[Paper Review] The Intrinsic Difficulties of Constructing Strongly Correlated States of Lattice Quantum Gases by Connecting Up Pre-engineered Isolated Atomic Clusters

Tin-Lun Ho|ArXiv.org|Aug 20, 2008

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper demonstrates that connecting pre-engineered isolated atomic clusters to form strongly correlated quantum states in optical lattices inevitably leads to severe heating due to intrinsic non-adiabaticity and lack of inter-cluster correlations. The final state reaches a temperature $T_f \gg E^*$, where $E^*$ is the characteristic energy scale of the target correlated state, rendering the desired quantum order unachievable via this direct construction method.

ABSTRACT

Suppose one engineers an artificial antiferromagnet on an array of {\em isolated} wells, and then increases the tunneling between wells slowly, will the system finally become an {\em equilibrium} antiferromagnet? Here, we show that due to the intrinsic non-adiabaticity at the start of this process, and that the atoms in the initial state in different wells are completely uncorrelated, the final equilibrium state will have a temperature $T_{f}$ far above the Neel temperature. Constructing other strongly correlated states (with characteristic energy per particle $E^{\ast}$ comparable to the hopping or the virtual hopping scale) with the same method will suffer the same problem, i.e. $T_{f}>>E^{\ast}$.

Motivation & Objective

• To investigate the feasibility of constructing strongly correlated quantum states in optical lattices by connecting pre-engineered isolated atomic clusters.
• To identify fundamental obstacles in achieving low-temperature, equilibrium correlated states using this direct construction approach.
• To analyze the role of non-adiabaticity and initial state uncorrelatedness in causing excessive energy and temperature in the final state.
• To quantify the temperature rise $T_f$ relative to the characteristic energy scale $E^*$ of the target correlated state.
• To demonstrate that this heating mechanism invalidates the direct construction method for states like antiferromagnets and RVB liquids.

Proposed method

• Model the initial state as an artificial antiferromagnet with alternating spin-up and spin-down fermions in isolated lattice sites, prepared in a finite-radius array.
• Use a time-dependent Hamiltonian $\mathcal{H}_J = \hat{H}_J(U) + \hat{V} + \hat{\Gamma}$, where $J(t)$ increases from zero to a finite value to connect the wells.
• Include a harmonic trap potential $V_{\bf r}(\omega) = \frac{1}{2}M\omega^2 r^2$ and a small environmental perturbation $\hat{\Gamma}$ to model decoherence.
• Analyze the energy difference $E^{\text{int}}_i - E^{\text{int}}_f \sim N E^*$ between initial and final equilibrium states due to missing inter-cluster correlations.
• Apply a generalized Sommerfeld expansion to the Fermi distribution to compute particle number and energy shifts under temperature effects.
• Derive the final temperature $T_f$ from the particle fluctuation energy at the surface, showing $T_f \gg E^*$.

Experimental results

Research questions

• RQ1 Can a system of pre-engineered isolated atomic clusters evolve adiabatically into a strongly correlated equilibrium state when tunneling is gradually increased?
• RQ2 What is the origin of the severe heating observed in the final state when connecting isolated clusters?
• RQ3 Why does the direct construction method fail to achieve low-temperature correlated states like antiferromagnets or RVB liquids?
• RQ4 How does the lack of inter-cluster correlations in the initial state lead to an energy excess proportional to $N E^*$?
• RQ5 What is the quantitative relationship between the final temperature $T_f$ and the characteristic energy scale $E^*$ of the target correlated state?

Key findings

• The final equilibrium state after connecting isolated clusters has a temperature $T_f$ far exceeding the Neel temperature $T_N$, invalidating the formation of an antiferromagnetic order.
• The energy difference between the initial and final states is $\sim N E^*$, where $N$ is the number of particles and $E^*$ is the characteristic energy per particle of the target state.
• This energy excess arises from the absence of inter-cluster correlations in the initial state and the intrinsic non-adiabaticity during the tunneling ramp-up.
• Particle fluctuations at the surface generate significant entropy, leading to $T_f \gg E^*$, even if the process is slow.
• The condition $T_f \ll E^*$ requires $\frac{E^{\text{int}}_i - E^{\text{int}}_f}{N} \sim \zeta^2 E^{*2}/\mu \ll E^*$, which is extremely stringent due to $\mu \gg E^*$.
• The heating mechanism is generic and applies to all strongly correlated states characterized by a small energy scale $E^*$, including RVB liquids and other quantum spin liquids.