Coherent State Path Integral Research Articles

Field theory of enzyme-substrate systems with restricted long-range interactions.

Enzyme-substrate kinetics form the basis of many biomolecular processes. The interplay between substrate binding and substrate geometry can give rise to long-range interactions between enzyme binding events. Here we study a general model of enzyme-substrate kinetics with restricted long-range interactions described by an exponent -γ. We employ a coherent-state path integral and renormalization group approach to calculate the first moment and two-point correlation function of the enzyme-binding profile. We show that starting from an empty substrate the average occupancy follows a power law with an exponent 1/(1-γ) over time. The correlation function decays algebraically with two distinct spatial regimes characterized by exponents -γ on short distances and -(2/3)(2-γ) on long distances. The crossover between both regimes scales inversely with the average substrate occupancy. Our work allows associating experimental measurements of bound enzyme locations with their binding kinetics and the spatial conformation of the substrate.

Lefschetz thimble quantum Monte Carlo for spin systems

Monte Carlo simulations are useful tools for modeling quantum systems, but in some cases they suffer from a sign problem, leading to an exponential slow down in their convergence to a value. While solving the sign problem is generically NP hard, many techniques exist for mitigating the sign problem in specific cases; in particular, the technique of deforming the Monte Carlo simulation's plane of integration onto Lefschetz thimbles (complex hypersurfaces of stationary phase) has seen significant success in the context of quantum field theories. We extend this methodology to spin systems by utilizing spin coherent state path integrals to reexpress the spin system's partition function in terms of continuous variables. Using some toy systems, we demonstrate its effectiveness at lessening the sign problem in this setting, despite the fact that the initial mapping to spin coherent states introduces its own sign problem. The standard formulation of the spin coherent path integral is known to make use of uncontrolled approximations; despite this, for large spins they are typically considered to yield accurate results, so it is somewhat surprising that our results show significant systematic errors. Therefore, possibly of independent interest, our use of Lefschetz thimbles to overcome the intrinsic sign problem in spin coherent state path integral Monte Carlo enables a novel numerical demonstration of a breakdown in the spin coherent path integral.

Nonperturbative renormalization of quantum thermodynamics from weak to strong couplings

By solving the exact master equation of open quantum systems, we formulate the quantum thermodynamics from weak to strong couplings. The open quantum systems exchange matters, energies and information with their reservoirs through quantum particles tunnelings that are described by the generalized Fano-Anderson Hamiltonians. We find that the exact solution of the reduced density matrix of these systems approaches a Gibbs-type state in the steady-state limit for both the weak and strong system-reservoir coupling strengths. When the couplings become strong, thermodynamic quantities of the system must be renormalized. The renormalization effects are obtained nonperturbatively after exactly traced over all reservoir states through the coherent state path integrals. The renormalized system Hamiltonian is characterized by the renormalized system energy levels and interactions, corresponding to the quantum work done by the system. The renormalized temperature is introduced to characterize the entropy production counting the heat transfer between the system and the reservoir. We further find that only with the renormalized system Hamiltonian and other renormalized thermodynamic quantities, can the exact steady state of the system be expressed as the standard Gibbs state. Consequently, the corresponding exact steady-state particle occupations in the renormalized system energy levels obey the Bose-Einstein and the Fermi-Dirac distributions for bosonic and fermionic systems, respectively. Thus, the conventional statistical mechanics and thermodynamics are thereby rigorously deduced from quantum dynamical evolution.

Effective dynamics of weak coupling loop quantum gravity

By taking the limit that Newton's gravitational constant tends to zero, the weak coupling loop quantum gravity can be formulated as a $U(1{)}^{3}$ gauge theory instead of the original $SU(2)$ gauge theory. In this paper, a parametrization of the $SU(2)$ holonomy-flux variables by the $U(1{)}^{3}$ holonomy-flux variables is introduced, and the Hamiltonian operator based on this parametrization is obtained for the weak coupling loop quantum gravity. It is shown that the effective dynamics obtained from the coherent state path integrals in $U(1{)}^{3}$ and $SU(2)$ loop quantum gravity respectively are consistent to each other in the weak coupling limit, provided that the expectation values of the Hamiltonian operators on the coherent states in these two theories coincide with their classical expressions respectively.

The importance of the metaplectic correction in Kähler quantization: a coherent-state path integral perspective

The metaplectic correction in Kahler quantization is a result of the introduction of the half-form structure when considering geometric quantization over Kahler manifolds. This correction is a very interesting topic, since no mathematical argument can prove or reject its necessity, and only a physical consideration can point to an answer. As a result, no universally accepted reason for considering such a structure has been identified in the context of quantization theory. In this letter, we view this topic not in a quantization context, but as a means to understand an exact connection between the canonical and path integral formulation of quantum mechanics. More specifically, we investigate whether or not the contribution of the metaplectic correction is related to the recent results found in the context of coherent-state path integrals, regarding the “correct” identification of the continuum limit. We then use the theory of coherent-state path integrals as a template, and our results indeed point to the necessity of this correction.

On the Interaction Between a Time-Dependent Field and a Two-Level Atom: Path Integral Treatment

We use the coherent state path integral and a Schwinger model for the spin to solve the generalized Jaynes-Cummings model with a time-dependent coupling parameter of the photons-field. The propagators are given explicitly as perturbation series. These are summed up exactly. The wave functions are deduced.

Star product representation of coherent state path integrals

In this paper, we determine the star product representation of coherent path integrals. By employing the properties of generalized delta functions with complex arguments, the Glauber–Sudarshan P-function corresponding to a non-diagonal density operator is obtained. Then, we compute the Husimi–Kano Q-representation of the time evolution operator in terms of the normal star product. Finally, the optical equivalence theorem allows us to express the coherent state path integral as a star exponential of the Hamiltonian function for the normal product.

Bicoherent-state path integral quantization of a non-hermitian hamiltonian

We introduce, for the first time, bicoherent-state path integration as a method for quantizing non-hermitian systems. Bicoherent-state path integrals arise as a natural generalization of ordinary coherent-state path integrals, familiar from hermitian quantum physics. We do all this by working out a concrete example, namely, computation of the propagator of a certain quasi-hermitian variant of Swanson’s model, which is not invariant under conventional PT-transformation. The resulting propagator coincides with that of the propagator of the standard harmonic oscillator, which is isospectral with the model under consideration by virtue of a similarity transformation relating the corresponding hamiltonians. We also compute the propagator of this model in position space by means of Feynman path integration and verify the consistency of the two results.

Hubbard–Stratonovich transformation and consistent ordering in the coherent state path integral: insights from stochastic calculus

Recently, doubts have been cast on the validity of the continuous-time coherent state path integral. This has led to controversies regarding the correct way of performing calculations with path integrals, and to several alternative definitions of what should be their continuous limit. Furthermore, the issue of a supposedly proper ordering of the Hamiltonian operator, entangled with the continuous-time limit, has led to considerable confusion in the literature. Since coherent state path integrals are at the basis of the modern formulation of many-body quantum theory, it should be laid on solid foundations.Here, we show that the issues raised above are coming from the illegitimate use of the (standard) rules of calculus, which are not necessarily valid in path integrals. This is well known in the context of stochastic equations, in particular in their path integral formulation. This insight allows for solving these issues and addressing the correspondence between the various orderings at the level of the path integral. We also use this opportunity to address the proper calculation of a functional determinant in the presence of a Hubbard–Stratonovich field, which shares in the controversies.

Bosonic Schwinger model for spinning Feshbach–Villars particle in step potential

Propagator for spinning particle described in the Feshbach–Villars formalism is set up using the bosonic coherent state path integral. The boson model for the charge-spin symmetry is presented following the Schwinger recipe. The calculations are explicitly evaluated in the case of the step potential. The perturbation technique is used and the series are exactly summed.

Effective dynamics from coherent state path integral of full loop quantum gravity

A new routine is proposed to relate Loop Quant Cosmology (LQC) to Loop Quantum Gravity (LQG) from the perspective of effective dynamics. We derive the big-bang singularity resolution and big bounce from the first principle of full canonical LQG. Our results are obtained in the framework of the reduced phase space quantization of LQG. As a key step in our work, we derive with coherent states a new discrete path integral formula of the transition amplitude generated by the physical Hamiltonian. The semiclassical approximation of the path integral formula gives an interesting set of effective equations of motion (EOMs) for full LQG. When solving the EOMs with homogeneous and isotropic ansatz, we reproduce the LQC effective dynamics in $\mu_0$-scheme. The solution replaces the big-bang singularity by a big bounce. In the end, we comment on the possible relation between the $\bar{\mu}$-scheme of effective dynamics and the continuum limit of the path integral formula.

Fermionic Schwinger model for spinning Feshbach–Villars particle in magnetic field

Propagator for spinning particle described in the Feshbach–Villars formalism is set up using fermionic coherent state path integral. The fermionization of the charge-spin symmetry is presented following the Schwinger recipe. The explicit calculations are done in the free case and magnetic interaction using the Foldy–Wouthuysen tranformation. The spectrum and wave functions are deduced.

Statistics of heat transport across a capacitively coupled double quantum dot circuit

We study heat current and the full statistics of heat fluctuations in a capacitively-coupled double quantum dot system. This work is motivated by recent theoretical studies and experimental works on heat currents in quantum dot circuits. As expected intuitively, within the (static) mean-field approximation, the system at steady-state decouples into two single-dot equilibrium systems with renormalized dot energies, leading to zero average heat flux and fluctuations. This reveals that dynamic correlations induced between electrons on the dots is solely responsible for the heat transport between the two reservoirs. To study heat current fluctuations, we compute steady-state cumulant generating function for heat exchanged between reservoirs using two approaches : Lindblad quantum master equation approach, which is valid for arbitrary coulomb interaction strength but weak system-reservoir coupling strength, and the saddle point approximation for Schwinger-Keldysh coherent state path integral, which is valid for arbitrary system-reservoir coupling strength but weak coulomb interaction strength. Using thus obtained generating functions, we verify steady-state fluctuation theorem for stochastic heat flux and study the average heat current and its fluctuations. We find that the heat current and its fluctuations change non-monotonically with the coulomb interaction strength ($U$) and system-reservoir coupling strength ($\Gamma$) and are suppressed for large values of $U$ and $\Gamma$.

Reply to “Comment on ‘Coherent-state path integrals in the continuum’”

In this Reply we briefly clarify the main points of our method to construct coherent-state path integrals in the continuum and we reply to the critique raised by in the preceding Comment by Kochetov [Phys. Rev. A 99, 026101 (2019)]. By using definite examples, we prove that our approach is capable of resolving the inconsistencies accompanying the standard coherent-state path-integral representation of interacting systems.

Comment on “Coherent-state path integrals in the continuum”

The authors of the recent paper [Phys. Rev. A 90, 032104 (2014)] claim to have established a time continuous formulation of path integration in the CS basis free from the mentioned inconsistency. Since a few recent investigations consider the approach advocated in [Phys. Rev. A 90, 032104 (2014)] as providing a "remedy" for the CS path-integral anomaly , I feel it is appropriate to show that the "recipe" proposed does not eliminate the anomaly.

Path integral and instanton calculus for stochastic jump processes with a finite number of states

We develop a method to analyze stochastic jump processes with a finite number of states ${0,1,...,N}$. It is known that the system can be analyzed using su(2) Lie algebra and the spin coherent state path integral. First, we analyze a linear susceptible-infected-susceptible epidemic model by using the Lie algebraic method and path integral. Then, we apply the instanton approximation to more general models where stochastic noise can induce bimodal distributions. By using density variables, we show that instanton solution can be interpreted as a solution of the equation of motion of the deterministic system corresponding to the stochastic system. It is also shown that instanton approximation correctly reproduces the noise-induced bimodality of the system.

Global Estimates of Errors in Quantum Computation by the Feynman\u2013Vernon Formalism

The operation of a quantum computer is considered as a general quantum operation on a mixed state on many qubits followed by a measurement. The general quantum operation is further represented as a Feynman–Vernon double path integral over the histories of the qubits and of an environment, and afterward tracing out the environment. The qubit histories are taken to be paths on the two-sphere S^2 as in Klauder’s coherent-state path integral of spin, and the environment is assumed to consist of harmonic oscillators initially in thermal equilibrium, and linearly coupled to to qubit operators hat{S}_z. The environment can then be integrated out to give a Feynman–Vernon influence action coupling the forward and backward histories of the qubits. This representation allows to derive in a simple way estimates that the total error of operation of a quantum computer without error correction scales linearly with the number of qubits and the time of operation. It also allows to discuss Kitaev’s toric code interacting with an environment in the same manner.

Generalized generating functional for mixed-representation Green’s functions: Coherent-state path integral representation

The aim of this paper is to study mixed-representation Green’s functions from the point of view of coherent-state path integrals. This is achieved by using the machinery of generalized generating functionals of Green’s functions, previously introduced by the present authors in the context of standard phase-pace path integrals. The obtained results are illustrated in the context of the linear harmonic oscillator.

Path Integral for an Effective Two Level Atom in a Kerr-Like Medium and Stark Shift with a Pseudo Hermitian Hamiltonian

We use the coherent state path integral and a angular model for the spin to solve the generalized Jaynes-Cummings model with a pseudo-hermitian Hamiltonian and nonlinear Kerr cavity. The propagators are given explicitly as perturbation series. These are summed up exactly. The energy spectrum and the bi-orthonormal basis of states are deduced.

Coherent-state path integrals in the continuum: The [formula omitted] case

We define the time-continuous spin coherent-state path integral in a way that is free from inconsistencies. The proposed definition is used to reproduce known exact results. Such a formalism opens new possibilities for applying approximations with improved accuracy and can be proven useful in a great variety of problems where spin Hamiltonians are used.