Quadratic Hamiltonians Research Articles

Differential formalism and the thermodynamic description of multimode Gaussian equilibrium states

A differential formalism for the covariance matrix of thermal equilibrium states is obtained when the Hamiltonian of the system is defined by quadratic forms. First, the cases of one- and two-modal Gaussian states are taken into account and the differential equations for the corresponding covariance matrices of these states in terms of the temperature are obtained. Later, the generalization of the differential equation to any number of modes is demonstrated by using the Wigner quasi-probability distribution. Some examples related to standard quadratic Hamiltonians are given and several thermodynamic properties as the free and internal energies, the entropy, and the heat capacity of different systems are listed.

Observability for Schrödinger equations with quadratic Hamiltonians

We consider time dependent harmonic oscillators and construct a parametrix to the corresponding Schrödinger equation using Gaussian wavepackets. This parametrix of Gaussian wavepackets is precise and tractable. Using this parametrix we prove L^2 and L^2-L^{infty } observability estimates on unbounded domains omega for a restricted class of initial data. This data includes a class of compactly supported piecewise C^1 functions which have been extended from characteristic functions. Initial data of this form which has the bulk of its mass away from omega ^c=Omega , a connected bounded domain, is observable, but data centered over Omega must be very nearly a single Gaussian to be observable. We also give counterexamples to established principles for the simple harmonic oscillator in the case of certain time dependent harmonic oscillators.

Quartic Hamiltonians, and higher Hamiltonians at next-to-leading order, for the affine Gaudin model

In this work we will use a general procedure to construct higher local Hamiltonians for the affine Gaudin model. We focus on the first non-trivial example, the quartic Hamiltonians. We show by direct calculation that the quartic Hamiltonians commute amongst themselves and with the quadratic Hamiltonians which define the model. We go on to introduce a certain next-to-leading-order semi-classical limit of the model. In this limit, we are able to write down the full hierarchy of higher local Hamiltonians and prove that they commute.

Solving Anderson Impurity Model by the Effective Hamiltonian Theory

We apply the Self-Consistent Effective Hamiltonian Theory (SCEHT), which uses a general variational Fermionic many-body wave function to generate an effective Hamiltonian in a quadratic form, to the Anderson impurity model. The chiral symmetry-breaking quadratic effective Hamiltonian is solved exactly for the single Fermion excitation spectrum. We validate the theory by numerically solving a model problem. The solution shows the correct Kondo resonance in the quasi-particle density of states.

Notes on Hardy's uncertainty principle for the Wigner distribution and Schrödinger evolutions

For Schrödinger equations with real quadratic Hamiltonians, it is known that the Wigner distribution of the solution at a given time equals, up to a linear coordinate transformation, the Wigner distribution of the initial condition. Based on Hardy's uncertainty principle for the joint time-frequency representation, we present a general uniqueness result for such Schrödinger equations, where the solution cannot have strong decay at two distinct times. This approach gives new proofs to known, sharp Hardy type estimates for the free Schrödinger equation, the harmonic oscillator and uniform magnetic potentials, as well as new uniqueness results.

Characterization of modulation spaces by symplectic representations and applications to Schrödinger equations

In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ-Wigner representations in [11].In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A∈Sp(2d,R). We find conditions on A for which the related symplectic time-frequency representation WA can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding WA are members of the Cohen class.Finally, we show that symplectic time-frequency representations WA can be efficiently employed in the study of Schrödinger equations. In fact, modulation spaces and WA representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro-singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.

Hartree method for molecular polaritons

The formation of the composite photonic-excitonic particle, known as a polariton, is a phenomenon emerging in materials possessing strong coupling to light. The organic-based materials besides the strong light-matter coupling also demonstrate strong interaction of electronic and vibrational degrees of freedom. We study the vibration-assisted polariton wave-function evolution treating both types of interactions as equally strong. Using the multiconfiguration Hartree approach we derive the equations of motion for the polariton wave function, where the vibration degrees of freedom interact with the polariton quantum field through the mean-field Hartree term. For the conventional quadratic polariton Hamiltonian and the Holstein-type vibration Hamiltonian (Tavis-Cummings-Holstein model), the obtained equations are in one-to-one correspondence with the original Schr\"odinger equation. In the second part of the paper, we show that our theory reproduces the physical properties of the polariton light emission spectrum. In particular, the theory explains experimental observations of the molecular Stokes shift in the polariton fluorescence spectra in the systems with strong light-matter coupling. We also investigate the behavior of the polariton wave function in the vicinity of the anticrossing point and demonstrate that the Hartree term can produce an infinite potential barrier of a dynamical origin, which is responsible for the formation of the mixed upper-lower polariton states. The nonlinear nature of the polariton theory reflects their collective behavior. We expect that the multiconfiguration Hartree approach being applied to polaritons and similar systems will result in a manifestation of new physical phenomena.

Free energy of a diluted spin glass model with quadratic Hamiltonian

We study a diluted mean-field spin glass model with a quadratic Hamiltonian. Our main result establishes the limiting free energy in terms of an integral of a family of random variables that are the weak limits of the quenched variances of the spins in the system with varying edge connectivity. The key ingredient in our argument is played by the identification of these random variables as the unique solution to a recursive distributional equation. Our results in particular provide the first example of the diluted Shcherbina–Tirozzi model, whose limiting free energy can be derived at any inverse temperature and external field.

Quantum Liouvillian exceptional and diabolical points for bosonic fields with quadratic Hamiltonians: The Heisenberg-Langevin equation approach

Equivalent approaches to determine eigenfrequencies of the Liouvillians of open quantum systems are discussed using the solution of the Heisenberg-Langevin equations and the corresponding equations for operator moments. A simple damped two-level atom is analyzed to demonstrate the equivalence of both approaches. The suggested method is used to reveal the structure as well as eigenfrequencies of the dynamics matrices of the corresponding equations of motion and their degeneracies for interacting bosonic modes described by general quadratic Hamiltonians. Quantum Liouvillian exceptional and diabolical points and their degeneracies are explicitly discussed for the case of two modes. Quantum hybrid diabolical exceptional points (inherited, genuine, and induced) and hidden exceptional points, which are not recognized directly in amplitude spectra, are observed. The presented approach via the Heisenberg-Langevin equations paves the general way to a detailed analysis of quantum exceptional and diabolical points in infinitely dimensional open quantum systems.

On the conservative character of discretizations to Itô-Hamiltonian systems with small noise

In this paper, we consider stochastic Hamiltonian systems of Itô type driven by a multiplicative small noise. It is well-known, indeed, that stochastic Hamiltonian problems are suitable candidates to reconcile the Hamiltonian classical mechanics with the non-differentiability of the Wiener process, which provides the innovative character of stochastic effects. In particular, in this work, we provide a characterization of the behavior of averaged Hamiltonians arised in such systems, with more emphasis on the separable and quadratic Hamiltonians. Next, we show that, in general, first order approximations to such systems are not able to retain the same behavior discovered for the exact averaged Hamiltonian. Hence, the analysis for the specific case of ϑ-methods is performed. Finally, numerical evidence is depicted in order to confirm theoretical results.

Semiclassical Asymptotics of Oscillating Tunneling for a Quadratic Hamiltonian on the Algebra $$\operatorname{su}(1,1)$$

Semiclassical Asymptotics of Oscillating Tunneling for a Quadratic Hamiltonian on the Algebra $$\operatorname{su}(1,1)$$

Quantum-jump vs stochastic Schrödinger dynamics for Gaussian states with quadratic Hamiltonians and linear Lindbladians

The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schrödinger dynamics for initially Gaussian states. While both unravellings converge to the same Lindblad dynamics when averaged, the individual dynamics can differ qualitatively. For the stochastic Schrödinger equation, Gaussian states remain Gaussian during the evolution, with stochastic differential equations governing the evolution of the phase-space centre and a deterministic evolution of the covariance matrix. In contrast to this, individual pure-state dynamics arising from the quantum-jump evolution do not remain Gaussian in general. Applying results developed in the non-Hermitian context for Hagedorn wavepackets, we formulate a method to generate quantum-jump trajectories that is described entirely in terms of the evolution of an underlying Gaussian state. To illustrate the behaviours of the different unravellings in comparison to the Lindblad dynamics, we consider two examples in detail, which can be largely treated analytically, a harmonic oscillator subject to position measurement and a damped harmonic oscillator. In both cases, we highlight the differences as well as the similarities of the stochastic Schrödinger and the quantum-jump dynamics.

Optimal rate of condensation for trapped bosons in the Gross–Pitaevskii regime

We study the Bose-Einstein condensates of trapped Bose gases in the Gross-Pitaevskii regime. We show that the ground state energy and ground states of the many-body quantum system are correctly described by the Gross-Pitaevskii equation in the large particle number limit, and provide the optimal convergence rate. Our work extends the previous results of Lieb, Seiringer and Yngvason on the leading order convergence, and of Boccato, Brennecke, Cenatiempo and Schlein on the homogeneous gas. Our method relies on the idea of 'completing the square', inspired by recent works of Brietzke, Fournais and Solovej on the Lee-Huang-Yang formula, and a general estimate for Bogoliubov quadratic Hamiltonians on Fock space.

Tight-binding billiards.

Recent works have established universal entanglement properties and demonstrated validity of single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians. However, a common property of all quantum-chaotic quadratic Hamiltonians studied in this context so far is the presence of random terms that act as a source of disorder. Here we introduce tight-binding billiards in two dimensions, which are described by noninteracting spinless fermions on a disorder-free square lattice subject to curved open (hard-wall) boundaries. We show that many properties of tight-binding billiards match those of quantum-chaotic quadratic Hamiltonians: The average entanglement entropy of many-body eigenstates approaches the random matrix theory predictions and one-body observables in single-particle eigenstates obey the single-particle eigenstate thermalization hypothesis. On the other hand, a degenerate subset of single-particle eigenstates at zero energy (i.e., the zero modes) can be described as chiral particles whose wave functions are confined to one of the sublattices.

Brane order and quantum magnetism in modulated anisotropic ladders

Two-leg spin-$1/2$ ladders with anisotropy and two different dimerization patterns are analyzed at zero temperature. This model is equivalent to a modulated interacting (Kitaev) ladder. The Hartree-Fock mean-field approximation reduces the model to a sum of two quadratic effective Majorana Hamiltonians, which are dual to two quantum transverse XY chains. The mapping between the effective Hamiltonian of the ladder and a pair of chains considerably simplifies calculations of the order parameters and analysis of the hidden symmetry breaking. The ground-state phase diagram of the staggered ladder contains nine phases, four of them are conventional antiferromagnets, while the other five possess nonlocal brane orders. Using the dualities and the newly found exact results for the local and string order parameters of the transverse XY chains, we were able to find analytically all the magnetizations and the brane order parameters for the staggered case, as functions of the renormalized couplings of the effective Hamiltonian. The columnar ladder has three ground-state phases and does not possess magnetic long-ranged order. The brane order parameters for these three phases are calculated numerically from the Toeplitz determinants. All brane-ordered phases are spin liquids with identified distinct order parameters, winding numbers, and sets of the Majorana edge modes. Disorder lines and the special points of disentanglement are found in both dimerization patterns. We expect this study to motivate the search for the real spin-Peierls anisotropic ladder compounds which manifest predicted properties.

Operator complexity for quantum scalar fields and cosmological perturbations

We calculate the operator complexity for the displacement, squeeze and rotation operators of a quantum harmonic oscillator, assuming equal computational cost for the corresponding fundamental gates. The complexity of the time-dependent displacement operator is constant, equal to the magnitude of the coherent state parameter, while the complexity of unitary evolution by a generic quadratic Hamiltonian is proportional to the amount of squeezing and is sensitive to the time-dependent phase of the unitary operator. We apply these results to study the complexity of a free massive scalar field, finding that the complexity has a period of rapid linear growth followed by a saturation determined by the UV cutoff and the number of spatial dimensions. We also study the complexity of the unitary evolution of quantum cosmological perturbations in de Sitter space, which can be written as time-dependent squeezing and rotation operators on individual Fourier mode pairs. The complexity of a single mode pair at late times grows linearly with the number of $e$-folds, while the complexity at early times oscillates rapidly due to the sensitivity of operator complexity to the phase of unitary time evolution. Integrating over all modes, the total complexity of cosmological perturbations scales as the square root of the (exponentially) growing volume of de Sitter space, suggesting that inflation leads to an explosive growth in complexity of the Universe.

Supercanonical coordinates for orthosymplectic evolution operators

A time-dependent self-adjoint even Hamiltonian is defined by a linear combination of generators of the semidirect sum osp(m′/n′)⊕sne(n′+1), of the orthosymplectic plus the even Heisenberg algebra by computing the supercommutator of odd binary forms Π, given as linear combinations of odd bilinear generators of the odd Heisenberg algebra no(m′+1) elements times ne(n′+1) elements, establishing a relationship between entangled boson systems and entangled fermion systems. This approach leads to the concept of intertwining, defined through the resulting quadratic Hamiltonians of bosons and, separately, of fermions with coefficients given in terms of the same coefficients of Π. Intertwining is invariant under transformations of Π, which leave certain binary forms of the coefficients of Π in the Hamiltonian unchanged. Alternatively, the coefficients can be interpreted as simultaneous time-dependent (super-) control parameters for both spin-statistics. Time-dependent inhomogeneous linear supercanonical transformations of wave vectors leave invariant the Heisenberg superalgebra n(m′/n′) and belong to the semidirect product Osp(m′/n′) ⋉ Ne(n′ + 1) of the orthosymplectic supergroup with the even Heisenberg group. The unitary time evolution operator is constructed using the adjoint map in canonical coordinates determined by the supercanonical transformation. The method is a generalization of an Inönu–Wigner contraction procedure and a Wei–Norman method for superalgebras with a selection of subalgebras associated with the root space decomposition of the Lie superalgebra. Analogously, this is a separation of variables method for quantum mechanical problems in systems with bosons and fermions. The standard Floquet theory leads to new results concerning stability for locally periodic coefficients. The lowest dimensional cases are explicitly computed. The intertwining of boson and fermions systems and the Hamiltonians considered here are of interest in quantum control theory for systems including fermions and bosons, in quantum optics, and quantum computation.

Generalized para-Bose states

In this paper, we construct integrals of motion in a para-Bose formulation for a general time-dependent quadratic Hamiltonian, which, in its turn, commutes with the reflection operator. In this context, we obtain generalizations for the squeezed vacuum states (SVS) and coherent states (CS) in terms of the Wigner parameter. Furthermore, we show that there is a completeness relation for the generalized SVS owing to the Wigner parameter. In the study of the probability transition, we found that the displacement parameter acts as a transition parameter by allowing access to odd states, while the Wigner parameter controls the dispersion of the distribution. We show that the Wigner parameter is quantized by imposing that the vacuum state has even parity. We apply the general results to the case of the time-independent para-Bose oscillator and find that the mean values of the coordinate and momentum have an oscillatory behavior similarly to the simple harmonic oscillator, while the standard deviation presents corrections in terms of the squeeze, displacement, and Wigner parameters.

Effective Heisenberg equations for quadratic Hamiltonians

In this paper, we discuss effective quantum dynamics obtained by averaging projector with respect to free dynamics. For unitary dynamics generated by quadratic fermionic Hamiltonians, we obtain effective Heisenberg dynamics. By perturbative expansions, we obtain the correspondent effective time-local Heisenberg equations. We also discuss a similar problem for bosonic case.

On the SQH method for solving optimal control problems with non-smooth state cost functionals or constraints

In this study, we discuss how the sequential quadratic Hamiltonian (SQH) scheme can be used to solve optimal control problems with non-smooth state constraints or non-smooth state cost terms. Theoretical and numerical aspects are both considered. Further, we connect to other recent techniques to solve such optimal control problems. The capability to handle non-smooth state terms extends the possibilities of modeling costs of the state and the dynamics of a system, e.g. to weight the deviation of the state from a desired state in an L1-norm. In this work, the non-smooth state terms are transformed into a bilinear structure on which the SQH scheme proved to solve optimal control problems fast promising an efficient numerical solution also in the non-smooth case. The novelty of the presented approach is that the non-smoothness is transformed into a bilinear structure. It is proved that the SQH method, which is based on the Pontryagin maximum principle, converges to an optimal solution of the corresponding transformed optimal control problem without any globalization techniques.