temperature., 40 pages, 6 figures, v2: references added, v3: typos corrected and references added

We study nonlinear dynamics of the flux tube between an external quark-antiquark pair in $\mathcal{N}=4$ super Yang-Mills theory using the AdS/CFT duality. In the gravity side, the flux tube is realized by a fundamental string whose endpoints are attached to the AdS boundary. We perturb the endpoints in various ways and numerically compute the time evolution of the nonlinearly oscillating string. As a result, cusps can form on the string, accompanied by weak turbulence and power law behavior in the energy spectrum. When cusps traveling on the string reach the boundary, we observe the divergence of the force between the quark and antiquark. Minimal amplitude of the perturbation below which cusps do not form is also investigated. No cusp formation is found when the string moves in all four AdS space directions, and in this case an inverse energy cascade follows a direct cascade., 38 pages, 23 figures, v2: version to appear in JHEP

An electric field quench, a suddenly applied electric field, can induce nontrivial dynamics in confining systems which may lead to thermalization as well as a deconfinement transition. In order to analyze this nonequilibrium transitions,we use the AdS/CFT correspondence for $\mathcal{N}=2$ supersymmetric QCD that has a confining meson sector. We find that the electric field quench causes the deconfinement transition even when the magnitude of the applied electric field is smaller than the critical value for the static case (which is the QCD Schwinger limit for quark-antiquark pair creation). The time dependence is crucial for this phenomenon, and the gravity dual explains it as an oscillation of a D-brane in the bulk AdS spacetime. Interestingly, the deconfinement time takes only discrete values as a function of the magnitude of the electric field. We advocate that the new deconfinement phenomenon is analogous to the exciton Mott transition., 43 pages, 21 figures

A massless scalar field exhibits an instability at the event horizon of an extreme black hole. We study numerically the nonlinear evolution of this instability for spherically symmetric perturbations of an extreme Reissner-Nordstrom (RN) black hole. We find that generically the endpoint of the instability is a non-extreme RN solution. However, there exist fine-tuned initial perturbations for which the instability never decays. In this case, the perturbed spacetime describes a time-dependent extreme black hole. Such solutions settle down to extreme RN outside, but not on, the event horizon. The event horizon remains smooth but certain observers who cross it at late time experience large gradients there. Our results indicate that these dynamical extreme black holes admit a C^1 extension across an inner (Cauchy) horizon., Comment: 33 pages plus appendices

Aretakis has proved that a massless scalar field has an instability at the horizon of an extreme Reissner-Nordstr\"om black hole. We show that a similar instability occurs also for a massive scalar field and for coupled linearized gravitational and electromagnetic perturbations. We present numerical results for the late time behaviour of massless and massive scalar fields in the extreme RN background and show that instabilities are present for initial perturbations supported outside the horizon, e.g.\ an ingoing wavepacket. For a massless scalar we show that the numerical results for the late time behaviour are reproduced by an analytic calculation in the near-horizon geometry. We relate Aretakis' conserved quantities at the future horizon to the Newman-Penrose conserved quantities at future null infinity., Comment: 44 pages, 19 figures

We study linearized gravitational perturbations of extreme black hole solutions of the vacuum Einstein equation in any number of dimensions. We find that the equations governing such perturbations can be decoupled at the future event horizon. Using these equations, we show that transverse derivatives of certain gauge invariant quantities blow up at late time along the horizon if the black hole solution satisfies certain conditions. We find that these conditions are indeed satisfied by many extreme Myers-Perry solutions, including all such solutions in five dimensions., Comment: 19 pages

We study the gravitational, electromagnetic and scalar field perturbations on the near-horizon geometries of the even-dimensional extremal Myers-Perry black holes. By dimensional reduction, the perturbation equations are reduced to effective equations of motion in AdS$_2$. We find that some modes in the gravitational perturbations violate the Breitenl\"ohner-Freedman bound in AdS$_2$. This result suggests that the even-dimensional (near-)extremal Myers-Perry black holes are unstable against gravitational perturbations. We also discuss implications of our results to the Kerr-CFT correspondence., Comment: 26 pages; v2: minor typo corrected

The higher-dimensional vacuum Einstein equation admits translationally non-uniform black string solutions. It has been argued that infinitesimally non-uniform black strings should be unstable in 13 or fewer dimensions and otherwise stable. We construct numerically non-uniform black string solutions in 11, 12, 13, 14 and 15 dimensions. Their stability is investigated using local Penrose inequalities. Weakly non-uniform solutions behave as expected. However, in 12 and 13 dimensions, strongly non-uniform solutions appear to be stable and can have greater horizon area than a uniform string of the same mass. In 14 and 15 dimensions all non-uniform black strings appear to be stable., Comment: 26 pages, 11 figures. V2: reference added, matches published version

Various higher-dimensional black holes have been shown to be unstable by studying linearized gravitational perturbations. A simpler method for demonstrating instability is to find initial data that describes a small perturbation of the black hole and violates a Penrose inequality. An easy way to construct initial data is by conformal rescaling of the unperturbed black hole initial data. For a compactified black string, we construct initial data which violates the inequality almost exactly where the Gregory-Laflamme instability appears. We then use the method to confirm the existence of the "ultraspinning" instability of Myers-Perry black holes. Finally we study black rings. We show that "fat" black rings are unstable. We find no evidence of any rotationally symmetric instability of "thin" black rings., 35 pages, 12 figures. v2: typos corrected, matches published version

It has been conjectured that a near-extreme Kerr black hole is described by a 2d CFT. Previous work has shown that CFT operators dual to axisymmetric gravitational perturbations have integer conformal weights. In this paper, we study the analogous problem in 5d. We consider the most general near-extreme vacuum black hole with two rotational symmetries. This includes Myers-Perry black holes, black rings and Kaluza-Klein black holes. We find that operators dual to gravitational (or electromagnetic or massless scalar field) perturbations preserving both rotational symmetries have integer conformal weights, the same for all black holes considered., Comment: 19 pages

We study the non-equilibrium condensation process in a holographic superconductor. When the temperature T is smaller than a critical temperature T_c, there are two black hole solutions, the Reissner-Nordstrom-AdS black hole and a black hole with a scalar hair. In the boundary theory, they can be regarded as the supercooled normal phase and the superconducting phase, respectively. We consider perturbations on supercooled Reissner-Nordstrom-AdS black holes and study their non-linear time evolution to know about physical phenomena associated with rapidly-cooled superconductors. We find that, for T, Comment: 20 pages, 7 figures

The Kerr/CFT correspondence has been recently broadened to the general extremal black holes under the assumption that the central charges from the non-gravitational fields vanish. To confirm this proposal, we derive the expression of the conserved charges in the Einstein-Maxwell-scalar theory with topological terms in four and five dimensions and check that the above assumption was correct. Combining the computed central charge with the expected form of the temperature, the Bekenstein-Hawking entropy of the general extremal black holes in four and five dimensions can be reproduced by using the Cardy formula., 20 pages, v2:references added, published in JHEP

We study evolution of gravitational perturbations of black strings. It is well known that for all wavenumber less than some threshold value, the black string is unstable against scalar type of gravitational perturbations, which is named the Gregory-Laflamme instability. Using numerical methods, we find the quasinormal modes and time-domain profiles of the black string perturbations in the stable sector and also show the appearance of the Gregory-Laflamme instability in the time domain. The dependence of the black string quasinormal spectrum and late time tails on such parameters as the wave vector and the number of extra dimensions is discussed. There is a numerical evidence that in the threshold point of instability the static solution of the wave equation is dominant. For wavenumbers slightly larger than the threshold value, in the region of stability, we see tiny oscillations with very small damping rate. While, for wavenumbers slightly smaller than the threshold value, in the region of the Gregory-Laflamme instability, we observe tiny oscillations with very small growth rate. We also find the level crossing of imaginary part of quasinormal modes between the fundamental mode and the first overtone mode, which accounts for the peculiar time domain profiles., 7 pages, 4 figures, RevTex

The squashed Kaluza-Klien (KK) black holes differ from the Schwarzschild black holes with asymptotic flatness or the black strings even at energies for which the KK modes are not excited yet, so that squashed KK black holes open a window in higher dimensions. Another important feature is that the squashed KK black holes are apparently stable and, thereby, let us avoid the Gregory-Laflamme instability. In the present paper, the evolution of scalar and gravitational perturbations in time and frequency domains is considered for these squashed KK black holes. The scalar field perturbations are analyzed for general rotating squashed KK black holes. Gravitational perturbations for the so called zero mode are shown to be decayed for non-rotating black holes, in concordance with the stability of the squashed KK black holes. The correlation of quasinormal frequencies with the size of extra dimension is discussed., 12 pages, 6 figures