We have presented evidence for a many-body localised phase in the one- dimensional disordered XYZ spin chain, which corresponds to a system of fermions hopping on a lattice with disordered on-site potentials in the absence of fermion number conservation. As such, this takes us beyond systems such as the XXZ spin chain that can be thought of in terms of the dynamics of a fixed number of strongly-interacting particles. The XYZ model does not fall under the scope of the original paper by Basko, Aleiner, and Altshuler [14], which relies on the conservation of particle number, and recent contructions of LIOMs which rely on the same assumptions [43, 44].
The heavy-tailed distributions of off-diagonal matrix elements, and the re- sults of the accompanying TEBD study [66], indicate that there are regions of anomalous energy transport in the ergodic phase of this model. However, we have no true understanding of the mechanism for this behaviour, and this would be an interesting direction for further study.
Chapter 4
Correlated noise in few-level
quantum systems
In this chapter we study the effect of correlated, classical noise on the fidelity of a time-dependent state preparation protocol in a few-level quantum sys- tem. More specifically, we analyse the probability of performing an adiabatic transformation on the quantum system in the presence of perfectly correlated, time-dependent fluctuations in multiple parameters of the Hamiltonian. We show that the quantum state of the system can become trapped in a small region of the state space, rather than becoming truly randomised, and that this results in a plateau in the state-preparation fidelity as a function of the time for which the system is exposed to the noise. We demonstrate the effect in a simple two-level system where the Bloch sphere picture offers a clear visualisation of the process.
4.1
The model
We consider a two-level quantum system with a time-dependent Hamilto- nian,H(t), which changes from an initial configuration at timet = 0to a final configuration att=T. The instantaneous eigenstates of the system are:
H(t)g(t) = Eg(t) g(t); (4.1) H(t)e(t) = Ee(t) e(t), (4.2)
where|g(t)iis the ground state and|e(t)iis the excited state, and the eigen- states never become degenerate (i.e.Ee(t)> Eg(t)at all times). We consider a state-preparation protocol in which the system is initialised in the ground state of the initial HamiltonianH(0),|ψ(0)i=|g(0)i, and the aim is to evolve the system adiabatically and find the system in its ground state att = T, |g(T)i. We define the fidelity,F(t), as the probability of finding the system in the instantaneous ground state ofH(t)at a timet:
F(t) =hg(t)|ψ(t)i2. (4.3)
Our measure of adiabaticity is the probability of finding the system in its ground state at the end of the process,F(T).
As was discussed in Section 2.1.3, if the Hamiltonian changes smoothly with time a higher fidelity can generally be achieved by increasing the duration of the process, soF(T)→1asT → ∞. However, if the Hamiltonian contains a fast, fluctuating noise term, then increasingT will not generally lead to an increasedF(T)as the eigenbasis will still fluctuate rapidly. In fact, one might expect a largeT to be detrimental to adiabatic evolution, as the noise will have more time in which to randomise the state of the system (exclud- ing, of course, the possibility that the operators that couple to the noise are diagonal in the eigenbasis of the noise-free system). For anN-level system, the average fidelity of a completely randomised state isF = 1/N, where the bar denotes an average over realisations of the noise.
We consider a quantum system which has a Hamiltonian that can be split into a clean part and a noisy part:
H(t) =H0(t) + η(t)Hn(t), (4.4)
whereH0andHnare operators which change smoothly and deterministically with time,is a parameter that defines the strength of the noise, andη(t)is the noise. We take the noise to be a Gaussian-distributed stochastic variable
with zero mean,η(t) = 0, and a two-time correlator
η(t)η(t+τ) =δ(τ). (4.5)
We will refer toT as the exposure time, as it is the length of time for which the system is exposed to the noise. The Hamiltonian (4.4) could correspond to a system described by the HamiltonianH0that, for timest∈[0, T], is exposed
to a field which couples to the operatorHnand fluctuates in intensity. This could be, for example, a qubit being transported through a region with two spatially varying magnetic fields; one with a smoothly changing intensity and the other with a fluctuating intensity.
Although the eigenbasis ofH(t)fluctuates quickly and randomly with the noise, the operatorsH0(t)andHn(t)evolve smoothly and deterministically. As a result we can define their instantaneous eigenstates and eigenenergies, which also evolve smoothly and deterministically:
H0(t) µ0(t) = Eµ0(t)µ0(t) ; (4.6) Hn(t) µn(t) = Eµn(t)µn(t) , (4.7)
whereµmay begore. We define the initial state and the fidelity in terms of the eigenstates of the noise-free Hamiltonian:
ψ(0)=g0(0) , (4.8) F(t) =g0(t) ψ(t)2. (4.9)