Measuring the temperature of cold many-body quantum systems

Precise low-temperature thermometry is a key requirement for virtually any quantum technological application. Unfortunately, as the temperature $T$ decreases, the errors in its estimation diverge very quickly. In this paper, we determine exactly how quickly this may be. We rigorously prove that the “conventional wisdom” of low-$T$ thermometry being exponentially inefficient is limited to local thermometry on translationally invariant systems with short-range interactions, featuring a nonzero gap above the ground state. This result applies very generally to spin and harmonic lattices. On the other hand, we show that a power-law-like scaling is the hallmark of local thermometry on gapless systems. Focusing on thermometry on one node of a harmonic lattice, we obtain valuable physical insight into the switching between the two types of scaling. In particular, we map the problem to an equivalent setup, consisting of a Brownian thermometer coupled to an equilibrium reservoir. This mapping allows us to prove that, surprisingly, the relative error of local thermometry on gapless harmonic lattices does not diverge as $T\to 0$; rather, it saturates to a constant. As a useful by-product, we prove that the low-$T$ sensitivity of a harmonic probe arbitrarily strongly coupled to a bosonic reservoir by means of a generic Ohmic interaction, always scales as $T^2$ for $T\to 0$. Our results thus identify the energy gap between the ground and first excited states of the global system as the key parameter in local thermometry, and ultimately provide clues to devising practical thermometric strategies deep in the ultracold regime.

Figure 1

Schematic diagram of a 2D translationally invariant lattice. The system $S$ and the boundary $B$ (of length $L$) appear depicted in orange and red, respectively. Although not shown, the nodes (gray dots) of the lattice are are connected by short-ranged interactions. It is also assumed that the many-body system is away from criticality.

Figure 2

(a) Log-log plog of the QFI as a function of the inverse temperature $\beta =1/T$ (open circles) for a probe with bare frequency ${\omega }_0>0$. The low-temperature asymptotic behavior ${\mathcal{F}}_T\sim T^2$ appears superimposed in red. (b) Same as in (a) but for a free probe (${\omega }_0\to 0$). In this case, the low-$T$ scaling is ${\mathcal{F}}_T\sim 1/T^2$ (red). In (a) ${\omega }_0=1$, while in (b), ${\omega }_0={10}^{-3}$. (Inset) Best-case relative error $\delta T/T=1/T\sqrt{M{\mathcal{F}}_T}$ for $M=1$ as a function of $T$ for the same parameters as in (b). In the two panels, we chose $\gamma =0.1$ and ${\omega }_c=100$. Note that, in both cases, the power-law-like scalings are maintained well beyond the bounds in Eqs. (20).

Figure 3

(a) Log plot of the QFI as a function of the inverse temperature $\beta$ (open circles) for a gapped translationally invariant harmonic chain with gap $\mathrm{\Delta }=0.01$. The exponential asymptotic scaling ${\mathcal{F}}_T\sim e^{-\mathrm{\Delta }/T}$ has been superimposed in red. In this case, $N=100$ and the interactions have been chosen as $G_n=G/n^t$, with $G=1$ and $t=2.5$. For these parameters, the corresponding bare oscillator frequency is ${\mathrm{\Omega }}^2\simeq 1.73425$. (b) Log-log plot of the QFI versus the inverse temperature $\beta$. All parameters are the same as in (a) except for ${\mathrm{\Omega }}^2\simeq 1.73435$, chosen so that the TIHC is gapless. The power-law-like divergence ${\mathcal{F}}_T\sim 1/T^2$ (or constant relative error $\sqrt{M}\delta T/T$) of Eq. (20b) has been plotted in red.

Figure 4

(a) Sketch of a translationally invariant harmonic chain with periodic boundary conditions and $2N+1$ nodes of frequency $\mathrm{\Omega }$. Node $\#1$ appears highlighted as the probe $P$. This corresponds to Eq. (21). (b) Equivalent “star” system, described by Eq. (27), where node $\#1$ couples to $N$ of the twofold-degenerate normal modes of the inaccessible part of the chain, with frequencies ${\omega }_i$.

Figure 5

Effective spectral densities after partitioning an $N=100$ TIHC as “node $\#1$” versus the rest. The interactions $G_n$ are the same as in Fig. 3. We plot both the gapless (solid black line) and the gapped case (dashed red line), with gap $\mathrm{\Delta }=0.5$ (i.e., ${\mathrm{\Omega }}^2\simeq 1.98425$). Note that in both situations $J\left(\omega \right)$ shows an approximately linear growth for low ${\omega }_n$ and features a cutoff. In the gapped case, however, the lowest sample mode coupled to the probe has a finite frequency and hence, the resulting spectral density is not Ohmic.

Figure 6

(a) Log-log plot of the coupling constants $G_n$ (open circles) corresponding to a discretized Caldeira-Leggett model with Ohmic spectral density and Lorentz-Drude cutoff, as a function of the internode distance $n$. In particular, the spectral density is characterized by $\gamma =0.1$ and ${\omega }_c=2$. A total of $N=2000$ frequencies were uniformly picked within the interval ${\omega }_n\in (0,{\omega }_{max})$, with ${\omega }_{max}=100$. The frequency of the probe was chosen as ${\omega }_0=0.2$. A linear fit $G_n\sim n^{-c}$ with $c\simeq 1.99469$ (red) has been added for comparison (see text for further details). (b) Coefficients of the expansion $q_0={\sum }_a{d}_a{Q}_a$ of the position of the probe in the CLM as a linear combination of the positions of the oscillators in the corresponding TIHC. All parameters are the same as in (a).