Few-fermion thermometry

Potential realization of a quantum thermometer operating in the nanokelvin regime, formed by a few-fermionic mixture confined in a one-dimensional harmonic trap, is proposed. Thermal states of the system are studied theoretically from the point of view of fundamental sensitivity to temperature changes. It is pointed out that the ability to control the interaction strength in such systems allows obtaining high-temperature sensitivity in the regime where the temperature is much lower than the characteristic temperature scale determined by a harmonic confinement. This sensitivity is very close to the fundamental bound that involves optimal engineering of level separations. The performance of practical measurement schemes and the possible experimental coupling of the thermometer to the probe are discussed.

Figure 1

Energy spectrum of the many-body Hamiltonian (6) for different numbers of particles as a function of interaction strength $g$. Note the quasidegeneracy of the many-body states for strong repulsions. The energies and the interaction strengths are measured in units of $\hslash \mathrm{\Omega }$ and ${({\hslash }^3\mathrm{\Omega }/m)}^{1/2}$, respectively.

Figure 2

Quantum Fisher information for a balanced system of $N=4$ fermions, as a function of temperature $T$, for different interaction strengths $g$. The blue envelope corresponds to optimization of $g$ for each value of $T$ and approaches very closely the fundamental bound (5) for $M=6$, which is the quasidegeneracy of the ground state. For $T\ge 0.1$ (shaded area), when $k_{B}T$ becomes comparable to $\hslash \mathrm{\Omega }$, higher energy levels start to contribute and hence the envelope may surpass the fundamental bound where only $M=6$ levels are considered. The QFI and temperature are measured in units of ${(\hslash \mathrm{\Omega }/k_B)}^{-1/2}$ and $\hslash \mathrm{\Omega }/k_B$, respectively.

Figure 3

Different Fisher information as a function of temperature $T$ calculated for a strongly interacting system ($g=14$) of $N_{\uparrow }=N_{\downarrow }=2$. The solid thick line corresponds to the quantum Fisher information ${\mathcal{F}}_Q\left(T\right)$, which defines an upper limit for other Fisher information obtained via particular measurement schemes. Other lines represent different variants of the Fock basis measurement: ${\mathcal{F}}_{\text{Fock}}\left(T\right)$, the complete Fock state projection (solid thin line); ${\mathcal{F}}_{\text{coarse}}\left(T\right)$, measurement of the total occupation of the two lowest single-particle orbitals (dotted line); ${\mathcal{F}}_{\text{Fermi}}\left(T\right)$, binary measurement of the presence of a particle just above the Fermi level (dash-dotted line). See the text for details. The Fisher information and temperature are measured in units of ${(\hslash \mathrm{\Omega }/k_B)}^{-1/2}$ and $\hslash \mathrm{\Omega }/k_B$, respectively.