Unitary Time Evolution Research Articles

Cosmological Cutting Rules

Primordial perturbations in our universe are believed to have a quantum origin, and can be described by the wavefunction of the universe (or equivalently, cosmological correlators). It follows that these observables must carry the imprint of the founding principle of quantum mechanics: unitary time evolution. Indeed, it was recently discovered that unitarity implies an infinite set of relations among tree-level wavefunction coefficients, dubbed the Cosmological Optical Theorem. Here, we show that unitarity leads to a systematic set of “Cosmological Cutting Rules” which constrain wavefunction coefficients for any number of fields and to any loop order. These rules fix the discontinuity of an n-loop diagram in terms of lower-loop diagrams and the discontinuity of tree-level diagrams in terms of tree-level diagrams with fewer external fields. Our results apply with remarkable generality, namely for arbitrary interactions of fields of any mass and any spin with a Bunch-Davies vacuum around a very general class of FLRW spacetimes. As an application, we show how one-loop corrections in the Effective Field Theory of inflation are fixed by tree-level calculations and discuss related perturbative unitarity bounds. These findings greatly extend the potential of using unitarity to bootstrap cosmological observables and to restrict the space of consistent effective field theories on curved spacetimes.

Entanglement Entropies of Equilibrated Pure States in Quantum Many-Body Systems and Gravity

We develop a universal approximation for the Renyi entropies of a pure state at late times in a non-integrable many-body system, which macroscopically resembles an equilibrium density matrix. The resulting expressions are fully determined by properties of the associated equilibrium density matrix, and are hence independent of the details of the initial state, while also being manifestly consistent with unitary time-evolution. For equilibrated pure states in gravity systems, such as those involving black holes, this approximation gives a prescription for calculating entanglement entropies using Euclidean path integrals which is consistent with unitarity and hence can be used to address the information loss paradox of Hawking. Applied to recent models of evaporating black holes and eternal black holes coupled to baths, it provides a derivation of replica wormholes, and elucidates their mathematical and physical origins. In particular, it shows that replica wormholes can arise in a system with a fixed Hamiltonian, without the need for ensemble averages.

Manifestly causal in-in perturbation theory about the interacting vacuum

In-In perturbation theory is a vital tool for cosmology and nonequilibrium physics. Here, we reconcile an apparent conflict between two of its important aspects with particular relevance to De Sitter/inflationary contexts: (i) the need to slightly deform unitary time evolution with an iϵ prescription that projects the free (“Bunch-Davies”) vacuum onto the interacting vacuum and renders vertex integrals well-defined, and (ii) Weinberg’s “nested commutator” reformulation of in-in perturbation theory which makes manifest the constraints of causality within expectation values of local operators, assuming exact unitarity. We show that a modified iϵ prescription maintains the exact unitarity on which the derivation of (ii) rests, while nontrivially agreeing with (i) to all orders of perturbation theory.

Detecting Bulk Topology of Quadrupolar Phase from Quench Dynamics.

Direct measurement of a bulk topological observable in topological phase of matter has been a long-standing issue. Recently, detection of bulk topology through quench dynamics has attracted growing interests. Here, we propose that topological characters of a quantum quadrupole insulator can be read out by quench dynamics. Specifically, we introduce a quantity, a quadrupole moment weighted by the eigenvalues of the chiral operator, which takes zero for the trivial phase and finite for the quadrupolar topological phase. By utilizing an efficient numerical method to track the unitary time evolution, we elucidate that the quantity we propose indeed serves as an indicator of topological character for both noninteracting and interacting cases. The robustness against disorders is also demonstrated.

Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles

Based on conservation laws for surface layer integrals for critical points of causal variational principles, it is shown how jet spaces can be endowed with an almost-complex structure. We analyze under which conditions the almost-complex structure can be integrated to a canonical complex structure. Combined with the scalar product expressed by a surface layer integral, we obtain a complex Hilbert space $\mathfrak{h}$. The Euler-Lagrange equations of the causal variational principle describe a nonlinear time evolution on $\mathfrak{h}$. Rewriting multilinear operators on $\mathfrak{h}$ as linear operators on corresponding tensor products and using a conservation law for a nonlinear surface layer integral, we obtain a linear norm-preserving time evolution on bosonic Fock spaces. The so-called holomorphic approximation is introduced, in which the dynamics is described by a unitary time evolution on the bosonic Fock space. The error of this approximation is quantified. Our constructions explain why and under which assumptions critical points of causal variational principles give rise to a second-quantized, unitary dynamics on Fock spaces.

Supercanonical transformations and time-dependent Hamiltonians

Starting from general self-adjoint linear combinations of generators of the superalgebra a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp. The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients.

Probabilistic cellular automata for interacting fermionic quantum field theories

A classical local cellular automaton can describe an interacting quantum field theory for fermions. We construct a simple classical automaton for a particular version of the Thirring model with imaginary coupling. This interacting fermionic quantum field theory obeys a unitary time evolution and shows all properties of quantum mechanics. Classical cellular automata with probabilistic initial conditions admit a description in the formalism of quantum mechanics. Our model exhibits interesting features as spontaneous symmetry breaking or solitons. The same model can be formulated as a generalized Ising model. This euclidean lattice model can be investigated by standard techniques of statistical physics as Monte Carlo simulations. Our model is an example how quantum mechanics emerges from classical statistics.

Topological entanglement properties of disconnected partitions in the Su-Schrieffer-Heeger model

We study the disconnected entanglement entropy, S^\mathrm{D}SD, of the Su-Schrieffer-Heeger model. S^\mathrm{D}SD is a combination of both connected and disconnected bipartite entanglement entropies that removes all area and volume law contributions and is thus only sensitive to the non-local entanglement stored within the ground state manifold. Using analytical and numerical computations, we show that S^\mathrm{D}SD behaves like a topological invariant, i.e., it is quantized to either 00 or 2\log(2)2log(2) in the topologically trivial and non-trivial phases, respectively. These results also hold in the presence of symmetry-preserving disorder. At the second-order phase transition separating the two phases, S^\mathrm{D}SD displays a finite-size scaling behavior akin to those of conventional order parameters, that allows us to compute entanglement critical exponents. To corroborate the topological origin of the quantized values of S^\mathrm{D}SD, we show how the latter remain quantized after applying unitary time evolution in the form of a quantum quench, a characteristic feature of topological invariants associated with particle-hole symmetry.

Gibbs States, Algebraic Dynamics and Generalized Riesz Systems

In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita–Takesaki theory in our context.

Probing early universe with a generalized action

Possibly, the most general action in the background of isotropic and homogeneous space–time has been considered to study the quantum evolution of the early universe, apart from a cosmological constant. The hermiticity of the effective Hamiltonian operator in the presence of curvature squared terms suggests unitary time evolution of the quantum states, assuring conservation of probability. The oscillatory behaviour of the semi-classical wavefunction around a de-Sitter solution signals that the theory is classically allowed, and the universe enters an inflationary regime just after Planck’s era. In view of a hierarchy of Hubble flow parameters, and using a redefined effective potential, the complicated classical field equations in the presence of several coupling parameters, reduce to standard general-relativistic equations with a single scalar field. As a result, inflation has been studied without considering any additional flow parameters. Inflationary parameters lie very much within the presently available Planck’s data, and the model admits graceful exit from inflation.

Conditional Action and Quantum Versions of Maxwell\u2019s Demon

We propose a new way of looking at the quantum Maxwell’s demon problem in terms of conditional action. A “conditional action” on a system is a unitary time evolution, selected according to the result of a previous measurement, which can reduce the entropy of the system. However, any conditional action can be realized by an (unconditional) unitary time evolution of a larger system and a subsequent Lüders measurement, whereby the entropy of the entire system is either increased or remains constant. We give some examples that illustrate and confirm our proposal, including the erasure of N qubits and the Szilard engine, thus relating the present approach to the Szilard principle and the Landauer principle that have been discussed as possible solutions of the Maxwell’s demon problem.

Positivity of the assignment map implies complete positivity of the reduced dynamics

Consider the set $\mathcal{S}=\lbrace\rho_{SE}\rbrace$ of possible initial states of the system-environment. The map which assigns to each $\rho_{S}\in \mathrm{Tr}_{E}\mathcal{S}$ a $\rho_{SE}\in \mathcal{S}$ is called the assignment map. The assignment map is Hermitian, in general. In this paper, we restrict ourselves to the case that the assignment map is, in addition, positive and show that this implies that the so-called reference state is a Markov state. Markovianity of the reference state leads to existence of another assignment map which is completely positive. So, the reduced dynamics of the system is also completely positive. As a consequence, when the system $S$ is a qubit, we show that if $\mathcal{S}$ includes entangled states, then either the reduced dynamics is not given by a map, for, at least, one unitary time evolution of the system-environment $U$, or the reduced dynamics is non-positive, for, at least, one $U$.

Necessary and sufficient condition for the reduced dynamics of an open quantum system interacting with an environment to be linear

The dynamics of a closed quantum system, under a unitary time evolution $U$, is, obviously, linear. But, the reduced dynamics of an open quantum system $S$, interacting with an environment $E$, is not linear, in general. Dominy et al. [Quant. Inf. Process. 15, 465 (2016)] considered the case that the set $\mathcal{S}=\lbrace\rho_{SE}\rbrace$, of possible initial states of the system-environment, is convex and, also, possesses another property, which they called $U$-consistency. They have shown that, under such circumstances, the reduced dynamics of the system $S$ is linear. Whether the Dominy-Shabani-Lidar framework is the most general one is the subject of this paper. We assume that the reduced dynamics is linear and show that this leads us to their framework. In other words, the reduced dynamics of the system is linear if and only if it can be formulated within the Dominy-Shabani-Lidar framework.

Blind Witnesses Quench Quantum Interference without Transfer of Which-Path Information.

Quantum computation is often limited by environmentally-induced decoherence. We examine the loss of coherence for a two-branch quantum interference device in the presence of multiple witnesses, representing an idealized environment. Interference oscillations are visible in the output as the magnetic flux through the branches is varied. Quantum double-dot witnesses are field-coupled and symmetrically attached to each branch. The global system—device and witnesses—undergoes unitary time evolution with no increase in entropy. Witness states entangle with the device state, but for these blind witnesses, which-path information is not able to be transferred to the quantum state of witnesses—they cannot “see” or make a record of which branch is traversed. The system which-path information leaves no imprint on the environment. Yet, the presence of a multiplicity of witnesses rapidly quenches quantum interference.

Controlling arbitrary observables in correlated many-body systems

Here we present an expanded analysis of a model for the manipulation and control of observables in a strongly correlated, many-body system, which was first presented in [McCaul et al., eprint: arXiv:1911.05006]. A field-free, non-linear equation of motion for controlling the expectation value of an essentially arbitrary observable is derived, together with rigorous constraints that determine the limits of controllability. We show that these constraints arise from the physically reasonable assumptions that the system will undergo unitary time evolution, and has enough degrees of freedom for the electrons to be mobile. Furthermore, we give examples of multiple solutions to generating target observable trajectories when the constraints are violated. Ehrenfest theorems are used to further refine the model, and provide a check on the validity of numerical simulations. Finally, the experimental feasibility of implementing the control fields generated by this model is discussed.

Guide to Exact Diagonalization Study of Quantum Thermalization

Exact diagonalization is a powerful numerical method to study isolated quantum many-body systems. This paper provides a review of numerical algorithms to diagonalize the Hamiltonian matrix. Symmetry and the conservation law help us perform the numerical study efficiently. We explain the method to block-diagonalize the Hamiltonian matrix by using particle number conservation, translational symmetry, particle-hole symmetry, and spatial reflection symmetry in the context of the spin-1/2 XXZ model or the hard-core boson model in a one-dimensional lattice. We also explain the method to study the unitary time evolution governed by the Schr\"odinger equation and to calculate thermodynamic quantities such as the entanglement entropy. As an application, we demonstrate numerical results that support that the eigenstate thermalization hypothesis holds in the XXZ model.

Validity of many-mode Floquet theory with commensurate frequencies

Many-mode Floquet theory [T.-S. Ho, S.-I. Chu, and J. V. Tietz, Chem. Phys. Lett. 96, 464 (1983)] is a technique for solving the time-dependent Schr\"odinger equation in the special case of multiple periodic fields, but its limitations are not well understood. We show that for a Hamiltonian consisting of two time-periodic couplings of commensurate frequencies (integer multiples of a common frequency), many-mode Floquet theory provides a correct expression for unitary time evolution. However, caution must be taken in the interpretation of the eigenvalues and eigenvectors of the corresponding many-mode Floquet Hamiltonian, as only part of its spectrum is directly relevant to time evolution. We give a physical interpretation for the remainder of the spectrum of the Hamiltonian. These results are relevant to the engineering of quantum systems using multiple controllable periodic fields.

Decaying quantum turbulence in a two-dimensional Bose-Einstein condensate at finite temperature

We numerically model decaying quantum turbulence in two-dimensional disk-shaped Bose--Einstein condensates, and investigate the effects of finite temperature on the turbulent dynamics. We prepare initial states with a range of condensate temperatures, and imprint equal numbers of vortices and antivortices at randomly chosen positions throughout the fluid. The initial states are then subjected to unitary time-evolution within the c-field methodology. For the lowest condensate temperatures, the results of the zero temperature Gross--Pitaevskii theory are reproduced, whereby vortex evaporative heating leads to the formation of Onsager vortex clusters characterised by a negative absolute vortex temperature. At higher condensate temperatures the dissipative effects due to vortex--phonon interactions tend to drive the vortex gas towards positive vortex temperatures dominated by the presence of vortex dipoles. We associate these two behaviours with the system evolving toward an anomalous non-thermal fixed point, or a Gaussian thermal fixed point, respectively.

Decoherence of a singlet-triplet superposition state under dipolar interactions of an uncorrelated environment

Recently it was shown that by means of an STM it is experimentally possible to stimulate clock transitions between the singlet and the non-magnetic triplet state of a Heisenberg-coupled spin dimer (Bae et al., Science Advances 4, eaau4159). This leads to more strongly protected clock transitions while ordinary ones only provide first-order protection against magnetic noise. However, large decoherence times of clock like states normally refer to ensembles of spins which do not dephase. In the cited experiment only one single dimer is manipulated and not an ensemble. For this reason we simulate how a single dimer behaves in an environment of other spins which couple to the dimer via dipolar interactions. We perform unitary time evolutions in the complete Hilbert space including dimer and a reasonably large environment. We will see that for a weak environment this approach confirms long decoherence times for the clock like state, but with stronger couplings this statement does not hold. As a reference we compare the behavior of the dimer with other, non-clock like, superposition states. Furthermore, we show that the internal dynamics of the bath plays an important role for the decoherence time of the system. In a regime where the system is weakly coupled to the bath, stronger interactions among environmental spins worsen the decoherence time up to a certain degree, while if system and bath are strongly coupled, stronger interactions in the environment improve decoherence times.

Time-dependent PT -symmetric quantum mechanics in generic non-Hermitian systems

$\mathcal{PT}$-symmetric quantum mechanics has been considered an important theoretical framework for understanding physical phenomena in $\mathcal{PT}$-symmetric systems, with a number of $\mathcal{PT}$-symmetry related applications. This line of research was made possible by the introduction of a time-independent metric operator to redefine the inner product of a Hilbert space. To treat the dynamics of generic non-Hermitian systems under equal footing, we advocate in this work the use of a time-dependent metric operator for the inner-product between time-evolving states. This treatment makes it possible to always interpret the dynamics of arbitrary (finite-dimensional) non-Hermitian systems in the framework of time-dependent $\mathcal{PT}$-symmetric quantum mechanics, with unitary time evolution, real eigenvalues of an energy observable, and quantum measurement postulate all restored. Our work sheds new lights on generic non-Hermitian systems and spontaneous $\mathcal{PT}$-symmetry breaking in particular. We also illustrate possible applications of our formulation with well-known examples in quantum thermodynamics.