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[Jun-ichi Inoue](https://orcid.org/0000-0001-6743-4258)

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[Upper Limit of Pure Dephasing Time of a Paired-Qubit with Anisotropic Heisenberg Coupling](https://mdr.nims.go.jp/datasets/4ad2d748-68b1-4012-ba3f-deca702d1900)

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Upper Limit of Pure Dephasing Time of aPaired-Qubit with Anisotropic Heisenberg CouplingJun-ichi Inoue∗National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, JapanJune 13, 2024Robustness against noise is essential for qubits used in applications suchas quantum computing [1] and sensing [2, 3, 4]. One measure of this propertyis their relaxation times, which have typically been used in Bloch equations[5, 6, 7]. Systems or materials with longer relaxation times are generallymore valued. This tendency is particularly notable in quantum sensing,where operation at room temperature is standard [8]. Consequently, ex-tensive research has been conducted across various fields to achieve longerrelaxation times.The immense importance of long relaxation times in quantum sensingis due to their impact on sensing resolution [8]. Thus, the natural expecta-tion that high-quality samples, such as those with fewer inclusions, defects,and dislocations, are more promising has driven the development of variouscrystal-growth techniques [9]. However, the situation is not so straightfor-ward, as shown by the counter-intuitive case where intentional phosphorusdoping resulted in the longest relaxation time [10].The pure dephasing time, T[p]2 , which is the main focus of this work,characterizes the dephasing process induced by inhomogeneity that causesthe energy levels of qubits to fluctuate [11, 12]. This time is a relevant com-ponent of the inhomogeneous transverse relaxation time. The dephasingtime typically describes an exponential decay of an in-plane Bloch vector,corresponding to free induction decay (FID). The reason for focusing solelyon T[p]2 in this study is its particular importance in DC-response measure-ments [8] and the theoretical interest in obtaining its analytical form for amodel that is both well-tailored for theoretical studies and experimentallyaccessible.∗inoue.junichi@nims.go.jp1The essence of pure dephasing time can be captured in a two-level sys-tem, i.e., a single spin 1/2 in a magnetic field with an amplitude that fluc-tuates according to a Gaussian random process. For later convenience, theresult and outline of the derivation using the Lindblad master equationare summarized here [13, 15, 14]. This equation describes the time evolu-tion of the density operator ρ(t) in an open system. For a single two-levelsystem exposed to a pure dephasing or phase-damping process [1], it isdρ/dt = −i[Ωσz, ρ] + (2LρL†, L†Lρ − ρL†L)/2, where h̄ = 1. In this equa-tion, σz is the z component of a usual 2×2 Pauli matrix σ⃗ = (σx, σy, σz), theconstant Ω denotes the energy difference between the two levels, L =√Γzσzis the Lindblad operator for the pure dephasing process, and Γz characterizesGaussian noise in the amplitude of the fictitious magnetic field. In this proce-dure, the pure dephasing time T[p]2 = 1/(2Γz) is given by one component of anin-plane Bloch vector ⟨σx(t)⟩ ≡ Tr[ρ(t)σx] = exp[−2Γzt] cos(2Ωt). To derivethis solution without losing generality, the initial state ρ(t = 0) = (1+σx)/2was used, corresponding to a state prepared using the π/2 pulse in usualFID experiments.In this study, we propose a theoretical and experimentally feasible ideato prolong the pure dephasing time in a qubit system and derive a formulafor this time, with NV− centers in diamond [3, 9, 8, 16] in mind as a testbed. The core idea is to use the interaction between qubits. When aninferior qubit (one with a shorter pure dephasing time) interacts with asuperior one, the former benefits and experiences a prolonged dephasingtime. Although this scenario seems intuitively plausible, several questionsmust be addressed to support it: (i) what kinds of interaction would serve,(ii) whether the prolongation has an upper limit, and (iii) how the prolongeddephasing time depends on the interaction. The following sections answerthese questions through an analytical expression of the pure dephasing timeobtained by solving a Lindblad master equation, which will be introducedshortly. Subsequently, the significance of the findings is also discussed.Consider two qubits α and β. Each is assumed to connect with its ownreservoir, Rα and Rβ, respectively. Reservoir Rα(β) causes the connected-qubit to dephase at its pure dephasing rate Γαz(βz). These qubits are nonequiv-alent in the sense that unequality of Γαz > Γβz is imposed. The two qubitsinteract with each other in an anisotropic Heisenberg or XXZ manner andtheir in-plane and longitudinal magnitudes are denoted by J// and Jz, respec-tively. Schematics of this system are shown in Fig.1. Hereafter, qubit α(β)is referred to as the surface (bulk) qubit.The Lindblad master equation for the total density operator ρ(t) of the2ΩΩJ/ / , Jzσαzσβz༎βz༎αzRαRβFigure 1: A system considered in this study consists of two qubits, α andβ, each of which is described by a Pauli operator σαz(βz), respectively. Eachconnects with its own reservoir Rα and Rβ, which induces pure dephasingin the connected-qubit with its rate Γαz > Γβz. The two qubits interact viaanisotropic Heisenberg coupling, J// and Jz. A solid horizontal line indicatesa surface of the sample.two qubits is given bydρ(t)dt=− i[H, ρ(t)]+∑µ=α, β(Lµρ(t)L†µ −12L†µLµρ(t) −12ρ(t)L†µLµ), (1)where the Hamiltonian H is expressed asH =Ω(σαz ⊗ 1β + 1α ⊗ σβz)+ J//(σαx ⊗ σβx + σαy ⊗ σβy) + Jz(σαz ⊗ σβz), (2)and a set of Lindblad operators {Lµ} describing the independent pure de-phasing process is expressed as{Lµ=α,β} ={√Γαzσαz ⊗ 1β,√Γβz1α ⊗ σβz}, (3)with the identity operator 1α(β) used for the surface (bulk) qubit. To addressquestions (i) – (iii), one component of the in-plane Bloch vector for a surfacequbit⟨σαx(t)⟩ = Tr [ρ̃α(t)σαx] , (4)is calculated. In this equation, the partial density operator for the surfacequbit ρ̃α(t) ≡ Trβ [ρ(t)] is introduced, where Trβ denotes the partial traceof the bulk qubit. Assume that π/2 pulse is injected equivalently into both3qubits and we take ρ(t = 0) = (1α + σαx)⊗ (1β + σβx)/4, as an initial statewithout loss of generality.⟨σαx(t)⟩ = e−(Γαz+Γβz)t cos(2Ωt)×[cos(2Jzt) cos(Λ(J//)t)+2J// sin(2Jzt) − (Γαz − Γβz) cos(2Jzt)Λ(J//)sin(Λ(J//)t)], (5)is obtained after a lengthy, albeit straightforward, calculation, whereΛ(J//) ≡√(2J//)2 − (Γαz − Γβz)2 is defined. To obtain an in-depth understanding ofEq.(5), examinations of three canonical cases, Ising-, XY-, and isotropicHeisenberg-interactions, are helpful.In case of Ising, we set J// = 0 making Λ(J// = 0) purely imaginary and itis represented as Λ(J// = 0) = i(Γαz− Γβz) because we assume that Γαz > Γβz.Using this reduces Eq. (5) to⟨σαx(t)⟩ =e−2Γαzt cos(2Ωt) cos(2Jzt). (6)The decay of ⟨σαx(t)⟩ follows a single exponential function with a pure de-phasing time T[p]2 = 1/(2Γαz). This result is the same as that of the singlequbit case. Note that the dephasing time is unrelated to the bulk qubitand their interaction, albeit the two qubits interact. Instead, the effect ofJz appears in the angular frequencies of the oscillation, which are bi-color2Ω± 2Jz.Next, for XY case, substitution of Jz = 0 yields Eq. (5) into⟨σαx(t)⟩ =e−(Γαz+Γβz)t cos(2Ωt)×[cos(Λ(J//)t) −Γαz − ΓβzΛ(J//)sin(Λ(J//)t)]. (7)In contrast to the Ising case, the pure dephasing time of ⟨σαx(t)⟩ is influencedby the bulk qubit, the manner of which is distinct depending on J// ≶ (Γαz−Γβz)/2 ≡ J//c > 0.When J// > J//c, Λ(J//) is real; thus, the time profile of ⟨σαx(t)⟩ is con-sidered at its face value, as shown in Eq.(7). The envelope exhibits a singleexponential decay with a time constant 1/(Γαz + Γβz), which now dependson the bulk qubit decay rate Γβz, but not on the interaction strength J//.This dephasing rate should be read as 2× (Γαz + Γβz)/2, which is analogousto the result for a single qubit 2 × Γz. Thus, the surface qubit is thoughtto effectively acquire a smaller dephasing rate, (Γαz + Γβz)/2 < Γαz, because4of XY-interaction with the bulk qubit. This effective value seems reasonablebecause it is the arithmetic mean of dephasing rates of the two qubits. Fromthis result, we propose that the two qubits for J// > J//c form a similar stateto the strongly coupled state in quantum optics [6, 7, 12]. Furthermore,the bulk qubit contributes to the angular frequencies of the oscillations. Itis bi-color, similar to the Ising case, but its angular frequencies are now2Ω±Λ(J//) which depend on Γβz and the interaction strength.For J// < J//c, Λ(J//) is purely imaginary; the envelope of ⟨σαx(t)⟩ con-sists of two exponential decay terms, each of whose rates is Γαz + Γβz ±√(Γαz − Γβz)2 − (2J//)2, respectively. Among these two rates, the larger one,or the faster decay channel, dominates the overall dephasing characteristicsof the surface qubit, so that it is a reasonable pure dephasing rate of ⟨σαz(t)⟩in this study. In contrast to the previous cases, this pure dephasing rate de-pends on both Γβz and J// in a nontrivial manner. Meanwhile, the oscillationis monochromatic, with the intrinsic angular frequency 2Ω. This result isthe same as that for a single isolated qubit, although the surface qubit stillinteracts with the bulk qubit through J//.Finally, in Heisenberg case, J// = Jz ≡ JH, the decay profile of ⟨σαx(t)⟩ isimmediately found to be the same as that in the XY case for the entire rangeof JH because Λ does not include Jz. An explicit form of the pure dephasingrate is obtained by replacing J// with JH in the XY case. On the other hand,in the oscillation property, one feature is observed: it is a combination ofthe previous two cases. For JH > J//c, the oscillation is characterized by fourangular frequencies 2Ω± 2JH±Λ(JH), whereas for JH < J//c, by two angularfrequencies 2Ω± 2JH, which is similar to the Ising case.Having examined these three cases, we can now fully characterize ourmain result for the anisotropic Heisenberg interaction case in Eq. (5). Thein-plane Bloch vector of the surface qubit interacting with the bulk qubitexhibits a single or double exponential decay accompanied by multi-coloredoscillations. The pure dephasing time T[p]2 can be formulated as follows:1T[p]2= Γαz + Γβz + Im[√(2J//)2 − (Γαz − Γβz)2]. (8)By increasing J// from the non-interacting case J// = 0, the pure dephasingtime of the surface qubit is monotonically prolonged from its bare value1/(2Γαz), and finally reaches a maximum of 1/(Γαz+ Γβz) at J//c, which valueis maintained for J// > J//c. In addition, Jz does not contribute to the puredephasing time. On the other hand, the angular frequencies of the oscillation52πF have a parallel form to Eq. (8)2πF = 2Ω± 2Jz ± Re[√(2J//)2 − (Γαz − Γβz)2]. (9)This multi-colored character of the oscillation also changes qualitatively atJ// = J//c from the angular frequencies 2Ω± 2Jz to 2Ω± 2Jz ± Λ(J//). Eachof Eqs.(8) and (9) are plotted as a function of J///J//c in Figs.2 (a) and(b), respectively. It is evident that the signs of J// and Jz are irrelevant aslong as Γαz > Γβz > 0. Thus, the results are valid for both of ferro- andantiferro-magnetic interactions.These analytical results address the previously raised questions: Thepure dephasing time of a qubit with a shorter dephasing time is prolongedthrough interaction with another qubit with a longer dephasing time. Theprolongation increases monotonically but nontrivially with the strength ofthe interaction between the two qubits. The benefit of the interaction comessolely from the XY component J//, and not from the longitudinal componentJz. The prolonged dephasing time eventually saturates at J// = J//c, beyondwhich the time remains constant. The upper limit of pure dephasing timeis given by the reciprocal of the reciprocal average of the dephasing timesof the two qubits, T[p]α(β)2 = 1/(2Γα(β)z). Therefore,T[p]α2 ≤ T[p]2 ≤2T[p]α2T[p]β2T[p]α2 + T[p]β2, (10)as described by Eq.(8). Although the form of the right-hand side or upperlimit can be intuitively guessed as it looks familiar, we claim that the noveltyof this work is that it explicitly shows how T[p]2 depends on J// and Γαz(βz)and that T[p]2 is free from Jz. This insensitivity to Jz is understandable,because the pure dephasing process of a qubit is caused by the fluctuationof z component of a fictitious magnetic field characterized by Γαz(βz), andthus manifests itself in the in-plane motion of the qubit [17]. It is J//, not Jz,that influences the in-plane motion. The saturation of the pure dephasingtime, whose region is represented in gray in Fig. 2, suggests that the twoqubits are considered to form a strongly coupled state. This state couldalso be monitored in terms of the number of oscillation frequencies. Theyare doubled from one to two or two to four, depending on Jz = 0 or not,respectively, with increasing J// for the given constants Γαz(βz).Now let us consider feasible experimental tests of the results obtainedabove. A candidate for real systems corresponding to Fig.1 is NV− centers in60 1(a)༎αz + ༎βz2༎αz2Ω2Ω + 2Jz2Ω − 2Jz(b)J‖/ J‖cFigure 2: Thick solid lines represent (a) the pure dephasing rate, Eq. (8)and (b) the angular frequencies, Eq. (9), as a function of J///J//c with fixedconstants, Γαz > Γβz, and Jz. The gray region indicates a strong couplingregion. Dashed lines are guides to the eyes.diamond. The origin of J// in Eq.(8) can be traced back to a magnetic dipoleinteraction between the surface and bulk qubits [18, 19]. With advancementsin fabrication techniques, these centers can now be accurately located andoriented in diamond [20]. Thus, the following experiments would be possible.Let r be the distance between two qubits. The dipole interaction strengthcan be simply represented by J// = J//c(rc/r)3 for J// < J//c, with a certainnormalization length rc. The details of rc, which depend on material char-acteristics, are beyond the scope of the current study. In this usage, Eq.(8)takes the form1T[p]2= ReΓαz + Γβz + (Γαz − Γβz)√1−(rrc)−6 , (11)and is plotted in Fig.3 as a function of r/rc. The inset shows a double-log plot of |1/T[p]2 − 2Γαz|, highlighting power dependence with respect tor in the vicinity of r/rc ≳ 1, which is accompanied by the dashed linelimr→∞ |1/T[p]2 − 2Γαz| = (Γαz − Γβz) (r/rc)−6 /2. Thus, for several sampleswith different r values but fixed orientations of NV−s, measurements of pure70 1 r/rc1 r/rc 2༎αz + ༎βz2༎αz1/T[p]22༎αz − 1/T[p]2༎αz − ༎βz12(༎αz − ༎βz)∼ r−6Figure 3: Distance between the two qubits dependence of the pure dephasingrate in a dipole interaction case (main) and double-log plot of |1/T[p]2 − 2Γαz|in the vicinity of r/rc ≳ 1 (inset, solid line), with a guide to the eyes (Γαz −Γβz)/2× (r/rc)−6 (inset, dashed line).dephasing times can test the present theory. In particular, when the puredephasing time becomes saturated when r is less than a certain value, thissuggests a strongly coupled state of the two qubits.Another test uses the relative orientation of the two qubits while main-taining the distance r. The in-plane interaction strength J// which originatesfrom the dipole interaction, depends on the relative angle θ of the two dipolesas J// ∝ cos θ. In the case of NV−, the orientation of the dipole vector isconsidered parallel to the direction from N- to the neighboring vacancy sites.As NV− has C3v point-group symmetry, where the threefold rotation axisis along the N-V direction, we have finite choices for θ, including the optionof ferro- or antiferro-like, although some of them are equivalent. However,considering θ as a continuous variable, we let J// = J//c cos θ, then Eq.(8)takes the form 1/T[p]2 = Γαz + Γβz + (Γαz − Γβz) sin θ. Hence, when this angledependence is observed for samples with different θ values and a commonr, it further supports this study.One of the challenges in the application of NV− centers in diamondfor quantum sensing is regarding the locations of the NV−s. To achieve afiner resolution, a shorter distance between the measured object and theNV− centers, or in other words, NV− centers located closer to the diamondsurface, is more advantageous. However, this would result in serious damageto the resolution because of the drastically shortened relaxation times of thequbits caused by various types of fluctuations intrinsic to the surface [21].The concept presented in this paper offers a solution to this problem.8Before concluding the paper, a brief description of previous studies isprovided. Studies of an open quantum system with a two-spin have so farassumed mainly semiconductor double quantum dots, whose major interestshave been, to name a few, decoherence dynamics, entanglement dynamics,and singlet-triplet conversion rate [22, 23, 24, 25, 26, 27, 28, 29]. In dotsystems, two electron spins are considered equivalent and thus treated onan equal footing, which allows the introduction of a sum-operator for thetwo spins and describes them in terms of singlet and triplet states. TheirHamiltonian has the sum-operator squared, which restricts the interactionof the two spins to an antiferromagnetically isotropic Heisenberg coupling.By contrast, this study naturally introduces an anisotropy into the inter-action, Jz ̸= J// [18, 19], between the two qubits. Until this anisotropy isconsidered, the important finding that the pure dephasing time is free fromthe longitudinal component of the interaction is not provided.A possible extension of the current study is to increase the number ofqubits. This interest is inspired by a report that a triple nitrogen-vacancycenter in diamond was fabricated by ion implantation [30]. This systemwould introduce a new ”degree of freedom” that a surface qubit is locatedeither at the end or the center of the row. As the number of qubits increases,a semi-infinite length chain of qubits becomes conceivable, aside from feasi-bility in real systems [32, 33, 34, 31]. The dynamics of a qubit attached tothe edge of the chain has been discussed using the Lindblad equation, focus-ing on non-Hermiticity, specifically exceptional points, PT symmetry, andthe eigenvalues distribution of the Liouville operator in conjunction withthe language of topology. Using a chain also allows for the introduction ofvarious kinds of interactions between qubits, such as Su-Schrieffer-Heeger,Kitaev, and others, which are expected to unveil further rich physics.In conclusion, we propose a theoretical scheme to prolong the pure de-phasing time of a qubit by coupling it with another qubit with a longerpure dephasing time. From the analytical solution obtained here, we haveshown that there is an upper limit to the time, below which the time is amonotonically but nontrivially increasing function of the in-plane couplingJ// but is insensitive to Jz. This theoretical idea could address the challengesof achieving longer dephasing times for improved sensor resolution. 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