Floquet Spin Amplification

Detection of weak electromagnetic waves and hypothetical particles aided by quantum amplification is important for fundamental physics and applications. However, demonstrations of quantum amplification are still limited; in particular, the physics of quantum amplification is not fully explored in periodically driven (Floquet) systems, which are generally defined by time-periodic Hamiltonians and enable observation of many exotic quantum phenomena such as time crystals. Here we investigate the magnetic-field signal amplification by periodically driven ${}^{129}\mathrm{Xe}$ spins and observe signal amplification at frequencies of transitions between Floquet spin states. This “Floquet amplification” allows us to simultaneously enhance and measure multiple magnetic fields with at least one order of magnitude improvement, offering the capability of femtotesla-level measurements. Our findings extend the physics of quantum amplification to Floquet spin systems and can be generalized to a wide variety of existing amplifiers, enabling a previously unexplored class of “Floquet spin amplifiers”.

Figure 1

(a) A periodically driven two-level system can be described by a series of Floquet states ${|\uparrow ⟩}_n$, ${|\downarrow ⟩}_m$. We study the transitions between these Floquet states. (b) Schematic of experimental setup. The key element is a cubic vapor cell containing 5 torr ${}^{129}\mathrm{Xe}$, 250 torr ${\mathrm{N}}_2$, and a droplet of isotopically enriched ${}^{87}\mathrm{Rb}$. ${}^{87}\mathrm{Rb}$ atoms are polarized by a circularly polarized laser at 795 nm and probed by a linearly polarized light blue-detuned 110 GHz from the D2 transition at 780 nm. ${}^{129}\mathrm{Xe}$ spins are polarized by spin-exchange collisions with polarized ${}^{87}\mathrm{Rb}$ atoms. An oscillating magnetic field as a periodic driving on ${}^{129}\mathrm{Xe}$ is applied along $\widehat{z}$. The driven ${}^{129}\mathrm{Xe}$ can amplify the magnetic field that oscillates at frequencies of transitions between Floquet states [see (a)]. The amplified field then is in situ read out by the ${}^{87}\mathrm{Rb}$ magnetometer.

Figure 2

(a) Plot of amplification as a function of the scanning test-field frequency. The bias field and periodic driving field are set as $B_0\approx 853\mathrm{nT}$, $B_{\mathrm{ac}}\approx 397\mathrm{nT}$, ${\nu }_{\mathrm{ac}}\approx 1.500\mathrm{Hz}$. The corresponding modulation index is $u\approx 3.12$. The interval between amplification frequencies is 1.500 Hz that is equal to ${\nu }_{\mathrm{ac}}$. The amplification at each resonance frequency satisfies ${\eta }_{k,0}$ [see Eq. (1)]. The amplification profile is asymmetric due to the Fano interference described in the text. (b) Spectrum of the amplification signal induced by a test field. The test signal is set at the frequency of the first-order sideband (see star) and there simultaneously exist other sideband signals. The amplification satisfies ${\eta }_{k,l}$ [see Eq. (2)]. (c) Plot of Floquet amplification as a function of the modulation index $u$. A periodic driving field of ${\nu }_{\mathrm{ac}}\approx 3.000\mathrm{Hz}$ is introduced. By scanning the amplitude $B_{\mathrm{ac}}$, the amplification of the test signal at 7.039 Hz (red points) follows $J_1^2(u)$ (red line). The amplification at 10.039 Hz (blue points) follows $J_1(u)J_0(u)$ (blue line) and at 4.039 Hz (green points) follows $J_1(u)J_2(u)$ (green line).

Figure 3

The magnetic sensitivity of the Floquet-amplification-based magnetometer. The bias field and periodic driving field are set as $B_0\approx 853\mathrm{nT}$, $B_{\mathrm{ac}}\approx 178\mathrm{nT}$, and ${\nu }_{\mathrm{ac}}\approx 1.500\mathrm{Hz}$. The corresponding modulation index is $u\approx 1.4$. The sensitivity is about $20\mathrm{fT}/{\mathrm{Hz}}^{1/2}$, $25\mathrm{fT}/{\mathrm{Hz}}^{1/2}$, $18\mathrm{fT}/{\mathrm{Hz}}^{1/2}$ at ${\nu }_0-1.5$, ${\nu }_0$, ${\nu }_0+1.5$, respectively. The line shape is asymmetric due to the Fano resonance.

Figure 4

Fano resonance in spin amplification. Black points denote the measured amplification of the test-field signal at 2.970 Hz with the parameters of ${\nu }_0\approx 10.039\mathrm{Hz}$ and ${\nu }_{\mathrm{ac}}\approx 13.000\mathrm{Hz}$. The signal amplitude is suppressed by a factor of 5 when the modulation index is $u\approx 3.497$. In the inset, the ${}^{129}\mathrm{Xe}\text{−}{}^{87}\mathrm{Rb}$ signals interfere with each other and their interference results in suppressing amplification.