Low-control and robust quantum refrigerator and applications with electronic spins in diamond

We propose a general protocol for low-control refrigeration and thermometry of thermal qubits, which can be implemented using electronic spins in diamond. The refrigeration is implemented by a probe, consisting of a network of interacting spins. The protocol involves two operations: (i) free evolution of the probe; and (ii) a swap gate between one spin in the probe and the thermal qubit we wish to cool. We show that if the initial state of the probe falls within a suitable range, and the free evolution of the probe is both unital and conserves the excitation in the $z$ direction, then the cooling protocol will always succeed, with an efficiency that depends on the rate of spin dephasing and the swap-gate fidelity. Furthermore, measuring the probe after it has cooled many qubits provides an estimate of their temperature. We provide a specific example where the probe is a Heisenberg spin chain, and suggest a physical implementation using electronic spins in diamond. Here, the probe is constituted of nitrogen vacancy (NV) centers, while the thermal qubits are dark spins. By using a novel pulse sequence, a chain of NV centers can be made to evolve according to a Heisenberg Hamiltonian. This proposal allows for a range of applications, such as NV-based nuclear magnetic resonance of photosensitive molecules kept in a dark spot on a sample, and it opens up possibilities for the study of quantum thermodynamics, environment-assisted sensing, and many-body physics.

Figure 1

Schematic of the cooling process. The probe (red spheres) is a network of spin-half systems, coupled through a Heisenberg interaction. The black spheres are a collection of thermal qubits that are initially in the state $\chi \left(T\right)$, with temperature $T>0$. The protocol cools the $k\text{th}$ thermal qubit by (i) first allowing the probe to evolve freely, for a time ${\tau }_k$, so that the target spin is prepared in the state $\chi \left(T^{\left(k\right)}\right)$, where $T^{\left(k\right)}\le T$; and (ii) subsequently, swapping the target spin of the probe with the $k\text{th}$ thermal qubit, thus cooling it.

Figure 2

The ideal cooling protocol, using a Heisenberg spin chain. (a), (b) Show, respectively, the dependence of the cooling efficiency of the $k\mathrm{th}$ thermal qubit ${\eta }_k$ on the length of the chain $N$ and temperature of the thermal qubits $T$. Here, $k_B$ is Boltzmann's constant, and $\omega$ is the spectral gap of the thermal qubit's Hamiltonian.

Figure 3

Pseudothermalization of a Heisenberg spin chain as a result of the time dynamics, where $D({\rho }_{\mathcal{P}}^{\left(k\right)},\chi {\left(T\right)}^{\otimes N})$ is the trace distance between the state of the probe and the pseudothermal state, after the $k\mathrm{th}$ thermal qubit has been cooled. In all cases, we set $J{\tau }_k=1$. (a), (b) Show, respectively, the dependence of pseudothermalization on the chain length $N$ and temperature $T$, for the ideal case. Here, $k_B$ is Boltzmann's constant, and $\omega$ is the spectral gap of the thermal qubit's Hamiltonian.

Figure 4

(a)–(c) Show the performance of the cooling protocol with a Heisenberg spin chain probe in the presence of local dephasing of strength $\mathrm{\Gamma }$. All waiting times $J{\tau }_k\in [0,N]$ are calculated for the ideal case with $N=10$ and $k_{B}T/\omega =5$. (a) Shows the dependence of the cooling efficiency of the $k\mathrm{th}$ thermal qubit ${\eta }_k$ on the dephasing strength. (b), (c) Show, respectively, how the dephasing strength affects the total entropy reduction of the thermal qubits $\mathrm{\Delta }{S}_{\mathcal{Q}}^{\mathrm{total}}\left(k\right)$ and the entropy of the chain $S\left({\rho }_{\mathcal{P}}^{\left(k\right)}\right)$, after the $k\mathrm{th}$ thermal qubit has been cooled. $S_T$ is the entropy of the thermal state $\chi \left(T\right)$. (d) Shows the effect of dephasing on pseudothermalization, where $D({\rho }_{\mathcal{P}}^{\left(k\right)},\chi {\left(T\right)}^{\otimes N})$ is the trace distance between the state of the probe and the pseudothermal state, after the $k\mathrm{th}$ thermal qubit has been cooled. Here, we set $J{\tau }_k=1$.

Figure 5

(a)–(c) Show the performance of the cooling protocol with a Heisenberg spin chain probe, with the swaps effected by a time-dependent Heisenberg interaction of strength $J_I$. All waiting times ${\tau }_k$ are calculated for the ideal case with $N=10$ and $k_{B}T=5$. (a) Shows the dependence of the cooling efficiency of the $k\mathrm{th}$ thermal qubit ${\eta }_k$ on $J_I$. (b), (c) Show, respectively, how $J_I$ affects the total entropy reduction of the thermal qubits $\mathrm{\Delta }{S}_{\mathcal{Q}}^{\mathrm{total}}\left(k\right)$ and the entropy of the chain $S\left({\rho }_{\mathcal{P}}^{\left(k\right)}\right)$, after the $k\mathrm{th}$ thermal qubit has been cooled. $S_T$ is the entropy of the thermal state $\chi \left(T\right)$. (d) Shows the effect of $J_I$ on the rate of pseudothermalization, where $D({\rho }_{\mathcal{P}}^{\left(k\right)},\chi {\left(T\right)}^{\otimes N})$ is the trace distance between the state of the probe and the pseudothermal state, after the $k\mathrm{th}$ thermal qubit has been cooled. Here, we set $J{\tau }_k=1$.

Figure 6

(a) Dark spin cooling with an NV spin chain for single-molecule NMR. Black spheres represent dark spins while red ones do NVs. (b) Orientation of NVs in the spin chain. To obtain a uniform interaction strength with optimal yield, (110)-cut diamond is presented. The magnetic field is aligned into the $\left[\overline{1}1\overline{1}\right]$ direction (marked as black arrows), and the NV should be oriented into one of the other three directions: [111], $\left[1\overline{1}\overline{1}\right]$, or $\left[\overline{1}\overline{1}1\right]$ (marked as blue arrows). Nearest-neighbor spins should have different directions. (c) Pulse sequences for the NV–dark spin interaction. For the dark spin, all five rf transitions are driven. For the spin-1 ${}^{14}\mathrm{N}$ hyperfine axis parallel with magnetic field, the hyperfine splitting is ${\mathrm{A}}_{\parallel }=114$ MHz, while for the other three axes, ${\mathrm{A}}_{\nparallel }=90$ MHz. (d) Bloch sphere representation of NV spin during pulse sequence. After ${\left(\frac{\pi }{2}\right)}_x$ pulse, the spin is locked into the $y$ direction (marked as a gray dotted arrow). (e) Dressed-state resonant coupling. In the laboratory frame, energy difference between two spins prohibits energy exchange (spin flip-flop). In the double-rotating frame with dressed states, energy can be exchanged between the two.

Figure 7

Variant-wahuha sequence for Heisenberg interaction and the whole pulse sequence. Cooling dark spins and thermalizing spin chains are alternated multiple times. Environment-assisted sensing can be applied after this cooling step. Note that variant-wahuha is not in a toggling frame, while wahuha is in a toggling frame.