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A cascade of two-level superconducting artificial atoms -- a source and a probe -- strongly coupled to a semi-infinite waveguide is a promising tool for observing nontrivial phenomena in quantum nonlinear optics. The probe atom can scatter an antibunched radiation emitted from the source, thereby generating a field with specific properties. We experimentally demonstrate wave mixing between nonclassical light from the coherently cw-pumped source and another coherent wave acting on the probe. We observe unique features in the wave mixing stationary spectrum which differs from mixing spectrum of two classical waves on the probe. These features are well described by adapting the theory for a strongly coupled cascaded system of two atoms. We further analyze the theory to predict non-classical mixing spectra for various ratios of atoms' radiative constants. Both experimental and numerical results confirm the domination of multi-photon scattering process with only a single photon from the source. We evaluate entanglement of atoms in the quasistationary state and illustrate the connection between the expected second-order correlation function of source's field and wave mixing side peaks corresponding to a certain number of scattered photons., Comment: 10 pages, 10 figures, RevTex 4.2

We demonstrate non-equilibrium steady-state photon transport through a chain of five coupled artificial atoms simulating the driven-dissipative Bose-Hubbard model. Using transmission spectroscopy, we show that the system retains many-particle coherence despite being coupled strongly to two open spaces. We show that system energy bands may be visualized with high contrast using cross-Kerr interaction. For vanishing disorder, we observe the transition of the system from the linear to the nonlinear regime of photon blockade in excellent agreement with the input-output theory. Finally, we show how controllable disorder introduced to the system suppresses this non-local photon transmission. We argue that proposed architecture may be applied to analog simulation of many-body Floquet dynamics with even larger arrays of artificial atoms paving an alternative way to demonstration of quantum supremacy, Comment: 5 pages, 4 figures, supplemental material

In recent years, there has been a significant progress in the development of digital quantum processors. The state-of-the-art quantum devices are imperfect, and fully-algorithmic fault-tolerant quantum computing is a matter of future. Until technology develops to the state with practical error correction, computational approaches other than the standard digital one can be used to avoid execution of the most noisy quantum operations. We demonstrate how a hybrid digital-analog approach allows simulating dynamics of a transverse-field Ising model without standard two-qubit gates, which are currently one of the most problematic building blocks of quantum circuits. We use qubit-qubit crosstalks (couplings) of IBM superconducting quantum processors to simulate Trotterized dynamics of spin clusters and then we compare the obtained results with the results of conventional digital computation based on two-qubit gates from the universal set. The comparison shows that digital-analog approach significantly outperforms standard digital approach for this simulation problem, despite of the fact that crosstalks in IBM quantum processors are small. We argue that the efficiency of digital-analog quantum computing can be improved with the help of more specialized processors, so that they can be used to efficiently implement other quantum algorithms. This indicates the prospect of a digital-to-analog strategy for near-term noisy intermediate-scale quantum computers., Comment: 19 pages, to be published in Physical Review A

In a view of recent proposals for the realization of anisotropic light-matter interaction in such platforms as (i) non-stationary or inductively and capacitively coupled superconducting qubits, (ii) atoms in crossed fields and (iii) semiconductor heterostructures with spin-orbital interaction, the concept of generalized Dicke model, where coupling strengths of rotating wave and counter-rotating wave terms are unequal, has attracted great interest. For this model, we study photon fluctuations in the critical region of normal-to-superradiant phase transition when both the temperatures and numbers of two-level systems are finite. In this case, the superradiant quantum phase transition is changed to a fluctuational region in the phase diagram that reveals two types of critical behaviors. These are regimes of Dicke model (with discrete $\mathbb{Z}_2$ symmetry), and that of (anti-) and Tavis-Cummings $U(1)$ models. We show that squeezing parameters of photon condensate in these regimes show distinct temperature scalings. Besides, relative fluctuations of photon number take universal values. We also find a temperature scales below which one approaches zero-temperature quantum phase transition where quantum fluctuations dominate. Our effective theory is provided by a non-Goldstone functional for condensate mode and by Majorana representation of Pauli operators. We also discuss Bethe ansatz solution for integrable $U(1)$ limits., Comment: 17 pages, 7 figures, 1 table. Substantial revisions made, the title changed. The version accepted in Phys. Rev. A

We point out that realization of quantum communication protocols in programmable quantum computers provides a deep benchmark for capabilities of real quantum hardware. Particularly, it is prospective to focus on measurements of entropy-based characteristics of the performance and to explore whether a "quantum regime" is preserved. We perform proof-of-principle implementations of superdense coding and quantum key distribution BB84 using 5- and 16-qubit superconducting quantum processors of IBM Quantum Experience. We focus on the ability of these quantum machines to provide an efficient transfer of information between distant parts of the processors by placing Alice and Bob at different qubits of the devices. We also examine the ability of quantum devices to serve as quantum memory and to store entangled states used in quantum communication. Another issue we address is an error mitigation. Although it is at odds with benchmarking, this problem is nevertheless of importance in a general context of quantum computation with noisy quantum devices. We perform such a mitigation and noticeably improve some results., Comment: 25 pages

We point out that superconducting quantum computers are prospective for the simulation of the dynamics of spin models far from equilibrium, including nonadiabatic phenomena and quenches. The important advantage of these machines is that they are programmable, so that different spin models can be simulated in the same chip, as well as various initial states can be encoded into it in a controllable way. This opens an opportunity to use superconducting quantum computers in studies of fundamental problems of statistical physics such as the absence or presence of thermalization in the free evolution of a closed quantum system depending on the choice of the initial state as well as on the integrability of the model. In the present paper, we performed proof-of-principle digital simulations of two spin models, which are the central spin model and the transverse-field Ising model, using 5- and 16-qubit superconducting quantum computers of the IBM Quantum Experience. We found that these devices are able to reproduce some important consequences of the symmetry of the initial state for the system's subsequent dynamics, such as the excitation blockade. However, lengths of algorithms are currently limited due to quantum gate errors. We also discuss some heuristic methods which can be used to extract valuable information from the imperfect experimental data., Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1710.09659

We implement several quantum algorithms in real five-qubit superconducting quantum processor IBMqx4 to perform quantum computation of the dynamics of spin-1/2 particles interacting directly and indirectly through the boson field. Particularly, we focus on effects arising due to the presence of entanglement in the initial state of the system. The dynamics is implemented in a digital way using Trotter expansion of evolution operator. Our results demonstrate that dynamics in our modeling based on real device is governed by quantum interference effects being highly sensitive to phase parameters of the initial state. We also discuss limitations of our approach due to the device imperfection as well as possible scaling towards larger systems., Comment: 11 pages

We consider a dissipative evolution of parametrically-driven qubits-cavity system under the periodical modulation of coupling energy between two subsystems, which leads to the amplification of counterrotating processes. We reveal a very rich dynamical behavior of this hybrid system. In particular, we find that the energy dissipation in one of the subsystems can enhance quantum effects in another subsystem. For instance, optimal cavity decay assists to stabilize entanglement and quantum correlations between qubits even in the steady state and to compensate finite qubit relaxation. On the contrary, energy dissipation in qubit subsystem results in the enhanced photon production from vacuum for strong modulation, but destroys both quantum concurrence and quantum mutual information between qubits. Our results provide deeper insights to nonstationary cavity quantum electrodynamics in context of quantum information processing and might be of importance for dissipative quantum state engineering., Comment: 18 pages, 6 figures. Version accepted for publication in Physical Review A

We consider dynamics of a disordered ensemble of qubits interacting with single mode photon field, which is described by exactly solvable inhomogeneous Dicke model. In particular, we concentrate on the crossover from few-qubit systems to the system of many qubits and analyze how collective behavior of coupled qubits-cavity system emerges despite of the broadening. We show that quantum interference effects survive in the mesoscopic regime -- dynamics of an entangled Bell state encoded into the qubit subsystem remains highly sensitive to the symmetry of the total wave function. Moreover, relaxation of these states is slowed down due to the formation of collective dark states weakly coupled to light. Dark states also significantly influence dynamics of excitations of photon subsystem by absorbing them into the qubit subsystem and releasing quasiperiodically in time. We argue that predicted phenomena can be useful in quantum technologies based on superconducting qubits. For instance, they provide tools to deeply probe both collective and quantum properties of such artificial macroscopic systems., Comment: 18 pages, accepted for publication in Phys. Rev. A

We consider a superconducting qubit coupled to the nonstationary transmission line cavity with modulated frequency taking into account energy dissipation. Previously, it was demonstrated that in the case of a single nonadiabatical modulation of a cavity frequency there are two channels of a two-level system excitation which are due to the absorption of Casimir photons and due to the counterrotating wave processes responsible for the dynamical Lamb effect. We show that the parametric periodical modulation of the resonator frequency can increase dramatically the excitation probability. Remarkably, counterrotating wave processes under such a modulation start to play an important role even in the resonant regime. Our predictions can be used to control qubit-resonator quantum states as well as to study experimentally different channels of a parametric qubit excitation., Comment: 6 pages, 5 figures; revised version accepted in Physics Letters A

Superconducting circuits provide a new platform to study nonstationary cavity QED phenomena. An example of such a phenomenon is a dynamical Lamb effect which is a parametric excitation of an atom due to the nonadiabatic modulation of its Lamb shift. This effect has been initially introduced for a natural atom in a varying cavity, while we suggested its realization in a superconducting qubit-cavity system with dynamically tunable coupling. In the present paper, we study the interplay between the dynamical Lamb effect and the energy dissipation, which is unavoidable in realistic systems. We find that despite of naive expectations this interplay can lead to unexpected dynamical regimes. One of the most striking results is that photon generation from vacuum can be strongly enhanced due to the qubit relaxation, which opens a new channel for such a process. We also show that dissipation in the cavity can increase the qubit excited state population. Our results can be used for the experimental observation and investigation of the dynamical Lamb effect and accompanying quantum effects., Comment: 14 pages

A natural atom placed into a cavity with time-dependent parameters can be parametrically excited due to the interaction with the quantized photon mode. One of the channels of such a process is the dynamical Lamb effect, induced by a nonadiabatic modulation of atomic level Lamb shift. However, in experiments with natural atoms it is quite difficult to isolate this effect from other mechanisms of atom excitation. We point out that a transmission line cavity coupled with a superconducting qubit (artificial macroscopic atom) provides a unique platform for the observation of the dynamical Lamb effect. A key idea is to exploit a dynamically tunable qubit-resonator coupling, which was implemented quite recently. By varying nonadiabatically the coupling, it is possible to parametrically excite a qubit through a nonadiabatic modulation of the Lamb shift, even if the cavity was initially empty. A dynamics of such a coupled system is studied within the Rabi model with time-dependent coupling constant and beyond the rotating wave approximation. An efficient method to increase the effect through the periodic and nonadiabatic switching of a qubit-resonator coupling energy is proposed., Comment: 16 pages, 8 figures

We analyze conditions of applicability of grand-canonical mean-field Bardeen-Cooper-Schrieffer theory to the evaluation of an interaction energy in the ground state of small-sized superconductors. We argue that this theory fails to describe correctly an interaction energy, when an average distance between energy levels near the Fermi energy due to the size quantization becomes of the order of the single-pair binding energy. In conventional superconductors, this quantity is much smaller than the superconducting gap., Comment: 5 pages, no figures

We address a generalized Richardson model (Russian doll BCS model), which is characterized by the breaking of time-reversal symmetry. This model is known to be exactly solvable and integrable. We point out that the Russian doll BCS model, on the level of Hamiltonian, is also particle-hole symmetric. This implies that the same state can be expressed both in the particle and hole representations with two different sets of Bethe roots. We then derive exact relations between Bethe roots in the two representations, which can hardly be obtained staying on the level of Bethe equations. In a quasi-classical limit, similar identities for usual Richardson model, known from literature, are recovered from our results. We also show that these relations for Richardson roots take a remarkably simple form at half-filling and for a symmetric with respect to the middle of the interaction band distribution of one-body energy levels, since, in this special case, the rapidities in the particle and hole representations up to the translation satisfy the same system of equations., Comment: 15 pages, 2 figures

Richardson approach provides an exact solution of the pairing Hamiltonian. This Hamiltonian is characterized by the electron-hole pairing symmetry, which is however hidden in Richardson equations. By analyzing this symmetry and using an additional conjecture, fulfilled in solvable limits, we suggest a simple expression of the ground state energy for an equally-spaced energy-level model, which is applicable along the whole crossover from the superconducting state to the pairing fluctuation regime. Solving Richardson equations numerically, we demonstrate a good accuracy of our expression., Comment: 9 pages, 1 figure; accepted for publication in Eur. Phys. J. B

Richardson equations can be mapped on the classical electrostatic problem in two dimensions. We have recently suggested a new analytical approach to these equations in the thermodynamical limit, which is based on the `probability' of the system of charges to be in a given configuration at the effective temperature equal to the interaction constant. In the present paper, we apply this approach to excited states of the Richardson pairing model. We focus on the equally-spaced situation and address arbitrary fillings of the energy layer, where interaction acts. The `partition function' for the classical problem on the plane, which is given by Selberg-type integral, is evaluated exactly. Three regimes for the energy gap are identified, which can be treated as the dilute regime of pairs, BCS regime, and dilute regime of holes., Comment: 14 pages, 1 figure

We consider two possible mechanisms for single-vortex fluctuative entry/exit through the surface barrier in ultrathin superconducting disk-shaped nanoislands made of Pb and consisting of just few monoatomic layers, which can be fabricated using modern techniques. We estimate tunnelling probabilities and establish criteria for the crossover between these two mechanisms depending on magnetic field and system sizes. For the case of vortex entry, quantum tunnelling dominates on the major part of the temperature/flux phase diagram. For the case of vortex exit, thermal activation turns out to be more probable. This nontrivial result is due to the subtle balance between the barrier height and width, which determine rates of the thermal activation and quantum tunneling, respectively., Comment: 5 pages, 2 figures

The Bogoliubov approach to superconductivity provides a strong mathematical support to the wave function ansatz proposed by Bardeen, Cooper and Schrieffer (BCS). Indeed, this ansatz --- with all pairs condensed into the same state --- corresponds to the ground state of the Bogoliubov Hamiltonian. Yet, this Hamiltonian only is part of the BCS Hamiltonian. As a result, the BCS ansatz definitely differs from the BCS Hamiltonian ground state. This can be directly shown either through a perturbative approach starting from the Bogoliubov Hamiltonian, or better by analytically solving the BCS Schr\"{o}dinger equation along Richardson-Gaudin exact procedure. Still, the BCS ansatz leads not only to the correct extensive part of the ground state energy for an arbitrary number of pairs in the energy layer where the potential acts --- as recently obtained by solving Richardson-Gaudin equations analytically --- but also to a few other physical quantities such as the electron distribution, as here shown. The present work also considers arbitrary filling of the potential layer and evidences the existence of a super dilute and a super dense regime of pairs, with a gap \emph{different} from the usual gap. These regimes constitute the lower and upper limits of density-induced BEC-BCS cross-over in Cooper pair systems., Comment: 15 pages, no figures

It is known that solutions of Richardson equations can be represented as stationary points of the "energy" of classical free charges on the plane. We suggest to consider "probabilities" of the system of charges to occupy certain states in the configurational space at the effective temperature given by the interaction constant, which goes to zero in the thermodynamical limit. It is quite remarkable that the expression of "probability" has similarities with the square of Laughlin wave function. Next, we introduce the "partition function", from which the ground state energy of the initial quantum-mechanical system can be determined. The "partition function" is given by a multidimensional integral, which is similar to Selberg integrals appearing in conformal field theory and random-matrix models. As a first application of this approach, we consider a system with the constant density of energy states at arbitrary filling of the energy interval, where potential acts. In this case, the "partition function" is rather easily evaluated using properties of the Vandermonde matrix. Our approach thus yields quite simple and short way to find the ground state energy, which is shown to be described by a single expression all over from the dilute to the dense regime of pairs. It also provides additional insights into the physics of Cooper-paired states., Comment: 20 pages, 1 figure; final version, accepted for publication in Journal of Physics: Condensed Matter

We extend the one-pair Cooper configuration towards Bardeen-Cooper-Schrieffer (BCS) model of superconductivity by adding one-by-one electron pairs to an energy layer where a small attraction acts. To do it, we solve Richardson's equations analytically in the dilute limit of pairs on the one-Cooper pair scale. We find, through only keeping the first order term in this expansion, that the $N$ correlated pair energy reads as the energy of $N$ isolated pairs within a $N(N-1)$ correction induced by the Pauli exclusion principle which tends to decrease the average pair binding energy when the pair number increases. Quite remarkably, extension of this first-order result to the dense regime gives the BCS condensation energy exactly. This leads us to suggest a different understanding of the BCS condensation energy with a pair number equal to the number of pairs feeling the potential and an average pair binding energy reduced by Pauli blocking to half the single Cooper pair energy - instead of the more standard but far larger superconducting gap., Comment: 14 pages, 2 figures

In the recent experiment by Cren \textit{et al.} [Phys. Rev. Lett. \textbf{102}, 127005 (2009)], no hysteresis for vortex penetration and expulsion from the nano-island of Pb was observed. In the present paper, we argue that this effect can be associated with the thermoactivated surmounting of the surface barrier by a vortex. The typical entrance (exit) time is found analytically from the Fokker-Planck equation, written in the form suitable for the extreme vortex confinement. We show that this time is several orders of magnitude smaller than 1 second under the conditions of the experiment considered. Our results thus demonstrate a possibility for the thermal suppression of the surface barrier in nanosized low-$T_{c}$ superconductors. We also briefly discuss other recent experiments on vortices in related structures., Comment: 12 pages, 2 figures

Thermal fluctuations of the order parameter in an ultrathin triangular shaped superconducting structure are studied near $T_{c}$, in zero applied field. We find that the order parameter is prone to much larger fluctuations in the corners of the structure as compared to its interior. This geometry-induced localization of thermal fluctuations is attributed to the fact that condensate confinement in the corners is characterised by a lower effective dimensionality, which favors stronger fluctuations., Comment: 9 pages, 5 figures

While the one-Cooper pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for N=2 already. The two-Cooper pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to $N$ pairs, but no precise description of this continuous change has been given so far. Using Richardson procedure, we here show that Pauli blocking increases the free part of the two-pair ground state energy, but decreases the binding part when compared to two isolated pairs - the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease of the pair binding while the gap increases strongly indicates that, at odd with common belief, the average pair binding energy cannot be of the order of the gap., Comment: 9 pages, no figures, final version

We extend the well-known Cooper's problem beyond one pair and study how this dilute limit is connected to the many-pair BCS condensate. We find that, all over from the dilute to the dense regime of pairs, Pauli blocking induces the same "moth-eaten effect" as the one existing for composite boson excitons. This effect makes the average pair binding energy decrease linearly with pair number, bringing it, in the standard BCS configuration, to half the single-pair value. This proves that, at odds with popular understanding, the BCS gap is far larger than the broken pair energy. The increase comes from Pauli blocking between broken and unbroken pairs. Possible link between our result and the BEC-BCS crossover is also discussed., Comment: 5 pages, 1 figure, to be published in Pis'ma v ZhETF (JETP Letters)

The influence of random pinning on the vortex dynamics in a periodic square potential under an external drive is investigated. Using theoretical approach and numerical experiments, we found several dynamical phases of vortex motion that are different from the ones for a regular pinning potential. Vortex transfer is controlled by kinks and antikinks, which either preexist in the system or appear spontaneously in pairs and then propagate in groups. When kinks and antikinks collide, they annihilate., Comment: 4 pages, 4 figures

We study theoretically the simultaneous effect of a regular and a random pinning potentials on the vortex lattice structure at filling factor of 1. This structure is determined by a competition between the square symmetry of regular pinning array, by the intervortex interaction favoring a triangular symmetry, and by the randomness trying to depin vortices from their regular positions. Both analytical and molecular-dynamics approaches are used. We construct a phase diagram of the system in the plane of regular and random pinning strengths and determine typical vortex lattice defects appearing in the system due to the disorder. We find that the total disordering of the vortex lattice can occur either in one step or in two steps. For instance, in the limit of weak pinning, a square lattice of pinned vortices is destroyed in two steps. First, elastic chains of depinned vortices appear in the film; but the vortex lattice as a whole remains still pinned by the underlying square array of regular pinning sites. These chains are composed into fractal-like structures. In a second step, domains of totally depinned vortices are generated and the vortex lattice depins from regular array., Comment: 13 pages, 9 figures

We have designed and investigated a contactless magnetic phase shifter for flux-based superconducting qubits. The phase shifter is realized by placing a perpendicularly magnetized dot at the centre of a superconducting loop. The flux generated by this magnetic dot gives rise to an additional shielding current in the loop, which, in turn, induces a phase shift. By modifying the parameters of the dot an arbitrary phase shift can be generated in the loop. This magnetic phase shifter can, therefore, be used as an external current source in superconducting circuits, as well as a suitable tool to study fractional Josephson vortices., Comment: to be published in Physical Review Letters

We have investigated the nucleation of superconductivity in an Al mesoscopic disk, with a magnetic dot on the top. The dot is magnetized perpendicularly, and its magnetic field is inhomogeneous. The Al disk and magnetic dot are separated by an insulating layer, which ensures that there is only magnetic interaction between them. This hybrid superconductor/ferromagnet structure exhibits the maximum critical temperature for a finite value of the perpendicular applied magnetic field, which is parallel to the magnetization of magnetic dot. We have found a good agreement between the experimental data for the superconductor/normal metal phase boundary and the theoretical predictions based on the Ginzburg-Landau theory., Comment: to be published in Europhysics Letters

We have studied the nucleation of superconductivity in a mesoscopic Al loop, enclosing magnetic dot with perpendicular magnetization. The superconducting phase boundary Tc(B), determined from transport measurements, is asymmetric with respect to the polarity of an applied magnetic field. The maximum critical temperature has been found for a finite applied magnetic field, which is antiparallel to the magnetization of the dot. Theoretical phase boundary shows a good agreement with the experimental data., Comment: to be published in Phys. Rev. B - Brief Reports

We have studied the nucleation of superconductivity in a mesoscopic Al disk wit a Co/Pd magnetic dot placed on the top by measuring the normal superconducting phase boundary Tc(B). The measurements have revealed a pronounced asymmetry with respect to the direction of the applied magnetic field, indicating an enhancement of the critical field when an applied magnetic field is oriented parallel to the magnetization of the magnetic dot. The theoretical Tc(B) curve is in a good agreement with the experimental data., Comment: 4 pages, 4 figures. To be published in Applied Physics Letters

The vortex patterns stabilized by the square array of artificial pinning sites with a tunable pinning strength are studied by using a phenomenological approach in the London limit. The transitions between pinned and deformed triangular lattices are analyzed as a function of the amplitude of the vortex-pinning site interaction and the characteristic length-scale of this interaction. The critical current and different phases of vortex lattice are studied in presence of external transport current., Comment: 13 pages, 14 figures, submitted to Phys. Rev. B

Vortex structures in mesoscopic cylinder placed in external magnetic field are studied under the general de Gennes boundary condition for the order parameter corresponding to the suppression of surface superconductivity. The Ginzburg-Landau equations are solved based on trial functions for the order parameter for vortex-free, single-vortex, multivortex, and giant vortex phases. The equilibrium vortex diagrams in the plane of external field and cylinder radius and magnetization curves are calculated at different values of de Gennes "extrapolation length" characterizing the boundary condition for the order parameter. The comparison of the obtained variational results with some available exact solutions shows good accuracy of our approach., Comment: RevTex, 11 pages, 10 figures

Superconducting state in mesoscopic cylinder placed in the external magnetic field is analyzed at general de Gennes boundary condition for the order parameter. The lower and the surface critical fields of the cylinder are calculated at different values of de Gennes "extrapolation length" $b$. For the case of standard boundary condition (corresponding to $b\to \infty $) we use a variational method to calculate the phase transitions between different superconducting states. Based on the trial function for the order parameter, we find an approximate solutions to the Ginzburg-Landau equations for the Meissner, the single-vortex, and the giant-vortex states. The magnetization curves of thin cylinders are calculated., Comment: 3 pages, 5 figures, submitted to IWCC 2001

The vibrational contribution $\mu_{v}(H)$ to the dynamic magnetic permeability is calculated allowing for the spatial variation of the order parameter in the vortex cores. The behavior of calculated dependence $\mu _{v}(H)$ is analyzed for the case when the vortices oscillate near their positions corresponding to the equilibrium vortex lattice. It is shown that taking into account the suppression of the order parameter in the vortex cores leads to appreciable difference in the behavior of $\mu_{v}(H)$ as compared to the similar dependence obtained in the London approximation. The main reason of this difference is a change of the effective potential well in which the vibrational motion of the vortices takes place., Comment: 3 pages, 3 figures, submitted to IWCC 2001

A simple variational model is proposed to analyze the superconducting state in long cylindrical type-II superconductor placed in the external magnetic field. In the framework of this model, it is possible to solve the Ginzburg-Landau equations for the states with axially symmetric distributions of the order parameter. Phase transitions between different superconducting states are studied in the presence of external magnetic field and an equilibrium phase diagram of thin cylinder is obtained. The lower critical field of the cylindrical type-II superconductor with arbitrary values of radius and Ginzburg-Landau parameter is found. The field dependence of the magnetization of thin cylinder, which can carry several magnetic flux quanta, is calculated., Comment: 10 pages, 5 figures, submitted to Physica C

A variational model is proposed to describe the magnetic properties of type-II superconductors in the entire field range between $H_{c1}$ and $H_{c2}$ for any values of the Ginzburg-Landau parameter $\kappa>1/\sqrt{2}$. The hexagonal unit cell of the triangular flux-line lattice is replaced by a circle of the same area, and the periodic solutions to the Ginzburg-Landau equations within this cell are approximated by rotationally symmetric solutions. The Ginzburg-Landau equations are solved by a trial function for the order parameter. The calculated spatial distributions of the order parameter and the magnetic field are compared with the corresponding distributions obtained by numerical solution of the Ginzburg-Landau equations. The comparison reveals good agreement with an accuracy of a few percent for all $\kappa$ values exceeding $\kappa \approx 1$. The model can be extended to anisotropic superconductors when the vortices are directed along one of the principal axes. The reversible magnetization curve is calculated and an analytical formula for the magnetization is proposed. At low fields, the theory reduces to the London approach at $\kappa \gg 1$, provided that the exact value of $H_{c1}$ is used. At high fields, our model reproduces the main features of the well-known Abrikosov theory. The magnetic field dependences of the reversible magnetization found numerically and by our variational method practically coincide. The model also refines the limits of some approximations which have been widely used. The calculated magnetization curves are in a good agreement with experimental data on high-T$_c$ superconductors., Comment: 8 pages, RevTex, 6 figures, submitted to Phys. Rev. B

The first integrals of the Ginzburg-Landau equations for a vortex-free state of superconductors with different mixed symmetries of the order parameter are found. The general boundary conditions for the order parameter at the ideal interface between the superconductor and vacuum are derived. Based on these integrals and boundary conditions, we analyze the stability criteria for vortex-free state in unconventional superconductors. The threshold field above which the Abrikosov vortices can enter the superconductor is found to be higher or equal to the thermodynamic critical field for all states under study., Comment: 8 pages, pdf file, no figures