Topology effects


	New Topological effects in QTM: Interacting spins


	Enrique del Barco group and collaborators at UF and UCSD report in NATURE PHYSICS a new dimeric molecular nanomagnet which reveals novel topological (Berry phase) effects in the quantum tunneling of interacting magnetic systems.

Tunable electron spins in solid media are among the most promising candidates for qubits in quantum computing. The synthetic flexibility of molecular nanomagnets allows one to systematically produce samples with desirable properties such as those with entangled spin states for implementation in quantum logic gates. A new molecular nanowheel, composed of two coupled halves with the same spin value, represents a substantial advance in this direction. A magnetic field modulates the coupling between states of different spin length leading to the observation of quantum interference—an effect that can be used to tune the entanglement of a prototypical molecular quantum device


	Berry phase

	To easily understand the effect of topology (~shape) on the dynamics of a system let’s look at the parallel transport of a vector in a closed path (from point 1 to point 7) along a curved surface, as shown in the figure. When the vector comes back to the original position, it points in a different direction and it’s said that the system has acquired a topological phase (90 degrees in this case), also known as Berry phase.


	Berry phase in precessing spins

	The case of a spin rotating under the action of a fixed magnetic field can also be explained in similar terms. As the spin vector rotates from the initial position towards being aligned with the applied field, it precesses around the magnetic field acquiring a phase which is proportional to the rotation angle covered during the precession.


	Effect of topology in QTM

In nanoscale size magnets, such us single-molecule magnets (SMMs), the spin can switch between opposite projections along the axis of application of the magnetic field without following a classical precession path in the real space. In this case the spin switches direction quantum mechanically by a process known as quantum tunneling. It turns out that this process can be understood in terms of tunneling trajectories occurring in an “imaginary time space” in a very similar way that the one exposed above. Under this perspective, the spin can also acquire a topological phase during its quantum mechanical switching. The novelty is that, under certain conditions and due to the quantum mechanical nature of the system, different quantum tunneling trajectories can combine and give rise to interference effects that may lead to the vanishing of the quantum tunneling in the case of destructive interference, in a very similar way of interference light beams or water waves. This was predicted in 1992 separately by theorists Daniel Loss and J. von Delft and first observed experimentally in SMM Fe8 by Wernsdorfer and Sessoli in 1999.

In a parallel plane, the effect of topology on the dynamics of two interacting spins is of great interest in different disciplines. Sjöqvist was the first to study this phenomenon in 2000 assuming two entangled spins precessing around a fixed magnetic field. He figured out that the topological phase acquired by the entangled system could not be trivially reduced to the sum of the Berry phases acquired by each individual spin. On the contrary, the effect of topology in entangled systems hindered a large range of novel physics to be explored.

There is not any formal theory yet to explain the effect of topology on the quantum tunneling of two interacting nanoscale magnetic systems. However, one could trace a clear equivalency between two interacting spins switching directions by quantum mechanical tunneling and the situation described by Sjöqvist of two entangled spins “classically” precessing around a fixed field, since the first could also be understood as quantum tunneling trajectories of both spins in an “imaginary time space”.


	Berry phase in tunneling interacting spins

Along these lines, we present the first experimental observation of quantum interference effects on the quantum mechanical tunneling of two exchange-coupled spins of a SMM. This behavior is a consequence of the unique characteristics of a molecular Mn₁₂ wheel which behaves as a (weak) ferromagnetically exchange-coupled molecular dimer in itself. The molecule is composed of 12 manganese ions arranged in an annular structure (wheel) giving rise to a ground spin state S=7 that points along the direction of the wheel axis. The total spin results from ferromagnetic coupling between the two spin S=7/2 halves of the wheel, as can be seen in the figure. This is due to a weak coupling between two pairs of manganese ions at opposite sides of the wheel, which breaks the wheel into two rigid magnetic units entangled by exchange interaction.

The observations reveal how the interaction between the two halves makes their tunneling trajectories to interfere and, under certain conditions, the tunneling probability vanishes as a result. This effect is similar to the destructive interference between two light beams, from which one can obtain darkness. It’s like two ice skaters turning with respect to each other. They are both rotating, but the couple forms the same figure after half a rotation, although the skaters switch positions.

This work is published in NATURE PHYSICS. This work is timely in that a number of theoretical groups have recently discussed the possibility of utilizing the spin states of molecular magnets to realize a quantum logic device (e.g. Leuenberger and Loss, Nature 2000). Furthermore, quantum interference could be employed to turn on and off the entanglement between two qubits (which in our case are integral part of a rigid molecule which e.g. could be extracted from the crystal while maintaining its unique properties) in a future molecular quantum device. In addition, it is worth pointing out the potential of some of the phenomena observed in this system. Specifically, competing Berry phases of interacting spins and coupling between states of opposite symmetry could lead to a better understanding of the role of the topology and symmetry of nanoscale systems in the manifestation of quantum phenomena

References:

C. M. Ramsey, E. del Barco, S. Hill, S. J. Shah, C. Beedle and D. N. Hendrikson
"Quantum Interference of Tunnel Trajectories between States of Different Spin Length in a Dimeric Molecular Nanomagnet"
Nature Physics. Advance online publication (2 March 2008).