Microwave spectroscopy of radio-frequency dressed $^{87}$Rb

TL;DR

This paper investigates the hyperfine spectrum of $^{87}$Rb atoms under radio-frequency dressing across various trapping conditions, revealing resonant side bands and demonstrating precise control and measurement capabilities.

Contribution

It provides a comprehensive experimental and theoretical analysis of RF-dressed $^{87}$Rb spectra in different trapping scenarios, including a semi-classical model and practical applications.

Findings

Observation of resonant side bands spaced by the dressing frequency

Theoretical explanation using semi-classical model and Rotating Wave Approximation

Demonstration of accurate determination of dressing configuration and microwave field

Abstract

We study the hyperfine spectrum of atoms of $^{87}$Rb dressed by a radio-frequency field, and present experimental results in three different situations: freely falling atoms, atoms trapped in an optical dipole trap and atoms in an adiabatic radio-frequency dressed shell trap. In all cases, we observe several resonant side bands spaced (in frequency) at intervals equal to the dressing frequency, corresponding to transitions enabled by the dressing field. We theoretically explain the main features of the microwave spectrum, using a semi-classical model in the low field limit and the Rotating Wave Approximation for alkali-like species in general and $^{87}$Rb atoms in particular. As a proof of concept, we demonstrate how the spectral signal of a dressed atomic ensemble enables an accurate determination of the dressing configuration and the probing microwave field.

Tables

Table 1. Table 1: Values of the components of AC and DC magnetic fields obtained from a a fit to the data in Fig.  9

Field component	Fitted value
$B_{\text{MW},x}$	$2.01\pm 0.07$mG
$B_{\text{MW},y}$	$1.33\pm 0.08$mG
$B_{\text{RF},x}$	$1.114\pm 0.001$G
$B_{\text{DC},x}$	$0.21\pm 0.02$G
$B_{\text{DC},y}$	$0.25\pm 0.02$G
$B_{\text{DC},z}$	$3.162\pm 0.008$G

Table 2. Table 2: Couplings between RF-dressed states induced by a π 𝜋 \pi -polarised MW field. The colour of the text in the H M ​ W π / ( ℏ ​ Ω 0 ) superscript subscript 𝐻 𝑀 𝑊 𝜋 Planck-constant-over-2-pi subscript Ω 0 H_{MW}^{\pi}/(\hbar\Omega_{0}) columns corresponds to the colours in Figure  13 c, with red, black and blue for groups n = 2 , 0 , − 2 𝑛 2 0 2 n=2,0,-2 respectively. The σ − subscript 𝜎 \sigma_{-} polarization is defined by taking ϕ z = 0 subscript italic-ϕ 𝑧 0 \phi_{z}=0 , B MW , z > 0 subscript 𝐵 MW 𝑧 0 B_{\textrm{MW},z}>0 and B MW , x = B MW , y = 0 subscript 𝐵 MW 𝑥 subscript 𝐵 MW 𝑦 0 B_{\textrm{MW},x}=B_{\textrm{MW},y}=0 .

Coupled	$\Delta \overline{m}$	$\frac{H_{\text{MW}}^{\pi ,n=2}}{\hslash {\Omega }_0}$	$\frac{H_{\text{MW}}^{\pi ,n=0}}{\hslash {\Omega }_0}$	$\frac{H_{\text{MW}}^{\pi ,n=-2}}{\hslash {\Omega }_0}$
pair
$|1,1⟩\leftrightarrow |2,-2⟩$	-1	\colorred$-\sqrt{2}\left(1-\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$-2\sqrt{2}\sin \left({\theta }_1\right){\sin }^2\left({\theta }_2\right)$	\colorblue $-\sqrt{2}\left(1+\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$
$|1,1⟩\leftrightarrow |2,-1⟩$	0	\colorred $-\sqrt{2}\left(1-\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	$2\sqrt{2}\sin \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	\colorblue$\sqrt{2}\left(1+\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$
$|1,1⟩\leftrightarrow |2,0⟩$	1	\colorred$\sqrt{3}\left(1-\cos \left({\theta }_1\right)\right)\sin \left(2{\theta }_2\right)$	$\frac{4}{\sqrt{3}}\sin \left({\theta }_1\right)\left(1-3{\cos }^2\left({\theta }_2\right)\right)$	\colorblue$-\sqrt{3}\left(1+\cos \left({\theta }_1\right)\right)\sin \left(2{\theta }_2\right)$
$|1,1⟩\leftrightarrow |2,1⟩$	2	\colorred$\sqrt{2}\left(1-\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	$-2\sqrt{2}\sin \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	\colorblue $-\sqrt{2}\left(1+\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$
$|1,1⟩\leftrightarrow |2,2⟩$	3	\colorred $\sqrt{2}\left(1-\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$-2\sqrt{2}\sin \left({\theta }_1\right)\sin {\left({\theta }_2\right)}^2$	\colorblue $\sqrt{2}\left(1+\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$
$|1,0⟩\leftrightarrow |2,-2⟩$	-2	\colorred$2\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$4cos\left({\theta }_1\right){\sin }^2\left({\theta }_2\right)$	\colorblue$-2\sin \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$
$|1,0⟩\leftrightarrow |2,-1⟩$	-1	\colorred$2\sin \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	$-4cos\left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	\colorblue$2\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$
$|1,0⟩\leftrightarrow |2,0⟩$	0	\colorred$-\sqrt{6}\sin \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	$-4\sqrt{\frac{2}{3}}\cos \left({\theta }_1\right)\left(1-3cos{\left({\theta }_2\right)}^2\right)$	\colorblue$-\sqrt{6}\sin \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$
$|1,0⟩\leftrightarrow |2,1⟩$	1	\colorred$-2\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	$4cos\left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	\colorblue$2\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)+2{\cos }^2\left({\theta }_2\right)\right)$
$|1,0⟩\leftrightarrow |2,2⟩$	2	\colorred$-2\sin \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$4cos\left({\theta }_1\right){\sin }^2\left({\theta }_2\right)$	\colorblue$2\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$
$|1,-1⟩\leftrightarrow |2,-2⟩$	-3	\colorred $-\sqrt{2}\left(1+\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$2\sqrt{2}\sin \left({\theta }_1\right){\sin }^2\left({\theta }_2\right)$	\colorblue$-\sqrt{2}\left(1-\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$
$|1,-1⟩\leftrightarrow |2,-1⟩$	-2	\colorred$-\sqrt{2}\left(1+\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	$-2\sqrt{2}\sin \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	\colorblue$\sqrt{2}\left(1-\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$
$|1,-1⟩\leftrightarrow |2,0⟩$	-1	\colorred$\sqrt{3}\left(1+\cos \left({\theta }_1\right)\right)\sin \left(2{\theta }_2\right)$	$-\frac{4}{\sqrt{3}}\sin \left({\theta }_1\right)\left(1-3co{s}^2\left({\theta }_2\right)\right)$	\colorblue $-\sqrt{3}\left(1-\cos \left({\theta }_1\right)\right)\left(\sin \left(2{\theta }_2\right)\right)$
$|1,-1⟩\leftrightarrow |2,1⟩$	0	\colorred $\sqrt{2}\left(1+\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	$2\sqrt{2}\sin \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	\colorblue$-\sqrt{2}\left(1-\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$
$|1,-1⟩\leftrightarrow |2,2⟩$	1	\colorred$\sqrt{2}\left(1+\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$2\sqrt{2}\sin \left({\theta }_1\right)\sin {\left({\theta }_2\right)}^2$	\colorblue $\sqrt{2}\left(1-\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$

Table 3. Table 3: Couplings between RF-dressed states induced by a σ − subscript 𝜎 \sigma_{-} -polarised MW field. The colour of the text in the H M ​ W − / ( ℏ ​ Ω − 1 ) superscript subscript 𝐻 𝑀 𝑊 Planck-constant-over-2-pi subscript Ω 1 H_{MW}^{-}/(\hbar\Omega_{-1}) columns corresponds to those found in Figure  14 c, with red, black and blue for groups n = 3 , 1 , − 1 𝑛 3 1 1 n=3,1,-1 respectively. The σ − subscript 𝜎 \sigma_{-} polarization is defined by taking ϕ x = 0 subscript italic-ϕ 𝑥 0 \phi_{x}=0 , ϕ y = π / 2 subscript italic-ϕ 𝑦 𝜋 2 \phi_{y}=\pi/2 , B MW , z = 0 subscript 𝐵 MW 𝑧 0 B_{\textrm{MW},z}=0 and B MW , x = B MW , y > 0 subscript 𝐵 MW 𝑥 subscript 𝐵 MW 𝑦 0 B_{\textrm{MW},x}=B_{\textrm{MW},y}>0 .

Coupled	$\Delta \overline{m}$	$\frac{{\overline{H}}_{\text{MW}}^{-,n=3}}{\hslash {\Omega }_{-1}}$	$\frac{{\overline{H}}_{\text{MW}}^{-,n=1}}{\hslash {\Omega }_{-1}}$	$\frac{{\overline{H}}_{\text{MW}}^{-,n=-1}}{\hslash {\Omega }_{-1}}$
pairs
$|1,1⟩\leftrightarrow |2,-2⟩$	-1	\colorred$\left(1-\cos \left({\theta }_1\right)\right){\left(1+\cos \left({\theta }_2\right)\right)}^2$	$2\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	\colorblue$\left(1+\cos \left({\theta }_1\right)\right){\sin }^2\left({\theta }_2\right)$
$|1,1⟩\leftrightarrow |2,-1⟩$	0	\colorred $2\left(1-\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$2\sin \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	\colorblue$-\left(1+\cos \left({\theta }_1\right)\right)\sin \left(2{\theta }_2\right)$
$|1,1⟩\leftrightarrow |2,0⟩$	1	\colorred $\sqrt{6}\left(1-\cos \left({\theta }_1\right)\right){\sin }^2\left({\theta }_2\right)$	$-\sqrt{6}\sin \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	\colorblue $-\sqrt{\frac{2}{3}}\left(1+\cos \left({\theta }_1\right)\right)\left(1-3{\cos }^2\left({\theta }_2\right)\right)$
$|1,1⟩\leftrightarrow |2,1⟩$	2	\colorred $2\left(1-\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$-2\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	\colorblue$\left(1+\cos \left({\theta }_1\right)\right)\sin \left(2{\theta }_2\right)$
$|1,1⟩\leftrightarrow |2,2⟩$	3	\colorred$\left(1-\cos \left({\theta }_1\right)\right){\left(1-\cos \left({\theta }_2\right)\right)}^2$	$-2\sin \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	\colorblue $\left(1+\cos \left({\theta }_1\right)\right){\sin }^2\left({\theta }_2\right)$
$|1,0⟩\leftrightarrow |2,-2⟩$	-2	\colorred$-\sqrt{2}\sin \left({\theta }_1\right){\left(1+\cos \left({\theta }_2\right)\right)}^2$	$-2\sqrt{2}\cos \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	\colorblue $\sqrt{2}\sin \left({\theta }_1\right){\sin }^2\left({\theta }_2\right)$
$|1,0⟩\leftrightarrow |2,-1⟩$	-1	\colorred$-2\sqrt{2}\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$-2\sqrt{2}\cos \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	\colorblue$-\sqrt{2}\sin \left({\theta }_1\right){\sin }^2\left(2{\theta }_2\right)$
$|1,0⟩\leftrightarrow |2,0⟩$	0	\colorred$-2\sqrt{3}\sin \left({\theta }_1\right){\sin }^2\left({\theta }_2\right)$	$2\sqrt{3}\cos \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	\colorblue$-\sqrt{\frac{4}{3}}\sin \left({\theta }_1\right)\left(1-3{\cos }^2\left({\theta }_2\right)\right)$
$|1,0⟩\leftrightarrow |2,1⟩$	1	\colorred$-2\sqrt{2}\sin \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$2\sqrt{2}\cos \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	\colorblue$\sqrt{2}\sin \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$
$|1,0⟩\leftrightarrow |2,2⟩$	2	\colorred$-\sqrt{2}\sin \left({\theta }_1\right){\left(1-\cos \left({\theta }_2\right)\right)}^2$	$2\sqrt{2}\cos \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	\colorblue$\sqrt{2}\sin \left({\theta }_1\right){\sin }^2\left({\theta }_2\right)$
$|1,-1⟩\leftrightarrow |2,-2⟩$	-3	\colorred$\left(1+\cos \left({\theta }_1\right)\right){\left(1+\cos \left({\theta }_2\right)\right)}^2$	$-2\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	\colorblue $\left(1-\cos \left({\theta }_1\right)\right){\sin }^2\left({\theta }_2\right)$
$|1,-1⟩\leftrightarrow |2,-1⟩$	-2	\colorred $2\left(1+\cos \left({\theta }_1\right)\right)\left(1+\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$-2\sin \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	\colorblue$-\left(1-\cos \left({\theta }_1\right)\right)\sin \left(2{\theta }_2\right)$
$|1,-1⟩\leftrightarrow |2,0⟩$	-1	\colorred$\sqrt{6}\left(1+\cos \left({\theta }_1\right)\right){\sin }^2\left({\theta }_2\right)$	$\sqrt{6}\sin \left({\theta }_1\right)\sin \left(2{\theta }_2\right)$	\colorblue $-\sqrt{\frac{2}{3}}\left(1-\cos \left({\theta }_1\right)\right)\left(1-3{\cos }^2\left({\theta }_2\right)\right)$
$|1,-1⟩\leftrightarrow |2,1⟩$	0	\colorred $2\left(1+\cos \left({\theta }_1\right)\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	$2\sin \left({\theta }_1\right)\left(1+\cos \left({\theta }_2\right)-2{\cos }^2\left({\theta }_2\right)\right)$	\colorblue$\left(1-\cos \left({\theta }_1\right)\right)\sin \left(2{\theta }_2\right)$
$|1,-1⟩\leftrightarrow |2,2⟩$	1	\colorred $\left(1+\cos \left({\theta }_1\right)\right){\left(1-\cos \left({\theta }_2\right)\right)}^2$	$2\sin \left({\theta }_1\right)\left(1-\cos \left({\theta }_2\right)\right)\sin \left({\theta }_2\right)$	\colorblue $\left(1-\cos \left({\theta }_1\right)\right){\sin }^2\left({\theta }_2\right)$

Equations

$$\hat{H}=\frac{A}{{\hslash }^2}\hat{\mathbf{I}}\cdot \hat{\mathbf{J}}+\frac{{\mu }_{\text{B}}}{\hslash }(g_I\hat{\mathbf{I}}+g_J\hat{\mathbf{J}})\cdot \mathbf{B}(t),$$

$$\mathbf{B}(t)=B_{\text{DC}}{\mathbf{e}}_z+{\mathbf{B}}_{\text{RF}}(t)+{\mathbf{B}}_{\text{MW}}(t).$$

$${\hat{H}}_0=\sum _{\text{F}}\left(E_F+{\mu }_{\text{B}}{g}_F{\hat{F}}_z{B}_{\text{DC}}\right){\widehat{\text{mathds}1}}_F,$$

$${\widehat{\text{mathds}1}}_F=\sum _{m_{\text{F}}=-F}^F|F,m_{\text{F}}⟩⟨F,m_{\text{F}}|,$$

$$E_F=\frac{1}{2}A\left(F(F+1)-I(I+1)-J(J+1)\right),$$

$$g_F$$

$${\mathbf{B}}_{\text{AC}}(t)$$

$${\mathcal{B}}_{\text{AC}}$$

$${\mathbf{e}}_0$$

$${\mathbf{e}}_{\pm }$$

$${\hat{H}}_{\text{RF}}(t)$$

$${\hat{H}}_{\text{MW}}(t)$$

$$\hat{H}={\hat{H}}_0+{\hat{H}}_{\text{RF}}(t),$$

$${\hat{U}}_z({\omega }_{\text{RF}}t)$$

$$\hat{\tilde{H}}$$

$${\hat{U}}_y$$

$${\theta }_F=\frac{\pi }{2}-{\tan }^{-1}\left(\frac{B_{\text{DC}}-\hslash {\omega }_{\text{RF}}/({\mu }_B|g_F|)}{\sqrt{2}{B}_{\text{RF},\text{sgn}(g_F)}}\right).$$

$$\hat{\bar{H}}=\sum _{\text{F}}{\widehat{\text{mathds}1}}_F\left[E_F+\hslash {\Omega }_{\text{RF}}^F{\hat{F}}_z\right],$$

$$\hslash {\Omega }_{\text{RF}}^F={\mu }_{\text{B}}{g}_F\sqrt{{\left(B_{\text{DC}}-\frac{g_F}{|g_F|}\frac{\hslash {\omega }_{\text{RF}}}{{\mu }_{\text{B}}{g}_F}\right)}^2+2{\left|B_{\text{RF},\text{sgn}(g_F)}\right|}^2}.$$

$$|F,\bar{m}⟩=\sum _{m=-F\dots F}{e}^{-i\text{sgn}(g_F)\bar{m}{\omega }_{\text{RF}}t}{d}_{m\bar{m}}^F({\theta }_F)|F,m⟩,$$

$$d_{m^{\prime },m}^F(\theta )=⟨F,m^{\prime }|e^{-i\theta {\hat{F}}_y}|F,m⟩,$$

$${\hat{U}}^{†}\hat{V}\hat{U}$$

$${\hat{\bar{H}}}_{\text{MW}}^{\sigma }={\hat{U}}_y^{†}({\theta }_{F+1},{\theta }_F){\hat{U}}_z^{†}(t){\hat{H}}_{\text{MW}}^{\sigma }{\hat{U}}_z(t){\hat{U}}_y({\theta }_{F+1},{\theta }_F),$$

$$⟨F+1,{\bar{m}}^{\prime }|{\hat{\bar{H}}}_{\text{MW}}^{\sigma }|F,\bar{m}⟩$$

$$B_{\text{MW}}^{\ell ,\sigma }(t)=\left(B_{\text{MW},\sigma }\left(\frac{1+\sigma \ell }{2}\right)+B_{\text{MW},\sigma }^{\ast }\left(\frac{1-\sigma \ell }{2}\right)\right)e^{-i\sigma \ell {\omega }_{\text{MW}}t}+(1-|\sigma |){\delta }_{\ell ,0}\left(B_{\text{MW},\sigma }{e}^{-i{\omega }_{\text{MW}}t}+B_{\text{MW},\sigma }^{\ast }{e}^{i{\omega }_{\text{MW}}t}\right).$$

$${\hat{\bar{H}}}_{\text{MW}}=\sum _{n,\sigma }{\hat{\bar{H}}}_{\text{MW}}^{\sigma ,n}{e}^{-i({\omega }_{\text{MW}}+n{\omega }_{\text{RF}})t}+{\hat{\bar{H}}}_{\text{MW}}^{\sigma ,n†}{e}^{i({\omega }_{\text{MW}}+n{\omega }_{\text{RF}})t},$$

$$⟨F+1,{\bar{m}}^{\prime }|{\hat{\bar{H}}}_{\text{MW}}^{\sigma ,n}|F,\bar{m}⟩$$

$${\omega }_{\text{MW}}+n{\omega }_{\text{RF}}={\omega }_{\text{hfs}}+\bar{m}{\Omega }_{\text{RF}}^F+{\bar{m}}^{\prime }{\Omega }_{\text{RF}}^{F+1},$$

$$S_2\propto n_{\bar{m}}(F(F+1)-3{\bar{m}}^2)$$

$$A\propto \frac{{\Omega }^2}{{\Omega }^2+(\Delta -{\Delta }_c{)}^2}{\sin }^2\frac{\sqrt{{\Omega }^2+(\Delta -{\Delta }_c{)}^2}t}{2}$$

$$V_{|F,\bar{m}⟩}\left(\mathbf{r}\right)=s\left(I+\frac{1}{2}\right)\frac{\hslash {\omega }_{\text{hfs}}}{2}+s\bar{m}\hslash \sqrt{{\delta }_F^2+{\Omega }_{\text{RF}}^2\left(\mathbf{r}\right)}+Mgz,$$

$$d^1({\theta }_1)=\left(\begin{array}{cccc}\frac{\cos \left({\theta }_1\right)+1}{2}& -\frac{\sin \left({\theta }_1\right)}{\sqrt{2}}& \frac{1-\cos \left({\theta }_1\right)}{2}& \\ \frac{\sin \left({\theta }_1\right)}{\sqrt{2}}& \cos \left({\theta }_1\right)& -\frac{\sin \left({\theta }_1\right)}{\sqrt{2}}& \\ \frac{1-\cos \left({\theta }_1\right)}{2}& \frac{\sin \left({\theta }_1\right)}{\sqrt{2}}& \frac{\cos \left({\theta }_1\right)+1}{2}& \end{array}\right)$$

Full text

I Introduction

The recent developments from the precise control of cold atoms Chu (1998); Phillips (1998); Cornell and Wieman (2002); Ketterle (2002) have paved the way to many breakthrough experimental and theoretical results Arndt et al. (2012); Amico et al. (2017). These span a range which runs from fundamental to applied physics, including quantum simulation Bloch et al. (2008), atom interferometry Cronin et al. (2009); Navez et al. (2016), high precision atomic clocks Treutlein et al. (2006); Sárkány et al. (2014) and sensitive compact quantum sensors Böhi and Treutlein (2012); Becker et al. (2018). Amongst these developments, radio-frequency (RF) and microwave (MW) dressing Garraway and Perrin (2016); Perrin and Garraway (2017) have provided the means to generate new types of control and trapping potentials for cold atoms. By combining magnetic fields at different frequencies, from DC to RF and MW, one can create highly non-trivial potential landscapes. These can have complex geometries that are robust against low-frequency environmental noise Treutlein et al. (2006); Morizot et al. (2008) and can also be transformed and manipulated adiabatically Garraway and Perrin (2016); Böhi et al. (2009). This provides a versatile platform to investigate the physics of non-trivial topologies, e.g. shell potentials Colombe et al. (2004), multiple nested shell potentials Harte et al. (2018), toroidal surfaces Fernholz et al. (2007) and ring-shaped structures Fernholz et al. (2007); Sherlock et al. (2011); Navez et al. (2016); Pandey et al. (2019); Lesanovsky and von Klitzing (2007). The dressed manifolds of different hyperfine states can often be coupled and manipulated independently Sinuco-León and Garraway (2016). This together with the robustness to temporal and spatial noise Trebbia et al. (2007), makes dressed potentials an ideal candidate for an interferometric, or general atomtronic, platform Böhi et al. (2009); Hofferberth et al. (2007); Berrada et al. (2013); Stevenson et al. (2015); Amico et al. (2017). However, the complexity of these potentials means that when additional fields are used to probe an atom, many new transition lines are found. This rich spectral panorama forms the subject of this paper.

We present an experimental and theoretical study of the response of RF-dressed atoms of 87Rb to MW radiation for the full range of relevant microwave frequencies. We identify qualitatively and quantitatively how the microwave spectrum emerges from probing the RF-dressing, and observe the signatures of the spectrum in three common experimental situations. In the following Sec. II there is a theoretical description of the internal dynamics of alkali-like atomic systems driven by one radio-frequency and one microwave field in the limit of a linear Zeeman shift and a weak RF field. We then present experimental results corresponding to three different scenarios: freely falling atoms (Sec. III.1), atoms in an optical dipole trap (Sec. III.2), and atoms in an RF-dressed shell trap (Sec. III.3). In each case, we describe the main features of the microwave spectrum and compare them with our theoretical model. Finally, in our closing section (Sec. IV), we provide a general outlook of our findings and comment on future applications.

II Interaction of an alkali atom with radio-frequency and microwave magnetic fields

The internal dynamics of an alkali atom in its electronic ground state interacting with a weak, time-dependent magnetic field $\mathbf{B}(t)$ are governed by the Hamiltonian:

[TABLE]

where $A$ is a hyperfine structure constant, and $\mu_{\text{B}}$ is the Bohr magnetron. The factors $g_{I}$ and $g_{J}$ are the nuclear and electronic $g$-factors, respectively They have the corresponding angular momentum operators $\hat{\mathbf{I}}$ and $\hat{\mathbf{J}}$.

Here we consider a magnetic field with three contributions: a time-independent (DC) part and two harmonically oscillating components at radio-frequency (RF) and microwave (MW) frequencies:

[TABLE]

Without loss of generality, we choose a quantization axis (unit vector $\mathbf{e}_{z}$) along the direction of the static field of strength $B_{\textrm{DC}}$.

For zero external magnetic field, the coupling between the nuclear and electronic magnetic moments (with quantum numbers $I$ and $J=L\pm S$) defines two hyperfine manifolds with different total angular momentum and corresponding quantum number $F=I\pm J$, which are split by an energy gap of $\Delta E_{\text{hfs}}=AJ(2I+1)$. The static component of the field, $B_{\textrm{DC}}$, lifts the degeneracy within each hyperfine manifold (Zeeman splitting). When the hyperfine splitting is much larger than the energy associated with the applied magnetic fields, that is, $\Delta E_{\text{hfs}}\gg\mu_{\text{B}}(B^{2}_{\text{DC}}+|\mathbf{B}_{\textrm{RF}}|^{2}+|\mathbf{B}_{\text{MW}}|^{2})^{1/2}$, the total angular momentum $F$ remains a good quantum number, and the atomic spectrum can be conveniently described in the basis $\{\left|F=I+J,m_{\text{F}}\right\rangle\}\oplus\left|F=I-J,m_{\text{F}}\right\rangle\}$, with $m_{\text{F}}={-F,...,F}$ Mischuck et al. (2012).

In this basis, the static part of the Hamiltonian Eq. (1) can be linearly approximated as

[TABLE]

where we have defined partial identity operators to project onto the hyperfine manifolds,

[TABLE]

and we have used the property $[\hat{\mathds{1}}_{F},\hat{F}_{z}]=0$. Energies and $g_{F}$-factors for the two manifolds are given by

[TABLE]

and

[TABLE]

(e.g. see Steck and Refs. [17,20,25] therein).

The arrangement of energy levels and coupling fields is illustrated for the 87Rb ground state in Figure 1 for the example of a $\pi$-polarized MW field. In the case of 87Rb ($I=3/2$), the two $g_{F}$-factors given by Eq. (5) are $g_{1}=-0.50182671$ for the lower manifold and $g_{2}=0.49983642$ for the upper manifold.

The two time-dependent terms in Eq. (2) oscillate at frequencies close to the resonance condition and cause two different types of transitions written as $\left|F^{\prime},m_{\text{F}}^{\prime}\right\rangle\leftrightarrow\left|F,m_{\text{F}}\right\rangle$. The radio-frequency field, $\mathbf{B}_{\textrm{RF}}(t)$ oscillates at a frequency of the order of the Zeeman splitting, $\omega_{\textrm{RF}}\sim|g_{F}|\mu_{\text{B}}B_{\textrm{DC}}/\hbar$, which is typically in the range of $10$s kHz – MHz. It is convenient to represent the corresponding atom-field interaction term in the basis of total angular momentum ${\left|F,m_{\text{F}}\right\rangle}$. In this basis, the hyperfine interaction $A\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}$ splits the energy spectrum in two blocks of Zeeman sub-states of total angular momentum $F=|I\pm J|$, and transitions within each block, corresponding to $|F-F^{\prime}|=0$, are near-resonantly coupled by the RF field. This part of the Hamiltonian can be approximated by a term $\hat{H}_{\textrm{RF}}\propto g_{F}\mathbf{B}_{\textrm{RF}}\cdot\hat{\mathbf{F}}$.

The microwave field $\mathbf{B}_{\text{MW}}(t)$ oscillates at a frequency of the order of the hyperfine splitting: $\omega_{\text{MW}}\sim|E_{I+J}-E_{|I-J|}|/\hbar$, which for alkali atoms ranges between $0.2$ - $10$ GHz Ginges et al. (2017); Steck . In this case, the couplings between blocks of states defined by the hyperfine coupling are resonant and the MW field leads to transitions between states belonging to different hyperfine manifolds such that $|F-F^{\prime}|=1$. For this part of the Hamiltonian, we neglect the small nuclear magnetic moment due to $g_{I}\ll g_{J}$, and approximate it by a term of the form $\hat{H}_{\text{MW}}\propto g_{J}\mathbf{B}_{\text{MW}}\cdot\hat{\mathbf{J}}$, see below.

The two oscillating fields can be expressed in spherical polarization components defined with respect to the direction of the static field Perrin and Garraway (2017) as

[TABLE]

Here we let $\textrm{AC}\rightarrow\textrm{RF,MW}$ and we have used the definitions Perrin and Garraway (2017)

[TABLE]

where $\phi_{i}$ represent the phases of the $i$th component of the AC field. Using this parametrisation of the fields and taking into account the range of frequencies of each component, the RF and MW interaction Hamiltonians are given by:

[TABLE]

where the raising and lowering angular momentum operators are defined by $\hat{F}_{\pm}=(\hat{F}_{x}\pm i\hat{F}_{y})$, with similar expressions for the electronic angular momentum $\hat{J}_{\pm}$. The factors $\eta_{+1}=-1/\sqrt{2}$, $\eta_{-1}=1/\sqrt{2}$, $\eta_{0}=1$ follow from our definitions in Eq. (8).

In the next section, we describe how the Rotating Wave Approximation (RWA) leads to an approximate description of the internal dynamics of alkali atoms subjected to this bi-chromatic field.

II.1 RF-dressing in the Rotating Wave Approximation

Let us first consider the case where there is no microwave field, i.e. $\mathbf{B_{\text{MW}}}(t)=0$. Then the Hamiltonian becomes,

[TABLE]

where $\hat{H}_{0}$ is defined in Eq. (3) and $\hat{H}_{\textrm{RF}}$ is given by Eq. (9). The resulting dynamics can be described in the dressed basis, i.e. by moving to a rotating frame where the most relevant component of the field becomes time-independent, and diagonalisation of the resulting Hamiltonian becomes analytically tractable. More specifically, we describe the driven atom in the rotating frame of reference that follows from the unitary transformation

[TABLE]

which corresponds to geometric rotations about the $z$-axis at frequency $\omega_{\textrm{RF}}$, but in opposite directions due to the opposite sign of the $g_{F}$ factors. In the rotating frame, the Hamiltonian (11) becomes:

[TABLE]

where we have neglected inter-manifold couplings and applied the Rotating Wave Approximation (RWA), which consists of neglecting time-dependent terms oscillating at angular frequency $\pm 2\omega_{\textrm{RF}}$. This procedure is valid as long as the processes associated with these terms are far from being resonant. The RF-dressed states are defined as the eigenstates of Eq. (LABEL:eq:H_RF_Rot), which can be obtained by performing a second (time-independent) rotation within each hyperfine manifold,

[TABLE]

where

[TABLE]

The resonance condition $\mu_{\text{B}}|g_{F}|B_{DC}=\hbar\omega_{\textrm{RF}}$ depends on $F$ and is shifted by $(g_{2}+g_{1})\mu_{\text{B}}B_{\text{DC}}$ (or $h\cdot 2.78666$ kHz/G in the case of 87Rb), which causes a small difference in the shape of the dressed potentials, as we will see below.

In the basis of RF dressed states, the Hamiltonian $\hat{\bar{H}}=\hat{U}_{y}^{\dagger}\hat{\tilde{H}}\hat{U}_{y}$ becomes

[TABLE]

with the Rabi frequencies, $\Omega_{\textrm{RF}}^{F}$, defined by:

[TABLE]

With this construction, the dressed states are defined as a time-dependent superposition of Zeeman states, i.e. they can be expressed in the bare basis as:

[TABLE]

where $d^{F}_{m\bar{m}}(\theta)$ is the Wigner $d$-matrix,

[TABLE]

which represents the rotation of the operator $\hat{U}_{y}(\theta)$. In the case of 87Rb, the nuclear angular momentum $I=3/2$ implies that the ground state manifold splits into two hyperfine manifolds of total angular momentum $F=1,2$, with Hilbert space dimensions $3$ and $5$, respectively. Values for the $d^{F}_{m^{\prime},m}(\theta)$ for rotations about the $y$-axis are presented in matrix form in Appendix A. In combination with time-dependent factors in Eq. (18), matrices (30) and (31) give us the time-dependent relation between the bare and dressed representations.

When dealing with problems restricted to a single hyperfine manifold a simpler treatment is possible Perrin and Garraway (2017). The unitary transformation to the basis of RF dressed states can then be expressed in terms of separate spatial rotational matrices, exploiting the equivalence between spin and spatial rotations for interactions of the form $\hat{V}=\boldsymbol{\mu}\cdot\hat{\mathbf{F}}$. More concretely, in a rotating frame reached by the unitary transformation $\hat{U}=\exp(-\theta\hat{n}\cdot\hat{\mathbf{F}})$, the interaction can be obtained using the Baker-Campbell-Hausdorff Lemma:

[TABLE]

where $R_{\hat{n}}(-\theta)$ is a $3\times 3$ matrix corresponding to the rotation by an angle $-\theta$ around the axis aligned in the direction of $\hat{n}$ Fernholz et al. (2007); Jammi et al. (2018). Here, we are concerned with couplings between RF dressed manifolds with different total angular momentum and therefore it is more convenient to use the transformation between the Zeeman and dressed bases as given by Eqs. (18)-(19).

II.2 MW coupling of RF-dressed states in the Rotating Wave Approximation

RF-dressed states of the electronic ground state of an alkali atom can be prepared by starting in bare states and adiabatically tuning into resonance with the dressing field. The resonance frequency is given by the Zeeman splitting, which corresponds to \omega_{\text{RF}}\sim 2\pi\times 0.70\leavevmode\nobreak\kHz per Gauss for 87Rb. In this section, we study how a coherent superposition of RF-dressed states of the two hyperfine manifolds can be prepared by a applying a second field with a frequency set by the hyperfine splitting, which corresponds to \omega_{\text{MW}}\sim 2\pi\times 6.834\leavevmode\nobreak\GHz for 87Rb.

This problem can be studied in the context of the response of continuously driven quantum systems, which has been the subject of theoretical and experimental study over several decades Jammi et al. (2018); Autler and Townes (1955); Series (1978). The experimental observations of the spectrum of off-resonant RF-dressed states made by Haroche and Cohen-Tannoudji can be understood using perturbative expansions of driven two-level systems (TLS) Haroche et al. (1970); Cohen-Tannoudji and Haroche (1966); Haroche and Cohen-Tannoudji (1969); Zanon-Willette et al. (2012); Beaufils et al. (2008). In addition, more recent experiments demonstrate that the modified response of resonantly RF-dressed alkali atoms to MW fields enables the encoding and manipulation of qudits exploiting the full complexity of the hyperfine manifold Mischuck et al. (2012); Smith et al. (2013), and going beyond the TLS paradigm. In this section we explain how the response of RF-dressed 87Rb to a MW field can be obtained by applying a second rotating wave approximation (for the MW field), which allows us to calculate selection rules, resonant conditions and coupling strengths.

Similar to the RF case, the interaction with the MW field has contributions from both the nuclear and electronic magnetic moments. However, since the nuclear gyromagnetic factor ($g_{I}=-0.000995$) is three orders of magnitude smaller than the electronic one ($g_{J}=2.002319$), within the RWA it is sufficient to consider only the electronic coupling in Eq. (10). When the atoms are continuously dressed by an RF field, the microwave field induces transitions between the dressed states defined by Eq. (18), which can be obtained by expressing the interaction $\hat{H}_{\text{MW}}$ in the dressed basis. Explicitly, this calculation corresponds to finding Sinuco-León and Garraway (2016)

[TABLE]

where $\hat{H}^{\sigma}_{MW}$ is the contribution of the field component with polarization $\sigma$ to the MW interaction Eq. (10), and the rotations are defined for each of the hyperfine manifolds. After some algebraic manipulation (see Appendix B), the matrix elements of the MW coupling are given in the dressed basis by

[TABLE]

where $\eta_{+1}=-1/\sqrt{2}$, $\eta_{-1}=1/\sqrt{2}$, $\eta_{0}=1$, and with the standard notation for the 3-j Wigner coefficients. We also use the Wigner $d$-matrix defined in Eq. (19), and the definition

[TABLE]

Due to the transformation to the (counter) rotating frame(s), a single frequency microwave field will appear modulated, which gives rise to fictitious sidebands. According to Eq. (22) the MW driving between dressed states causes coupling terms with angular frequencies equal to $\omega_{\text{MW}}$ plus multiples of the RF dressing frequency, $n\omega_{\textrm{RF}}$. This lets us split the interaction into contributions from each MW polarization ($\sigma$) at different frequencies in the form Sinuco-León and Garraway (2016)

[TABLE]

with $n\in[-2I,2I]$, $\sigma\in[-1,1]$ and the matrix elements defined by Eqs. (22,23).

The coefficients $B_{\textrm{MW}}^{\ell,\sigma}$ defined in Eq. (23) lead to several relations between the matrix elements that depend on the polarization of the MW field but not on the RF dressing configuration. They give rise to a structure that reproduces the bare microwave spectrum. The $\pi$-polarised component of the MW field enables coupling at even sidebands, i.e. for oscillatory terms of MW frequency plus even multiples of $\omega_{\textrm{RF}}$. Similarly, the $\sigma_{\pm}$-polarised components enable coupling at the MW frequency plus odd multiples of $\omega_{\textrm{RF}}$, but not for the respective extremal $\omega_{\textrm{MW}}\pm(2F+1)\omega_{\textrm{RF}}$. (Note that an apparent positive sideband allows for red-detuned driving in the laboratory frame.)

In general, the coupling between dressed states depends on the RF dressing configuration via the Wigner $d$-matrices. However, in agreement with symmetry considerations and conservation of the angular momentum of the atom plus radiation system, the matrix elements of each contribution to Eq. (24) satisfy the relation:

[TABLE]

The MW couplings in the RF dressing configuration must meet the resonance conditions

[TABLE]

with $n\in[-2(F+1)-1,2F+1]$ and $\bar{m},\bar{m}^{\prime}\in Z$ and $\Omega_{\textrm{RF}}^{F}$ defined in Eq. (17). On the left hand side of Eq. (26) we have the oscillating frequency of the MW field observed in the dressed frame of reference, while on the right hand side we have written the quasi-energy difference between pairs of dressed states {$\left|F\bar{m}\right\rangle$, $\left|F+1,\bar{m}^{\prime}\right\rangle$}.

In Figure 2 we depict schematically the MW spectrum of resonantly RF-dressed 87Rb, considering as the initial state each one of the dressed sub-levels of the lower hyperfine manifold $F=1$, and the three possible MW polarizations. In this case, there are $105$ potential transition frequencies corresponding to $3\times 5=15$ different pairs of states in the lower and upper hyperfine manifolds, coupled by terms oscillating at the $7$ different frequencies $\omega_{\text{MW}}+n\omega_{\textrm{RF}}$ with $n\in-3,3$. Resonant frequencies are given by Eq. (26) and the MW couplings are calculated with Eq. (22), considering resonant RF-dressing and neglecting the difference between gyromagnetic factors. An explicit form of the couplings for 87Rb is presented in extended form in Appendix C.

Groups of resonant transitions between RF-dressed states can be labelled by the integer multiplier $n$ of the RF angular frequency in the resonant condition Eq. (26). As a consequence of the conservation of angular momentum (see Eq. (42) in Appendix C), transitions in the even and odd groups are induced by $\pi$- and $\sigma_{\pm}$-polarised MW radiation, which is reminiscent of the MW transitions of bare atoms.

The analysis presented above applies to the electronic ground state of alkali-metal atoms and alkali-metal-like ions Ginges et al. (2017), with the total number of possible transitions and groups defined by the nuclear total spin (and then the ranges of $\bar{m},\bar{m}^{\prime}$ and $n$ in Eq. (26)). For instance, the MW spectrum of the RF-dressed bosonic species 87Rb, 39K, 23Na and 7Li, present the same number of resonances since the ground state of all of them is split in the manifold $F=2$ and $F=1$, though the resonant frequencies are determined by their fine and hyperfine constants.

III Microwave spectroscopy of RF-dressed rubidium-87

A typical experimental sequence describing the general outline for all three experimental scenarios presented in this section is shown in Fig. 3, with the eigenenergies of the 87Rb hyperfine sub-levels at different stages of the sequence. We first examine the MW spectrum of freely falling clouds prepared selectively in one of the three dressed states of the $F=1$ manifold (Section III.1). In a second experiment (Section III.2), the spectrum is obtained for 87Rb atoms in the dressed $|1,-1\rangle$ state trapped in an optical potential, with particular focus on the group of transitions corresponding to $n=1$, as defined in Eq. (26). The third experimental configuration studies the MW spectrum of atoms confined in an RF-dressed shell trap (Section III.3), where effects of the inhomogeneity of the field distribution play an important role. The experimental details for the different dressing configurations are presented in the following sections III.1–III.3, including analysis and discussion of the observed spectroscopic measurements.

III.1 Free-falling atoms in homogeneous fields

Using free-falling ensembles of 87Rb atoms released from a magneto-optical trap (MOT) allows us to apply nearly homogeneous magnetic fields to otherwise unaffected atoms. By preparing pure dressed states and using a dispersive detection method to obtain state-dependent signals Jammi et al. (2018) we are able to attribute spectroscopic features to individual transitions. The state preparation sequence, shown in Fig. 4, is performed after optical molasses cooling and optical hyperfine pumping with initial atomic population in all five Zeeman sub-levels of $F=2$. We apply a MW $\pi$-pulse in a weak, homogeneous magnetic field ($\approx 1\leavevmode\nobreak\ \mathrm{G}$) by driving coherent Rabi cycles on one of the bare $\pi$-transitions. These transitions are non-degenerate due to the opposite sign of the $g$-factors in the two hyperfine states and thus frequency selective. This allows us to populate a single Zeeman sub-level in the $F=1$ manifold, i.e. only one of the states $|F=1,m=\pm 1,0\rangle$, see the example in Fig. 4a. For each of the three $\pi$-transitions, we adjust the pulse duration to maximise population in the target state. Subsequently, the population in the $F=2$ manifold is removed by shining a resonant laser beam tuned to the $F=2\rightarrow F^{\prime}=3$ transition of the $D_{2}$-line (Fig. 4b). Multiple photon scattering on this closed transition accelerates atoms away from the observed volume. Finally, the remaining atoms in the pure bare state are adiabatically dressed by ramping up the RF-field amplitude and tuning the atomic Larmor frequency near resonance using the static field amplitude, see Fig. 4c. For this set of experiments, we work in the weak field regime using a dressing field amplitude of $B_{\mathrm{RF}}\approx 10\leavevmode\nobreak\ \mathrm{mG}$ at a frequency of $\omega_{\textrm{RF}}=2\pi\times 180\leavevmode\nobreak\ \mathrm{kHz}$, resonant for a static field of $B_{\mathrm{DC}}\approx 257\leavevmode\nobreak\ \mathrm{mG}$. The final dressed state is typically populated by $n_{m}\approx 3\times 10^{7}$ atoms.

The spectroscopy is performed by first applying a weak MW pulse, typically a few ms long, which may couple the prepared initial dressed state in the $F=1$ manifold to one of the five dressed states in the $F=2$ manifold, depending on the frequency of the MW pulse. The atomic response is then recorded by observing the AC-modulated linear birefringence of the ensemble, which we can measure separately for both hyperfine states using two laser beams and a balanced polarimeter Jammi et al. (2018). An ensemble of atoms in a (bare) Zeeman state $|F,m\rangle$ will exhibit a linear birefringence $S$ proportional to atom number $n_{m}$. The birefringence depends quadratically on the magnetic quantum number $m$ and may change sign according to $S\propto n_{m}(F(F+1)-3m^{2})$. Adiabatic dressing of the atoms modulates the linear birefringence of the ensemble, and depending on laser detuning and experimental geometry, we can detect a signal

[TABLE]

at the second harmonic of the dressing frequency, where sign and amplitude now depend on the adiabatic quantum number $\bar{m}$.

Depending on the polarization of the MW field, we observe up to seven main groups of dressed hyperfine transitions. As can be seen in Fig. 5, each group is centred around one of the bare hyperfine transition frequencies, which are separated by the dressing frequency of \omega_{\textrm{RF}}=2\pi\times 180\leavevmode\nobreak\kHz. The appearance of the groups depends on the polarization of the MW field and resembles the bare scenario, with three groups emerging for $\pi$-polarization (i.e. $B_{\textrm{MW}}$ aligned with the static field $B_{\textrm{DC}}$), and four groups for linear $\sigma$-polarization (i.e. $B_{\textrm{MW}}$ orthogonal to $B_{\textrm{DC}}$). The frequencies of individual transitions are in good agreement with the theoretical prediction from Eq. (26). The individual peak heights and widths of the experimental data are not a direct reflection of the transitions’ coupling strengths due to their dependence on various experimental settings. The widths of these lines are determined by a combination of MW power broadening, residual field inhomogeneities and magnetic field noise. The experimental data shows some transitions that are predicted to vanish according to the approximation $g_{1}=-g_{2}$ that was used to produce the theoretical spectrum shown in Fig. 2. These transitions are observable because the small difference in the magnitude of the Landé factors $g_{1}$ and $g_{2}$ and detuning(s) from RF resonances lead to non-zero coupling coefficients, see Eq. (22).

In our experiment, the population signals from the different $F=2$ levels scale relative to each other by a factor given by Eq. (27). The peak heights are not directly indicative of the transition strengths as the MW pulse of fixed duration (0.4 ms) induces Rabi cycles of differing frequencies for each transition and results in a different population fraction in $F=2$ depending on the number of Rabi cycles on each transition. The data for the three initial states differ in strength due to variations in the experimental state preparation efficiency, and the $\pi$- and $\sigma$- polarization data sets may be subject to variations in external experimental conditions as these were taken at different times.

The set of transitions corresponding to the group of resonances in the vicinity of $\omega_{\textrm{hfs}}+3\times\omega_{\textrm{RF}}$ is shown in Fig. 6a. As before, atoms prepared in each of the initial three states give rise to five resonant transitions separated in frequency by the RF Rabi frequency (\approx 10\leavevmode\nobreak\kHz). The strength of the signal reflects not only the MW transition strength, but also carries a signature of the populated target state in $F=2$ according to Eq. (27), which explains why signals from transitions to $|F=2,m=\pm 2\rangle$ are negative in sign. This spectrum was acquired with low MW power in order to significantly reduce the effect of power broadening on the transition peaks. The width of the spectral lines in this case is a consequence of homogeneous broadening due to field noise and inhomogeneous broadening due to magnetic field gradients for all but the three sharpest peaks. The sharp resonances in each group, shown in Fig. 6b, correspond to the transitions $|1,-1\rangle\rightarrow|2,1\rangle$, $|1,0\rangle\rightarrow|2,0\rangle$ and $|1,1\rangle\rightarrow|2,-1\rangle$. These transitions are least affected by the fields, because states in each pair experience (almost) equal magnetic shifts due to near identical factors $g_{F}m_{F}$ for the involved states. A small frequency splitting between these transitions remains due to the marginal difference in magnitude of the $g_{F}$ factors. As a result, these lines are coherently driven, with theoretical line shapes of the form

[TABLE]

where $A$ is the $F=2$ signal amplitude, $\Omega$ is the MW Rabi frequency, $\Delta_{c}$ is the centre frequency, $t$ is the pulse duration of 5 ms and $\Delta=\omega_{\textrm{MW}}-\omega_{\text{hfs}}$ Foot (2004).

In principle, the Rabi frequencies extracted from the least squares fit using Eq. (28) should allow for a comparison with the theory. Under the approximation $g_{1}=-g_{2}$, the theoretically predicted ratio of resonant coupling strengths is 1 : $-\sqrt{3}$ : 1 for the pairs with $\bar{m}=\mp 1,0,\pm 1$, respectively, see outer groups in Fig. 2. These ratios are qualitatively reflected by the experimental data. However, experimental uncertainties in the relative populations of the initial states as well as in the signal scale prohibit an accurate determination from just the line shapes.

III.2 87Rb in an optical dipole trap

In the second set of experiments we confine the atoms in a crossed-beam optical dipole trap, which allows us to work at high field strengths and address all dressed states in a trapped scenario. The preparation sequence begins by loading an atom cloud from a MOT into a magnetic quadrupole trap, where it is compressed and evaporatively cooled. This is followed by further compression and evaporation in the crossed-beam dipole trap ($\lambda=1064$ nm, P=1.8 W , final axial and radial trapping frequencies $\omega_{\text{z}}/2\pi\approx 180$ Hz and $\omega_{\rho}/2\pi\approx 30$ Hz). This yields a fully polarized sample of approximately $3\times 10^{5}$ atoms at 50 nK in the bare state $|1,-1\rangle$. At this stage, a vertical bias field $B_{\textrm{DC}}\mathbf{e}_{z}$ is ramped from zero up to $|g_{F}|\mu_{\text{B}}B_{\textrm{DC}}\approx 1.4\,\hbar\omega_{\textrm{RF}}$, where $\omega_{\textrm{RF}}$ is the RF frequency of the dressing field that will be applied. We then switch on an RF-dressing field of frequency $\omega_{\textrm{RF}}$, which is linearly polarised along $\mathbf{e}_{x}$, and subsequently dress the cloud by adiabatically ramping down $B_{\textrm{DC}}$ until a near-resonant condition $|g_{F}|\mu_{\text{B}}B_{\textrm{DC}}\approx\hbar\omega_{\textrm{RF}}$ is reached in \Delta t=200\leavevmode\nobreak\ms. The spectroscopy is performed by shining a microwave pulse of duration $\Delta t_{\textrm{MW}}=0.7$ ms, followed by a short free-expansion of typically 5 ms, right after all AC fields are switched off. This is followed by absorption imaging adapted for simultaneous recording of the atoms transferred to the $F=2$ manifold and atoms remaining in the $F=1$ manifold.

The RF-fields are produced by a pair of Helmholtz coils such that the generated magnetic field points along $\mathbf{e}_{x}$. We generate the MW field with a tuned dipole antenna placed in the $x$-$y$ plane, forming an angle of $45^{\circ}$ with the $\mathbf{e}_{x}$ axis as we sketch in Fig. 7. The antenna was aligned to produce a MW-field linearly polarized in the $x$-$y$ plane, at $45^{\circ}$ from the $x$-axis and orthogonal to $B_{DC}$. The finite amplitude of the even groups in the MW spectroscopy results (Fig. 8) suggest that the MW-field polarization is not exactly orthogonal to $B_{\text{DC}}$ because of reflections from neighbouring metallic surfaces. The duration of the MW radiation pulse $\Delta t_{\textrm{MW}}$ was chosen to be much shorter than a $\pi/2$ pulse for the strongest transition. This allows direct comparison with the theoretical predictions for weak MW-fields from Section II.2.

As in the case of the free-falling atoms (Section III.1), when the atoms are dressed and trapped in a crossed dipole potential, we observe seven groups of five transitions (for the initial state $|1,-1\rangle$) with variable couplings that depend on the configuration of the magnetic fields. Fig. 8 shows the full measured spectrum starting with a cloud prepared in the dressed state $|1,-1\rangle$ together with the theoretical prediction from Eqs. (22) and (26). The measured spectrum is for the field configuration described above. In this case, the MW antenna is oriented such that it produces a MW field that lies in the plane of the RF-field, mostly orthogonal to the static magnetic field. As a result of this MW-polarization, when we scan the MW frequency, the number of atoms transferred to the upper hyperfine manifold for the even groups is significantly smaller compared to the number of atoms transferred for the odd groups.

The vertical scale of Fig. 8 shows the fraction of atoms transferred to the upper states starting from $F=1$. This is calculated from a separate measurement of the total atom number in the sample, with $\omega_{\textrm{RF}}/2\pi=2.27$ MHz. Quantitative agreement between the experimental results and the theoretical values is limited by other experimental factors not considered in this analysis: e.g. atomic losses, and drifts in the RF amplitude or in the homogeneous magnetic fields. Nevertheless, there is a good agreement between the theoretical predictions of the transition frequencies in Eqs. (22) and (26) with our experimental results. In particular, the peaks corresponding to the $\pi$-polarised component of the MW field are well reproduced by our theory, with qualitative agreement for the the circularly polarized components.

These findings motivate the use of MW spectroscopy as a tool to determine the field configuration driving the atomic cloud. In order to test this idea, we took a spectrum of the group of resonances in the vicinity of $\omega_{\text{hfs}}+\omega_{\textrm{RF}}$ using $\omega_{\textrm{RF}}/2\pi=2.26341$ MHz and $\Delta t_{\textrm{MW}}=0.7$ ms. Scanning the microwave frequency, we directly determine the transition probability by measuring the population of both hyperfine manifolds after the MW pulse. Calculating the numerically exact atomic time-evolution Sinuco-Leon and Garraway (2019), we adjust the components of all applied fields to get the best fit to the experimental results. We also adjust all three components of the static field since the Earth’s magnetic field adds components in the $x$-$y$ plane in our set-up. We fit the $x$ and $y$ components of the microwave field because they produce significant couplings in the range of frequencies tested. The RF antennas are oriented to produce a RF field linearly polarised in the $x$ direction. The Table 1 shows the value of the parameters adjusted and Fig. 9 shows a comparison of the experimental data with the fit. This procedure yields a measurement of the MW field amplitude $B_{\textrm{MW,x}}$ with a precision of approximately $10^{-2}$ and the error on $B_{\textrm{RF,x}}$ is of order $10^{-3}$. These errors depend on the knowledge of the DC-magnetic fields and the precision of the transition frequency measurement, which becomes worse for broader and more noisy line-shapes.

III.3 87Rb in an RF-dressed shell trap

We produce an RF-dressed shell trap Garraway and Perrin (2016); Merloti et al. (2013) by modifying the current in the DC coils of Fig. 7 so that it is now in an anti-Helmholtz configuration as in Fig. 10. We apply an RF field as before. When such an RF-dressed shell trap and a dipole trap are spatially matched through the resonant condition $|g_{F}|\mu_{\text{B}}B_{\textrm{DC}}=\hbar\omega_{\textrm{rf}}$, then the atom cloud in the dipole trap can be transferred to the shell trap by ramping up a quadrupole magnetic gradient and slowly ($\Delta t=0.5$ s) ramping down to zero the power of the dipole beams (see Fig. 10). With this method, atoms can be loaded in the dressed $|1,-1\rangle$ state adiabatically, with non-measurable atom loss or heating. The shell trap potential can be written as Zobay and Garraway (2001)

[TABLE]

with $g$ the gravitational acceleration, $M$ the atomic mass of 87Rb, and the detuning $\delta_{F}=\Omega_{\text{L}}^{F}\left(\mathbf{r}\right)-\omega_{F}$ (with $|\hbar\Omega_{\text{L}}^{F}(\mathbf{r})|=|g_{F}|\mu_{B}|B_{\text{DC}}(\mathbf{r})|$, where $\Omega_{\text{L}}(\mathbf{r})$ is the Larmor frequency) and $\Omega_{\textrm{RF}}(\mathbf{r})$ is the spatially dependent Rabi coupling Perrin and Garraway (2017). The parameter $s$ in Eq. (29) is given by $s=g_{F}/|g_{F}|$ so that $s=1$ for $F=2$ and $s=-1$ for $F=1$.

Trappable states are those where $g_{F}\bar{m}>0$. As we present in Fig. 11, this leads to state dependent traps, not only with regards to the RF-polarization coupling $g_{F}$-dependence, but also on the quadrupole-field induced $\bar{m}$-dependent force. Concretely, in Fig. 11a we show the trapping potentials for the three trappable states $|1,-1\rangle$, $|2,1\rangle$, $|2,2\rangle$. One can readily see that the traps have different curvatures and minima. In addition, in Fig. 11 we show the differences in energy $\Delta E_{1}=V_{|1,-1\rangle}-V_{|2,1\rangle}$ and $\Delta E_{2}=V_{|1,-1\rangle}-V_{|2,2\rangle}+\Omega_{0}$, which serve as an illustration of the inhomogeneous broadening related to the mismatch of the traps that a cloud of size $\Delta z$ would experience if such transitions were driven (with $\Omega_{0}$ as the Rabi frequency at the centre of the shell trap). One observes that, at the trap position $z_{0}$ of $V_{|1,-1\rangle}$ (the initial state), the curve $\Delta E_{1}$ is sloped, which is a direct result of the different $g_{F}$ factors. One can also see how the parabola-shaped curve $\Delta E_{2}$ is, firstly, not centred at $z_{0}$ (this is, again, due to the different $g_{F}$ factor); and, secondly, shows a larger curvature as $|z|$ diverges from the trap centre (this is a result of the different $\bar{m}$).

In the RF-dressed shell trap we observe the same MW spectrum structure found in Fig. 8. In this case, the trap geometry, its spatial location and the trapping frequencies are directly determined by the resonant condition of the RF-field and the DC magnetic quadrupole field, although the gravitational sag may become non-negligible. This results in state-dependent traps for any pair of initial and final states, which are in general different for different (trappable) states, as we showed in Fig. 11. As a consequence, the transition line-width may increase in the magnetic trap (compared to the optical trap) and the transferred atoms will experience higher heating rates as they are coupled via MW radiation if the traps of the initial and final states lie in different positions. Moreover, any homogeneous magnetic DC field simply translates the quadrupole in space and thus does not shift transition frequencies. This is a consequence of the fact that the trap position is fundamentally determined by the resonant condition of the quadrupole field with the RF-dressing frequency: i.e. $|g_{F}|\mu_{B}\alpha z=\hbar\omega_{\textrm{RF}}$, where $\alpha$ is the quadrupole field gradient. In Fig. 12, we show experimental measurements of the three central pairs of transitions ($\Delta t_{\textrm{MW}}=2.5$ ms) from $|1,-1\rangle$ to $|2,0\rangle$, $|2,1\rangle$ and $|2,2\rangle$ for an adiabatic magnetic potential with Rabi frequency $\Omega_{\textrm{RF}}/2\pi\approx 423\pm 2$ kHz (in $\mathbf{e}_{x}$), quadrupole gradient $\alpha=45$ G/cm and $\omega_{\textrm{RF}}/2\pi=2.22$ MHz. We have fitted simple Lorentzian curves to the spectral data after 1 ms hold time (blue). We have furthermore measured the peak optical density after 95 ms hold time for each of the transitions and we observe how the transitions from $|1,-1\rangle$ at $\omega_{\textrm{hfs}}+(n\omega_{\textrm{RF}}-\Omega_{\textrm{RF}})$ lead to the non-trapped state $|2,0\rangle$, at $\omega_{\textrm{hfs}}+n\omega_{\textrm{RF}}$ to $|2,1\rangle$ with 100 ms lifetime and at $\omega_{\textrm{hf}}+(n\omega_{\textrm{RF}}+\Omega_{\textrm{RF}})$ to $|2,2\rangle$ with a 60 ms lifetime. In both trapped states we observe significant heating due to the mismatch of the traps, being higher in the $|2,2\rangle$ case.

The line-widths are remarkably different for the three pairs of transitions because of the different overlap between the initial and final adiabatic potentials. In this experiment, the transition from $|1,-1\rangle$ to $|2,1\rangle$ is narrower ($1.5$ kHz) than the other two transitions to $|2,0\rangle$ and $|2,2\rangle$, which are broader and more noisy ($20$ kHz).

IV Conclusions

In this work we presented a complete theoretical and experimental study of the hyperfine spectrum of 87Rb dressed by an RF field. The theoretical analysis of the spectrum considers the regime of weak static and RF dressing fields. In all three experimental situations discussed, the overall features of the spectrum are well described by this analytic treatment. In particular, we found the relative position of the resonant frequencies and various selection rules associated with the polarization of the microwave probing field. In the case of free-falling atomic ensembles, the strengths of the applied fields are in the weak field regime and we identify all possible microwave transitions between pairs of radio-frequency dressed states. In this case, using the AC-modulated linear birefringence of the atomic ensemble prepared in fully polarised atomic states allows us to unambiguously assign quantum numbers and confirm the predicted value of the relative coupling strengths for all observed resonances. In the cases of atomic ensembles in the crossed-dipole and adiabatic shell traps, we used relatively strong DC ($\sim 3.1$ G) and RF fields ($\sim 0.5$ G). The number and distribution of allowed transitions remains the same as in our first experiment. However, the line-spacing is modified due to non-linear Zeeman shifts, which we include when fitting the measured spectrum. Finally, in the case of the ensemble trapped in an adiabatic shell, the nature of the RF-dressed adiabatic potential leads to small spin-dependent discrepancies in the size and curvature of the trapping potential. Even though the spectrum remains unchanged, the lifetimes and heating rates in the shell trap depend strongly on the spin-states involved in the transition.

The study and experimental observation of the MW spectroscopy in RF-dressed states is a first step towards the characterisation and implementation of several quantum optics and atom interferometry schemes, such as the matter-wave interferometry in ring traps Pandey et al. (2019); Stevenson et al. (2015) and atomic-clocks Kazakov and Schumm (2015). The experimental situations tested in this work have potential advantages for such applications. For example, trapped atomic ensembles permit interferometric sequences with long interrogation times, whereas collisions in free-falling ensembles can be exploited to increase the coherence time using spin self-rephasing Deutsch et al. (2010). In all cases, it should be possible to find optimal dressing configurations that enable robust coherent manipulations between dressed states. Also, the sensitivity of the microwave spectrum to the polarization of the RF and MW fields can be used for precise measurements. Finally, applications similar to those discussed here are currently being developed with a great variety of atomic and solid-state alkali-like systems (e.g. alkali-metals Garraway and Perrin (2016), alkali-metal-like ions Randall et al. (2015) and NV centres Golter et al. (2014)), where similar spectral signatures can be observed and explained using the theoretical framework we have presented.

The datasets generated for this paper are accessible at 10.17639/nott.7002 Nottingham Research Data Management Repository Sinuco Leon et al. (2019).

Acknowledgements

The UK authors thank the UK Engineering and Physical Sciences Research Council for support (Grant EP/M013294/1). The Greek part of this work is supported by the project HELLAS-CH (MIS 5002735), which is implemented under the Action for Strengthening Research and Innovation Infrastructures, funded by the Operational Programme Competitiveness, Entrepreneurship and Innovation (NSRF 2014-2020) and co-financed by Greece and the European Union (European Regional Development Fund). GV received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 750017. SP acknowledges financial support from the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat and Technology (GSRT), under the HFRI PhD Fellowship grant (4823).

BG, WK and TF conceived the main ideas and devised the project and WK initiated the collaboration. HM, SP, BF, SJ and KP carried out the experiments. GS, HM, SP, WK, BF, SJ, and KP contributed to the data analysis. GS, BG, HM and TF are responsible for the theoretical work. HM, SP, VB, WK, BF, SJ, KP and TF contributed to building the experiments. GS, BG, HM, SP, GV, WK, BF, SJ, KP and TF contributed to the result discussion and manuscript writing.

Appendix A Matrix representation of the operator $\hat{U}_{y}=\exp(-i\theta\hat{F}_{y})$ for total angular momentum F=1,2

For 87Rb, the RF-dressed states described in Sec. II.1 become linear superpositions of the Zeeman split states as in Eq. (18). The coefficients of such a superposition involve the Wigner $d$-matrix of Eq.(19). In the case of 87Rb these are explicitly given by:

[TABLE]

with $F=1$, and

[TABLE]

with

[TABLE]

These expressions simplify in case of resonant RF dressing where $\theta_{F}=\pi/2$.

Appendix B Matrix elements of the MW coupling in the basis of RF-dressed states

In the lab frame of reference, the polar decomposition of the MW coupling has the form

[TABLE]

with

[TABLE]

where we used the definition $\eta_{+1}=-1/\sqrt{2}$, $\eta_{-1}=1/\sqrt{2}$ and $\eta_{0}=1$. Expressed as a sum of spherical angular momentum operators, these components of the Hamiltonian can be written as

[TABLE]

with

[TABLE]

[TABLE]

with

[TABLE]

For concreteness, let’s consider an element that couples states in different hyperfine manifolds:

[TABLE]

in which we have used the identity operator of each hyperfine manifold in the lab frame $\hat{\mathds{1}}_{F}=\sum_{m}\left|F,m\right\rangle\left\langle F,m\right|$. Since the time-dependent rotation operator is diagonal in this basis we obtain

[TABLE]

Now, using the the matrix representation of the rotation $\hat{U}_{y}$ given by the Wigner $d$-matrix Messiah (2003) and rearranging the exponential factors we obtain:

[TABLE]

Now we use the matrix elements of the electronic angular momentum operators, $J_{\ell}$, defined in terms of 3-j symbols Messiah (2003) to obtain:

[TABLE]

The 3j-symbols are different from zero if and only if $-m^{\prime}+\ell+m=0$, which help us to reduce one of the sums in the following way:

[TABLE]

Putting this result together with Eq. (36), we obtain

[TABLE]

as in Eq. (22).

We can also obtain explicit expressions for the couplings associated to each polar component of the microwave field oscillating at different frequencies ($\omega_{\textrm{MW}}+n\omega_{\textrm{RF}}$), following the factorisation of the coupling matrices in Eq. (24):

[TABLE]

with

[TABLE]

Appendix C Microwave coupling of RF dressed states of 87Rb

In the limit of weak static magnetic fields, the microwave couplings between RF-dressed states are given by Eq. (22), which indicates that it is convenient to group the couplings between dressed states according to the polarization of the MW field. Taking into account the difference between gyromagnetic factors of the two ground state hyperfine manifolds we obtain the results presented below.

The RF-field is taken to be linearly polarised and perpendicular to the static field $B_{\textrm{DC}}$. The value $\Delta\bar{m}$ given in the table indicates, for 87Rb, the value of $\bar{m}+\bar{m}^{\prime}$ in Eq. (26), such that for nearly equal RF Rabi frequencies $\Omega_{\textrm{RF}}^{F}$ and $\Omega_{\textrm{RF}}^{F+1}$, we see an indication of the location of five spectral components within one of the seven groups determined by the index $n$ in Eq. (26). Following Eq. (24), the superscripts of the label $\hat{H}^{\sigma,n}_{\textrm{MW}}$ indicate the corresponding polarization ($\sigma$) and the shift of the angular frequency of oscillation of the coupling as observed in the dressed frame, i.e. $\omega_{\textrm{MW}}+n\omega_{\textrm{RF}}$. With this, the couplings with $n>0$ ($n<0$) lead to resonances red (blue) detuned with respect to the hyperfine splitting.

The tables below display the coupling between RF-dressed states normalised to the factor $\hbar\Omega_{\sigma}=\frac{1}{16}\sqrt{\frac{3}{2}}|\eta_{\sigma}|\mu_{\text{B}}g_{J}B_{\textrm{MW},\sigma}$, for the $\pi$ and $\sigma_{-}$ polar components of the MW field. The couplings associated with the $\sigma_{+}$ polarization can be obtained using relation Eq. (LABEL:eq:relations).

$\pi$ polarised MW field

$\sigma_{-}$ polarised MW field