Quantum state tomography of a Bose–Einstein condensate

In the laboratory we manipulate the internal states of atoms in a Bose–Einstein condensate (BEC) in such a way that they become quantum-mechanically entangled. It is essential that we then analyze the joint quantum state of the BEC in as much detail as possible, in order to characterize the operations performed on the atoms, as well as to gauge the usefulness of entangled states for quantum technology applications such as quantum metrology.

The quantum-mechanical state (wavefunction or density matrix) of a system (for example our BEC) cannot be measured directly, as it contains too much information for a single measurement. Instead, it is usually found by mathematically processing a large number of measurements in a process called a state tomography. The main obstacles to a straightforward state tomography in a BEC are the large number of particles (which generally require an exponentially large number of coefficients for their exact description) and the lack of experimental precision (there are unavoidable errors associated with all measurements) and atomic resolution (we do not have single-atom resolution in our atom-counting measurements). We have developed a special tomography procedure which is particularly suited for analyzing BECs. There are two factors which make our tomography possible: first, the fact that we use indistinguishable bosons in our experiments means that we can restrict the state reconstruction to the totally symmetric subspace of wave-functions. And second, by reconstructing the Wigner function instead of the density matrix, we are handed a natural way of distinguishing more significant (less noisy) from less significant (more noisy) components of the state description.

Figure 1: Reconstructed Wigner function of a spin-squeezed Bose–Einstein condensate. The Bloch sphere is viewed from the top (along the z-axis); the state is centered at the north pole. Reconstruction noise due to insufficient experimental data is visible away from the north pole. The axis of minimal variance is tilted by 6.7° away from the x-axis. Data from Ref. [2]; figure from Ref. [1].

The internal state of each atom, describing an effective two-level system, is represented by a pseudospin-½ variable s_i. The total pseudospin S=∑_is_i of a set of N atoms is a quantum-mechanical angular momentum of length ⟨S·S⟩=S(S+1) with S=N/2, since the state must be fully symmetric under particle exchange. The Wigner function describes the orientation of this total pseudospin, similar to the phase-space density of classical mechanics. However, unlike a phase-space density it can take negative values, and thus it cannot be directly interpreted as a density. Our specific Wigner function is most naturally plotted on the surface of a sphere (see figure 1), and mathematically written as a sum over spherical harmonics: W(θ,φ)=∑_kqρ_kqY_kq(θ,φ). As it may be expected, the main large-scale structure of the Wigner function (small angular momenta k) is robust to experimental noise, while the fine structure (tiny wiggles, large angular momenta k) is much more susceptible to noise and can likely not be reconstructed faithfully. Luckily it is precisely this large-scale structure which contains the necessary information to evaluate many experimentally relevant expectation values.

The result from our state tomography, described in detail in Ref. [1], is a Wigner function on the Bloch sphere. It can be used to evaluate many observables which cannot be measured directly in an experiment. It is also a very appealing and intuitive graphical representation of the joint quantum-mechanical state (see figure 1).

References

[1] Tomographic reconstruction of the Wigner function on the Bloch sphere: R. Schmied and P. Treutlein, New J. Phys. 13, 065019 (2011).
[2] Atom-chip-based generation of entanglement for quantum metrology: M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, Nature 464,1170 (2010).