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Controlled Phase Gate Based on an Electron Floating on Helium
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CHIN. PHYS. LETT. Vol. 28, No. 5 (2011) 050302
Controlled Phase Gate Based on an Electron Floating on Helium
*
SHI Yan-Li(石艳丽), MEI Feng(梅峰), YU Ya-Fei(於亚飞), ZHANG Zhi-Ming(张智明)** Key laboratory of Photonic Information Technology of Guangdong Higher Education Institutes, SIPSE and LQIT, South China Normal University, Guangzhou 510006
(Received 30 December 2010) We propose a scheme to generate the controlled phase gate by using an electron floating on liquid helium. The electron is also driven by a classical laser beam and by an oscillating magnetic field. In the process, the vibration of the electron is used as the qubus to couple the energy level qubit (1D Stark-shifted hydrogen) and spin qubit. Ultimately, the controlled phase gate can be generated.
PACS: 03.67.−a, 42.50.Ct, 73.20.−r
DOI: 10.1088/0256-307X/28/5/050302
In current schemes for quantum information processing with trapped ions, one can use laser-induced interactions to implement single-qubit rotations and multi-qubit entangled gates through coupling the internal states of ions to their external states.[1] Recently, Platzman et al.[2,3] proposed that the electrons floating on liquid helium could be used to implement quantum computation. Zhang et al.[4] proposed an implementation of this model by using a single classical laser beam to drive an electron floating on liquid helium. Two energy levels and quantized vibration of the trapped electron can be coupled together by using a classical laser field. We consider that the electronic spin can also act as qubit. As discussed in Refs. [5– 8] by applying a static magnetic field gradient, the qubit addressing and state coupling of vibration and spin states can be realized. This is required for twoion entangling gates. In this Letter, we use a single mode classical laser beam and an oscillating magnetic field to drive the electron floating on the surface of liquid helium. We can realize the coupling between the internal energy level, spin and vibration of the trapped electron. With the vibration degree of freedom as sharing bus, we can find the controlled phase gate based on the internal degree of freedom and the spin degree of freedom. We consider the system in which one electron floats on the surface of liquid helium. This trapped electron can be regarded as a 1D Stark-shifted hydrogen atom moving in a harmonic potential.[4] We assume that the electron has one ground state |𝑔⟩ and two excited states |𝑒⟩ and |𝑒0 ⟩. The states |𝑒⟩ and |𝑒0 ⟩ are nondegenerate and the transition frequency of |𝑔⟩ → |𝑒0 ⟩ is very different from that of |𝑔⟩ → |𝑒⟩ and therefore the state |𝑒0 ⟩ is not affected during the |𝑔⟩ → |𝑒⟩ interaction. The quantum information is stored in the states |𝑔⟩ and |𝑒0 ⟩.[4] The 𝑥-direction harmonic oscillator is initially prepared in the Fock state |𝑛⟩ (𝑛 is its occupation number). In order to couple the two
degrees of freedom |𝑒⟩ (|𝑔⟩) and |𝑛⟩, we apply a classical laser beam, propagating along the 𝑥 direction (see Fig. 1). This is similar to the trapped-ion system with coupling between the external and internal degrees of freedom.[4] The oscillating magnetic field can be used to couple the electron spin | ↑⟩ (| ↓⟩) and |𝑛⟩ and can be utilized to implement single-qubit rotations and multi-qubit entangled gates.[1] The applied laser beam (of wave vector 𝑘𝑙 , amplitude 𝐸0 , frequency 𝜔𝑙 and initial phase 𝜙) and magnetic field (of ˜ ′ on the 𝑥 direction) take the forms of field gradient 𝐵 𝑥 ˜𝑥′ cos(𝜔𝑡 + 𝜑), 𝐸 = 𝑒𝑧 𝐸0 cos(𝑘𝑙 𝑥 − 𝜔𝑙 𝑡 + 𝜙), 𝐵 = 𝑒𝑥 𝑥𝐵 [1] respectively.
z x h
Helium
Fig. 1. A sketch of one electron trapped on the surface of liquid helium. Lower: the microelectrode submerged by the depth ℎ ∼ 0.5 µm beneath the helium surface. Upper: the calculated one-electron potential energy.[3]
The Hamiltonian of the system can be written as[4] (~ = 1) 𝐻 = 𝐻0 + 𝐻𝐸 + 𝐻𝐵 , 𝑗 𝑗 𝐻0 = 𝜔𝐸0 𝜎𝐸𝑧 + 𝜔𝐵0 𝜎𝐵𝑧 + 𝜐𝑎† 𝑎,
𝐻𝐸 = − 𝑑 · 𝐸 (︀ + )︀ [︀ (︀ )︀ ]︀ − = 2Ω𝐸 𝜎𝐸 + 𝜎𝐸 cos 𝜂 𝑎+ + 𝑎 − 𝜔𝑙 𝑡 + 𝜑 , 𝐻𝐵 = − 𝜇 · 𝐵 (︀ + )︀ (︀ + )︀ − = 2Ω𝐵 𝜎𝐵 + 𝜎𝐵 𝑎 + 𝑎 cos (𝜔𝑡 + 𝜙) ,
(1)
where 𝜔𝐸0 is the frequency of the transition |𝑒⟩ → |𝑔⟩, 𝜔𝐵0 is the frequency of the transition | ↑⟩ → | ↓⟩, 𝑗 𝑗 𝜎𝐸𝑧 = 12 (| ↑⟩𝑗 ⟨↑ |−| ↓⟩𝑗 ⟨↓ |), = 21 (|𝑒⟩𝑗 ⟨𝑒|−|𝑔⟩𝑗 ⟨𝑔|), 𝜎𝐵𝑧
* Supported by the National Natural Science Foundation of China under Grant No 60978009 and the National Basic Research Program of China under Grant Nos 2009CB929604 and 2007CB925204. ** Corresponding author. Email:
[email protected] © 2011 Chinese Physical Society and IOP Publishing Ltd
050302-1
CHIN. PHYS. LETT. Vol. 28, No. 5 (2011) 050302
𝑎+ and 𝑎 are the creation and annihilation operators with vibration frequency 𝜐, Ω𝐸 (Ω𝐵 ) is the so-called Rabi frequency describing the strength of coupling between the applied √︀laser (magnetic) field and the electron. Here 𝜂 = 𝑘 ~/2𝑚𝜐 is the so-called Lamb–Dicke + − parameter; 𝜎𝐸 = |𝑒⟩ ⟨𝑔| and 𝜎𝐸 = |𝑔⟩ ⟨𝑒| are the en+ ergy level raising and lowering operators, 𝜎𝐵 = | ↑⟩ ⟨↓| − and 𝜎𝐵 = | ↓⟩ ⟨↑| are the spin raising and lowering operators, respectively; 𝜇 is the electron’s magnetic moment. In the interaction-picture, the Hamiltonian is + 𝑖𝜈𝑡
−𝑖𝜈𝑡
+ 𝑖𝜔𝐸0 𝑡 − −𝑖𝜔𝐸0 𝑡 ) 𝐻𝐼 = Ω𝐸 (𝜎𝐸 𝑒 + 𝜎𝐸 𝑒 )(𝑒𝑖𝜂(𝑎 𝑒 +𝑎𝑒 + 𝑖𝜈𝑡 −𝑖𝜈𝑡 ) 𝑒𝑖(𝜔𝑙 𝑡−𝜙) ) × 𝑒−𝑖(𝜔𝑙 𝑡−𝜙) + 𝑒−𝑖𝜂(𝑎 𝑒 +𝑎𝑒 + 𝑖𝜔𝐵0 𝑡 − −𝑖𝜔𝐵0 𝑡 + Ω𝐵 (𝜎𝐵 𝑒 + 𝜎𝐵 𝑒 )(𝑎+ 𝑒𝑖𝜈𝑡 + 𝑎𝑒−𝑖𝜈𝑡 )
× (𝑒𝑖(𝜔𝑡+𝜑) + 𝑒−𝑖(𝜔𝑡+𝜑) ).
(2)
Considering the behavior of the trapped electron in the Lamb–Dicke regime, we keep the terms only to the first order in 𝜂. If the frequencies of the applied laser beam and magnetic field are tuned to the first red sideband transition, the Jaynes–Cummings model could be effectively realized.[4] In the rotating-wave approximation, the Hamiltonian is given by + + 𝐻𝐼 = 𝑖𝜂Ω𝐸 𝑒𝑖𝜑 𝜎𝐸 𝑎 + Ω𝐵 𝑒−𝑖𝜙 𝜎𝐵 𝑎 + H.c.
(3)
If we drive the electron with the laser field and magnetic field respectively, we can have the evolution
This system has received much attention in recent years for quantum information processing due to its scalability, easy manipulation, and relative long coherence time. The scheme proposed here is feasible in experiments.[10] First, trapping an electron in a small region on the surface of liquid helium by setting suitable microelectrode has been experimentally realized.[11] Second, the liquid helium provides a sufficiently low-temperature surrounding for the system. We know that this coupling between the electron and outside world is very weak at low temperature.[12] Supposing that the liquid helium is cooled to 𝑇 = 0.01 K,[3] then the electronic vibration will be well limited to the vacuum state. The value of the Lamb– Dicke parameter 𝜂 of the electron is much smaller than the trapped ion,[4] and trapping many electrons in this way is easier than trapping many ions. This makes the trapped-electron system more suitable for the large-scale quantum information processing than the trapped-ion system. In summary, we have proposed a scheme for preparing the controlled phase gate based on an electron floating on liquid helium. In the process we use the vibration degree of freedom as the qubus to couple the internal degree of freedom and the spin degree of freedom. We anticipate that the scheme has possible applications in quantum information processing in the future.
𝑈𝐸 |𝑒⟩|0⟩ = cos (𝜂Ω𝐸 𝑡) |𝑒⟩|0⟩ − 𝑒−𝑖𝜑 sin (𝜂Ω𝐸 𝑡) |𝑔⟩|1⟩, 𝑈𝐸 |𝑔⟩|1⟩ = cos (𝜂Ω𝐸 𝑡) |𝑔⟩|1⟩
References
+ 𝑒𝑖𝜑 sin (𝜂Ω𝐸 𝑡) |𝑒⟩|0⟩, 𝑈𝐵 | ↑⟩|0⟩ = cos (Ω𝐵 𝑡) | ↑⟩|0⟩
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− 𝑖𝑒𝑖𝜙 sin (Ω𝐵 𝑡) | ↓⟩|1⟩,
[2] Platzman P M and Dykman M I 1999 Science 284 1967
𝑈𝐵 | ↓⟩|1⟩ = cos (Ω𝐵 𝑡) | ↓⟩|1⟩ − 𝑖𝑒−𝑖𝜙 sin (Ω𝐵 𝑡) | ↑⟩|0⟩,
(4)
where 𝑈𝐸 and 𝑈𝐵 are the evolution operators produced by the laser field and magnetic field, respectively. We choose 𝜂Ω𝐸 𝑡 = 𝜋, Ω𝐵 𝑡 = 𝜋2 , thus we can obtain the controlled phase gate[9] 𝑈
𝐵 | ↓⟩|𝑔⟩|0⟩ −→ | ↓⟩|𝑔⟩|0⟩
𝑈
𝑈
[10] Zhang M, Jia H Y, Huang J S and Wei L F 2010 Opt. Lett. 35 1686
𝑈
𝐵 | ↑⟩|𝑔⟩|0⟩ −→ −𝑖| ↓⟩|𝑔⟩|1⟩
[11] Papageorgiou G, Glasson P, Harrabi K, Antonov V, Collin E, Fozooni P, Frayne P G, Lea M J and Rees D G 2005 Appl. Phys. Lett. 86 153106
𝑈
𝐸 𝐵 −→ 𝑖| ↓⟩|𝑔⟩|1⟩ −→ | ↑⟩|𝑔⟩|0⟩,
𝑈
𝐵 | ↑⟩|𝑒0 ⟩|0⟩ −→ −𝑖| ↓⟩|𝑒0 ⟩|1⟩
𝑈
𝑈
[6] Ciaramicoli G, Marzoli I and Tombesi P 2003 Phys. Rev. Lett. 91 017901
[9] Cirac J I and Zoller P 1995 Phys. Rev. Lett. 74 4091
𝐸 𝐵 −→ | ↓⟩|𝑒0 ⟩|0⟩ −→ | ↓⟩|𝑒0 ⟩|0⟩,
𝐸 𝐵 −→ −𝑖| ↓⟩|𝑒0 ⟩|1⟩ −→ −| ↑⟩|𝑒0 ⟩|0⟩.
[5] Mintert F and Wunderlich C 2001 Phys. Rev. Lett. 87 257904
[8] Wang S X, Labaziewicz J, Ge Y F, Shewmon R and Chuang I L 2009 Appl. Phys. Lett. 94 094103
𝑈
𝐵 | ↓⟩|𝑒0 ⟩|0⟩ −→ | ↓⟩|𝑒0 ⟩|0⟩
𝑈
[4] Zhang M, Jia H Y and Wei L F 2009 Phys. Rev. A 80 055801
[7] Johanning M, Braun A, Timoney N, Elman V, Neuhauser W and Wunderlich Chr 2009 Phys. Rev. Lett. 102 073004
𝑈
𝐸 𝐵 −→ | ↓⟩|𝑔⟩|0⟩ −→ | ↓⟩|𝑔⟩|0⟩,
𝑈
[3] Dykman M I, Platzman P M and Seddighrad P 2003 Phys. Rev. B 67 155402
(5)
[12] Dykman M I and Platzman P M 2000 Fortschr. Phys. 48 1095
050302-2
