Generation of Einstein-Podolsky-Rosen pairs and interconversion of static and flying electron spin qubits

We propose a method of generating fully entangled electron spin pairs using an open static quantum dot and a moving quantum dot, realized by the propagation of a surface acoustic wave (SAW) along a quasi-one-dimensional channel in a semiconductor heterostructure. In particular, we consider a static dot (SD) loaded with two interacting electrons in a singlet state and demonstrate a mechanism which enables the moving SAW dot to capture and carry along one of the electrons, hence yielding a fully entangled static-flying pair. We also show how with the same mechanism we can load the SD with one or two electrons which are initially carried by a SAW-induced dot. The feasibility of realizing these ideas with existing semiconductor technology is demonstrated and extended to yield flying or static pairs that are fully entangled and arbitrary interconversion of static and flying electron spin qubits.

Figure 1

(Color online) Schematic illustration of the proposed system to generate EPR electron spin pairs. (a) An empty static dot (SD) with a confining potential $V_{\mathit{SD}}$ is formed in a depleted $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ one-dimensional channel. A surface acoustic wave (SAW) induces a sine-wave potential $V_{\mathit{SAW}}$, which propagates along the channel (from left to right) forming moving quantum dots. A SAW dot is adjusted to contain two electrons which, in general, may not be in a singlet state. (b) The total potential $V=V_{\mathit{SAW}}+V_{\mathit{SD}}$ enables tunneling whose degree can be efficiently controlled using a time-dependent gate voltage which tunes the depth of the SD. (c) Within a regime of parameters, the two electrons can tunnel from the SAW dot into the SD and can remain bound as the SAW propagates (alternatively, the electrons can tunnel into the SD sequentially). Then, the electrons are allowed to relax to their singlet ground state. (d) The SD contains two electrons in a singlet ground state while the SAW dot is empty. (e) Due to the time-dependent gate voltage, controlled tunneling can take place from the SD into the SAW dot. (f) One electron can tunnel from the SD into the SAW dot, and it can be carried along by the SAW leaving the second electron trapped in the SD, hence generating a static-flying EPR pair. The SD electron can then be ejected into a SAW dot that follows generating a flying EPR pair, or the SAW electron can be captured by a nearby SD generating a static EPR pair.

Figure 2

(a) Variation of the SD depth, which is due to the time-dependent gate voltage, when electrons are transferred from the SD to the SAW dot. (b) The same as in (a) but when electrons are transferred from the SAW dot to the SD. In both cases, $V_w^o$ is the initial depth of the SD, $V_w^f$ is the final depth of the SD, and the tuning takes place from $t_a$ to $t_b$. These quantities are not, in general, the same for both processes. (c) Total potential for four different times when electrons are transferred from the SD to the SAW dot. Note that in this case, the electrons are bound at $t_o=0$ in the SD $(x=0)$ whose depth decreases in time from $t_a$ to $t_b$. On the other hand, when electrons are transferred from the SAW dot to the SD, they are bound at $t_o=0$ in the SAW dot $(x=-0.5\mu \mathrm{m})$ and the SD potential increases in time from $t_a$ to $t_b$. The SAW propagates always from left to right.

Figure 3

(a) Occupation probabilities at the final time $t_f=T=0.37\mathrm{ns}$ as a function of the final depth of the SD when the initial depth of the SD is $V_w^o=13\mathrm{meV}$ and $t_a=0.45T=0.17\mathrm{ns}$, $t_b=0.65T=0.24$, and at the initial time $t_o=0$, the two electrons are trapped in the SD. $P_{\mathit{SD},\mathit{SD}}$ is the probability of finding the two electrons in the SD, $P_{\mathit{SAW},\mathit{SAW}}$ is the probability of finding the two electrons in the SAW dot, and $P_{\mathit{SD},\mathit{SAW}}$ is the probability of finding one electron in the SD and the other in the SAW dot. The inset shows the initial $(t_o=0)$ two-electron distribution (in arbitrary units) in the SD when $V_{\mathit{SAW}}=0$. (b) Total potential and two-electron distribution at final time $t_f=T=037\mathrm{ns}$ when $V_w^f=6\mathrm{meV}$, and at $t_o=0$, the two electrons are trapped in the SD. (c) The same as in (a) but as a function of time when $V_w^f=6\mathrm{meV}$.

Figure 4

Occupation probabilities at final time $t_f=T=0.37\mathrm{ns}$ as a function of the initial depth of the SD when the final depth of the SD is $V_w^f=11\mathrm{meV}$ and $t_a=0.39T=0.14\mathrm{ns}$, $t_b=0.58T=0.21\mathrm{ns}$, and at the initial time $t_o=0$, one electron is carried by the SAW dot. $P_{\mathit{SD}}$ is the probability of loading the SD with a single electron, and $P_{\mathit{SAW}}$ is the probability of keeping the electron in the SAW dot. The inset shows a typical example of the time evolution when $V_w^o=5\mathrm{meV}$.

Figure 5

(a) Occupation probabilities at the final time $t_f=T=0.37\mathrm{ns}$ as a function of the initial depth of the SD when the final depth of the SD is $V_w^f=14.5\mathrm{meV}$ and $t_a=0.42T=0.16\mathrm{ns}$, $t_b=0.7T=0.26\mathrm{ns}$, and at the initial time $t_o=0$, the two electrons are carried by the SAW dot. $P_{\mathit{SD},\mathit{SD}}$ is the probability of loading with two electrons the SD, $P_{\mathit{SAW},\mathit{SAW}}$ is the probability of keeping the two electrons in the SAW dot, and $P_{\mathit{SD},\mathit{SAW}}$ is the probability of loading with a single electron the SD, while the second electron remains in the SAW dot. The inset shows the initial $(t_o=0)$ two-electron distribution (in arbitrary units) in the SAW dot. (b) Total potential and two-electron distribution at final time $t_f=T=037\mathrm{ns}$ when $V_w^o=2\mathrm{meV}$, and at $t_o=0$, the two electrons are carried by the SAW dot. (c) The same as in (a) but as a function of time when $V_w^o=2\mathrm{meV}$.