Shot noise and spin-orbit coherent control of entangled and spin-polarized electrons

We extend our previous work on shot noise for entangled and spin polarized electrons in a beam-splitter geometry with spin-orbit (SO) interaction in one of the incoming leads (lead 1). In addition to accounting for both the Dresselhaus and the Rashba spin-orbit terms, we present general formulas for the shot noise of singlet and triplets states derived within the scattering approach. We determine the full scattering matrix of the system for the case of leads with two orbital channels coupled via weak SO interactions inducing channel anticrossings. We show that this interband coupling coherently transfers electrons between the channels and gives rise to an additional modulation angle—dependent on both the Rashba and Dresselhaus interaction strengths—which allows for further independent coherent control of the electrons traversing the incoming leads. We derive explicit shot noise formulas for a variety of correlated pairs (e.g., Bell states) and lead spin polarizations. Interestingly, the singlet and each of the triplets defined along the quantization axis perpendicular to lead 1 (with the local SO interaction) and in the plane of the beam splitter display distinctive shot noise for injection energies near the channel anticrossings; hence, one can tell apart all the triplets, in addition to the singlet, through noise measurements. We also find that spin-orbit induced backscattering within lead 1 reduces the visibility of the noise oscillations, due to the additional partition noise in this lead. Finally, we consider injection of two-particle wavepackets into leads with multiple discrete states and find that two-particle entanglement can still be observed via noise bunching and antibunching.

Figure 1

(Color online) (a) Spin-entangled electrons injected into a beam-splitter setup with spin-orbit interactions, Rashba and Dresselhaus, within a finite region $L$ of lead 1. The strength $\alpha$ of the Rashba interaction can, in principle, be controlled via a top gate so as to be equal or unequal to the Dresselhaus coupling $\beta$. For two orbital channels in lead 1 and $\alpha =\beta$, no SO-induced band mixing occurs, right panel (b). For $\alpha \ne \beta$ (or when either $\alpha =0$ or $\beta =0$) the bands anti cross, left panel (b). Only a single spin rotation ${\theta }_{\mathrm{SO}}=2m\sqrt{{\alpha }^2+{\beta }^2}L∕{\hslash }^2$ is present for $\alpha =\beta$, while an additional “mixing” spin rotation ${\theta }_d$ modulates the electron transport in lead 1 for $\alpha \ne \beta$ and impinging energies near the crossing $\epsilon \approx {\epsilon }_c$. This modulation appears in the current fluctuations (shot noise) measured in lead 3. In particular, each of the triplets—for a quantization axis along the $y$ direction—exhibits a distinctive noise as a function of $({\theta }_{\mathrm{SO}},{\theta }_d)$.

Figure 2

Schematic of the quantum wire energy dispersions ${\epsilon }_{s,s^{\prime }}\left(k\right)$ [Eq. (16)] for $\alpha \ne \beta$. The blowup shows the band anticrossing for $d\ne 0$ in more detail. The crossing thin solid lines represent the uncoupled case $d_{ab}=0$. The curves with circles are obtained from Eq. (29) $\left[{\epsilon }_{\pm }\left(k\right)\right]$ and are good approximation for the actual dispersions near crossing point $k_c^0$. The wave vectors $k_{c1}$, $k_{c2}$, and $k_2$, used to expand an incoming plane wave within the SO region [Eq. (35)], are also shown in the inset.

Figure 3

(Color online) Fano factors as a function of ${\theta }_R$ and ${\theta }_d$ for triplet (a) and singlet (b) pairs (defined along the $y$ axis) with injection energies near the channel anticrossings. We assume the injected pairs to be initially in channel $a$ of the non-Rashba region in lead 1. In addition to the usual Rashba-induced spin rotation ${\theta }_R$, the spin-orbit interaction induces a further modulation on the Fano factors via the coherent transfer of electrons between the channels (mixing angle ${\theta }_d$). Interestingly, the triplets display distinct Rashba SO modulations: the unentangled spin-up pair $T_{u_{\uparrow }y}$ is not sensitive to spin-orbit effects and shows full antibunching, the unentangled spin-down triplet $T_{u_{\downarrow }y}$ depends on only ${\theta }_d$ and oscillates between the full anti-bunching and the classical value for distinguishable pairs, the entangled triplet $T_{e_y}$ displays sizable oscillations between full bunching and antibunching as ${\theta }_d$ and ${\theta }_R$ are varied. For the sake of clarity, we have omitted in (a) the ${\theta }_R$ dependence of Fano factors corresponding to $T_{u_{\uparrow }y}$ and $T_{u_{\downarrow }y}$.

Figure 4

(Color online) Two cases for a beam splitter with backscattering. (a) Backscattering in arm 1 only, induced by the SO region, with $a$ the backscattering amplitude, and $r$ and $t$ the original amplitudes of the beam splitter. (b) Backscattering present in all leads (amplitude $a$), with also cross-backscattering (c). Here $r$ and $t$ are the modified amplitudes satisfying the normalization ${\mid a\mid }^2+{\mid c\mid }^2+{\mid r\mid }^2+{\mid t\mid }^2=1$. This case will be discussed in the Appendix .

Figure 5

(Color online) Noise in the presence of backscattering in the lead with SO interaction, for backscattering probabilities given by $A=0$, 0.1, and 0.4. The transmission probabilities are $R=1-T=0.5$ (a), (b); $R=0.9$ (c), (d); and $R=0.1$ (e), (f). The left (right) column shows the case of the singlet $S$ (unentangled triplet $T_{u,z}$) with ${ϵ}_1={ϵ}_2$.

Figure 6

(Color online) Injection of electrons into equidistant discrete lead states $n=0,\pm 1,\pm 2,\dots$, with spacing $\delta$ at center energies ${\epsilon }_{1,2}$ separated by $\Delta ={\epsilon }_2-{\epsilon }_1$ with distributions $g({\epsilon }_{1,2},\epsilon )$ of width $\gamma$. The distributions $g({\epsilon }_{1,2},\epsilon )$ are not drawn to scale, as their normalizations depend on $q$.

Figure 7

(Color online) The function ${\mid h(\Delta ,\delta ,\gamma )\mid }^2$ as expressed in Eq. (131), plotted versus the dimensionless quantities $\Delta ∕\delta$ and $\delta ∕\gamma$, where $\Delta ={\epsilon }_2-{\epsilon }_1$ denotes the mean energy difference between the two injected electrons, $\gamma$ the width of their energy distributions, and $\delta$ the level spacing in the leads. The case of matching energies ${\epsilon }_1\approx {\epsilon }_2$ or $\Delta ⪡\delta ,\gamma$ (Sec. 3E1) corresponds to the edge indicated as “small $\Delta$”; here we find ${\mid h\mid }^2=1$, irrespective of the ratio $\gamma ∕\delta$. The case of a broad energy distribution (fast injection) $\gamma ⪢\Delta ,\delta$ (Sec. 3E4) corresponds to the edge indicated as “large $\gamma$”; along this edge ${\mid h\mid }^2=1$, regardless of $\Delta ∕\delta$. The red arrow follows a line of constant $\gamma$ and $\Delta$ with variable $\delta$. For large $\delta$, we find again ${\mid h\mid }^2=1$ for all $\Delta ∕\gamma$, whereas for small $\delta$ (continuum limit, Sec. 3E3), the limit ${\mid h\mid }^2=0$ or ${\mid h\mid }^2=1$ is reached, depending on whether $\Delta ∕\gamma ⪢1$ or $\Delta ∕\gamma ⪡1$.