NNote: This is the preprint of an article to appear in the
Proceedings of the International School of Physics “E. Fermi”Course 204:
Nanoscale Quantum Optics eds. M. Agio, I. D’Amico, C. Toninelli, and R. Zia
Quantum optics with single spins
Lee C. Bassett
Quantum Engineering LaboratoryDepartment of Electrical & Systems EngineeringUniversity of PennsylvaniaPhiladelphia, PA, USA
A course presented at the International School of Physics “Enrico Fermi”at Villa Monastero, Varenna, Lake Como, Italy in July 2018.
Summary. — Defects in solids are in many ways analogous to trapped atoms ormolecules. They can serve as long-lived quantum memories and efficient light-matterinterfaces. As such, they are leading building blocks for long-distance quantum net-works and distributed quantum computers. This chapter describes the quantum-mechanical coupling between atom-like spin states and light, using the diamondnitrogen-vacancy (NV) center as a paradigm. We present an overview of the NVcenter’s electronic structure, derive a general picture of coherent light-matter in-teractions, and describe several methods that can be used to achieve all-opticalinitialization, quantum-coherent control, and readout of solid-state spins. Thesetechniques can be readily generalized to other defect systems, and they serve as thebasis for advanced protocols at the heart of many emerging quantum technologies. c (cid:13) Lee C. Bassett a r X i v : . [ qu a n t - ph ] A ug Lee C. Bassett
1. – Introduction
Solid-state spins are among the most versatile platforms for quantum science andtechnology. Select semiconductor defects — exemplified by the nitrogen-vacancy (NV)center in diamond — exhibit spin coherence at room-temperature and intrinsic opticalspin-readout mechanisms that underly their remarkable capabilities as room-temperaturequbits and quantum sensors. When used in this way, quantum-coherent control is per-formed using microwaves that couple resonantly to the qubit’s electron spin Hamilto-nian. Optical pumping and fluorescence are used for spin initialization and readout,respectively, but these processes rely on dissipation through nonradiative and vibronictransitions that involve coupling to phonons in the crystal and are therefore incoherent.When the crystal is cooled down, however, the optical transitions between different or-bital states become coherent, and they can be manipulated using resonant optical fieldsjust as the spin is controlled with microwaves. Moreover, spin-orbit coupling mediatesinteractions between optical fields and spins, enabling all-optical (i.e., microwave free)spin control, robust spin initialization and readout, and various schemes for generatingspin-photon entanglement.In this chapter based on lectures from the 2018 Enrico Fermi Summer School on
Nanoscale Quantum Optics , we introduce a general picture for coherent light-matterinteractions based on coherent, dispersive interactions with spin-selective optical tran-sitions based on the Jaynes-Cummings Hamiltonian for quantum electrodynamics. Thechapter is organized as follows. In Section , we introduce the spin and optical finestructure of the NV center, including the role of various perturbations. Subsequently, inSection we summarize key concepts of quantum optics including the Jaynes-CummingsHamiltonian, and show how coherent, dispersive, light-matter interactions give rise to theoptical Stark effect and the Faraday effect, which can be used respectively to control andmeasure NV-center spin states. In Section we generalize the treatment to include morecomplex dynamics exhibited by an optical Λ configuration, including coherent populationtrapping and stimulated Raman transitions, and in Section we describe an alternate,non-dispersive technique to probe and control quantum dynamics using ultrafast opticalpulses. Section summarizes the chapter and highlights future directions for the ap-plication of these techniques to address other spin-qubit platforms, and to enable moreadvanced schemes for quantum control within quantum networks. Much of the materialis adapted from Buckley et al. [1], Yale et al. [2], and Bassett et al. [3], and moreinformation regarding the experiments and models can be found in those references.
2. – Electronic structure of the diamond nitrogen-vacancy center
The NV center in diamond has been an object of fascination since the 1950s asone of the predominant color centers in diamond, and the focus of intense study inquantum information science since the turn of the 21 st century [4]. Its popularity andimportance in quantum science stem from several key characteristics, including long spincoherence of the triplet ground state, which persists to room temperature and above, and uantum optics with single spins Room Temperature m s = 0 m s = ±1 m s = 0 m s = ±1 ISC
637 nm γ nr±1 γ nr0 γ r γ r TripletSinglet1042 nm κ ±1 κ PL Low Temperature δ Transverse strain / electric field |A 〉|A 〉|E x,y 〉|E 〉 | U,m s 〉| L,m s 〉 } A E (a) (b) Figure 1. –
Electronic structure of the diamond NV center . (a) At room temperature,rapid phonon transitions within the orbitals of the E excited state lead to an effective spin-triplet with spin-dependent non-radiate decay channels, γ nr m s , through the ISC as shown. Thesedynamics produce the NV center’s spin-dependent PL. (b) At low temperature, the orbital finestructure within E is resolved. The unperturbed spin-orbit states evolve into two separatedorbital branches as a function of the transverse strain or dc Stark shift, δ . Eigenstates in (b)are calculated according to Eq. (2) for B = 0 G and α s = 0. efficient, stable, visible photoluminescence (PL) that can be used to measure the spinstate populations. The latter property stems from the NV center’s specific electroniclevel structure, which at room temperature takes the effective form shown in Fig. 1(a).The spin-triplet ground state and optically excited state — which is responsible for thevisible PL — is connected to manifold of intermediate singlet states through an inter-system crossing (ISC). The nonradiative ISC is mediated by phonons and the spin-orbit interaction, and the rates in both the upper and lower branches depend on thetriplet spin projection. In particular, the upper ISC transition from the triplet excitedstate occurs predominantly for the m s = ± S z projection, where z is along the defect’s symmetry axis). These intrinsic, spin-dependentoptical dynamics provide the mechanisms for optical spin initialization and PL-basedspin readout that are used in a majority of NV-center applications, especially at roomtemperature [5].While the A ground state is an orbital singlet, the E excited state is an orbitaldoublet. At room temperature, rapid phonon-mediated transitions between the orbitalbranches result in an effective spin-triplet Hamiltonian as shown in Fig. 1(a). At temper-atures below ≈
20 K, however, phonon-induced transitions are suppressed, and the fine
Lee C. Bassett structure associated with the full six-dimensional excited-state Hamiltonian emerges inthe optical transitions, as shown in Fig. 1(b) [6, 7]. At these temperatures, and in purediamond samples, the zero-phonon-line (ZPL) transitions become spectrally narrow, insome cases approaching the lifetime limit, such that coherent Rabi oscillations can beobserved between the ground and excited-state orbitals [8]. .1. The electronic Hamiltonian . – The form of the NV center’s electronic Hamiltoniancan be derived and understood using group theory [9, 10]. The ground-state Hamiltonianis given by(1) H gs = D gs (cid:18) S z − (cid:19) + g gs µ B S · B , where D gs is the reduced matrix element (RME) for the axial spin-spin interaction, andthe second term describes the Zeeman interaction in terms of the Land´e g -factor, g gs ,Bohr magneton µ B , electron spin operator S (where S = 1, S ± = S x ± iS y ), andmagnetic field B . We generally set h = 1 such that terms in the Hamiltonian can bewritten in either energy or frequency units. Similarly, the excited-state Hamiltonian canbe written as a sum of terms due to intrinsic (spin-orbit, spin-spin) interactions andextrinsic (magnetic, strain/electric) fields,(2) H es = H so + H ss + H Z + H L + H s . Below we give explicit matrix expressions for these terms in the product basis | ε, m s (cid:105) ∈{ ( | X (cid:105) , | Y (cid:105) ) ⊗ ( |− (cid:105) , | (cid:105) , | +1 (cid:105) ) } , where ( | X (cid:105) , | Y (cid:105) ) are E -symmetry orbital states trans-forming like the vectors ( x, y ) in the NV-center coordinate system. The spin sublevels | m s (cid:105) are eigenstates of the S z operator, whereas the orbital part of the Hamiltonian canbe written in terms of the Pauli matrices σ es i ( i = x, y, z ) and σ es ± = σ es z ± iσ es x , which op-erate on the two-dimensional orbital excited-state degree of freedom, i.e. , σ es z | X (cid:105) = | X (cid:105) and σ es z | Y (cid:105) = −| Y (cid:105) . Note that σ es ± are not the standard raising and lowering operators.The only spin-orbit coupling allowed by symmetry is the axial one ( i.e. , proportionalto S z [10]), with RME λ . In the product basis, this takes the form:(3) H so = − λσ es y S z = − iλ iλiλ − iλ . The spin-spin interaction has three symmetry-allowed RMEs, corresponding to one axial uantum optics with single spins ( D es ) and two transverse couplings (∆ , ∆ ). This takes the form: H ss = D es (cid:18) S z − (cid:19) − ∆ (cid:0) S σ es − + S − σ es+ (cid:1) + ∆ √ (cid:0) { S + , S z } σ es+ + { S − , S z } σ es − (cid:1) = D es / − ∆ / − ∆ / i ∆ / − i ∆ / − ∆ / − D es / / − i ∆ / − i ∆ / − ∆ / / D es / i ∆ / i ∆ / i ∆ / − i ∆ / D es / / / − i ∆ / − i ∆ / / − D es / − ∆ / i ∆ / i ∆ / / − ∆ / D es / . (4)Here, { A, B } ≡ AB + BA denotes the anticommutator.The Zeeman ( H Z ), diamagnetic ( H L ) and strain/dc-Stark ( H s ) perturbations affectonly the spin or orbital degrees of freedom individually. The E symmetry of the excitedstate allows different effective g -factors ( g (cid:107) es , g ⊥ es ) for axial and transverse components ofthe Zeeman interaction, such that H Z = g ⊥ es µ B ( B x S x + B y S y ) + g (cid:107) es µ B B z S z = I ⊗ µ B − g (cid:107) es B z g ⊥ es ( B x + iB y ) 0 g ⊥ es ( B x − iB y ) 0 g ⊥ es ( B x + iB y )0 g ⊥ es ( B x − iB y ) g (cid:107) es B z . (5)Similarly, the axial diamagnetic shift is given by the orbital operator(6) H L = µ B L z B z σ es y = µ B L z B z (cid:18) − ii (cid:19) ⊗ I , where L z µ B is the z component of the orbital magnetic moment. Symmetry implies thattransverse diamagnetic components are zero. The orbital magnetic moment is knownto be relatively small from measurements of circular dichroism [11, 12], with a value L z = 0 .
05 that corresponds to a frequency shift of only L z µ B B/h ≈
50 MHz at B =100 G. This value is comparable to typical optical linewidths and smaller than most otherterms in the Hamiltonian, hence the diamagnetic shift is often ignored for measurementsperformed at relatively low magnetic fields.Finally, the perturbation due to transverse strain or electric fields is given by(7) H s = − δ x σ es z + δ y σ es x = δ (cid:18) − cos( α s ) sin( α s )sin( α s ) cos( α s ) (cid:19) ⊗ I , where δ x = δ cos( α s ) and δ y = δ sin( α s ) are the strain (or dc Stark) perturbation com-ponents in crystallographic x and y directions with units of energy, where the totaltransverse perturbation has an effective angle α s (note the total energy splitting betweenthe orbital branches is 2 δ ). Lee C. Bassett .2. Low- and high-strain regimes . – An arbitrary crystal strain tensor can be decom-posed into components that transform according to the C v irreducible representations A (transforming like the vector z ), and E (with components { E x , E y } that transformlike the vectors { x, y } , respectively). Similarly, the dc Stark perturbation due to electricfields applied along z transform like A whereas transverse fields transform like E . Sincethe perturbations affect the excited-state Hamiltonian in exactly the same way, the dcStark effect can be used to compensate an uncontrolled local strain [13]. Whereas trans-verse strain/Stark fields shift the orbital energies and eigenstates according to Eq. (7),the longitudinal perturbation is proportional to the orbital identity operator, amount-ing to an overall shift of the optical transition frequency between the ground and excitedstate, but no variations of the eigenstates within the excited-state manifold. The longitu-dinal shift can be important when multiple NV centers need to be tuned to interact withindistinguishable photons [14], however for control of individual defects it is generallypossible to compensate this shift by tuning the laser, so this term is neglected here.Near δ = 0, it is convenient to use the spin-orbit basis in which the Hamiltonian isnearly diagonal [9, 10], aside from the small spin-spin coupling ∆ . The spin-orbit basisstates can be written as follows in terms of the product basis: | A (cid:105) = − i | X, − (cid:105) + | X, +1 (cid:105) + i | Y, − (cid:105) − i | Y, +1 (cid:105) )(8a) | A (cid:105) = 12 ( | X, − (cid:105) − | X, +1 (cid:105) + i | Y, − (cid:105) + i | Y, +1 (cid:105) )(8b) | E (cid:105) ≡ | E ± ,x (cid:105) = − i | X, − (cid:105) + | X, +1 (cid:105) − i | Y, − (cid:105) + i | Y, +1 (cid:105) )(8c) | E (cid:105) ≡ | E ± ,y (cid:105) = −
12 ( | X, − (cid:105) − | X, +1 (cid:105) − i | Y, − (cid:105) − i | Y, +1 (cid:105) )(8d) | E x (cid:105) ≡ | E ,x (cid:105) = −| Y, (cid:105) (8e) | E y (cid:105) ≡ | E ,y (cid:105) = | X, (cid:105) . (8f)In this basis, the states are labeled according to the symmetry of the tensor product ofspin and orbital states, which can be obtained from tables of group-theoretic couplingcoefficients [15]. For example, the state | A (cid:105) transforms like the irreducible representation A . The arrangement of these levels at zero strain and zero magnetic field is shown inFig. 1(b). It is important to work in this low-strain regime for some applications. Forexample, spin-orbit optical selection rules that link particular spin states with circularpolarization states are present when | A (cid:105) and | A (cid:105) are excited-state eigenstates, andthese selection rules can be used to generate spin-photon entanglement [16] or to mapphoton states onto spin states [17].On the other hand, when the transverse strain/Stark perturbation is large, the excited-state manifold splits into two orbital branches, each with (spin-independent) linear po-larization optical selection rules for transitions to the ground state. This situation occurswhen the strain splitting, 2 δ , dominates over the other coupling terms between the or-bital branches, the most important being the spin-orbit parameter λ = 5 .
33 GHz [3]. uantum optics with single spins Since strain splittings observed for NV centers in high-quality bulk diamond typicallyrange between 5–50 GHz, this is often the natural situation for experiments, and it canbe useful when one wishes to isolate the role of a single orbital branch.Below, we use the Schrieffer-Wolff transformation to derive approximate expressionsfor the Hamiltonian in each orbital branch in this regime. Rotating the basis in orbitalspace by the angle α s enclosed by the crystallographic x axis and the direction of thetransverse perturbation, we rewrite the Hamiltonian in the form˜ H = e − iα s σ es y H es e iα s σ es y = gµ B BS z − λσ es y S z − δσ es z + D es (cid:18) S z − (cid:19) − ∆ (cid:0) e − iα s S σ es − + e iα s S − σ es+ (cid:1) + ∆ √ (cid:0) e iα s { S + , S z } σ es+ + e − iα s { S − , S z } σ es − (cid:1) , (9)where the strain term is block diagonal. Note that in this expression we have assumedthat the magnetic field is applied along z , and we ignore the orbital diagmagnetic shift.In this basis, the states are labeled | ε, m s (cid:105) , where ε ∈ { L, U } are the lower-energy andhigher-energy states, respectively, and m s ∈ {− , , +1 } .Provided that 2 δ > λ , we can treat the inter-branch coupling as a perturbation,dividing the Hamiltonian into(10) ˜ H = H + V, with the inter-branch coupling term(11) V = (cid:20) − λS z + i ∆ (cid:0) e − iα s S − e iα s S − (cid:1) + i ∆ √ (cid:0) e iα s { S + , S z } − e − iα s { S − , S z } (cid:1)(cid:21) σ es y . Starting from this model, we apply quasi-degenerate perturbation theory in the form ofa Schrieffer-Wolff transformation H eff = e G ˜ He − G = ˜ H + (cid:104) G, ˜ H (cid:105) + 12 (cid:104) G, (cid:104) G, ˜ H (cid:105)(cid:105) + O ( G )= H + V + [ G, H ] + [ G, V ] + 12 [ G, [ G, H ]] + · · · , (12)where the generator G is defined such that G † = − G in order to eliminate the couplingsbetween the two strain-split branches in lowest order. The condition for this to work is[ G, H ] = − V , because it implies a transformed effective Hamiltonian(13) H eff = H + 12 [ G, V ] Lee C. Bassett which is second order in the couplings λ , ∆ , and ∆ . This effective Hamiltonian isblock-diagonal, i.e. , it can be split up into a lower and upper branch component, eachcontaining the contributions due to virtual transitions via the other branch up to linearorder in the couplings.The effective Hamiltonian takes the general form:(14) H eff = D es (cid:18) S z − (cid:19) + gµ B BS z − δσ es z + (cid:18) H L H U (cid:19) . Within the lengthy expressions for H L and H U , we assume the strain splitting 2 δ is thedominant energy scale, and expand to lowest order in 1 /δ to obtain(15) H L (cid:39) − λ δ − ∆ δ − e − iα s ∆ f + ( α s ) e − iα s ∆ (cid:0) λδ − (cid:1) − e iα s ∆ f + ( α s ) ∗ e − iα s ∆ f + ( α s ) e iα s ∆ (cid:0) λδ − (cid:1) e iα s ∆ f + ( α s ) ∗ − λ δ − ∆ δ + O (cid:18) δ (cid:19) , for the lower branch and(16) H U (cid:39) λ δ + ∆ δ e − iα s ∆ f − ( α s ) e − iα s ∆ (cid:0) λδ + 1 (cid:1) e iα s ∆ f − ( α s ) ∗ − e − iα s ∆ f − ( α s ) e iα s ∆ (cid:0) λδ + 1 (cid:1) − e iα s ∆ f − ( α s ) ∗ λ δ + ∆ δ + O (cid:18) δ (cid:19) for the upper branch. Here, we have introduced the expression f ± ( α s ) = ∆ δ +2 e iα s (cid:0) ± λδ (cid:1) ,which leads to an oscillation with the strain angle α s of the splitting between the S z = 0and S z = ±
3. – Coherent light-matter coupling
Experiments that probe spin-light coherence [1, 18], and related protocols for all-optical coherent control [2, 3] of NV-center spins draw on a rich history in quantumoptics and atomic physics. For general background in this subject, we refer the readerto excellent textbooks such as those by Gerry and Knight [19] or Cohen-Tannoudji andGu´ery-Odelin [20]. In this section, we give a brief introduction to the concept of coher-ent coupling between a light field and atomic transitions, using the Jaynes-CummingsHamiltonian to derive expressions for the optical (ac) Stark effect and the Faraday effect.This derivation naturally captures the correspondence between these two effects, whichboth result from the polariton energy shifts due to the interactions between the light fieldand atomic transitions. We discuss how the concept was applied by Buckley et al. [1] todemonstrate dispersive optical measurements of the spin state and all-optical coherentspin rotations. uantum optics with single spins .1. The Jaynes-Cummings Hamiltonian . – The Jaynes-Cummings Hamiltonian de-scribes the interaction between light and matter in the rotating wave approximation (see, e.g. , Chapter 4 of Gerry and Knight [19] for a full derivation). It is typically used in thecontext of cavity quantum electrodynamics to describe coherent coupling of an atom-likesystem to the optical field in a cavity, however it can be applied more generally evenwhen cavities are not involved. For example, in the experiments by Buckley et al. [1]and Yale et al. [2], the ‘cavity’ is defined by the duration of a laser pulse, τ , which isassumed to propagate in a single spatial mode that we can treat as a coherent state oflight with a well-defined electric-field amplitude and phase. We assume that the turn-onand turn-off of this pulse is smooth, such that the interaction with the NV center isadiabatic. We also neglect spontaneous emission and other forms of decoherence such asspectral hopping and laser noise. A treatment of these effects can be found in Ref. [1].Our starting point is the dipole interaction Hamiltonian(17) H int = (cid:112) F DW (cid:126)µ · (cid:126)E, where (cid:126)µ is the NV-center electric dipole, (cid:126)E is the local electric field, and F DW = 0 . ± .
01 is the Debye-Waller factor, which empirically accounts for the reduced resonantcoupling between NV center ground and excited states due to displacement of the nuclearcoordinates during optical transitions [21]. The dipole magnitude is directly related tothe NV center’s spontaneous decay rate γ r = 1 /
13 ns − [22] through(18) | (cid:126)µ | = 3 πε (cid:126) c γ r E n D , where E ph = 1 .
945 eV is the photon energy and n D = 2 . n in the pulse and the effective equal-intensity optical mode area atthe NV center A eff through the classical irradiance(19) I = cn D ε (cid:12)(cid:12) (cid:126)E (cid:12)(cid:12) = nE ph τ A eff , such that(20) (cid:12)(cid:12) (cid:126)E (cid:12)(cid:12) = (cid:114) nE ph n D ε A eff cτ . By introducing the operators ˆ (cid:126)E = i (cid:12)(cid:12) (cid:126)E (cid:12)(cid:12) (ˆ a † − ˆ a ) and ˆ (cid:126)µ = | (cid:126)µ | (ˆ σ + + ˆ σ − ) for the electricfield and optical dipole, respectively, we cast H int into the form(21) ˆ H int (cid:39) i (cid:126) Ω (cid:0) ˆ a † ˆ σ − − ˆ a ˆ σ + (cid:1) , Lee C. Bassett where ˆ a † (ˆ a ) and ˆ σ + (ˆ σ − ) are creation (annihilation) operators for optical photons andatomic excitations, respectively. If the atomic ground and excited states are | g (cid:105) and | e (cid:105) , then ˆ σ + = | e (cid:105)(cid:104) g | and ˆ σ − = | g (cid:105)(cid:104) e | . Here we neglect energy-nonconserving terms (cid:8) ˆ a ˆ σ − , ˆ a † ˆ σ + (cid:9) in the rotating wave approximation. The quantity Ω is the on-resonanceoptical Rabi frequency, given by(22) Ω = √ F DW | (cid:126)µ | (cid:12)(cid:12) (cid:126)E (cid:12)(cid:12) cos( θ ) (cid:126) , where θ is the angle between the optical dipole and the light’s linear polarization axis.With the addition of the non-interacting Hamiltonian for the spin and light fields, givenby(23) ˆ H = E ph (cid:18) ˆ a † ˆ a + 12 (cid:19) + E j ˆ σ ( j ) z , where E j is the transition energy for the spin state with m s = j and ˆ σ ( j ) z = | e j (cid:105)(cid:104) e j | −| g j (cid:105)(cid:104) g j | describes the NV center orbital excitation, we obtain the Jaynes-CummingsHamiltonian describing the light-matter system when the spin is in state j ,ˆ H ( j )JC = ˆ H ( j )0 + ˆ H ( j )int (24) = E ph ˆ a † ˆ a + E j ˆ σ ( j ) z (cid:126) Ω (cid:16) ˆ a ˆ σ ( j )+ + ˆ a † ˆ σ ( j ) − (cid:17) , (25)where we have set the optical zero-field energy to zero for simplicity.The Hamiltonian ˆ H JC is naturally expressed in the basis of non-interacting polaritonstates,(26) (cid:40) | ψ ( n,j )0 (cid:105) = | g j (cid:105) ⊗ | n + 1 (cid:105)| ψ ( n,j )1 (cid:105) = | e j (cid:105) ⊗ | n (cid:105) , where | g j (cid:105) ( | e j (cid:105) ) are the bare ground (excited) states of the NV-center orbital transitionfor m s = j , and | n (cid:105) is a photon-number Fock state of the electromagnetic field. Bydiagonalizing ˆ H JC in this basis, we obtain the eigenenergies(27) E ± ( n, ∆ j ) = E ph (cid:18) n + 12 (cid:19) ± (cid:126) (cid:113) ∆ j + Ω ( n ) , where ∆ j = ( E ph − E j ) / (cid:126) is the detuning of the laser from the unshifted NV-centertransition frequency and the n -dependence of Ω (implicit through (cid:12)(cid:12) (cid:126)E (cid:12)(cid:12) in Eq. 22) isshown explicitly. These eigenenergies take the form of an anticrossing about ∆ j = 0.Since the atom is initially in the ground state and we assume that the onset of thelight field is adiabatic, the occupied state during the pulse will be the polariton eigenstate uantum optics with single spins ε ( M H z ) Δ (GHz) ε −Δ Laserenergy | E 〉| G 〉| e 〉| g 〉 m s = − m s =0 | L,m s 〉| GS,m s 〉 Laser energy ( G Hz) 020-20 Σ S ( M H z ) Φ F ( μ r ad ) Φ F Σ S m s = − m s =0 (a) (b) (c) Figure 2. –
Light-matter coupling in the diamond NV center . (a) The interaction betweenan atomic transition with ground and excited states {| g (cid:105) , | e (cid:105)} and a near-resonant laser field isdescribed by the Jaynes Cummings Hamiltonian in terms of polariton states {| G (cid:105) , | E (cid:105)} with anenergy shift ε . (b) Energy-resolved transitions for different spin sublevels in the NV center’soptical fine structure produce spin-dependent interactions, which manifest (c) as optical Starkrotations with frequency Σ S and a Faraday phase shift, Φ F as a function of laser energy. Panels(a) and (c) are adapted from Ref. [1] and reprinted with permission from AAAS. having maximum overlap with | ψ (cid:105) , which has energy E g = E ± for ∆ j ≷
0. The observedenergy shift of this ‘ | g j (cid:105) -like’ state relative to its non-interacting energy(28) E g = E ph ( n + 1) − E j ε g ( n, ∆ j ) = E g − E g = (cid:126) ∆ j (cid:34)(cid:115) ∆ j − (cid:35) , and is plotted in Fig. 2(a). This energy shift, present for the duration of the laser pulse,adds a net phase to the polariton given by(30) Φ( n, ∆ j ) = τ ε g (cid:126) which in the far-detuned limit | ∆ j | (cid:29) Ω reduces to(31) Φ( n, ∆ j ) (cid:39) τ Ω j = D n ∆ j , Lee C. Bassett where(32) D = | µ | F DW E ph cos ( θ )2 (cid:126) cn D ε A eff . In typical experiments using a high-NA free-space objective to focus on a single NV centerthrough a planar, (100)-oriented, diamond surface, D/ π ≈
10 kHz , so the accumulatedphase per photon is only D/ ∆ j ≈ − rad for typical detunings in the GHz range.Nonetheless, an optical pulse with power ≈ µ W and duration ≈ µ s contains ≈ photons, so we can still obtain an observable signal from the total accumulated phase. .2. The Faraday and optical Stark effects . – In order to obtain expressions for theFaraday and optical Stark effects using this model, we need to resolve the resultingpolariton state into its spin and optical components. For that purpose, we calculate thereduced density matrices(33) (cid:26) ˆ ρ light = Tr spin (ˆ ρ )ˆ ρ spin = Tr light (ˆ ρ )in terms of the full density matrix ˆ ρ for polariton states, which we derive below. Whereasthe polariton states are naturally written in terms of the Fock basis of photon numberstates, the laser field is best described by an optical coherent state, | α (cid:105) , defined by(34) ˆ a | α (cid:105) = α | α (cid:105) . The coherent state can be expanded in the Fock basis using the relation(35) | α (cid:105) = e − | α | (cid:88) n α n √ n ! | n (cid:105) , which describes a Poisson distribution of Fock states, characterized by mean photonnumber (cid:104) n (cid:105) = | α | and with uncertainty ∆ n = | α | = (cid:112) (cid:104) n (cid:105) . An initial polariton statedescribed by(36) | Ψ (cid:105) = (cid:88) j β j | g j (cid:105) ⊗ | α (cid:105) therefore evolves to the state(37) | Ψ (cid:105) = (cid:88) j β j e − | α | (cid:88) n α n √ n ! e i Φ( n, ∆ j ) | g j (cid:105) ⊗ | n (cid:105) uantum optics with single spins after an interaction involving n photons. Using Eq. (31) in the limit | ∆ j | (cid:29) Ω we recastthis as | Ψ (cid:105) = (cid:88) j β j e − | α | (cid:88) n (cid:0) αe iφ j (cid:1) n √ n ! | g j (cid:105) ⊗ | n (cid:105) (38) = (cid:88) j β j | g j (cid:105) ⊗ (cid:12)(cid:12) αe iφ j (cid:11) , (39)where φ j = D/ ∆ j is the phase per photon accumulated by the state | g j (cid:105) ⊗ | α (cid:105) . The fulldensity matrix of the resulting spin-light system is then given by ρ = | Ψ (cid:105)(cid:104) Ψ | .We first consider the Faraday effect, which describes the observable properties of thelaser light following the interaction. The reduced density matrix for the optical field isreadily evaluated as ˆ ρ light = (cid:88) k (cid:104) g k | ˆ ρ | g k (cid:105) (40) = (cid:88) j | β j | (cid:12)(cid:12) αe iφ j (cid:11)(cid:10) αe iφ j (cid:12)(cid:12) . (41)Thus the optical field is in the state (cid:12)(cid:12) αe iφ j (cid:11) with a probability | β j | equal to the initialoccupation probability of the spin state | g j (cid:105) . The observable quantity in this case isthe sinusoidal phase of the electric field, which for a coherent state α = | α | e iγ has anexpectation value given by(42) (cid:104) ˆ E ( (cid:126)x, t ) (cid:105) α = −√ E | α | (cid:126)u ( (cid:126)x ) sin( ωt − γ ) , where (cid:126)u ( (cid:126)x ) describes the spatial mode and E is the vacuum electric field [19]. Thecomplex phase of the coherent state | α (cid:105) is therefore reflected as the phase of the electricfield. In the experiment by Buckley et al. [1], only one linear polarization of light iscoupled to the transition j . Its phase is shifted relative to the non-interacting polarizationstate by an amount φ j , which rotates the linear polarization angle of the transmittedlight. Figure 3 shows a schematic of the experimental setup.The experiment is performed in the intermediate-strain regime (2 δ ≈
17 GHz) wherethe excited-state orbitals are energetically separated and can be individually addressed.The approximate level structure of the ground state and lower-branch excited state isshown in Fig. 2(b); a relatively large axial magnetic field of B z = 1920 G ensures thatthe ˆ S z eigenstates are a good spin basis for the excited state, however the spin-spin andspin-orbit interactions shift the energies relative to the ground state as shown by Eq. (15).Thus, the optical resonance for different spin sublevels occur at different frequencies, withthe m s = − m s = 0 transition.We define the Faraday phase Φ F as the difference in phase between the m s = 0 and Lee C. Bassett cryostat Dichroic mirror 1Dichroic mirror 2Single-photon detectorObjective Half-wave plateFast steering mirrorLens pairQuarter-wave plate Polarizing beamsplitterTunable 637nm laserGreen532 nm laser AOM SBCCollimating lensLong wave pass filterAOM PhotodiodebridgeDiamond sampleSolid immersion lensPermanent magnet Long wave pass filter
Figure 3. –
Measurement setup . Schematic of the experimental setup used to measure Fara-day and optical Stark effects. A tunable laser near the NV-center ZPL at 637 nm provides thecoherent optical pulses. A second laser at 532 nm is used to initialize the NV-center spin andcharge state. [AOM: Acousto-optic modulator; SBC: Soleil-Babinet compensator]. Adaptedfrom Ref. [1] and reprinted with permission from AAAS. m s = − F = φ − φ − = D (cid:18) − − (cid:19) = − D ω s ∆ ∆ − , where ω s = ( E − − E ) / (cid:126) is the frequency spacing between the resonances. uantum optics with single spins Similarly, the reduced density matrix for the spin is given byˆ ρ spin = (cid:104) α | ˆ ρ | α (cid:105) (44) = (cid:88) j,k β ∗ k β j exp (cid:2) −| α | (cid:0) − e iφ j − e − iφ k (cid:1)(cid:3) | g j (cid:105)(cid:104) g k | , (45)where we have used the identity(46) (cid:104) α | α (cid:48) (cid:105) = exp (cid:20) − (cid:0) | α | + | α (cid:48) | − α ∗ α (cid:48) (cid:1)(cid:21) . Since φ j (cid:28)
1, we can approximate(47) ˆ ρ spin (cid:39) (cid:88) j,k β ∗ k β j e i (cid:104) n (cid:105) ( φ j − φ k ) | g j (cid:105)(cid:104) g k | , from which we identify the effective spin states(48) | spin (cid:105) = (cid:88) j β j e i (cid:104) n (cid:105) φ j | g j (cid:105) = (cid:88) j β j e i τ Ω204∆ j , such that ˆ ρ spin = | spin (cid:105)(cid:104) spin | . Physically, this shows that the spin states acquire rela-tive phases due to their different detunings from the light field, producing an effectivespin rotation. In the experiment [1], this relative optical-Stark-effect phase is directlyproportional to the corresponding Faraday-effect phase through the photon number:(49) Φ OSE = n Φ F . The optical field in the experiment consists of two polarization modes, each with photonnumber n , of which only one is coupled to the NV-center optical transitions, so the totallaser power is given by P L = 2 nE ph /τ , and the corresponding optical Stark frequencyshift is(50) Σ S = Φ OSE πτ = P L πE ph Φ F . This proportionality in the far-detuned regime allows the two measurements to be showntogether on the same graph as in Fig. 2(c).Although the expressions above were derived assuming the limit of large detuning( | ∆ j | (cid:29) Ω ), the full expressions across the absorption resonance are known from otherarguments. The Faraday effect results from the real part of the frequency-dependentrefractive index of the atomic transition near an absorption resonance. As a consequence Lee C. Bassett of the Kramers-Kronig relation between the refractive index’s real and imaginary parts,the full Faraday effect lineshape is known to be an odd Lorentzian of the form(51) φ j = F j ∆ j ∆ j + Γ j , where Γ j is the width of absorption resonance j and F j is the Faraday amplitude. Bycomparing this expression with Eq. (43) in the far-detuned limit we see that the constant D takes the place of the Faraday amplitude F . Likewise, the shift in the Larmor preces-sion rate due to the optical Stark effect is a direct consequence of the polariton energyshift of Eq. (29), and so is given across all detunings by(52) S j = ∆ j π (cid:34)(cid:115) ∆ j − (cid:35) . In comparing measurements to these expressions, we can extract experimental values for F j , Γ j , and Ω for the appropriate transitions. For the data in Fig. 2(c) from Ref. [1],we obtain F = 2 π × . µ rad · GHz, Γ = 2 π ×
140 MHz, F − = 2 π × . µ rad · GHz,Γ − = 2 π ×
300 MHz and Ω = 2 π ×
70 MHz. The asymmetry in the curve mainly resultsfrom the different absorption widths for the m s = 0 and m s = − .3. Discussion and implications . – The preceding derivation illustrates how coher-ent light-matter interactions give rise to observable spin-dependent optical phase shifts(the Faraday effect) and coherent, optical-power-dependent spin rotations (the opticalStark effect). In principle, the Faraday effect provides a means to measure the spinstate nondestructively, i.e. , without exciting the optical transition and re-initializing thestate. This is possible since the absorption resonance has a Lorentzian lineshape, varyingas 1 / ∆ for large ∆, whereas the Faraday phase shift is an odd Lorentzian, varying as1 / | ∆ | . Nondestructive measurements are important for certain applications in quantuminformation processing, and similar dispersive measurements are used extensively in thecircuit quantum electrodynamics paradigm of superconducting qubits [23]. In practice,the Faraday phase shifts on the order of 10 − rad are too small to allow high-fidelity, non-destructive measurements of individual NV centers without an optical cavity to amplifythe interaction. Although it remains a challenge to fabricate nanophotonic optical cavi-ties containing NV centers while maintaining stable optical transitions, such a platformhas recently been achieved for silicon-vacancy (SiV) centers in diamond [24], where dis-persive interactions analogous to those we have discussed above can also serve to mediateinteractions between two SiV spins within the same cavity [25].The optical Stark effect, meanwhile, provides a means to perform operations on aspin qubit using light rather than microwaves, which can allow addressing of individualqubits within optical networks. With enhanced interactions from an optical cavity, theoptical Stark effect can provide a means for generating spin-photon entanglement orquantum operations between remote spins. Whereas the spin rotations that result from uantum optics with single spins the optical Stark shifts in a level structure like Fig. 2(b) generate precession about thequbit’s polar axis, variations in the energy level structure and experimental design canenable rotations about arbitrary axes on the Bloch sphere, in addition to general protocolsfor qubit readout and initialization [2], as we discuss in the next section.
4. – All-optical coherent spin control
In the previous section, the optical Stark effect — viewed as the spin-like componentof the coherent polariton dynamics as in Eq. (44) — manifests as a relative energy shiftbetween spin sublevels, with no change in the spin eigenstates. This is analogous to theapplication of a magnetic field along the defect’s symmetry axis. When treating two ofthe triplet spin sublevels as a qubit, this amounts to a light-induced rotation about the z axis in the Bloch sphere. In order to achieve arbitrary unitary operations on a qubit,however, rotations about two noncollinear axes are required. One can therefore ask if itis possible to realize optical Stark effects that perturb the ground-state Hamiltonian inmore complex ways, e.g. , to generate an effective magnetic field pointing along x or y .Indeed, this is possible if one can engineer the electronic structure and optical transitiondiagram to enable light-induced mixing of the spin eigenstates.Such mixing occurs naturally in a level configuration known as a lambda (Λ) system,where two lower-energy states (the qubit manifold) couple coherently to a single excitedstate, as shown in Fig. 4(a). Lambda configurations occur in a variety of quantum systemsincluding atoms [26, 27], trapped ions [28], quantum dots [29], and superconductingqubits [30]. As we will show below, the concept of optical Stark rotations as applied to aΛ system can be extended to realize arbitrary qubit operations; in this context they areknown as stimulated Raman transitions. Furthermore, the Λ configuration is the basisfor many well-known effects in quantum optics, including coherent population trapping(CPT) [26], electromagnetic induced transparency [31], slow light [32], atomic clocks [33],and spin-photon entanglement [18]. .1. Dark states and coherent population trapping . – The essential feature of a Λsystem is the appearance of “dark resonances” that occur when two light fields coherentlydrive both transitions to the excited state. When the light fields are tuned such thattheir frequency difference exactly matches the resonance frequency of the ground-statesublevels, the atom is no longer pumped to the excited state and therefore becomesdark. This phenomenon can be simply understood from the following argument [20]. Ifthe atom is initially in a superposition of ground states,(53) | ψ ( t = 0) (cid:105) = c | g (cid:105) + c | g (cid:105) , and it interacts with two laser fields characterized by instantaneous Rabi frequencies(here assumed to be complex quantities),(54) Ω i = (cid:126)µ i · (cid:126)E i (cid:126) , Lee C. Bassett Ω | g 〉 | g 〉| e 〉 Ω Δ |0 g 〉 |+1 g 〉 Ωsin( θ /2) e iω L t + iω g t + ɸ Ωcos( θ /2) e iω L t δ e | R e 〉| L e 〉 | S 〉 Δ (a) (b) Figure 4. –
Physics of Λ configurations . (a) Three levels arranged in a Λ configuration. (b)Realization of a Λ system for the NV center from an excited-state avoided level crossing. then the amplitude for a transition to occur from state | g i (cid:105) to the excited state, | e (cid:105) , isproportional to the product c i Ω i . If a ground-state superposition | ψ (cid:105) exists such that(55) c Ω + c Ω = 0 , then the amplitudes for the transitions from both ground states interfere destructively,and the atom cannot be excited. This is called a dark state . Since both the probabilityamplitudes c i and the electric field amplitudes (cid:126)E i are functions of time, the atom is notguaranteed to stay in a dark state indefinitely; however, it is straightforward to showthat the condition of Eq. (55) is maintained continuously if(56) ε − ε = (cid:126) ( ω − ω ) , where ε i and ω i are the ground-state energies and laser frequencies, respectively, i.e. , ifthe detuning of the two light fields matches the ground-state energy splitting.The existence of a persistent dark state results in the phenomena of CPT and electro-magnetic induced transparency. Starting from an arbitrary ground-state configurationand subject to light fields satisfying Eq. (56), the atom is transiently excited and relaxesuntil it is trapped in the dark state and no longer interacts with the optical fields. Onecan think of this dissipative process as a generalization of traditional optical pumping, i.e. , where only one arm of the Λ system is driven. Intuitively, if only transition 1 isdriven, the system will quickly relax into a steady state with | g (cid:105) fully populated, uncou-pled to the optical field. This scenario is a special case of Eq. (55) with Ω = 0, wherethe dark state is | D (cid:105) = | g (cid:105) . In fact, a dark state satisfying Eq. (55) is guaranteed toexist for any values of Ω i , and there will always be a corresponding bright state , | B (cid:105) , that uantum optics with single spins is orthogonal to | D (cid:105) and couples maximally to the optical field. Thus, by choosing theamplitude and phases of the optical fields that define Ω i , one can initialize the systeminto an arbitrary superposition of qubit ground states.While the CPT process is necessarily dissipative ( i.e. , non-unitary), coherent evolu-tion in the ground state can be achieved using dispersive interactions in analogy withthe optical Stark effect. When the optical fields satisfying Eq. (56) are simultaneouslydetuned from the resonance condition with | e (cid:105) as shown in Fig. 4(a), the resulting lightshift occurs only for the state | B (cid:105) and not | D (cid:105) . In the qubit manifold, this manifests as alight-induced rotation about the axis pointing from | D (cid:105) to | B (cid:105) in the Bloch sphere. Theaxis can be chosen arbitrarily, including configurations on the equator when | Ω | = | Ω | that result in complete population transfer between the qubit eigenstates. The effect inthis context is usually known as stimulated Raman transitions (SRTs), drawing inspira-tion from an alternative picture of the process in terms of virtual transitions through theexcited state. However, it is important to understand that SRTs are a dispersive effectthat do not involve absorption. Again, whereas CPT varies with 1 / ∆ , where ∆ is thedetuning from the optical resonance(s), the effective Rabi frequency of SRTs scales with1 / ∆, so it can be substantial even when absorption is negligible.On a practical note, it is important to recognize that the condition to have a dark statecan only be sustained if the two optical fields have a deterministic phase relationship. Ifthe fields are derived from different lasers, they must be frequency and phase stabilizedto a suitable reference. Alternatively, if the required frequency difference occurs in theradiofrequency or microwave spectrum, the two fields can be derived from a single laserusing an optical modulator to generate frequency sidebands. This is often the easiestapproach, and it is the one adopted by Ref. [2]. .2. Forming a Λ system from the NV center . – NV centers in diamond have long beenknown to exhibit electromagnetic-induced transparency and CPT [11, 34–36], evidencethat Λ configurations can be realized under certain conditions. At zero strain and zeromagnetic field, the spin-orbit eigenstates | A (cid:105) and | A (cid:105) are equal superpositions of the m s = ± |± g (cid:105) , with circular-polarization optical selection rules thatfacilitate the generation of spin-photon entanglement [18] and CPT in the {| +1 g (cid:105) , |− g (cid:105)} ground-state subspace [16]. However, it is often more convenient to work with a ground-state qubit defined in a manifold including the m s = 0 sublevel, | g (cid:105) , since this state isnaturally prepared by off-resonant optical pumping, and at low temperature it featuresoptical cycling transitions that facilitate robust, high-fidelity readout [37].As is apparent from the spin-spin terms in the excited-state Hamiltonian, Eq. (4),and the approximate spin-triplet representations in the high-strain regime, Eqs. (15)and (16), the excited-state m s = 0 states are weakly admixed with m s = ± . However, this parameter is rather small (∆ = 150 MHz [3]), so themixing only becomes apparent near an avoided level crossing, when the m s = 0 sublevelbecomes nearly degenerate with m s = +1 or m s = −
1. Such a situation is depicted inFig. 4(b), where the applied magnetic field is tuned such that a crossing occurs betweenthe m s = +1 and m s = 0 spin sublevels of the lower orbital branch, {| +1 e (cid:105) , | e (cid:105)} . (This Lee C. Bassett particular crossing can only occur when the strain is relatively small, since for largetransverse perturbations the | +1 e (cid:105) state is higher in energy than | e (cid:105) even when B = 0;see Fig. 1(b).) At the closest approach, the anticrossing levels are separated by an energy δ e ≈ ∆ , and the eigenstates become | R e (cid:105) = 1 √ | e (cid:105) + | +1 e ) (cid:105) (57a) | L e (cid:105) = − √ | e (cid:105) − | +1 e ) (cid:105) . (57b)Either of these states can serve as the upper state of a Λ system connecting the {| +1 e (cid:105) , | e (cid:105)} qubit states.Yale et al. [2] explored this situation by tuning to an excited-state avoided levelcrossing as shown in Fig. 4(b) and modulating a tunable laser near 637 nm using anelectro-optic phase modulator in order to generate sidebands separated by the ground-state resonance frequency, ω gs . This also allows for control of the relative phase betweenthe two optical fields and their relative amplitude through the power and phase of themicrowave signal applied to the modulator. These parameters determine the azimuthal( φ ) and polar ( θ ) angles of the dark state formed in the ground-state Bloch sphere. .3. All-optical initialization, control, and readout . – To describe the dynamics of theNV-center spin under optical excitation as shown in Fig. 4(b), we construct a modelincluding five energy levels: two out of the three ground-state levels | g (cid:105) , | + 1 g (cid:105) , the twomixed excited states | L e (cid:105) and | R e (cid:105) , as well as the intermediate singlet | S (cid:105) , which plays arole in mediating unintentional ISC transitions that cause dissipation. The Hamiltonian,in the rotating frame, for the subspace spanned by these five basis states can be expressedas(58) H = ∆ L θ/
2) Ω cos( θ/
2) 00 ∆ L Ω sin( θ/ e iφ − Ω sin( θ/ e iφ
0Ω cos( θ/
2) Ω sin( θ/ e − iφ θ/ − Ω sin( θ/ e − iφ − δ e
00 0 0 0 (cid:15) S where the ordering of the states is {| + 1 g (cid:105) , | g (cid:105) , | R e (cid:105) , | L e (cid:105) , | S (cid:105)} , ∆ L is the detuning ofthe laser frequency ( ω L ) from resonance to the | R e (cid:105) Λ system, δ e is the separation ofthe excited state levels, Ω is the optical Rabi frequency, φ is the relative phase betweenthe two coherent light fields, and tan( θ/
2) is the relative amplitude between the drivingfields.The time evolution of the system includes both coherent and dissipative processes.These can be captured using the Lindblad master equation,(59) ˙ ρ = i [ ρ, H ] + (cid:88) α,α (cid:48) Γ αα (cid:48) (cid:18) σ α (cid:48) α ρσ αα (cid:48) − σ αα ρ − ρσ αα (cid:19) ≡ W ρ. uantum optics with single spins The first term describes unitary evolution of the density matrix due to the Hamiltonianof Eq. (58), whereas the second term captures dissipative processes, with the Lindbladoperators σ αα = | α (cid:105)(cid:104) α | = σ † α (cid:48) α σ α (cid:48) α and σ α (cid:48) α = σ † αα (cid:48) = | α (cid:48) (cid:105)(cid:104) α | . For n = 5 levels, the den-sity matrix ρ is a Hermitian 5x5 matrix and can be described by n = 25 real parameters( n − ρ ) = 1). The superoperator W can thus be viewed as a 25x25 matrix with rank 24.The Lindblad operators describe incoherent, spontaneous transitions between states.We denote the decay rate from the excited states ( E = L e , R e ) to the ground states ( G =0 ,
1) with Γ = Γ
E,G g , the rate for ISC from the excited states to the singlet Γ i = Γ E,S ,and the inverse ISC rate from | S (cid:105) to one of the ground state levels as Γ (cid:48) i = Γ S,G g . Thespin relaxation rate in the ground state is Γ = 1 /T . At T ≈
10 K, the thermalfrequency k B T /h ≈
200 GHz exceeds the relevant NV level splittings ≈ +1 g , g = Γ g , +1 g = Γ /
2. Pure dephasing between the twoground state levels is approximated by adding a term γ = 1 /T = Γ g , g . All other ratesare set to zero.The state of the system after optical excitation during time t is obtained by integratingEq. (59),(60) ρ ( t ) = e W t ρ (0) , where the initial state, ρ (0), is typically one of the ground states. Equation (60) admitssimple analytical solutions only for special cases, so in general we simulate the dynamicsnumerically. Depending on the parameters, this model can describe both CPT and SRT.In the idealized case Γ = γ = Γ i = 0, and with only one of the excited levels included,the system reduces to the three-level Λ system of Fig. 4(a), and the stationary state ¯ ρ in the long-time limit t (cid:29) / Γ obtained from ˙ ρ = 0 as the null space of W is the darkstate:(61) | D (cid:105) = cos( θ/ | g (cid:105) − exp( ∓ iφ ) sin( θ/ | + 1 g (cid:105) where the upper (lower) sign holds for the single excited state level being E = R ( E = L ).With realistic parameters, the evolution is not so simple, since the ISC and spindecoherence tend to disspate the system away from the ideal dark state. Furthermore,we notice from Eq. (61) that the dark states corresponding to the different excited states | L e (cid:105) and | R e (cid:105) have opposite phases. When these states lie on the equator ( θ = 0), theyare orthogonal, such that the dark states from one Λ system is actually the bright statefrom the other. Since the separation between these states is small ( δ e /h ∼
180 MHz inRef. [2]), there exists a tradeoff between the speed of the operations, set by the laserpower, and the competition between these two orthogonal Λ systems, which becomesmore prevalent as the laser power increases.In any case, the time-dynamics of the Bloch vector representing the qubit densitymatrix can be obtained from(62) b ( t ) = Tr ( σ ρ ( t )) , Lee C. Bassett
Figure 5. –
All-optical control via coherent dark states . Experiments (points) and sim-ulations (curves) of quantum dynamics in the NV-center ground state driven by optical pulsesdesigned to achieve CPT (a) and SRT (b). Orange (top) and blue (bottom) trajectories corre-spond to situations where the initial state is | g (cid:105) or | +1 g (cid:105) , respectively. Adapted from Ref. [2]and reprinted with permission from the National Academy of Sciences. where the components of σ are the Pauli matrices in the ground-state subspace, σ x = | + 1 g (cid:105)(cid:104) g | + | g (cid:105)(cid:104) +1 g | , (63) σ y = i ( | + 1 g (cid:105)(cid:104) g | − | g (cid:105)(cid:104) +1 g | ) , (64) σ z = | g (cid:105)(cid:104) g | − | + 1 g (cid:105)(cid:104) +1 g | . (65)The fidelity of an operation can be calculated by comparing the final density matrix toa target state, e.g. for initialization via CPT in the dark state | D (cid:105) ,(66) F ( t ) = (cid:104) D | ρ ( t ) | D (cid:105) . Figure 5 shows examples of experimental CPT and SRT trajectories from Yale etal. [2] alongside simulations performed using this model. The measurements (points)are acquired by performing Bayesian quantum state tomography to reconstruct the statevector from experiments where the NV-center spin is repeatedly initialized, subjected to aparticular optical pulse, and then measured in one of three orthogonal bases. In additionto arbitrary-basis initialization and coherent control via
CPT and SRT, respectively, Yale et al. [2] also demonstrated how the intrinsic fluorescence contrast between the brightand dark state can be used to perform projective readout of the spin state in an arbitrary uantum optics with single spins basis. It is thus possible to perform full quantum operations, for example Rabi, Ramsey,or Hahn-echo spin coherence measurements, using light fields alone. Crucially, thesemethods do not rely on the NV center’s intrinsic level structure and spin-dependent ISCdynamics; they can be adapted to any system where a Λ configuration can be realizedthrough tuning of external electric or magnetic fields. Indeed, the methods have recentlybeen adapted to study the quantum properties of spin defects that do not exhibit anISC, for example the negatively-charged SiV in diamond [38, 39] and transition-metalimpurities in silicon carbide [40].
5. – Ultrafast control
The versatile concepts of light-matter coupling presented in Sections underliemany applications in quantum optics and quantum information science. In particular,dispersive effects such as the Faraday phase shift, the optical Stark shift, and stimu-lated Raman transitions provide a means to perform coherent quantum operations onindividual spins and to generate quantum correlations between light and matter. How-ever, practical limitations mean they are not always the most efficient method to controlsolid-state defects. Although the technique is all-optical in the sense that only light fieldsinteract with the spin, generation of the requisite phase-locked optical fields demands sta-ble, tunable laser sources, optical modulators, and corresponding microwave equipment.Moreover, the CPT and SRT trajectories shown in Fig. 5 exhibit several drawbacks ofthis technique as applied to the NV center specifically. The CPT trajectories do notterminate on the surface of the Bloch sphere, indicating a partially mixed initializedstate, and the SRT trajectories rapidly spiral inwards towards a totally mixed state atthe Bloch-sphere center. These nonidealities result from various experimental factorssuch as laser noise and spectral drift of the NV-center optical resonances, and from in-trinsic properties of the NV center. One key limitation is the small spin-spin couplingparameter, ∆ /h ≈
150 MHz, responsible for the excited-state anticrossing that formsa pair of Λ systems for the {| (cid:105) , | +1 (cid:105)} spin sublevels as in Fig. 4(b). Since the brightstate from one Λ system is the dark state for the other, competing dynamics betweenthe two Λ systems limit the fidelity of CPT initialization and add decoherence to SRToperations. This dual-Λ configuration also limits the effective speed of SRT operations( i.e. . the ground-state Rabi frequency, Ω g ) such that (cid:126) Ω g (cid:28) ∆ . For the NV center, thepractical limit is Ω g / π ≈
10 MHz, whereas traditional microwave control of the groundstate can facilitate high-fidelity operations at Rabi frequencies approaching 1 GHz [41]. .1. Quantum control with ultrafast optical pulses . – In this section, we introducean alternate approach to achieving all-optical quantum control using ultrafast opticalpulses that mitigates some of these limitations [3]. This approach abandons the disper-sive approximation of negligible optical excitation; rather, we directly leverage dynamicsgenerated by the excited-state Hamiltonian to achieve desired unitary operations on thespin. Figure 6(a) shows the NV center’s orbital structure in the intermediate-to-highstrain regime. As described in Section .2, transverse strain splits the excited state into Lee C. Bassett +1
110 G 155 G 190 G 260 G 400 G E v o l u t i on t i m e ( n s ) La s e r f r eq . ( G H z ) −10 100 200 300Magnetic Field (G) P L ( k C t s / s )
010 200 400−101 Magnetic Field (G) −1 +1 −1 +1 L f i ne s t r u c t u r e ( G H z ) | U 〉 | L 〉| G 〉 V ˆ H ˆ (a) (b) (c)(d) Figure 6. –
Coherent spin control with ultrafast pulses . (a) Orbital structure of theNV center at intermediate-to-high transverse strain. (b) Fine structure as a function of axialmagnetic field in the | L (cid:105) orbital branch when δ = 6 . | L, (cid:105) and | L, +1 (cid:105) around B = 110 G. (d) Trajectories of the ground-state spin qubit as a function evolution timebetween two optical pulses, for different settings of the magnetic field. Adapted from Ref. [3]and reprinted with permission from the AAAS. two orbital manifolds, each of which are connected to the ground state via orthogonal,linear-polarization selection rules. Whereas previously we considered optical pulses de-rived from a continuous-wave laser with durations measured in nanoseconds, which canresolve the NV center’s gigahertz-scale fine structure, an optical pulse with duration (cid:46) (cid:38) h = 1 in the {| G (cid:105) , | X (cid:105) , | Y (cid:105)} basis with a strain δ indirection α S is given by(67) H orb = f − δ cos( α S ) δ sin( α S )0 δ sin( α S ) f + δ cos( α S ) , where f = c/λ is the optical transition frequency. Each pulse corresponds to a unitaryoperation on the orbital states, with parameters determined by the pulse intensity, shape, uantum optics with single spins and polarization. We parameterize the electric field of the optical pulses by(68) E ( α E , β E ) = (cid:18) cos( α E ) cos( β E ) − i sin( α E ) sin( β E )sin( α E ) cos( β E ) + i cos( α E ) sin( β E ) (cid:19) , where α E is the angle of the linearly-polarized component (major axis) in the NV center’s( x, y ) plane, and β E ∈ [ − π , π ] defines the ellipticity, such that β E = 0 and β E = ± π correspond to linearly and circularly polarized light, respectively. Using the dipole matrixelements (cid:104) X | ˆ y | G (cid:105) = −(cid:104) Y | ˆ x | G (cid:105) (other combinations vanish), we find that a pulse ofpolarization E ( α E , β E ) couples | G (cid:105) to the orbital state(69) | E (cid:105) = −E y E x , leaving the orthogonal ES basis state,(70) | E (cid:48) (cid:105) = E ∗ x E ∗ y , unaffected.In the experiments by Bassett et al. [3], pairs of pulses were derived from a single seedlaser using beamsplitters and a delay line, so they were nominally identical. In this case,we can treat the pulses as instantaneous unitary operators parameterized by a rotationangle, θ , and with a relative phase, φ : U FP1 = | E (cid:48) (cid:105)(cid:104) E (cid:48) | + cos (cid:18) θ (cid:19) (cid:0) | E (cid:105)(cid:104) E | + | G (cid:105)(cid:104) G | (cid:1) + sin (cid:18) θ (cid:19) (cid:0) | E (cid:105)(cid:104) G | − | G (cid:105)(cid:104) E | (cid:1) , (71a) U FP2 = | E (cid:48) (cid:105)(cid:104) E (cid:48) | + cos (cid:18) θ (cid:19) (cid:0) | E (cid:105)(cid:104) E | + | G (cid:105)(cid:104) G | (cid:1) + sin (cid:18) θ (cid:19) (cid:0) e iφ | E (cid:105)(cid:104) G | − e − iφ | G (cid:105)(cid:104) E | (cid:1) . (71b)Between the pulses, the system freely evolves according to the system Hamiltonian. Theevolution can include both unitary and dissipative processes, e.g., following a Lindbladmaster equation similar to Eq. (59).Even though the pulses only act on the orbital degrees of freedom directly, spin-orbitinteractions in the excited state naturally induce spin dynamics during the free evolu-tion period. Depending on the pulse parameters, this scheme can be adapted to probeboth orbital and spin dynamics on timescales spanning femtoseconds to nanoseconds,and to realize deterministic control over the spin. For example, a pair of phase-locked Lee C. Bassett optical pulses can be designed to perform a generalized Ramsey sequence on the three-dimensional orbital Hamiltonian, where the first pulse generates a coherent superpositionof ground and excited states that proceeds to evolve, and the second pulse projects theresulting state onto the measurement basis of excited states (which emit PL) and theground state (which is dark). This scheme can be adapted to probe orbital coherencebetween the ground state and excited states or (by tuning the polarization to excite asuperposition of | L (cid:105) and | U (cid:105) ) coherence within the excited-state manifold. Alternatively,by setting θ = π , the optical pulses can be designed to achieve full population transferfrom | G (cid:105) to a desired excited state orbital, and vice versa . From the point of view of thespin, this manifests as an instantaneous change in the Hamiltonian. For a pair of suchpulses that populates and subsequently depopulates the excited state after a time, t , theexcited-state evolution generates a deterministic unitary operation on the spin. .2. Applications . – This novel approach to generating coherent spin rotations by uti-lizing free evolution in the excited state has several applications. As a time-domain spec-troscopy technique, measurements of the spin dynamics that result from pairs of opticalpulses provide the means to map an arbitrary excited-state Hamiltonian. The techniqueis termed time-domain quantum tomography (TDQT). In contrast to frequency-domainspectroscopies which typically yield only the Hamiltonian eigenvalues, TDQT yields boththe eigenvalues and eigenvectors, from which it is possible to construct the full Hamilto-nian matrix. TDQT also provides time-domain information about various non-unitary,dissipative processes. Bassett et al. applied the TDQT technique to extract the spin-orbit and spin-spin parameters of the NV center’s excited state Hamiltonian, and to studythe role of decoherence due to spontaneous photon emission, spectral diffusion, phonon-mediated orbital relaxation, hyperfine-induced spin dephasing, and the state-selectiveISC transitions.As a quantum control technique, the pulse timings can be chosen to achieve a desiredunitary quantum operation on the ground-state spin. If we are interested in the evolutionwithin a qubit subspace (and assuming we can effectively isolate the evolution to thatsubspace within the excited state), we can view the effect of a pair of such pulses as atemporary change in the effective magnetic field. With appropriate control over the pulsetimings and excited-state Hamiltonian, this all-optical, and microwave-free technique canbe applied to generate rotations for the ground-state spin qubit.Consider for example the situation of the double-Λ configuration of Fig. 4(b) thatis formed near an excited-state anticrossing of the | L, (cid:105) and | L, +1 (cid:105) eigenstates. Bytuning the polarization of the optical pulses following Eq. (69) such that the optically-coupled excited state is | E (cid:105) = | L (cid:105) , we can isolate most of the unitary dynamics to thefour-dimensional subspace spanned by {| G, (cid:105) , | G, +1 (cid:105)} and {| L, (cid:105) , | L, +1 (cid:105)} . To modelthis, we start from a diagonal ground-state Hamiltonian(72) H gs = ω gs s z , describing precession of the effective spin-1/2 qubit about the z axis due to the effective uantum optics with single spins external magnetic field with frequency ω gs . Here, s z is a spin-1/2 Pauli- z operatoracting on the {| G, (cid:105) , | G, +1 (cid:105)} spin subspace. Similarly, the effective excited-state qubitHamiltonian describes a precession about an axis tilted by an angle η relative to theground state, and with a different frequency ω es ,(73) H es = ω es η s x + cos η s z ) = ω es s (cid:48) z . Here we have set the complex phase of the off-diagonal matrix element to zero, sincein experiments this phase is convolved with the constant but unspecified relative timingbetween the optical pulses and the microwaves used to address the ground-state spin.The full four-dimensional Hamiltonian of this effective model is(74) H = (cid:18) H gs H es + ω opt I (cid:19) = 12 (1 − σ z ) H gs + 12 (1 + σ z ) ( H es + ω opt I ) , where ω opt is the optical frequency difference between | G (cid:105) and | L (cid:105) , and σ z is a Paulioperator for the orbital GS-ES degree of freedom, i.e., σ z | GS (cid:105) = −| GS (cid:105) and σ z | ES (cid:105) = | ES (cid:105) . The action of a resonant ultrafast pulse with polarization ˆ H (see Fig. 6) is describedby the unitary operator of Eq. (71), which reduces to(75) U FP ( θ, φ ) = cos (cid:18) θ (cid:19) − i sin (cid:18) θ (cid:19) (cos( φ ) σ x + sin( φ ) σ y ) , corresponding to a coherent rotation in the {| G (cid:105) , | L (cid:105)} orbital basis by an angle θ aboutan axis defined by | G (cid:105) + e − iφ | L (cid:105) (i.e., an equatorial axis in the orbital Bloch sphere).The excited-state Hamiltonian parameters η and ω es can be tuned by the externalmagnetic, electric, and strain fields. The effective expression, Eq. (15), for the | L (cid:105) -branchHamiltonian is useful for identifying regimes in which unwanted mixing with other spinand orbital states are minimized. Figure 6(b) shows the fine structure of | L (cid:105) as a func-tion of B z corresponding to the strain configuration ( δ/h = 6 . α s = − .
08 rad)from Ref. [3]. The configuration is similar to the one we considered in Section , wherean avoided level crossing occurs between | L, (cid:105) and | L, +1 (cid:105) around B z = 110 G. Theexistence of such an anticrossing is confirmed using standard photoluminescence excita-tion spectroscopy as in Fig. 6(c). However, frequency-domain spectroscopy only providesinformation about the energy eigenvalues, not the eigenstates.According to the Hamiltonian, the excited-state spin eigenstates are fully hybridizedat the center of the anticrossing; hence the effective precession axis in the excited stateis orthogonal to that of the ground state, lying in the equatorial plane of the qubit Blochsphere. At other field values, the precession axis is tilted by an angle η that approacheszero far from the level anticrossing. These eigenstates are directly revealed by TDQTmeasurements of the spin evolution between two ultrafast optical pulses, as shown inFig. 6(d). The figures show trajectories that begin from an initialized state in either | (cid:105) or | +1 (cid:105) (and, at B = 400 G, from a spin superposition state). The trajectories are Lee C. Bassett fits to the raw TDQT data using an analytical model that captures both unitary anddissipative dynamics [3].At the center of the anticrossing where η = π/
2, a full π -pulse on the spin qubitcan be achieved using a single pair of optical pulses. For sequences of multiple single-qubit operations, the relative phase between pulses is deterministically set by the pulsetimings. In this way, universal quantum operations on the spin can be achieved usingpairs of identical optical pulses. Furthermore, whereas the dispersive SRT technique islimited in this configuration to a Rabi frequency Ω g (cid:46)
10 MHz (cid:28) ∆ , direct evolutionin the excited state occurs at the bare coupling rate, Ω g ∼ ω e s ∼ ∆ . In the data ofFig. 6(d), Ω g = 260 MHz, corresponding to a π rotation in only 1 .
6. – Conclusions and future directions
The purpose of this chapter is to provide an introduction to quantum optics in thecontext of solid-state spins like the diamond NV center. However, the methods andtechniques we describe only scratch the surface of quantum optics and its potential ap-plications for quantum information science. For example, the CPT and SRT techniquesdescribed in Section have been applied to realize alternate forms of robust quantumcontrol employing geometric phases [42–44]. Whereas we focused on the diamond NV cen-ter, the techniques are general and are now routinely applied to other quantum systemsincluding quantum dots [45] and other defect systems [38–40, 46–48]. As the numberof materials platforms and applications for spin-based quantum technologies expands[49, 50], the importance of these techniques will only continue to grow. ∗ ∗ ∗ The original work described in Refs. [1–3] was supported by the Air Force Officeof Scientific Research, the Army Research Office, and the Defense Advanced ResearchProjects Agency. Preparation of these notes and the associated lectures was supportedby the U. S. National Science Foundation under CAREER award ECCS-1553511. Theauthor thanks David Hopper and S. Alex Breitweiser for their critical reading of thismanuscript; Mario Agio, Irene D’Amico, Rashid Zia, and Costanza Toninelli for organiz-ing the Enrico Fermi Summer School on
Nanoscale Quantum Optics ; the Italian PhysicalSociety for hosting the course at the Villa Monastero in Varenna; and all the studentsand lecturers who attended the school for many enjoyable discussions.
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