# Simulation of quantum systems

Some fundamental questions in statistical mechanics such as under which conditions a system coupled to a reservoir equilibrates and how the canonical distribution emerges from the interaction between the system and the reservoir, have only been partially resolved. Roughly and more generally speaking one could say that the main unresolved question is how the basic equations of physics, which are all deterministic and time-reversible, can give rise to the time-irreversible (thermodynamic) phenomena that we observe.

## Equilibration of nano-scale systems at finite temperature

As well for classical as quantum systems it is well-known that if the interaction between a system and a much larger reservoir having a large number of degrees of freedom and a dense distribution of energy levels, is weak, the system is described by a canonical ensemble when the composite system is described by the microcanonical ensemble with a given total energy.

In the case of quantum systems, it has recently been shown that the microcanonical mixed state for the composite system is not a required starting point for the system to be described by a canonical ensemble, but that the composite system being initially in a randomly picked pure state is sufficient.

Using approximation-free simulation methods we have studied the equilibration of systems of 4 spin-1/2 particles coupled to a reservoir of 18 to 31 spin-1/2 particles, both the system and the reservoir being described by general quantum spin-1/2 Hamiltonians. The initial state of the composite system was taken to be a product state of a pure state of the system and a pure state of the reservoir, representing the reservoir at a given temperature in the canonical ensemble. We solved the time-dependent Schrödinger equation, governing the time-evolution of the closed composite quantum system, numerically and then analyzed the behavior of the reduced density matrix of the system, obtained by tracing out the degrees of freedom of the reservoir. As a function of time we have calculated the variance of the set of eigenvalues of the reduced density matrix, the entropy, the degree of decoherence of the system and the difference between the reduced density matrix and the canonical distribution. Our simulation results show that, independent of the strength of the interaction between the system and the reservoir and the initial temperature of the reservoir, the system evolves to a stationary state of which the properties strongly depend on the initial temperature of the reservoir. This equilibration is remarkable given the relative small size of the reservoir, since usually in equilibration studies the hypothesis of having a large reservoir is essential. We show that for sufficiently large initial temperatures of the reservoir, the stationary state of the system is represented by a canonical ensemble density matrix at some finite effective temperature. For decreasing temperatures, the reduced density matrix of the system deviates from the canonical density matrix. The deviation increases for decreasing values of the interaction strength between the system and the reservoir.

**Read more**

[1] F. Jin, H. De Raedt, S. Yuan, M.I. Katsnelson, S. Miyashita, and K. Michielsen, Decoherence and relaxation in nano-scale magnets at finite temperature, J. Phys. Soc. Jpn. 79, 124005 (2010).

## Dynamics and manipulation of quantum spin systems

Molecular magnets are generally considered as potential candidates for realizing scalable quantum information processing. Crucial for quantum information applications is that the qubits (i.e. spin-1/2 particles) in these magnets exhibit coherence over a sufficiently long period of time. For instance, the V15 molecular magnet is an assembly of 15 spin-1/2 electrons that has been shown to display Rabi oscillations, indicating that it may be a suitable system for obtaining long coherence times.

Experiments indicate that the conventional Bloch equations are insufficient to fully describe the quantum dynamical response of these systems to applied fields, which is essential for quantum information applications. For instance, empirically one finds that the observed coherence time depends on the applied microwave power, among other things. This observation has been associated with ad-hoc stochastic noise in the applied microwave field but the origin of this noise remains elusive.

We study this problem starting from a realistic model of the magnetic properties of a collection of molecular magnets, including local anisotropic fields, dipolar interactions etc. By solving the time-dependent Schrödinger equation of the interacting spin system directly and by adopting the same procedure as used in pulsed electron-spin-resonance experiments, we can follow the time evolution of the spins explicitly and extract the information that is necessary to disentangle the different processes that give rise to the observed phenomena. Note that in the simulation model it is essential to account for the (long-range) dipole-dipole interactions that are always present in real magnetic materials.