Manipulating transverse modes of photons for quantum cryptography

Several schemes have been proposed to extend quantum key distribution protocols aimed at improving their security or at providing new physical substrates for qubit implementation. We present a toolbox to jointly create, manipulate, and measure qubits stored in polarization and transverse-modes degrees of freedom of single photons. The toolbox includes local operations on single qubits, controlled operations between the two qubits, and projective measurements over a wide variety of nonlocal bases in the four-dimensional space of states. We describe how to implement the toolbox to perform an extended version of the BB84 protocol for this Hilbert space (ideally transmitting two key bits per photon). We present the experimental implementation of the measurement scheme both in the regimes of intense light beams and with single photons. Thus, we show the feasibility of implementing the protocol, providing an interesting example of a method for quantum information processing using the polarization and transverse modes of light as qubits.

Figure 1

Sagnac interferometer with a PBS, which implements a controlled operation from polarization to transverse modes. The Dove prism applies a rotation to the transverse wave-front distribution, being the sign of this rotation is determined by the direction of propagation.

Figure 2

Mach-Zehnder interferometer with an extra mirror, which can discriminate and states. The states with symmetry against specular reflection along the $\widehat{y}$ axis remain invariant after mirror reflections, and the states with antisymmetry get a $\pi$ phase. The superposition on the MZEM output generates destructive and constructive interference on the complementary exits for states with different parity eigenvalues.

Figure 3

Scheme of Alice's setup to prepare a general state using single-qubit operations for each qubit (UPR and UTR) and a controlled operation implemented with the Sagnac interferometer and a Dove prism.

Figure 4

Scheme of Bob's setup, with which any basis projection can be obtained. The “basis selection” part consists of a reversed copy of Alice's setup and an extra $\pi$ converter, which can map any basis onto the canonical one. The “canonical basis projection” part consists of the combination of a TM discriminator (MZEM), two polarization discriminators (PBSs), and four single-photon detectors that implement a projective measurement of the canonical basis.

Figure 5

Scheme of the TEM builder used to test the MZEM. The lower part shows a simplified picture of the beam profile obtained at different stages of the process.

Figure 6

The geometrical properties of the double mirror enable it to generate a change over the optical path length. A displacement of $\Delta d$ over the symmetry axis produces a difference of $\sqrt{2}\Delta d$ on the optical path.

Figure 7

Geometry considerations for the beam propagation in the MZEM. In contrast to a normal Mach-Zehnder, a lateral shift on the incident beam produces a relative displacement of the two beams at the input of the second beam splitter, which limits the interference effect to the overlap area between the two displaced beams. At the scheme, an aligned (green) and a displaced (dotted red/orange) beam are shown. The F symbol depicts the transformation of the transverse intensity pattern of each beam by the mirror reflections.