The Entanglement Dynamics Of Two Equivalent Atoms In An Infinite Rectangular Waveguide

In recent years,people have done a lot of research on controlling the interaction between quantum emitters and electromagnetic fields.One of the main driving force in this is quantum information.The development of efficient quantum in-formation devices requires the control of light-matter interaction at a quantum level and,in particular,the precise handling of the entanglement resource.Many promising applications of entangled light-matter states have been proposed,such as quantum teleportation,quantum cryptography and quantum computing.The fundamental system on which all these applications rely is the two-qubit ensemble.Specifically,much interest is focused on generating entanglement between qubits by coupling them to a common electromagnetic environment.In order to realize this objective,the property of the surrounding electromagnetic modes are critical.As the interaction between emitters in free space is not strong enough for entangle-ment purpose,additional structures have to be included.These structures modify the density of the electromagnetic states through the Purcell effect,giving rise to collective phenomena.A waveguide is one of these structures.It can not only cause interaction between two distant quantum emitters coupled with it,but also generating long-lived entanglement.Waveguides are good candidate for large-scale quantum devices.This thesis investigates the entanglement dynamics of the two identical two-level systems（TLSs）interacting with a one-dimension（1D）reservoir of guided modes confined in a rectangular waveguide of a cross section.The interaction between the TLSs with both single transverse mode and double transverse modes of the waveguide are considered respectively,and the effects of the inter-TLS distance on the time evolution of the concurrence of the TLSs are examined.Our main research methods and results are as follows:1.A waveguide with a rectangular cross section is considered to investigate the interaction of two TLSs with multiple transverse modes in the waveguide.Two TLSs are fixed in the waveguide and their electric dipole moments are along the direction of electromagnetic field propagating in the waveguide.So only the transverse magnetic modes of the waveguide interact with the TLSs.Besides,we assume the TLSs is weakly coupled to the 1D rectangular waveguide and the atom size（i.e.,the magnitude of the electric dipole moment）is much smaller than the field wavelength in the waveguide.For the energy separation of the identical TLSs far away from the cutoff frequencies of transverse modes,the Schrodinger equation for the wave function with single excitation initial in the TLSs is reduced to the delay differential equations for the probability amplitudes of two TLSs.By comparing the equations for the probability amplitudes of the anti-symmetric state and the symmetric state,it is found that that the symmetric state and the anti-symmetric state are uncoupled.2.The analytical solutions of the probability amplitudes of symmetric and antisymmetric states and the numerical evolution of the degree of concurrency with the dimensionless time t/τ₁in the unit of delay timeτ₁are compared.If only one transverse mode are considered and the initial state is symmetric state,when time required for photons to travel from one TLS to another TLS is much smaller than the TLS decay time,the dynamic of the entanglement are dramatically af-fected by phases,leading to an enhanced and inhibited exponential decay of the concurrence,the delay time can be ignored.As the inter-TLS distance increases,both phases and delay times affect the concurrence.The interference produced by multiple reemissions and reabsorptions of photon results in an oscillatory entangle-ment.When the inter-TLS distance is so large that the Concurrence is reduced to almost zero before the photon radiated by one TLS reaches another TLS,the phase does not make any sense.As soon as the emitted photon returns to the TLSs,the entanglement is created.Then with the photon re-emitted by TLSs,the decrease of entanglement begins.The entanglement of the TLSs exhibits peaks due to the iteration of the process where a photon emitted by one of the atoms is reabsorbed by another atom,but its periodic maxima are reduced in magnitude as time in-creases.There are collapses and revivals of the concurrence.Until the probability amplitude of the TLSs excited state decays to zero,the entanglement is no longer recovered.When the TLSs coupled to two transverse modes and the initial state is anti-symmetric state,in order to analyze the role of the second transverse mode,the phase of the first transverse mode was fixed as 2nπ,so that the entanglement between the two TLSs is suppressed in the first transverse mode.It is found that increasing number of the transverse modes results in an enhanced exponential de-cay of the Concurrence.With the increase of the inter-TLS distance,the energy radiated by TLSs first experiences delay in the transverse mode with higher cutoff frequency.When the inter-TLS distance increases to the point where the photons radiated by the TLSs from the two shortest time intervals no longer interfere,the phase ceases to matter.