Proposal of a Quantum Measurement of Boltzmann constant

Abstract

The recent redefinition of the International System of Units in 2019 fixes the numerical values of seven defining constants, and links to them the definition of the seven fundamental units. Specifically, the fundamental constant related to the definition of the measurement unit for thermodynamic temperature (the kelvin) is the Boltzmann constant, $k_B$. Measurement of the Boltzmann constant is now done with classical methods. We propose an alternative method based on the measurement of the work done on a quantum system: we will take advantage of the Crooks-Tasaki fluctuation theorem and we will exploit an interferometric method to reduce the uncertainty in the measurement of $k_B$ in specific temperature regimes, i.e. below 100 mK, that are challenging for the realization of the thermometric scale. The work $W$ done on a system of finite size and in contact with a thermal reservoir is a stochastic variable. Every time we repeat the protocol we will measure, in general, a different value for $W$, following a probability distribution $P(W)$. The system considered is a qubit in contact with a thermal reservoir and we perform work on the open system varying a parameter $lambda(t)$ of the system Hamiltonian $H(lambda(t))$. We will use an interferometric technique that takes advantage of an ancilla qubit for measuring the work $W$. We collect the $W$ statistics to obtain $P_{forward}(W)$. The information on $k_B$ is embedded in the magnitude of the inteferometric oscillations at precisely known frequencies. If, starting from the final state, we perform work by applying the inverse protocol $H(lambda(t)), with $lambda’(t)=lambda(-t)$, we will have, in general, a different probability distribution $P_{backward}(W)$. The Crooks-Tasaki theorem connects $P_{forward}(W)$ with $P_{backward}(W)$: $frac{P_{forward}(W)}{P_{backward}(-W})=e^{frac{W-Delta F}{k_BT}}$ where $T$ is the temperature of the reservoir and $W-Delta F$ is the energy dissipated in the forward protocol, being $Delta F$ the difference between the initial and final free energy of the system. Eventually, we evaluate $k_B$ by plotting the logarithm of the left-hand-side of the Crooks-Tasaki theorem equation for different values of $W$ at fixed $T$. By evaluating the slope of the line $y= ln{frac{P_{forward}(W)}{P_{backward}(-W)}$ at fixed $T$, we are able to estimate $k_B$. We note that larger slope values, corresponding to lower temperature, allow better estimation of $k_B$. We propose to use the electronic spin of a Nitrogen-Vacancy centre in diamond as the system qubit, and as an ancillary qubit the nuclear spin of the neighbouring Nitrogen atom. We perform work on the Nitrogen-Vacancy centre varying the intensity and direction of an oscillating magnetic field, taking advantage of the discretization of system energy levels and the not-adiabaticity of the process. This method will establish a quantitative link between the work done and the energy dissipated by a quantum task. Furthermore, this measurement reaches its optimum for very low temperatures (5-80 mK) typical of solid-state quantum computation, thus providing a primary thermometer for such applications.