This gives rise to the question of how the hole density influences the diffusion constant. In Fig. 5.5 we compare all 1D measurements for the lowest possible hole probability (the same data as shown in Fig. 5.3) with data obtained for larger hole probabilities. The higher hole probability is obtained by heating the initial MI state. Our data shows a clear trend towards increasing diffusion constant with increasing hole probability, consistent with numerical predic- tions based on the t-J model. A linear increase can be indeed expected in 1D as each hole – localized during the preparation – introduces a fixed phase defect.
5.8 Summary
In conclusion, we have studied far-from-equilibrium spin transport in the Heisenberg model using high-energy-density spin spiral states in 1D and 2D. A numerical analysis explained the observed diffusion-like behavior in inte- grable 1D chains on a microscopic level. We found that the main features of
0 2 4 0 1.5 3 Hole probability Diffusion constant, D (ħ/m) ~ 0 0.1 0.2 0.3 0.4 0 0.1 0.2 Q2 (alat-2) Rate, Γ (J ex /ħ)
Figure 5.5: Dependence of the diffusion constant in 1D on the hole density. The diffusion constant D increases approximately linearly with hole proba- bility. The gray area is the numerical result of the t-J model with its 95 % confidence interval. The inset shows the decay rate 1/τof the spin spiral ver-
sus the squared wave vector Q2 for the lowest (blue) and highest (red) hole probability.
the magnetic spin transport are robust against a small number of mobile hole defects in the system. In contrast to the diffusive behavior in 1D we observed anomalous super-diffusion in 2D Heisenberg magnets where integrability is broken. For future studies it would be interesting to explore long-time behav- ior which in 1D might shed light on the question of a residual ballistic trans- port [221–233], while in 2D it could unveil a possible crossover from a super- diffusive behavior to sub-diffusive behavior [217]. Especially in 1D, it would be valuable to study spirals prepared with a wave vector close toQ∼π/alat,
where a transformation to the antiferromagnetic Heisenberg Hamiltonian is possible. Thus one can expect that the dynamics can be described with a Luttinger liquid formalism and predictions of Reference [227, 228] could be tested. Furthermore, it would be interesting to study the absence of trans- port in interacting, many-body localized spin systems subject to quenched disorder [243] using for instance local interferometric techniques [244, 245].
In contrast to the previous ballistic experiments, the spin distribution equi- librates which is also expected for a thermalizing system and no information about the initial state is conserved. This is expected for a thermalizing sys- tem, but further studies would be needed to verify if the final state is in fact thermal.
ultracold quantum gases
It is a typical textbook exercise to calculate the thermal state for closed classi- cal systems. However it is much more difficult to understand how and under which conditions thermalization occurs for isolated quantum systems. This chapter provides a demonstrative explanation how closed quantum systems which evolve under uniform transformation can be interpreted as thermal. Thereafter, examples of non-thermalizing systems are given and many-body localization is introduced as a more generic setting where ergodicity is bro- ken.
6.1 Closed Quantum Systems
a b environment system isolation system S sub- systems A
Figure 6.1: Isolated quantum system. a, Quantum gases in optical lattices are very well isolated from the environment and thus do not have an external heat bath. b, For quantum systems, local subsystems will look thermal even though the total system evolves under the unitary time evolution given by the Hamiltonian.
The evolution of a system can be strongly affected by its coupling to the environment. Identifying the properties of an uncoupled system is challeng- ing if the connection to the outside world is not small compared to the rele- vant energy scales which are investigated. This is especially challenging for quantum phenomena as here the relevant energy scales are very small and the quantum nature is only visible if systems are cooled down such that the ground state acquires a macroscopic occupation. Coupling to external bath or noise source can lead to decoherence, which can qualitatively change the quantum evolution.
Additionally to classical closed systems where only the coupling to exter- nal thermal reservoirs needs to be suppressed, measurements play an impor- tant role in quantum mechanics and closed quantum systems are not allowed to have external observers. Too frequent observations of a system can even lead to complete suppression of quantum evolution [246, 247]. Hence to ob- serve out-of-equilibrium dynamics in quantum systems, it is required that these disturbing factors are negligible. If this is fulfilled, the time evolution of a density matrix ρ in the Schrödinger representation under the Hamilto-
nianHis given as:
i¯hdρ
dt =[Hˆ,ρ] (6.1)
ρ(t) =e−iHh¯ tρ(t=0)eiH¯h t
The expectation valuehOˆi(t)for any observable ˆOat time t is obtained by:
hOˆi(t) =TrOˆρ(t) (6.2) Note that here only the density matrix is time dependent but not the operator
ˆ
O.
Besides many other experiments, the coherent spin propagation in Chapter 4 demonstrates that ultracold atoms are well suited to realize versatile closed quantum systems. The atom inside the vacuum chamber are well decou- pled from their environment and additionally are only susceptible to certain frequencies as there energy spectrum is very discrete. Chapter 4 includes ex- emplary observations of coherent evolutions of quantum systems illustrating ballistic expansion of the wave function which indicates such a condition of isolation.