It is well known that entanglement is invariant to local unitary transformations. An example of such is the well-known fact that the degree of entanglement remains unmodified upon free-space propagation. Crucially, nonseparability, the fundamental aspect of entanglement, is not exclusive to quantum systems, and this principle applies to both nonlocal and local entanglement at the single- and multiphoton levels. The former is observed between photons that simultaneously exist in physically separated locations, and the latter between the internal degrees of freedom of photons. Classical states of light such as complex vectorial light fields [1] also exhibit local entanglement, attributed to the nonseparability between their spatial and polarization degrees of freedom (DoFs). Here it is worth clarifying what is meant by such a statement, and to this end we follow the terminology of several seminal works [2–5]. The definition of entanglement is simply the nonseparability of sums of product states that exist in different vector spaces. While there are many options for the DoFs to use with optical fields, e.g., spatial, temporal, frequency, mode index, and even more exotic choices such as path and trajectory [6], here we are interested in the vector spaces made of the polarization and spatial modes, defined by our two DoFs, the former two dimensional and the latter infinite dimensional. The resulting Hilbert space describing the field is the tensor product of the two, i.e., an infinite number of two-dimensional spaces. We will consider the dynamics of one pure state vector in such a Hilbert space. We adopt conventional notation in the literature and call this state a “vector beam” if the polarization is nonhomogeneously distributed in space and a “scalar beam” if homogeneously distributed. Our interest here is in the former: vector beams nonseparable in the polarization and spatial mode DoFs, controversially called “classically entangled” [5,7–9]. Despite the controversy, it is becoming clear that some quantum systems can be tested and probed with classically entangled light [10–12]. For example, classical entanglement has been exploited in quantum error correction [13], quantum state tomography [14], optical communications [15–17], and optical metrology [18–20]. Additionally, the tightly focusing properties of complex vector modes have been exploited in the field of optical tweezers [21–27], micromachining [28], as well as in super-resolution microscopy [29–31].

Recently, there has been an increasing interest in the engineering of vector light beams whose polarization state varies upon free-space propagation. Most of these have focused on the generation of pure vector beams that oscillate from one vector state to another while keeping a constant degree of nonseparability [32–37]. A more interesting case, which encloses the previous cases, reported the generation of light beams whose degree of nonseparability oscillates between scalar and vector modes, offering a tool for the on-demand delivery of specific states to desired positions [38]. Such oscillating modes are generated from the superposition of two counterpropagating vector beams, whose implementation can be cumbersome. While these studies have only considered cylindrical vector vortex modes, the use of new symmetries (spatial shapes), such as parabolic or elliptical, could potentially allow us to unveil properties of vector modes that up to now have remained hidden.

Here we demonstrate a new class of vector beam whose degree of nonseparability features interesting dynamics as it propagates, evolving from a nonhomogeneously polarized vector beam to a quasi-homogeneously polarized beam. Such modes are generated from a nonseparable superposition of orthogonal parabolic beams, which are natural solutions to the Helmholtz equation in parabolic cylindrical coordinates [39–44], and orthogonal polarization states. The entanglement dynamics are quantified through a modified measurement of concurrence C, commonly used to measure the degree of nonseparability in vector modes [45–49]. It is noteworthy that, while it is tempting to view these beams as separable in their two degrees of freedom, vindicated by the local measure of C, their global concurrence remains unchanged.

The engineered vector beams are constructed from a superposition of traveling parabolic-Gaussian (TPG±) beams. Mathematically, TPG beams are given by superposition of the even and odd parabolic-Gaussian (PG) beams as [39]TPG±(r;a)=PGe(r;a)±iPGo(r;a).(1)

Here the functions PGe,o(·) are the even and odd PG modes of the parabolic cylindrical coordinates r=(η,ξ,z), given by [44]$$\begin{gathered} PGe(r;a)=exp(−ikt22kzμ)GB(r)|Γ1|2π2Pe(2ktμξ;a) \\ ×Pe(2ktμη;−a), \end{gathered}$$(2)$$\begin{gathered} PGo(r;a)=exp(−ikt22kzμ)GB(r)2|Γ3|2π2Po(2ktμξ;a) \\ ×Po(2ktμη;−a), \end{gathered}$$where Pe(·) and Po(·) are the even and odd solutions of the parabolic cylindrical differential equation [d2/dx2+(x2/4−a)]P(x;a)=0, and a∈(−∞,∞) represents the continuous order of the beam. Importantly, the solutions Pe(·) and Po(·) can be written in terms of Kummer confluent hypergeometric functions [50], which are available in many numerical libraries. Furthermore, Γ1=Γ[(1/4)+(1/2)ia] and Γ3=Γ[(3/4)+(1/2)ia], with Γ(·) the gamma function; and kt is the transverse component of the wave vector k, whose magnitude is related to the longitudinal component kz as k2=kt2+kz2. Additionally, GB(r) is the fundamental Gaussian beam given by GB(r)=exp(ikz)μexp(−r2μω02),(3)where μ=μ(z)=1+iz/zr with zr=kω02/2 being the usual Rayleigh range of a Gaussian beam. The parabolic coordinates r=(η,ξ,z) are related to the Cartesian coordinates as x=(η2−ξ2)/2 and y=ηξ, where η∈[0,∞) and ξ∈(−∞,∞). For values γ=ktωo≫1, the PG beam propagates in a nondiffracting way within the range [−zmax,zmax], where zmax=ω0k/kt.

The traveling parabolic-Gaussian vector (TPGV) beams are generated as a nonseparable weighted superposition of the polarization and spatial degrees of freedom. Here the polarization degree of freedom is encoded in the circular polarization basis while the spatial degree of freedom is precisely encoded in the TPG±(·) modes. Mathematically, such superposition can be written as TPGV(r;a)=12[TPG+(r;a)e^R+TPG−(r;a)exp(iϕ)e^L],(4)where the unitary vectors e^R and e^L represent the right and left circular states of polarization, respectively. Finally, the term exp(iϕ) (ϕ∈[−π/4,π/4]) is a phase difference between both constituting modes.

To quantify the degree of nonseparability we relied on a well-known measure from quantum mechanics, the concurrence C, which assigns a value in the range [0,1] to the degree of entanglement [45–49]. The concurrence C is measured by integrating the Stokes parameters Si,i=0,1,2,3 over the entire transverse plane through the relation [46,47]C=1−(S1S0)2−(S2S0)2−(S3S0)2,(5)where Si=∬R2SidA. The Stokes parameters Si are computed from a set of four intensity measurements as [54] $$\begin{gathered} S0=I0,S1=2IH−S0, \\ S2=2ID−S0,S3=2IR−S0, \end{gathered}$$(6)where I0 is the total intensity of the mode and IH, ID, and IR the intensity of the horizontal, diagonal, and right-handed polarization components, respectively. As illustrated in Fig. 3(a), such intensity measurements were acquired by a CCD camera through the combination of a linear polarizer (P) and a quarter-wave plate (QWP2) [55]. Specifically, the intensities of the horizontal (IH) and diagonal (ID) polarization components were obtained by passing the beam through a linear polarizer at 0° and 45°, respectively, whereas intensity corresponding to the RCP component (IR) was obtained by passing the beam simultaneously through a QWP at 45° and a linear polarizer at 90°. As an example, Fig. 3(b) shows the experimental Stokes parameters S0, S1, S2, and S3 for the specific mode TPGV(r,3) at z=0. Such parameters were used to reconstruct the transverse polarization distribution on a 20×20 grid as shown in Fig. 3(c). Here, for the sake of clarity, we also display the transverse intensity profile. For this specific example, we have S1/S0=0.12, S2/S0=0.09, and S3/S0=−0.02, which upon substitution in Eq. (5) yield the value C=0.98, which, as expected, corresponds to a maximally entangled mode.

With the idea of better visualizing the evolution of polarization upon propagation, we mapped the different states of polarization acquired at each plane onto the well-known Poincaré sphere (PS), in which the different polarization states are associated to unique points on the surface of the sphere [54] as shown in Figs. 4(c) and 4(d) for experiment and numerical simulation, respectively. Here the coordinate axes are given in terms of the Stokes parameters S1, S2, and S3. For z=0, all the states of polarization in the transverse plane are linear and therefore mapped to points along the equator. For z>0, such linear polarization states gradually evolve from linear into elliptical and finally to circular. In the PS, this is seen as points along a spiral trajectory connecting the north and south poles, in the intermediate planes, and in the far field as points on the north and south poles. Notice that even though the polarization structure evolves from completely mixed to completely unmixed, the amount of right- and left-elliptically polarized photons remains in equilibrium.

There are several mechanisms by which the concurrence of an initially pure state with maximum nonseparability could decay, but perhaps the most prominent in the context of spatial modes is an evolution toward a mixed state (by interacting with an open system, for example) or by modal scattering and subsequent subspace concatenation (ignoring the full expansion in the vector space and considering only the initial subspace). Neither happens here. Instead, our carefully constructed example highlights the counterintuitive dynamics at play: the local concurrence changes dramatically with propagation, eventually decreasing to zero everywhere; yet the global concurrence remains unchanged with propagation, consistent with the fact that free space is a unitary channel, and thus the state vector remains pure, i.e., there is no evolution to a mixed state and no modal scattering. Colloquially, it appears as if the initial beam is split into two homogeneously polarized parts, and so appears separable, it yet remains fully nonseparable in our Hilbert space, as it remains one coherent mode. Our work suggests the need for new or adapted parameters to quantify the nonhomogeneous distribution of polarization states for vector beams. It is intriguing to wonder if our test case may be viewed as a Young’s experiment in reverse, with our initial beam mimicking interference to the final beam reminiscent of two slits. Further, this interesting behavior may benefit from recent insights into polarization coherence, where an exact equality linking concurrence, visibility, and duality has been obtained [4]. These and other open questions are exciting challenges for follow-up work.

In this work we demonstrated a novel kind of complex light field that upon free-space propagation evolves from maximally mixed and locally nonseparable to completely unmixed and locally separable. More precisely, we generated a vector beam with a nonhomogeneous polarization distribution that upon free-space propagation evolves into a homogeneously polarized mode. Such behavior is directly observed at various propagation distances through a reconstruction of the transverse polarization distribution performed via Stokes polarimetry. This is further evinced by mapping the entire polarization distribution at each plane onto the Poincaré sphere, which exhibits an evolution of the state of polarization from the equator to the poles. Such evolution happens along spirals connecting the north and south poles. A quantification of this free-pace propagation dynamics was performed through the concurrence C, which takes the value C=1 for vector states and C=0 for scalar beams. We noted that a measure of the global concurrence yields a constant value, while when measuring this in smaller sections of the beam, what we called the “local concurrence,” it clearly shows a decrease as a function of the propagation distance, reaching the value C=0 in the far field. In other words, by measuring the local concurrence we observe what appears to be a decay in nonseparability, even though the entire state is coherent and nonseparable. This evinces the need for an alternative definition of that takes into account the local variations of the nonseparability, but this lies beyond the scope of this research. Importantly, the nonseparability dynamics reported here cannot be observed with cylindrical vector beams, as they are an intrinsic property of parabolic vector beams. It is also worth mentioning that the beam’s dynamics can be adjusted through the parameter zmax, which depends on ω0 and kt. Finally, these novel states of light offer a new tool for a wide variety of applications in fields such as optical communications, optical metrology, and optical tweezers, to mention a few.

Acknowledgment. The authors acknowledge fruitful discussions with Prof. Andrea Aiello from the Max Planck Institute.