Tarek Khalil

Propagation of local excitations through strongly correlated quantum chains

Abstract The propagation of an external transverse magnetic signal acting locally on a 1d chain of spins generates a disturbance which runs through the system. This quantum effect can be interpreted as a classical travelling wave which contains a superposition of a large set of frequencies depending on the size of the chain. Its local amplitude fixes the size of the z-component of the spins at any location in the chain. The average and maximum value of the group velocity are determined and compared with the transmission velocity fixed by the Lieb–Robinson upper bound inequality.

Low-energy properties of non-perturbative quantum systems: A space reduction approach

We propose and test a renormalization procedure which acts in Hilbert space. We test its efficiency on strongly correlated quantum spin systems by working out and analyzing the low-energy spectral properties of frustrated quantum spin systems in different parts of the phase diagram and in the neighbourhood of quantum critical points.

Study of the spectral properties of spin ladders in different representations via a renormalization procedure

We implement an algorithm which is aimed to reduce the dimensions of the Hilbert space of quantum many-body systems by means of a renormalization procedure. We test the role and importance of different representations on the reduction process by working out and analyzing the spectral properties of strongly interacting frustrated quantum spin systems.

Water as a Quantum Computing Device

We propose a new approach in Nuclear Magnetic Resonance (NMR) technology for quantum computing. Two basic elements for quantum computation are qubits and interaction. Traditionnally in NMR, qubits are obtained by considering the spin states of certain atoms of a specific molecule. The interaction that is necessary to create qubit gates such as Control-NOT is then made possible by the spin-spin coupling that exists in such a molecule. One of the drawbacks of this method is the low scalability. More qubits usually means finding an entire different molecule to label the qubits. We take a different view on the NMR approach. We use a water tube as quantum computer where qubits are defined by the spins which have the same frequency resonance in a small interval defined by the NMR linewidth. Two fundamental roadblocks have to be crossed before this method can even be considered as a possible quantum computation technique: single qubits need to be identified and adressed and an interaction between these qubits has to exist to create two-qubit gates. We settle the first of these problems by using a magnetic field gradient applied in the main magnetic field direction. The application of a magnetic field gradient during the RF pulse induces a rotation only of those spins whose resonant frequency is equal to the bandwidth of the RF pulse, therefore a qubit can be defined by its resonant frequency and manipulated by selective RF pulses. The main advantage of creating qubits in this way is scalability. As qubits are no longer atoms of a specific molecule but segments of our water tube, increasing the number of qubits would hypothetically just mean increasing the number of segments by applying a stronger magnetic field gradient. Another potential advantage can be obtained during the initialisation phase of the qubits. The second roadblock, the problem of creating interaction between qubits, is work in progress. As for now we are investigating the use of the dipole-dipole interaction between the spins to generate a coupling between the spins in order to create entanglements.