Quantum atom optics lab
Treutlein group

Atom-chip-based generation of entanglement for quantum metrology

We have for the first time generated multi-particle entanglement on an atom chip by controlling elastic collisional interactions with a state-dependent potential. We employ this technique to generate spin-squeezed states of a two-component Bose-Einstein condensate that are a useful resource for quantum metrology. The observed reduction in spin noise combined with the spin coherence imply four-partite entanglement between the condensate atoms and could be used to improve an interferometric measurement by about a factor of 2 over the standard quantum limit.

These results where published in Nature [1].

In the currently emerging field of quantum metrology, multi-particle entangled states such as spin-squeezed states are investigated as a means to improve measurement precision beyond the ‘standard quantum limit’. Atom chips combine exquisite coherent control with a compact and robust setup, suggesting their use for quantum metrology with portable atomic clocks and interferometers. Several techniques to create entangled states on atom chips have been proposed, but none had been experimentally realized so far.

The so called ‘one-axis twisting’ scheme [2-4] in principle allows to create a huge amount of entanglement in a two-component BEC. In this scheme, an initially separable (non-entangled) state, where each atom is in a superposition of two internal states |0> and |1>, dynamically evolves into a spin-squeezed state in which the condensate atoms are entangled. This is due to atomic interactions that provide a nonlinear term in the Hamiltonian for the BEC internal state.

In the experiments described here, we realize this scheme for the first time on an atom chip [1]. A notable feature is that we control the interactions through the wave function overlap of the two states in a state-dependent microwave potential [5]. This is a new and versatile technique for tuning of interactions in a BEC which also works in magnetic traps and for atomic state pairs where no convenient Feshbach resonances exist. We use such a pair, the hyperfine states |0> = |F = 1, m_F = -1> and |1> = |F = 2, m_F = 1> of ⁸⁷Rb, which is also employed in chip-based atomic clocks with magnetically trapped atoms.

Figure 1: Sequence used for entanglement generation. A pi/2-pulse creates a superposition of |0> and |1>. When the microwave potential is applied, the two wavefunctions separate and the squeezing dynamics takes place. After one complete oscillation, the microwave is switched off and the squeezing dynamics and relative atomic motion stops. A second pulse is used for turning the state before readout.

Our experimental sequence for squeezing is shown in figure 1. It starts with a resonant pi/2-pulse to prepare a coherent spin state. After the pulse, we squeeze the state by turning on the microwave near-field potential for a well-defined time and thus spatially splitting and recombining the two components of the BEC in the following way: turning on the microwave potentials results in an abrupt separation of the trap minima for |0> and |1>. The two components of the BEC start to perform one oscillation in their respective potentials. During the oscillation, which is strongly influenced by mean-field effects, the mode functions Φ₀ and Φ₁ of the two states almost completely separate.
After a time T = 12.7 ms the states overlap again, the microwave potential is switched off, and the squeezing dynamics, as well as the relative atomic motion, stops.

We analyze the produced state by performing state tomography. With the mean spin along x, we measure the transverse spin components along any angle θ by rotating the state vector in the yz-plane by that angle prior to detection of S_z. Figure 2a shows the noise in S_θ obtained from a large number of such measurements as a function of θ. Data for a squeezed state are shown in comparison with data for a coherent spin state where the traps were not separated during the sequence (reference measurement). We plot the normalized variance Δ_nS_θ² = 4ΔS_θ²/< N >, so that Δ_nS_θ² = 0 dB corresponds to the standard quantum limit. In the squeezed state, the spin noise falls significantly below the standard quantum limit, reaching a minimum of Δ_nS_θ² = -3.7 dB at θ_min = 6°. The corresponding interference contrast is 88%. This results in a squeezing parameter [3] of ξ = -2.5 dB, proving that the state is a useful resource for quantum metrology and that the condensate atoms are entangled. The reference measurement, by contrast, stays above the standard quantum limit for all values of θ.

Figure 2: Spin tomography and reconstructed Wigner function.
(a) Normalized Variance in S_θ as a function of θ. Filled circles are the results for the squeezed state, white circles for the reference measurement. The blue (red) line is the theoretically predicted squeezing from the simulation without (with) technical noise. (b) Reconstructed Wigner Function of the squeezed state. Squeezed and ‘anti-squeezed’ quadratures are clearly visible. For comparison, the circular contour of an ideal coherent spin state is shown.

We compare our measurements with a sophisticated simulation that was performed by our collaborators Dr. Alice Sinatra and Li Yun from ENS Paris. The spin noise reduction obtained from this simulation is shown in Fig. 2a along with the data. The blue line shows the expected squeezing taking into account atom loss but no technical noise. The maximal reduction in variance is -12.8 dB, significantly larger than observed. The red line, which describes our data well, additionally includes technical fluctuations, which have been measured and characterized independently.

The measured histograms of S_θ for different angles θ are tomographic data that allow us to reconstruct the Wigner function of the squeezed BEC. Figure 2b shows the reconstruction. The two contour lines indicate where the Wigner functions of our squeezed state and of an ideal coherent spin state have fallen to 1/√e of their maximum. The squeezing along the direction θ_min as well as the ‘anti-squeezing’ in the perpendicular direction can be clearly seen.

Entanglement in the BEC has been defined as the non-separability of the N-particle density matrix. An intriguing question concerns the depth of entanglement: How large must the clusters of entangled atoms be at least in order to produce the observed squeezing? In [6], a procedure to determine the depth of entanglement from the measured spin noise reduction and mean spin length is described. Our data imply that the condensate atoms are entangled in clusters of at least 4 ± 1 particles.

We envisage the implementation of our technique in portable atomic clocks and interferometers operating beyond the standard quantum limit. Furthermore, it is a valuable tool for experiments on many-body quantum physics and could enable quantum information processing on atom chips.

For more details about these experiments, see our paper [1].