Testing Quantum Dissipation Theory with Electron Diffraction

Anglin and Zurek’s model [10] give the decoherence time for an electron interacting with a resistive semiconductor wall,

where $h$ is Planck’s constant, $k_{\mathrm{B}}$ is Boltzmann’s constant, $T=300$ $\mathrm{K}$ is the temperature, $\rho$ is the resistivity of the wall, $z=h_{P}$ is the electron height above the surface, $\Delta x$ is the transverse coherence length or in other words, the distance between two interfering paths. For the proposed method, a correction factor is needed. The image charge associated with one electron path can overlap with the image charge distribution associated with another electron path. The overlap removes “which-way” information by a factor $C=(z/\Delta x)^{2}$ [19]. The decoherence amount is given by $R_{dec}=\int{dt(C\tau_{dec}^{Zurek})^{-1}}$. The dissipation energy loss is given by [11]:

The energy loss $\Delta E$ is $P\times t_{f}$ (for $\Delta E<<E$), where $t_{f}$ is the flight time of the electron over the surface, and the kinetic energy is $E=mv^{2}/2$. For completeness, we note that the decoherence amount

has a fixed relation to the energy loss and can thus be measured.

In a reversal of the traditional role of a light wave diffracting from a material structure, Kapitza and Dirac’s [20] proposal to diffract electrons from a standing wave of light was realized in 2001 [17]. To find the interaction potential it suffices to study the classical motion of an electron in a standing electromagnetic wave. Consider an electron with velocity $\vec{v}=v\hat{z}$ and a standing wave vector potential

The fields are given by

and magnetic component

The phase differences between the fields rectify the oscillatory forces so that the electron experiences a time averaged ponderomotive potential

where $I=\epsilon_{0}cE^{2}_{0}/2$ is the intensity of the laser. This potential writes a periodic phase pattern on the electron wave which is used in the electron diffraction calculation.

The diffraction pattern due to the Kapitza-Dirac effect can be obtained using Feynman’s path integral approach [21]. This method provides convenient control over the amount of coherence present in the electron’s quantum mechanical motional state.

We start with the initial wave function described by a source point,

where $x^{\prime\prime\prime}$ represents the position at the source plane, $\chi$ represents the source point. Next we propagate from the source plane to the second slit using the kernel

The wave function becomes

where $x^{\prime\prime}$ represents a position at the slit plane, and the kernel is given by with $\ell_{x^{\prime\prime\prime}\to x^{\prime\prime}}$ being the distance between the source point and the location at the second slit. The second collimating slit has a Gaussian transmission profile to avoid edge effects. The propagation from the second slit plane to right before the laser region is given by

While crossing the laser region, there electron wave after interacting with the laser is

where the phase is

where $q$ the charge of the electron, $V_{p}$ is given by (7), and $t_{\ell}=D/v_{e}$ is the time it takes the electron with velocity $v_{e}$ to cross the laser region of width $D$. Finally, the electron wave is propagated to the detection screen plane,

With $x$ representing the position at the screen. The probability on the detection screen is given by

which gives the diffraction pattern produced by a single source point at $\chi$ (eq. (8)).

To control the amount of coherence, the diffraction pattern probability distribution is integrated over the source point distribution. This incoherent sum reduces the contrast of the diffraction pattern. Because the incoherent sum can be performed at any intermediate plane, the source width controls the amount of coherence at any location. The source shape is chosen to be a Gaussian profile with standard deviation $\sigma_{e}$. A finite detection size is modeled by convolution with a detection slit of width $\sigma_{d}$,

The FWHM of the anti-diagonal of the density matrix is used as the measure of the transverse coherence length. In the position representation, the density matrix for one source point is given by $\rho_{P}=\langle x_{i}|\Psi\rangle\langle\Psi|x_{j}\rangle$. At the location right before the laser, the incoherent sum is given by

As the source width is increased, the FWHM of the anti-diagonal decreases, consistent with Heisenberg uncertainty principle. The initial source width (that is the first slit width) sets the coherence length of the electron’s density matrix $\rho_{in}(x_{i},x_{j})$ just before it enters the standing wave of light. This density matrix is affected by the source width to model the effect of decoherence as described by the “decohered” density matrix $\rho_{dec}(x_{i},x_{j})$. The ratio of the coherence terms in these density matrices at a location $(x_{i},x_{j})=(x+\frac{1}{4}\sqrt{2}\Delta x,x+\frac{1}{4}\sqrt{2}\Delta x)$ is given by

The transverse coherence length, $x_{coh}$, is found by solving

The parameters are chosen within the range achieved in previous experiments (Fig. 1). The $2.5\ keV$ electron beam passes through a first slit of width $w_{1}$ (varied from 6.7, 22 to 100 microns for the results in Figs. 2 and 3) and a second collimating slit of 1 micron placed 24 $cm$ downstream. A gallium arsenide plate of 40 microns of length placed 1 $mm$ after the second slit is illuminated with $<800\ nm$ light is located underneath of the diffracted electron beam at a variable height $h_{p}$. The intensity of the illumination is used to excite the band gap, populate the conduction band and thus decrease its resistivity to a value of $144\ \Omega m$. Note that not-illuminated GaAs has a resistivity exceeding $10^{6}\ \Omega m$ (vertical-gradient-freeze grown undoped GaAs crystals with 110-crystalline orientation is commercially available with 2.5-5.1 $10^{6}\ \Omega m$) and several $mW$ of light can easily reduce the resistivity to $5\ \Omega m$ [22]. This provides a convenient continuous control of the resistivity. The time of flight is $t_{f}=1.3\times 10^{-12}\ s$. Then, after a downstream distance of $1\ cm$ the electron beam interacts with a standing wave of light from a $532\ nm$ wavelength laser with a laser beam waist of 125 microns placed perpendicular to the initial direction of the beam. After a further distance of $24\ cm$ a 5 micron detection slit was used in the simulation.

A diffraction pattern was obtained for a laser intensity of $I=1\times 10^{14}\ W/m^{2}$, which is shown in the curve in blue in Fig. 2. The decoherence amount was varied by changing the incoherent source width $w_{1}$ (see above) to simulate a loss of coherence due to the interaction of the electron beam with the metal plate placed 1 or 2 microns underneath the beam (Fig. 2). The time it takes the electron to cross the laser region is $t_{\ell}=4.2\times 10^{-12}\ s$ for $h_{p}=2\ \mu m$ and $t_{\ell}=4.8\times 10^{-12}\ s$ for $h_{p}=1\ \mu m$. The decoherence amount was measured using the FWHM of the gaussian distribution of the density matrix anti-diagonal. For the incoherent source with $w_{1}$ of $6.9\ \mu m$, the FWHM occurred at $\pm 200\ \mu m$. After changing the width $w_{1}$ to $22\ \mu m$, the value of the gaussian at $\pm 200\mu m$ was approximately $\exp{(-2.185)}$ smaller, i.e. there was a decoherence amount of 2.185. The energy of the beam was set to lower value according to eq. (2) in order to take into account both the decoherence and the energy loss after the interaction of the beam with the metallic plate. This is shown in the same figure with the curve in maroon. This curve shows how the lateral peaks of the diffraction pattern are barely visible due to decoherence. This was done again, but now with a decoherence amount of 70 caused by the plate now being placed at a height $h_{p}$ of 1 micron underneath the beam, which is shown in the same figure yellow. This curve now shows how the diffraction pattern has been completely lost.
Fig. 3 shows the same procedure described before, but now with a laser intensity of $I=18\ W/m^{2}$. This was done in order to be able to see more diffracting orders, as the energy loss is more visible as the diffraction order increases. Indeed, the 13th peak of the diffraction pattern with a 2.185 decoherence amount shifts by less than 2%, while the pattern obtained with a decoherence amount of 69.9, there is a shift of almost 35%.

Not only can the dissipation rate and decoherence rate be measured in one physical system, but one measurement (Fig. 3) suffices to test the Caldeira-Leggett prediction. This is experimentally advantageous, as the effect of rate changes or other experimental drifts are ameliorated. Assuming that the proposed parameters regime can be reached and that dissipation and decoherence play their predicted role, the physical system can be varied in interesting ways. If the surface chosen is a semi-conductor then electrons in the valence band can be excited to the conduction band to lower the resistivity and thus develop decoherence-free surfaces to act as a wave guide for electrons [22]. For thin surfaces the excitation can proceed through back-illumination [23].

An alternative way to the use of the Kapitza-Dirac effect is the use of nano-fabricated gratings. They have been show to support the detection up to the $20^{th}$ diffraction order [24]. However, the detection rate at higher orders was low for the typical $0.5$ open fraction gratings ($50\ nm$ slit, $100\ nm$ periodicity). To overcome that problem, nano-fabricated grating with narrower slits would be needed. The single slit diffraction pattern acts as the envelope for the grating diffraction pattern. For narrower slits the envelop broadens and enhances the higher diffraction orders. It appears that $10\ nm$ wide slit is within current nano-fabrication capabilities, and our proposed method, encourages such work to be performed.

The choice of placing the surface before or after the light grating can also be considered. It may appear natural to place the surface after the light grating when considering two paths starting at the grating that interfere in the detection plane. However as each point source at the collimating slit illuminates the grating coherently, which in turn recombines the electron paths, the thought experiment is an interferometer with separate paths. At the location where the paths are separated the most is where the surface causes most decoherence, regardless if that is before are after the grating.

The proposed method could fail to test the predicted relation between the dissipation and the decoherence rate, if the classical dissipation rate estimate is too high. This could be compensated by a closer proximity to the wall. On the other hand the length scale could become so small that for all practical purposes the quantum regime is reached and an estimate of a quantum dissipation rate is needed [25, 26].

Finally, we note that it may be of interest to theoretically investigate the decoherence of ion beams by nearby conducting surface to establish if that will also make the testing of quantum dissipation accessible.