Schwinger Keldysh Research Articles

Schwinger–Keldysh Path Integral Formalism for a Quenched Quantum Inverted Oscillator

In this work, we study the time-dependent behavior of quantum correlations of a system of an inverted oscillator governed by out-of-equilibrium dynamics using the well-known Schwinger–Keldysh formalism in the presence of quantum mechanical quench. Considering a generalized structure of a time-dependent Hamiltonian for an inverted oscillator system, we use the invariant operator method to obtain its eigenstate and continuous energy eigenvalues. Using the expression for the eigenstate, we further derive the most general expression for the generating function as well as the out-of-time-ordered correlators (OTOCs) for the given system using this formalism. Further, considering the time-dependent coupling and frequency of the quantum inverted oscillator characterized by quench parameters, we comment on the dynamical behavior, specifically the early, intermediate and late time-dependent features of the OTOC for the quenched quantum inverted oscillator. Next, we study a specific case, where the system of an inverted oscillator exhibits chaotic behavior by computing the quantum Lyapunov exponent from the time-dependent behavior of OTOCs in the presence of the given quench profile.

Anisotropic Mass of a Magnetic Skyrmion due to Its Interaction with Conduction Electrons: A Schwinger–Keldysh Field Theory Approach

Anisotropic Mass of a Magnetic Skyrmion due to Its Interaction with Conduction Electrons: A Schwinger–Keldysh Field Theory Approach

Graviton-mediated polarisation–polarisation entanglement of photons by means of the Schwinger–Keldysh and Kadanoff–Baym formalisms and quantum Boltzmann equations

Graviton-mediated polarisation–polarisation entanglement of photons by means of the Schwinger–Keldysh and Kadanoff–Baym formalisms and quantum Boltzmann equations

Big Steve and the State of the Universe

I share some reminiscences of the late Steven Weinberg. Then I discuss a topic in quantum field theory that he taught me: the role of state wave functionals in deriving the iϵ term of the Feynman propagator when using functional formalism. This is perhaps a curiosity for in–out scattering amplitudes on flat-space backgrounds, but it has much greater significance for the in–in amplitudes of the Schwinger–Keldysh formalism in cosmology. It also touches on the fate, about which Weinberg wondered, of the large logarithms one sometimes finds in quantum corrections from inflationary particle production.

Decoherence and entropy generation in an open quantum scalar-fermion system with Yukawa interaction

We have studied the decoherence mechanism in a fermion and scalar quantum field theory with the Yukawa interaction in the Minkowski spacetime, using the non-equilibrium effective field theory formalism appropriate for open systems. The scalar field is treated as the system whereas the fermions as the environment. As the simplest realistic scenario, we assume that an observer measures only the Gaussian 2-point correlator for the scalar field. The cause of decoherence and the subsequent entropy generation is the ignorance of information stored in higher-order correlators, Gaussian and non-Gaussian, of the system and the surrounding. Using the 2-loop 2-particle irreducible effective action, we construct the renormalised Kadanoff–Baym equation, i.e., the equation of motion satisfied by the 2-point correlators in the Schwinger–Keldysh formalism. These equations contain the non-local self-energy corrections. We then compute the statistical propagator in terms of the 2-point functions. Using the relationship of the statistical propagator with the phase space area, we next compute the von Neumann entropy, as a measure of the decoherence or effective loss of information for the system. We have obtained the variation of the entropy with respect to various relevant parameters. We also discuss the qualitative similarities and differences of our results with the scenario when both the system and the environment are scalar fields.

Density Matrix Formalism for Interacting Quantum Fields

We provide a description of interacting quantum fields in terms of density matrices for any occupation numbers in Fock space in a momentum basis. As a simple example, we focus on a real scalar field interacting with another real scalar field, and present a practicable formalism for directly computing the density matrix elements of the combined scalar–scalar system. For deriving the main formula, we use techniques from non-equilibrium quantum field theory like thermo-field dynamics and the Schwinger–Keldysh formalism. Our results allow for studies of particle creation/annihilation processes at finite times and other non-equilibrium processes, including those found in the theory of open quantum systems.

Dephasing effects on the low-energy dynamics of ϕ4-model

In this paper, we investigate the Lindblad dynamics arising due to single particle loss and local dephasing in a d-dimensional nonrelativistic quantum system. Using Schwinger–Keldysh functional techniques, we derive the static mean field equation for the symmetry broken phase. Fluctuations over mean field are studied within the Bogoliubov approximation, and the momentum distribution and effective temperature are calculated. An effective temperature arises which depends on the dephasing rate [Formula: see text] and mean field particle density. Furthermore, a perturbative renormalization group (RG) upto one loop order is used to study the underlying dynamics of the model. The conditions for the system to be perturbative are obtained.

Dynamical chemistry: non-equilibrium effective actions for reactive fluids

We present two approaches for describing chemical reactions taking place in fluid phase. The first method mirrors the usual derivation of the hydrodynamic equations of motion by relating conserved—or to account for chemical reactions, non-conserved—currents to local-equilibrium parameters. The second method involves a higher-brow approach in which we attack the same problem from the perspective of non-equilibrium effective field theory (EFT). Non-equilibrium effective actions are defined using the in–in formalism on the Schwinger–Keldysh contour and are therefore capable of describing thermal fluctuations and dissipation as well as quantum effects. The non-equilibrium EFT approach is especially powerful as all terms in the action are fully specified by the symmetries of the system; in particular the second law of thermodynamics does not need to be included by hand, but is instead derived from the action itself. We find that the equations of motion generated by both methods agree, but the EFT approach yields certain advantages. To demonstrate some of these advantages we construct a quadratic action that is valid to very small distance scales—much smaller than the scales at which ordinary hydrodynamic theories break down. Such an action captures the full thermodynamic and quantum behavior of reactions and diffusion at quadratic order. Finally, taking the low-frequency and low-wavenumber limit, we reproduce the linearized version of the well-known reaction–diffusion equations as a final coherence check.

Lattice Lindblad simulation

Abstract We perform real-time lattice simulation of an open quantum system based on the Schwinger–Keldysh path integral representation of the Lindblad formalism. Although the real-time simulation generally suffers from the sign problem, a few exceptional cases exist. We focus on a sign-problem-free system of a non-relativistic spinless fermion and analyze the time evolution under driving and dissipation.

Electromagnetic coupling effects in natural inflation

In this work we study the effects of the electromagnetic coupling in natural inflation in a systematic manner using the Schwinger–Keldysh formalism. The corresponding influence functional is evaluated in the one-loop level. It can be interpreted as due to a single stochastic force. The equation of motion of the inflaton field is therefore given in the form of a Langevin equation. Lastly, the two-point and the three-point correlation functions of the inflaton field are worked out. They are related to the power spectrum and the nongaussianity of the inflaton field, respectively.

Cosmological correlators, In–In formalism and double copy

We are investigating if the double copy structure as product of scattering amplitudes of gauge theories applies to cosmological correlators computed, in a class of theories for inflation, by the operatorial version of the In–In formalism of Schwinger–Keldysh. We consider tree-level momentum–space correlators involving primordial gravitational waves with different polarizations and the scalar curvature fluctuations on a three-dimensional fixed spatial slice. The correlators are sum of terms factorized in a time-dependent scalar factor, which takes into account the curved background where energy is not conserved, and in a so-called tensor factor, constructed by polarization tensors. In the latter, we recognize scattering amplitudes in four-dimensional Minkowski space spanned by three points gravitational amplitudes related by double copy to those of gauge theories. Our study indicates that gravitational waves are double copy of gluons and the primordial scalar curvature is double copy of a scalar with Higgs-like interactions.

“Old” conformal bootstrap on AdS: O(N) symmetric scalar model

Bootstrap equations for conformal correlators that mimic the early theory of conformal bootstrap are written down in frames of the AdS/CFT approach. The simplified version of these equations, that may be justified if Schwinger–Keldysh formalism is used in AdS/CFT instead of conventional Feynman–Witten diagrams technique, permits to calculate values of conformal dimensions in the [Formula: see text] symmetric model with conformal or composite Habbard–Stratonovich field.

Non-thermal fixed points: Universal dynamics far from equilibrium

In this article we give an overview of the concept of universal dynamics near non-thermal fixed points in isolated quantum many-body systems. We outline a non-perturbative kinetic theory derived within a Schwinger–Keldysh closed-time path-integral approach, as well as a low-energy effective field theory which enable us to predict the universal scaling exponents characterizing the time evolution at the fixed point. We discuss the role of wave-turbulent transport in the context of such fixed points and discuss universal scaling evolution of systems bearing ensembles of (quasi) topological defects. This is rounded off by the recently introduced concept of prescaling as a generic feature of the evolution towards a non-thermal fixed point.

Discrete-time construction of nonequilibrium path integrals on the Kostantinov–Perel’ time contour

Rigorous nonequilibrium actions for the many-body problem are usually derived by means of path integrals combined with a discrete temporal mesh on the Schwinger–Keldysh time contour. The latter suffers from a fundamental limitation: the initial state on this contour cannot be arbitrary, but necessarily needs to be described by a non-interacting density matrix, while interactions are switched on adiabatically. The Kostantinov–Perel’ contour overcomes these and other limitations, allowing generic initial-state preparations. In this article, we apply the technique of the discrete temporal mesh to rigorously build the nonequilibrium path integral on the Kostantinov–Perel’ time contour.

Correlation functions in Schwarzian liquid

We study a class of SYK-type models in large N limit from the gravity dual side in terms of Schwarzian action analytically. The quantum correction to two point correlation function due to the Schwarzian action produces transfer of degree of freedom from the quasiparticle peak to Hubbard band in density of states (DOS), a signature strong correlation. In Schwinger-Keldysh (SK) formalism, we calculate higher point thermal out-of-time order correlation (OTOC) functions, which indicate quantum chaos by having Lyapunov exponent. Higher order local spin-spin correlations are also calculated, which can be related to the dynamical local susceptibility of quantum liquids such as spin glasses, disordered metals.

Nonequilibrium field theory for dynamics starting from arbitrary athermal initial conditions

Schwinger Keldysh field theory is a widely used paradigm to study non-equilibrium dynamics of quantum many-body systems starting from a thermal state. We extend this formalism to describe non-equilibrium dynamics of quantum systems starting from arbitrary initial many-body density matrices. We show how this can be done for both Bosons and Fermions, and for both closed and open quantum systems, using additional sources coupled to bilinears of the fields at the initial time, calculating Green's functions in a theory with these sources, and then taking appropriate set of derivatives of these Green's functions w.r.t. initial sources to obtain physical observables. The set of derivatives depend on the initial density matrix. The physical correlators in a dynamics with arbitrary initial conditions do not satisfy Wick's theorem, even for non-interacting systems. However our formalism constructs intermediate ``n-particle Green's functions'' which obey Wick's theorem and provide a prescription to obtain physical correlation functions from them. This allows us to obtain analytic answers for all physical many body correlation functions of a non-interacting system even when it is initialized to an arbitrary density matrix. We use these exact expressions to obtain an estimate of the violation of Wick's theorem, and relate it to presence of connected multi-particle initial correlations in the system. We illustrate this new formalism by calculating density and current profiles in many body Fermionic and Bosonic open quantum systems initialized to non-trivial density matrices. We have also shown how this formalism can be extended to interacting many body systems.

Bell inequality in the holographic EPR pair

We study the Bell inequality in a holographic model of the casually disconnected Einstein–Podolsky–Rosen (EPR) pair. The Clauser–Horne–Shimony–Holt (CHSH) form of Bell inequality is constructed using holographic Schwinger–Keldysh (SK) correlators. We show that the manifestation of quantum correlation in Bell inequality can be holographically reproduced from the classical fluctuations of dual accelerating string in the bulk gravity. The violation of this holographic Bell inequality supports the essential quantum property of this holographic model of an EPR pair.

Comments on the adiabatic theorem

We consider the simplest example of a nonstationary quantum system which is quantum mechanical oscillator with varying frequency and [Formula: see text] self-interaction. We calculate loop corrections to the Keldysh, retarded/advanced propagators and vertices using Schwinger–Keldysh diagrammatic technique and show that there is no physical secular growth of the loop corrections in the cases of constant and adiabatically varying frequency. This fact corresponds to the well-known adiabatic theorem in quantum mechanics. However, in the case of nonadiabatically varying frequency we obtain strong IR corrections to the Keldysh propagator which come from the “sunset” diagrams, grow with time indefinitely and indicate energy pumping into the system. It reveals itself via the change in time of the level population and of the anomalous quantum average.

COSMOS- $$e'$$ e ′ -soft Higgsotic attractors

In this work, we have developed an elegant algorithm to study the cosmological consequences from a huge class of quantum field theories (i.e. superstring theory, supergravity, extra dimensional theory, modified gravity, etc.), which are equivalently described by soft attractors in the effective field theory framework. In this description we have restricted our analysis for two scalar fields – dilaton and Higgsotic fields minimally coupled with Einstein gravity, which can be generalized for any arbitrary number of scalar field contents with generalized non-canonical and non-minimal interactions. We have explicitly used R^2 gravity, from which we have studied the attractor and non-attractor phases by exactly computing two point, three point and four point correlation functions from scalar fluctuations using the In-In (Schwinger–Keldysh) and the delta mathcal{N} formalisms. We have also presented theoretical bounds on the amplitude, tilt and running of the primordial power spectrum, various shapes (equilateral, squeezed, folded kite or counter-collinear) of the amplitude as obtained from three and four point scalar functions, which are consistent with observed data. Also the results from two point tensor fluctuations and the field excursion formula are explicitly presented for the attractor and non-attractor phase. Further, reheating constraints, scale dependent behavior of the couplings and the dynamical solution for the dilaton and Higgsotic fields are also presented. New sets of consistency relations between two, three and four point observables are also presented, which shows significant deviation from canonical slow-roll models. Additionally, three possible theoretical proposals have presented to overcome the tachyonic instability at the time of late time acceleration. Finally, we have also provided the bulk interpretation from the three and four point scalar correlation functions for completeness.

Control of noisy quantum systems: Field-theory approach to error mitigation

We consider the quantum-control task of obtaining a target unitary operation via control fields that couple to the quantum system and are chosen to best mitigate errors resulting from time-dependent noise. We allow for two sources of noise: fluctuations in the control fields and those arising from the environment. We address the issue of error mitigation by means of a formulation rooted in the Martin-Siggia-Rose (MSR) approach to noisy, classical statistical-mechanical systems. We express the noisy control problem in terms of a path integral, and integrate out the noise to arrive at an effective, noise-free description. We characterize the degree of success in error mitigation via a fidelity, which characterizes the proximity of the sought-after evolution to ones achievable in the presence of noise. Error mitigation is then accomplished by applying the optimal control fields, i.e., those that maximize the fidelity subject to any constraints obeyed by the control fields. To make connection with MSR, we reformulate the fidelity in terms of a Schwinger-Keldysh (SK) path integral, with the added twist that the `forward' and `backward' branches of the time-contour are inequivalent with respect to the noise. The present approach naturally allows the incorporation of constraints on the control fields; a useful feature in practice, given that they feature in real experiments. We illustrate this MSR-SK approach by considering a system consisting of a single spin $s$ freedom (with $s$ arbitrary), focusing on the case of $1/f$ noise. We discover that optimal error-mitigation is accomplished via a universal control field protocol that is valid for all $s$, from the qubit (i.e., $s=1/2$) case to the classical (i.e., $s \to \infty$) limit. In principle, this MSR-SK approach provides a framework for addressing quantum control in the presence of noise for systems of arbitrary complexity.