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Physical Review Letters

Experimental Demonstration of Three Mutually Orthogonal Polarization States of Entangled Photons

Tedros Tsegaye, Jonas Söderholm, Mete Atatüre, Alexei S. Trifonov, Gunnar G. Bjork, Alexander V. Sergienko, Bahaa E. A. Saleh, Malvin Carl Teich

Published Dec 11, 2000• 46 citations
Publisher

Abstract

Linearly polarized classical light can be expressed in a vertical and a horizontal component. Geometrically rotating vertically polarized light by 90 degrees will convert it to the orthogonal horizontal polarization. We have experimentally generated a two-photon state of light which evolves into an orthogonal state upon geometrical rotation by 60 degrees. Rotating this state by an additional 60 degrees will yield a state which is mutually orthogonal to the first two states. Generalizing this procedure, one can generate N+1 mutually orthogonal N-photon states that cyclicly evolve from one to another upon a geometric rotation by 180/(N+1) degrees.

DOI
10.1103/physrevlett.85.5013

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Linearly polarized classical light can be expressed in a vertical and a horizontal component. Geometrically rotating vertically polarized light by 90 degrees will convert it to the orthogonal horizontal polarization. We have experimentally generated a two-photon state of light which evolves into an orthogonal state upon geometrical rotation by 60 degrees. Rotating this state by an additional 60 degrees will yield a state which is mutually orthogonal to the first two states. Generalizing this procedure, one can generate N+1 mutually orthogonal N-photon states that cyclicly evolve from one to another upon a geometric rotation by 180/(N+1) degrees.
We show that any pure, two-mode, NNN-photon state with NNN odd or equal to two can be transformed into an orthogonal state using only linear optics. According to a recently suggested definition of polarization degree, this implies that all such states are fully polarized. This is also found to be true for any pure, two-mode, energy eigenstate belonging to a two-dimensional SU(2) orbit. Complete two- and three-photon bases whose basis states are related by only phase shifts or geometrical rotations are also derived.
Quantum polarization properties of two-mode states have been discussed in the context of quantum states with ``hidden polarization''. In this paper we demonstrate that all pure two-photon and three-photon states have unit degree of quantum polarization. We also show that pure states belonging to groups of discrete symmetry have unit degree of quantum polarization for arbitrary number of photons. For a certain class of two-photon states, we substantiate the possibility of generating the full basis set (three orthogonal states) by using only combinations of the SU(2) generators, i.e., by using geometrical rotations and differential phase shifts. Finally, we generate the whole basis set for pure reflection-symmetrical three-photon states by means of SU(2) generators.
We consider polarization states of three photons, propagating collinearly and having equal given frequencies but with arbitrary distributed horizontal or vertical polarizations of photons. A general form of such states is a superposition of four basic three-photon polarization modes, to be referred to as the three-photon polarization ququarts (TPPQ). All such states can be considered as consisting of one- and two-photon parts, which can be entangled with each other. The degrees of entanglement and polarization, as well as the Schmidt decomposition and Stokes vectors of TPPQ are found and discussed.
We report on two quantum polarization experiments. The first one shows that three geometrical rotations can in principle be distinguished with certainty using two photons. The second experiment shows quantum polarization properties of classically unpolarized light. We outline a new treatment of polarization in quantum optics. Our formalism differs from the usual quantum description, which uses the Stokes operators. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Multi-particle entangled states are becoming more and more interesting for different applications in quantum information, such as quantum multi-particle teleportation, entangled based cryptography and quantum communication schemes based on multi-particle states, but also for fundamental reasons such as testing local realism by means of the Greenberger-Horne-Zeilinger states (GHZ) arguments . The advantage of multiparticle entanglement versus the most common two particle one lies in the fact that the greater the number of particles involved in the entangled state, the more clearly the quantum effects are exhibited. Our final goal is to exploit the multi-photon entangled states for quantum metrology. By the interaction of those states with nanostructures (mostly nanoholes, but in a future also nanospheres), we expect to reach sub-wavelength sensitivity phase measurements beating the Standard Quantum Limit as well as to detect plasmonic effects. In this thesis we present our work consisting on designing and building an experimental setup for the generation of states of 4-photon entangled in the transverse linear momentum. In the first part we explain some basics in the form of a technical introduction. We then adapt a general theoretical model to our particular case, explaining the physics of the process of generation of entangled states of photons. We proceed explaining the details of our setup, as well as the procedure of measurement. We then present the results of our tests in a "standard" 2-photon regime to show the correct functioning of the setup itself. Some more sophisticated measurements using structured light in the form of a first-order Hermite-Gaussian mode has been performed to generate entanglement in higher spatial modes are also presented. We conclude with theoretical and experimental details regarding the 4-photon experiments feasible with our setup. Our source is based on Spontaneous Parametric Down-Conversion (SPDC). In the last two decades it has been extensively studied for the generation of two-photon entangled states, since Kwiat et al. published in 1995 the new scheme based on polarization entangled photons. Most of the studies regarding entangled states involving more than two photons are indeed based on quantum correlations in the polarization or in the path of the down converted beams. The great advantage of entangling a spatial degree of freedom (as we do) is related to the dimensionality of the Hilbert space in which the state is defined. In the particular case of linear momentum, which is a continuous variable, we are dealing with infinite dimensions. Since in each dimension of the state one quanta of information can be encoded, the advantage of having multidimension entanglement for the field of quantum communication and computing is clear. The specs of our setup include a periodically-poled KTP crystals offering high conversion efficiency and cut for type-0 SPDC. The photons generated are well known to be inherently entangled in different degrees of freedom such as polarization, energy-time, linear momentum, and orbital angular momentum. Since each single SPDC event generates one photon pair, in order to reach four photon we entangle two consecutive SPDC events per each state. Furthermore, it is possible to manipulate (to enhance) the dimensionality of the entanglement by properly tailoring the pump spatial profile. The key point for the generation of four-photon entangled states is that, under some conditions, two consecutive pairs generated via SPDC cannot be described as two independent pairs but necessarily as a four-photon state showing genuine entanglement. This can happen when using a pulsed laser to generate the entangled photons; in particular, in the regime in which the coherence time of the down-converted photons τc is (much) longer than the temporal length of the pumping pulse Δt. In this condition the emission of a second pair is stimulated by the presence of the first one. That effect can be ultimately seen as a constructive multi-particle interference effect, the deep origin of which has to be sought in the boson statistic of photons.
We present a scheme to produce an entangled four-photon W state by using linear optical elements. The symmetrical setup of linear optical elements consists of four beam splitters, four polarization beam splitters, and four mirrors. A photon Einstein-Podolsky-Rosen pair and two single photons are required as the input modes. The projection on the W state can be made by a four-photon coincidence measurement. Further, we show that by means of a horizontally oriented polarizer in front of one detector, the W state of three photons can be generated.