Dispersive response of atoms trapped near the surface of an optical nanofiber with applications to quantum nondemolition measurement and spin squeezing

We study the strong coupling between photons and atoms that can be achieved in an optical nanofiber geometry when the interaction is dispersive. While the Purcell enhancement factor for spontaneous emission into the guided mode does not reach the strong-coupling regime for individual atoms, one can obtain high cooperativity for ensembles of a few thousand atoms due to the tight confinement of the guided modes and constructive interference over the entire chain of trapped atoms. We calculate the dyadic Green's function, which determines the scattering of light by atoms in the presence of the fiber, and thus the phase shift and polarization rotation induced on the guided light by the trapped atoms. The Green's function is related to a full Heisenberg-Langevin treatment of the dispersive response of the quantized field to tensor polarizable atoms. We apply our formalism to quantum nondemolition (QND) measurement of the atoms via polarimetry. We study shot-noise-limited detection of atom number for atoms in a completely mixed spin state and the squeezing of projection noise for atoms in clock states. Compared with squeezing of atomic ensembles in free space, we capitalize on unique features that arise in the nanofiber geometry including anisotropy of both the intensity and polarization of the guided modes. We use a first-principles stochastic master equation to model the squeezing as a function of time in the presence of decoherence due to optical pumping. We find a peak metrological squeezing of $\sim 5$ dB is achievable with current technology for $\sim 2500$ atoms trapped 180 nm from the surface of a nanofiber with radius $a=225$ nm.

Figure 1

Cooperativity and mode matching for various atom-light geometries. (a) The beam area at the waist of a tightly focused beam is closely matched with the atomic scattering cross section, but the scattered light of a single atom is poorly mode-matched with the probe. (b) A paraxial beam probing a rarefied atomic cloud whose scattered radiation interferes constructively in the forward direction. (c) Atoms trapped in a 1D optical lattice near the surface of an optical nanofiber interacting with a fiber-guided probe. The tight confinement and automatic mode matching that accompanies scattering into the guided mode leads to strong cooperativity in the atom-photon interaction.

Figure 2

Quantum interface for spin-polarization coupling of two one-dimensional lattices of cold, trapped atoms and the guided modes of an optical nanofiber. (a) Schematic of the interface. A linearly polarized probe is launched into the nanofiber and the output light is analyzed in a polarimeter. The atoms (green circles), trapped in the $x\text{−}z$ plane, couple to the evanescent portion of the guided $H$ and $V$ modes. Contours of the $H$- and $V$-mode intensities in (b) the $x$ direction and (c) the transverse $x\text{−}y$ plane show the mode anisotropy at the atomic positions.

Figure 3

Parameters of the atom-light interface using the clock states of ${}^{133}\mathrm{Cs}$. (a) Energy level structure for atoms probed with one of two magic frequencies on the D1-line. (b) Magnitude of the magic detunings at which the clock states are equally light-shifted, $|{\tilde{\mathrm{\Delta }}}_f|/2\pi \equiv |{\omega }_0-{\tilde{\omega }}_f|/2\pi$ (in units of MHz). There are two solutions shown as blue (dashed) and red lines. (c) The coupling strength, as measured by the magnitude of the polarization rotation angle on the Poincaré sphere, $|{\chi }_{{}_{J_3}}|$, for an atom trapped in the $x\text{−}z$ plane at a distance $r_{\perp }^{\prime }=1.8a$ from the fiber center. In both (b) and (c) we plot the parameter as the direction of the clock-state quantization axis is varied in the $x\text{−}y$ plane, for the two possible choices of magic detunings. See text for details.

Figure 4

Squeezing on the clock states as a function of time in units of the scattering rate ${\gamma }_s$ for 2500 atoms trapped in the $x\text{−}z$ plane at a distance $r_{\perp }^{\prime }=1.8a$ from the axis of the nanofiber. The optimal quantization axis (red) is compared to the quantization along the $x$ axis (blue). Dashed lines indicate simulations without optical pumping, i.e., no decoherence. (a) Metrological spin squeezing parameter ${\xi }^{-2}$, Eq. (51), in dB. (b) Collective mean spin $\langle {\widehat{J}}_1\rangle$ and decaying atom number in the clock states, $N_C$ (inset). (c) Conditional squeezed variance $\mathrm{\Delta }{J}_3^2$. The inset shows the decomposition at the optimal quantization axis of $\mathrm{\Delta }{J}_3^2$ (red dashed) into the single-body variance, $N_A{\left(\mathrm{\Delta }{j}_3^{\left(1\right)}\right)}^2$ (green) and the two-body covariance, $N_A(N_A-1)〈\mathrm{\Delta }{j}_3^{\left(1\right)}\mathrm{\Delta }{j}_3^{\left(2\right)}〉$ (black), as given by Eq. (75).

Figure 5

Dependence of the parameters that determine metrological squeezing on the direction of the quantization axis that defines the clock states, ${\mathbf{e}}_{\tilde{z}}$. In all cases we consider 2500 atoms trapped in the $x\text{−}z$ plane at distance $r_{\perp }^{\prime }=1.8a$ from the axis of the nanofiber. In (b)–(d) ${\mathbf{e}}_{\tilde{z}}$ is confined to the $x\text{−}y$ plane, where the optimal peak squeezing occurs. (a),(b) Peak achievable squeezing at the maximum time, measured in dB, as a function of the direction of ${\mathbf{e}}_{\tilde{z}}$. (c) OD/$N_A={\sigma }_0{A}_{in}/A_{J_3}^2$, Eq. (78). (d) Rates of atom loss ${\gamma }_{00}$ and depolarization ${\gamma }_{11}$ relative to the characteristic scattering rate ${\gamma }_s$. See Eqs. (C4a) and (C4e).

Figure 6

Parameters that define squeezing as a function of trapping distance $r_{\perp }^{\prime }$ and initial atom number $N_A$. (a) OD/$N_A$. Inset is the corresponding magic detuning in units of MHz (see text). (b) Optimal quantization axis orientation angle ${\mathbf{e}}_{\tilde{z}}$ in the $x\text{−}y$ plane for different atom numbers. The line with 500 atoms terminates when the squeezing effect is too weak to be observed $\left(r_{\perp }^{\prime }>2.3a\right)$. (c) Peak metrological squeezing at the optimal ${\mathbf{e}}_{\tilde{z}}$ for different atom numbers.