- Email: [email protected]
Deterministic controlled-phase gate and SWAP gate with dipole-induced transparency in the weak-coupling regime
Optics Communications 379 (2016) 19–24
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Deterministic controlled-phase gate and SWAP gate with dipoleinduced transparency in the weak-coupling regime A-Peng Liu a,n, Liu-Yong Cheng b, Shou Zhang c, Yu Zhao d, Xiao-Zhen Gao e, Yan-Hong Chang e, Ai-Ping Wang e a
Laboratory of Quantum Optics, Shanxi Institute of Technology, Yangquan, Shanxi 045000, China School of Physics and Information Engineering, Shanxi Normal University, Linfen, Shanxi 041004, China c Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China d Personnel Department, Shanxi Institute of Technology, Yangquan, Shanxi 045000, China e Department of Basic Curriculum, Shanxi Institute of Technology, Yangquan, Shanxi 045000, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 14 January 2016 Received in revised form 17 May 2016 Accepted 18 May 2016
We present a scheme to construct a controlled phase-flip (CPF) gate deterministically with the dipole induced transparency (DIT) of a diamond nitrogen-vacancy center embedded in a photonic crystal cavity coupled to two waveguides. Further more, a SWAP gate between a photon and an NV center in cavity is presented with the same quantum system by using the CPF gate. We then show a quantum teleportation scheme between two remote NV centers. The fidelities and efficiencies of the gates can reach relatively high values even if cavity decay and leakage are considered. & 2016 Elsevier B.V. All rights reserved.
Keywords: Quantum logic gate Quantum teleportation Dipole induced transparency Nitrogen-vacancy center
1. Introduction Quantum logic gates are key ingredients for quantum information processing (QIP). Two-qubit controlled phase-flip (CPF) gate together with single-qubit gates are sufficient for universal quantum computing [1]. Among various quantum information carriers, stationary qubits are usually encoded in stationary quantum systems such as stationary atoms, quantum dots, semiconductors, or nitrogen-vacancy (NV) centers, since these systems are suitable for quantum information storing; while flying qubits are usually encoded in flying photons. In this respect, it is significant to construct a quantum SWAP gate between stationary and flying qubits for “the ability to interconvert stationary and flying qubits” [2]. Many proposals have been proposed to realize quantum logic gate with different systems [3–8]. Among all these systems, cavity quantum electrodynamics (QED), as it holds great promise for photon-photon, photon-dipole interactions, has attracted lots of attentions and is regarded as a promising candidate for QIP. Usually, strong-coupling regime is needed to realize efficient light-dipole interaction in cavity QED [9,10]. While in a bad cavity regime (Purcell regime) [11], i.e., the cavity decay rate much n
Corresponding author. E-mail addresses: [email protected] (A.-P. Liu), [email protected] (S. Zhang). http://dx.doi.org/10.1016/j.optcom.2016.05.054 0030-4018/& 2016 Elsevier B.V. All rights reserved.
bigger than the dipole decay rate, the interesting nonlinear optical property can also be observed with a much smaller coupling strength g. So in the weak-coupling regime (low-Q regime), the dipole induced transparency (DIT) can be used for quantum information processing with the Purcell effect [10,12–14]. Recently, NV center in diamond [15] has attracted great interests since the electronic spin state in NV center has long lifetime even at room temperature [16–21]. When an NV center coupled to a microcavity, the zero phonon line emissions relevant to the emitted photons from the single NV center can be significantly enhanced [13,22,23]. Therefore cavity-NV-center system is emerging as one of the most promising candidates for QIP recently. For the past few years, many theoretical [4,24-30] and experimental [23,32,33] efforts have been devoted to QIP based on NV centers. Recently, NV centers in diamond coupled to micro- or nanoresonators with either a strong-coupling strength [33] or a weakcoupling one [34] have been reported. Entanglement generation based on DIT in a cavity-waveguide system has been reported [14,35,36]. Wei et al. [29,30] proposed schemes for universal quantum logic gates based on diamond NV centers coupled to resonators. Recently, Ren et al. [31] proposed a hybrid hypercontrolled-NOT gate and a hybrid hyper-Toffoli gate with DIT of a diamond NV center embedded in a photonic crystal cavity coupled to two waveguides. In this paper, inspired by the above ideas, by using the DIT of a
20
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
diamond NV center embedded in a photonic crystal cavity coupled to two waveguides, we propose deterministic CPF gate between a photon and an NV center. Then a SWAP gate between one NV center and one photon is realized using the CPF gate. As an application, we introduce a scheme to realize quantum teleportation between two remote NV centers with the above two gates. Finally, we analyse the effect of system decoherence on the fidelities and efficiencies. The calculation results illustrate that our schemes are efficient and feasible under appropriate parameter conditions.
2. DIT for doubled-sided cavity-NV-center system
∑
(1)
out
in
a^out′ = a^in′ +
η a^,
ωk (k ¼ 71) is the frequency of the transition between † | + 1〉 ( | − 1〉) and |A2 〉; ωc is the cavity mode; a^ and a^ are the creation and annihilation operators of the cavity field, respectively; σ^+ , σ^− and σ^z are used, respectively, to denote raising, lowering and inversion operators of the NV center between the two corresponding transition levels. Here we suppose a photon with frequency ω is injected into the cavity. Consider the situation that the reservoir temperature is low and neglecting the vacuum input field. The DIT of the cavity-NVcenter system (shown in Fig. 1(a)) can be calculated by the Heisenberg equations of motion for the cavity field operator a^ and the dipole operator σ^− [10,12,38]
η1
need to set the decay rates of the cavity field into two waveguides very close to get approximately the same fidelity for both directions (η1 ≅ η2 = η). Under the assumption of weak excitation limit, i.e., 〈σz 〉 ¼ 1, the reflection coefficient r (ω) and transmission coefficient t (ω) can be described as [10,31]
r (ω) = 1 + t (ω), ⎡ γ⎤ −η ⎢ i (ωk − ω) + ⎥ ⎣ 2⎦ . t (ω) = ⎡ γ ⎤⎡ κ⎤ 2 ⎢⎣ i (ωk − ω) + 2 ⎥⎦ ⎢⎣ i (ωc − ω) + η + 2 ⎥⎦ + g
a′out
(3)
On condition that ωc = ωk = ω , the reflection and transmission coefficients become
2Fp + r=
λ 2 λ 2
−1
t=
λ 2
2Fp + 1 +
,
. (4)
Here Fp = g 2/(ηγ ) is the Purcell factor ( κ ≈ 0) and λ = κ /η. And for an uncoupled cavity system (g ¼0), the reflection and transmission coefficients are
λ 2
r0 (ω) =
λ 2
1+ −1
t0 (ω) =
1+
λ 2
,
. (5)
The dynamics of the photon-NV system can be described as
a out
A2
z
L /R
σ+ η2
(2)
where γ is the decay rate of the emitter; η and κ are the decay rates into waveguide channel modes and cavity intrinsic loss modes, ^ ^ respectively; H and G are the noise operators; a^in ( a^in′) and a^out ^ ( a ′) are the input and output field operators, respectively. We
where
a in
^ η a^in + H ,
⎡ dσ− γ⎤ ^ = − ⎢ i (ωk − ω) + ⎥ σ− − gσz a^ + G, dt 2 ⎦ ⎣ a^ = a^ + η a^,
2Fp + 1 +
⎤ ⎡ ⎢ ωk σ^z, k + ig (a^σ^+, k − a^†σ^−, k ) + ωc a^†a^⎥, ⎥⎦ ⎢ 2 k =+ 1, −1 ⎣
η a^in′ −
out
The system we consider here is a singly NV center embedded in a photonic crystal cavity coupled to two waveguides as shown in Fig. 1. NV center in diamond consists of a substitutional nitrogen atom and an adjacent vacancy, which have trapped an additional electron in it. The energy structure of the system is shown in Fig. 1 (b) with the ground state of electronic spin triplet |3A2 〉. Here |3A2 〉 = |E0 〉 ⊗ |ms = 0, ± 1〉 with spin S ¼1, where |E0 〉 is the orbital state with zero angular momentum projection along the NV axis (z axis in Fig. 1(b)). The excited state is chosen as |A2 〉 = (|E−〉|ms =+ 1〉 + |E+〉|ms = − 1〉) [37], where |E±〉 are orbital states with angular momentum projection 7 1 along the NV axis. The cavity resonantly couples to the transition | ± 1〉 ↔ |A2 〉 for the NV center with coupling constant g. This system is governed by the following hamiltonian (ℏ ¼1) [4,25]
H=
⎡ da^ κ⎤ = − ⎢ i (ωc − ω) + η + ⎥ a^ − gσ− − dt 2 ⎣ ⎦
a′in
−1
L /R
σ−
+1
Fig. 1. The optical transitions of an NV center with circularly polarized lights. (a) A double-sided cavity-waveguide-NV-center system. (b) The optical transitions of an NV center. The photon in the state |R↑〉 or |L↓〉 corresponds to s þ , and the photon in the state |R↓〉 or |L↑〉 corresponds to s-. |R↑〉 ( |R↓〉) and |L↑〉 ( |L↓〉) represent the right- and leftcircularly polarized lights with their input (output) directions parallel (antiparallel) to the z direction.
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
follows [10,31]:
|R↑, − 1〉 → |L↓, − 1〉,
|L↑, − 1〉 → − |L↑, − 1〉,
|R↓, − 1〉 → − |R↓, − 1〉,
|L↓, − 1〉 → |R↑, − 1〉,
|R ↑ ,
|L↑, + 1〉 → |R↓, + 1〉,
+ 1〉 → −
|R ↑ ,
+ 1〉 ,
|R↓, + 1〉 → |L↑, + 1〉,
|L↓, − 1〉 → − |L↓, − 1〉.
(6)
Here, in the left side of “ → ” in Eq. (6), superscript ↑( ↓ ) denotes that the photon is injected into the cavity-NV-center system from down (upper) spatial mode; for the right side of the equations, superscript ↑( ↓ ) denotes that the photon exits from upper (down) spatial mode of cavity-NV-center system. Taking advantage of this double-sided cavity-NV-center system, based on the rules in Eq. (6), we can realize our CPF gate and SWAP gate in a deterministic way.
21
photon in the state |L↑〉 emitted into upper spatial mode (path 3), and the photon in the state |R↓〉 into down spatial mode (path 4). Before the photon coming from the spatial mode 4 reaches the one-way mirror, a polarization-flip operation ( |R〉 ↔ |L〉) is performed with a half-wave plate X. Then after the two wavepackets from spatial modes 5 and 2 pass though PBS2 simultaneously, the system consists of a photon and an NV center involves into
|Ψ 〉2 = αp αc |R, − 1〉 + αp βc |R, + 1〉 − βp αc |L, − 1〉 + βp βc |L, + 1〉.
(9)
From Eq. (9), one can see that when the photon is in the state |L〉 and the NV center is in the state | − 1〉, the phase will be flipped. This result shows that a CPF gate between an NV center and a photon is constructed in a deterministic way.
4. SWAP gate between photon and NV center Based on the above CPF gate, a quantum SWAP gate between one NV center and one photon can be realized. Such a SWAP gate can be written as
3. Controlled phase-flip gate A deterministic CPF gate can be written as
CPF Ucp
⎛1 ⎜ 1 ⎜0 = 2 ⎜⎜ 0 ⎝0
0 1 0 0
0 0 ⎞ ⎟ 0 0 ⎟ 1 0 ⎟ ⎟ 0 − 1⎠
SWAP Ucp
(7)
on the four basis {|R, + 1〉|R, − 1〉, |L, + 1〉 and |L, − 1〉}. As shown in Fig. 2, our device performs a phase flip operation if and only if the photon polarization qubit is in the state |L〉 and the NV center is in the state | − 1〉. Suppose the flying photon and the NV center in the cavity are initially prepared in superposition states |ψ 〉p = (αp |R〉 + βp |L〉), |ψ 〉c = (αc | − 1〉 + βc | + 1〉), respectively. Here
|αp |2 + |βp |2 = |αc |2 + |βc |2 = 1. The subscripts p and c denotes the photon and the NV center, respectively. At first, the injecting photon passes PBS1 which transmits a photon in the polarization state |R〉 while reflects a photon in the state |L〉. After the control photon passes though PBS1, the part in the state |L〉 is injected into the cavity and interacts with the NV center while the part in the state |R〉 does not interact with the cavity. The nonlinear interaction between the photon and the NV center makes the state of the photon-NV combined system evolve to
|Ψ 〉1 = αp αc |R, − 1〉 + αp βc |R, + 1〉 − βp αc |L↑, − 1〉 + βp βc |R↓, + 1〉.
(8)
It can be seen that the doubled-sided cavity-NV-center system acts essentially as a entanglement beam splitter, it leads the
PBS1
PBS2
DL 2
in
out
4 M
0 1 0 0
0⎞ ⎟ 0⎟ , 0⎟ ⎟ 1⎠
(10)
Hp = Hc =
1 2
⎛1 1 ⎞ ⎟, ⎜ ⎝ 1 − 1⎠
(11)
thus we have
Hp |R〉 =
1 (|R〉 + |L〉), 2
Hc | + 1〉 =
Hp |L〉 =
1 (| + 1〉 + | − 1〉), 2
1 (|R〉 − |L〉); 2
Hc | − 1〉 =
OWM X M
Fig. 2. The quantum circuit for constructing a deterministic CPF gate with a flying photon polarization as the control qubit and a confined NV center as the target qubit. PBS: polarized beam splitter; X: half-wave plate which used to perform a polarization bit-flip operation X = |R〉〈L| + |L〉〈R|; OWM: one way mirror; DL: optical delay lines.
(12)
1 (| + 1〉 − | 2
− 1〉).
(13)
Suppose the photon is in the state |φ〉p = (αp |R〉 + βp |L〉) and the NV center is in the state |φ〉c = (αc | − 1〉 + βc | + 1〉). There are three steps needed to realize the SWAP gate as shown in Fig. 3: (1) At the beginning of the scheme, K1 and K2 are adjusted to transmit and reflect photons, respectively. Here Ki (i¼1,2) is an optical switch which can be controlled exactly as needed to transmit or reflect photons. After the photon passes through K1, an Hp operation is applied on the photon by HWP1. K1 needs to be adjusted to reflect photons after the photon transmits it while K2 reflects photons still. The state of the system changes to
1 (αp αc |R, ↑ 〉 + αp αc |L, ↑ 〉 + αp βc |R, ↓ 〉 + αp βc |L, ↓ 〉 2 + βp αc |R, ↑ 〉 − βp αc |L, ↑ 〉 + βp βc |R, ↓ 〉 − βp βc |L, ↓ 〉).
5 3
0 0 1 0
on the four basis { |R, + 1〉, |L, + 1〉, |R, − 1〉, |L, − 1〉}. For conciseness, we define single qubit Hadamard operations Hp and Hc for one photon and one NV center respectively:
|ϕ〉1 =
1
⎛1 ⎜ 0 =⎜ ⎜0 ⎜ ⎝0
(14)
Then the photon is injected into the CPF gate again. After the photon is reflected by K1, K2 and the mirrors, another Hp operation is applied on the photon when the photon passes through HWP1. The state of the system evolves to
|ϕ〉2 = αp αc |L, − 1〉 + αp βc |R, + 1〉 + βp αc |R, − 1〉 + βp βc |L, + 1〉.
(15)
(2) Secondly, for the NV center, two Hc operations are applied by two lasers before and after the photon passes through the CPF gate, respectively. The state of the system evolves to
|ϕ〉3 = αp αc |L, + 1〉 + αp βc |R, + 1〉 + βp αc |R, − 1〉 + βp βc |L, − 1〉.
(16)
22
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
Fig. 4. Schematic setup for teleportation of the state of an NV center. Photon1 and photon2 are prepared in polarization-entangled states.
|Φ〉1 =
1 (α| − 1〉|R〉1|R〉2 + α| − 1〉|L〉1|L〉2 + β| + 1〉|L〉1|R〉2 + β| 2 + 1〉|R〉1|L〉2 ).
(19)
(2) An Hc operation is applied on NV center1 by a classical laser. The state of the NV-photon-photon combined system evolves to 1 (α | + 1〉|R〉1|R〉2 − α | − 1〉|R〉1|R〉2 + α | + 1〉|L〉1|L〉2 − α | − 1〉|L〉1|L〉2 2 + β| + 1〉|L〉1|R〉2 + β| − 1〉|L〉1|R〉2 + β| + 1〉|R〉1|L〉2 + β| − 1〉|R〉1|L〉2 ) 1 = (| + 1〉|R〉1σ x |φ〉p2 + | − 1〉|R〉1σ x σ z |φ〉p2 + | + 1〉|L〉1|φ〉p2 + | − 1〉|L〉1σ z |φ〉p2 ) , (20) 2
|Φ〉2 =
Fig. 3. Schematic setup to realize a SWAP gate between one NV center and one photon. Ki (i¼ 1,2): optical switch.
(3) Subsequently, the photon needs to go through HWP1 and the CPF gate once again. The state of the system evolves to
|ϕ〉4 =
1 (αp αc |R, + 1〉 − αp αc |L, + 1〉 + αp βc |R, + 1〉 + αp βc |L, + 1〉 2
+ βp αc |R, − 1〉 − βp αc |L, − 1〉 + βp βc |R, − 1〉 + βp βc |L, − 1〉). (17) Then K2 is adjusted to transmit photons and a Hp operation should be applied on the photon again by HWP2, the state of the system involves eventually into
|ϕ〉5 = αp αc |L, + 1〉 + αp βc |R, + 1〉 + βp αc |L, − 1〉 + βp βc |R, − 1〉 = (αc |L〉 + βc |R〉)(αp | − 1〉 + βp | + 1〉).
(18)
It seems from the result that we have realized a quantum SWAP gate between one photon and one NV center. It should be pointed out that the time of needed to complete the above three steps should be less than the decoherence time of an NV center. Fortunately, long decoherence time (∼ms) [39,40] and nanosecond manipulation time [41] have been realized, which make our schemes achievable.
5. Teleportation between two remote NV centers
where |φ〉p2 = (α|L〉 + β|R〉). (3) Alice performs measurements on the photon and the NV center at her side, the state of photon2 will collapse to one of the corresponding components. Based on the measurement results from NV center1 and photon1, Bob can deterministically recover the unknown state of photon2 by local operations as listed in Table 1. Here an optical delay line for photon2 to be stored is required during that time when photon1 and NV center1 undergo certain processes (not shown in the figure). After above steps, photon2 maps its information to NV center2 via the SWAP gate and thus the teleportation between two remote NV centers is accomplished.
6. Discussion and summary As the CPF gate is the basic construction, the fidelities of the SWAP gate and the quantum teleportation rely on the fidelity of the CPF gate, in this part we briefly discuss the fidelity of the CPF gate. Recently, higher fidelity of an operation in bad cavity conditions ( FP > 1) rather than the strong-coupling regime has been demonstrated [42], where the maximal fidelity is obtained at the point κ /η = 0.1 in their experiment. Prior works [10,13,31] have demonstrated that the dipole interaction between the photon and the NV center leads to different reflection and transmission coefficients between the right- and left-circularly polarized lights with their input (output) directions parallel (antiparallel) to the z direction. Accordingly, in the resonant condition ( ωc = ωk = ω ), the transmission and reflection rules for a photon become [10,31]
|L↑, − 1〉 → − |t0 ||L↑, − 1〉 − |r0 ||R↓, − 1〉,
|L↑, + 1〉 → |r||R↓,
+ 1〉 + |t||L↑, + 1〉. Our schematic setup to realize teleportation between two remote NV centers is shown as Fig. 4. A pair of maximally polarization-entangled photons is needed in this scheme, e.g., |ψ 〉p = (|L〉1|R〉2 + |R〉1|L〉2 ) / 2 , meanwhile assume that the state of the NV center1 is |φ〉c = (α| − 1〉 + β| + 1〉), here |α|2 þ |β|2 ¼ 1. Suppose photon1 and NV center1 belong to Alice, while photon2 and NV center2 are attached to Bob. Our scheme is comprised of the following steps: (1) Photon1 goes through the HWP1 before it enters into the CPF gate. After the photon experienced the CPF gate, the photon will go through HWP2, HWP1 and HWP2 apply Hp operations on the photon. Then the state of the NV-photon-photon combined system evolves to:
(21)
Substituting Eq. (21) for Eq. (6), and combing the arguments made in Section 3, we find that the state of the system described by Eq. (9) becomes Table 1 Correspondence between Alice's Measurement and Bob's Local Operation in Teleportation. Alice's Measurement
| | | |
+ − + −
1〉|R〉1 1〉|R〉1 1〉|L〉1 1〉|L〉1
Bob's State
Local Operation
α|R〉 + β|L〉 −α|R〉 + β|L〉 α|L〉 − β|R〉 −α|L〉 + β|R〉
sx σz σx I sz
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
|Ψ 〉′4 = αp αc |R, − 1〉 + αp βc |R, + 1〉 − βp αc |t0 ||L, − 1〉 −βp αc |r0 ||L, − 1〉 + βp βc |r||L, + 1〉+βp βc |t||L, + 1〉,
(22)
the terms with underlines indicate the states which take the bitflip error. We can calculate the average fidelity of the CPF gate as
FCPF =
1 4π 2
∫0
2π
dθ1
∫0
2π
dθ 2 |ideal 〈ψ |ψ 〉real |2 ,
(23)
where |ψ 〉ideal refers to the output state of the photon-spin system in the ideal case and |ψ 〉real represents the actual output state. Without loss of generality, we set αp = sin θ1, βp = cos θ1, αc = sin θ 2, βc = cos θ 2 (θ1, θ 2 ∈ [0, 2π ]). Therefore, the fidelity of our CPF gate in the two-qubit hybrid system discussed in Section 3 can be written as
FCPF =
1 4π 2
∫0
2π
dθ1
∫0
2π
dθ2[sin2θ1sin2θ2 + sin2θ1cos2θ2
+ cos2θ1sin2θ2|t0| + cos2θ1sin2θ2|r0| + cos2θ1cos2θ2|t| + cos2θ1cos2θ2|r|]2 = 1.
(24)
It can be seen that, if the effect of the optical elements can be ignored, fidelity of the CPF gate is 1, this increases the fidelities of the SWAP gate and the quantum teleportation directly. Meanwhile we define the efficiency of a quantum gate as the ratio of the number of the outputting photons to the inputting photons. The efficiency of the CPF gate therefore can be expressed as
ξCPF =
1 4π 2
∫0
2π
dθ1
∫0
2π
dθ 2 [sin2 θ1 sin2 θ 2 + sin2 θ1 cos2 θ 2
+ cos2 θ1 sin2 θ 2 |t0 |2 + cos2 θ1 cos2 θ 2 |r|2 ] 1 = [2 + |t0 |2 + |r|2 ], 4
ξSWAP =
1 [2 + |t0 |6 + |r|6 ]. 4
(25)
The efficiencies of our universal quantum gates are shown in Fig. 5. For our schemes, in the weak coupling regime, if λ ¼0.1, Fp = 5, ξCPF = 93.4% and ξSWAP = 82.8% , while ξCPF = 95.7% and ξSWAP = 89.1% when λ = 0; in the strong coupling regime, if λ = 0.5, FP = 11.52, ξCPF = 89.0% and ξSWAP = 82.8% . If the cavity intrinsic loss can be neglected, the efficiencies can reach near unity ( ξCPF = 99.6% and ξSWAP = 98.7% ). It is worth pointing out that FP and λ do not affect the fidelities of the gates and they remain at unity.
23
An NV center in diamond is an appropriate dipole emitter in cavity QED to obtain the high-fidelity reflection-transmission property in the Purcell regime. In the gate operation process, it is necessary to note that the relaxation time of the electron spin in NV center must longer than the photon interval time and operation time [19–21]. In terms of experimental progress, the relaxation time of the electron spin ground states of an NV center ranges from ∼ms at room temperature to ∼s at low temperature [17,21,43]. When an NV center coupled to a photonic crystal resonator (typically, Q∼3000), the zero phonon line emissions relevant to the emitted photons from the single NV center can be significantly enhanced [23,44] with the coupling strength as a few GHz. Strong coupling case FP = 325 in a silicon nitride photonic crystal nanocavity with a Q factor of 1.4 × 106 and a mode volume Vm ∼ 2.5 × 10−20 m3 has been reported [45]. For the present schemes, the cavity resonator mode is coupled to the two waveguides simultaneously, so approximately equal coupling constants are needed. In experiment, the deviation between the decay rates of the cavity field into waveguide channel modes is about Δη ∼ 0.2η, which leads to approximately the same fidelity for both transmission and reflection directions [42]. On the other hand, cavity mirror absorption, scattering, diffraction and inefficient detection may cause photon loss, which will affect the efficiencies of the schemes. It is noteworthy that the present schemes work in a repeat-until-success way, so it merely decrease the success probability but does not affect the fidelities once the photon lost. Furthermore, a highly efficient single-photon server (30,000 high-quality single photons are delivered within 30 s) has been reported [46], this promises a short time to implement our schemes. With the in crease in the number of logic qubits, our schemes would be less efficient, but they are still scalable in principle and are experimentally feasible. In summary, we have proposed a controlled CPF gate with the dipole induced transparency of a diamond NV center embedded in a photonic crystal cavity coupled to two waveguides. Based on the CPF gate and a series of single qubit operations, a SWAP gate between photon and spin in the cavity was proposed. Furthermore, a quantum teleportation scheme was proposed as an example of applications. The fidelities and efficiencies of the gates have been discussed. The results show that our schemes are insensitive to decoherence caused by cavity decay, and photon loss affect merely the efficiency but the fidelity. Our schemes work well in the bad cavity regime (the Purcell regime). In the ideal case (cavity leakage is much lower than the cavity loss), the fidelities and the efficiencies of our gates can reach near unity in the strong coupling regime.
Fig. 5. The efficiency of our CPF gate (a) and SWAP gate (b) versus the Pucell factor FP and λ.
24
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
Acknowledgements A-Peng Liu thank Qi Guo, Zhao Jin and Shi-Lei Su for enlightening discussions. Shou Zhang is supported by the National Natural Science Foundation of China under Grant no. 61465013.
[22] [23] [24] [25] [26]
References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. [2] D.P. DiVicenzo, Fortschr. Phys. 48 (2000) 771. [3] L.M. Liang, C.Z. Li, Phys. Rev. A 72 (2005) 024303. [4] L.Y. Cheng, H.F. Wang, S. Zhang, K.H. Yeon, Opt. Express 21 (2013) 5988. [5] H.R. Wei, F.G. Deng, Phys. Rev. A 87 (2013) 022305. [6] H.R. Wei, F.G. Deng, Sci. Rep. 4 (2014) 7551. [7] M.Z. Zhu, C.R. Zhao, L. Ye., Opt. Commun. 285 (2012) 1576. [8] P. Xue, Chin. Phys. Lett. 27 (2010) 060301. [9] L.M. Duan, H.J. Kimble, Phys. Rev. Lett. 92 (2004) 127902. [10] C.Y. Hu, W.J. Munro, J.L. O'Brien, J.G. Rarity, Phys. Rev. B 80 (2009) 205326. [11] E.M. Purcell, H.C. Torrey, R.V. Pound, Phys. Rev. 69 (1946) 37. [12] C. Bonato, F. Haupt, S.S.R. Oemrawsingh, J. Gudat, D.P. Ding, M.P. van Exter, D. Bouwmeester, Phys. Rev. Lett. 104 (2010) 160503. [13] A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, J. Vuckovic, Opt. Express 16 (2008) 12154. [14] E. Waks, J. Vuckovic, Phys. Rev. Lett. 96 (2006) 153601. [15] M. Stoneham, Physics 2 (2009) 34. [16] T. van der Sar, E.C. Heeres, G.M. Dmochowski, G. de Lange, L. Robledo, T. H. Oosterkamp, R. Hanson, Appl. Phys. Lett. 94 (2009) 173104. [17] P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, J. Wrachtrup, Science 320 (2008) 1326. [18] F. Jelezko, T. Gaebel, I. Popa, A. Gruber, J. Wrachtrup, Phys. Rev. Lett. 92 (2004) 076401. [19] F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, J. Wrachtrup, Phys. Rev. Lett. 93 (2004) 130501. [20] L. Childress, M.V. Gurudev Dutt, J.M. Taylor, A.S. Zibrov, F. Jelezko, J. Wrachtrup, P.R. Hemmer, M.D. Lukin, Science 314 (2006) 281. [21] T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J.
[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
[41] [42] [43] [44] [45] [46]
R. Rabeau, N. Stavrias, A.D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, J. Wrachtrup, Nat. Phys. 2 (2006) 408. J. Wolters, A.W. Schell, G. Kewes, N. Nüsse, M. Schoengen, H. Döscher, T. Hannappel, B. Löchel, M. Barth, O. Benson, Appl. Phys. Lett. 97 (2010) 141108. A. Faraon, P.E. Barclay, C. Santori, K.M.C. Fu, R.G. Beausoleil, Nat. Photonics 5 (2011) 301. Y.J. Zhao, X.M. Fang, F. Zhou, K.H. Song, Chin. Phys. Lett. 30 (2013) 050304. Q. Chen, W.L. Yang, M. Feng, J.f. Du, Phys. Rev. A 83 (2011) 054305. A.P. Liu, L.Y. Cheng, L. Chen, S.L. Su, H.F. Wang, S. Zhang, Opt. Commun. 313 (2014) 180. I.M. Mirza, S.J. van Enk, H.J. Kimble, J. Opt. Soc. Am. B 10 (2013) 2640. I.M. Mirza, S.J. van Enk, Opt. Commun. 343 (2015) 172. H.R. Wei, G.L. Long, Sci. Rep. 5 (2015) 12918. H.R. Wei, G.L. Long, Phys. Rev. A 91 (2015) 032324. B.C. Ren, G.Y. Wang, F.G. Deng, Phys. Rev. A 91 (2015) 032328. Y.S. Park, A.K. Cook, H. Wang, Nano Lett. 6 (2006) 2075. P.E. Barclay, K.M. Fu, C. Santori, R.G. Beausoleil, Opt. Express 17 (2009) 9588. P.E. Barclay, K.M.C. Fu, C. Santori, R.G. Beausoleil, Appl. Phys. Lett. 95 (2009) 191115. I.M. Mirza, Phys. Lett. A 379 (2015) 1643. I.M. Mirza, J. Mod. Opt 62 (2015) 1048. E. Togan, Y. Chu, A.S. Trifonov, L. Jiang, J. Maze, L. Childress, M.V. Gurudev Dutt, A.S. Sørensen, P.R. Hemmer, A.S. Zibrov, M.D. Lukin, Nature 466 (2010) 730. D.F. Walls, G.J. Milburn, Quantum Optics, Springer-Verlag, Berlin, 1994. B.B. Buckley, G.D. Fuchs, L.C. Bassett, D.D. Awschalom, Science 330 (2010) 1212. G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P.R. Hemmer, F. Jelezko, J. Wrachtrup, Nat. Mater. 8 (2009) 383. G.D. Fuchs, V.V. Dobrovitski, D.M. Toyli, F.J. Heremans, D.D. Awschalom, Science 326 (2009) 1520. D. O'Shea, C. Junge, J. Volz, A. Rauschenbeutel, Phys. Rev. Lett. 111 (2013) 193601. J. Harrison, M.J. Sellars, N.B. Manson, Diam. Relat. Mater. 15 (2006) 586. P.E. Barclay, K.M.C. Fu, C. Santori, A. Faraon, R.G. Beausoleil, Phys. Rev. X 1 (2011) 011007. M.W. McCutcheon, M. Lončar, Opt. Express 16 (2008) 19136. M. Hijlkema, B. Weber, H.P. Specht, S.C. Webster, A. Kuhn, G. Rempe, Nat. Phys. 3 (2007) 253.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Deterministic controlled-phase gate and SWAP gate with dipoleinduced transparency in the weak-coupling regime A-Peng Liu a,n, Liu-Yong Cheng b, Shou Zhang c, Yu Zhao d, Xiao-Zhen Gao e, Yan-Hong Chang e, Ai-Ping Wang e a
Laboratory of Quantum Optics, Shanxi Institute of Technology, Yangquan, Shanxi 045000, China School of Physics and Information Engineering, Shanxi Normal University, Linfen, Shanxi 041004, China c Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China d Personnel Department, Shanxi Institute of Technology, Yangquan, Shanxi 045000, China e Department of Basic Curriculum, Shanxi Institute of Technology, Yangquan, Shanxi 045000, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 14 January 2016 Received in revised form 17 May 2016 Accepted 18 May 2016
We present a scheme to construct a controlled phase-flip (CPF) gate deterministically with the dipole induced transparency (DIT) of a diamond nitrogen-vacancy center embedded in a photonic crystal cavity coupled to two waveguides. Further more, a SWAP gate between a photon and an NV center in cavity is presented with the same quantum system by using the CPF gate. We then show a quantum teleportation scheme between two remote NV centers. The fidelities and efficiencies of the gates can reach relatively high values even if cavity decay and leakage are considered. & 2016 Elsevier B.V. All rights reserved.
Keywords: Quantum logic gate Quantum teleportation Dipole induced transparency Nitrogen-vacancy center
1. Introduction Quantum logic gates are key ingredients for quantum information processing (QIP). Two-qubit controlled phase-flip (CPF) gate together with single-qubit gates are sufficient for universal quantum computing [1]. Among various quantum information carriers, stationary qubits are usually encoded in stationary quantum systems such as stationary atoms, quantum dots, semiconductors, or nitrogen-vacancy (NV) centers, since these systems are suitable for quantum information storing; while flying qubits are usually encoded in flying photons. In this respect, it is significant to construct a quantum SWAP gate between stationary and flying qubits for “the ability to interconvert stationary and flying qubits” [2]. Many proposals have been proposed to realize quantum logic gate with different systems [3–8]. Among all these systems, cavity quantum electrodynamics (QED), as it holds great promise for photon-photon, photon-dipole interactions, has attracted lots of attentions and is regarded as a promising candidate for QIP. Usually, strong-coupling regime is needed to realize efficient light-dipole interaction in cavity QED [9,10]. While in a bad cavity regime (Purcell regime) [11], i.e., the cavity decay rate much n
Corresponding author. E-mail addresses: [email protected] (A.-P. Liu), [email protected] (S. Zhang). http://dx.doi.org/10.1016/j.optcom.2016.05.054 0030-4018/& 2016 Elsevier B.V. All rights reserved.
bigger than the dipole decay rate, the interesting nonlinear optical property can also be observed with a much smaller coupling strength g. So in the weak-coupling regime (low-Q regime), the dipole induced transparency (DIT) can be used for quantum information processing with the Purcell effect [10,12–14]. Recently, NV center in diamond [15] has attracted great interests since the electronic spin state in NV center has long lifetime even at room temperature [16–21]. When an NV center coupled to a microcavity, the zero phonon line emissions relevant to the emitted photons from the single NV center can be significantly enhanced [13,22,23]. Therefore cavity-NV-center system is emerging as one of the most promising candidates for QIP recently. For the past few years, many theoretical [4,24-30] and experimental [23,32,33] efforts have been devoted to QIP based on NV centers. Recently, NV centers in diamond coupled to micro- or nanoresonators with either a strong-coupling strength [33] or a weakcoupling one [34] have been reported. Entanglement generation based on DIT in a cavity-waveguide system has been reported [14,35,36]. Wei et al. [29,30] proposed schemes for universal quantum logic gates based on diamond NV centers coupled to resonators. Recently, Ren et al. [31] proposed a hybrid hypercontrolled-NOT gate and a hybrid hyper-Toffoli gate with DIT of a diamond NV center embedded in a photonic crystal cavity coupled to two waveguides. In this paper, inspired by the above ideas, by using the DIT of a
20
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
diamond NV center embedded in a photonic crystal cavity coupled to two waveguides, we propose deterministic CPF gate between a photon and an NV center. Then a SWAP gate between one NV center and one photon is realized using the CPF gate. As an application, we introduce a scheme to realize quantum teleportation between two remote NV centers with the above two gates. Finally, we analyse the effect of system decoherence on the fidelities and efficiencies. The calculation results illustrate that our schemes are efficient and feasible under appropriate parameter conditions.
2. DIT for doubled-sided cavity-NV-center system
∑
(1)
out
in
a^out′ = a^in′ +
η a^,
ωk (k ¼ 71) is the frequency of the transition between † | + 1〉 ( | − 1〉) and |A2 〉; ωc is the cavity mode; a^ and a^ are the creation and annihilation operators of the cavity field, respectively; σ^+ , σ^− and σ^z are used, respectively, to denote raising, lowering and inversion operators of the NV center between the two corresponding transition levels. Here we suppose a photon with frequency ω is injected into the cavity. Consider the situation that the reservoir temperature is low and neglecting the vacuum input field. The DIT of the cavity-NVcenter system (shown in Fig. 1(a)) can be calculated by the Heisenberg equations of motion for the cavity field operator a^ and the dipole operator σ^− [10,12,38]
η1
need to set the decay rates of the cavity field into two waveguides very close to get approximately the same fidelity for both directions (η1 ≅ η2 = η). Under the assumption of weak excitation limit, i.e., 〈σz 〉 ¼ 1, the reflection coefficient r (ω) and transmission coefficient t (ω) can be described as [10,31]
r (ω) = 1 + t (ω), ⎡ γ⎤ −η ⎢ i (ωk − ω) + ⎥ ⎣ 2⎦ . t (ω) = ⎡ γ ⎤⎡ κ⎤ 2 ⎢⎣ i (ωk − ω) + 2 ⎥⎦ ⎢⎣ i (ωc − ω) + η + 2 ⎥⎦ + g
a′out
(3)
On condition that ωc = ωk = ω , the reflection and transmission coefficients become
2Fp + r=
λ 2 λ 2
−1
t=
λ 2
2Fp + 1 +
,
. (4)
Here Fp = g 2/(ηγ ) is the Purcell factor ( κ ≈ 0) and λ = κ /η. And for an uncoupled cavity system (g ¼0), the reflection and transmission coefficients are
λ 2
r0 (ω) =
λ 2
1+ −1
t0 (ω) =
1+
λ 2
,
. (5)
The dynamics of the photon-NV system can be described as
a out
A2
z
L /R
σ+ η2
(2)
where γ is the decay rate of the emitter; η and κ are the decay rates into waveguide channel modes and cavity intrinsic loss modes, ^ ^ respectively; H and G are the noise operators; a^in ( a^in′) and a^out ^ ( a ′) are the input and output field operators, respectively. We
where
a in
^ η a^in + H ,
⎡ dσ− γ⎤ ^ = − ⎢ i (ωk − ω) + ⎥ σ− − gσz a^ + G, dt 2 ⎦ ⎣ a^ = a^ + η a^,
2Fp + 1 +
⎤ ⎡ ⎢ ωk σ^z, k + ig (a^σ^+, k − a^†σ^−, k ) + ωc a^†a^⎥, ⎥⎦ ⎢ 2 k =+ 1, −1 ⎣
η a^in′ −
out
The system we consider here is a singly NV center embedded in a photonic crystal cavity coupled to two waveguides as shown in Fig. 1. NV center in diamond consists of a substitutional nitrogen atom and an adjacent vacancy, which have trapped an additional electron in it. The energy structure of the system is shown in Fig. 1 (b) with the ground state of electronic spin triplet |3A2 〉. Here |3A2 〉 = |E0 〉 ⊗ |ms = 0, ± 1〉 with spin S ¼1, where |E0 〉 is the orbital state with zero angular momentum projection along the NV axis (z axis in Fig. 1(b)). The excited state is chosen as |A2 〉 = (|E−〉|ms =+ 1〉 + |E+〉|ms = − 1〉) [37], where |E±〉 are orbital states with angular momentum projection 7 1 along the NV axis. The cavity resonantly couples to the transition | ± 1〉 ↔ |A2 〉 for the NV center with coupling constant g. This system is governed by the following hamiltonian (ℏ ¼1) [4,25]
H=
⎡ da^ κ⎤ = − ⎢ i (ωc − ω) + η + ⎥ a^ − gσ− − dt 2 ⎣ ⎦
a′in
−1
L /R
σ−
+1
Fig. 1. The optical transitions of an NV center with circularly polarized lights. (a) A double-sided cavity-waveguide-NV-center system. (b) The optical transitions of an NV center. The photon in the state |R↑〉 or |L↓〉 corresponds to s þ , and the photon in the state |R↓〉 or |L↑〉 corresponds to s-. |R↑〉 ( |R↓〉) and |L↑〉 ( |L↓〉) represent the right- and leftcircularly polarized lights with their input (output) directions parallel (antiparallel) to the z direction.
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
follows [10,31]:
|R↑, − 1〉 → |L↓, − 1〉,
|L↑, − 1〉 → − |L↑, − 1〉,
|R↓, − 1〉 → − |R↓, − 1〉,
|L↓, − 1〉 → |R↑, − 1〉,
|R ↑ ,
|L↑, + 1〉 → |R↓, + 1〉,
+ 1〉 → −
|R ↑ ,
+ 1〉 ,
|R↓, + 1〉 → |L↑, + 1〉,
|L↓, − 1〉 → − |L↓, − 1〉.
(6)
Here, in the left side of “ → ” in Eq. (6), superscript ↑( ↓ ) denotes that the photon is injected into the cavity-NV-center system from down (upper) spatial mode; for the right side of the equations, superscript ↑( ↓ ) denotes that the photon exits from upper (down) spatial mode of cavity-NV-center system. Taking advantage of this double-sided cavity-NV-center system, based on the rules in Eq. (6), we can realize our CPF gate and SWAP gate in a deterministic way.
21
photon in the state |L↑〉 emitted into upper spatial mode (path 3), and the photon in the state |R↓〉 into down spatial mode (path 4). Before the photon coming from the spatial mode 4 reaches the one-way mirror, a polarization-flip operation ( |R〉 ↔ |L〉) is performed with a half-wave plate X. Then after the two wavepackets from spatial modes 5 and 2 pass though PBS2 simultaneously, the system consists of a photon and an NV center involves into
|Ψ 〉2 = αp αc |R, − 1〉 + αp βc |R, + 1〉 − βp αc |L, − 1〉 + βp βc |L, + 1〉.
(9)
From Eq. (9), one can see that when the photon is in the state |L〉 and the NV center is in the state | − 1〉, the phase will be flipped. This result shows that a CPF gate between an NV center and a photon is constructed in a deterministic way.
4. SWAP gate between photon and NV center Based on the above CPF gate, a quantum SWAP gate between one NV center and one photon can be realized. Such a SWAP gate can be written as
3. Controlled phase-flip gate A deterministic CPF gate can be written as
CPF Ucp
⎛1 ⎜ 1 ⎜0 = 2 ⎜⎜ 0 ⎝0
0 1 0 0
0 0 ⎞ ⎟ 0 0 ⎟ 1 0 ⎟ ⎟ 0 − 1⎠
SWAP Ucp
(7)
on the four basis {|R, + 1〉|R, − 1〉, |L, + 1〉 and |L, − 1〉}. As shown in Fig. 2, our device performs a phase flip operation if and only if the photon polarization qubit is in the state |L〉 and the NV center is in the state | − 1〉. Suppose the flying photon and the NV center in the cavity are initially prepared in superposition states |ψ 〉p = (αp |R〉 + βp |L〉), |ψ 〉c = (αc | − 1〉 + βc | + 1〉), respectively. Here
|αp |2 + |βp |2 = |αc |2 + |βc |2 = 1. The subscripts p and c denotes the photon and the NV center, respectively. At first, the injecting photon passes PBS1 which transmits a photon in the polarization state |R〉 while reflects a photon in the state |L〉. After the control photon passes though PBS1, the part in the state |L〉 is injected into the cavity and interacts with the NV center while the part in the state |R〉 does not interact with the cavity. The nonlinear interaction between the photon and the NV center makes the state of the photon-NV combined system evolve to
|Ψ 〉1 = αp αc |R, − 1〉 + αp βc |R, + 1〉 − βp αc |L↑, − 1〉 + βp βc |R↓, + 1〉.
(8)
It can be seen that the doubled-sided cavity-NV-center system acts essentially as a entanglement beam splitter, it leads the
PBS1
PBS2
DL 2
in
out
4 M
0 1 0 0
0⎞ ⎟ 0⎟ , 0⎟ ⎟ 1⎠
(10)
Hp = Hc =
1 2
⎛1 1 ⎞ ⎟, ⎜ ⎝ 1 − 1⎠
(11)
thus we have
Hp |R〉 =
1 (|R〉 + |L〉), 2
Hc | + 1〉 =
Hp |L〉 =
1 (| + 1〉 + | − 1〉), 2
1 (|R〉 − |L〉); 2
Hc | − 1〉 =
OWM X M
Fig. 2. The quantum circuit for constructing a deterministic CPF gate with a flying photon polarization as the control qubit and a confined NV center as the target qubit. PBS: polarized beam splitter; X: half-wave plate which used to perform a polarization bit-flip operation X = |R〉〈L| + |L〉〈R|; OWM: one way mirror; DL: optical delay lines.
(12)
1 (| + 1〉 − | 2
− 1〉).
(13)
Suppose the photon is in the state |φ〉p = (αp |R〉 + βp |L〉) and the NV center is in the state |φ〉c = (αc | − 1〉 + βc | + 1〉). There are three steps needed to realize the SWAP gate as shown in Fig. 3: (1) At the beginning of the scheme, K1 and K2 are adjusted to transmit and reflect photons, respectively. Here Ki (i¼1,2) is an optical switch which can be controlled exactly as needed to transmit or reflect photons. After the photon passes through K1, an Hp operation is applied on the photon by HWP1. K1 needs to be adjusted to reflect photons after the photon transmits it while K2 reflects photons still. The state of the system changes to
1 (αp αc |R, ↑ 〉 + αp αc |L, ↑ 〉 + αp βc |R, ↓ 〉 + αp βc |L, ↓ 〉 2 + βp αc |R, ↑ 〉 − βp αc |L, ↑ 〉 + βp βc |R, ↓ 〉 − βp βc |L, ↓ 〉).
5 3
0 0 1 0
on the four basis { |R, + 1〉, |L, + 1〉, |R, − 1〉, |L, − 1〉}. For conciseness, we define single qubit Hadamard operations Hp and Hc for one photon and one NV center respectively:
|ϕ〉1 =
1
⎛1 ⎜ 0 =⎜ ⎜0 ⎜ ⎝0
(14)
Then the photon is injected into the CPF gate again. After the photon is reflected by K1, K2 and the mirrors, another Hp operation is applied on the photon when the photon passes through HWP1. The state of the system evolves to
|ϕ〉2 = αp αc |L, − 1〉 + αp βc |R, + 1〉 + βp αc |R, − 1〉 + βp βc |L, + 1〉.
(15)
(2) Secondly, for the NV center, two Hc operations are applied by two lasers before and after the photon passes through the CPF gate, respectively. The state of the system evolves to
|ϕ〉3 = αp αc |L, + 1〉 + αp βc |R, + 1〉 + βp αc |R, − 1〉 + βp βc |L, − 1〉.
(16)
22
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
Fig. 4. Schematic setup for teleportation of the state of an NV center. Photon1 and photon2 are prepared in polarization-entangled states.
|Φ〉1 =
1 (α| − 1〉|R〉1|R〉2 + α| − 1〉|L〉1|L〉2 + β| + 1〉|L〉1|R〉2 + β| 2 + 1〉|R〉1|L〉2 ).
(19)
(2) An Hc operation is applied on NV center1 by a classical laser. The state of the NV-photon-photon combined system evolves to 1 (α | + 1〉|R〉1|R〉2 − α | − 1〉|R〉1|R〉2 + α | + 1〉|L〉1|L〉2 − α | − 1〉|L〉1|L〉2 2 + β| + 1〉|L〉1|R〉2 + β| − 1〉|L〉1|R〉2 + β| + 1〉|R〉1|L〉2 + β| − 1〉|R〉1|L〉2 ) 1 = (| + 1〉|R〉1σ x |φ〉p2 + | − 1〉|R〉1σ x σ z |φ〉p2 + | + 1〉|L〉1|φ〉p2 + | − 1〉|L〉1σ z |φ〉p2 ) , (20) 2
|Φ〉2 =
Fig. 3. Schematic setup to realize a SWAP gate between one NV center and one photon. Ki (i¼ 1,2): optical switch.
(3) Subsequently, the photon needs to go through HWP1 and the CPF gate once again. The state of the system evolves to
|ϕ〉4 =
1 (αp αc |R, + 1〉 − αp αc |L, + 1〉 + αp βc |R, + 1〉 + αp βc |L, + 1〉 2
+ βp αc |R, − 1〉 − βp αc |L, − 1〉 + βp βc |R, − 1〉 + βp βc |L, − 1〉). (17) Then K2 is adjusted to transmit photons and a Hp operation should be applied on the photon again by HWP2, the state of the system involves eventually into
|ϕ〉5 = αp αc |L, + 1〉 + αp βc |R, + 1〉 + βp αc |L, − 1〉 + βp βc |R, − 1〉 = (αc |L〉 + βc |R〉)(αp | − 1〉 + βp | + 1〉).
(18)
It seems from the result that we have realized a quantum SWAP gate between one photon and one NV center. It should be pointed out that the time of needed to complete the above three steps should be less than the decoherence time of an NV center. Fortunately, long decoherence time (∼ms) [39,40] and nanosecond manipulation time [41] have been realized, which make our schemes achievable.
5. Teleportation between two remote NV centers
where |φ〉p2 = (α|L〉 + β|R〉). (3) Alice performs measurements on the photon and the NV center at her side, the state of photon2 will collapse to one of the corresponding components. Based on the measurement results from NV center1 and photon1, Bob can deterministically recover the unknown state of photon2 by local operations as listed in Table 1. Here an optical delay line for photon2 to be stored is required during that time when photon1 and NV center1 undergo certain processes (not shown in the figure). After above steps, photon2 maps its information to NV center2 via the SWAP gate and thus the teleportation between two remote NV centers is accomplished.
6. Discussion and summary As the CPF gate is the basic construction, the fidelities of the SWAP gate and the quantum teleportation rely on the fidelity of the CPF gate, in this part we briefly discuss the fidelity of the CPF gate. Recently, higher fidelity of an operation in bad cavity conditions ( FP > 1) rather than the strong-coupling regime has been demonstrated [42], where the maximal fidelity is obtained at the point κ /η = 0.1 in their experiment. Prior works [10,13,31] have demonstrated that the dipole interaction between the photon and the NV center leads to different reflection and transmission coefficients between the right- and left-circularly polarized lights with their input (output) directions parallel (antiparallel) to the z direction. Accordingly, in the resonant condition ( ωc = ωk = ω ), the transmission and reflection rules for a photon become [10,31]
|L↑, − 1〉 → − |t0 ||L↑, − 1〉 − |r0 ||R↓, − 1〉,
|L↑, + 1〉 → |r||R↓,
+ 1〉 + |t||L↑, + 1〉. Our schematic setup to realize teleportation between two remote NV centers is shown as Fig. 4. A pair of maximally polarization-entangled photons is needed in this scheme, e.g., |ψ 〉p = (|L〉1|R〉2 + |R〉1|L〉2 ) / 2 , meanwhile assume that the state of the NV center1 is |φ〉c = (α| − 1〉 + β| + 1〉), here |α|2 þ |β|2 ¼ 1. Suppose photon1 and NV center1 belong to Alice, while photon2 and NV center2 are attached to Bob. Our scheme is comprised of the following steps: (1) Photon1 goes through the HWP1 before it enters into the CPF gate. After the photon experienced the CPF gate, the photon will go through HWP2, HWP1 and HWP2 apply Hp operations on the photon. Then the state of the NV-photon-photon combined system evolves to:
(21)
Substituting Eq. (21) for Eq. (6), and combing the arguments made in Section 3, we find that the state of the system described by Eq. (9) becomes Table 1 Correspondence between Alice's Measurement and Bob's Local Operation in Teleportation. Alice's Measurement
| | | |
+ − + −
1〉|R〉1 1〉|R〉1 1〉|L〉1 1〉|L〉1
Bob's State
Local Operation
α|R〉 + β|L〉 −α|R〉 + β|L〉 α|L〉 − β|R〉 −α|L〉 + β|R〉
sx σz σx I sz
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
|Ψ 〉′4 = αp αc |R, − 1〉 + αp βc |R, + 1〉 − βp αc |t0 ||L, − 1〉 −βp αc |r0 ||L, − 1〉 + βp βc |r||L, + 1〉+βp βc |t||L, + 1〉,
(22)
the terms with underlines indicate the states which take the bitflip error. We can calculate the average fidelity of the CPF gate as
FCPF =
1 4π 2
∫0
2π
dθ1
∫0
2π
dθ 2 |ideal 〈ψ |ψ 〉real |2 ,
(23)
where |ψ 〉ideal refers to the output state of the photon-spin system in the ideal case and |ψ 〉real represents the actual output state. Without loss of generality, we set αp = sin θ1, βp = cos θ1, αc = sin θ 2, βc = cos θ 2 (θ1, θ 2 ∈ [0, 2π ]). Therefore, the fidelity of our CPF gate in the two-qubit hybrid system discussed in Section 3 can be written as
FCPF =
1 4π 2
∫0
2π
dθ1
∫0
2π
dθ2[sin2θ1sin2θ2 + sin2θ1cos2θ2
+ cos2θ1sin2θ2|t0| + cos2θ1sin2θ2|r0| + cos2θ1cos2θ2|t| + cos2θ1cos2θ2|r|]2 = 1.
(24)
It can be seen that, if the effect of the optical elements can be ignored, fidelity of the CPF gate is 1, this increases the fidelities of the SWAP gate and the quantum teleportation directly. Meanwhile we define the efficiency of a quantum gate as the ratio of the number of the outputting photons to the inputting photons. The efficiency of the CPF gate therefore can be expressed as
ξCPF =
1 4π 2
∫0
2π
dθ1
∫0
2π
dθ 2 [sin2 θ1 sin2 θ 2 + sin2 θ1 cos2 θ 2
+ cos2 θ1 sin2 θ 2 |t0 |2 + cos2 θ1 cos2 θ 2 |r|2 ] 1 = [2 + |t0 |2 + |r|2 ], 4
ξSWAP =
1 [2 + |t0 |6 + |r|6 ]. 4
(25)
The efficiencies of our universal quantum gates are shown in Fig. 5. For our schemes, in the weak coupling regime, if λ ¼0.1, Fp = 5, ξCPF = 93.4% and ξSWAP = 82.8% , while ξCPF = 95.7% and ξSWAP = 89.1% when λ = 0; in the strong coupling regime, if λ = 0.5, FP = 11.52, ξCPF = 89.0% and ξSWAP = 82.8% . If the cavity intrinsic loss can be neglected, the efficiencies can reach near unity ( ξCPF = 99.6% and ξSWAP = 98.7% ). It is worth pointing out that FP and λ do not affect the fidelities of the gates and they remain at unity.
23
An NV center in diamond is an appropriate dipole emitter in cavity QED to obtain the high-fidelity reflection-transmission property in the Purcell regime. In the gate operation process, it is necessary to note that the relaxation time of the electron spin in NV center must longer than the photon interval time and operation time [19–21]. In terms of experimental progress, the relaxation time of the electron spin ground states of an NV center ranges from ∼ms at room temperature to ∼s at low temperature [17,21,43]. When an NV center coupled to a photonic crystal resonator (typically, Q∼3000), the zero phonon line emissions relevant to the emitted photons from the single NV center can be significantly enhanced [23,44] with the coupling strength as a few GHz. Strong coupling case FP = 325 in a silicon nitride photonic crystal nanocavity with a Q factor of 1.4 × 106 and a mode volume Vm ∼ 2.5 × 10−20 m3 has been reported [45]. For the present schemes, the cavity resonator mode is coupled to the two waveguides simultaneously, so approximately equal coupling constants are needed. In experiment, the deviation between the decay rates of the cavity field into waveguide channel modes is about Δη ∼ 0.2η, which leads to approximately the same fidelity for both transmission and reflection directions [42]. On the other hand, cavity mirror absorption, scattering, diffraction and inefficient detection may cause photon loss, which will affect the efficiencies of the schemes. It is noteworthy that the present schemes work in a repeat-until-success way, so it merely decrease the success probability but does not affect the fidelities once the photon lost. Furthermore, a highly efficient single-photon server (30,000 high-quality single photons are delivered within 30 s) has been reported [46], this promises a short time to implement our schemes. With the in crease in the number of logic qubits, our schemes would be less efficient, but they are still scalable in principle and are experimentally feasible. In summary, we have proposed a controlled CPF gate with the dipole induced transparency of a diamond NV center embedded in a photonic crystal cavity coupled to two waveguides. Based on the CPF gate and a series of single qubit operations, a SWAP gate between photon and spin in the cavity was proposed. Furthermore, a quantum teleportation scheme was proposed as an example of applications. The fidelities and efficiencies of the gates have been discussed. The results show that our schemes are insensitive to decoherence caused by cavity decay, and photon loss affect merely the efficiency but the fidelity. Our schemes work well in the bad cavity regime (the Purcell regime). In the ideal case (cavity leakage is much lower than the cavity loss), the fidelities and the efficiencies of our gates can reach near unity in the strong coupling regime.
Fig. 5. The efficiency of our CPF gate (a) and SWAP gate (b) versus the Pucell factor FP and λ.
24
A.-P. Liu et al. / Optics Communications 379 (2016) 19–24
Acknowledgements A-Peng Liu thank Qi Guo, Zhao Jin and Shi-Lei Su for enlightening discussions. Shou Zhang is supported by the National Natural Science Foundation of China under Grant no. 61465013.
[22] [23] [24] [25] [26]
References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. [2] D.P. DiVicenzo, Fortschr. Phys. 48 (2000) 771. [3] L.M. Liang, C.Z. Li, Phys. Rev. A 72 (2005) 024303. [4] L.Y. Cheng, H.F. Wang, S. Zhang, K.H. Yeon, Opt. Express 21 (2013) 5988. [5] H.R. Wei, F.G. Deng, Phys. Rev. A 87 (2013) 022305. [6] H.R. Wei, F.G. Deng, Sci. Rep. 4 (2014) 7551. [7] M.Z. Zhu, C.R. Zhao, L. Ye., Opt. Commun. 285 (2012) 1576. [8] P. Xue, Chin. Phys. Lett. 27 (2010) 060301. [9] L.M. Duan, H.J. Kimble, Phys. Rev. Lett. 92 (2004) 127902. [10] C.Y. Hu, W.J. Munro, J.L. O'Brien, J.G. Rarity, Phys. Rev. B 80 (2009) 205326. [11] E.M. Purcell, H.C. Torrey, R.V. Pound, Phys. Rev. 69 (1946) 37. [12] C. Bonato, F. Haupt, S.S.R. Oemrawsingh, J. Gudat, D.P. Ding, M.P. van Exter, D. Bouwmeester, Phys. Rev. Lett. 104 (2010) 160503. [13] A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, J. Vuckovic, Opt. Express 16 (2008) 12154. [14] E. Waks, J. Vuckovic, Phys. Rev. Lett. 96 (2006) 153601. [15] M. Stoneham, Physics 2 (2009) 34. [16] T. van der Sar, E.C. Heeres, G.M. Dmochowski, G. de Lange, L. Robledo, T. H. Oosterkamp, R. Hanson, Appl. Phys. Lett. 94 (2009) 173104. [17] P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, J. Wrachtrup, Science 320 (2008) 1326. [18] F. Jelezko, T. Gaebel, I. Popa, A. Gruber, J. Wrachtrup, Phys. Rev. Lett. 92 (2004) 076401. [19] F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, J. Wrachtrup, Phys. Rev. Lett. 93 (2004) 130501. [20] L. Childress, M.V. Gurudev Dutt, J.M. Taylor, A.S. Zibrov, F. Jelezko, J. Wrachtrup, P.R. Hemmer, M.D. Lukin, Science 314 (2006) 281. [21] T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J.
[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
[41] [42] [43] [44] [45] [46]
R. Rabeau, N. Stavrias, A.D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, J. Wrachtrup, Nat. Phys. 2 (2006) 408. J. Wolters, A.W. Schell, G. Kewes, N. Nüsse, M. Schoengen, H. Döscher, T. Hannappel, B. Löchel, M. Barth, O. Benson, Appl. Phys. Lett. 97 (2010) 141108. A. Faraon, P.E. Barclay, C. Santori, K.M.C. Fu, R.G. Beausoleil, Nat. Photonics 5 (2011) 301. Y.J. Zhao, X.M. Fang, F. Zhou, K.H. Song, Chin. Phys. Lett. 30 (2013) 050304. Q. Chen, W.L. Yang, M. Feng, J.f. Du, Phys. Rev. A 83 (2011) 054305. A.P. Liu, L.Y. Cheng, L. Chen, S.L. Su, H.F. Wang, S. Zhang, Opt. Commun. 313 (2014) 180. I.M. Mirza, S.J. van Enk, H.J. Kimble, J. Opt. Soc. Am. B 10 (2013) 2640. I.M. Mirza, S.J. van Enk, Opt. Commun. 343 (2015) 172. H.R. Wei, G.L. Long, Sci. Rep. 5 (2015) 12918. H.R. Wei, G.L. Long, Phys. Rev. A 91 (2015) 032324. B.C. Ren, G.Y. Wang, F.G. Deng, Phys. Rev. A 91 (2015) 032328. Y.S. Park, A.K. Cook, H. Wang, Nano Lett. 6 (2006) 2075. P.E. Barclay, K.M. Fu, C. Santori, R.G. Beausoleil, Opt. Express 17 (2009) 9588. P.E. Barclay, K.M.C. Fu, C. Santori, R.G. Beausoleil, Appl. Phys. Lett. 95 (2009) 191115. I.M. Mirza, Phys. Lett. A 379 (2015) 1643. I.M. Mirza, J. Mod. Opt 62 (2015) 1048. E. Togan, Y. Chu, A.S. Trifonov, L. Jiang, J. Maze, L. Childress, M.V. Gurudev Dutt, A.S. Sørensen, P.R. Hemmer, A.S. Zibrov, M.D. Lukin, Nature 466 (2010) 730. D.F. Walls, G.J. Milburn, Quantum Optics, Springer-Verlag, Berlin, 1994. B.B. Buckley, G.D. Fuchs, L.C. Bassett, D.D. Awschalom, Science 330 (2010) 1212. G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P.R. Hemmer, F. Jelezko, J. Wrachtrup, Nat. Mater. 8 (2009) 383. G.D. Fuchs, V.V. Dobrovitski, D.M. Toyli, F.J. Heremans, D.D. Awschalom, Science 326 (2009) 1520. D. O'Shea, C. Junge, J. Volz, A. Rauschenbeutel, Phys. Rev. Lett. 111 (2013) 193601. J. Harrison, M.J. Sellars, N.B. Manson, Diam. Relat. Mater. 15 (2006) 586. P.E. Barclay, K.M.C. Fu, C. Santori, A. Faraon, R.G. Beausoleil, Phys. Rev. X 1 (2011) 011007. M.W. McCutcheon, M. Lončar, Opt. Express 16 (2008) 19136. M. Hijlkema, B. Weber, H.P. Specht, S.C. Webster, A. Kuhn, G. Rempe, Nat. Phys. 3 (2007) 253.