Measurement of Geometric Phase for Mixed States Using Single Photon Interferometry

Geometric phase may enable inherently fault-tolerant quantum computation. However, due to potential decoherence effects, it is important to understand how such phases arise for mixed input states. We report the first experiment to measure mixed-state geometric phases in optics, using a Mach-Zehnder interferometer, and polarization mixed states that are produced in two different ways: decohering pure states with birefringent elements; and producing a nonmaximally entangled state of two photons and tracing over one of them, a form of remote state preparation.

Figure 1

Mixed-state generation and interferometer to measure geometric phase. Mixed states are prepared via two methods: (1) tracing over the polarization of one photon of a nonmaximally entangled polarization state and, (2) using an initial pure polarization state with birefringent decoherers that couple polarization to photon arrival time (see dashed box) [16]. In the latter case, tracing over this time prepares a mixed state. Half-wave plates at ${\theta }_1$ and ${\theta }_2$ generate geometric phase but do not otherwise alter the transmitted polarization state. Two crossed wave plates in the lower interferometer arm give no geometric phase, but compensate the optical path difference between the arms to achieve high visibility. The interferometer shape minimizes unwanted polarization changes arising from non-normal mirror and beam splitter reflections.

Figure 2

The solid angle $\Omega$ enclosed by the cyclic path of one eigenvector of the density matrix. The other eigenvector traces the same path but in the opposite direction, thus enclosing the solid angle $-\Omega$. $\Omega$ can be varied by adjusting ${\theta }_1-{\theta }_2$, the relative angle between the optic axes of the two HWPs in the geometric phase arm of the interferometer.

Figure 3

The mixed-state geometric phases and visibilities as a function of the half-wave plate angle ${\theta }_1$. (a),(b) The photons in the mixed-polarization state were produced with decohering quartz elements (see text). (c),(d) The mixed-polarization state photons were produced by tracing over one photon in a nonmaximally entangled state. (e),(f) The classical laser was decohered with an imbalanced polarizing interferometer. The error bars are derived from the fit of the raw data to Eq. (5). The error in the theoretical curves shown results from uncertainties in the determination of $r$, due to photon counting statistics [or intensity fluctuations for (e) and (f)]. The visibility theory curves are normalized to the average visibility when ${\theta }_1=0:$ 95% for (b), 98% for (d), and 93% for (e). The slightly imperfect visibility is largely due to imperfect interferometer mode matching. Typical data is shown inset in (a) (${\theta }_1=0°$ and ${\theta }_1=15°$ for $r=0.81$) and (c) (${\theta }_1=0°$ and ${\theta }_1=15°$, and $r=0.57$). Note: In (a) and (c) the curves are flipped along the $x$ axis: in the first setup the input states possessed larger right-circular polarization eigenvalues, while in the second setup, left-circular polarization was dominant [23].