[Paper Review] Optimal nonequilibrium thermometry in finite time

Pavel Sekatski, Martí Perarnau-Llobet|arXiv (Cornell University)|Jul 9, 2021

Quantum Information and Cryptography66 references4 citations

TL;DR

This paper establishes fundamental bounds on the precision of nonequilibrium quantum thermometry using a weakly coupled probe governed by a quantum Markovian master equation. It shows that measurement uncertainty scales inversely with time due to dissipative dynamics, and that adaptive control strategies can saturate these bounds, with the Lamb shift enabling polynomial decay of uncertainty at low temperatures and collective coupling yielding N² scaling in multi-qubit probes.

ABSTRACT

What is the minimum time required to take the temperature? In this paper, we solve this question for any process where temperature is inferred by measuring a probe (the thermometer) weakly coupled to the sample of interest, so that the probe's evolution is well described by a quantum Markovian master equation. Considering the most general control strategy on the probe (adaptive measurements, arbitrary control on the probe's state and Hamiltonian), we provide bounds on the achievable measurement precision in a finite amount of time, and show that in many scenarios these fundamental limits can be saturated with a relatively simple experiment. We find that for a general class of sample-probe interactions the scaling of the measurement uncertainty is inversely proportional to the time of the process, a shot-noise like behaviour that arises due to the dissipative nature of thermometry. As a side result, we show that the Lamb shift induced by the probe-sample interaction can play a relevant role in thermometry, allowing for finite measurement resolution in the low-temperature regime (more precisely, the measurement uncertainty decays polynomially with the temperature as $T ightarrow 0$, in contrast to the usual exponential decay with $T^{-1}$). We illustrate these general results for (i) a qubit probe interacting with a bosonic sample, where the role of the Lamb shit is highlighted, and (ii) a collective superradiant coupling between a $N$-qubit probe and a sample, which enables a quadratic decay with $N^2$ of the measurement uncertainty.

Motivation & Objective

• To determine the minimum time required to achieve a given temperature measurement precision in nonequilibrium quantum thermometry.
• To identify fundamental limits on measurement precision when using a weakly coupled quantum probe governed by a Markovian master equation.
• To explore whether these precision bounds can be saturated using realistic control strategies on the probe.
• To investigate the role of the Lamb shift in enabling finite resolution at low temperatures.
• To analyze how collective coupling in multi-qubit probes enhances measurement precision beyond single-particle scaling.

Proposed method

• Formulates the probe-sample interaction using a general quantum Markovian master equation to describe the probe's evolution under weak coupling.
• Applies optimal control theory to the probe, including adaptive measurements and arbitrary Hamiltonian and state control, to maximize information gain per unit time.
• Derives fundamental bounds on measurement uncertainty using quantum Fisher information and the Cramér-Rao inequality in the finite-time regime.
• Identifies the dissipative nature of thermometry as the origin of shot-noise-like scaling (uncertainty ∝ 1/time).
• Analyzes the Lamb shift induced by the probe-sample interaction as a mechanism enabling non-exponential (polynomial) decay of uncertainty at low temperatures.
• Considers two concrete models: a single qubit coupled to a bosonic bath, and a collective N-qubit superradiant coupling to the sample, to demonstrate N² scaling of precision.

Experimental results

Research questions

• RQ1 What is the fundamental lower bound on measurement uncertainty in finite-time quantum thermometry with a weakly coupled probe?
• RQ2 Can this bound be saturated using adaptive control and arbitrary probe manipulation?
• RQ3 How does the Lamb shift arising from the probe-sample interaction affect measurement resolution at low temperatures?
• RQ4 What is the scaling of measurement uncertainty with the number of qubits in a collective superradiant probe?
• RQ5 Does the dissipative nature of thermometry lead to a shot-noise-like uncertainty scaling with time?

Key findings

• The measurement uncertainty scales inversely with the process time, exhibiting a shot-noise-like behavior due to the dissipative nature of the thermometry process.
• The fundamental precision bounds derived from quantum Fisher information can be saturated using adaptive measurements and optimal control of the probe.
• The Lamb shift induced by the probe-sample interaction enables a polynomial decay of uncertainty as temperature approaches zero, in contrast to the typical exponential decay with T⁻¹.
• In the case of a collective N-qubit superradiant probe, the measurement uncertainty decays quadratically with N, achieving N² scaling of precision.
• The results hold for a general class of sample-probe interactions, making them broadly applicable to various quantum thermometry setups.
• The analysis reveals that even simple experimental implementations can achieve near-optimal precision when the control strategy is properly designed.