Schrodinger conceived his wave equation having in mind de Broglie’s famous relation from which we learnt to attribute complementary behavior to quantum objects depending on the experimental situation in question. He also thought of a wave in the sense of classical waves, like electromagnetic waves and others. However, the space-time asymmetry of the equation with governs quantum phenomena lead the scientific community to investigate the new physics this specific wave was about to unveil. It turns out that in certain experimental condition classical light has its behavior dictated by a bidimensional Schrodinger equation for a free particle. This fact is well known for several years (Yariv, 1991; Snyder & Love, 1991; Berman, 1997; Marte & Stenholm, 1997). For this special kind of waves it is possible to define the analog of a Hilbert space and operators which do not commute (as reviewed in section 2) in such a way that the mathematical analogy becomes perfect. A natural question emerging in this context, and the case of the present investigation is the following: how far, in the sense of leaning new physics, can we take this analogy ?
We have been able to show that the generalized uncertainty relation by Robertson and Schrodinger, naturally valid for paraxial waves, can shed new light on the physical context of a beautiful phenomenon, long discovered by Gouy (Gouy, 1890; 1891) which is an anomalous phase that light waves suffer in their passage by spatial confinement. This famous phase is directly related to the covariance between momentum and position and since for the "free particles" we are considering tJxxtJpp — c%p = constant we see that Gouy phase can be indirectly measured from the coordinate and momentum variances, quantities a lot easier to measure than covariance between x and p. On the other hand, as far as free atomic particles are concerned, experiments elaborated to test the uncertainty relation (Nairz et al., 2002) will reveal to us the matter wave equivalence of Gouy phase. Unfortunately the above quoted experiment was not designed to determine the phase and that is the reason why, so far, we have only an indirect evidence of the compatibility of theory and experiment. The last aim of our research is to try to encourage laboratories with facilities involving microwave cavities and atomic beams to perform an experiment to obtain the Gouy phase for matter waves. We believe that Gouy phase for matter waves could have important applications in the field of quantum information. The transversal wavefunction of an atom in a beam state can be treated not only as a continuous variable system, but also as an infinite-dimensional discrete system.
The atomic wavefunction can be decomposed in Hermite-Gaussian or Laguerre-Gaussian modes in the same way as an optical beam (Saleh & Teich, 1991), which form an infinite discrete basis. This basis was used, for instance, to demonstrate entanglement in a two-photon system (Mair et al., 2001). However, it is essential for realizing quantum information tasks that we have the ability to transform the states from one mode to another, making rotations in the quantum state. This can be done using the Gouy phase, constructing mode converters in the same way as for light beams (Allen et al., 1992; Beijersbergen et al., 1993). In a recent paper is discussed how to improved electron microscopy of magnetic and biological specimens using a Laguere-Gauss beam of electron waves which contains a Gouy phase term (McMorran et al., 2011).
Analogy between paraxial equation and Schrodinger equation
One of the main differences in the dynamical behavior of electromagnetic and matter waves relies in their dispersion relations. Free electromagnetic wave packets in vacuum propagate without distortions while, e.g., an initially narrow gaussian wave function of a free particle tends to increase its width indefinitely. However, the paraxial approximation to the propagation of a light wave in vacuum is formally identical to Schrodinger’s equation. In this case they are bound to yield identical results.
We start our analysis by taking the simple route of a direct comparison between the Gaussian solutions of the paraxial wave equation and the two-dimensional Schrodinger equation. Consider a stationary electric field in vacuum
The paraxial approximation consists in assuming that the complex envelope function A(r) varies slowly with z such that 92 A/9z2 may be disregarded when compared to kdA/9z. In this condition, the approximate wave equation can be immediately obtained and reads (Saleh & Teich, 1991)
where Al is the light wavelength.
Consider now the two-dimensional Schrodinger equation for a free particle of mass m
Here, ^(x, y, t) stands for the wave function of the particle in time t. Assuming that the longitudinal momentum component pz is well-defined (Viale et al., 2003), i.e., Apz ^ pz, we can consider that the particle’s movement in the z direction is classical and its velocity in this direction remains constant. In this case one can interpret the time variation At as proportional to Az, according to the relation t = z/vz. Now using the fact that Ap = h/pz and substituting in Equation (3) we get
where Ap is the wavelength of particle. As we can see the Equations (2) and (4) are formally identical.
The analogy between classical light waves and matter waves is more apparent if we use the formalism of operators in the classical approach introduced by Stoler (Stoler, 1981). In this formalism, the function A (x, y, z) is represented by the ket vector | A(z)). If we take the inner product with the basis vectors |x,y), we obtain A(x,y,z) = (x,y|A(z)). The differential operators —i(d/dx) and —i(9/9y) acting on the space of functions containing A(x,y,z) are represented in the space of abstract ket by the operators kx and ky. The algebraic structure of operators kx, ky, x and y is specified by the following commutation relations
The generalized uncertainty relation for light waves
The analogy between the above equations in what concerns the uncertainty relation can be immediately constructed given the formal analogy between the equations.
Consider the plane wave expansion of the normalized wave u(x, t) in one dimension (Jackson, 1999)
The amplitudes A(kx) are determined by the Fourier transform of the u(x,0) (t = 0 for simplicity)
The averages of functions f (x, kx) of x and kx are evaluated as (Stoler, 1981)
Thus, we can write the variances
and the covariance
Equation (12) is the equivalent of generalized Schrodinger-Robertson uncertainty relation but for paraxial waves. It is also true in this context that the evolution given by Equation (2) preserves this quantity. This fact allows us to experimentally assess the covariance axkx by the measurements of axx and ^kxkx, which are quite simple to perform. Moreover, as we show next, axkx is directly related to the Rayleigh length and Gouy phase.
Next, we show one important result which is a consequence of this analogy – the Gouy phase for matter waves. The free time evolution of an initially Gaussian wave packet
according to Schrodinger’s equation is given by (da Paz, 2006)
The comparison with the solution of the wave equation in the paraxial approximation with the same condition at z = 0 yields
The parameter B(t) (w(z)) is the width of the particle beam (of light beam), the parameter R(t) (R(z)) is the radius of curvature of matter wavefronts (wavefront of light), ^(t) (£(z)) is the Gouy phase for matter waves (for light waves). The parameter T0 is only related to the initial condition and is responsible for two regimes of growth of the beam width B(t) (da Paz, 2006; Piza, 2001), in complete analogy with the Rayleigh length which separates the growth of the beam width w(z) in two different regimes as is well known in optics (Saleh & Teich, 1991). The above equations show that the matter wave propagating in time with fixed velocity in the propagation direction and the stationary electric field in the paraxial approximation are formally identical [if one replaces t = z/vz in the Equations (15-17)].
Next we show that (t) is directly related to the Schrodinger-Robertson generalized uncertainty relation. For quadratic unitary evolutions (as the free evolution in the present case) the determinant of the covariance matrix is time independent and for pure Gaussian states saturates to its minimum value,
Since the covariance axp is non-null if the Gaussian state exhibits squeezing (Souza et al., 2008), if one measures &xp, from the above relation it is possible to infer the Gouy phase for a matter wave which can be described by an evolving coherent wave packet. For light waves this is a simple task as can be seen below.
The Gouy phase for light waves
The generalized uncertainty relation for the Gaussian light field can be immediately obtained.
satisfy the equality
Analogue expressions can be found for the second moments of the y transverse component. The saturation at the value 1/4 allows for the determination of the covariance axkx. From Equation (25) and using the expressions (22) and (23) we get
which is a function of z/z0 just like expression (17) for the Gouy phase.
The connection between the Gouy phase and the covariance axkx is of purely kinematical nature. As pointed by Simon and Mukunda (Simon & Mukunda, 1993), the parameter space of the gaussian states has a hyperbolic geometry, and the Gouy phase has a geometrical interpretation related to this geometry.
Note that axkx can be positive or negative according to the Equation (26). However, the Equation (26) was deduced assuming that the focus of the beam is z = 0. If we shift the focus to any position zc , as in the experiment, we must take this into account. The plus and minus sign in Equation (26) can be better understood if we look at the Equation (24)
which agrees with the experimental data as we show in what follows. Here we can see that for light waves propagating in the direction of focus (z < zc) the covariance is negative, on the other hand, for light waves propagating after focus (z > zc) the covariance is positive. Now note Equation (26) suggests that by measuring the beam width w(z) we can indirectly infer the value of axkx and thus the value of the Gouy phase by Equation (24). Next, we describe a simple experiment to measure w(z). To experimentally obtain the beam width as a function of the propagation distance, we use the following experimental arrangement shown in Figure 1 (Laboratory of Quantum Optics at UFMG), where L\ represents a divergent lens, L2 a convergent lens and D is a light detector. With this arrangement we can measure the width of the beam as a function of z. The width of the beam in position z is the width of the intensity curve, adjusted by a Gaussian function. In Figure 2, we show the width of the beam for different distances z, along with the corresponding result for axkx.
Fig. 1. Sketch of experimental arrangement used to indirectly measure the Gouy phase of a focused light beam.
Fig. 2. On the left, the width of Gaussian beam w(z) as a function of propagation direction z. Solid curve corresponds to the Equation (15) and the points were obtained of experiment. On the right, covariance axkx as a function of z — zc. Solid curve corresponds to the Equation (27) and the points were obtained of experiment through the equation (26).
The determination of axkx or w(z) allows us to determine £(z) (see Figure 3).
Macromolecules diffraction and indirect evidence for the Gouy phase for matter waves
Recent experiments involving the diffraction of fullerene molecules and the uncertainty relation are shown to be quantitatively consistent with the existence of a Gouy phase for matter waves (da Paz et al., 2010). In Ref. (Nairz et al., 2002) an experimental investigation of the uncertainty relation in the diffraction of fullerene molecules is presented. In that experiment, a collimated molecular beam crosses a variable aperture slit and its width is measured as a function of the slit width. Before reaching the slit diffraction the molecular beam passes through a collimating slit whose width is fixed at ag = 10 y.m, producing a correlated beam (see Figures 1 and 3 in Ref. (Nairz et al., 2002)).
The wave function of the fullerene molecules that leave the slit of width bg, in the transverse direction, is given by
Fig. 3. Gouy phase for Gaussian light beam as a function of propagation direction z — zc. Solid curve corresponds to the Equation (17) and the points were obtained of experiment through the Equation (24).
where kx is the transverse wave number. The wave function on the screen is given by
and t = z/vz is the propagation time from slit to detector, vz is the most probable speed on the z direction. After some algebraic manipulations we obtain, for the normalized wave function at the detector, the following result
where B(t), R(t) and ^(t) are given by the Equations (15), (16) and (17), respectively. As discussed in Ref. (Viale et al., 2003), given the way the fullerene molecules are produced, it is reasonable to assume that the outgoing beam after the diffraction slit has a random transverse momentum. Due to the thermal production the beam contains different components kx, although it has been collimated (Viale et al., 2003). The beam is an incoherent mixture of wave functions with wavenumber kx randomly distributed according to probability distribution g(0) (kx). This distribution depends on the geometry of the collimator, secondary source, which reduces the beam width in the direction x. The index 0 represents the plane of the secondary source (the plane of the collimator), which means that the loss of coherence of the beam is due to the production mechanism only. It is not physically reasonable to assume that a coherent wave packet leaves the diffraction slit due to the thermal production of the fullerene molecules as discussed above. Therefore, in order to introduce some incoherence along the spatial transverse direction, where the quantum effects occur, we use the formalism of density matrices (Gase, 1994; Ballentine, 1998; Scully & Zubairy, 1997; Fano, 1957). The density matrix of the beam at time t is given by
This allows us to obtain for the density matrix Equation (32), the following result
We observe that the density matrix Equation (34) is a mixed state due to the incoherence of the source. The bar has been used to differentiate the parameters of the pure Gaussian state of matter waves of the respective parameters from a mixed Gaussian state. The quantity Mp is the quality factor of the particle beam. The quantity T0 is a generalization of the definition of time aging (Piza, 2001) (timescale) for partially coherent Gaussian state of matter waves. We see that this quantity is always smaller than the aging time of Gaussian pure states, T0, and in this case, a mixed Gaussian state will spread faster with time than the pure Gaussian states. In the coherent limit 5k ^ 0 (ideal collimation), we obtain the parameters of pure Gaussian state, Equations (15), (16) and (17). In the limit t ^ 0 (the plane of source), we have
where the last exponential term of this equation make the role of the spectral degree of coherence defined in the theory of optical coherence (Mandel & Wolf, 1995). We see that the dependence of this term with the transverse position appears as the difference between the positions and, in this case, the source of fullerenes is a source of type Schell (Mandel & Wolf, 1995). Again, the source of fullerenes we refer to here is the collimation slit and not the oven. With the density matrix, we obtain the intensity at the detector by using x = x’ e t = z/vz, i.e.,
Next, we calculate the new elements of the covariance matrix and obtain the following results
With these new elements, we obtain the following result for the determinant of covariance matrix
This result shows that the determinant of the covariance matrix remains time independent, but has a different value from ^ because now we have an incoherent state. The experimental result for the width Wfwhm (full width at half maximum) at the detector, realized by the group of A. Zeilinger in Ref. (Nairz et al., 2002) is shown in Figure 4 and compared with our theoretical calculation, Equation (39) (where Wfwhm = 2\/2ln2cxx). The points are experimental data extracted from Ref. (Nairz et al., 2002), the dashed curve is the beam width with incoherence effect and without convolution with the detector and the solid curve takes into account both effects. These curves show that to adjust the experimental points with theoretical model, we take into account the convolution with the detector and the partial coherence of the fullerenes source. To take into account the convolution with the detector, we use a detector width FWHM of order of 12 ^m, where we took as reference the value quoted in (Nairz et al., 2002). The parameter that measures the partial coherence in the transverse direction of the beam that best fits the experimental data is given by Skx = 9.0 x 106 . With this value of 5kx we calculate the initial transverse coherence length, i.e., £qx = £x (t = 0) and we obtain £qx = (Skx/v^)-1 ~ 1.3 x 10~7 m. As we do not take into account the coupling with the environment in our model, the initial coherence length remains constant in time, i.e., £x (t) = £qx . To compare the value of the coherence length with the value of the wavelength, we calculate Ap through the equation Ap ^ Az = h/mvz (where vz k, 200 m/s is the most probable speed) and we obtain Ap ~ 2.5 pm. Thus, we have £gx ^ Ap, and the condition discussed in Ref. (Mandel & Wolf, 1995) for a locally coherent source is guaranteed. Because the source size is much larger than transverse coherence length, i.e., oq ^ £gx, the angle of beam divergence of fullerenes produced in the secondary source (collimation slit) is given by a value consistent with the experimental value quoted in Ref. (Nairz et al., 2000) (2 < d < 10 ^rad).
Fig. 4. Beam width of fullerene molecules C70 as a function of slit width. Solid and dashed curves correspond to our calculation, Equation (35), and the points are the experimental results obtained in Ref. (Nairz et al., 2002). Dashed curve corresponds to the incoherent case without convolution with the detector and solid curve corresponds to the case where both effects were taken into account. To adjust the theoretical calculation with the experimental data we use Skx = 9.0 x 106 and t = z/vz = 6.65 ms.
The range of wavelengths along the direction x is given by
The value obtained for the range of wavelengths is the same order of magnitude of the transverse coherence length £qx, what justifies the existence of quantum effects along this direction. The component of the wave vector in the direction z has the value kz = mvz/hi ~ 2.24 x 1012 m_1. The values found for kz and Akx show that kz ^ Akx and thus, paraxial approximation is guaranteed for the partially coherent matter wave beam.