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Register a new userLoschmidt echo in the Bose-Hubbard model: turning back time in an optical lattice
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Added by
Fernando M. Cucchietti
Publication date
2006
fields
Physics
and research's language is
English
Authors
Fernando M. Cucchietti
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I show how to perform a Loschmidt echo (time reversal) in the Bose-Hubbard model implemented with cold bosonic atoms in an optical lattice. The echo is obtained by applying a linear phase imprint on the lattice and a change in magnetic field to tune the boson-boson scattering length through a Feshbach resonance. I discuss how the echo can measure the fidelity of the quantum simulation, the intensity of an external potential (e.g. gravity), or the critical point of the superfluid-insulator quantum phase transition.
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We present an approach using quantum walks (QWs) to redistribute ultracold atoms in an optical lattice. Different density profiles of atoms can be obtained by exploiting the controllable properties of QWs, such as the variance and the probability distribution in position space using quantum coin parameters and engineered noise. The QW evolves the density profile of atoms in a superposition of position space resulting in a quadratic speedup of the process of quantum phase transition. We also discuss implementation in presently available setups of ultracold atoms in optical lattices.
We study the decay rate of the Loschmidt echo or fidelity in a chaotic system under a time-dependent perturbation $V(q,t)$ with typical strength $hbar/tau_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length $xi_0$ and a correlation time $tau_0$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi Golden Rule, $1/tilde{tau}= tau_{c}/tau_{V}^2$, where typically $tau_{c} sim min[tau_{0},xi_0/v]$ with $v$ the particle velocity. Whenever the rate $1/tilde{tau}$ is larger than the Lyapunov exponent of the system, a perturbation independent Lyapunov decay regime arises. We also find that by speeding up the fluctuations (while keeping the perturbation strength fixed) the fidelity decay becomes slower, and hence, one can protect the system against decoherence.
Environment--induced decoherence causes entropy increase. It can be quantified using, e.g., the purity $varsigma={rm Tr}rho^2$. When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo $bar M(t)$. It is given by the average squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of $varsigma(t)$ and $bar M(t)$. In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of $varsigma$ and $bar M$ has a regime where it is dominated by the classical Lyapunov exponents
Evaluating the role of perturbations versus the intrinsic coherent dynamics in driving to equilibrium is of fundamental interest to understand quantum many-body thermalization, in the quest to build ever complex quantum devices. Here we introduce a protocol that scales down the coupling strength in a quantum simulator based on a solid-state nuclear spin system, leading to a longer decay time T2, while keeping perturbations associated to control error constant. We can monitor quantum information scrambling by measuring two powerful metrics, out-of-time-ordered correlators (OTOCs) and Loschmidt Echoes (LEs). While OTOCs reveal quantum information scrambling involving hundreds of spins, the LE decay quantifies, via the time scale T3, how well the scrambled information can be recovered through time reversal. We find that when the interactions dominate the perturbation, the LE decay rate only depends on the interactions themselves, T3 ~ T2, and not on the perturbation. Then, in an unbounded many-spin system, decoherence can achieve a perturbation-independent regime, with a rate only related to the local second moment of the Hamiltonian.
We introduce a new method for analysing the Bose-Hubbard model for an array of bosons with nearest neighbor interactions. It is based on a number-theoretic implementation of the creation and annihilation operators that constitute the model. One of the advantages of this approach is that it facilitates computation with arbitrary accuracy, enabling nearly perfect numerical experimentation. In particular, we provide a rigorous computer assisted proof of quantum phase transitions in finite systems of this type. Furthermore, we investigate properties of the infinite array via harmonic analysis on the multiplicative group of positive rationals. This furnishes an isomorphism that recasts the underlying Fock space as an infinite tensor product of Hecke spaces, i.e., spaces of square-integrable periodic functions that are a superposition of non-negative frequency harmonics. Under this isomorphism, the number-theoretic creation and annihilation operators are mapped into the Kastrup model of the harmonic oscillator on the circle. It also enables us to highlight a kinship of the model at hand with an array of spin moments with a local anisotropy field. This identifies an interesting physical system that can be mapped into the model at hand.
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