Odinger Picture Research Articles

What Is the Gravitational Field of a Mass in a Spatially Nonlocal Quantum Superposition?

The study of the gravitational field produced by a spatially nonlocal, superposed quantum state of a massive particle is an interesting and active area of research. One outstanding issue is whether the gravitational field behaves like the classical superposition of the gravitational field of two particles separated by a spatial distance with half the mass at each position. Alternatively, does the gravitational field behave as a quantum superposition with a far more interesting and subtle behavior than a simple classical superposition? Quantum field theory is ideally suited to probe exactly this kind of question. We study the scattering of a massless scalar on a spatially nonlocal quantum superposition of a massive particle. We compute the differential scattering cross section corresponding to one-graviton exchange. We find that the scattering cross section disagrees with the Newton-Schrödinger picture of potential scattering from two localized sources with half the mass at each source. This suggests that experimental observation of gravitational scattering could inform the viability of the semiclassical treatment of the gravitational field, as in the Newton-Schrödinger description, vs the fully quantum mechanical treatment adopted here. We comment on the experimental feasibility of observing such effects in systems with many particles such as Bose-Einstein condensates.

Langevin dynamics of generalized spins as SU( N ) coherent states

Quantum spin states often contain additional structure beyond the usual angular momentum dipole. The classical Landau-Lifshitz spin dynamics may be generalized to model the coupled dynamics of dipoles, quadrupoles, and higher order multipoles. This paper introduces temperature-dependent Langevin noise and damping terms to the generalized spin dynamics. High numerical efficiency is possible by working with coherent states in the Schr\"odinger picture. The method is illustrated via simulation of magnetic skyrmion dynamics both in and out of equilibrium.

Quantum speed limits for observables

In the Schr{\"o}dinger picture, the state of a quantum system evolves in time and the quantum speed limit describes how fast the state of a quantum system evolves from an initial state to a final state. However, in the Heisenberg picture the observable evolves in time instead of the state vector. Therefore, it is natural to ask how fast an observable evolves in time. This can impose a fundamental bound on the evolution time of the expectation value of quantum mechanical observables. We obtain the quantum speed limit time-bound for observable for closed systems, open quantum systems and arbitrary dynamics. Furthermore, we discuss various applications of these bounds. Our results can have several applications ranging from setting the speed limit for operator growth, correlation growth, quantum thermal machines, quantum control and many body physics.

Quantum regression theorem for multi-time correlators: A detailed analysis in the Heisenberg picture

The quantum regression theorem is one of the central results in open quantum systems and is extensively used for computing multi-point correlation functions. Traditionally it is derived for two-time correlators in the Markovian limit employing the Schr\"odinger picture. In this paper we make use of the Heisenberg picture to derive the quantum regression theorems for multi-time correlation functions, which in the special limit reduce to the well-known two-time regression theorem. For the multi-time correlation function we find that the regression theorem takes the same form as it takes for the two-time correlation function with a mild restriction that one of the times should be greater than all other time variables. Interestingly, the Heisenberg picture also allows us to derive an analog of regression theorem for out-of-time-ordered correlators. We further extend our study for the case of non-Markovian dynamics and report the modifications to the standard quantum regression theorem. We illustrate all of the above results using the paradigmatic dissipative spin-boson model.

Improving the efficiency of open-quantum-system simulations using matrix product states in the interaction picture

Modeling open quantum systems---quantum systems coupled to a bath---is of value in condensed-matter theory, cavity quantum electrodynamics, nanosciences, and biophysics. The real-time simulation of open quantum systems was advanced significantly by the recent development of chain mapping techniques and the use of matrix product states that exploit the intrinsic entanglement structure in open quantum systems. The computational cost of simulating open quantum systems, however, remains high when the bath is excited to high-lying quantum states. We develop an approach to reduce the computational costs in such cases. The interaction representation for the open quantum system is used to distribute excitations among the bath degrees of freedom so that the occupation of each bath oscillator is ensured to be low. The interaction picture also causes the matrix dimensions to be much smaller in a matrix product state of a chain-mapped open quantum system than in the Schr\"odinger picture. Using the interaction representation accelerates the calculations by one to two orders of magnitude over the existing matrix-product-state method. In the regime of strong system-bath coupling and high temperatures, the speedup can be as large as three orders of magnitude. The approach developed here is especially promising to simulate the dynamics of open quantum systems in high-temperature and strong-coupling regimes.

Nontrivial damping of quantum many-body dynamics

Understanding how the dynamics of a given quantum system with many degrees of freedom is altered by the presence of a generic perturbation is a notoriously difficult question. Recent works predict that, in the overwhelming majority of cases, the unperturbed dynamics is just damped by a simple function, e.g., exponentially as expected from Fermi's golden rule. While these predictions rely on random-matrix arguments and typicality, they can only be verified for a specific physical situation by comparing to the actual solution or measurement. Crucially, it also remains unclear how frequent and under which conditions counterexamples to the typical behavior occur. In this work, we discuss this question from the perspective of projection-operator techniques, where exponential damping of a density matrix occurs in the interaction picture but not necessarily in the Schrödinger picture. We show that a nontrivial damping in the Schrödinger picture can emerge if the dynamics in the unperturbed system possesses rich features, for instance due to the presence of strong interactions. This suggestion has consequences for the time dependence of correlation functions. We substantiate our theoretical arguments by large-scale numerical simulations of charge transport in the extended Fermi-Hubbard chain, where the nearest-neighbor interactions are treated as a perturbation to the integrable reference system.

Trinity of relational quantum dynamics

The problem of time in quantum gravity calls for a relational solution. Using quantum reduction maps, we establish a previously unknown equivalence between three approaches to relational quantum dynamics: 1) relational observables in the clock-neutral picture of Dirac quantization, 2) Page and Wootters' (PW) Schr\"odinger picture formalism, and 3) the relational Heisenberg picture obtained via symmetry reduction. Constituting three faces of the same dynamics, we call this equivalence the trinity. We develop a quantization procedure for relational Dirac observables using covariant POVMs which encompass non-ideal clocks. The quantum reduction maps reveal this procedure as the quantum analog of gauge-invariantly extending gauge-fixed quantities. We establish algebraic properties of these relational observables. We extend a recent clock-neutral approach to changing temporal reference frames, transforming relational observables and states, and demonstrate a clock dependent temporal nonlocality effect. We show that Kucha\v{r}'s criticism, alleging that the conditional probabilities of the PW formalism violate the constraint, is incorrect. They are a quantum analog of a gauge-fixed description of a gauge-invariant quantity and equivalent to the manifestly gauge-invariant evaluation of relational observables in the physical inner product. The trinity furthermore resolves a previously reported normalization ambiguity and clarifies the role of entanglement in the PW formalism. The trinity finally permits us to resolve Kucha\v{r}'s criticism that the PW formalism yields wrong propagators by showing how conditional probabilities of relational observables give the correct transition probabilities. Unlike previous proposals, our resolution does not invoke approximations, ideal clocks or ancilla systems, is manifestly gauge-invariant, and easily extends to an arbitrary number of conditionings.

Exact N -point-function mapping between pairs of experiments with Markovian open quantum systems

We formulate an exact spacetime mapping between the $\mathcal{N}$-point correlation functions of two different experiments with open quantum gases. Our formalism extends a quantum-field mapping result for closed systems [Phys. Rev. A \textbf{94}, 043628 (2016)] to the general case of open quantum systems with Markovian property. For this, we consider an open many-body system consisting of a $D$-dimensional quantum gas of bosons or fermions that interacts with a bath under Born-Markov approximation and evolves according to a Lindblad master equation in a regime of loss or gain. Invoking the independence of expectation values on pictures of quantum mechanics and using the quantum fields that describe the gas dynamics, we derive the Heisenberg evolution of any arbitrary $\mathcal{N}$-point function of the system in the regime when the Lindblad generators feature a loss or a gain. Our quantum field mapping for closed quantum systems is rewritten in the Schr\"odinger picture and then extended to open quantum systems by relating onto each other two different evolutions of the $\mathcal{N}$-point functions of the open quantum system. As a concrete example of the mapping, we consider the mean-field dynamics of a simple dissipative quantum system that consists of a one-dimensional Bose-Einstein condensate being locally bombarded by a dissipating beam of electrons in both cases when the beam amplitude or the waist is steady and modulated.

Matrix product states approaches to operator spreading in ergodic quantum systems

We review different tensor network approaches to study the spreading of operators in generic nonintegrable quantum systems. As a common ground to all methods, we quantify this spreading by means of the Frobenius norm of the commutator of a spreading operator with a local operator, which is usually referred to as the out of time order correlation (OTOC) function. We compare two approaches based on matrix-product states in the Schr\"odinger picture: the time dependent block decimation (TEBD) and the time dependent variational principle (TDVP), as well as TEBD based on matrix-product operators directly in the Heisenberg picture. The results of all methods are compared to numerically exact results using Krylov space exact time evolution. We find that for the Schr\"odinger picture the TDVP algorithm performs better than the TEBD algorithm. Moreover the tails of the OTOC are accurately obtained both by TDVP MPS and TEBD MPO. They are in very good agreement with exact results at short times, and appear to be converged in bond dimension even at longer times. However the growth and saturation regimes are not well captured by both methods.

Constructing an entangled state in the Heisenberg picture for inflationary cosmology

The effects of the entanglement of the inflaton field which initially entangled with those of another field on observables like power spectrum are known in the context of the Schr\"odinger field theory. To clarify this effect in the Heisenberg picture, there were some attempts to construct the initial entangled state by making use of an entangled transformation (like Bogoliubov transformation) between the Bunch-Davies vacuums and squeezed states. We study the role of the time-dependent entangled transformation in the Schr\"odinger field theory. We derive the relation between two vacuum states which their mode functions are transformed by the entangled transformation in Schr\"odinger picture. We discuss that the time-dependency of the entanglement parameter is inevitable and only in the first order of the entanglement parameter perturbation this time-dependency vanishes. We study the entangled transformation in the Heisenberg picture in term of the entangled parameter which appears in the entangled state in Schr\"odinger picture.

Exact wave packet dynamics of singlet fission in unsubstituted and substituted polyene chains within long-range interacting models

Singlet fission (SF) is a potential pathway for significant enhancement of efficiency in organic solar cells (OSC). In this paper, we study singlet fission in a pair of polyene molecules in two different stacking arrangements employing exact many-body wave packet dynamics. In the non-interacting model, the SF yield is absent. The individual molecules are treated within Hubbard and Pariser-Parr-Pople (PPP) models and the interaction between them involves transfer terms, intersite electron repulsions and site-charge--bond-charge repulsion terms. Initial wave packet is constructed from excited singlet state of one molecule and ground state of the other. Time development of this wave packet under the influence of intermolecular interactions is followed within the Schr\"odinger picture by an efficient predictor-corrector scheme. In unsubstituted Hubbard and PPP chains, $2{}^1A$ excited singlet state leads to significant SF yield while the $1{}^1B$ state gives negligible fission yield. On substitution by donor-acceptor groups of moderate strength, the lowest excited state will have sufficient $2{}^1A$ character and hence results in significant SF yield. Because of rapid internal conversion, the nature of the lowest excited singlet will determine the SF contribution to OSC efficiency. Furthermore, we find the fission yield depends considerably on the stacking arrangement of the polyene molecules.

Intensified antibunching via feedback-induced quantum interference

We numerically show that time delayed coherent feedback controls the statistical output characteristics of driven quantum emitters. Quantum feedback allows to enhance or suppress a wide range of classical and nonclassical features of the emitted quantum light. As exemplary quantum system, we use a pumped cavity containing two emitters. By applying phase-selective feedback, we demonstrate that photon antibunching and bunching can be increased in orders of magnitude due to intrinsically and externally controllabe quantum interferences. Our modelling is based on a fully non-Markovian quantum simulation of a structured photon continuum. We show that an approximative method in the Schr\"odinger picture allows a very good estimate for quantum feedback induced features for low pump rates.

Cosmic decoherence: massive fields

We study the decoherence of massive fields during inflation based on the Zurek's density matrix approach. With the cubic interaction between inflaton and massive fields, the reduced density matrix for the massive fields can be calculated in the Schr\"odinger picture which is related to the variance of the non-Gaussian exponent in the wave functional. The decoherence rate is computed in the one-loop form from functional integration. For heavy fields with $m\gtrsim \mathcal{O}(H)$, quantum fluctuations will easily stay in the quantum state and decoherence is unlikely. While for light fields with mass smaller than $\mathcal{O}(H)$, quantum fluctuations are easily decohered within $5\sim10$ e-folds after Hubble crossing. Thus heavy fields can play a key role in studying problems involving inflationary quantum information.

A model of the two-dimensional quantum harmonic oscillator in an $$AdS_3$$ A d S 3 background

In this paper we study a model of the two-dimensional quantum harmonic oscillator in a 3-dimensional anti-de Sitter background. We use a generalized Schr\"odinger picture in which the analogs of the Schr\"odinger operators of the particle are independent of both the time and the space coordinates in different representations. The spacetime independent operators of the particle induce the Lie algebra of Killing vector fields of the $AdS_3$ spacetime. In this picture, we have a metamorphosis of the Heisenberg's uncertainty relations.

Weak-value amplification beyond the standard quantum limit in position measurements

In a weak measurement with postselection, a measurement value, called the weak value, can be amplified beyond the eigenvalues of an observable. However, there are some controversies whether weak-value amplification is practically useful in increasing the sensitivity of the measurement, in which fundamental quantum noise dominates. In this paper, we investigate the sensitivity limit of an optical interferometer when weak-value amplification is implemented, properly accounting for quantum shot noise and radiation-pressure noise. To do so, we formulate weak-value amplification in the Heisenberg picture instead of in the Schr\"odinger picture, which is conventionally used. This formulation enables us to understand intuitively what happens when the measurement outcome is postselected and the weak value is amplified. As a result, we find that the sensitivity limit is given by the standard quantum limit that is the same as in a standard interferometry. We also discuss a way to circumvent the standard quantum limit.

Geometric phase carried by the observables and its application to quantum computation

We investigate geometric phases in terms of Heisenberg equation. We find that, equivalently to Schr\"odinger picture with a memory of its motion in terms of the geometric phase factor contained in the wave function, the observales carry with the geometric message under their evolutions in the Heisenberg picture. Such an intrinsic geometric feature may be particularly useful to implement the multi-time correlation geometric quantum gate in terms of the observables, which leads to a possible reduction in experimental errors as well as gate timing. An application is discussed for nuclear-magnetic-resonance system, where the geometric quantum gate is proposed.

Unitarity and ultraviolet regularity in cosmology

Quantum field theory in curved space-times is a well developed area in mathematical physics which has had important phenomenological applications to the very early universe. However, it is not commonly appreciated that on time dependent space-times ---including the simplest cosmological models--- dynamics of quantum fields is not unitary in the standard sense. This issue is first explained with an explicit example and it is then shown that a generalized notion of unitarity does hold. The generalized notion allows one to correctly pass to the Schr\"odinger picture starting from the Heisenberg picture used in the textbook treatments. Finally, we indicate how these considerations can be extended from simple cosmological models to general globally hyperbolic space-times

Relativistic dynamical collapse model

A model is discussed where all operators are constructed from a quantum scalar field whose energy spectrum takes on all real values. The Schr\"odinger picture wave function depends upon space and time coordinates for each particle, as well as an inexorably increasing evolution parameter $s$ which labels a foliation of space-like hypersurfaces. The model is constructed to be manifestly Lorentz invariant in the interaction picture. Free particle states and interactions are discussed in this framework. Then, the formalism of the CSL (Continuous Spontaneous Localization) theory of dynamical collapse is applied. The collapse-generating operator is chosen to to be the particle number space-time density. Unlike previous relativistically invariant models, the vacuum state is not excited. The collapse dynamics depends upon two parameters, a parameter $\Lambda$ which represents the collapse rate/volume and a scale factor $\ell$. A common example of collapse dynamics, involving a clump of matter in a superposition of two locations, is analyzed. The collapse rate is shown to be identical to that of non-relativistic CSL when the GRW-CSL choice of $\ell=a=10^{-5}$cm, is made, along with $\Lambda=\lambda/a^{3}$ (GRW-CSL choice $\lambda=10^{-16}s^{-1}$). However, it is also shown that the change of mass of a nucleon over the age of the universe is then unacceptably large. The case where $\ell$ is the size of the universe is then considered. It is shown that the collapse behavior is satisfactory and the change of mass over the age of the universe is acceptably small, when $\Lambda= \lambda/\ell a^{2}$.

Evolution of quantum field, particle content, and classicality in the three stage universe

We study the evolution of a quantum scalar field in a toy universe which has three stages of evolution, viz., (i) an early (inflationary) de Sitter phase (ii) radiation dominated phase and (iii) late-time (cosmological constant dominated) de Sitter phase. Using Schr\"odinger picture, the scalar field equations are solved separately for the three stages and matched at the transition points. The boundary conditions are chosen so that field modes in the early de Sitter evolves from Bunch-Davies vacuum state. We determine the (time-dependent) particle content of this quantum state for the entire evolution of the universe and describe the various features both numerically and analytically. We also describe the quantum to classical transition in terms of a {\it classicality parameter} which tracks the particle creation and its effect on phase space correlation of the quantum field.

Simulation of Multipartite Cavity Quantum Electrodynamics

Cavity quantum electrodynamics of multipartite systems is studied in depth, which consist of an arbitrary number of emitters in interaction with an arbitrary number of cavity modes. The governing model is obtained by taking the full field-dipole and dipole-dipole interactions into account, and is solved in the Schr\"odinger picture without assumption of any further approximation. An extensive code is developed which is able to accurately solve the system and track its evolution with high precision in time, while maintaining sufficient degrees of arbitrariness in setting up the initial conditions and interacting partitions. Using this code, we have been able to numerically evaluate various parameters such as probabilities, expectation values (of field and atomic operators), as well as the concurrence as the most rigorously defined measure of entanglement of quantum systems. We present and discuss several examples including a seven-partition system consisting of six quantum dots interacting with one cavity mode. We observe for the first time that the behavior of quantum systems under ultrastrong coupling is significantly different than the weakly and strongly coupled systems, marked by onset of a chaos and abrupt phase changes. We also discuss how to implement spin into the theoretical picture and thus successfully simulate a recently reported spin-entanglement experiment.