Temperature effect on a kicked Tonks-Girardeau gas

This paper demonstrates that many-body dynamical localization in a kicked Tonks-Girardeau gas persists even at high finite temperatures, albeit with degraded coherence and modified thermalization, while also revealing a localization-delocalization transition at intermediate temperatures.

The Gist

Imagine a crowded dance floor where everyone is trying to move to a beat. In the quantum world, this "dance floor" is a gas of atoms, and the "beat" is a series of rhythmic kicks.

This paper explores what happens to this quantum dance when two things change:

1. The dancers are extremely stubborn: They refuse to pass each other (this is the "Tonks-Girardeau" gas, where atoms act like hard-core billiard balls).
2. The dancers are a bit "warm": Instead of being perfectly frozen in a quantum state (absolute zero), they have some thermal energy, like a room that's slightly warm rather than an ice cube.

Here is the breakdown of their findings using simple analogies:

1. The Setup: The Quantum Dance Floor

In a normal world, if you keep kicking a ball, it gains speed and flies away forever. But in the quantum world, there's a weird trick called Dynamical Localization. It's like the ball gets stuck in a specific spot on the dance floor, refusing to gain more energy no matter how hard you kick it. This happens because the quantum waves of the particles interfere with each other, creating a "traffic jam" that stops them from moving.

Scientists already knew this happened at absolute zero (the coldest possible temperature). But they wondered: What happens if the room gets warm? Usually, heat destroys quantum magic, turning the organized dance into a chaotic mess (thermalization).

2. The Periodic Kick: The Metronome

First, the researchers kicked the atoms in a perfect, repeating rhythm (like a metronome).

• The Surprise: Even when the gas was "warm" (at finite temperatures), the traffic jam (localization) still held! The atoms didn't fly off the dance floor.
• The Catch: While they stayed stuck, the "dance" got messier. The perfect quantum coordination between the atoms was degraded. It's like a choir singing in perfect harmony at zero temperature; when it gets warm, they are still singing the same song in the same spot, but they are slightly out of tune with each other.
• The "Effective" Heat: The researchers found a way to describe this stuck state as if it were a hot gas. They created a "thermometer" for this stuck state, showing that even though the atoms are stuck, they behave as if they have a specific, higher temperature than they started with.

3. The Quasiperiodic Kick: The Chaotic DJ

Next, they changed the rhythm. Instead of a perfect metronome, they used a "chaotic DJ" who played kicks in a pattern that never quite repeated (quasiperiodic).

• The Result: This is where things got interesting. At low temperatures, the atoms stayed stuck (localized). At very high temperatures, they started moving freely (delocalized).
• The Middle Ground: At an intermediate temperature, the system underwent a phase transition. It was like a switch flipping. The atoms went from being stuck in a traffic jam to suddenly flowing freely across the dance floor.
• Why it matters: This shows that temperature isn't just a background noise; it can actually control whether the quantum system stays stuck or starts moving.

4. The Big Picture: Why This Matters

Think of this research as a guide for building future quantum computers or sensors using cold atoms.

• The Problem: Real-world experiments can never reach absolute zero. There is always some "heat" or noise.
• The Solution: This paper proves that even with that "heat," the cool quantum effects (like the traffic jam) are surprisingly robust. You don't need a perfect ice-cold environment to see these effects.
• The Takeaway: The researchers showed us exactly how to predict the behavior of these atoms when they are warm. They provided a "rulebook" (scaling laws) that tells scientists how the atoms will behave whether they are stuck in a jam or flowing freely, depending on how hot the room is and how hard the kicks are.

In a nutshell:
The paper proves that quantum "traffic jams" are tougher than we thought. They survive even when the system gets warm. However, if you turn up the heat just right and change the rhythm of the kicks, you can force the traffic jam to dissolve, turning a stuck quantum system into a flowing one. This gives scientists a new way to control and understand the quantum world in realistic, non-perfect conditions.

Technical Summary

1. Problem Statement

The paper addresses the interplay between finite temperature and many-body dynamical localization (MBDL) in a strongly interacting quantum system.

• Context: Conventional wisdom suggests that injecting energy into a system leads to thermalization and the loss of initial state memory. However, quantum systems like the Quantum Kicked Rotor (QKR) can exhibit Dynamical Localization (DL), where energy growth saturates due to quantum interference.
• Gap: While MBDL has been studied extensively in the Lieb-Liniger (LL) model (specifically the Tonks-Girardeau or TG limit of infinite interaction) at zero temperature, the effects of finite temperature remain largely unexplored.
• Core Question: Does many-body dynamical localization persist at finite temperatures? How does initial temperature affect the coherence, energy saturation, and the transition between localized and delocalized phases in both periodic and quasiperiodic kicked systems?

2. Methodology

The authors employ a combination of analytical mapping and numerical simulation using the Grand-Canonical Ensemble.

• Model: The system is modeled as $N$ bosons in 1D with contact interactions (Lieb-Liniger model) subjected to pulsed cosine potentials (kicks). The study focuses on the Tonks-Girardeau (TG) limit ($g \to \infty$), where bosons behave as impenetrable hard-core particles.
• Hamiltonian: The dimensionless Hamiltonian includes the free kinetic term, the interaction term (infinite repulsion), and a kick term $K(t)$. The kick can be periodic ($\epsilon=0$) or quasiperiodic ($\epsilon \neq 0$, involving incommensurate frequencies).
• Numerical Approach:
  • Lattice Discretization: The continuous system is mapped to a hard-core Bose-Hubbard model on a lattice.
  • Jordan-Wigner Transformation: Hard-core bosons are mapped to non-interacting spinless fermions. This allows the use of free-fermion techniques to solve the many-body problem exactly.
  • Thermal State: The initial state is prepared as a thermal density matrix $\hat{\rho}_{LL} = e^{-(\hat{H}_{LL}-\mu N)/T}/Z$ within the grand-canonical ensemble.
  • Observables: The authors compute the One-Particle Density Matrix (OPDM), momentum distributions ($n(k)$), correlation functions ($g_1(r)$), and kinetic energy growth.
  • Effective Parameters: They extract an effective temperature ($T_{eff}$) and effective chemical potential ($\mu_{eff}$) by fitting the evolved fermionic momentum distribution to the Fermi-Dirac distribution.

3. Key Contributions

1. Persistence of MBDL at High Temperatures: The study demonstrates that MBDL persists even at high initial temperatures ($T_0 \sim \epsilon_F$, where $\epsilon_F$ is the Fermi energy), although the coherence of the localized state is degraded.
2. Nonlinear Effective Thermalization: The authors establish a modified scaling law for the effective thermalization of the localized state. Unlike the zero-temperature case, the relationship between the localization length and effective temperature becomes nonlinear when the initial temperature is non-negligible.
3. Temperature-Induced Transition: In the quasiperiodic case, the paper identifies a many-body dynamical localization-delocalization transition occurring at intermediate temperatures, distinct from the zero-temperature phase diagram.
4. Scaling Laws: The work quantifies different dynamical phases (localized, critical, delocalized) using one-parameter scaling laws for momentum distributions, extending these concepts to the finite-temperature regime.

4. Key Results

A. Periodically Kicked TG Gas (Periodic Case)

• Localization Persistence: For $\epsilon=0$, the kinetic energy saturates for all initial temperatures $T_0$, confirming MBDL persists.
• Coherence Degradation: As $T_0$ increases, the coherence of the localized state decreases further. The correlation length ($r_c$) of the bosonic correlation function $g_1(r)$ is reduced compared to the $T=0$ case.
• Effective Thermalization:
  • The steady state can be described by an effective thermal density matrix.
  • Low Temperature ($T \ll \epsilon_F$): The relationship between the localization momentum $p_{loc}$ and $T_{eff}$ follows a specific scaling derived from Sommerfeld expansion.
  • High Temperature ($T \sim \epsilon_F$): The relationship becomes nonlinear. The data collapses onto a unified curve when plotted against $T_{eff}/\epsilon_F$, deviating from the linear $T=0$ prediction.
  • Tan's Contact: The high-momentum tail of the bosonic momentum distribution follows the algebraic decay $n_B(k) \propto k^{-4}$, consistent with Tan's contact for a thermal gas.

B. Quasiperiodically Kicked TG Gas (Quasiperiodic Case)

• Phase Transition: By varying the kick strength $K$ and anisotropy $\epsilon$, the system exhibits three phases: Localized ($\gamma \approx 0$), Critical ($\gamma \approx 2/3$, anomalous diffusion), and Delocalized ($\gamma = 1$, normal diffusion).
• Temperature Independence in Delocalization: In the delocalized phase, the system absorbs energy until it reaches an "infinite temperature" state. Consequently, at late times, the initial temperature $T_0$ becomes negligible, and the dynamics of finite-temperature systems converge to those of zero-temperature systems.
• Scaling Functions:
  • The momentum distribution $n(k)$ obeys a scaling law $n(k) = t^{-\alpha} f(kt^{-\alpha})$, where $\alpha$ depends on the phase ($1/2$ for delocalized, $1/3$ for critical).
  • The algebraic tail ($k^{-4}$) follows a separate scaling law $n(k) = t^{-2\alpha} F(kt^{-\alpha})$.
• Effective Thermalization in Delocalized Phase: In the delocalized regime, the effective chemical potential $\mu_{eff}$ becomes negative at late times, causing the distribution to transition from Fermi-Dirac to Maxwell-Boltzmann statistics. The effective temperature scales with energy growth as $T_{eff} \sim t$.

5. Significance

• Experimental Guidance: The results provide crucial benchmarks for cold-atom experiments involving strongly interacting gases (Tonks-Girardeau regime). Since real experiments always occur at finite temperatures, this work validates that MBDL is a robust phenomenon observable even when $T > 0$.
• Theoretical Extension: It extends the theory of dynamical localization from the ground state to the finite-temperature regime, clarifying how thermal fluctuations compete with quantum interference.
• Universality: The findings suggest that the underlying free-fermion mapping (via Jordan-Wigner) remains a powerful tool for understanding thermal many-body dynamics in the TG limit, provided the interactions are infinite.
• Future Directions: The paper highlights the need for further research into finite-interaction regimes ($g < \infty$), where non-integrability and long-range correlations might induce true thermalization, potentially destroying MBDL even at low temperatures.

In conclusion, the paper establishes that while finite temperature degrades coherence and modifies the quantitative scaling of localization, the fundamental phenomenon of many-body dynamical localization in the Tonks-Girardeau gas is robust against thermal effects, persisting even at high temperatures.