Harmonic fine tuning and triaxial spatial anisotropy of dressed atomic spins
Giuseppe Bevilacqua, Valerio Biancalana, Antonio Vigilante, Thomas Zanon-Willette, Ennio Arimondo
HHarmonic fine tuning and triaxial spatial anisotropy of dressed atomic spins
Giuseppe Bevilacqua, ∗ Valerio Biancalana, Antonio Vigilante,
1, 2
Thomas Zanon-Willette, and Ennio Arimondo
4, 5 Dept. of Information Engineering and Mathematics - DIISM,University of Siena – Via Roma 56, 53100 Siena, Italy Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LERMA, F-75005, Paris, France Dipartimento di Fisica E. Fermi, Università of Pisa – Lgo. B. Pontecorvo 3, 56127 Pisa, Italy INO-CNR, Via G. Moruzzi 1, 56124 Pisa, Italy (Dated: May 22, 2020)The addition of a weak oscillating field modifying strongly dressed spins enhances and enriches thesystem quantum dynamics. Through low-order harmonic mixing the bichromatic driving generatesadditional rectified static field acting on the spin system. The secondary field allows for a fine tuningof the atomic response and produces effects not accessible with a single dressing field, such as a spatialtriaxial anisotropy of the spin coupling constants and acceleration of the spin dynamics. This tuning-dressed configuration introduces an extra handle for the system full engineering for quantum controlapplications. Tuning amplitude, harmonic content, spatial orientation and phase relation are controlparameters. A theoretical analysis, based on perturbative approach, is experimentally validated byapplying a bichromatic radiofrequency field to an optically pumped Cs atomic vapour. We measurethe resonance shifts produced by tuning fields up to the third harmonic.
Dressing of a quantum system by a non-resonant elec-tromagnetic field represents an important tool withinthe quantum control. Energies and electromagnetic re-sponse are modified by the dressing. The seminal workof Cohen-Tannoudji and Haroche (CTH) [1, 2] derivedthe modifications of the spin precession frequency in astatic magnetic field in presence of a strong radiofre-quency (rf) dressing field, off-resonant and linearly po-larised orthogonally to the static one. A key dressingsignature is the J zero-order Bessel function dependenceof the eigenenergies. The dressing produces as additionalfeature a cylindrical spatial anisotropy for the evolutionof the quantum coherences [3]. The J eigenenergy col-lapse was examined for atoms in [4–9], for a Bose-Einsteincondensate in [10], for an artificial atom in [11]. Ref. [12]investigated the generalization to a dressing with a peri-odic arbitrary waveform. The close connection of the J collapse with the tunneling suppression was pointed outin [13, 14], and with the dynamical localization freez-ing in optical lattices reviewed in [15]. The oscillatingfield driving and the J Bessel response were describedas a frequency modulation in [16], and extended to thepresence of dissipation in [17]. Critical dressing basedon the simultaneous dressing of two spin species to thesame effective Larmor precession frequency was exploredin [18–20]. A variety of microwave and rf dressings wasexplored in recent years, with those based on the J re-sponse for cold atoms in [21, 22], for a two-dimensionalelectron gas in [23], for high resolution magnetometryin [20], and for the control of a strong spin-exchange re-laxation in [24]. The dressing applied by Haroche [25]to compensate an inhomogeneous spin distribution, wasextended determining magic values based on high order ∗ [email protected] corrections to the J response [26], or applying an inho-mogeneous dressing field [27].This work introduces a flexible quantum handle allow-ing a continuous control between collapse and enhance-ment of the quantum response. The tuning tool is aweak non-resonant additional rf field operating in thesplit biharmonic driving configuration, i.e., oscillating ata low order harmonic of the dressing frequency and ap-plied along a direction orthogonal to the dressing one.This configuration demonstrates control performance un-matched by the analog single-harmonic system. A quan-tum coupling more versatile than the well known J eigenenergy dependence and a triaxial spatial anisotropyof the quantum response are the tuning-dressed signa-tures. The interaction with the additional electromag-netic field produces a modification of the eigenenergiesdepending on the spatial direction of the applied mag-netic field, namely an undressed response along the dress-ing field direction and a fully tunable one in the orthog-onal plane. As basic mechanism, the harmonic mixingof the biharmonic driving originates a rectified field thatmodifies the system eigenenergies and eigenstates. Thisnonlinear rectification process borrows strength from thedressing field and is associated with high order light-shifts due to the biharmonic driving. Within the topo-logical description of [28, 29], our process represents afrequency conversion between two low-order commensu-rable rf fields. In magnetic resonance double irradia-tions, e.g., rotary saturation, spin tickling, etc., representa powerful tool to disentangle complex spectra. In ourdouble irradiation the commensurable and low harmonicdrive are elements of the fine tuning quantum control. Inthe quantum information language, our quantum handlerepresents an additional storage resource. For the quan-tum simulation applications, the anisotropy of the qubitinteractions allows the exploration of a wider range of a r X i v : . [ qu a n t - ph ] M a y Hamiltonians.Our experimental test is based on an optical magne-tometric apparatus operating on a gas-buffered caesiumgas in the regime of µ T magnetic fields [27]. We monitorthe quantum system spatial anisotropy by measuring thespin effective Larmor precession frequency under differ-ent geometries. The theoretical predictions are preciselyconfirmed by measurements performed with tuning fre-quencies up to the third harmonic.Our analytical theory is based on a perturbative treat-ment for the quantum coupling to static and tuning fieldsof a strongly dressed quantum system, not treated withinthe rotating wave approximation. Ref. [30] contains adescription appropriate for the high spin atomic systemexplored in the experiment. Here we consider a spin 1/2system (either real or artificial atom) interacting with astatic magnetic field having components B j along the j = ( x, y, z ) axes. For an atomic system with g Landéfactor and µ B the Bohr magneton, the spin-field cou-pling is determined by the gyromagnetic ratio γ = gµ B and characterized by the energies ω j = γB j , assuming (cid:126) = 1. The quantum system is driven by two magneticfields oscillating at frequencies ω and ω t = pω , dressingand tuning respectively, B d oriented along the x axis and B t along the y axis. The corresponding Rabi frequenciesare Ω d = γB d and Ω t = γB t .Using the adimensional time τ = ωt the Hamiltonianis H = X j =( x,y,z ) ω j ω σ j + Ω d ω cos( τ ) σ x + Ω t ω cos( pτ + Φ) σ y , (1)where σ j are the Pauli matrices and Φ the phase differ-ence between the two oscillating fields.We operate in a strong dressing regime, with ξ =Ω d /ω (cid:29) Ω t /ω, ω /ω . Within a perturbative analysis wefactorize the time evolution operator as U = U U I , the U dressing evolution is given by U = e − iϕ ( τ ) σ x / , (2)where ϕ ( τ ) = Ω d sin ( τ ) / ( ω ). The U I interaction evolu-tion is given by the following equation i ˙ U I = h ω x ω σ x + g y ( τ ) σ y + g z ( τ ) σ z i U I = (cid:15)A ( τ ) U I , (3)where g y = + ω y ω cos( ϕ ) + ω z ω sin( ϕ ) + Ω t ω cos( ϕ ) cos( pτ + Φ) ,g z = − ω y ω cos( ϕ ) + ω z ω cos( ϕ ) − Ω t ω sin( ϕ ) cos( pτ + Φ) , and (cid:15) ≈ ω o /ω, ≈ Ω t /ω is a bookkeeper for the perturba-tion orders.As the A matrix is periodic, we use the Floquet theo-rem to write U I ( τ ) = e − iP ( τ ) e − i Λ τ (4) with P (0) = 0 and P ( τ + 2 π ) = P ( τ ). The Floquetmatrix Λ is a time-independent matrix with spin eigen-values λ ± . Applying to U I the Floquet-Magnus expan-sion [31, 32] we obtain P = (cid:15)P + . . . and Λ = (cid:15) Λ + . . . ,where the first order terms areΛ = 12 π Z π A ( τ )d τP ( τ ) = Z τ A ( τ )d τ − τ Λ . (5)Exploiting the J n Bessel expansione i z sin θ = + ∞ X n = −∞ J n ( z ) e i n θ (6)and the cos( ϕ ( τ )) and sin( ϕ ( τ )) time integrals of [30], weobtain Λ = 12 ω h · σ. (7)We introduce here the effective rectified magnetic field h driving the spin evolution. For p even, h measured inenergy units is h = ω x J ( ξ ) ω y + J p ( ξ )Ω t cos(Φ) J ( ξ ) ω z . (8)For p odd, the J p term is added to the z component withcos(Φ) replaced by sin(Φ). The excitation with severalharmonic frequencies and arbitrary orientations for thetuning field presented in [30] leads to an extended quan-tum control. However it does not modify the geometryof the rectified fields generated in the yz plane orthog-onal to the dressing field direction. We verify that thesecond order perturbative expansion generates an extraeffective field oriented along the direction of the dressedfield, enabling an independent control of the three axes,not reached within the first order expansion. The recti-fication strengths depends on Ω t , ξ and on the Φ phaserelation between the rf fields. The h fields are also in-terpreted as light-shifts produced by the tuning-dressingfields with the spatial inversion symmetry preserved bythe role of the p order and of the Φ phase.Generalizing the analysis of [19], we derive in [30] thetemporal evolution of the atomic coherences. Within theperturbation treatment and for an initial state preparedin a σ x eigenstate, the spin x component is h σ x ( t ) i = (1 − h x ω ) cos(Ω L t ) + h x ω . (9)The modulated x axis response, as explored in the exper-iment, contains only a precession at the Ω L frequency.Instead the spin components along the other axes con-tains oscillations also at harmonics of the ω frequency.The effective field modifies the spin spatial componentsboth in mean and oscillating components. B x ( T) L R a t i o Figure 1. (Color online) B x calibration data obtained forthe dressed and B t = 0 non-tuned case at ω z / π = 5 . ω/ π = 30 kHz. From the Ω measurements vs B x at ξ = 0 and ξ = 1 . B x scale at the fourpercent precision level given by the horizontal error bars. The B x = 0 data point provides the ξ calibration. From the λ ± eigenvalues we derive that the rectifiedmagnetic field produces an energy splitting described byan effective Ω L Larmor precession frequencyΩ L = q ω ox + e ω oy + e ω z , (10)where for p even e ω oy = J ( ξ ) ω y + J p ( ξ )Ω t cos(Φ) , e ω oz = J ( ξ ) ω oz , (11)and for p odd e ω oy = J ( ξ ) ω y , e ω oz = J ( ξ ) ω z + J p ( ξ )Ω t sin(Φ) . (12)Eqs. (8) and (10) evidence the triaxial spatial responseto the external drivings, equivalent to an anisotropic non-linear gyromagnetic ratio.The tuning-dressed precession frequencies are testedusing the optical magnetometric apparatus of ref. [33].The atomic sample is caesium in a gas-buffered cell,pumped to the F g = 4 ground hyperfine state by theD1 line and optically probed on the D2 line. The pumplaser propagates along the x direction of the oscillatingdressing field, and the probe laser along the same direc-tion monitors the atomic evolution given by Eq. (9). Thepolarization of the transmitted probe laser is analyzedby a balanced polarimeter. We operate in a Bell-Bloom-like configuration by applying to the D1 pumping laser awide-range periodic modulation with frequency ω M . Thismodulation creates also the repumper from the F g = 3Cs ground state. By scanning ω M around Ω L , the po-larimetric signal is analyzed in order to derive the atomicmagnetic resonance with a 20 Hz HWHM linewidth due to spin-exchange relaxation and to probe perturbations.This system used for a frequency measurement reachesan accuracy at the Hz level [12].A static magnetic field is applied in a direction of the xz plane at a variable angle from the z axis. Essen-tial components are three large size, mutually orthogonalHelmholtz pairs, here used to to lock the B x and B z field components to desired values, in the range 1 − µ T ( ω x / π, ω z / π in the range 3-15 kHz), and to com-pensate the y component of the environmental magneticfield. In addition five quadrupoles coils compensate thefield gradients at the nT/cm level.We operate with dressing frequency ω/ π = 9 −
30 kHz,and p = 1 − B d rf dressing field is generated by a longsolenoidal coil external to the magnetometer core. The B t rf tuning field is produced by a separate Helmholtzcoil pair. While B d and B t values may be derived fromgeometry and current of the coils at the few percent level.For a higher precision determination of B d and B x weuse the following precession law in the B t = 0 case:Ω = γ q ω x + ω z J ( ξ ) . (13)For B x = 0, a fit of the Ω vs ξ data determines thedressing parameter at the three per thousand precisionlevel. In order to determine B x , we measure Ω vs thistransverse static field for the ξ values 0 and ≈ .
83 max-imising the J slope. A fit of their ratio to the above pre-cession predictions, as in Fig. 1, allows us to derive the B x value at the four percent precision level. In additionthe fit determines that the applied B x field contains athree percent component along the z axis.We perform experiments with the ω x = ω y = 0 con-figuration in order to verify the validity of Eqs. (10)for the Ω L dependence on the experimental handles.The objective is to evidence the quantum control flex-ibility associated to the tuned-dressing. In the experi-mental configuration the precession frequency becomesΩ L = ω z J ( ξ ) + Ω t J p ( ξ ) sin(Φ) for p odd, and Ω L = q (Ω t J ( ξ ) cos(Φ)) + ( ω z J ( ξ )) for p even. The threepanels of Fig. 2 report the measured (black dots) andtheoretical (continuous lines) Ω L absolute values vs the ξ driving parameter for different combinations of the p ,Φ and Ω t parameters. Their values are chosen in orderto maximise the atomic response tuning. Panels (a) and(c) deal with p = 1 , J and J Bessel functions play the key role for the ξ dependencePanel (b) dealing with the p = 2 (even) case evidencesthe J function role for Ω L . An important result of the p odd (a) plot is the possibility of increasing the Larmorfrequency, a feature not accessible to the single irradi-ation configuration. The odd harmonic cases, plots (a)and (c), allow for a sign change for the Larmor frequency,showing up as a slope change in the measured absolutevalue. A sign change occurs also in the single dressingcase, with its J dependence and cylindrical symmetry. (a) L / ( k H z ) (b) (c) Figure 2. (Color online) Absolute Ω L frequency vs ξ = Ω d /ω scanning Ω d / π in the (0 , ω z / π =2 . ω x = ω y = 0. Parameters p , Φ, ω/ π and Ω t / π , both in kHz: in (a) [1 , π/ , , . , , , . , π/ , , . ξ at the three per thousand level, both smaller than thedot size. Theory in continuous blue lines from perturbativetreatment, black line from numerical analysis and dashed redone in (b) for a refined Ω t / π value, as presented in text. The tuning-dressed configuration producing the triaxialanisotropy allows for an independent control of sign andamplitude for both the components orthogonal to thedressing field direction. Notice that in Fig. 2 the pertur-bative treatment is not valid for the ξ ≤ ω z /ω, Ω t /ω values, ≈ . , ≈ . ξ = 0 value isdetermined by treating Ω t as the dressing field. A nu-merical analysis of the spin evolution, as in black line ofthe panel (a) inset, leads to a better agreement with thedata. Fig. 3 reports the measured Ω L dependence onthe Φ phase and compares it to the theoretical predic-tions given by the continuous lines. In panels (a) and (c)for odd harmonics, the data follows a sine profile, withamplitudes given by the J ( ξ ) and J ( ξ ) predictions. Inpanel (b) for an even harmonic, the variation follows thesquared-cosine profile with amplitude set by the valueof J ( ξ ). These results confirm the usefulness of the Φphase difference as an additional parameter for the con-trolled tuning of the dressed system. The agreement levelbetween data of Figs. 2 and 3 and the theoretical predic-tions relies on the precise determination of the tuningfield amplitude. The theoretical analysis shows that forthe p odd cases the fit quality remains constant for Ω t variations within the error bar. Instead for the p = 2even case of panel (b) in both figures, a Ω t scaling up by (a) L / ( k H z ) (b) (deg) (c) Figure 3. (Color online) Ω L vs the Φ phase difference, for ω x = ω y = 0 and ω z / π = 2 . p , ξ , ω/ π and Ω t / π , both in kHz: in (a) [1 , . , , . , . , , . , . , , . t / π value, as presented in the text. four percent produces the red dashed lines of the figureswith a better theory-experiment agreement.In order to test the full triaxial anisotropy Larmor fre-quencies, exploiting our experimental pump/probe paral-lel to the x axis, we modify the spin spatial evolution ap-plying different B x magnetic fields for a fixed B z value.With a tilted static field axis we explore the triaxial spa-tial dependence probed by the Ω L of Eqs. (10). Fig. 4reports the experimental measurements for the Larmorfrequency as a function of B x value. A precise theoreti-cal analysis of the data requires the determination of theapplied Ω t field. The B x = 0 measurement with staticfield applied only along the z axis, as explored within theprevious Figures allow us to derive Ω t using Eq. (10).The continuous line of figure shows the excellent com-parison with the p = 1 theory for the whole B x = 0values. This result confirms the anisotropy properties ofthe tuned quantum system.The rectification and harmonic mixing [34], split bihar-monic driving [35] widely investigated within the quan-tum ratchet topic [36] present features similar to our in-vestigation. Those systems deal with the external de-grees of freedom, while our work examines the internalones. However the symmetries widely applied in quan-tum ratchets could represent a tool for exploring the gen-eration of rectified magnetic fields. x /2 (kHz) L / ( k H z ) Figure 4. (Color online) Ω L vs ω x as a test of the spinanisotropy, for an applied p = 1 tuning field, Ω t / π =0 . ω x scale. The tuning dressed mechanism is the interference inthe excitation produced by the two rf fields in a harmonicfrequency ratio, enhanced by the low-harmonic order.Such interference between the absorption/emission pro-cesses of both fields was examined in [37] within a Greenfunction approach. The regime of double strong dressingwith low order commensurable frequencies will enhancethe interferences and may lead to interesting topologicalfeatures as those explored for incommensurable multipleexcitations [28, 29].The introduction of a secondary field into a dressedsystem enriches the control on the spin dynamics and in-troduces features useful to several quantum control areas.For interferometric investigations with artificial or natu-ral atoms [38, 39], the fine tuning of the gyromagneticratio with a controlled collapse along different spatial di-rections may be applied to reach higher precisions. Formagnetic resonance, the condition of Ω L exceeding theunperturbed one, not obtainable using a single dressing field, shifts the spin resonant frequency to higher fre-quencies where the detection sensitivity increases. Thecritical dressing condition, which improves the sensitivityto small magnetic frequency shifts, may be reached withno first-order dependence on the ξ parameter. For thatdressing, the identification of tuning parameters produc-ing a reduced d Ω L /dξ sensitivity together with an ar-bitrary value of Ω L , attenuates the detrimental effectscaused by dressing field inhomogeneities [20]. In addi-tion, the required condition of a large d Ω L /dξ derivativeis easily reached relaxing the single dressing constraintof a Ω L reduction. This will help also the magnetometryapplications where Ω L was made deliberately position-dependent by means of an inhomogeneous ξ [27, 40]. Re-markably, a space dependent Ω L may be introduced inour scheme by means of a B t inhomogeneity, easier toimplement and control since B t (cid:28) B d . The Φ phase de-pendence on the tuning rf field should be applied to com-plement the amplitude dependence in the magnetometricdetection of weakly conductive material targets [41, 42].The spin individual spatial components and their signsare not accessible to our experimental investigation.However a direct test of the spatial anisotropy can beobtained in a critical dressing experiment as in [18] withspin exchange of the transverse magnetization along the x, y axes. Playing with the different tuning responsefor the two investigated spins, the spin collapse in onedirection and the enhancement in a different directionwill find their perfect testbed and also their application.This controlled spatial anisotropy of the spin representsan additional flexible handle with applications in quan-tum simulation and spintronics. 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Supplemental Material: Harmonic tuning of an atomic dressed system in magneticresonance
I. SPIN 1/2 SYSTEMA. x,y coherences
For the evolution of the x and y atomic coherences, we approximate e − iP ( τ ) by the identity matrix (see theAppendix). The time evolution operator becomes U ( τ ) ≈ e − iϕ ( τ ) σ x / e − i Λ τ . Therefore introducing the dimensionaltime, we obtain for the σ x operator σ x ( t ) = U ( t ) † σ x U ( t )= e i Λ ωt e iϕ ( t ) σ x / σ x e − iϕ ( t ) σ x / e − i Λ ωt = e i Λ ωt σ x e − i Λ ωt = cos(Ω L t ) σ x − sin(Ω L t ) (cid:18) h z ω σ z − h y ω σ y (cid:19) +(1 − cos(Ω L t )) h x ω (cid:18) h x ω σ x + h y ω σ y + h z ω σ z (cid:19) , where we use the Λ expansion of the main text and the Pauli matrix exponentiation. If the initial state is preparedin a σ x eigenstate, the x axis coherence becomes that reported within the main text.Repeating the derivation for the y axis we obtain h σ y ( t ) i = (cid:20) h y ω sin( ϕ ) + h z ω cos( ϕ ) (cid:21) sin(Ω L t )+ (cid:20) h x h y ω cos( ϕ ) − h x h z ω sin( ϕ ) (cid:21) (1 − cos(Ω L t )) . For the z axis we obtain h σ z ( t ) i = (cid:20) h z ω sin( ϕ ) − h y ω cos( ϕ ) (cid:21) sin(Ω L t )+ (cid:20) h x h z ω cos( ϕ ) + h x h y ω sin( ϕ ) (cid:21) (1 − cos(Ω L t )) . B. Tuning field in arbitrary direction
The section discusses the case of a tuning rf field oriented in an arbitrary direction having ( X t , Y t , Z t ) spatialcomponents with ( q, p, r ) harmonic temporal dependencies, with the index p assigned to the y axis as in the maintext. The interaction with the spin 1/2 system is described by the following Hamiltonian: H = Y t ω cos( pτ + Φ y ) σ y + X t ω cos( qτ + Φ x ) σ x + Z t ω cos( rτ + Φ z ) σ z . We repeat the Floquet-Magnus expansion for the interaction operator. The analysis for the h first order effectivemagnetic field introduced by Eq. (8) of the main text leads to the following components associated to differenteven/odd values of the harmonic coefficients h x = ω x ,h y = J ( ξ ) ω y + J p ( ξ ) Y t cos(Φ y ) ( p, r ) even , − J r ( ξ ) Z t sin(Φ z ) ( p, r ) odd , + J p ( ξ ) Y t cos(Φ y ) − J r ( ξ ) Z t sin(Φ z ) , p even , r odd , +0 p odd , r even ,h z = J ( ξ ) ω z + J r ( ξ ) Z t cos(Φ z ) ( p, r ) even , + J p ( ξ ) Y t cos(Φ y ) ( p, r ) odd , +0 p even , r odd , + J p ( ξ ) Y t sin(Φ y ) + J r ( ξ ) Z t cos(Φ z ) p odd , r even . II. SYSTEMS WITH A HIGHER SPIN
While the main text examines the simple case of a two level atom, the present analysis confirms the validity alsofor a higher spin system, as for the caesium ground state experimentally tested.The evolution of atomic magnetization M , i.e., the mean value of a quantum operator, in an external field B isdescribed by the Bloch equations ˙M = γ B × M . (S1)By examining the B x = B y = 0 case the equation for the atomic magnetization in presence of a magnetic field is d M d τ = (cid:20) Ω d ω cos( τ ) L x + Ω t ω cos( pτ + Φ) L y + ω z ω L z (cid:21) M where L j with ( j = x, y, z ) are the L = 1 angular momentum operator matrices.Using the perturbation theory we factorize the time evolution operator U ( τ ), i.e., M ( τ ) ≡ U ( τ ) M (0), in theinteraction representation as U ( τ ) = e [ ξ sin τ A ] U I ( τ )= ϕ ( τ ) − sin ϕ ( τ )0 sin ϕ ( τ ) cos ϕ ( τ ) U I ( τ ) , and we obtain the following dynamical equation for U I ( τ ): d U I d τ = (cid:20)(cid:18) ω z ω sin( ϕ ) + Ω t ω cos( ϕ ) cos( pτ + Φ) (cid:19) L y + (cid:18) ω z ω cos( ϕ ) − Ω t ω sin( ϕ ) cos( pτ + Φ) (cid:19) L z (cid:21) U I ≡ (cid:15)A ( τ ) U I . Because the matrix A ( τ ) is periodic A ( τ + 2 π ) = A ( τ ), we follow the main text steps to write U I ( τ ) as in Eq. (4),and we introduce the Floquet-Magnus expansion to calculate the lowest order terms. For the first order perturbationwe make use of Eqs. (S5) of the Appendix to obtainΛ = ( ω z ω J ( ξ ) L z + Ω t ω J p ( ξ ) cos Φ L y p even , (cid:2) ω z ω J ( ξ ) + Ω t ω J p ( ξ ) sin Φ (cid:3) L z p odd , (S2)and P = (cid:18) ω z ω f ( τ ) + Ω t ω f ( τ ) (cid:19) L y + (cid:18) Ω t ω f ( τ ) − Ω t ω f ( τ ) (cid:19) L z . (S3)The functions f i ( τ ) are reported in the Appendix.From Eq. (S2) we calculate the eigenvalues ( − Ω L , , Ω L ) with Larmor frequencyΩ L = (q ω z J ( ξ ) + Ω t J p ( ξ ) cos Φ p even , | ω z J ( ξ ) + Ω t J p ( ξ ) sin Φ | p odd . (S4)These equations are equivalent to those derived within the main text for a two-level system. The present approachcan be extended to the Zeeman structure for higher spin systems. III. APPENDIX
The Λ derivation of the main text and the above Λ and P derivation are based on the following integrals: Z τ cos( ϕ ( τ )) dτ = J ( ξ ) τ + f ( τ ) , (S5a) Z τ sin( ϕ ( τ )) dτ = f ( τ ) , (S5b) Z τ cos( ϕ ( τ )) cos( pτ + Φ) dτ = 1 + ( − p τ J p ( ξ ) cos Φ+ f ( τ ) , (S5c) Z τ sin( ϕ ( τ )) cos( pτ + Φ) dτ = − − p τ J p ( ξ ) sin Φ+ f ( τ ) . (S5d)These auxiliary f i functions are defined as f ( τ ) = ∞ X n =1 J n ( ξ ) n sin(2 nτ ) ,f ( τ ) = 4 ∞ X n =0 J n +1 ( ξ )2 n + 1 sin (( n + 1 / τ ) ,f ( τ ) = < ( g ( τ )) ,f ( τ ) = = ( g ( τ )) , (S6)where g ( τ ) = e i Φ X n = − p J n ( ξ ) i ( n + p ) (cid:16) e i ( n + p ) τ − (cid:17) + e − i Φ X n = p J n ( ξ ) i ( n − p ) (cid:16) e i ( n − p ) τ − (cid:17) . (S7)These functions, required in the evaluation of e − iP ( τ ) , have a limited and oscillating behaviour. One can see byinspection that e − iP ≈1