Improvement of an Atomic Clock using Squeezed Vacuum

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt

Phys. Rev. Lett. 117, 143004 – Published 28 September 2016

Abstract

Since the pioneering work of Ramsey, atom interferometers are employed for precision metrology, in particular to measure time and to realize the second. In a classical interferometer, an ensemble of atoms is prepared in one of the two input states, whereas the second one is left empty. In this case, the vacuum noise restricts the precision of the interferometer to the standard quantum limit (SQL). Here, we propose and experimentally demonstrate a novel clock configuration that surpasses the SQL by squeezing the vacuum in the empty input state. We create a squeezed vacuum state containing an average of 0.75 atoms to improve the clock sensitivity of 10000 atoms by ${2.05}_{- 0.37}^{+ 0.34} dB$ . The SQL poses a significant limitation for today’s microwave fountain clocks, which serve as the main time reference. We evaluate the major technical limitations and challenges for devising a next generation of fountain clocks based on atomic squeezed vacuum.

Received 25 May 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.143004

Physics Subject Headings (PhySH)

Atom interferometry Atomic, optical & lattice clocks

Atomic, Molecular & Optical

Authors & Affiliations

I. Kruse¹, K. Lange¹, J. Peise¹, B. Lücke¹, L. Pezzè², J. Arlt³, W. Ertmer¹, C. Lisdat⁴, L. Santos⁵, A. Smerzi², and C. Klempt¹

¹Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
²QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy
³Institut for Fysik og Astronomi, Aarhus Universitet, Ny Munkegade 120, DK-8000 Århus C, Denmark
⁴Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany
⁵Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, D-30167 Hannover, Germany

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Vol. 117, Iss. 14 — 30 September 2016

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Images

Figure 1
The three-mode interferometer. (a) Hyperfine ground states of ${}^{87}{Rb}$ . (I) A Bose-Einstein condensate in the state $| 0 ⟩ = | F = 1, m_{F} = 0 ⟩$ is coupled to the states $| 1, \pm 1 ⟩$ by a resonant radio-frequency pulse. The second-order Zeeman shift is compensated by a detuned microwave dressing to the state $| 2, - 2 ⟩$ . (II) A microwave pulse with detuning $δ$ couples the states $| 0 ⟩ = | 1, 0 ⟩$ and $| e ⟩ = | 2, 0 ⟩$ . The radio-frequency and microwave pulses form an effective $π / 2$ pulse between $| 1, \pm 1 ⟩$ and $| e ⟩$ . (III) After a Ramsey evolution time $τ_{R}$ , a second detuned microwave pulse is applied. (IV) A final radio-frequency pulse coupling $| 0 ⟩$ and $| 1, \pm 1 ⟩$ closes the interferometer. (b) The interferometer corresponds to two three-mode beam splitters and a phase shift in between. The total number of atoms $N$ enters the central state $| 0 ⟩$ . A measurement of all three components $N_{- 1, 0, 1}$ after the interferometer allows for an estimation of the phase shift $θ$ . (c) The interferometer can be simplified by introducing the symmetric state $| g ⟩$ and the antisymmetric state $| h ⟩$ . The three-mode beam splitters only couple to the symmetric state, thus yielding an effective two-mode interferometer with an unchanged antisymmetric state.
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Figure 2
Output of the interferometer. (a) The microwave detuning $δ$ is varied for different Ramsey times $τ_{R} = 250$ (open circles) and $τ_{R} = 1000 μ s$ (open diamonds). The phase shift, set by the microwave detuning ( $x$ axis) and the evolution time $τ = τ_{R} + τ_{mw}$ , results in the Ramsey fringes in the transferred fraction. The solid lines (dark and light) represent the results of our model. (b) The phase adjustment time $t$ before the interferometer sequence is varied and the corresponding variance of the transferred fraction is recorded with respect to projection noise (open circles). The data are well reproduced by a sinusoidal fit (light blue line) and reaches clearly below projection noise (solid black line). Result of the classical interferometer (shaded dashed area). The error bars and shaded areas represent the statistical uncertainty of 100 repetitions.
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Figure 3
Phase estimation uncertainty. (a) Mean values of the transferred fraction $\bar{f}$ (black open circles) for five different microwave detunings (bottom $x$ axis) and corresponding phase shifts (top $x$ axis) close to the midfringe position. The slope of the linear fit (blue solid line) is close to the optimum, as represented by our model (black dashed line). The error bars represent the (sub-projection noise) statistical uncertainty of 390 repetitions. (b) The phase estimation uncertainties $Δ^{2} θ$ normalized to the SQL (blue open circles) reach well below the SQL (black dashed line). The shaded area represents an upper bound of the uncertainty of the atom number calibration of 1% [34]. The small detuning dependency of the recorded sensitivity is reproduced by our single-atom model including magnetic field noise (black dotted line). The best phase estimation uncertainty of $- 1.5 6_{- 0.45}^{+ 0.41} dB$ below the SQL is reached at a detuning of $- 5.9 kHz$ . Error bars are the statistical uncertainty (1 standard deviation) of 390 repetitions.
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Figure 4
Reconstructed Wigner function and Bloch sphere representation. (a) The data presented in Fig. 2 are used to reconstruct the two-dimensional Wigner function in the $X − P$ space. Here, the Wigner function of the state after the optimal phase adjustment time of $t = 300 μ s$ is shown. (b) The interferometer is illustrated on the multiparticle Bloch sphere for the states $| g ⟩$ and $| e ⟩$ , in terms of the pseudospin operators $J_{x} = (1 / 2) (e^{†} g + g^{†} e)$ , $J_{y} = 1 / (2 i) (e^{†} g - g^{†} e)$ , $J_{z} = (1 / 2) (e^{†} e - g^{†} g)$ . The employed squeezed vacuum corresponds to an elliptical uncertainty disk with variable orientation angle depending on the phase adjustment time $t$ . An optimal orientation angle, as shown, allows for a measurement of the transferred fraction with a sub-projection-noise uncertainty. The two radio-frequency pulses generate rotations around the $J_{x}$ axis (I/III). The phase shift corresponds to a rotation around the $J_{z}$ axis (II).
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