Stochastic wave-function approach to the calculation of multitime correlation functions of open quantum systems

We continue our study of the doubled Hilbert space formalism of double-scaled SYK model initiated in Okuyama (2024) [7]. We show that the 1-particle Hilbert space introduced by Lin and Stanford (2023) [6] is related to the doubled Hilbert space H0⊗H0 of the 0-particle Hilbert space H0 by some linear isomorphism C. It turns out that the entangler and disentangler appeared in our previous study are given by the dual (or adjoint) of C−1 and C.

This thesis contributes to the study of quantum chaos and thermalization of closed quantum systems by applying tools from quantum information theory. We study scrambling – the delocalization of quantum information – in one-dimensional quantum many-body systems using the observable-independent tripartite information 𝐼3. We consider closed quantum systems that undergo unitary time evolution governed by a Hamiltonian, and compare them to the strong scrambling caused by random unitary operators (Haar scrambling). The tripartite information is a tool from quantum information theory that allows us to quantify correlations in space and time by considering the unitary operator evolving the system as a state |𝑈 (𝑡)⟩ in a doubled Hilbert space H ⊗ H. We found a ballistic spreading of information in translation invariant systems of spinless fermions that causes Haar scrambling in interacting models. For strongly disordered versions of the XXZ chain and the transverse-field Ising chain we found a complete breakdown of in- formation transport in non-interacting Anderson localized systems, as well as a logarithmically slow spreading of the information signal in systems that are said to be many-body localized (MBL).

We investigate the algebraic properties of the quantum counterpart of the classical canonical transformations using the symbol calculus approach to quantum mechanics. In this framework we construct a set of pseudodifferential operators which act on the symbols of operators, i.e. on functions defined over phase space. They act as operatorial left and right multiplication and form a W∞×W∞ algebra which contracts to its diagonal subalgebra in the classical limit. We also describe the Gel’fand-Naimark-Segal (GNS) construction in this language and show that the GNS representation space (a doubled Hilbert space) is closely related to the algebra of functions over phase space equipped with the star product of the symbol calculus.

The pseudomode model effectively captures the nonperturbative dynamics of open quantum systems with significantly reduced degrees of freedom. However, it is limited by the exponential growth of the Hilbert space dimension. To overcome the computational challenges, we propose a novel method that combines the multiple Davydov Ansatz with the Choi-Jamiolkowski isomorphism. Within this framework, the Lindblad equation is transformed into the non-Hermitian Schrödinger equation in a double Hilbert space, with its dynamics determined using the time-dependent variational principle. Three cases are calculated to demonstrate the effectiveness of the proposed method. We first discuss how the Davydov Ansatz works for the model with a single pseudomode. Extending the method to multiple pseudomodes, we show that the Ansatz effectively circumvents the exponential growth of the Hilbert space. Additionally, the method is also capable of addressing potential intersections that emerge in multibath scenarios. This approach offers potential applicability to various types of pseudomode models and other dissipative systems, providing a promising tool for the studies of open quantum dynamics.

We present a novel protocol for teleportation of arbitrary bipartite pure and mixed state with shared cluster entanglement in this paper. By employing Bell-state measurement on the teleported state and the shared cluster state twice, a sender could transmit the arbitrary bipartite state to a distant receiver. We show the good feature of the cluster state channel, with which it can realize the deterministic teleportation rather than probabilistic one. Moreover, since we require less particles to be shared and need no auxiliary qubit in our protocol, it is more efficient and applicable than the previous schemes.

On the calculation of correlation functions in nuclear matter

The cosmic microwave background can measure the inflaton potential only if\ninflation lasts sufficiently long before the time of horizon crossing of\nobservable fluctuations, such that non-linear effects in the time evolution of\nGreen's functions lead to a loss of memory of initial conditions for the\nultraviolet tail of the spectrum. Within a derivative expansion of the quantum\neffective action for an interacting scalar field we discuss the most general\nsolution for the correlation function, including arbitrary pure and mixed\nquantum states. In this approximation no loss of memory occurs - cosmic\nmicrowave observations see the initial spectrum at the beginning of inflation,\nprocessed only mildly by the scale-violating effects at horizon crossing\ninduced by the inflaton potential.\n

A method for the quantification of synchrony and oscillatory properties of neuronal activity

The Schwinger-Keldysh diagram technique is usually involved in the calculation of real-time in-in correlation functions. In the case of a thermal state, one can analytically continue imaginary-time Matsubara correlation functions to real times. Nevertheless, not all real-time correlation functions usually can be obtained by such procedure. Moreover, numerical analytic continuation is an ill-posed problem. Thus, even in the case of a thermal state one may need for the Schwinger-Keldysh formalism. If the potential of a system admits degenerate minima, instantonic effects enter the game, so one should also integrate over the instantonic moduli space, including the one, corresponding to the imaginary time translational invariance. However, the Schwinger-Keldysh closed time contour explicitly breaks such invariance. We argue, that this invariance must be recovered, and show, how it can be done. After that, we construct an extension of the Schwinger-Keldysh diagram technique to instantonic systems and demonstrate it on the example of the first few-point correlation functions.

The viscoelasticity of a stretched polymer chain with its end particles exposed to oppositely acting forces is studied via collisional molecular dynamics and analytically. A simple model according to which a polymer molecule is a chain of particles linked through freely jointed elastic bonds is adopted. The analytical theory is in good agreement with the results of the computer simulation of time correlation functions in the range of large-scale motions of a polymer molecule. It is found that the decay of correlation functions K αβμν of fluctuations of the microscopic stress tensor of a chain, K zzzz, K zαzα , = K zαzα , (α = x, y; z is the axis along which the forces act), is slowed down, and their value increases relative to the respective correlation functions of a chain with fixed ends. The greater the force, the higher this difference. The correlation functions that are transverse with respect to the z axis do not differ from those for chains with fixed ends. The results show that, in the calculation of time correlation functions of strongly stretched polymer chains, different statistical ensembles are not equivalent; this must be taken into account in the dynamic theories of heavily deformed polymers.

The multifractal formalism [1,2], which originated in the context of turbulence and chaotic systems [3], has become a standard tool to analyze phenomena observed in fractal aggregates, semiconductors, disordered systems and so on [4]. To describe a multifractal set one usually divides it into small boxes (of size l say) and introduces a partition sum of order q (or a generalized correlation function), C q(l), defined as the sum over the box-probabilities, each raised to a power q. From the scaling behavior C q(l) ~ lτ(q), the multifractal f(α) spectrum then follows from a Legendre transform of τ(q) [1,2]. To obtain the entire f(α) spectrum the moment q must vary from -∞ to +∞. Sometimes this demands very cumbersome calculations and it can be hard to obtain a good convergence of the scaling exponents. Instead by using the concept of multiscaling it is possible to obtain a part of the f(α) spectrum by calculating only one moment of the partition sum (or one correlation function). The idea of multiscaling was introduced by Kadanoff and co-workers to account for the scaling behavior of power spectra of temperature signals obtained in convective turbulence [5] and to describe distribution functions computed in simple models for sandpiles [6]. It was found that standard finite size scaling given by a few exponents did not describe the data satisfactory. Instead, by the multiscaling ansatz, the data could be nicely fitted using an infinity of scaling exponents. Recently, it was shown that it is possible to relate the concepts of multiscaling and multifractality to each other in a very natural way, namely by introducing a lower cut-off in the calculation of the correlation functions [7]. As this cut-off varies, a cross-over in the scaling of the correlation function C q(l) appears at a certain length scale. Below this length scale the correlation function stops being a power law, and this cross-over point moves as the cut-off varies. Using the multiscaling idea one then rescales the various correlation curves and collapse then onto a single scaling function (apart a small correction) of which a section is given by the f(α) spectrum. Again, this means that we obtain the multifractal spectrum from calculating only one moment in the partition sum. In turbulence, this type of behavior appears very naturally, if a multifractal model is assumed [8]. Here one can relate various values of the scaling exponents to various dissipative cut-offs [4]. In order words, there will be different viscous cut-offs (for each value of the scaling exponent) occurring at different length scales as one moves below the Kolmogorov inner scale. This defines a kind of intermediate viscous regime, which again by a multiscaling transformation can be rescaled onto one curve, part of which is the singularity spectrum for turbulence [8].

Multiscaling is shown to be a consequence of multifractality when a lower cutoff \ensuremath{\varepsilon} is introduced in calculations of correlation functions. After a suitable rescaling, the correlation function data for different values of \ensuremath{\varepsilon} seem to fall onto a single curve. In the multiscaling regime, however, we show that there is not a unique functional form at varying \ensuremath{\varepsilon}, but a spread very close to a single curve. For each \ensuremath{\varepsilon}, this curve can be computed analytically in terms of the f(\ensuremath{\alpha}) spectrum which characterizes the multifractal. Part of this spectrum can thus be obtained by computing only one moment of the weights at \ensuremath{\varepsilon}\ensuremath{\ne}0.

A theory of magnetic nuclear relaxation, providing a calculation of the correlation functions of complex motion of methyl groups is presented. The complex motion consists of jumps over the barrier (classical motion) and jumps through the barrier (tunneling). The Schrödinger equation has been applied in the calculation of the rate constant of tunneling jumps through the barrier. The equations for the spectral densities (ω), where ω = (ωI ± ωT), and ω = (mωI), where m = 1 or 2, ωI is the resonance angular frequency, and ωT is the angular frequency of the tunnel splitting, are derived. These spectral densities concern the motion of spin pairs inside methyl groups ("intra") and outside methyl groups ("inter"). The calculated spectral densities are applied to analyze the temperature dependencies of the spin−lattice relaxation rate in solids containing methyl groups. A wide regime of temperatures from 0 K up to the melting point is considered. The temperature at which the tunneling process ceases is discussed. The theory proposed explains the different temperature dependencies of T1 for CH3COOK obtained in the experiments caused by small amounts of CH3COOH or water impurities. The theoretical equations derived in this paper are compared to those known in the literature.

We compute the partition function of mathcal{N} = 2 supersymmetric mixed dimensional QED on a squashed hemisphere using localization. Mixed dimensional QED is an abelian gauge theory coupled to charged matter fields at the boundary. The partition function is a function of the complex gauge coupling τ, the choice of R-symmetry and the squashing deformation. The superconformal R-symmetry is determined using the 3-dimensional F-maximization. The free energy as a function of squashing deformation allows computing correlation functions that contain the insertion of the energy-momentum tensor. We compute the 2-point correlation function of the energy-momentum tensor of 3-dimensional theory by differentiating the free energy with respect to the squashing parameter. We comment on the behaviour of the 2-point function as we change the complex coupling τ.

Plating and stripping of lithium protrusions in lithium metal symmetric cells containing a solid block copolymer electrolyte was studied as a function of time in 3D using time-resolved X-ray tomography. These measurements enabled determination of the spatial variation in current densities at the plating and stripping electrodes. The initial interelectrode distance was 27 μm. Correlation functions were calculated to reveal the relationships between current densities at the two electrodes and local electrolyte thickness. Current densities at opposing electrode locations during protrusion growth is uncorrelated until the local interelectrode distance decreases to less than 6 μm, just before the cell shorts. Mass balance was used to determine the area from which lithium ions that form a protrusion were stripped. Computational modeling of the plating and stripping process reveals the interplay between electrochemical and mechanical driving forces and their effect on nonuniform current distribution. Model predictions were compared with experiments without resorting to any adjustable parameters. The computed correlation functions were in qualitative agreement with experiments. Finally, the model was used to calculate contour plots of electrochemical potential within the electrolyte, shedding light on how geometry, salt concentration, interelectrode distance, and mechanical stress influence local rates of electrochemical reaction.