Spin thermometry and spin relaxation of optically detected ${\mathrm{Cr}}^{3+}$ ions in ruby ${\mathrm{Al}}_2{\mathrm{O}}_3$

Paramagnetic ions in solid state crystals form the basis for many advanced technologies such as lasers, masers, frequency standards, and quantum-enhanced sensors. One of the most-studied examples is the ${\mathrm{Cr}}^{3+}$ ion in sapphire (${\mathrm{Al}}_2{\mathrm{O}}_3$), also known as ruby, which has been intensely studied in the 1950s and 1960s. However, despite decades of research on ruby, some of its fundamental optical and spin properties have not yet been characterized at ultralow temperatures. In this paper, we present optical measurements on a ruby crystal in a dilution refrigerator at ultralow temperatures down to 20 mK. Analyzing the relative populations of its ${}^{4}A{}_2$ ground-state spin levels, we extract a lattice temperature of $143\pm 7\mathrm{mK}$ under continuous laser excitation. We perform spin-lattice relaxation $T_1$ measurements in excellent agreement with the direct, one-phonon model. Furthermore, we perform optically detected magnetic resonance measurements showing magnetically driven transitions between the ground-state spin levels for various magnetic fields. Our measurements characterize some of ruby's low-temperature spin properties, and lay the foundations for more advanced spin control experiments.

Figure 1

(a) Hexagonal crystal structure of ${\mathrm{Al}}_2{\mathrm{O}}_3$ containing a single interstitial ${\mathrm{Cr}}^{3+}$ ion. (b) Polarization dependence of the phonon sideband (PSB) emission with excitation laser in resonance with the ${|\pm 3/2\rangle }_{\mathrm{g}}\leftrightarrow \overline{E}$ transition at $B_0=0\mathrm{T}$. (c) Electronic level scheme of ${\mathrm{Cr}}^{3+}$ in ${\mathrm{Al}}_2{\mathrm{O}}_3$ at $B_0=0\mathrm{T}$ and $B_0\perp c$ axis. (d) Emission spectrum of ruby with 520 nm (off-resonant) excitation at room temperature (red curve) and $T_{\mathrm{MXC}}=70\mathrm{mK}$ (blue curve). Marked are the zero-phonon lines (ZPLs) used for resonant excitation and the PSB used for signal detection in all further experiments. (e) PLE spectrum as a function of magnetic field for $B_0\perp c$ axis. The overlaid, dashed blue lines correspond to the Hamiltonian in Eq. (1) with parameters $g=1.982, -2D=11.460\mathrm{GHz}$, and 693.587 nm as the splitting between the midpoint of ${|\pm 1/2\rangle }_{\mathrm{g}}$ and ${|\pm 3/2\rangle }_{\mathrm{g}}$, and $\overline{E}$. The quantity $\mathrm{\Delta }$ (top $x$ axis) denotes the detuning in laser excitation energy (in frequency units) from the midpoint of the ${}^{52}\mathrm{Cr} {|\pm 1/2\rangle }_{\mathrm{g}}\leftrightarrow \overline{E}$ and ${|\pm 3/2\rangle }_{\mathrm{g}}\leftrightarrow \overline{E}$ transitions. (f) Zero-field photoluminescence excitation (PLE) spectrum showing multiple peaks that are associated with the different $\mathrm{Cr}$ isotopes. Peaks are fitted with Lorentzian functions with their relative peak intensities fixed to correspond to the isotopes' natural abundance (4.3% for ${}^{50}\mathrm{Cr}$, 83.8% for ${}^{52}\mathrm{Cr}$, 9.5% for ${}^{53}\mathrm{Cr}$, and 2.4% for ${}^{54}\mathrm{Cr}$ [21]). (g) PLE spectrum at $B_0=0\mathrm{T}$ for varying mixing chamber (MXC) temperatures, where the temperature readings are given by the MXC thermometer. Increased PLE intensity from the ${|\pm 1/2\rangle }_{\mathrm{g}}\leftrightarrow \overline{E}$ transition is in accordance with the redistribution of spin population at increased temperature.

Figure 2

(a) Normalized PLE spectra of the $|0\rangle \leftrightarrow \overline{E}$ and $|1\rangle \leftrightarrow \overline{E}$ transitions at $T_{\mathrm{MXC}}=20$, 173, and 300 mK. (b) Ratio of PLE peak intensities (circles) extracted from (a) as a function of magnetic field. The solid lines are fits to the Boltzmann distribution, determining $T_{\mathrm{Ruby}}=143\pm 7, 202\pm 12$, and $285\pm 31\mathrm{mK}$ for $T_{\mathrm{MXC}}=20$, 173, and 300 mK, respectively. (c) Extracted $T_{\mathrm{Ruby}}$ vs $T_{\mathrm{MXC}}$.

Figure 3

(a) Pulse sequence for the all-optical $T_1$ measurement. (b) Rate model and transition rates describing the electron spin dynamics of the ${\mathrm{Cr}}^{3+}$ ground and excited state. See Appendix pp2 for more details. (c) $T_1$ spin decay curve measured at $T_{\mathrm{MXC}}=20\mathrm{mK}$ (base), $B_0=0\mathrm{T}$, and 1 $\mu \mathrm{W}$ laser excitation power. (d) Temperature dependence of $T_1$ as a function of lattice temperature $T_{\mathrm{Ruby}}$. The solid circles correspond to data measured with 1 $\mu \mathrm{W}$ laser power, and the open triangles are confirmation measurements with 15 nW laser power for low temperatures. The open circles and the open square are literature values from Standley et al. [9] and Bates et al. [20], respectively, shown for comparison. The literature value from Bates et al. [20] is additionally used to infer the $T_1$ at lower temperature assuming the one-phonon process described by Eq. (7) (solid line).

Figure 4

(a) Pulse sequence for implementing pulsed ODMR. The initialization laser pulse is 3 s long, and the second laser pulse is also 3 s long and acts as a readout and reinitialization pulse. Within the ${\tau }_{\mathrm{off}}=10$ ms dark time, a MW pulse of 10 $\mu \mathrm{s}-1$ ms length is applied, aligned to occur just before the readout pulse. The SPAD is gated to detect photons for the first 0.1 s of readout. The laser excitation power is 1–10 $\mu \mathrm{W}$. (b)–(f) ODMR transitions with linewidths [full width at half maximum (FWHM)] (b) $62.0\pm 4.3\mathrm{MHz}$, (c) $38.2\pm 4.7\mathrm{MHz}$, (d) $32.7\pm 4.9\mathrm{MHz}$, (e) $8.8\pm 2.0\mathrm{MHz}$, and (f) $52\pm 17\mathrm{MHz}$. (g) All ODMR transition frequencies measured, superimposed with expected transition frequencies given by Eq. (1). The data points obtained from (b)–(f) are emphasized with larger circles.

Figure 5

(a) Pulse sequence for measuring the initialization time constant. (b) PL signal as a function of laser on-time ${\tau }_{\mathrm{ON}}$ before detecting photons at $T_{\mathrm{MXC}}=20\mathrm{mK}$. (c) Extracted initialization time constants as a function of temperature $T_{\mathrm{MXC}}$. (d) Pulse sequence for measuring the optical relaxation time $T_{1,\mathrm{opt}}$. (e) Optical relaxation time $T_{1,\mathrm{opt}}$ measurement at $T_{\mathrm{MXC}}=20\mathrm{mK}$. (f) $T_{1,\mathrm{opt}}$ dependence on temperature $T_{\mathrm{MXC}}$.

Figure 6

(a) Root-mean-square errors of best fits to the 1-$\mu \mathrm{W}$, 4-K spin initialization curve as a function of laser pump rate ${\lambda }_{\mathrm{p}}$ and branching ratio $r$. The red line indicates the curve of the lowest fitting error. (b) Simulation of the spin initialization measurement using the rate model Eq. (B1), with ${\lambda }_{\mathrm{p}}=1.5\mathrm{Hz}$ and $r=0.5$. The populations $N_{{|\pm 1/2\rangle }_{\mathrm{g}}}$ (red line), $N_{{|\pm 3/2\rangle }_{\mathrm{g}}}$ (green line), and $N_{{|\pm 1/2\rangle }_{\overline{\mathrm{E}}}}$ (blue line) are superimposed with the experimental data in arbitrary units (black circles). (c) Excitation power dependence at $T_{\mathrm{MXC}}=20\mathrm{mK}$ of the PLE signal “S” (red circles), and the background “B” (blue circles) with fits in the form $a{x}^b$, where $x$ is the laser excitation power. The green line shows the expected signal with the background subtracted “S-B.” (d) Extracted exponential $b$ from the excitation power dependence in (c) performed at varying $T_{\mathrm{MXC}}$.

Figure 7

(a)–(f) Rabi frequencies ${\mathrm{\Omega }}_{\mathrm{R}}$ extracted from time evolution simulations for transitions (a) $|0\rangle \leftrightarrow |1\rangle$, (b) $|0\rangle \leftrightarrow |2\rangle$, (c) $|0\rangle \leftrightarrow |3\rangle$, (d) $|1\rangle \leftrightarrow |2\rangle$, (e) $|1\rangle \leftrightarrow |3\rangle$, and (f) $|2\rangle \leftrightarrow |3\rangle$ as a function of $B_0$ strength and $B_1$ direction as specified by $\varphi$.