Multiparticle State Tomography: Hidden Differences

We address the problem of completely characterizing multiparticle states including loss of information to unobserved degrees of freedom. In systems where nonclassical interference plays a role, such as linear-optics quantum gates, such information can degrade interference in two ways, by decoherence and by distinguishing the particles. Distinguishing information, often the limiting factor for quantum optical devices, is not correctly described by previous state-reconstruction techniques, which account only for decoherence. We extend these techniques and find that a single modified density matrix can completely describe partially coherent, partially distinguishable states. We use this observation to experimentally characterize two-photon polarization states in single-mode optical fiber.

Figure 1

Experimental implementation of state preparation and tomography protocols showing polarizing beamsplitters (PBS), beamsplitters (BS), nonlinear BBO crystals, a second harmonic generation crystal (SHG), quarter waveplates (QWP), half wave plates (HWP), polarization-maintaining fiber (PMF), single-mode fiber (SMF), and single photon counting modules (SPCM). A spontaneous parametric down-conversion (SPDC) crystal produces pairs of H and V photons. The separation between the H and V photons is controlled with movable quartz wedges and very long delays can be introduced by inserting a thick piece of BBO into the beam. Single-mode fiber and a 10 nm interference filter make the photons essentially indistinguishable in the spatiotemporal modes. Tomography is performed with a set of wave plates and a polarizing beam splitter. This system can implement all the measurements in Table .

Figure 2

The density matrices measured with quantum state tomography. The imaginary parts of (a) through (f), which are not shown, had all elements less than 0.05. White elements are inaccessible to measurement. (a) Indistinguishable H and V photons, (b) distinguishable H and V photons, (c) partially distinguishable H and V photons, (d) indistinguishable photons transformed to the 2-NOON state, (e) distinguishable photons with the same transformation applied, (f) the same state as it would be characterized by the technique of 12 (g) a 2-NOON state after passage through decohering, misaligned BBO crystal.