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Register a new userInvariant-based inverse engineering of time-dependent, coupled harmonic oscillators
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Two-dimensional systems with time-dependent controls admit a quadratic Hamiltonian modelling near potential minima. Independent, dynamical normal modes facilitate inverse Hamiltonian engineering to control the system dynamics, but some systems are not separable into independent modes by a point transformation. For these coupled systems 2D invariants may still guide the Hamiltonian design. The theory to perform the inversion and two application examples are provided: (i) We control the deflection of wave packets in transversally harmonic waveguides; and (ii) we design the state transfer from one coupled oscillator to another.
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We derive explicitly the thermal state of the two coupled harmonic oscillator system when the spring and coupling constants are arbitrarily time-dependent. In particular, we focus on the case of sudden change of frequencies. In this case we compute purity function, R{e}nyi and von Neumann entropies, and mutual information analytically and examine their temperature-dependence. We also discuss on the thermal entanglement phase transition by making use of the negativity-like quantity. Our calculation shows that the critical temperature $T_c$ increases with increasing the difference between the initial and final frequencies. In this way we can protect the entanglement against the external temperature by introducing large difference of initial and final frequencies.
By applying invariant-based inverse engineering in the small-oscillations regime, we design the time dependence of the control parameters of an overhead crane (trolley displacement and rope length), to transport a load between two positions at different heights with minimal final energy excitation for a microcanonical ensemble of initial conditions. The analogies between ion transport in multisegmented traps or neutral atom transport in moving optical lattices and load manipulation by cranes opens a route for a useful transfer of techniques among very different fields.
In the context of the de Broglie-Bohm pilot wave theory, numerical simulations for simple systems have shown that states that are initially out of quantum equilibrium - thus violating the Born rule - usually relax over time to the expected $|psi|^2$ distribution on a coarse-grained level. We analyze the relaxation of nonequilibrium initial distributions for a system of coupled one-dimensional harmonic oscillators in which the coupling depends explicitly on time through numerical simulations, focusing in the influence of different parameters such as the number of modes, the coarse-graining length and the coupling constant. We show that in general the system studied here tends to equilibrium, but the relaxation can be retarded depending on the values of the parameters, particularly to the one related to the strength of the interaction. Possible implications on the detection of relic nonequilibrium systems are discussed.
Exponential growth of thermal out-of-time-order correlator (OTOC) is an indicator of a possible gravity dual, and a simple toy quantum model showing the growth is being looked for. We consider a system of two harmonic oscillators coupled nonlinearly with each other, and numerically observe that the thermal OTOC grows exponentially in time. The system is well-known to be classically chaotic, and is a reduction of Yang-Mills-Higgs theory. The exponential growth is certified because the growth exponent (quantum Lyapunov exponent) of the thermal OTOC is well matched with the classical Lyapunov exponent, including their energy/temperature dependence. Even in the presence of the exponential growth in the OTOC, the energy level spacings are not sufficient to judge a Wigner distribution, hence the OTOC is a better indicator of quantum chaos.
We study the spontaneous decoherence of the coupled harmonic oscillators confined in a ring container, where the nearest-neighbor harmonic potentials are taken into consideration. Without any external symmetry breaking field or surrounding environment, the quantum superposition state prepared in the relative degrees of freedom gradually loses its quantum coherence spontaneously. This spontaneous decoherence is interpreted by the hidden couplings between the center-of-mass and relative degrees of freedoms, which actually originates from the symmetries of the ring geometry and corresponding nontrivial boundary conditions. Especially, such spontaneous decoherence completely vanishes at the thermodynamical limit because the nontrivial boundary conditions become trivial Born-von Karman boundary conditions when the perimeter of the ring container tends to infinity. Our investigation shows that a thermal macroscopic object with certain symmetries has chance to degrade its quantum properties even without applying an external symmetry breaking field or surrounding environment.
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