How to detect qubit-environment entanglement generated during qubit dephasing
HHow to detect qubit-environment entanglement generated during qubit dephasing
Katarzyna Roszak,
1, 2
Damian Kwiatkowski, and (cid:32)Lukasz Cywi´nski Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,Wroc(cid:32)law University of Science and Technology, 50-370 Wroc(cid:32)law, Poland Institute of Physics, Academy of Sciences of the Czech Republic, 18221 Prague, Czech Republic Institute of Physics, Polish Academy of Sciences,Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland (Dated: August 22, 2019)We propose a straightforward experimental protocol to test whether qubit-environment entan-glement is generated during pure dephasing of a qubit. The protocol is implemented using onlymeasurements and operations on the qubit – it does not involve the measurement of the system-environment state of interest, but the preparation and measurement of the qubit in two simplevariations. A difference in the time dependencies of qubit coherence between the two cases tes-tifies to the presence of entanglement in the state of interest. Furthermore, it signifies that theenvironment-induced noise experienced by the qubit cannot be modeled as a classical stochasticprocess independent of the qubit state. We demonstrate the operation of this protocol on a re-alistically modeled nitrogen vacancy center spin qubit in diamond interacting with a nuclear spinenvironment, and show that the generation of entanglement should be easily observable in this case.
I. INTRODUCTION
The interaction between a quantum system and its en-vironment leads to decoherence [1, 2] of superpositions ofa system’s pointer states [3]. This ubiquitous feature ofquantum open system dynamics has fundamental signif-icance for the realistic description of all possible devicesemploying truly quantum features of physical systems forvarious tasks, as well as for the understanding of thequantum-classical transition [1, 4, 5]. The sensitivity ofexperimentally investigated qubits to environmental in-fluence has also led to the development of a whole fieldof research devoted to the use of qubits to characterizetheir environments [6, 7].While any environment of a qubit should be in princi-ple described quantum mechanically, it is now clear thatenvironments relevant for the description of pure dephas-ing of qubits (for examples showing that pure dephasingis a very common dominant source of decoherence seee.g. [7–15]) can often be modeled as sources of noise, theproperties of which are independent of dynamics of thesequbits, and even of their existence, see [6, 7]. We stressthat this feature – the ability to correctly describe thedynamics of the environment that leads to dephasing ofthe qubit by looking only at the dynamical properties ofthe environment, or in other words the absence of visibleback-action of the qubit on the environment – is takenhere as the defining one of the “classical environmen-tal noise” model of dephasing. Environments that havetheir dynamics unaffected by presence of the qubit canbe modeled classically, by specifying all the multi-pointcorrelation functions that characterize the stochastic pro-cess [7, 16], but note that the converse is not necessarilytrue: it could be possible to describe qubit decoherencecaused by an environment by a model of external classicalnoise, while the actual joint qubit-environment evolutioninvolves nontrivial back-action of the qubit on the en-vironment and creation of quantum correlations between the two - in fact, pure dephasing of a freely evolving qubit(but, interestingly, not of a higher-dimensional system[17]) can always be effectively described by constructingan artificial model of external classical noise [18, 19]. Ourgoal here is not to show when one cannot come up withan effectively classical model of the environmental influ-ence on the qubit, but to devise a simple experiment, thepositive result of which clearly proves that treating theenvironment as independent of the qubit is impossible.The former is really a statement on the dynamics of thequbit (“can these dynamics of an open quantum systembe reconstructed by introducing classical noise acting onthe qubit”), while the latter is a statement on the physi-cal nature of the environment coupled to the qubit. It isthus somewhat surprising that the experiment describedhere relies only on control and measurement of the qubit.These features make it of course easy to implement.While the necessary conditions for applicability of sucha “classical noise” approximation are not known, a largesize of the environment and its high temperature are,as expected, positively correlated with “classicality” ofqubit dephasing. In the simplest - and very often realis-tic, as it arises for an environment consisting of many un-correlated sub-environments, each weakly coupled to thequbit - case of noise with Gaussian statistics, full charac-terization of the noise is contained in its spectral density.In this case, qubits can be straightforwardly used as noisespectrometers [6, 7]. It also should be noted that spec-troscopy of non-Gaussian noise has been theoretically putforward in [16], but the implementation of the protocolproposed there is definitely much more involved [20] thanin case of reconstruction of spectrum of Gaussian noise.However, while the classical noise model of qubit dephas-ing is believed to be widely applicable, a vexing funda-mental problem remains unsolved: how can one unam-biguously prove that decoherence of a given qubit is infact truly quantum , i.e. not amenable to description usingclassical environmental noise. a r X i v : . [ qu a n t - ph ] A ug The general issue of quantum vs. classical nature ofenvironmentally-induced qubit dephasing has anotherfacet. For an initially pure state of the environment, thedephasing is in one-to-one correspondence with qubit-environment entanglement (QEE) generation [1, 4, 21,22]. However, in the realistic case of a mixed initial stateof the environment, decoherence does not have to be ac-companied by generation of QEE [22–26]. This fact isnot strongly stressed in most of seminal papers on de-coherence. The reasons were twofold. First, their focuswas on the most strikingly quantum situation, in whichcoherence of one system is lost due to establishment ofentanglement with a larger system that is also in purestate - the state of the whole remains pure, but its coher-ent nature becomes inaccessible by measurements on thesystem only. Second, the theory of mixed state entangle-ment started to be developed years after the foundationsof decoherence theory (for historical perspective on themsee [4]) had been established. It is worth noting thatthe definition of separable mixed states (and the defini-tion of entangled mixed states that follows from it) wasgiven only in 1989 [27]. After the theory of entangle-ment of mixed states was developed to a sufficiently ad-vanced degree [28–31], the issue of system-environmententanglement generated during system’s dephasing wasrevisited. In Ref. [23] it was shown, that in the quan-tum Brownian motion model (an oscillator coupled to anenvironment of other oscillators), for a large class of ini-tially mixed qubit states no system-environment entan-glement was generated during decoherence, and this factwas described as “surprising”. The system-environmententanglement in thermal states in the quantum Brown-ian motion model was further investigated in [24], whereit was shown that it disappears above a certain tempera-ture. Qubit-environment entanglement in the case of anenvironment consisting of non-interacting bosons, cou-pled linearly to the qubit, was considered in [26], whereit was shown that for a qubit initialized in a pure stateand an environment in a thermal equilibrium state at fi-nite temperature, decoherence is always accompanied bynonzero QEE - but not necessarily so for a qubit beinginitially in a mixed state.Clearly, the fact that the issue of correlation betweena system’s decoherence and the generation of system-environment entanglement is a nontrivial one in a generalsetting, has been a subject of intense attention. How-ever, a simple theoretical criterion showing when qubitpure dephasing is accompanied by generation of QEE(for a qubit initialized in a pure state interacting with any finite-dimensional environment) has been formulatedonly quite recently [22, 32]: QEE is generated if and onlyif the evolutions of the environment conditioned on twopointer states of the qubit lead to distinct states of theenvironment. Obviously such a situation is incompatiblewith treating the environment as an entity that evolvesindependently of the qubit.In this paper we propose a very simple experimentalscheme, which can be used to test the generation of QEE during the joint evolution of the qubit and its the en-vironment initially in a product state, where the qubitis in a superposition of its pointer states, and the inter-action leads to its pure dephasing. The scheme relieson the fact that only for entangling evolutions the envi-ronment behaves in a distinct way depending on whichpointer state the qubit is in. Hence, only if an evolutionis entangling can there be a difference in the evolution ofqubit coherence when the environment has been allowedto evolve for a finite time in the presence of qubit state | (cid:105) or | (cid:105) before a qubit superposition state was created.The observation of distinct evolutions of qubit coherencefor the two preparation procedures is therefore a QEEwitness. Furthermore, the fact that the evolution of theenvironment does depend on the state of the qubit is in-compatible with the assumption that the environment isan entity that evolves independently of the qubit [33].Sensing the capability of the system to generate nonzeroQEE during a qubit’s dephasing also proves then that theenvironmental influence cannot be described as externalclassical noise.Critically, unlike the scheme proposed in [22], here onlymeasurements on the qubit, not on the environment, arerequired, making the scheme completely straightforwardto implement. Although it is common knowledge thatdetection of entanglement between two systems requires,in general, measurements on both of the systems, herewe need to measure only one of the systems (the qubit),because the problem is constrained: we are interestedin entanglement generated during pure dephasing of thequbit interacting with an environment. We illustratethe concept with a calculation performed for a nitrogen-vacancy (NV) center in diamond, a spin qubit coupled toa nuclear spin environment that is widely used for noisespectroscopy and nanoscale nuclear magnetic resonancepurposes [6, 34].The paper is organized in the following way. In Sec. IIwe discuss the pure dephasing model of decoherence andits applicability. In Sec. III we describe the protocol forthe detection of the system’s capacity to generate qubit-environment entanglement which is the central result ofthis paper. The discussion of the significance of this pro-tocol for sensing the non-classical nature of the environ-mental noise is given in Sec. IV. Then, in Sec. V we pre-dict the performance of the protocol applied to an theNV center spin qubit interacting with a partially polar-ized nuclear environment. Sec. VI concludes the paper. II. PURE DEPHASING HAMILTONIAN
The system under study is composed of a qubit andan environment of arbitrary size. The interaction be-tween the two is such that the effect of the environmenton the qubit can only lead to its pure dephasing, so pro-cesses which affect the occupations of the qubit are notallowed. Such a class of Hamiltonians can be simply de-fined, since the condition for decoherence to be limitedto pure dephasing amounts to the fact that the free qubitHamiltonian must commute with the interaction terms.We choose states | (cid:105) and | (cid:105) to be qubit pointer states,which allows us to write an explicit general form of thepure-dephasing Hamiltonian,ˆ H = (cid:88) i =0 , ε i | i (cid:105)(cid:104) i | + ˆ H E + (cid:88) i =0 , | i (cid:105)(cid:104) i | ⊗ ˆ V i . (1)Here the first term describes the free evolution of thequbit and ε i are the energies of the qubit states, thesecond term describes the environment, while the lastterm describes the qubit-environment (QE) interaction.The environmental operators ˆ V and ˆ V are arbitrary, thesame as the environment Hamiltonian, ˆ H E .QE evolution can be formally solved for Hamiltoni-ans of this class and the QE evolution operator ˆ U ( t ) =exp( − i ˆ Ht ) can be written asˆ U ( t ) = | (cid:105)(cid:104) | ⊗ ˆ w ( t ) + | (cid:105)(cid:104) | ⊗ ˆ w ( t ) , (2)where the operators which describe the evolution of theenvironment conditional on the state of the qubit aregiven by ˆ w i ( t ) = exp( − i ˆ H i t ) , (3)with i = 0 ,
1. The operators ˆ H i = ˆ H E + ˆ V i contain thefree Hamiltonian of the environment and the appropriatepart of the interaction.The above Hamiltonian is not only a paradigmaticmodel of decoherence (as it describes the simplest set-ting in which environment causes dephasing of superpo-sitions of pointer states [1, 3, 4], but it also describesthe dominant decoherence process for most types of cur-rently researched qubits, e.g. spin qubits in quantum dots[12, 15, 35–37], spin qubits based on NV centers [36, 38–40] and electrons bound to donors [41], trapped ions[10, 42], and exciton-based qubits [43–46]. In all thesesystems the dephasing of a superposition of | (cid:105) and | (cid:105) state occurs on timescales orders of magnitude shorterthan timescale on which energy is exchanged betweenthe qubit and the environment, and consequently popu-lations of these states are modified.It is also worth noting that the absence of transversecouplings ∝ ˆ σ x ˆ V x + ˆ σ y ˆ V y in the QE Hamiltonian is notnecessary for pure dephasing to be the process that limitsthe coherence of the qubit. It is enough for the energyscale ∆ E ≡ (cid:15) − (cid:15) of the qubit’s Hamiltonian to bemuch larger than the energy scales associated with thesetransverse terms, i.e. ∆ E (cid:29) [Tr E ( ˆ R (0) ˆ V x/y )] / , whereˆ R (0) is the density matrix of the environment. If thespectral density of the ˆ V x,y fluctuations of the environ-ment does not overlap very strongly with frequency rangearound ∆ E , the energy exchange with the environmentwill be weak, and the qubit’s quantization axis will beonly slightly tilted away from the z direction by ˆ V x,y terms. In this situation, one can use a Schrieffer-Wolff canonical transformation and obtain an effective pure de-phasing Hamiltonian containing the terms ∝ ˆ σ z ˆ V x,y / III. PROTOCOL OF DETECTION OFSYSTEM’S CAPACITY TO GENERATEQUBIT-ENVIRONMENT ENTANGLEMENT
We begin with the main result of Ref. [22], which pro-vides a criterion to distinguish between entangling andnon-entangling QE evolutions for pure-dephasing pro-cesses. The criterion works only for product initial statesof the qubit and the environment, | φ (cid:105) (cid:104) φ | ⊗ ˆ R (0). Ad-ditionally the initial state of the qubit has to be pure | φ (cid:105) = α | (cid:105) + β | (cid:105) (for QEE to be generated in a pure-dephasing process, obviously the qubit has to be in a su-perposition of its pointer states, hence α, β (cid:54) = 0). Thereare no constraints on the initial state of the environ-ment, neither on its size or purity, and it is describedby the density matrix ˆ R (0). The criterion states thatQEE is present at time τ after initialization, if and onlyif [ ˆ w † ( τ ) ˆ w ( τ ) , ˆ R (0)] (cid:54) = 0. This condition can be rewrit-ten [32] in the more physically meaningful formˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) (cid:54) = ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) . (4)Here the conditional evolution operators of the environ-ment are given by eq. (3). Note that ˆ w i ( τ ) ˆ R (0) ˆ w † i ( τ ), i = 0 ,
1, is the state of the environment at time τ condi-tional on the qubit being initialized in state | i (cid:105) . There-fore, if and only if QEE is not generated, the evolution ofthe environment in the presence of either qubit pointerstate will be the same, otherwise, it has to differ.This condition itself provides a straightforward QEEwitness, since any observable on the environment can beused to test it [22]. If the qubit would be prepared ini-tially in state | (cid:105) and the time dependence of an observ-able on the environment would be measured, and thenthe qubit would be prepared in state | (cid:105) (for the sameinitial state of the environment ˆ R (0)) which would befollowed by measuring the time dependence of the sameobservable, a discrepancy at time τ between the expec-tation values of said observable would mean that if thequbit were initially prepared in any superposition state,it would be entangled with the environment at time τ .Since the result would obviously depend on the choiceof observable, the same expectation value at time τ isinconclusive (as conclusive testing would require the fullknowledge of the conditional density matrices of the en-vironment at time τ ). The problem with such tests ofQEE is that it requires measurements to be performedon the environment which is usually hard to access.In the following, we use the fact that QEE genera-tion in the described evolutions always corresponds todifferent evolutions of the environment conditional on thepointer state of the qubit and propose a scheme for QEEdetection, which requires operations and measurements | i | i ⌧ t⌧ t p | i + | i )1 p | i + | i ) ⇣ ⇡ ⌘ y ⇣ ⇡ ⌘ y FIG. 1. Schematic representation of the protocol for the de-tection of a system’s capacity to generate qubit-environmententanglement. After a preparation time τ when the environ-ment evolves in the presence of state | (cid:105) or | (cid:105) , the qubit is(operationally instantaneously) prepared in a superpositionstate (the same in both cases). Then the evolution of the co-herence is measured and results in both cases are compared. on the qubit alone. The protocol is schematically de-picted in Fig. 1.The idea is to first prepare the qubit in state | (cid:105) andlet it and the environment evolve jointly for time τ .For pure dephasing evolutions this does not change thequbit state, but the environment evolves into ˆ R ( τ ) =ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) from its initial state ˆ R (0). Now,if at time τ the qubit state is changed to | ψ (cid:105) =1 / √ | (cid:105) + | (cid:105) ) by an appropriate unitary operation (theequal superposition state is chosen to maximize the vis-ibility of the effect, but any superposition would work),further evolution will lead to pure dephasing of the qubitand the coherence will evolve according to ρ (0)01 ( τ, t ) = 12 Tr (cid:16) ˆ w ( t ) ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) ˆ w † ( t ) (cid:17) , (5)where t is the time elapsed from time τ . This coherenceneeds to be measured. Next, if the same procedure isperformed with the qubit in state | (cid:105) between the initialmoment and time τ , the coherence of the superpositionqubit state (after time τ ) will evolve according to ρ (1)01 ( τ, t ) = 12 Tr (cid:16) ˆ w ( t ) ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) ˆ w † ( t ) (cid:17) . (6)Naturally if the separability condition of eq. (4) is ful-filled, the evolution given by eq. (5) would be the same asthe evolution given by eq. (6). Otherwise, if at any time t , ρ (0)01 ( τ, t ) (cid:54) = ρ (1)01 ( τ, t ), then there must be QEE at time τ in a system initially in a product of any qubit super-position state and environmental state ˆ R (0). Therefore,unless the two coherence decay signals, (5) and (6), arein perfect agreement, pure dephasing of a superposition of qubit states lasting for time τ must be accompaniedby QEE generation.The scheme outlined above is an entanglement wit-ness, since there exists one situation when QEE is gen-erated, which it does not detect. This is the casewhen [ ˆ w ( t ) , ˆ w ( t )] = 0 for all times t and t (suchcommutation also implies commutation when one orboth operators are Hermitian conjugated), resultingin ˆ ρ (0) / (1)01 ( τ, t ) = Tr[ ˆ R (0) ˆ w † ( t ) ˆ w ( t )]. This requires[ ˆ H , ˆ H ] = 0. Note that if we do not exactly know theform of ˆ H E and ˆ V i , we can check if [ ˆ H , ˆ H ] = 0 by per-forming a spin-echo experiment, in which a superpositionstate of the qubit is initialized, it interacts with the envi-ronment for time τ , is subjected then to a ˆ σ x operation,and the coherence read out after time τ elapses again isgiven by ρ echo01 ( τ, τ ) = 12 Tr (cid:16) ˆ w ( τ ) ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) ˆ w † ( τ ) (cid:17) . (7)Perfect recovery of initial coherence for any τ is thusequivalent to [ ˆ H , ˆ H ] = 0. IV. RELATION TO THE CLASSICAL NOISEMODEL OF THE ENVIRONMENTALINFLUENCES
Let us now connect the above QEE detection schemewith the question of the nature of noise that leads toqubit dephasing. If the dynamics of the environment iscompletely independent of the presence of the qubit, wecan think of it as a source of a field that evolves in time insome complicated way, essentially stochastic. This fieldcan couple to the two levels of the qubit in a distinct way,so that the Hamiltonian of the qubit exposed to it isˆ H ( t ) = (cid:88) i =0 , ε i | i (cid:105)(cid:104) i | + (cid:88) i =0 , | i (cid:105)(cid:104) i | ξ i ( t ) , = [∆ ε + ∆ ξ ( t )]ˆ σ z / ε + ¯ ξ ( t )] / , (8)where ξ i are stochastic fields coupling to the qubit state | i (cid:105) , ∆ ε = ε − ε , ¯ ε = ε + ε , and ∆ ξ ( t ) and ¯ ξ ( t ) aredefined in an analogous way. It is now crucial to beaware that the dependence of ξ i ( t ) on the qubit state | i (cid:105) does not mean that the actual dynamics of E dependson this state: both ξ i ( t ) are related to an underlyingdynamics of the environment itself, and the dependenceon i is due to the fact that the two states might couple tothe environmental noise in a distinct way (see below fora simple example). The density matrix describing theinitialized | i (cid:105) state of the qubit does not change underthe influence of the above Hamiltonian. The evolutionfor time τ that precedes the creation of superpositionstate of the qubit is thus absent, and ρ (0)01 ( τ, t ) = ρ (1)01 ( τ, t )while being also independent of τ . Furthermore, ¯ ε and¯ ξ ( t ) drop out from the expression for qubit coherence, ρ (0 / ( τ, t ) = e − i ∆ εt (cid:68) e − i (cid:82) t ∆ ξ ( t (cid:48) )d t (cid:48) (cid:69) , (9)where (cid:104) . . . (cid:105) denotes averaging over realizations of ∆ ξ ( t (cid:48) )noise. Therefore, the observation of ρ (0)01 ( τ, t ) (cid:54) = ρ (1)01 ( τ, t )means that the environment cannot be described as asource of external classical noise acting on the qubit.Note that the known result that any pure dephasingevolution of a qubit can be reproduced by replacing theenvironment by an artificially constructed source of clas-sical noise [17–19], has no relevance to the above reason-ing. We are interested here in making a statement aboutthe dynamics of the real environment of the given qubit,and the above test allows us to easily notice the situationin which the qubit-environment interaction modifies thedynamics of the environment. V. RESULT FOR NV CENTER INTERACTINGWITH PARTIALLY POLARIZED NUCLEARENVIRONMENT
We now present an example of a system in which thecreation of QEE, and the non-classicality of the environ-mental noise, can be detected using the scheme describedabove. We focus on a nitrogen-vacancy (NV) center spinqubit in diamond, which has been a subject of intense re-search aimed at using it as a nanoscale resolution sensorof magnetic field fluctuations [6, 13, 34, 55–57]. The lowenergy degrees of freedom of the NV center constitutean effective electronic spin S = 1, subjected to zero-fieldsplitting ∆( S z ) , with the direction of z axis determinedby the geometry of the center. The presence of a finitemagnetic field (assumed here to be along the z axis) leadsto a splitting of m s = ± S = 1 manifold can be used as a qubit con-trolled by microwave electromagnetic fields. We focus onthe most widely employed qubit based on m = 0 and 1levels.The relevant environment of this qubit consists of nu-clear spins of either C spinful isotope naturally presentin a diamond lattice, or nuclei of molecules attached tothe surface of the diamond crystal [13, 57]. Due to alarge value of the zero-field splitting (∆ = 2 .
87 GHz),and a large ratio of electronic and nuclear gyromagneticfactors, for almost all values of the magnetic field, theenergy exchange between the qubit and the environmentis very strongly suppressed, and we can safely use thepure dephasing approximation [38]. Crucially, the m = 0state of the qubit is completely decoupled from the nu-clear environment (and if the nuclei can be treated assource of classical noise, ξ ( t ) = 0 while ξ ( t ) (cid:54) = 0), so thatkeeping the qubit in this state between the measurementand re-initialization does not perturb the state of the en-vironment. The QE Hamiltonian is thus given byˆ H = (∆ + Ω) | (cid:105) (cid:104) | + ˆ H E + | (cid:105) (cid:104) | ⊗ ˆ V , (10)where Ω = − γ e B z with the electron gyromagnetic ratio γ e = 28 .
02 GHz/T, and ˆ H E = (cid:80) k γ n B z ˆ I zk + ˆ H nn , where k labels the nuclear spins, γ n = 10 .
71 MHz/T for C nuclei, ˆ I zk is the operator of the z component of the nuclear spin k , and ˆ H nn contains the internuclear magnetic dipolarinteractions. Finally, the interaction term describes thehyperfine NV-nucleus interaction, andˆ V = (cid:88) k (cid:88) j ∈ ( x,y,z ) A z,jk ˆ I jk . (11)This coupling contains two parts, the Fermi contact in-teraction corresponding to non-zero probability of findingan electron bound to an NV center on the location of agiven nucleus, and the dipolar coupling. Usually, the for-mer is omitted since the wavefunction of a deep defectis strongly localized and in fact, for an NV center it hasnon-negligible impact when the distance between the va-cancy and the nucleus is not greater than 0.5 nm [58].With both types of interaction, the coupling constantsare given by A z,jk = 8 πγ e γ n | ψ e ( r k ) | + µ π γ e γ n r k (cid:32) − r k · ˆ j )( r k · ˆ z ) r k (cid:33) , (12)where µ is the magnetic permeability of the vacuum, r k is a displacement vector between the k -th nucleus andthe qubit and ψ e ( r k ) is the wavefunction of an electronfrom NV center at position of the k -th nucleus.We use B z = 200 Gauss, which was employed in a fewrecent experiments on qubit-based characterization of thesmall groups of nuclei [13, 55, 56]. We consider an en-vironment of about 500 spins in a ball of 9 nm radiuswith the NV at its center. Using a well-established andsystematic procedure of Cluster-Correlation Expansion(CCE) [38, 59, 60], we have checked that neither increas-ing the size of the environment, nor including the interac-tions within the environment, ˆ H nn , gives any visible con-tribution to decoherence of a freely evolving qubit. Thisis because the coherence decays practically completelybefore the more remote nuclei can have an appreciableinfluence on the qubit and the inter-nuclear correlationscreated by interactions become significant. We can thenfocus on single-spin precession as the only source of dy-namics within the environment.We consider a dynamically polarized nuclear environ-ment. The justification is twofold: (1) without dynamicnuclear polarization (DNP), the density operator of theenvironment at low fields is ˆ R (0) ∝ , and according toEq. (4) there is no QEE for such initial states; (2) DNPof the environment of an NV center has been recentlymastered [61–69] and its presence is expected to enhancethe signal that the qubit experiences. We assume thatˆ R (0) does not contain any correlations between the nu-clei, i.e. ˆ R (0) = (cid:78) k ˆ ρ k , where ˆ ρ k is the density matrixof k -th nucleus, given in the case of spin-1 / ρ k = ( + p k ˆ I zk ), where p k ∈ [ − ,
1] is the polarization ofthe k -th nucleus. Below we show results for the case inwhich all the (spinful) nuclei within a radius r p from thequbit are fully polarized, while the remaining nuclei are ( , ) ( , ) | ( , ) | FIG. 2. Difference between real (upper panel) and imaginary(middle panel) parts, as well as the absolute value (lowerpanel) of ρ (0)01 ( τ, t ) and ρ (1)01 ( τ, t ) coherence signals (normal-ized by the maximum qubit coherence) of an NV center qubitinteracting with partially polarized nuclear environment fora single randomly generated spatial arrangement of environ-mental spins, plotted for t = τ at magnetic field B z = 200 G.Dashed, dot-dashed and solid lines correspond to polarizationradius r p = 0 . . . r p = 0 . one polarized spin, hence thelack of evolution in the upper panel, as follows from Eq. (13). in a completely mixed state. This mimics the experimen-tally relevant situation, in which the DNP is created byappropriate prior manipulations on the qubit, that leadto polarization of nuclei that are most strongly coupledto it.We work in the rotating frame where the phase accu-mulated due to the controlled energy splitting of qubitlevels is absent. The coherence signal ρ (0 / ( τ, t ) is thenexpressed as (cid:81) k L (0 / k ( τ, t ), where L (0 / k ( τ, t ) are signalsthat would be obtained if the environment consisted only of the k -th nuclear spin. While the difference of the twosignals, ∆ ρ ≡ ρ (0)01 − ρ (1)01 , is not easily expressed through∆ L k ≡ L (0) k − L (1) k , it is instructive to look at such quanti-ties, which describe the difference of the coherence decaysignals for an environment consisting of a single spin:∆ L k = − i p k A x sin (cid:0) ω xz t (cid:1) sin (cid:0) ω xz τ + t (cid:1) sin (cid:0) ω τ (cid:1) ω xz , (13) t [] ( , ) ( , ) FIG. 3. Real (left) and imaginary (right) parts of ∆˜ ρ ( τ, t )as functions of τ and t for all nuclei polarized within radius r p = 0 . B z = 200G (spatial arrangement of environmental spins as in Fig. 2).Dashed black line signifies t = τ , which corresponds to theresults shown in Fig. 2. in which for clarity we only kept the z and x cou-plings to the qubit ( A z = A z,zk and A x = A z,xk ), ω xz ≡ (cid:112) A x + ( A z + ω ) . The above expression vanishes wheneither p k = 0, A x = 0, or τ = 0. Since ∆ L k is purely imagi-nary, we should carefully inspect both real and imaginaryparts of ∆ ρ , not just its magnitude. The results for anNV center interacting with natural concentration envi-ronment of C spins in diamond, obtained for a singlerandomly generated spatial arrangement of these spins,are shown in Figs 2 and 3. Both figures show the real andimaginary parts of ∆˜ ρ ( τ, t ) = ∆ ρ ( τ, t ) /ρ ( τ,
0) (thedifference is normalized by the initial qubit coherence),while Fig. 2 additionally contains plots of the differencebetween the absolute values of ρ (0)01 ( τ, t ) and ρ (1)01 ( τ, t )(identically normalized). When for a given delay time τ any of these values is non-zero, this signifies that QEEwould be present at time τ during the joint evolution ofan initial product state of any superposition of the qubitand state ˆ R (0) of the environment. In Fig. 2 the resultsshown are for equal evolution and delay times, t = τ ,showing that QEE is present for an initial superpositionqubit state throughout the evolution.Fig. 4 contains the plots of the imaginary part of∆˜ ρ ( τ, τ ) for eight different random realizations of thenuclear environment supplemented by bar graphs illus-trating the number of spinful nuclei for a given polariza-tion radius. For half of the presented realizations of theenvironment, there are no C nuclei in the region (a ballof radius 0.5 nm around the vacancy), where the Fermicontact part has to be taken into account. In fact, thereis roughly a 45% probability to find a realization contain-ing a C nucleus whose Fermi contact coupling shouldbe included. For such realizations, the Fermi contactcoupling affects the coupling parallel to the quantizationaxes, which, according to Eq. (13), modifies denominatorof the expression, but also produces much faster oscilla-tions in difference of real as well as imaginary parts ofcoherence.Results shown in Fig. 4 provide additional evidence ( , ) (a) (b) (c) (d) o f s p i n s ( , ) (e) (f) (g) (h) o f s p i n s FIG. 4. Imaginary part of ∆˜ ρ ( τ, τ ) of an NV center qubit for 8 different random realizations of the nuclear environment atmagnetic field B z = 200 G. Dashed, dot-dashed, and solid lines correspond to polarization radius r pol = 0.5, 0.7 and 0.9 nm,respectively. The bar graphs show the number of C nuclei corresponding to a given r pol in the figure directly above. for the high magnitude of the QEE signal. The differ-ence in both real and imaginary parts (correspondingto measurement of ˆ σ x and ˆ σ y of the qubit initialized in( | (cid:105) + | (cid:105) ) / √ VI. DISCUSSION AND CONCLUSION
In conclusion, we have described a simple experimen-tal protocol that allows to check, if QEE generation ac-companies pure dephasing of a qubit. Importantly, thisprotocol requires operations to be performed only on thequbit. Although we focus on the fact that the proposedmethod allows for straightforward experimental verifica-tion, it is relevant to note that it is also a good theoreticaltool. The advantage over the method of Ref. [22] stemsfrom the fact that only the evolution of qubit coherence(for two different initial states of the environment) needs to be calculated and neither the whole QE state (as ingeneral methods) nor operators acting on the environ-ment have to be found. A positive result of such a testnot only certifies that QEE is created, but also that theinfluence of the environment cannot be described as clas-sical (i.e. independent of the existence of the qubit) noiseof either Gaussian or non-Gaussian statistics (note thatsome tests [70] aimed at detecting the non-classical na-ture of environmental noise were in fact detecting thenon-Gaussian statistics of it).We have presented theoretical results of the working ofthis protocol for an NV center coupled to a partially po-larized environment consisting of nuclear spins. We havepredicted a signal of magnitude comparable to the ob-served coherence, clearly showing that the protocol is ro-bust to single-qubit control errors that in principle coulddepend on the state of the environment. While quanti-fying the relation between the degree in which the zero-entanglement condition is broken and the magnitude ofthe signal observed in our protocol is beyond the scopeof this paper, the fact that the signal is clearly visiblemeans that the classical picture of environmental noise,while being widely adopted for analysis of data obtainedwith NV centers coupled to nanoscale nuclear environ-ments, is definitely not exact in the case of the NV centerinteracting with polarized nuclei.
ACKNOWLEDGEMENTS
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