# Photon Enhanced Interaction and Entanglement in Semiconductor Position-Based Qubits

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## Abstract

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## 1. Introduction

## 2. Statement of the Problem

## 3. Position-Based Semiconductor Qubits in the Frame of Semiconductor Photon Communication

#### 3.1. Rabi Flopping Frequency of a Position-Based Qubit

#### 3.2. Representation of the System in a Position Basis

## 4. Photon Emission Due to Transitions in a Semiconductor Position-Based Qubit—Description Based on a Jaynes–Cummings–Hubbard Formalism

#### 4.1. Description of the System of Coupled Position-Based Qubit with a Cavity

#### 4.2. Simulation Results

#### 4.3. Description of System of Two Entangled Coupled Position-Based Qubits with a Cavity

#### 4.4. Maximally Entangled States and Entanglement Entropy

#### 4.5. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bashir, I.; Asker, M.; Cetintepe, C.; Leipold, D.; Esmailiyan, A.; Wang, H.; Siriburanon, T.; Giounanlis, P.; Blokhina, E.; Pomorski, K.; et al. A mixed-signal control core for a fully integrated semiconductor quantum computer system-on-chip. In Proceedings of the IEEE European Solid-State Circuits Conference (ESSCIRC), Poland, Krakow, 23–26 September 2019; pp. 125–128. [Google Scholar]
- Leipold, D. Fully integrated quantum computing SOC in 22nm FD-SOI. In Proceedings of the International Workshop on Cryogenic Electronics for Quantum Systems, Fermilab, Chicago, IL, USA, 17–20 June 2019. [Google Scholar]
- Bluhm, H.; Schreiber, L.R. Semiconductor Spin Qubits-A Scalable Platform for Quantum Computing? In Proceedings of the 2019 IEEE International Symposium on Circuits and Systems (ISCAS), Sapporo, Japan, 26–29 May 2019; pp. 1–5. [Google Scholar]
- Lehmann, T. Cryogenic Support Circuits and Systems for Silicon Quantum Computers. In Proceedings of the 2019 IEEE International Symposium on Circuits and Systems (ISCAS), Sapporo, Japan, 26–29 May 2019; pp. 1–5. [Google Scholar]
- Pakkiam, P.; Timofeev, A.; House, M.; Hogg, M.; Kobayashi, T.; Koch, M.; Rogge, S.; Simmons, M. Single-shot single-gate rf spin readout in silicon. Phys. Rev. X
**2018**, 8, 041032. [Google Scholar] [CrossRef] - Ekanayake, S.R.; Lehmann, T.; Dzurak, A.S.; Clark, R.G.; Brawley, A. Characterization of SOS-CMOS FETs at low temperatures for the design of integrated circuits for quantum bit control and readout. IEEE Trans. Electron Devices
**2010**, 57, 539–547. [Google Scholar] [CrossRef] - Bonen, S.; Alakusu, U.; Duan, Y.; Gong, M.; Dadash, M.; Lucci, L.; Daughton, D.; Adam, G.; Iordănescu, S.; Pǎşteanu, M.; et al. Cryogenic Characterization of 22-nm FDSOI CMOS Technology for Quantum Computing ICs. IEEE Electron Device Lett.
**2018**, 40, 127–130. [Google Scholar] - Patra, B.; Incandela, R.M.; Van Dijk, J.P.; Homulle, H.A.; Song, L.; Shahmohammadi, M.; Staszewski, R.B.; Vladimirescu, A.; Babaie, M.; Sebastiano, F.; et al. Cryo-CMOS circuits and systems for quantum computing applications. IEEE J. Solid-State Circuits
**2018**, 53, 309–321. [Google Scholar] [CrossRef] - Lutchyn, R.M.; Sau, J.D.; Sarma, S.D. Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures. Phys. Rev. Lett.
**2010**, 105, 077001. [Google Scholar] [CrossRef] - Maurand, R.; Jehl, X.; Kotekar-Patil, D.; Corna, A.; Bohuslavskyi, H.; Laviéville, R.; Hutin, L.; Barraud, S.; Vinet, M.; Sanquer, M. A CMOS silicon spin qubit. Nat. Commun.
**2016**, 7, 13575. [Google Scholar] [CrossRef] - Crippa, A.; Maurand, R.; Bourdet, L.; Kotekar-Patil, D.; Amisse, A.; Jehl, X.; Sanquer, M.; Laviéville, R.; Bohuslavskyi, H.; Hutin, L.; et al. Electrical spin driving by g-matrix modulation in spin-orbit qubits. Phys. Rev. Lett.
**2018**, 120, 137702. [Google Scholar] [CrossRef] - Corna, A.; Bourdet, L.; Maurand, R.; Crippa, A.; Kotekar-Patil, D.; Bohuslavskyi, H.; Laviéville, R.; Hutin, L.; Barraud, S.; Jehl, X.; et al. Electrically driven electron spin resonance mediated by spin–valley–orbit coupling in a silicon quantum dot. Quantum Inf.
**2018**, 4, 6. [Google Scholar] [CrossRef] - Shinkai, G.; Hayashi, T.; Ota, T.; Fujisawa, T. Correlated Coherent Oscillations in Coupled Semiconductor Charge Qubits. Phys. Rev. Lett.
**2009**, 103, 056802. [Google Scholar] [CrossRef] - Mills, A.; Zajac, D.; Gullans, M.; Schupp, F.; Hazard, T.; Petta, J.R. Shuttling a single charge across a one-dimensional array of silicon quantum dots. Nat. Commun.
**2019**, 10, 1063. [Google Scholar] [CrossRef] - Hensgens, T.; Fujita, T.; Janssen, L.; Li, X.; Van Diepen, C.; Reichl, C.; Wegscheider, W.; Sarma, S.D.; Vandersypen, L.M. Quantum simulation of a Fermi–Hubbard model using a semiconductor quantum dot array. Nature
**2017**, 548, 70. [Google Scholar] [CrossRef] [PubMed] - Van der Wiel, W.G.; De Franceschi, S.; Elzerman, J.M.; Fujisawa, T.; Tarucha, S.; Kouwenhoven, L.P. Electron transport through double quantum dots. Rev. Mod. Phys.
**2002**, 75, 1. [Google Scholar] [CrossRef] - Liu, Y.Y.; Petersson, K.; Stehlik, J.; Taylor, J.M.; Petta, J.R. Photon emission from a cavity-coupled double quantum dot. Phys. Rev. Lett.
**2014**, 113, 036801. [Google Scholar] [CrossRef] - Bose, R.; Cai, T.; Choudhury, K.R.; Solomon, G.S.; Waks, E. All-optical coherent control of vacuum Rabi oscillations. Nat. Photonics
**2014**, 8, 858. [Google Scholar] [CrossRef] - Fink, J.; Göppl, M.; Baur, M.; Bianchetti, R.; Leek, P.; Blais, A.; Wallraff, A. Climbing the Jaynes–Cummings ladder and observing its nonlinearity in a cavity QED system. Nature
**2008**, 454, 315. [Google Scholar] [CrossRef] - Akopian, N.; Lindner, N.; Poem, E.; Berlatzky, Y.; Avron, J.; Gershoni, D.; Gerardot, B.; Petroff, P. Entangled photon pairs from semiconductor quantum dots. Phys. Rev. Lett.
**2006**, 96, 130501. [Google Scholar] [CrossRef] - Deppe, F.; Mariantoni, M.; Menzel, E.; Marx, A.; Saito, S.; Kakuyanagi, K.; Tanaka, H.; Meno, T.; Semba, K.; Takayanagi, H.; et al. Two-photon probe of the Jaynes–Cummings model and controlled symmetry breaking in circuit QED. Nat. Phys.
**2008**, 4, 686. [Google Scholar] [CrossRef] - Araneda, G.; Higginbottom, D.B.; Slodička, L.; Colombe, Y.; Blatt, R. Interference of single photons emitted by entangled atoms in free space. Phys. Rev. Lett.
**2018**, 120, 193603. [Google Scholar] [CrossRef] - Lee, C.J. Photon Emission Dynamics of a Two-Level Atom in a Cavity. arXiv
**2015**, arXiv:1512.08607. [Google Scholar] - Luo, Y.; Liu, N.; Li, X.; Hone, J.C.; Strauf, S. Single photon emission in WSe2 up 160 K by quantum yield control. 2D Mater.
**2019**, 6, 035017. [Google Scholar] [CrossRef] - Liu, Y.x.; Miranowicz, A.; Koashi, M.; Imoto, N. Realization of symmetric sharing of entanglement in semiconductor microcrystallites coupled by a cavity field. Phys. Rev. A
**2002**, 66, 062309. [Google Scholar] [CrossRef] [Green Version] - Giounanlis, P.; Blokhina, E.; Pomorski, K.; Leipold, D.; Staszewski, R.B. Modeling of semiconductor electrostatic qubits realized through coupled quantum dots. IEEE Access
**2019**, 7, 49262–49278. [Google Scholar] [CrossRef] - Giounanlis, P.; Blokhina, E.; Leipold, D.; Staszewski, R.B. Occupancy oscillations and electron transfer in multiple–quantum–dot qubits and their circuit representation. In Proceedings of the 25th IEEE International Conf. on Electronics Circuits and Systems (ICECS), Bordeaux, France, 9–12 December 2018; pp. 153–156. [Google Scholar]
- Gerry, C.; Knight, P. Introductory Quantum Optics; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Orlando, T. Two-Level System with Static and Dynamic Coupling; Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. 2004. Available online: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-728-applied-quantum-and-statistical-physics-fall-2006/study-materials/twolevel04.pdf (accessed on 20 July 2019).
- Dubbers, D.; Stöckmann, H.J. Quantum Physics: The Bottom-up Approach: From the Simple Two-Level System to Irreducible Representations; Springer Science & Business Media: Berlin, Germany, 2013. [Google Scholar]
- Terhal, B.M. Bell inequalities and the separability criterion. Phys. Lett. A
**2000**, 271, 319–326. [Google Scholar] [CrossRef] [Green Version] - Anaya-Contreras, J.A.; Moya-Cessa, H.M.; Zúñiga-Segundo, A. The von Neumann entropy for mixed states. Entropy
**2019**, 21, 49. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Cross-section of an FDSOI transistor-like ‘quantum’ device, without the source/drain diffused regions. Details of the specific technology are omitted. As the device dimensions decrease, one can achieve quantum operation at cryogenic temperatures. Various layers of the device facilitate different properties which are essential to the resulting potential energy profile. (

**b**) Each device can facilitate a double-quantum-dot (DQD). When interconnected via a waveguide, it can couple the entangled qubits. This can be achieved with the use of high-$\u03f5$ materials and proper isolation of the quantum core from the rest of the surrounding circuitry of a chip. (

**c**) Top-view of a layout structure of two coupled DQDs interconnected via a waveguide. (

**d**) 1D representation of two coupled (DQDs). Each DQD can facilitate a qubit. In the schematic, the two states of each qubit are denoted, as ${\left|0\right.\u232a}_{A,B}$, corresponding to a particle in the left QD of system-A or system-B (of the corresponding potential energy profile of a device), and as ${\left|1\right.\u232a}_{A,B}$, corresponding to a particle in the right QD. (

**e**) 1D schematic representation of a potential energy profile formed by a chain of devices forming a series of QDs. The absorption of a photon can cause a transition to a higher energy level. Similarly, the emission of a photon can occur when a particle transitions from a higher energy level to a lower energy level.

**Figure 2.**(

**a**) Finite element method (FEM) COMSOL simulations of the electrostatically shaped potential energy as a function of distance assuming carrier freezout operation for a chain of six devices for a quantum register. Appropriate voltage configuration allows one to construct a desired potential energy profile of a desired mode of operation. In the figures, “S/D” denote the source/drain, while “${I}_{1}$–${I}_{7}$”denote the imposers (gates). It can be seen that the potential energy profile can be approximated by an equivalent square-potential energy profile. The position-based qubit can be defined in a region of two potential wells separated by a barrier. For example, as shown, a qubit can be defined between the imposers “${I}_{3}$–${I}_{5}$”. The double well can be approximated by a DQD, The smaller the dimensions of the structure the more accurate this approximation. (

**b**) By manipulating ${\mathrm{V}}_{GS}$ (gate-source) and ${\mathrm{V}}_{DS}$ (drain-source) applied voltages, one can achieve various potential energy profiles. In the schematic, the potential well bottoms between the imposers “${I}_{3}$–${I}_{5}$” are raised, which is equivalent to a lowered energy barrier between them. With the use of such an electrostatic mechanism, one can manipulate the resulting tunneling probability between the wells. Such an alteration of a potential energy profile can be achieved in a specific implementation by keeping ${\mathrm{V}}_{GS}$, ${\mathrm{V}}_{DS}$ voltages constant and by applying pulses at the imposers at precisely controlled time instances, duration and magnitude. Depending on the magnitude of the pulse, one can pump enough energy into the system so the particle (assuming in the ground state) gets excited from the ground energy and jumps to a new energy level. These transitions between energy levels due to the perturbed driving field can cause photon emission, typically of the same energy as the perturbation. This mechanism is similar to the absorption of a photon from a cavity. In this case, if the photon’s energy is similar to the gap between the two energy levels in a potential well, the particle (assuming already in the ground state) can become excited to allow tunneling.

**Figure 3.**Potential energy profile as a piecewise approximation of a double-well with four energy-levels. This function is extracted from COMSOL electrostatic simulations, assuming carrier freezout operation. The number of bound states in the double well depends on a particular voltage configuration applied at the gates. Therefore, it is possible to electrostatically control the energy gaps between the different energy levels. In addition, the energy levels of a given potential energy double well determine the allowed energies of photons that are possible to emitte and absorb during the quantum operation between communicating qubits.

**Figure 4.**(

**a**) Evolution of the probability of the states $\left(\right)$ and $\left(\right)$. These states correspond to the same position (localized state $\left|0\right.\u232a$) but different energy levels (ground state $\left|g\right.\u232a$ and excited state $\left|e\right.\u232a$). It is observed that they are anti-correlated. (

**b**) Time Evolution of the probability of the states $\left(\right)$ and $\left(\right)$. Similarly, these states correspond to the same position (in this case the localized state $\left|1\right.\u232a$). Energy anti-correlation is also observed in this case. (

**c**) The time evolution of the probability of the states $\left(\right)$ and $\left(\right)$. These states correspond to the same ground energy level (energy state $\left|g\right.\u232a$) but different position (localized states $\left|0\right.\u232a$ and $\left|1\right.\u232a$). Therefore, they are anti-correlated. (

**d**) Time evolution of the probability of the states $\left(\right)$ and $\left(\right)$. These states correspond to the same excited energy level (in this case energy state $\left|e\right.\u232a$) but different position (localized states $\left|0\right.\u232a$ and $\left|1\right.\u232a$). They are also anti-correlated.

**Figure 5.**(

**a**) Simulated quantum path as a result of transitions from the localized ground state $\left|g\right.\u232a$, to the localized excited state $\left|e\right.\u232a$. The transitions take place in a probabilistic manner, following the time evolution determined from the solution of (15). (

**b**) Number of emitted/absorbed photons as a result of transitions between the ground $\left|g\right.\u232a$ and the excited $\left|e\right.\u232a$ states for the simulated quantum path. It is evident that transitions from the ground state $\left|g\right.\u232a$ to the excited state $\left|e\right.\u232a$ correspond to an absorption of a photon, while the transitions from the excited state $\left|e\right.\u232a$ to the ground state $\left|g\right.\u232a$ correspond to an emission of a photon.

**Figure 6.**(

**a**) Evolution of probability as a function of time of the ground state of system-A, ${P}_{\left(\right)}(t)+{P}_{\left(\right)}$, together with that of the ground state of system-B, ${P}_{\left(\right)}(t)+{P}_{\left(\right)}$. Remarkably, the energy states of the two qubits are anti-correlated. (

**b**) Evolution of the probability of the localized state $\left(\right)$ of system-A, ${P}_{\left(\right)}(t)+{P}_{\left(\right)}$, together with the evolution of probability of the localized state $\left(\right)$ of system-B, ${P}_{\left(\right)}(t)+{P}_{\left(\right)}$ as a function of time. In this case, the anti-correlation is not trivial, and the result will depend on initial conditions. This is expected, since we assumed that there is no electrostatic interaction between the two particles, i.e., the Coulomb force is negligible. Considering this, the position of each particle is not expected to be “strongly” affected by the position of the other, only their energy states.

**Figure 7.**(

**a**) Quantum paths of system-A and system-B. The methodology to obtain these graphs is similar to the one discussed for the system of a single qubit. It can be seen that the quantum trajectories of the two systems are anti-correlated. In other words, when system-A is in the ground state $\left(\right)$, system-B is in the excited state $\left(\right)$; (

**b**) The emitted and absorbed photons between the two systems are anti-correlated. When system-A emits a photon, system-B absorbs a photon, and vice versa.

**Figure 8.**Von Neumann entanglement entropy ${S}_{\mathrm{N}}$ defined between system-A and system-B. The maximum value of ${S}_{\mathrm{N}}$ for this system is $2ln2$. When this is the case, the quantum states are maximally entangled (inseparable). Thus, the wave-function cannot be represented as a product of the wave-functions of each particle. One can observe from the plot that ${S}_{\mathrm{N}}$ is a time-dependent quantity. For this system it reaches a maximum value $2ln2$. The states of the two coupled qubits via the waveguide are entangled.

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**MDPI and ACS Style**

Giounanlis, P.; Blokhina, E.; Leipold, D.; Staszewski, R.B.
Photon Enhanced Interaction and Entanglement in Semiconductor Position-Based Qubits. *Appl. Sci.* **2019**, *9*, 4534.
https://doi.org/10.3390/app9214534

**AMA Style**

Giounanlis P, Blokhina E, Leipold D, Staszewski RB.
Photon Enhanced Interaction and Entanglement in Semiconductor Position-Based Qubits. *Applied Sciences*. 2019; 9(21):4534.
https://doi.org/10.3390/app9214534

**Chicago/Turabian Style**

Giounanlis, Panagiotis, Elena Blokhina, Dirk Leipold, and Robert Bogdan Staszewski.
2019. "Photon Enhanced Interaction and Entanglement in Semiconductor Position-Based Qubits" *Applied Sciences* 9, no. 21: 4534.
https://doi.org/10.3390/app9214534