This is the submitted version of the article: De Moraes B.G., Cummings A.W., Roche S.. Emergence of intraparticle entanglement and time-varying violation of Bell's inequality in Dirac matter. Physical Review B, (2020). 102. 041403: - . 10.1103/PhysRevB.102.041403. Available at: https://dx.doi.org/10.1103/PhysRevB.102.041403 Emergence of Intra-Particle Entanglement and Time-Varying Violation of Bell’s Inequality in Dirac Matter Bruna Gabrielly de Moraes,1, 2 Aron W. Cummings,1 and Stephan Roche1, 3, ∗ 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain 2 Department of Physics, Universitat Autónoma de Barcelona, Campus UAB, Bellaterra, 08193 Barcelona, Spain 3 ICREA–Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain We demonstrate the emergence and dynamics of intra-particle entanglement in massless Dirac fermions. This entanglement, generated by spin-orbit coupling, arises between the spin and sublat- tice pseudospin of electrons in graphene. The entanglement is a complex dynamic quantity but is generally large, independent of the initial state. Its time dependence implies a dynamical violation of a Bell inequality, while its magnitude indicates that large intra-particle entanglement is a general feature of graphene on a substrate. These features are also expected to impact entanglement be- tween pairs of particles, and may be detectable in experiments that combine Cooper pair splitting with nonlocal measurements of spin-spin correlation in mesoscopic devices based on Dirac materials. The importance of quantum entanglement was Two-dimensional Dirac materials offer true nov- sparked by the seminal gedanken experiment pro- elty in this context. In addition to spin, electrons posed in 1935 by Einstein, Podolsky and Rosen [1], in Dirac materials such as graphene carry other de- who in an attempt to demonstrate the incomplete- grees of freedom, including sublattice (aka pseu- ness of quantum mechanics actually inspired the dospin), valley, and layer (in bilayer graphene) [33– study of nonlocal correlations between distant par- 35], which opens new possibilities for intra-particle ticles [1–3]. Beyond the fascinating fundamental de- entanglement. Sharing the mathematical structure bate, which found its resolution in the violation of of spin, these degrees of freedom can become entan- Bell’s inequality [4, 5], quantum entanglement be- gled with each other and with spin, such that ma- tween internal degrees of freedom of particles has nipulating one degree of freedom may result in an become key for understanding emerging topological effect on another. For example, when graphene is on phases in many-body physics [6, 7], as well as for a substrate, Rashba spin-orbit coupling (SOC) leads controlling quantum states in the quest to realize to entanglement between the spin and pseudospin. quantum computing applications [8–10]. Identified as a source of spin dephasing in pristine Spin is a quantity that can be entangled to create graphene [38–40], this is a clear example of the ef- nonlocal quantum correlations, and there have been fect of intra-particle entanglement on a measurable many proposals to generate spin-entangled electrons spin transport property. Finally, the eigenstates of [11–13] or electron-hole pairs [14–16] in solid state bilayer graphene also exhibit strong sublattice-layer systems. These have stimulated a variety of ex- entanglement near the charge neutrality point [41]. perimental studies, including attempts to generate However, there are currently no studies of the for- entanglement through the splitting of Cooper pairs mation and dynamics of intra-particle entanglement [17–23]. To date however, the verification of non- in 2D Dirac materials. local quantum correlations between fermions, via In this Communication we explore the origin, dy- a violation of Bell’s inequality, remains challenging namics, and magnitude of intra-particle entangle- [24–26]. In contrast, by taking advantage of in- ment between the spin and pseudospin degrees of tra-particle entanglement between the spatial and freedom of electrons propagating in graphene. We spin degrees of freedom of a single neutron, the vi- show that Rashba SOC drives the generation and olation of Bell’s inequality has been demonstrated evolution of this entanglement, which emerges re- in neutron interferometry experiments [27, 28]. Fi- gardless of the initial quantum state. Large intra- nally, there are many other potential combinations particle entanglement is thus a general feature of of intra- and inter-particle entanglement, as well as graphene, opening the door for its indirect detection proposals to swap between them [29–32]. via inter-particle correlations, and future research on its use in alternative quantum information process- ing schemes. ∗ stephan.roche@icn2.cat Hamiltonian and entanglement of eigenstates. We consider a continuum model of graphene with    (QHUJ\ PH9 Rashba SOC, which is induced by a perpendicular   &RQFXUUHQFH |s|| |   electric field or a substrate. The Hamiltonian is     Ĥ = ~vF (τ σ̂x kx + σ̂y ky ) ⊗ ŝ0             PH9  PH9 + λR (τ σ̂x ⊗ ŝy − σ̂y ⊗ ŝx ) , (1)    where vF is the Fermi velocity, τ = ±1 is the  valley index, ~k is the electron momentum, λR is      the Rashba SOC strength, and ŝ (σ̂) are the Pauli  PH9 matrices for the spin (pseudospin) degree of free- dom. The eigenenergies of Ĥ are εe,h ± = νε± , where FIG. 1. Concurrence of the eigenstates of the graphene- p Rashba system near the charge neutrality point for dif- ν = ±1 for electrons/holes, ε± = ε2 + λ2R ± λR , ferent strengths of Rashba SOC. The left inset shows and ε = ~vF |k|. A typical bandstructure is shown a typical band structure and the right inset shows the in the left inset of Fig. 1, with the conduction and magnitude of the spin and pseudospin of the eigenstates, valence bands split by 2λR . both for λR = 100λ = 3.75 meV. We limit ourselves to a single valley (τ = 1) and express the Hamiltonian in the basis {|A ↑i , |B ↑i , |A ↓i , |B ↓i}, with states of the form the charge neutrality point (Cφ → 1 as ε → 0),  T wA↑ wB↑ wA↓ wB↓ , which describes the rela- and the entanglement decays as 1/ε at finite dop- tive weights of the wave function on the A or B ing. This is shown in Fig. 1 for different values of sublattice, with ↑ (↓) denoting spin pointing along λR = nλ, with λ = 37.5 µeV typical of graphene +z (−z), perpendicular to the graphene plane. on SiO2 or hBN [44], and 10λ or 100λ typical of Like the spin, the pseudospin can point in an ar- graphene on high-SOC substrates [45]. The total bitrary direction on the Bloch sphere, given the spinpand pseudospin of the eigenstates, |s| = |σ| = proper distribution of the wave function between |ε|/ ε2 + λ2R , are shown in the right inset. Both the A and B sublattices. For example, the state disappear as ε → 0, a result of their becoming max- T √1 1 1 0 0 = √12 (|Ai + |Bi) ⊗ |↑i has pseu-  2 imally entangled. dospin pointing along +x and spin along +z, while Entanglement dynamics. We now examine the en- 1 T 1 −i −1 i = √12 (|Ai − i |Bi) ⊗ √12 (|↑i − |↓i) tanglement between the spin and pseudospin of an  2 has pseudospin along −y and spin along −x. In this arbitrary electronic state injected into graphene. basis the eigenstates of the Hamiltonian are This electron will be in an initial state |ψi =  T a b c d and will evolve in time as |ψ(t)i = E 1  −iθ T φe,h νγ± ±iγ± ±νieiθ , (2) P ± =p e Û (t) |ψi, where Û (t) = j exp(−iεj t/~) |φj i hφj | is N± the time evolution operator, with εj and |φj i the where θ = arctan(ky /kx ) is the direction of electron eigenenergies and eigenstates of Ĥ, described above. 2 The concurrence of this state will thus evolve in time momentum, γ± = ε± /ε, and N± = 2(1 + γ± ). To quantify entanglement between the spin and as Cψ (t) = 2|a(t)d(t) − b(t)c(t)|. pseudospin, we use the concurrence Cψ of a given We first highlight the entanglement dynamics of state |ψi [42]. The concurrence has a 1-to-1 rela- some specific initial states. Without loss of gen- tionship with the von Neumann entropy, equals 0 for erality, we assume transport along the x-axis such completely separable states, and equals 1 for max- that θ = 0. Four states are considered : ψx↑ = imally entangled states. This entanglement mea- T √1 1 1 0 0 , with pseudospin pointing along +x,  sure was originally defined for mixed two-qubit sys- 2 tems, and has been extended to systems of many parallel to the transport direction, and spin along T +z; ψy↑ = √12 1 i 0 0 , with pseudospin along  qubits [43]. Here we study pure states of the form T +y, perpendicular to the transport direction, and  |ψi = a b c d , where the two qubits are the T = √12 1 0 0 1 , a Bell 1  spin and pseudospin of the electron. In this case the spin along +z; ψBell concurrence is Cψ = 2|ad − bc|. state with maximal spin-pseudospin entanglement; Applying this definition to Eq. (2) gives Cφe,h = T = √12 0 1 1 0 , another Bell state 2  p ± and ψBell λR / ε2 + λ2R . The eigenstates of the graphene- with maximal entanglement. The entanglement dy- Rashba system are thus maximally entangled near namics of these states are shown in Fig. 2 at three 2 different energies ε = 0, λR , and 10λR , with λR = | x | y | 1 Bell | 2 Bell 37.5 µeV, depicted in the top three panels.  We first analyze the concurrence dynamics at the   1 charge neutrality point, ε = 0. The Bell state ψBell  2 maintains perfect entanglement while ψBell oscil- lates between perfect and zero entanglement. Inter-  ↑ estingly, ψx,y both develop nonzero entanglement  despite being initially separable, periodically reach-   R C (t) ing a maximum concurrence of 0.5. These behaviors  can be understood by examining the eigenenergies and eigenstates of Ĥ at ε = 0,   E 1  T   R φe,h + = √ 0 ν i 0 , with εe,h + = ±2λR , 2  E 1  T φe,h − = √ 1 0 0 −iν , with εe,h − = 0. (3)  2 2 | 1 The eigenstates are maximally entangled, but their Bell  PH9  linear combinations can yield different behavior.  | nr 1 The first Bell state can be written as ψBell =  | nr0 1−i e 1+i h √ 2 φ − + √ 2 φ − . These eigenstates are degen-        erate (εe,h 1 − = 0) and ψBell thus remains static with 7LPH SV its initial CψBell 1 = 1. Meanwhile, the second Bell state can be written as ψBell2 = 1−i √ 2 φe+ + 1+i √ 2 φh+ . FIG. 2. Entanglement dynamics in the graphene-Rashba These eigenstates are separated in energy by εe+ − system. The top three panels show the concurrence dy- ↑ 1,2 εh+ = 4λR , and ψBell 2 oscillates with CψBell 2 (t) = namics of the initial states ψx,y and ψBell at energies ε = 0, λR , and 10λR , with λR = 37.5 µeV. The bottom | cos(4λR t/~)|. The separable states can be writ- panel shows the time-dependent violation of the CHSH ten as ψx↑ = 21 φe+ − φh+ + 21 φe− + φh− 1 inequality for ψBell , one instance of |ψnr i, and one in- and ψy↑ = 2i φe+ − φh+ + 12 φe− + φh− . The   r0 stance of ψn at ε = 25 meV, also for λR = 37.5 µeV. first term contributes an oscillatory component while the second term is static. The static component has  T zero concurrence ( φe− + φh− ∝ 1 0 0 0 ), lim- the Hamiltonian in Eq. (1) and considering trans- iting the maximum concurrence that is reached, with port along x, these effective fields are [38] 1 Cψx,y ↑ (t) = 2 | sin(4λR t/~)|. Bseff (t) = λR (−hσ̂y i(t) , hσ̂x i(t) , 0) , Next we analyze the concurrence dynamics when approaching the high-energy limit, which is the Bσeff (t) = λR (hŝy i(t) , −hŝx i(t) , 0) (4) most likely experimental situation (third row in Fig. + ε (1 , 0 , 0) , 2). Here the concurrence of both Bell states os- cillates as Cψ1,2 (t) ≈ | cos(2λR t/~)|. Meanwhile, where hôi(t) ≡ hψ(t)| ô |ψ(t)i. In the presence Bell of Rashba SOC, the spin and pseudospin precess the concurrence of ψx↑ is small, approaching zero around one another with a frequency ωR = 2λR /~. as ε → ∞. In contrast, the concurrence of ψy↑ The pseudospin also precesses around a component is Cψy↑ (t) ≈ | sin(2λR t/~)|, and periodically reaches of Bσeff that is parallel to the momentum and has maximal entanglement. Remarkably, in graphene magnitude ε. In the high-energy limit (ε  λR ), with Rashba SOC, an electron that initially has no this term dominates the pseudomagnetic field. Thus, intra-particle entanglement can become maximally when the initial pseudospin points along the momen- entangled, and has the same average entanglement tum direction, as for ψx↑ , the pseudospin remains as the maximally-entangled Bell states. fixed along x and only the spin precesses, in the x-z To understand the origin of this, we now look at plane. The spin and pseudospin thus remain well- the problem from the perspective of spin and pseu- defined and separable at all times, and the entan- dospin dynamics. In the graphene-Rashba system, glement is negligible. On the other hand, when the the spin and pseudospin will precess around effective initial pseudospin is perpendicular to the momen- magnetic and pseudomagnetic fields. By examining tum direction, as for ψy↑ , the pseudospin precesses 3 rapidly in the y-z plane with frequency ωε = 2ε/~.  This then enables mutual precession between spin D | y and pseudospin driven by Rashba SOC, and the de- C (t) −1  velopment of entanglement on the time scale ωR . In this way, a separable state whose pseudospin is not  | x parallel to the direction of transport will develop fi-  nite intra-particle entanglement.  At intermediate energies where neither ε nor λR  are dominant, more complex dynamics can emerge. E | 1 | 2 Bell Bell This can be seen in second row of Fig. 2, where C (t) ε = λR . Here, multiple precession processes all co-  exist with similar weight and frequency, giving rise  to more complex dynamics. We finally note that  this behavior may be connected to the concept of  quantum synchronization. Recent work has found  that synchronization between pairs of spins certifies F the presence of entanglement [46], which resembles | r n C n(t) the situation at ε = 0, while the absence of syn-  chronization does not preclude the development of | r0 n entanglement, as we see in the high-energy limit. Time-averaged entanglement. Figure 2 shows that       the entanglement can vary significantly with energy while exhibiting complex dynamics. To quantify the  PH9 overall degree of entanglement of a given state at a FIG. 3. Time-averaged concurrence as a function of given energy, we consider the time-averaged concur- ↑ 1,2 energy for the initial states (a) ψx,y and (b) ψBell . rence hCψ (t)i. This is shown in Fig. 3, with panel (a) For each state there are three curves, corresponding to ↑ corresponding to the initial states ψx,y and panel Rashba SOC strengths of λR = λ, 10λ, and 100λ, with E 1,2 λ = 37.5 µeV. Vertical dashed lines mark local maxima (b) to ψBell . For each state there are three curves, or minima in the concurrence for the case λR = 100λ. corresponding to λR = λ, 10λ, and 100λ. Panel (c) shows the time- and state-averaged concur- ↑ At low energies, the average concurrences of ψx,y rence, averaged over 1000 initial states |ψnr i and ψnr0 . exhibit a local minimum, increasing away from ε = 0 The dashed line shows the average concurrence of the as more complex dynamics come into play, as seen initial states |ψnr i at t = 0. by comparing the top two panels of Fig. 2. At higher energies the average concurrence of ψy↑ continues to grow, approaching hCψ (t)imax = h| sin(ωR t)|i = 2/π. out of Fig. 3. For any ε 6= 0, the spin-pseudospin Meanwhile, for ψx↑ the pseudomagnetic field dom- dynamics limit the average concurrence to hCψ (t)i < inates at higher energies, fixing the pseudospin and 2/π. We also note that the Rashba SOC strength leading to a 1/ε decay of hCψ (t)i. This crossover oc- has no impact on the magnitude of hCψ (t)i; its only curs at |ε| = (2/3)λR , marked by the vertical dashed effect is to rescale the energy dependence, as shown lines in panel (a). In Fig. 3(b), both Bell states by the different curves in Fig. 3(a,b). have the same energy dependence (except exactly at Entanglement of a random state. We have studied ε = 0, see below). Around ε = 0, hCψ (t)i approaches four specific states and found they all exhibit dy- 2/π and decays with increasing energy. This can be namic spin-pseudospin entanglement. Particularly seen in the top two rows of Fig. 2, where the complex interesting is that the initially separable state ψy↑ spin-pseudospin dynamics at finite ε lead to an av- exhibits as much time-averaged intra-particle en- erage reduction of the concurrence from its maximal tanglement as the maximally-entangled Bell states. behavior. The minimal average concurrence of the This suggests that in order to generate intra-particle Bell states is reached at |ε| = λR , shown by the ver- entanglement in graphene, it is not necessary to pre- tical dashed lines. Above this energy, the Bell states pare entangled states. Instead, injecting appropriate converge back to hCψ (t)imax = h| cos(ωR t)|i = 2/π. “normal” states is sufficient. However, injecting an 1 Here we note that ψBell has a constant Cψ = 1 at initial state with well-defined spin and pseudospin ε = 0 (top row of Fig. 2). However, this occurs only is also challenging. The spin can be controlled to exactly at ε = 0, and we have left this data point some degree with ferromagnetic contacts, but pre- 4 cise control of the pseudospin, determined by the Thus, because the concurrence of a particular elec- weight and phase of the wave function on each sub- tron is time-dependent, its violation of the CHSH in- lattice, is more difficult. equality also varies in time. We show this explicitly Given these issues, it is useful to consider a sim- in Fig. 2(d), where we plot the time dependence of 1 pler situation, where a bulk metal contact injects an β for ψBell , one instance of |ψnr i, and one instance electrical current into graphene. This current will r0 of ψn at ε = 25 meV. In all cases, β oscillates consist of electrons with random initial states, which with frequency ωR , highlighting the point that the raises the question: what is the entanglement of an Bell inequality violation is a dynamic and periodic arbitrary state in graphene? This can be quantified quantity in the graphene-Rashba system. by averaging over many random initial states |ψn i, Discussion and conclusions. We have shown that PN such that hhCψn (t)ii ≡ N1 n=1 hCψn (t)i is the av- intra-particle entanglement in graphene is a complex erage intra-particle entanglement in graphene. We dynamic quantity, governed by the mutual preces- consider two sets of random initial states, sion of spin and pseudospin. For an arbitrary initial T state the average entanglement is large, with a con- |ψnr i = an bn cn dn ,  n o currence of 0.5–0.6 that corresponds to a 10–16% p ψnr0 = cos(θnp /2) |Ai + sin(θnp /2)eiφn |Bi maximal violation of the CHSH inequality. Most n o importantly, spin-orbit coupling drives the genera- s ⊗ cos(θns /2) |↑i + sin(θns /2)eiφn |↓i . (5) tion of entanglement in electrons that are initially separable. This suggests that intra-particle entan- Each element of |ψnr i is a random complex number glement in graphene may be robust to disorder and chosen from the normal distribution. This generates dephasing; after an entanglement-destroying inter- states that are equivalent to the action of a random action, spin and pseudospin dynamics will lead to re- unitary matrix on some reference state, which are generation of the intra-particle entanglement. The uniform over the four-dimensional Hilbert space [47], interplay between this entanglement re-generation with average initial concurrence hCψnr (0)i ≈ 0.59. and entanglement revival in a dephasing environ- Meanwhile, ψnr0 is a separable state, with (θnp , φpn ) ment [50] may also yield unique behavior. and (θns , φsn ) random spherical angles defining the Going beyond the single-particle picture, these re- orientations of the pseudospin and spin on the Bloch sults have important implications for inter-particle sphere. The set ψnr0 thus represents all two- entanglement between pairs of electrons in graphene. qubit states with zero initial entanglement. It has been shown that the dynamics of intra-particle The averages over these initial states are shown in entanglement can significantly impact inter-particle Fig. 3(c). For all possible initial states, the average entanglement, including transfer from one type of concurrence is independent of energy and equal to its entanglement to another [29]. Experimentally, this initial value, hhCψnr (t)ii = hCψnr (0)i ≈ 0.59, shown could be realized via the injection and splitting of by the magenta and the dashed black lines. Mean- spin-entangled Cooper pairs in graphene-based de- while, the average over separable initial states has an vices [18, 22]. The dynamical evolution of intra- energy dependence similar to hCψx↑ (t)i in Fig. 3(a), particle entanglement would then be reflected in the but with a slower decay. Importantly, hhCψnr0 (t)ii inter-particle spin-spin correlation, and could be de- remains large over the entire energy range, around tectable in conventional nonlocal setups [17–23], ei- 0.5. This indicates that large intra-particle entangle- ther using magnetic detectors [13, 26] or current ment is a general feature of graphene with spin-orbit correlation (noise) measurements [24, 51, 52], with coupling, even for states that are initially separable. further tuning of the dynamics via magnetic fields Time-dependent Bell inequality violation. As [53]. If these dynamical aspects of the Bell inequal- shown above, intra-particle entanglement between ity violation could be detected in such experiments, spin and pseudospin is generally large in graphene it would open a hitherto unexplored dimension con- with Rashba SOC. We have used the concurrence cerning generation, detection, and manipulation of to quantify entanglement, but experimentally this entanglement in Dirac and topological materials. is not directly measurable. Rather, entanglement is demonstrated through a violation of a Bell inequality [4, 5]. For pure states there is a direct connection ACKNOWLEDGMENTS between the concurrence and maximal violation of the Clauser-Horne-Shimony-Holt (CHSH) q variant of We thank L.E.F. Torres for fruitful com- Bell’s inequality [48], given by β = 1 + Cψ2 [49]. ments. ICN2 is funded by the CERCA Pro- 5 gramme/Generalitat de Catalunya, and is sup- ment Nos. 785219 and 881603 (Graphene Flagship). ported by the Severo Ochoa program from Spanish B.G. de M. has received funding from the European MINECO (Grant No. SEV-2017-0706). 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