Theory of Single-Shot Phase Contrast Imaging in Spinor Bose-Einstein Condensates

Abstract

We introduce a theoretical framework for single-shot phase contrast imaging (PCI) measurements of spinor Bose-Einstein condensates. Our model allows for the simple calculation of the quantum backaction resulting from the measurement, and the amount of information that is read out. We find that there is an optimum time Gτ ∼ 1/N for the light-matter interaction (G is the ac Stark shift frequency, N is the number of particles in the BEC), where the maximum amount of information can be read out from the BEC. A universal information-disturbance tradeoff law ǫF ǫG ∝ 1/N 2 is found where ǫF is the amount of backaction and ǫG is the estimation error. The PCI measurement can also be found to be a direct probe of the quantum fluctuations of the BEC, via the noise of the PCI signal.PACS numbers: 03.75. Dg, 37.25.+k, 03.75.Mn Quantum mechanics puts a limit on the amount of information that can be gained by performing a measurement of an unknown state. This concept, arising from the fact that it is impossible to measure an unknown state without disturbing it, has been studied in the context of quantum state estimation [1][2][3]. Quantum state estimation is typically concerned with how to optimally measure and estimate a given quantum state. In some contexts it is desirable to limit the amount of backaction the quantum state experiences, and perform an optimal state estimation under this constraint. The other extreme in this continuum is the notion of weak measurements [4,5], where it is in principle possible to perform a measurement without disturbing the system at all, given a large number of copies [6]. This informationdisturbance tradeoff has been studied in a variety of different classes of quantum states such as single qubits [7], multiple copies of qubits [8,9] Figure 1). By measuring the phase shift via interference, the state of the BEC can be inferred. Because the PCI measurement does not destroy the atomic condensate, it can be applied repeatedly [21,22] nique is a useful resource for information readout in several proposed applications of ultra-cold atomic systems to metrology [23,24], quantum information and processing [25,26]. Other methods [27][28][29] also use a feedback mechanism to achieve greater control over the backaction induced by the measurement.In this paper we formulate a theory of PCI measurements at the single-shot level. We introduce a simple model that allows for the calculation of the backaction on the BEC, and the amount of information that is available by measurement of the light. Prior to this paper most works have modeled the effect of the backaction via continuous quantum measurement framework [30-

Split spin-squeezed Bose–Einstein condensates

“…probable result according to (38). The behavior of the conditional Wigner function is consistent with that of marginal Wigner function.…”

“…Several theoretical studies have analyzed the entanglement created by a S z S z type interaction, pointing to a complex entanglement structure with fractal characteristics [33,34]. Numerous proposal have been made for generating such entanglement, based on cold atomic collisions [35], atom-light interactions [36][37][38][39][40], and Rydberg excitations [41]. Several proposals for quantum information applications have been made based on such entanglement [38,42,43].…”

Sagnac interferometry with coherent vortex superposition states in exciton-polariton condensates

“…However, the polaritons do possess the great advantage that their signal-to-noise ratio should be considerably larger than that of atomic BECs. In an atomic BEC the optical measurement has the limitation that it should not disturb the system [81], hence there are restrictions on how bright the probe light should be. In contrast, for polaritons they are themselves part light, hence illumination by laser light is intrinsic to the condensation process.…”

Macroscopic quantum information processing using spin coherent states

“…However, only one of the distributions of the coherent states can be fixed as can be seen from (20) and (21). While absorption imaging can be done with an accuracy of several atoms [28], this destroys the BEC hence non-destructive methods such as phase contrast imaging would be required [44]. Thus the effect of number fluctuations can be considered to be an effective gate error, which may be mitigated if the particle numbers are known.…”

“…The PCI measurement does not destroy the atomic condensate, it can be applied repeatedly [42,43] on the same atomic sample. This also amounts to an estimate of S z , but is not a strong measurement and instead results in some dephasing of the coherence between the levels [44].…”