Enhancing precision thermometry with nonlinear qubits

TL;DR

This work addresses achieving higher precision in quantum thermometry at ultralow temperatures by harnessing nonlinear quantum dynamics. It analyzes single- and two-qubit systems under nonlinear Schrödinger evolution, deriving how the quantum speed limit and the quantum Fisher information for temperature, $\mathcal{F}_{\beta\beta}$, behave when the Gibbs state is not invariant under the nonlinear terms. A central finding is that, although the nonlinear dynamics can modify state evolution (and even reduce the maximal speed in some open-system configurations), the temperature-estimation information can grow in time, leading to enhanced thermometric precision, with the strongest gains in two-qubit setups and for logarithmic nonlinearities. These results point to a potential route toward ultra-precise thermometry at ultracold temperatures and motivate extending nonlinear metrological concepts to more realistic many-body systems.

Abstract

Quantum thermometry refers to the study of measuring ultra-low temperatures in quantum systems. The precision of such a quantum thermometer is limited by the degree to which temperature can be estimated by quantum measurements. More precisely, the maximal precision is given by the inverse of the quantum Fisher information. In the present analysis, we show that quantum thermometers that are described by nonlinear Schrödinger equations allow for a significantly enhanced precision, that means larger quantum Fisher information. This is demonstrated for a variety of pedagogical scenarios consisting of single and two-qubits systems. The enhancement in precision is indicated by non-vanishing quantum speed limits, which originate in the fact that the thermal, Gibbs state is typically not invariant under the nonlinear equations of motion.

Figures (4)

• Figure 1: Quantum speed limit \ref{['eq:qsl_qubit']} for the nonlinear Landau-Zener model \ref{['eq:diffeq_LZ']} with a Gross-Pitaevskii nonlinearity, $\kappa(x)=g x^2$, (left panel) and a logarithmic nonlinearity, $\kappa(x)=g \ln(x^2)$, (right panel) with an initial Gibbs state \ref{['eq:Gibbs_LZ']}. Parameters are $\Delta=1$, $J=1$, $\beta=1$, for $g= 0$ (red, dotdashed line), $g=1$ (purple, dashed line), and $g= 2$ (blue, solid line).
• Figure 2: Quantum Fisher information for temperature, $\mathcal{F}_{\beta,\beta}$, \ref{['eq:fisher_qubit']} for the nonlinear Landau-Zener model \ref{['eq:diffeq_LZ']} with a Gross-Pitaevskii nonlinearity, $\kappa(x)=g x^2$, (left panel) and a logarithmic nonlinearity, $\kappa(x)=g \ln(x^2)$, (right panel) with an initial Gibbs state \ref{['eq:Gibbs_LZ']}. Parameters are $\Delta=1$, $J=1$, $\beta=1$, for $g= 0$ (red, dotdashed line), $g=1$ (purple, dashed line), and $g= 2$ (blue, solid line).
• Figure 3: Quantum speed limit \ref{['eq:qsl_qubit']} for the nonlinear thermometer \ref{['eq:thermo']} with a Gross-Pitaevskii nonlinearity, $\kappa(x)=g x^2$, (left panel) and a logarithmic nonlinearity, $\kappa(x)=g \ln(x^2)$, (right panel) with the initial state $\rho_0$\ref{['eq:rho0']}. Parameters are $\Delta=1$, $J=1$, $\beta=1$, for $g= 0$ (red, dotdashed line), $g=1$ (purple, dashed line), and $g= 2$ (blue, solid line).
• Figure 4: Quantum Fisher information for temperature, $\mathcal{F}_{\beta,\beta}$, \ref{['eq:fisher_qubit']} for the nonlinear thermometer \ref{['eq:thermo']} with a Gross-Pitaevskii nonlinearity, $\kappa(x)=g x^2$, (left panel) and a logarithmic nonlinearity, $\kappa(x)=g \ln(x^2)$, (right panel) with the initial state $\rho_0$\ref{['eq:rho0']}. Parameters are $\Delta=1$, $J=1$, $\beta=1$, for $g= 0$ (red, dotdashed line), $g=1$ (purple, dashed line), and $g= 2$ (blue, solid line).