Quantum-noise-limited interferometric measurement of atomic noise: Towards spin squeezing on the Cs clock transition

We investigate theoretically and experimentally a nondestructive interferometric measurement of the state population of an ensemble of laser-cooled and trapped atoms. This study is a step toward generation of (pseudo)spin squeezing of cold atoms targeted at the improvement of the cesium clock performance beyond the limit set by the quantum projection noise of atoms. We calculate the phase shift and the quantum noise of a near-resonant optical probe pulse propagating through a cloud of cold ${}^{133}\mathrm{Cs}$ atoms. We analyze the figure of merit for a quantum nondemolition (QND) measurement of the collective pseudospin and show that it can be expressed simply as a product of the ensemble optical density and the pulse-integrated rate of the spontaneous emission caused by the off-resonant probe light. Based on this, we propose a protocol for the sequence of operations required to generate and utilize spin squeezing for the improved atomic clock performance via a QND measurement on the probe light. In the experimental part we demonstrate that the interferometric measurement of the atomic population can reach a sensitivity of the order of $\sqrt{N_{\mathrm{at}}}$ in a cloud of $N_{\mathrm{at}}$ cold atoms, which is an important benchmark toward the experimental realization of the theoretically analyzed protocol.

Figure 1

(Color online) (a) Diagram of the Cs hyperfine levels included in the $D_2$ line. (b) Theoretically evaluated phase shift of the probe as a function of the detuning ${\Delta }_{45}$ from the $6S_{1∕2}(F=4)\to 6P_{3∕2}(F^{\prime }=5)$ transition.

Figure 2

(Color online) The Mach-Zehnder interferometer with loss sources and associated input fields indicated. The atoms are considered to be in the upper arm.

Figure 3

(Color online) The Bloch sphere scheme for the state preparation (a),(b), QND measurement (e),(f),(g), and the Ramsey spectroscopy for a coherent state (c),(d), and a spin-squeezed state (h),(i), with the corresponding simulations of the projection noise on the Ramsey fringe, (j) and (k), respectively.

Figure 4

(Color online) Experimental setup employed to lock the probe beam. The elements included in the sketch are BS, 50-50 beam splitter; FPD, fast photodetector; SA, spectrum analyzer; DBM, double balanced mixer; LPF, low-pass filter; Amp, amplifier.

Figure 5

(Color online) Sketch of the setup of the interferometer with following elements: BS, 50-50 beam splitter; C1 and C2, 50-50 fiber couplers; PC1, PC2, and PC3, fiber polarization controllers; L1 and L2, achromatic lenses; F1 and F2, interference filters transmitting at $852\mathrm{nm}$; D1 and D2, Hamamatsu low-noise, high-gain photodiodes; D3 and D4, photodetectors; and several half-wave plates $\lambda ∕2$ and collimating lenses for fiber coupling. $i_L$ is the locking signal, whereas $i_{-}=i_1-i_2$ is the probe signal.

Figure 6

(Color online) The two-point variance extracted from measurements done with pulse separation ${\tau }_0=20\mu s$. Lower trace: Interferometer in white-light position with probe laser locked and the variance increases on larger time scales due to the $50\mathrm{Hz}$ line noise. Upper trace: Interferometer out of white-light position with probe laser frequency modulated at $5\mathrm{kHz}$, which is directly reflected in the variance by a $200\mu s$ oscillation period. Additional phase noise from the laser raises the level of the minima (dotted line) with respect to the lower trace, because the laser is not phase locked and the interferometer is not in the white light position.

Figure 7

(Color online) Noise in the amplitude $x$ (엯) and phase $y$ (●) quadratures of the probe light. Fits to the data in log-log scale give slopes of $1.2\pm 0.2$ (---) for the amplitude and $1.2\pm 0.4$ (—) for the phase quadrature.

Figure 8

(Color online) dc phase shift due to dispersion by the atomic cloud with fit (—) to Eq. (3). A closer fit (---) is obtained by letting the line strengths be independent, and thus not given only by the value of the $6J$ symbols.

Figure 9

(Color online) Phase noise induced in the probe light from the interaction with cold atoms. The dc phase shift is used as a measure for the number of atoms.