External-level assisted cooling by measurement (2024)

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  Figure 1

  (a)Sketch of the magnon spectrum. The dipole peak (Kittel mode) can be spectrally separated from the other magnon peaks under certain conditions [58]. (b)Diagram of the model comprising a three-level system (the ancillary system) and a YIG sphere (the target system). $g_m$ is the coupling strength between them. The transition $|e\rangle \leftrightarrow |f\rangle$, driven by a laser field with strength $g_f$, is far off resonant from the Kittel mode. (c)A general circuit model for the ground-state cooling by measurement. Initially, the resonator (the upper line) is in a thermal-equilibrium state, while the ancillary system (the bottom line) is at the ground state. Then the total system undergoes a free evolution for a period of $\tau$, after which an instantaneous projective measurement in the form of $|g\rangle \langle g|$ is performed. If the ancillary system is detected to remain at its ground state, then the total system is allowed to evolve another period of $\tau$. The resonator is thus cooled down in a nondeterministic way and approaches its ground state $|0\rangle$ under the repetition of the preceding process.

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  Figure 2

  Cooling coefficients as functions of the Fock-state index $n$ by a single measurement and 10 measurements on the ground state of the ancillary systems. The blue solid line and the black solid line with circles indicate $|{\alpha }_n{\left(\tau \right)|}^{2N}$ in our protocol using an ancillary three-level system. The red dash-dotted line and the yellow dashed line indicate $|{\beta }_n{\left(\tau \right)|}^{2N}$ in the conventional protocol using an ancillary two-level system. The coupling between the resonator and the ancillary system is $g_m/{\omega }_m=0.0004$, the driving strength is $g_f/g_m=50$, and the measurement period is ${\omega }_m\tau =700$.

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  Figure 3

  The average magnon number $\overline{n}\left(N\right)$ as a function of measurement numbers $N$. The black solid lines marked with triangles and the red solid lines marked with circles indicate the conventional and external-level assisted cooling protocols, respectively. (a)$T=0.1$K, (b)$T=1.0$K, and (c)$T=2.2$K. The frequency of the Kittel mode is set as ${\omega }_m=15.6$GHz. $g_m=2\pi \times 1$MHz, $g_f/g_m=30$, and ${\omega }_m\tau =700$.

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  Figure 4

  Cooling performances at a higher temperature $T=10$K of both measurement-cooling protocols, including (a)the average magnon number $\overline{n}$, (b)the fidelity of the ground state $F\left(N\right)\equiv \langle 0\left|{\rho }_m\left(N\tau \right)\right|0\rangle$, and (c)the survival probability $P_g\left(N\right)$ of detecting the ancillary system at $|g\rangle$ and the resonator in ${\rho }_m\left(N\tau \right)$ as functions of $N$. ${\omega }_m=15.6$GHz, $g_m=2\pi \times 1$MHz, $g_f/g_m=30$, and ${\omega }_m\tau =700$.

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  Figure 5

  Population histograms for various states of the resonator over the Fock space. (a)The initial thermal state at $T=10$K. (b)and (c)The states after $N=60$ measurements by the conventional protocol and our protocol, respectively. ${\omega }_m=15.6$GHz, $g_m=2\pi \times 1$MHz, and ${\omega }_m\tau =700$.

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  Figure 6

  Average magnon numbers $\overline{n}\left(N\right)$ under various initial high temperatures. The black solid, blue dot-dashed, red dashed, and green dotted lines describe $T=90$, 100, 110, and 120K, respectively. ${\omega }_m=15.6$GHz, $g_m=2\pi \times 1$MHz, $g_f/g_m=50$, and ${\omega }_m\tau =220$.

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  Figure 7

  Cooling performance at $T=10$K, including (a)the average magnon number $\overline{n}$, (b)the ground-state fidelity $F$, and (c)the survival probability $P_g$ as functions of $N$ under various protocols. The black lines marked with triangles, the red lines marked with circles, and the blue lines marked with squares denote the conventional, external-level assisted, and hybrid cooling protocols, respectively. ${\omega }_m=15.6$GHz, $g_m=2\pi \times 1$MHz, and ${\omega }_m\tau =700$.

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  Figure 8

  (a)The average magnon number $\overline{n}$ and (b)the survival probability $P_g$ as functions of the normalized driving strength $g_f$ (in units of $g_m$). The number of measurements is fixed at $N=50$. ${\omega }_m=15.6$GHz, $g_m=2\pi \times 1$MHz, and ${\omega }_m\tau =700$.

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  Figure 9

  Cooling coefficients of (a)the driving-assisted protocol $|{\alpha }_n{|}^2$ or $|{\tilde{\alpha }}_n{|}^2$ and (b)the conventional protocol $|{\beta }_n{|}^2$ or $|{\tilde{\beta }}_n{|}^2$ as functions of the Fock-state index $n$ after a single measurement. $g_m/{\omega }_m=0.0004$, $g_f/g_m=50$, and ${\omega }_m\tau =700$.

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