An efficient and deterministic source of highly-entangled photons is needed in quantum informa-tion processing [see (i).–(iii). in Sec.4.1]. In this chapter, we examined a new scheme of entangled photon production, based on the emission of quantum vacuum radiation from the intersubband cavity system. The emission of photon pairs can be triggered by a nonadiabatic modulation of the ground state of the system, enabling deterministic photon generation. The ground state consists
4.5 Conclusions
0 0.5 1 1.5 2 2.5 3
0 0.2 0.4 0.6 0.8 1
k (107 m−1)
C(k,q)
(a)
q = 0.1 x 107 m−1 q = 0.5 x 107 m−1 q = 1.0 x 107 m−1 q = 2.5 x 107 m−1
0 0.1 0.2 0.3 0.4 0.5
0 0.2 0.4 0.6 0.8 1
k (107 m−1)
C(k,q)
(b)
Figure 4.11–Photon entanglement in the limit of a large cavity.(a) ConcurrenceC(k,q) in the case of large sample areas for GaAs/AlGaAs quantum wells. (Parameter values:ε = 10, m∗=0.067,neffqw=50,Leffc =2µm,~ω12=113 meV,N2deg=1012cm−2.) (b) Zoom in the range only up tok=0.5×107m−1.
of an infinite number of photonic and electronic states, and we propose a post-selective measure-ment to reduce the photonic system to an effective two-qubit system. Thereby, the computational basis is defined as two different in-plane wave vectors. We find an analytical expression for the concurrence to quantify the so-called mode entanglement of the photons, which depends on the absolute values of the chosen wave vectors. The concurrence, and thus the entanglement of the post-selected photons, is nonzero. In the limiting case of large sample areas, there exists a continuous set of mode pairs for which the concurrence is 1, i.e. the photons are maximally entangled. Furthermore, it turns out that for photon energies around the intersubband resonance, which is in the mid infrared of the electromagnetic spectrum, the photons are as well almost maximally entangled, the concurrence being close to unity. This is fundamentally important for the possible use in quantum information processing, e.g. to implement quantum teleportation. A high degree of entanglement can also be reached if the modes chosen in the post-selection are close to each other.
We also want to comment on the possibility of triggering the photon-pair emission by a systematic quench of the light-matter interaction in the microcavity and give a rough estimate on the efficiency of the source. The repetition rate for the photon production is limited by the switching times, which have to be fast enough to perturb the system nonadiabatically. The experiments today achieved switch-on times of the ultrastrong coupling regime of about 10 fs by ultrashort laser pulses [62,94,95]. The switch-off process, which is the relevant operation related to photon emission, has to be as fast as this timescale. This gives a rough estimate for a repetition rate of 1013−1014 full cycles (switch-on and switch-off ) per second. The probability to really measure a desired two-photon state is given by the probability of a successful post-selectionp(k,q), which depends on the chosen modes and is given by the expectation value of the projectorPlr, i.e.p(k,q) =hG(2)|Plr|G(2)i=NH/Nlr. The probability is largest for wave vectors corresponding to the resonance frequency of the intersubband transition and is on the order of 10−9. This value
47
Chapter 4: Entangled Photons in the Polariton Vacuum
can be significantly increased if not only single modes are allowed by the post-selection, but the photons can have energies within a predefined frequency range. However, the effect on the entanglement still needs to be worked out for this case.
Further studies are also required to model noninstantaneous switch-off processes, presumably using a time-dependent perturbation theory approach. Another open issue is the simultaneous emission of black body radiation at finite temperature, an effect which is expected to be small com-pared to the vacuum radiation [93], but which will to some extent reduce the average entanglement of the emitted photons.
©Parts of this chapter have been published by the American Physical Society (APS) in Ref.98:
A. Auer and G. Burkard,Entanglement purification with the exchange interaction, Physical Review A90, 022320 (2014).
5
Entanglement Purification with the Exchange Interaction
5.1 Introduction
A fundamental resource for the implementation of large-scale quantum communication networks [38] is the generation of long-distance entanglement between the network nodes. However, due to imperfect sources and the inevitable interaction of the entangled particles with their environment the degree of entanglement decreases. To overcome this issue, the concept of a quantum repeater was established for the practical generation of long-distance entanglement [8,9], as we describe in Sec.3.6. The centerpiece of such a device is entanglement purification [10,11], for which a quantum memory [39] is indispensable. The working principle of the quantum repeater is to divide the distance between the network nodes into smaller segments, create entangled states between them, and sequentially generate high-fidelity entanglement between the nodes through alternating purification and entanglement-swapping steps (Fig.3.6). Ideal candidates for the realization of stationary qubits acting as quantum memory are spins in solid-state systems, like electron spins in semiconductor quantum dots (QDs) [18] and nitrogen-vacancy centers in diamond [23] (see also Chap.6). Due to their long coherence times, their complete controllability by electrical or optical means [19,23,99], and the possibility of interfacing them with photons [100], the considerable potential of spins as quantum memory has been demonstrated. The main focus of this chapter is devoted to electron-spin qubits that are coupled by an isotropic Heisenberg-type interaction, which is the case, e.g., for spin qubits in lateral QDs [19] (Sec.5.2). The idea of recurrence protocols for entanglement purification is to use two or more imperfectly entangled qubit pairs to purify one of them with respect to a maximally entangled state. Using existing recurrence protocols involving symmetric local two-qubit operations, such as the bbpssw or the dejmps protocol (see Sec.3.5), turns out to be rather unpractical for spin qubits since an efficient implementation of the required cnot gates is challenging. Hence, we concentrate on an open problem in the development of quantum repeaters based on spin qubits, which is the lack of an efficient scheme for entanglement
Chapter 5: Entanglement Purification with the Exchange Interaction
purification.
Pioneering experiments of entanglement purification have been performed using photonic qubits [101–104]. Limitations of these schemes are the destructive measurements of the purified pairs and the requirement for pure input states [44,105], besides the impracticality of using photons as quantum memory. The original bbpssw protocol of C. H. Bennettet al.[10] based on cnot gates has been implemented with atomic qubits, but only locally in the same optical trap [106]. Previous theoretical proposals have demonstrated purification schemes for spin qubits, e.g. by replacing the cnot in the bilateral operation by the gate sequence that uses two-qubit gates directly generated from the interaction Hamiltonian [107], requiring additional single-qubit operations. Other procedures use three input pairs [108–110] or specifically work for two-spin singlet-triplet qubits [111]. The aim of our study is therefore to examine whether entanglement purification is feasible for characteristic coupling types of spin qubits, however, without single-qubit manipulation, which is too slow for the realization of a quantum repeater.
We present a simple purification protocol based solely on the one-time activation of a Heisen-berg exchange interaction leading to the
√
swap gate (Sec.5.3). The exchange interaction is readily available in many spin-based qubits systems, such as quantum dots [18]. Following the same ap-proach, we also find a similar result for qubits coupled via a XY-type interaction, which happens to be the interaction between superconducting qubits [112] as well as between optically coupled spin qubits [113], or for qubits coupled via magnetic dipole-dipole interaction (Sec.5.4). Our proposal works with two input pairs of spin-1/2 qubits and only requires a single two-qubit interaction. In comparison with existing protocols, we achieve an advantage by allowing fordifferenttwo-qubit manipulations locally in the bilateral operation. For a single purification step, the derived protocol requires no extra single-spin operations, making it much faster and less susceptible to gate errors.
The only needed operation, the
√
swap gate, has been implemented experimentally, with a gate time below 0.2 ns [114] (see also [48]), making the implementation of our proposal within reach of current technology. Furthermore, the required single-shot measurement of an electron spin state has also been successfully performed [115,116].