Quantum dynamics in strongly driven random dipolar magnets
M. Buchhold, C. S. Tang, D. M. Silevitch, T. F. Rosenbaum, G. Refael
QQuantum dynamics in strongly driven random dipolar magnets
M. Buchhold, C. S. Tang, D. M. Silevitch, T. F. Rosenbaum, and G. Refael Department of Physics and Institute for Quantum Information and Matter,California Institute of Technology, Pasadena, CA 91125, USA (Dated: February 20, 2020)The random dipolar magnet LiHo x Y − x F enters a strongly frustrated regime for small Ho + concentrationswith x < .
05. In this regime, the magnetic moments of the Ho + ions experience small quantum corrections tothe common Ising approximation of LiHo x Y − x F , which lead to a Z -symmetry breaking and small, degen-eracy breaking energy shifts between di ff erent eigenstates. Here we show that destructive interference betweentwo almost degenerate excitation pathways burns spectral holes in the magnetic susceptibility of strongly drivenmagnetic moments in LiHo x Y − x F . Such spectral holes in the susceptibility, microscopically described interms of Fano resonances, can already occur in setups of only two or three frustrated moments, for which thedriven level scheme has the paradigmatic Λ -shape. For larger clusters of magnetic moments, the correspondinglevel schemes separate into almost isolated many-body Λ -schemes, in the sense that either the transition matrixelements between them are negligibly small or the energy di ff erence of the transitions is strongly o ff -resonantto the drive. This enables the observation of Fano resonances, caused by many-body quantum corrections tothe common Ising approximation also in the thermodynamic limit. We discuss its dependence on the drivingstrength and frequency as well as the crucial role that is played by lattice dissipation. PACS numbers:
I. INTRODUCTION
Magnetic dipoles with Ising symmetry randomly dis-tributed on a lattice provide the opportunity to explorethe e ff ects of interactions , disorder , frustration , randomfields , entanglement , and quantum fluctuations , with theability to tune their interplay . When driven out of equilib-rium, new many-body states emerge, with characteristics thatare the magnetic analogues to optically driven atomic systems,but involving numerous quantum degrees of freedom . Whendecoupled from the thermal environment, the states are intrin-sically non-linear with very small linewidths .Investigating the magnetic phases and dynamics in thedisordered dipolar quantum magnet LiHo x Y − x F has beenthe focus of this class of research activity for severaldecades , yet explanations for several of its propertiesat low temperatures remain elusive . In LiHo x Y − x F ,the magnetic Ho + cations mainly interact via dipole-dipoleinteractions. At large concentrations x > . . Diluting the Ho + concentration below x < . , which leads tothe formation of an Ising spin glass for 0 . ≤ x ≤ .
25 at suf-ficiently low temperature T ≤ .
5K and transverse field ,while below concentrations of x < .
15 the nature of thelow temperature state can be manifestly classical or quan-tum depending upon the strength of the thermal link to a heatbath .Recently, attention has been drawn to the dilute limit x ≤ .
05 in which di ff erent experiments have observed aspects ofan Ising spin glass , a quantum disordered, so-called“antiglass” with spin liquid characteristics and iso-lated quantum degrees of freedom . The root of the irrec-oncilability of these observations seems to be found in thestrength of the dissipation experienced by the magnet, i.e., by the coupling of the sample to the environment, as evidencedby recent experiments that tune a LiHo x Y − x F sample froman Ising spin glass to an “antiglass” by reducing its thermalcoupling to the environment .A key signature of the antiglass behavior is “spectral holeburning”, i.e. the observation of a Fano resonance in themagnetic susceptibility χ ( ω ) in a LiHo x Y − x F sample, whichis strongly driven by a time-dependent magnetic field .Fano resonances are commonly a signature of quantum in-terference. In LiHo x Y − x F the resonances are observableat arbitrary transverse fields and surprisingly small drive fre-quencies ω d ≈ π ×
200 Hz and probe frequency detunings ω p − ω d ≈ π × − x Y − x F sample is well isolated from its environment and vanish if thecoupling to the environment is increased.The magnetic moments in LiHo x Y − x F form a compli-cated, disordered, and strongly interacting many-body prob-lem, which is hard to address theoretically even in the sim-plified Ising approximation . What is especially puzzlingin the hole burning experiments is the presence of several,strongly separated energy scales, and the apparent sensitiv-ity of hole burning to all of them. The dipole-dipole inter-action between two neighboring moments is of the order of ∆ V =
500 mK and falls o ff with a distance as ∼ / | (cid:126) r | .The LiHo x Y − x F sample is held at a temperature of about T = Ω d ≈ µ K, a drive frequency, whichcorresponds to ω d (cid:39) ω p − ω d (cid:39) . x Y − x F , incorporating the full magnetic dipole-dipoleinteraction and the crystal field for the J = + ion. Using exact diagonalization, weshow that the observation of Fano resonances can be explainedon a qualitative level already for a single pair of Ho + -ions.The resonances appear as a consequence of interference be- a r X i v : . [ c ond - m a t . d i s - nn ] F e b tween two quasi-degenerate excitation pathways, correspond-ing to a pair of quasi-degenerate quantum states, which can becoupled by applying an external, oscillating magnetic field.In order to generalize this observation to more realistic sam-ples with n ≥
10 magnetic degrees of freedom, we devise a toymodel of e ff ective spin- degrees of freedom, which capturesthe main ingredients for the observation of hole burning andreduces to the LiHo x Y − x F Hamiltonian at low energies andfor few magnetic moments. Exploring the dynamics of smallsamples shows that an external, oscillating magnetic field ad-dresses only a small fraction of the many-body Hilbert space,for a given set of driving parameters. The predicted magneticsusceptibility χ ( ω ) displays several spectral holes, which canbe explained in terms of quasi-degenerate many-body excita-tion pathways and which match quantitatively very well withthe experimental findings and energy scales. Within our sim-plified model, we can understand the origin and the impor-tance of the di ff erent energy scales and, in addition, can findan explanation why the Fano resonance is only observed inthe limit of very small coupling between the sample and theenvironment.Based on these findings, we propose an experimentalscheme to manipulate the Fano signals by an external, acous-tic drive of the lattice vibrations. The idea behind this ap-proach is to engineer the dissipation rate of the magnetic mo-ments by controlling their interactions with the phonon con-tinuum. The latter is controlled by the number of phonons thatare accessible for scattering at a given energy. Driving phononmodes explicitly generates a nonequilibrium phonon distribu-tion, which is peaked at the drive frequency and increases thedissipation rate at matching energies. This reduces or evendestroys the interference pattern of the Fano resonances. Ob-serving this reverse or “anti”-hole burning at the phonon drivefrequencies would confirm our present explanation of holeburning and open a path to control the magnetic propertiesof LiHo x Y − x F via both time-dependent magnetic fields andsound. II. MODEL
In this section we briefly review the microscopic model forthe magnetic degrees of freedom in LiHo x Y − x F compoundsand illustrate that several aspects of the long time dynamicsof the dilute material ( x (cid:28)
1) are not captured by an e ff ectiveIsing description. Instead, the strong dipole-dipole interactionbetween magnetic Ho + atoms induces non-trivial entangle-ment in the magnetic degrees of freedom and lifts the expected Z symmetry of an Ising magnet. We show that the deviationfrom a common, random Ising magnet becomes crucial at lowtemperatures or when the system is driven by an external field.For suitable driving frequencies, the latter resolves violationsof the Z symmetry and therefore the quantum nature of themagnet, which manifests itself via the absence of degener-ate energy levels and the presence of non-vanishing moments (cid:104) α | J z | β (cid:105) (cid:44) ff erent eigenstates | α (cid:105) , | β (cid:105) . A. Microscopic Hamiltonian
LiHo x Y − x F is a magnetic material because of the mag-netic Ho + ions, in which the 4 f electrons form an I elec-tronic ground state manifold . In this manifold, each Ho + ion l is described by a J = (cid:126) J l . The coupling of each Ho + ion to its non-magnetic neighbors via Coulomb interactions and ion-latticecoupling is described by a crystal field Hamiltonian H cf ( (cid:126) J l )which aims to polarize (cid:126) J l along the magnetic c -axis of thecrystal. Exchange interactions between neighboring Ho + arein general weak and become negligible in the dilute limit x (cid:28)
1. The remaining interaction between two di ff erentHo + ions is the magnetic dipole-dipole interaction, whichcan, however, become rather strong due to the large total an-gular momentum J = + .In addition to the mentioned terms, Ho-cations display asignificant hyperfine interaction between the nuclear magneticmoment ( I = /
2) and the electronic moments with a cou-pling constant A J = T < . ff ective low energydegrees of freedom and quantitatively modifies the phasediagram both for the ferromagnetic and for the spin glasstransition . Qualitatively, however, the localized nuclearmoments do not change the nature of the long-range coupledlow energy degrees of freedom of the magnetic dipoles (cid:126) J awayfrom a phase transition (as discussed in Sec. IV C). As weshow below, hole burning in LiHo x Y − x F can be very wellexplained without considering hyperfine interactions. We willdiscuss potential modifications due to hyperfine interactionsat the end of Sec. IV.We consider no static external magnetic field (cid:126) B =
0, whichyields the microscopic Hamiltonian H = (cid:88) l H cf ( (cid:126) J l ) + µ g L µ B π (cid:88) l (cid:44) m L αβ ( (cid:126) r lm ) J α m J β l . (1)Here, L αβ ( (cid:126) r ) = δ αβ | (cid:126) r | − r α r β | (cid:126) r | is the dipole-dipole matrix elementbetween two Ho + ions l , m , which is evaluated at their relativecoordinate (cid:126) r = (cid:126) r lm = (cid:126) r l − (cid:126) r m . The interaction strength dependson the Bohr magneton µ B =
23 KT , the vacuum magnetic perme-ability µ = π − and the Land´e g -factor g L = .LiHo x Y − x F has a tetragonal structure with lattice con-stants a = .
175 Å and c = .
75 Å and four possi-ble spots for Ho + , or Y + ions per unit cell . In termsof the unit cell coordinates ( a , a , c ) their positions are at(0 , , ) , (0 , , ) , ( , , ) , ( , , ∆ r min ≈ .
73Å between two Ho + ions. Thecorresponding magnetic interaction energy is A dip = . + ion the crystal field Hamiltonian H cf ( (cid:126) J l ) fea-tures a two-fold degenerate ground state doublet and an ex-cited singlet state separated by an energy ≈ .
5K from theground states. The remaining 14 eigenstates are separatedmore than 20K from these three states. At low tempera-tures T <
10K and zero magnetic field, thermal activationof excited states can be excluded and each magnetic momentis commonly projected onto the ground state manifold, i.e.,treated as an Ising degree of freedom .In the Ising approximation, each magnetic moment reducesto an Ising spin J α l = δ α, z C zz σ zl , whose orientation is de-scribed by the Pauli matrix σ z . Under this transformation, theHamiltonian (1) reduces to H → H Ising with H Ising = A dip C zz (cid:88) l (cid:44) m σ zl σ zm L zz ( R lm ) . (2)Here A dip is the coupling introduced above, C zz ≈ . ff ective magnetic moment in the z -direction and R lm is thedimensionless distance between spin l and m in units of ∆ r min .In the dense limit x ≥ .
25, the Hamiltonian (2) describesan Ising dipolar ferromagnet with ordering temperature T c = .
53K at x = . For stronger dilution, x < .
25, therandom positions of the Ho + ions induce frustration betweendi ff erent magnetic moments and the system enters a dipolarIsing spin glass phase at su ffi ciently low temperatures T < T c ≈ xT mf c = x · . .There was a debate in the literature, whether LiHo x Y − x F enters a glass state at very low concentrations x ≤ .
05 andtemperatures T ≤ ,which is supplemented by classical Monte Carlo simulations of the Ising Hamiltonian (2), another set of experiments re-ported evidence for an anti-glass state in which low energyquantum fluctuations prevent the spin glass freezing .A most recent experimental study showed that both, glass andanti-glass behavior can be realized in the same setup by chang-ing the systems interaction with the environment from strong(glass state) to weak (anti-glass state) coupling .Colloquially speaking, the Hamiltonian (1) can be well ap-proximated by the Ising Hamiltonian (2) for x ≤ .
05, if thedissipation rates, corresponding to dephasing and incoherentflips of the magnetic moments, are larger than the energy levelsplittings between quasi-degenerate states. In this case, dis-sipation dominates over coherent dynamics and the dynam-ics looks e ff ectively classical, i.e. Ising-like. This statementwill be made more quantitative below by showing that thehole burning, associated to the anti-glass dynamics, can beexplained on the basis of the Hamiltonian (1) without per-forming the Ising approximation. B. Dimer and Trimer level schemes
In order to point out the importance of quantum e ff ects inthe LiHo x Y − x F Hamiltonian in (1) for x ≤ .
05, we com-pare the low energy physics of H with H Ising for small spin’clusters’ of n = , n = , n positions for the Ho + atoms and diagonalize the 17 n × n Hamiltonian H with ex-act material parameters . The precise form of the crystal fieldHamiltonian for LiHo x Y − x F is discussed in the Appendix A.First, we consider a dimer setup of two Ho + ions with J =
8, which both experience the crystal field and mutual dipole-dipole interactions. The Hamiltonian of the two ions l = ,
100 200 300 400 500 600 700 800 900 100010 -5 -4 -3 -2 -1 data1data2data3data4data5data6
100 200 300 400 500 600 700 800 900 100010 -5 -4 -3 -2 -1 ✓ ⇡ ⇡ E ✓ ⇡ ⇡ { { Ising
LiHo x Y x F | E ± Ising | ✏ ✏ | E ± Ising | ✏ ✏ µ µ µ µ µ µ µ ↵ coupling matrix | i| i| i| i ( c ) level scheme ~J ~J ✓ dipole-dipole: ⇠ L zz ( ✓ ) J z J z | {z } Ising + L xz ( ✓ ) J x J z + ... | {z } quantum corrections ( a ) dimer configuration r min ( b )( d ) ( e ) L zz ( ✓ ) < L zz ( ✓ ) > | i| i| i| i µ ↵ = h ↵ | J z + J z | i Figure 1: A magnetic dimer is the most simple unit from which holeburning in LiHo x Y − x F can be understood. (a) It is formed by two J = z -axis and fea-tures an Ising-type ground state manifold associated with spin upand down. This singles out the Ising contribution ∼ J z J z as thedominant dipole-dipole interaction at low temperatures. Quantumcorrections, led by the terms ∼ J x J z , J z J x , are strongly suppressedby the crystal field but crucial for the understanding of hole burn-ing in driven LiHo x Y − x F samples. (b) The terms J z J x + ... arenot compatible with the Ising-symmetry. They lift the Ising degen-eracy and introduce small level splittings (cid:15) , between two quasi-degenerate eigenstates in the LiHo x Y − x F dimer. (c) Breaking theIsing symmetry also introduces small but non-zero transition ma-trix elements µ αβ = (cid:104) α | ( J z + J z ) | β (cid:105) between di ff erent dimer eigen-states | α (cid:105) , | β (cid:105) . (d + e) Quantitative analysis of the level spacings (d)and transition matrix elements (e) from exact diagonalization of aLiHo x Y − x F dimer described by H (2) in Eq. (3) with relative ori-entation, (cid:126) r = ∆ r min (sin θ, , cos θ ) (we set L αβ ( θ ) ≡ L αβ ( θ )( (cid:126) r ).The energies are compared to an equivalent Ising dimer, describedby Eq. (4). The colors in (e) match with the illustration in (c). At θ = arccos √ , π , the states | (cid:105) and | (cid:105) , | (cid:105) are degenerate. For some θ , there is one ”dark” state (dashed line) corresponding to an Isingsinglet, which does not couple to the other states via J z . The quasi-degenerate partner of the dark state, however, weakly couples to bothstates of the remaining quasi-degenerate pair. We refer to the partic-ular form of µ αβ in (c,e), i.e. the coupling of a quasi-degenerate pairof states to an energetically well separated state, as ” Λ -scheme”. Itis the basic building block for hole burning in driven LiHo x Y − x F . which are separated by a vector (cid:126) R (in units of ∆ r min ), is H (2) = H cf ( (cid:126) J ) + H cf ( (cid:126) J ) + A dip (cid:88) α,β = x , y , z L αβ ( (cid:126) R ) J α J β . (3)The corresponding Ising Hamiltonian is obtained by project-ing onto the ground state doublets of H cf ( (cid:126) J , ) and is H (2)Ising = A dip C zz L zz ( (cid:126) R ) σ z σ z . (4)It has eigenenergies ± E Ising = ±| A dip C zz L zz ( (cid:126) R ) | , each ofwhich are two-fold degenerate.In general, the dipole-dipole interaction in Eq. (3) does notfeature a compatible Z -symmetry and thus breaks the groundstate degeneracy of the crystal field Hamiltonian. This intro-duces splitting energies (cid:15) , as illustrated in Fig. 1(b). We in-troduce the projector P ( n ) , which projects onto the 2 n states oflowest energy of H ( n ) . For each dimer eigenbasis one finds P (2) H (2) P (2) = ∆ + (cid:15) , ∆ , (cid:15) , , (5) H (2)Ising = E Ising , E Ising , , . (6)Away from the special point L zz ( (cid:126) R ) =
0, where the ’classi-cal Ising’ interaction vanishes, the modifications of the eigen-values of P (2) H (2) P (2) compared to H (2)Ising seem rather small,i.e., (cid:15) , ∆ , (cid:12)(cid:12)(cid:12)(cid:12) ∆ − E Ising E Ising (cid:12)(cid:12)(cid:12)(cid:12) ∼ − − − for | (cid:126) R | =
1, see Fig. 1(d). For | (cid:126) R | >
1, one finds a very accurate scaling estimate (cid:15) , ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | (cid:126) R | > ≈ (cid:15) , ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | (cid:126) R | = | (cid:126) R | . (7)This anticipates that corrections of P (2) H (2) P (2) compared to H (2)Ising can be understood in terms of second order perturba-tion theory. The eigenvalues of the dipole matrix J α J β can,however, become very large. Using second order Brillouin-Wigner perturbation theory in the eigenbasis of H cf ( (cid:126) J ) + H cf ( (cid:126) J ) we find that convergence towards P (2) H (2) P (2) requiresto include more than N =
100 of the 17 =
289 eigenstates.This makes it di ffi cult to express the eigenstates of P (2) H (2) P (2) in the Ising basis analytically.A second modification caused by using H (2) instead of H (2)Ising is that the total z -axis magnetization J z tot = J z + J z isno longer diagonal in the basis of energy eigenstates. For theIsing Hamiltonian, J zl ∝ σ zl and [ σ zl , H Ising ] = J z tot , H Ising can be diagonal in the same basis. In contrast, all diagonalmatrix elements of J z tot vanish in the eigenbasis of H . We de-fine the matrix elements of the total magnetic moment in the z -direction between eigenstates | α (cid:105) , | β (cid:105) , µ αβ = (cid:104) α | J z tot | β (cid:105) = (cid:88) l (cid:104) α | J zl | β (cid:105) . (8)For Ising eigenstates, we find µ αβ ∼ δ αβ , while forLiHo x Y − x F clusters we find µ αβ ∼ (1 − δ αβ ). The absolutevalues | µ αβ | for the dimer setup are shown in Fig. 1(e).For non-commuting [ J z tot , H ] (cid:44)
0, an oscillating magneticdrive field h ( t ) = h d cos( ω d t ) in the z -direction, which is de-scribed by the Hamiltonian δ H ( t ) = h d cos( ω d t ) µ B µ J z tot , (9)induces transitions between di ff erent energy eigenstates | α (cid:105) ↔| β (cid:105) . The transition rates are proportional to | µ αβ | , see 1(e), andthe corresponding level schemes for the dimer setup are illus-trated in Fig. 1(c). The transition matrix µ αβ , which couplesa quasi-degenerate pair of states to another, energetically wellseparated state, has the shape of a (inverse) Λ and we refer to it as Λ -scheme. As we will discuss, it features a similar dy-namics as driven three-level systems in quantum optics, where Λ -schemes of this shape are common. The Λ -scheme is thebasic building block for the understanding of hole burning inLiHo x Y − x F and we will analyze it in detail in Sec. III.Adding more magnetic moments to the cluster either en-hances or suppresses corrections to the Ising approxima-tion and may lead to more involved coupling matrices µ αβ .We demonstrate this in the following by analyzing magnetictrimer configurations with at least one frustrated moment. Thecorresponding LiHo x Y − x F - and Ising- Hamiltonians are H (3) = (cid:88) l = H cf ( (cid:126) J l ) + A dip (cid:88) α,β = x , y , z (cid:88) l = , m > l L αβ ( (cid:126) R lm ) J α l J β m , (10) H (3)Ising = A dip C z (cid:88) l = , m > l L zz ( (cid:126) R lm ) σ zl σ zm . (11)The level scheme for two specific trimer configurations isshown in Fig. 2. The deviations of the exact level schemefrom the one predicted by the Ising approximation, i.e., thedegeneracy breaking energies (cid:15) , range from very small values (cid:15) ∼ − K to relatively large ones (cid:15) ∼ . µ αβ , shown inFig. 2 (right column), as compared to the diagonal structurepredicted by the Ising approximation.When the system is driven with an time-dependent mag-netic field, i.e., when adding δ H ( t ) in Eq. (9), the transitionmatrix µ αβ corresponding to H ( n ) enables coherent, magne-tization changing transitions between the states | α (cid:105) ↔ | β (cid:105) with Rabi frequency Ω αβ = h d µ B µ µ αβ . This is in contrastto the classical Ising Hamiltonian, which remains diagonal inthe presence of a magnetic field in the z -direction, i.e., doesnot induce coherent transitions between di ff erent Ising eigen-states. The consequences of the particular structure of µ αβ inLiHo x Y − x F for the response to external driving, in partic-ular, how it leads to hole burning, will be discussed in thefollowing section. III. HOLE BURNING IN DRIVEN
LiHo x Y − x F DIMERSAND TRIMERS
Including the quantum corrections to the Ising Hamilto-nian, the phenomenon of spectral hole burning, i.e., the emer-gence of a Fano resonance in the magnetic susceptibility χ ( ω ),can be explained theoretically even in the most simple dimerand trimer level schemes for a LiHo x Y − x F Hamiltonian dis-cussed above. The origin of the Fano resonance in a dimer ortrimer is quantum interference between two almost degenerateexcitation pathways. We will discuss this phenomenon on thebasis of the instructive dimer scheme and highlight the roleplayed by dissipation for the resonance in this section, beforewe discuss its generalization to the case of many moments inthe following section. E = 0 . E = 0 . E = 0 . E = 0 . E = 0 . states states E = 0 . E = 2 . · K E = 10 K E = 10 K E = 10 K E = 0 . ~R = (0 , , ~R = (2 , , L zz ( ~R ) < L zz ( ~R ) > L zz ( ~R ) < H (3)Ising configuration H (3) µ µ µ µ µ µ µ µ µ µ µ ↵ ~R = (0 , , ~R = (2 , , L zz ( ~R ) > L zz ( ~R ) > L zz ( ~R ) > ~R = (1 , , ~R = (0 , , | E ± Ising | | E ± Ising | | E ± Ising | | E ± Ising | ✏ ✏ ✏ ✏ ✏ Figure 2: Level scheme and transition matrix elements for the low en-ergy eigenstates of two generic LiHo x Y − x F trimer configurations.Among the 3 Ising couplings in the top (bottom) row, 2 (0) are fer-romagnetic and 1 (3) are antiferromagnetic. As for the LiHo x Y − x F dimer, the trimer energy scheme (3rd column) shows several quasi-degenerate level splittings compared to the degenerate Ising scheme(2nd column). The ratio of a quasi-degeneracies over its correspond-ing Ising energy ranges from | (cid:15)/ E ± Ising | ≈ . − − and covers awider range than a dimer with comparable distances between themagnetic moments. The transition matrix µ αβ = (cid:104) α | (cid:80) l = J zl | β (cid:105) (rightcolumn) for each trimer configurations can be decomposed into twodistinct Λ -schemes corresponding to solid and dashed arrows, eachof which corresponds to a characteristic energy di ff erence ∆ andquasi-degeneracy (cid:15) . The quasi-degeneracies in the spectrum and thecomposition of the transition matrix µ αβ from Λ -schemes are genuinefeatures of magnetic clusters in LiHo x Y − x F , originating from theweak breaking of an e ff ective Ising symmetry in the crystal field’sground state manifold. A. Magnetic dissipation rates
In order to study the dynamics of a dimer under externaldriving, we need some estimate on the environmental induceddissipation, i.e., the dissipative transition rates γ α → β betweentwo quantum states | α (cid:105) , | β (cid:105) . The major source of dissipationfor the magnetic moments in LiHo x Y − x F is the coupling ofthe Ho + ions to lattice vibrations, i.e., phonons. In order toestimate the associated dissipation rates, we consider acoustic,Debye type low energy phonon modes, which are describedby a wavevector (cid:126) k , dispersion ω (cid:126) k = c | (cid:126) k | , Debye frequency ω D and Debye temperature Θ D .Each phonon mode has a linewidth Γ , which describes therate at which the mode exchanges energy with other phononsand the environment. Due to weak phonon-phonon interac-tions, the linewidth is dominated by the coupling of the sam-ple to the environment. In recent experiments it has beenpointed out that having a weak sample-environment couplingis crucial for the observation of hole burning and antiglassdynamics in driven LiHo x Y − x F . Here, we consider thelattice-environment coupling in terms of an e ff ective phononlinewidth. Such a linewidth will set the lower bound for themagnetic dissipation rates, and enable or disable the emer- gence of a Fano resonance. This yields a phenomenologicalexplanation for the presence or absence of hole burning in sev-eral LiHo x Y − x F experiments.Acoustic phonons in the Debye model are described by theHamiltonian H D = (cid:88) (cid:126) k , c | (cid:126) k | <ω D c | (cid:126) k | b † (cid:126) k b (cid:126) k , (12)with bosonic ladder operators b † (cid:126) k , b (cid:126) k at momentum (cid:126) k . The lin-ear coupling between the phonons and the magnetic states istypically of the form H mag-ph = (cid:88) α,β,(cid:126) k g αβ ( (cid:126) k ) (cid:18) b † (cid:126) k + b − (cid:126) k (cid:19) ( | α (cid:105)(cid:104) β | + | β (cid:105)(cid:104) α | ) , (13)with coupling matrix elements g αβ ( (cid:126) k ) between di ff erent eigen-states | α (cid:105) , | β (cid:105) of the magnetic Hamiltonian (1).The transition rate γ α → β between two magnetic states | α (cid:105) →| β (cid:105) with energy di ff erence E αβ ≡ E α − E β can be estimated byFermi’s golden rule (see Appendix C). It yields an energy-dependent decay rate γ αβ = γ ( E αβ ) = γ ( E ) γ ( E ) = − i (cid:88) (cid:126) k | g ( E ,(cid:126) k ) | n (cid:126) k E + i + + c | (cid:126) k | + n (cid:126) k + E + i + − c | (cid:126) k | = | g ( | E | ) | (cid:16) n B ( | E | ) + δ sign( E ) , (cid:17) ρ ph ( | E | ) . (14)Here n B ( | E | ) is the Bose-Einstein distribution at temperature T and energy | E | , ρ ph ( E ) is the phonon density of states andwe used the shortcut g ( | E | ) ≡ g αβ ( c | (cid:126) k | = E αβ ).For energy di ff erences E (cid:28) k B T , the Bose function showsthe typical E -divergence n B ( | E | ) ≈ k B T | E | (cid:29)
1. For acous-tic phonons at low energies, g ( | E | ) = g √| E | , which yields γ ( E ) ≈ g k B T ρ ph ( | E | ) = γ D T ρ ph ( | E | ) Θ D ρ ph ( ω D ) , where γ D is the de-cay rate at the Debye frequency. For a linear dispersion withlinewidth Γ the density of states in d = ρ ph ( | E | ) = ρ E for E (cid:29) √ Γ ω D and a con-stant ρ ph ( | E | ) = ρ Γ ω D for E (cid:28) Γ . This yields the dissipationrate γ ( | E | ) ≈ γ D T Θ D ω D × (cid:40) E for √ Γ ω D < E Γ ω D π for √ Γ ω D > E . (15)One thus observes that for very small Γ , E , the dissipationrates, which push the system back towards its equilibriumstate can become very small, leading to an out-of-equilibriumstate under driving. For LiHo x Y − x F the parameters inEq. (15) are hard to quantify due to the lack of knowledge onthe interaction between phonons and the magnetic momentsin the material. In order to obtain a qualitative estimate, onemight consider the dissipation rates of spin vacancies in dia-mond, where the phonon-induced dissipation has been deter-mined very precisely to be γ D Θ D ω D = − − −
14 1Hz K 36,3749 .For the significantly small level spacings of E ∼ γ ≈ Γ T K with the Debye | i | i| i ! ! ! , ⌦ d cos( ! d t ) ⌦ p cos( ! p t ) dimer µ ↵ ⌘ = ! d E ⌘ + ⌫ = ! p E Figure 3: Illustration of an idealized, driven Λ -scheme, which is re-alized in an antiferromagnetic Ho + dimer or trimer subject to time-dependent magnetic drive and probe fields in the z -direction. Theinset shows a corresponding dimer transition matrix µ αβ extractedfrom Fig. 1 (c). The drive, probe fields here oscillate with frequency ω d , ω p and the strength of the couplings is described by the Rabi fre-quency Ω d , Ω p , which is proportional to the corresponding couplingmatrix elements and the strength of the magnetic drive, probe fields h p , h d , i.e., Ω p ∝ h p µ , Ω d ∝ h d µ . The drive scheme also includesdissipative transitions with rates γ α → β , corresponding to Stokes ( ↑ )and anti-Stokes ( ↓ ) transitions, which stem from the coupling of themagnetic moments to a low temperature phonon continuum. Adjust-ing the detuning η, ν + η of the drive and probe field from the energydi ff erences E , E in the Λ -scheme enables a Fano resonance, i.e.,hole burning, in the linear susceptibility χ . Note: This illustrationrepresents an idealization. In reality, both the drive and probe fieldscontribute to µ and µ at the same time. In linear response to theprobe field h p , however, the measured signal is very well approxi-mated by Λ -schemes linear in h d . In addition to this figure, this iscovered by a drive scheme with ( ω d , h d ) ↔ ( ω p , h p ). temperature of LiHoF being Θ D = K . The linewidth Γ sets a lower bound to the dissipation rate, indicating thatfor strong coupling to the environment, the system is hardlypushed away from its equilibrium. B. Magnetic susceptibility and Fano signal for spin dimers
The Λ -schemes found in magnetic dimers and trimers inFigs. 1(c) and 2 are common candidates for the observation ofinterference between di ff erent excitation pathways and Fanoresonances . In this section, we discuss the mechanism ofdestructive interference, which leads to a Fano resonance inthe magnetic susceptibility, for an idealized Λ -scheme. The Λ -scheme is illustrated in Fig. 3. It consists of three quan-tum states | , , (cid:105) , which are driven by two external fields.The | (cid:105) ↔ | (cid:105) transition is driven by a time-dependent drivingfield and the | (cid:105) ↔ | (cid:105) transition is driven by a time-dependentprobe field. The measured time-dependent magnetic suscep-tibility will be proportional to the coherences | (cid:105)(cid:104) | , | (cid:105)(cid:104) | .Their dynamics does not depend on whether the Λ -schemeis regular or inverted and without loss of generality, we dis-cuss an inverted scheme. The generalization to the situationof many magnetic moments follows in Sec. IV.The ideal Λ -scheme consists of three levels | l (cid:105) , l = , , ff erent dimer eigenstates, asshown in Fig. 3(c). An oscillating external magnetic field with Rabi frequency Ω d and drive frequency ω d drives the | (cid:105) ↔ | (cid:105) transition with a detuning η = ω d − E from resonance. At thesame time, an oscillating probe field with Rabi frequency Ω p and drive frequency ω p probes the | (cid:105) ↔ | (cid:105) transition withdetuning ν + η = ω p − E from resonance. In addition, inco-herent transitions are induced by the coupling of the states toa phonon continuum. The corresponding rates for the (anti-)Stokes γ → , ( γ , → ) processes can be estimated by Eq. (15).The time-dependent Hamiltonian for this Λ -scheme is H Λ ( t ) = E | (cid:105)(cid:104) | + E | (cid:105)(cid:104) | + Ω d cos( ω d t )( | (cid:105)(cid:104) | + h.c.) +Ω p cos( ω p t )( | (cid:105)(cid:104) | + h.c.) (16)with E , E >
0. Assuming small Rabi frequencies, Ω d , p ≤ ω d , p , one can perform a rotating wave approxima-tion (RWA), i.e., transform the H V ( t ) into a frame rotatingwith the drive and pump fields and discard all counterrotat-ing terms ∼ ω p , d . The corresponding unitary transformationis U ( t ) = exp (cid:104) it (cid:16) ω p | (cid:105)(cid:104) | + ω d | (cid:105)(cid:104) | (cid:17)(cid:105) and the transformedHamiltonian˜ H Λ = U † ( t ) H Λ ( t ) U ( t ) − iU † ( t ) ∂ t U ( t ) RWA = Ω d | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) + Ω p | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) + ν | (cid:105)(cid:104) | + ( ν + η ) | (cid:105)(cid:104) | . (17)In the last step, we added a constant energy shift ˜ H Λ → ˜ H Λ + ( ν + η ), which does not change the dynamics.In order to account for the dissipation, we use a densitymatrix description of the magnetic system. The density matrixˆ ρ = (cid:88) α,β = ρ αβ | α (cid:105)(cid:104) β | (18)is hermitian ρ αβ = ρ ∗ βα and has unit trace (cid:80) α ρ αα =
1. Itstime evolution is described by a quantum master equation inLindblad form ∂ t ˆ ρ = i [ ˆ ρ, ˜ H Λ ] + (cid:88) α = , ( L α → + L → α ) ˆ ρ. (19)The second term describes the dissipative transitions via thesuperoperators L α → β , which act linearly on ˆ ρ , L α → β ˆ ρ = γ α → β (cid:32) | β (cid:105)(cid:104) α | ˆ ρ | α (cid:105)(cid:104) β | − (cid:8) | α (cid:105)(cid:104) α | , ˆ ρ (cid:9)(cid:33) . (20)The linear response to the probe field ∼ Ω p is obtained fromthe stationary state ( ∂ t ˆ ρ =
0) of Eq. (19). To simplify notation,we assume one common rate γ ≡ γ α → β for all dissipative pro-cesses. This is justified for k B T > E , E . One finds ρ ≈ ρ = − ρ , (21) ρ = Ω d ρ − Ω p ρ i γ − ν ) , (22) ρ = Ω d (3 ρ − η + i γ , (23) ρ = i γ − ν + Ω d η + i γ Ω d − ( i γ − ν )(6 i γ − η + ν )) Ω p (3 ρ − . (24) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.27891011121314 w F ( ! p ) . . [arb. units] -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.27891011121314 w F ( ! p ) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.27891011121314 ⌫ p E ! p E . . [arb. units] | i | i| i ! ! ! , ⌦ d cos( ! d t ) ⌦ p cos( ! p t ) ⌘ = ! d E ⌘ + ⌫ = ! p E ⌫ p . ! p . Figure 4: Spectral hole (Fano resonance) in the susceptibility χ ( ω p )obtained in linear response in Ω p from the Λ -scheme in Fig. 3 (re-peated in inset). Both the real ( χ (cid:48) , red bold line) and the imaginarypart ( χ (cid:48)(cid:48) , grey dotted line) display an asymmetric line shape, indicat-ing a Fano resonance close to the resonance condition ω p − ω d = ≈ E − E . The dimensionless parameters for this figure are γ = . , Ω d = , ω d = , E = . , E = .
4. The dependence ofthe strength F , spectral width w and position ν p of the signal on thedrive parameters can be found in Tab. I. The time-dependent expectation of an arbitrary, time-independent operator ˆ O in the rotating frame is (cid:104) ˆ O (cid:105) ( t ) = Tr( U † ( t ) ˆ ρ U ( t ) ˆ O ) = (cid:88) αβ ( U † ( t ) ˆ ρ U ( t )) αβ O βα . (25)If the response is evaluated at the probe frequency ω p , onlyterms proportional to ρ , ρ ∼ e ± i ω p t contribute to Eq. (25).This yields the linear response of the generic operator ˆ O , ∂ (cid:104) ˆ O (cid:105) ω p ∂ Ω p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω p = = O (cid:18) i γ − ν + Ω d η + i γ (cid:19) Ω d − ( i γ − ν )(6 i γ − η + ν )) (3 ρ − . (26)For the specific choice of ˆ O = | (cid:105)(cid:104) | + H.c., i.e., measuring theoperator to which the probe field is coupled, ∂ (cid:104) ˆ O (cid:105) ω p ∂ Ω p ≡ χ O ( ω p )is the susceptibility.The real and imaginary part χ (cid:48) ( ω d ) = Re χ ( ω d ) and χ (cid:48)(cid:48) ( ω d ) ≡ Im χ ( ω d ) of the susceptibility are shown in Fig. 4for a suitable set of parameters. They display a pronouncedFano resonance, i.e., a spectral hole, whose strength dependson the dissipation rate γ and the Rabi frequency of the drive Ω d . For weak driving Ω d →
0, the signal reduces to the ex-pected Lorentzian χ (cid:48)(cid:48) ( ω p ) = γ γ + ω p − E ) , with a peak ∼ γ at ω p = E . C. Occurrence and strength of the Fano resonance
The analytical form of the susceptibility χ ( ω d ) in Eq. (26)appears rather complicated. Especially when the contributionof several Λ -type schemes to the susceptibility is expected,the total signal is hard to estimate from the form of each in-dividual χ ( ω d ) in Eq. (26). In order to make the Fano signal parameter weak γ strong γ F γ + (cid:18) η Ω d (cid:19) γ (cid:16) Ω d γ (cid:17) w η (cid:18) + Ω d η (cid:19) γ η ν p η (cid:16) Ω d η (cid:17) γ η Table I: Signal strength F , width w and position ν p of a Fano reso-nance in the Λ -scheme for weak and strong dissipation γ . Comparewith Fig. 4 for an illustration of the parameters. more theoretically accessible, we perform a Taylor expansionof the imaginary part χ (cid:48)(cid:48) ( ω d ) for strong and weak dissipationi.e., for γ (cid:29) Ω d , η and γ (cid:28) Ω d , η .Experimentally and theoretically well accessible parame-ters, which characterize a Fano resonance are illustrated inFig. 4 and consist of the strength of the resonance F , its spec-tral width w and its spectral peak position ν p . Their corre-sponding values obtained from Eq. (26) can be found in Ta-ble I. In general, the Fano resonance is most significant forsmall dissipation rates γ and small ratios η Ω d (cid:28)
1. The widthand position of the signal can be adjusted by tuning η .Transferring the present analysis to the dimer and trimerlevel schemes shows that hole burning, i.e., Fano reso-nances in the magnetic susceptibility, can be observed inLiHo x Y − x F already on the basis of magnetic dimers andtrimers. One crucial requirement for its observation, however,are su ffi ciently small dissipation rates γ , which are of the orderof the detunings ν, η and the Rabi frequency Ω d . According toEq. (15) this can be achieved if the lattice degrees of freedomhave a vanishing linewidth Γ →
0, i.e., in the limit of stronglyisolated systems, which is in accordance with experimentalfindings . IV. HOLE BURNING FOR MANY MOMENTS
The mechanism which leads to the emergence of a Fanoresonance in driven LiHo x Y − x F can be understood in termsof a Λ -scheme in a Ho + dimer configuration. A single dimerrepresents, however, an idealized setup, which completely ne-glects the many-body aspect of the magnetic moments in aLiHo x Y − x F sample. In this section, we aim to generalizethe previous findings to many interacting moments. Devis-ing a phenomenological, e ff ective spin- Hamiltonian, whichmodels the low-energy Hilbert space of LiHo x Y − x F includ-ing quantum corrections, we show that the driven system fea-tures an extensive number of many-body Λ -schemes, whichcan display Fano resonances under the above outlined condi-tions. Based on this finding, we argue that hole burning sur-vives also in the realistic, many-body setting and that many-body Λ -schemes are in fact required in order observe Fanoresonances at experimentally relevant conditions . A. E ff ective spin- quantum dipole Hamiltonian At low energies, the dynamics of the magnetic degrees offreedom in LiHo x Y − x F is dominated by collective magneticmoments rather than by dimer or trimer configurations. In or-der to investigate the driving schemes in the many-body setup,we introduce an e ff ective spin- quantum dipole Hamiltonian,which, on the one hand, is consistent with the results from theexact dimer and trimer analysis above and, on the other hand,recovers the common Ising approximation at large energies.In accordance with Eqs. (3) and (4), we propose a spin- Hamiltonian of the form H = A dip C zz (cid:88) l , m (cid:88) α,β = x , y , z σ α l σ β m g α ( (cid:126) R lm ) L αβ ( (cid:126) R lm ) g β ( (cid:126) R lm ) . (27)Here σ α l is the Pauli matrix α = x , y , z describing the orien-tation of spin l and A dip , C zz and L αβ ( (cid:126) R lm ) are the same as inSec. III. The position dependent and dimensionless g -factors g α ( (cid:126) R lm ) are chosen such that g z ( (cid:126) R lm ) = g x , y ( (cid:126) R lm ) (cid:28) , ∀ (cid:126) R lm . In the limit g x , y ( (cid:126) R lm ) →
0, the Hamiltonian H re-duces to the Ising Hamiltonian in Eq. (4).We introduce non-zero g x , y ( (cid:126) R lm ) to describe deviations ofthe true LiHo x Y − x F system from the Ising approximation.For the dimer configuration, this deviation vanishes at largedistances with ∼ | (cid:126) R lm | − , as discussed in Eq. (7). This sug-gests g x , y ( (cid:126) R lm ) ∼ | (cid:126) R lm | − in a similar fashion. For more thantwo magnetic moments (e.g. in trimer configurations) we find,however, that non-Ising corrections decay much slower in thedistance | (cid:126) R lm | . We attribute this behavior to strong contri-butions to the dipole-dipole interactions from highly excitedcrystal field eigenstates, which were observed in perturba-tion theory for the dimer setup. We expect the non-Isingcorrections therefore to become more pronounced for largerspin clusters and thus chose g x , y ( (cid:126) R lm ) ≡ g x , y independent ofthe distance. A similar e ff ective model has also been pro-posed to explain temperature dependence of the specific heatof LiHo x Y − x F .For the choice g x = . , g y = .
07, the eigenvalues of H match well with the behavior of the lowest order eigenvaluesof the full Hamiltonian H for n = , , the g -factors here are not isotropic. This isa necessary requirement in order to obtain the observed levelsplittings and a Λ -type driving scheme. This is consistent withthe anisotropy of the crystal field Hamiltonian .In order to simulate a realistic subsystem of LiHo x Y − x F with x ≤ .
05, we consider a three-dimensional volume of N = × × n = × N =
300 potential Ho + positions. Thiscorresponds to a dilution of x = .
04. The spins experiencedipole-dipole interactions, which are described by the Hamil-tonian H in Eq. (27). The eigenvalues and eigenstates of themany-body spin Hamiltonian are obtained via exact diagonal-ization.The sequence of eigenvalues can be understood in a similarway as for the trimer scheme. Consider the eigenvalues λ l of H with l = , ..., and sorted in ascending order, i.e.
600 610 620 630 640 650 660 670 680 690 7000123456 10 -4
607 608 609 610 611 612 613 614 615 616 61700.511.5 10 -4 eigenstate index l ⇥ eigenstate index l
610 613 616 . ⇥ | µ m,l | m = 200 m = 199 | µ m,l | | i | i | i| i | i | i | i (a) (b) | i | i | i | i | i | i| i -6 -5 -4 -3 -2 -1 RWA regime strongly coupled | µ ,l || µ ,l || µ ,l | ⇠ l | µ m,l | eigenstate index l (ordered) (c) -4 -2 l ⇠ l ✏ l ⇠ ✏ l ! p,d l ⇠ l eigenvalue index l Hz | ! p ! d | (d) Figure 5: The energy spectrum and the eigenstates of the e ff ectiveHamiltonian H in Eq. (27) confirm the observation from the dimerand trimer configurations: Due to the weakly broken Ising symmetryin H its eigenstates come in pairs, each consisting of two quasi-degenerate states with relative level splitting (cid:15) , and separated fromother pairs by an ”Ising” energy ∆ . For eigenvalues λ l of H in as-cending order, we define the quasi-degeneracies (cid:15) l ≡ λ l − λ l − andthe ”Ising” energies ∆ l = λ l + − λ l . These are shown in panel (d).For a cluster of n =
12 random magnetic moments, corresponding toa small LiHo x Y − x F crystal with x = .
04, one finds two well sepa-rated energy bands of quasi-degeneracies and Ising energies. The en-ergies of the Ising band and the quasi-degenerate band of the clustercorrespond well with the drive, probe frequency ω p , d ≈ π × δω = ω p − ω d ∼ . (a + b) The transition matrix elements µ m , l = (cid:104) m | J z tot | l (cid:105) between di ff erent eigenstates | l (cid:105) , | m (cid:105) establish a setof Λ -schemes, similar to the dimer and trimer configurations. In (a)this is shown for the absolute values of µ m , l for the quasi-degeneratepair m = ,
200 and for 500 < l < Λ -schemestructure is illustrated in (b) for the states in the inset. Thick arrowscorrespond to large matrix elements and thin arrows to small matrixelements. The whole set of coupling matrix elements for a giveneigenstate m = , , n =
12 spin cluster is shownin (c), where the | µ m , l | are sorted in descending order. Matrix ele-ments | µ l , m | < − correspond to Rabi frequencies Ω d = O (kHz)in the presence of a h d = . ? . For drive and probefrequencies ω d , p ≈ π × Ω d , p ≤ ω d , p and allows us to treat the response of the system inthe rotating wave approximation (RWA). The minority of stronglycoupled transitions has no observable influence on the dynamics. Inthis framework, hole burning, i.e., a Fano resonance in the linearmagnetic susceptibility, is observable when the detuning δω equalsthe energy (cid:15) l of one (or several) quasi-degeneracies. This leads todestructive interference between two di ff erent pathways in the Λ -scheme and to a Fano resonance as in Fig. 4. λ l + > λ l for all l . We define the ’Ising’ level spacings ∆ j and the ’quantum’ level spacings (cid:15) j according to (cid:15) j = λ j − λ j − , for j = , ... , (28) ∆ j = λ j + − λ j , for j = , ... − . (29)The level spacings ∆ j , (cid:15) j are both positive for all j and thevalues of the (cid:15) j ’s are a measure for the deviation of H from H Ising . For g x , y → (cid:15) j → H approaches H Ising . A characteristic distribution of { λ j , ∆ j , (cid:15) j } for a systemof n =
12 spins is shown in Fig. 5 (d).The importance of the level spacings (cid:15) j becomes apparentwhen the many-spin system is driven with a time dependentmagnetic field in the z -direction. As in the previous sec-tion, this is formally described by adding a Hamiltonian H → H ( t ) = H + δ H ( t ) with δ H ( t ) = h d cos( ω d t ) C zz g L µ B µ S z tot and S z tot = (cid:80) l = σ zl . The transition matrix elements between twoeigenstates | l (cid:105) , | m (cid:105) of the Hamiltonian H are µ l , m = (cid:104) l | S z tot | m (cid:105) . (30)As in the previous dimer and trimer configurations,[ S z tot , H Ising ] = S z tot , H ] (cid:44)
0, and thus the transitionmatrix can be chosen diagonal in the Ising basis but will benon-diagonal in the basis of H .Figure 5 (a) shows the matrix elements µ m , l for fixed m =
199 (red line) and m =
200 (grey line). One observes a gener-alization of the dimer and trimer drive schemes to the many-spin system. Both states m = ,
200 act as the base stateof a whole set of inverse Λ -schemes, which couple to pairsof states l = j − , j . The levels of each pair are sepa-rated by a ’quantum’ level spacing (cid:15) j (see Fig. 5 (b) for anillustration). Each Λ -scheme consists of one strong and oneweak transition matrix element, i.e. one generally finds either | µ , l | (cid:28) | µ , l + | or | µ , l | (cid:29) | µ , l + | . This order is ex-changed when going from m to m +
1, as can be seen from theinset of Fig. 5 (a) and the arrows in Fig. 5 (b). The pairs, inturn, are separated from the base state m = ,
200 by one orseveral Ising level spacings ∆ m .The complete set of transition matrix elements µ m , l for a se-lection of fixed m is plotted in Fig. 5 (c) in descending order.One observes a small number of about 10 matrix elements foreach m , which are ∼ O (1). For l ≥
10 one observes a signif-icant drop in the magnitude of | µ m , l | , which is followed by adecay ∼ l − . While for each m the states corresponding to agiven l are di ff erent (due to individual ordering) the magnitudeand decay of the matrix elements is very similar.Large matrix elements | µ m , l | = O (1) result from overlapsof nearly Ising or Z -reversed partners. Consider thereforea state α of the particular form | ⇑ α (cid:105) ≡ | ↑↑↓↑ ... (cid:105) and its Z -reversed partner | ⇓ α (cid:105) ≡ | ↓↓↑↓ ... (cid:105) , where α is the labelthat indicates which spins are pointing up and which ones arepointing down. Due to the smallness of g x , y , many eigenstateswill be of the form | ψ α ±(cid:105) = |⇑ α (cid:105)±|⇓ α (cid:105)√ + ... , where ... indicatesperturbative corrections due to non-zero g x , y . The largest tran-sition matrix elements result from overlaps (cid:104) ψ α + | S z tot | ψ α −(cid:105) = n ↑ − n ↓ + ... , which is the di ff erence in the number of up-spinsand down-spins and is O (1). All the remaining matrix ele-ments with | µ m , l | (cid:28) ... and, as wewill see, dominate the dynamics under driving.
10 20 30 40 50 60 70 80 90 1005.6015.6025.6035.6045.6055.6065.6075.608 10 -10 ! ( ! ) [Hz][arb. units] (a)
40 60 80 H d = 2MHz H d = 2 . H d = 3MHz -40 -30 -20 -10 0 10 20 30 40-3-2.5-2-1.5-1-0.500.5 10 -14 (b)(d) ( ! ) ! [mHz] ! [mHz] [arb. units] -50 -40 -30 -20 -10 0 10 20 30 403.85597398653.8559739873.85597398753.8559739883.85597398853.8559739893.85597398953.85597399 10 -5 ( ! ) [arb. units] -50 -40 -30 -20 -10 0 10 20 30 40 50-2.5-2-1.5-1-0.500.511.5 10 -14 -40 -30 -20 -10 0 10 20 30 40 502.2152.222.2252.232.2352.242.2452.252.255 10 -12 (c)(e) ( ! ) ! [mHz] ! [mHz] [arb. units] H d = 1 . H d = 2MHz H d = 2 . H d = 3MHz = 4mHz = 8mHz = 12mHz = 16mHz ( ! ) [arb. units] = 1Hz = 4mHz H d = 3MHz Figure 6: For suitable driving conditions, the combination of quasi-degenerate and Ising energy levels with the Λ -schemes in µ l , m causeobservable Fano resonances in the linear magnetic susceptibility χ ( ω ). The plots (a-e) show Fano resonances in the real part χ (cid:48) and theimaginary part χ (cid:48)(cid:48) of the susceptibility for di ff erent dissipation rates γ . The signal is obtained from n =
12 magnetic moments, whichare described by H and driven by a magnetic field with frequency ω d = π × H d . χ is probed at frequency ω p = δω + ω d . The drive strength H d = h d ≈ . γ andthe detuning δω and, in addition, the Rabi frequency Ω d for the tran-sition needs to be su ffi ciently large, | Ω d | (cid:29) δω , to cause interference.If γ or H d are changed considerably a given resonance vanishes andthe signal becomes flat until another resonance becomes accessible. B. Driving the many-spin system
In this section, we discuss the response of the many-spinsystem with its multiple Λ -schemes to external driving.The driving regime of interest is the one discussed inRefs. , where a clear Fano resonance has been observed.The setup consists of a LiHo x Y − x F sample, which is drivenby two di ff erent, time-dependent magnetic fields, a drivingfield ∼ h d cos( ω d t ) and a probe field ∼ h p cos( ω p t ) with smallamplitude h p (cid:28) h d . Typical experimental values for the driveand the probe frequency are ω d , p ≈ π × ≈ . ff erence δω = | ω p − ω d | ≤ π × h d ≈ . h p = . Ω l , m for transitions between spin eigenstates | l (cid:105) ↔ | m (cid:105) are Ω d , pl , m = h d , p g L C zz µ B µ (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ≡ H d , p µ l , m , (31)which amounts to Ω dl , m ≈ µ l , m × . Ω pl , m = Ω dl , m . Wedefined the e ff ective driving, probing field H d , p for brevity.We distinguish two di ff erent regimes for the Rabi frequen-cies Ω d , pl , m and the matrix elements µ l , m : (i) a regime of strongdriving with Ω d , pl , m > ω d , p and (ii) a rotating wave regime(RWA) for Ω d , pl , m < ω d , p . By definition, the conditions for thestrong coupling regime deny the application of the rotatingwave approximation and the corresponding transitions have tobe treated in the Floquet formalism . In the RWA regime,however, the rotating wave approximation is applicable andthe discussion of Sec. III can be generalized to the multi-spinsetup.For the above mentioned parameters , the two regimesare illustrated in Fig. 5 (c). It shows that this particular choiceof driving parameters leads to a clear separation between thestrong driving regime and the RWA regime, which is indicatedby a jump of µ l , m over at least one order of magnitude after es-caping the RWA regime and only a few matrix elements thatexceed slightly the RWA condition. This is further justifica-tion why we can treat the strong coupling and RWA regimeseparately. In Appendix B we show that very strongly driventransitions Ω d (cid:29) ω d will e ff ectively freeze out and need notbe considered. We will thus focus on the RWA regime.In the RWA regime, the analysis of Sec. III can be general-ized almost straightforwardly to the case of many Λ -schemes.One di ff erence between the idealized scheme and the realdriving scheme is, however, that both the drive and the probefield couple to the same transition matrix elements. Thisyields the time-dependent Hamiltonian H ( t ) = H + (cid:16) H d cos( ω d t ) + H p cos( ω p t ) (cid:17) (cid:88) l σ zl . (32)Considering a single Λ -scheme | l (cid:105) ↔ | m (cid:105) ↔ | l + (cid:105) , bothtransitions couple to the combined magnetic field, which givesrise to two meaningful ways of going to a rotating frame. Oneis obtained by performing the rotating transformation as inEq. (21) and yields H RWA = (cid:32) H d + H p e i δω t µ l , m | l (cid:105)(cid:104) m | + H.c. (cid:33) (33) + (cid:32) H p + H d e − i δω t µ l + , m | l + (cid:105)(cid:104) m | + H.c. (cid:33) + ν | l (cid:105)(cid:104) l | + ( ν + η ) | m (cid:105)(cid:104) m | . It still contains slowly varying terms with frequency δω = ω p − ω d . A similar transformation is obtained by exchanging l ↔ l + H RWA butwith H d ↔ H p and ν → δω − ν, η → η + ν − δω . Both Hamil-tonians yield the equivalent time evolution since in both trans-formations only the fast contributions ∼ ω p , ω d and ω p + ω d have been neglected. The ambiguity in choosing the transfor-mation reflects the fact that, when measuring at the frequency ω p , one can either probe the l ↔ m transition (correspondingto Eq. (33)) or the l + ↔ m transition corresponding to thesecond transformation. Per the Λ -scheme, one can thus probetwo di ff erent transitions, which we take into account individ-ually.For the many-body scheme in Fig. 5 (b), the magneticsusceptibility at the probe frequency χ ( ω p ) is given by thesum of all possible transitions, i.e., by the sum over all Λ -schemes with two di ff erent contributions per scheme. At ex-tremely long measurement times ∼ δω the experiment sin-gles out contributions at ω p and discards all other parts. Inlinear response, the dimensionless susceptibility is χ ( ω p ) = ∂ (cid:104) S z tot (cid:105) ω p ∂ H p (cid:12)(cid:12)(cid:12)(cid:12) H p = . For a single Λ -scheme from Fig. 5 (b) of theform | l (cid:105) ↔ | m (cid:105) ↔ | l + (cid:105) it acquires two contributions,one probing the | l (cid:105) ↔ | m (cid:105) -transition and one probing the | l + (cid:105) ↔ | m (cid:105) -transition, which yields˜ χ l , m ( ω p ) = | µ l , m | (cid:18) i γ − ν + H d | µ l + , m | η + i γ (cid:19) H d | µ l + , m | − ( i γ − ν )(6 i γ − η + ν )) (34) + | µ l + , m | (cid:18) i γ + ν − δω + H d | µ l , m | η + ν + δω ) + i γ (cid:19) H d | µ l , m | − ( i γ + ν − δω )(6 i γ − η + δω )) . Here, according to the definitions in Sec. III B, ν = λ l + − λ l + ω d − ω p and η = λ l − λ m − ω d . The signal correspond-ing to the smaller transition matrix element is strongly sup-pressed compared to the one corresponding to the larger ma-trix element due to the | µ | prefactor and rarely contributesto the susceptibility. As a consequence, the total susceptibil-ity χ ( ω p ) = (cid:80) l , m ˜ χ l − , m ( ω p ) is very well approximated as thesum of independent Λ -schemes.In order to contribute a Fano resonance to the magnetic re-sponse, a given Λ -scheme has to produce a significant sig-nal strength F at a small signal width w , as shown in Tab. I.For a transition | l (cid:105) ↔ | m (cid:105) ↔ | l + (cid:105) to contribute this re-quires a near resonant detuning from the drive frequency η = ω d − | λ l − λ m | = O (mHz) and in addition a transitionmatrix element µ l , m in the RWA regime and a detuning of theprobe frequency δ = (cid:15) l / − | ω d − ω p | = O (mHz).In Fig. 6, we show the magnetic susceptibility χ of a systemof n =
12 magnetic moments, which is described by H withrealistic parameters for LiHo x Y − x F . It is strongly driven byan external drive field H d = . − h d ≈ . − . δω = ω p − ω d whose emergence and visibility depend on thedissipation rate γ , the drive field strength H d and the drive fre-quency ω d , as predicted by the Λ -scheme analysis in Sec. III.The observed resonances correspond to Λ -schemes for whichthe detuning η , the Rabi frequency H d µ l , m and the dissipationare roughly of the same order of magnitude (mHz for Fig. 6(b-e) and Hz for Fig 6(a)).1
18 20 22 24 26 28 30 3210 -10 -5
15 25 l
20 40 60 80 100 12010 -12 -10 -8 -6 -4 -2 -15 -10 -5 0 5 10-2-1.5-1-0.500.511.5 10 -8 ( ! ) ( ! )10 arb.units ! [Hz] E [Hz] l
40 80 1200110 ✏ l l ( a )( b )( c ) ( d ) l +1 l [Hz] A J C zz = 215mK ~R = (1 , , ~ R = ( , , ) ~R = (0 , , = 4Hz ! d = 120kHz H d = 0 . | i | i | µ m,l | l | i Figure 7: The hole burning phenomenology persists after includinghyperfine interactions between the magnetic moments and the Ho nu-clear spins. Including the nuclear spin degrees of freedom in a mag-netic LiHo x Y − x F dimer configuration with relative orientation (cid:126) R confirms the established picture of quasi-degenerate pairs of eigen-states of the Hamiltonian H (2)full in Eq. (37) and Λ -schemes in theirtransition matrix elements. The alternating level structure of Ising-type level di ff erences ∆ (even l ) and quasi-degeneracies (cid:15) (odd l ) canbe found throughout the entire spectrum of H (2)full and is illustrated in(a) for the 30 lowest energy eigenstates. This leads to energeticallywell separated bands of level di ff erences ∆ l = λ l + − λ l and quasi-degeneracies (cid:15) l = λ l − λ l − , shown in (b) for di ff erent dimer orienta-tions (cid:126) R , as it was observed for magnetic clusters without hyperfineinteractions. The transition matrix elements µ m , l = (cid:104) m | J z + J z | l (cid:105) be-tween di ff erent dimer eigenstates | m , l (cid:105) reveal multiple Λ -schemes.For the quasi-degenerate pair m = ,
10 and 15 < l <
30, this isdemonstrated in (c), where the | (cid:105) ↔ | (cid:105) ↔ | (cid:105) transition is high-lighted in the inset. This combination of level spacings and transitionmatrix elements again enables Fano resonances in the linear suscep-tibility, which is demonstrated in (d) for a driven LiHo x Y − x F dimerfor a specific set of drive parameters. C. The e ff ect of hyperfine interactions The spin − toy model in Eq. (27) predicts the observa-tion of hole burning at quantitatively correct energy scales inLiHo x Y − x F under experimentally realistic conditions. Theshape of H in Eq. (27) is motivated by the microscopicHamiltonian in Eq. (1), which predicted hole burning by the same mechanism as for H but only for very di ff erent energyscales, which correspond to the flipping of a single magneticmoment. Throughout this discussion, we have completely ne-glected the hyperfine interaction of the electron magnetic mo-ment (cid:126) J with the nuclear moments (cid:126) I of the Ho atoms. Thee ff ect of hyperfine interactions in LiHo x Y − x F has been ad-dressed by several papers and, in accordance with theirfindings, we argue that the hyperfine interactions do not mod-ify our hole burning phenomenology for su ffi ciently smalltransverse magnetic fields.The microscopic hyperfine interaction is described by theHamiltonian H hf = A J (cid:88) l (cid:34) I zl J zl + (cid:16) I + l J − l + I − l J + l (cid:17)(cid:35) (35)with A J =
39 mK and a nuclear spin I = . The longitudinalpart ∼ A J J zl I zl splits each electronic angular momentum stateinto a multiplet with eight nuclear spin states m J = − , ... .In the Ising approximation, the hyperfine interaction reducesto H hf-Ising = A J C zz (cid:88) l σ zl J zl (36)and each state is separated from its adjacent states m J ± ∼ A J C zz =
215 mK.Both H hf and H hf-Ising are invariant under ( J zl , I zl ) → ( − J zl , − I zl ) or ( σ zl , I zl ) → ( − σ zl , − I zl ), respectively, and thusrespect the Ising symmetry of the ground state manifold ofthe crystal field Hamiltonian. The leading order correctionsto the Ising approximation are thus again arising from thedipole-dipole interactions between electronic magnetic mo-ments of the form ∼ J xl J zm (or ∼ σ xl σ zm in H ). Comparedto the situation without nuclear moments one, however, ex-pects the hyperfine interactions to further suppress the corre-sponding quantum corrections. Qualitatively, this is due tothe energy cost associated with changing the orientation ofthe electronic spin by applying J xl to the electronic magneticmoment while at the same time leaving the nuclear spin ori-entation unchanged.At low temperatures T = O (0 . ∼ . ∼ J xl J zm again induce transi-tions only inside the ground state manifold of the crystal fieldHamiltonian | ↑(cid:105) l ↔ | ↓(cid:105) , which now experiences an addi-tional energetic suppression given by the di ff erence in the hy-perfine interaction energy ∆ E hf . According to Eq. (36) it isapproximately ∆ E hf ≈ A J C zz | m J | and for a given electronicmagnetic moment l the additional suppression of quantumcorrections may be expected to be proportional to its nuclearspin orientation.As a consequence, the phenomenology of hole burning aris-ing from Ising symmetry breaking dipole-dipole interactions,which lift the degeneracy between Ising-reversed partners andintroduce small but finite transition matrix elements µ αβ = (cid:104) α | (cid:80) l J zl | β (cid:105) would survive, with quantitative corrections, also2in the presence of hyperfine interactions. In order to test thisassumption, we determine the eigenenergies and eigenstatesof a microscopic dimer configuration in LiHo x Y − x F includ-ing hyperfine interactions. Each electronic magnetic moment (cid:126) J , then experiences the crystal field, the nuclear spin of theHo + ion and the mutual magnetic dipole-dipole interaction.This extends the dimer Hamiltonian H (2) in Eq. (3) to H (2)full = H (2) + A J (cid:88) l = (cid:126) J l · (cid:126) I l . (37)We diagonalize this Hamiltonian numerically in the (17 × -dimensional Hilbert space and then inspect the (2 × -dimensional subspace of low energy eigenstates.The results obtained from the diagonalization confirm theabove picture and support our phenomenology of hole burn-ing. As for the dimer and trimer schemes without hyper-fine interactions, each eigenstate of H (2)full comes with a quasi-degenerate partner. In Fig. 7 this is demonstrated for a dimerwith relative orientation (cid:126) R = ( a / , , c /
4) where a , c arethe LiHo x Y − x F lattice constants. Figure 7(a) shows thelevel di ff erences λ l + − λ l for the m =
32 lowest energystates in the dimer, whose alternating pattern reveals the quasi-degeneracies. This represents an extension of the dimer en-ergy levels without nuclear spins shown in Fig. 1 (b,d).Grouping the di ff erences of adjacent energies in the dimerspectrum into quasi-degenerate level splittings (cid:15) l ≡ λ l − λ l − and ”Ising”-splittings ∆ l = λ l + − λ l , each dimer configu-ration now gives rise to a whole band of splittings, shown inFig. 7 (b) for di ff erent configurations (cid:126) R . Compared with-Fig. 1, the hyperfine interactions generally suppress both (cid:15) l and ∆ l . A trend towards stronger suppression for an increas-ing polarization of the nuclear moments, P (2) z = |(cid:104) I z (cid:105)| + |(cid:104) I z (cid:105)| ,in a given set of quasi-degenerate states is observed.As for the dimer setup without nuclear spins, the degener-acy breaking dipole-dipole interactions induce non-zero tran-sition matrix elements µ α,β ≡ (cid:104) α | (cid:80) l J zl | β (cid:105) between di ff erenteigenstates | α, β (cid:105) of H (2)full . In Fig. 7 (c), these are shown forfixed α = ,
10 (two quasi-degenerate partners from the spec-trum) and β = , ...,
30 for the same configuration as in (a).It implies that in the presence of a time dependent externalmagnetic field ∼ (cid:80) l J zl two quasi-degenerate partners buildout several Λ -schemes with alternating strong and weak tran-sitions, very similar to the scheme observed in Fig. 5 (b) forclusters of magnetic moments without hyperfine interactions.In the presence of hyperfine interactions, a single dimerscheme thus already contributes a whole set of many-body Λ -schemes, which can support Fano resonances over a muchlarger frequency range than a dimer scheme without hyper-fine interactions. For example, the magnetic susceptibility χ ( ω ) for the dimer configuration with (cid:126) R = ( a / , , c /
4) isshown in Fig. 7 (d) for a dissipation rate γ = h d = ω d = ω p = ω d + δω , with the use of Eq. (34) and by summing overthe m =
256 low energy eigenstates.In conclusion, the consideration of hyperfine interactionsextends the Λ -scheme of a dimer configuration of Ho + mag-netic moments to several, many-body Λ -schemes, each of which has the potential to establish a Fano resonance in themagnetic susceptibility when driving the system with a strongexternal magnetic field. Although the hyperfine interactionssuppress the quasi-degenerate splittings (cid:15) α and the corre-sponding transition matrix elements µ α,β in the dimer, theirnumerical values are still larger than what is observed in ex-perimental measurements . This indicates that the true,experimentally observed hole burning actually results froman interplay of dipole-dipole interactions between many elec-tronic magnetic moments on the one hand and hyperfine in-teractions on the other hand. The basic phenomenology re-mains the same in the presence of hyperfine interactions, butthey suppress quantum e ff ects, which e ff ectively decreases thenecessary size of magnetic clusters in order to observe Fanoresonances at the millihertz scale. D. Inferring dissipation scales from experimental data
The analysis of the e ff ective Hamiltonian H motivatedthe assumption of isolated, many-body Λ -schemes, for whichEq. (34) is applicable and shows low-energy Fano resonancesin the magnetic susceptibility comparable with the experimen-tally observed amplitude and frequency scales. In order to ob-serve resonances, we had, however, to guess a suitable valuefor the magnetic dissipation rates ad hoc. In this section, wewill fit the prediction for χ ( ω ) from Eq. (34) to experimentallymeasured susceptibilities at varying temperatures. The goodagreement between experimental data and the theoretical fit isin support of our phenomenological theory for hole burningand confirms a linear-in- T growths of the magnetic dissipa-tion, as it is predicted from a phonon bath (c.f. Eq. (15)). Inaddition, the resulting fitting parameters confirm that the res-onances are cause by small quantum corrections to the Isingapproximation of the order a few microhertz.The experimental data was taken from hole burning exper-iments on a LiHo x Y − x F crystal with x = . . The measure-ments were taken at di ff erent temperatures increasing from T = T = h d = . ω d = π × h p = δω = ω p − ω d ∈ [ − , Λ -scheme, whichreduces Eq. (34) to three states. Without loss of generality weset l = , m = χ ( δω ) = − β + α | µ , | (cid:18) i γ − ν + H d | µ , | η + i γ (cid:19) H d | µ , | − ( i γ − ν )(6 i γ − η + ν )) + | µ , | (cid:18) i γ + ν − δω + H d | µ , | η + ν + δω ) + i γ (cid:19) H d | µ , | − ( i γ + ν − δω )(6 i γ − η + δω )) . (38)3 -5 -4 -3 -2 -1 0 1 2 3 4 5-6-4-202468 [arb. units] ( ! )04 ! [mHz] T = 150mK T = 200mK T = 250mK T = 300mK T = 350mK ↵ = (247 , , , , = (5 , . , . , . , . ⌫ = ✏ = 22mHz ⌘ = 21 . µ = 1 . h d µ = 0 . h d = 0 . ⇥ T ( ⌫ + ⌫ ) = ↵ µ ⇣ i ⌫ + h d µ ⌘ +6 i ⌘ h d µ ( i ⌫ )(6 i ⌘ + ⌫ )) + µ ⇣ i + ⌫ + h d µ ⌘ + ⌫ )+6 i ⌘ h d µ ( i + ⌫ )(6 i ⌘ ) + ! d = 2 ⇡ ⇥ h d = 0 . h p = 10mOe Figure 8: Comparing experimental data for the imaginary part of themagnetic susceptibility from a LiHo x Y − x F sample with x = . Λ -scheme in Eq. (34) yieldsvery good agreement. The experimental data is represented by mark-ers (circles, diamonds and squares) and was taken for varying probefield detuning δω = ω p − ω d ∈ π × [ − , T = T = Λ -scheme (without loss of generality l = m = µ , = . × − , µ , = . × − , quasi-degeneracy (cid:15) = E = η = ω d − ∆ = . T -linear dissipation rate γ = . × T . The comparison demon-strates, that the experimentally observed signal is very well explainedalready on the basis of a single Λ -scheme, and with energy levels andtransition matrix elements, which agree well with our predictionsfor small magnetic clusters in LiHo x Y − x F . The linear tempera-ture dependence of the dissipation rate is in agreement with acousticphonons at very small energy di ff erences ∼ ω d . The parameters α and β are added in order to take into ac-count the experimental measurement procedure, in which theasymptotic behavior (at large detunings δω ) of the Fano sig-nal is normalized and isolated from a temperature dependentbackground signal. We model the dissipation rates to increaselinearly with temperature γ = γ T and insert ν = δω − (cid:15) , η = ω d − ∆ . The energies (cid:15), ∆ again correspond to the quasi-degenerate, quantum energy splitting and the Ising level split-ting, respectively. The drive field amplitude H d = . h d = . Λ -scheme. All curves share the same transition matrix el-ements µ , = . × − , µ , = . × − , energy lev-els (cid:15) = ω p − ∆ = . γ = . × T . The parameters α, β display a nonlinear temperature dependence and we find α = (247 , , , , β = (5 , . , . , . , . T = (150 , , , , V. ANTI-HOLE BURNING VIA DRIVEN LATTICEVIBRATIONS
The dissipation experienced by the magnetic momentsin the LiHo x Y − x F samples is not easy to controlexperimentally . The dissipation rate depends not onlyon the density of states and the (thermal) occupation of thephonon modes but is also strongly a ff ected by the system-environment coupling, see Eq. (14). Here we suggest amechanism to manipulate dissipation, which is experiencedby the magnetic degrees of freedom, in a more controllableand purposeful way by energy resolved heating. The basicidea behind this approach is to drive the lattice vibrations,i.e., the phonon modes, in a LiHo x Y − x F crystal monochro-matically with frequency ν d . In the low frequency regime ν d (cid:104) . − ff ectivetemperature T e ff ( E ) = T + ∆ T δ ( E − ν d ), which is peaked atthe drive frequency but otherwise flat and given by the initialtemperature of the sample T .For the magnetic degrees of freedom, this nonequilibriumstate of the lattice translates towards energy dependent dis-sipation rates γ ( E ), which are as well peaked at ν d . Mag-netic transitions at energy E = ν d will therefore experiencemuch stronger dissipation that other transitions at higher orlower energies. In our LiHo x Y − x F level scheme, this allowsone to target the explicit suppression or elimination of those Λ -schemes, which display transitions at ν d . For su ffi cientlystrong phonon driving, the spectral holes at the correspond-ing frequency will disappear completely. The observation ofthis “anti-hole burning” would be strongly supportive of ourtheory and yields a further knob to manipulate the low energyphysics in LiHo x Y − x F samples.The dependence of the Fano signal on the phonon degreesof freedom has been observed in previous experiments . Aswe pointed out, reducing the phonon linewidth via decou-pling the lattice from the environment is crucial for observ-ing Fano resonances. The coupling to the environment, how-ever, is not an easily tunable parameter. Similarly, the de-pendence of the magnetic susceptibility on the temperatureof the sample, which is a measure of the total phonon occu-pation, has been studied and a strong reduction of the Fanoresonances has been observed for increasing temperature (seeFig. 8). Temperature, however, increases the dissipation rateuniformly without frequency resolution.In order to estimate the e ff ect of acoustic driving on thelattice degrees of freedom, we consider a simple toy modelfor phonon modes subject to external driving, which is givenby the Hamiltonian H ph = (cid:88) (cid:126) k c | (cid:126) k | b † (cid:126) k b (cid:126) k + A cos( ν d t )( b (cid:126) k + b † (cid:126) k ) . (39)Assuming linear sound absorption with amplitude A , the co-4herent drive couples linearly to the bosonic phonon creationand annihilation operators b † (cid:126) k , b (cid:126) k and for weak driving A ≤ ν d one can apply the rotating wave approximation, which yields˜ H ph = (cid:88) (cid:126) k ˜ ω (cid:126) k b † (cid:126) k b (cid:126) k + A b (cid:126) k + b † (cid:126) k ) , (40)with ˜ ω (cid:126) k = c | (cid:126) k | − ν d and A = Fu . The force of the drive F = ma is the product of acceleration of the atoms by the sound waves a and their mass m . Realistic values are between a = . − g . Together with the phonon matrix element u = √ m ω and the mass of Ho atoms, one reaches Rabi frequencies of A = − ρ ph of the phonons according to ∂ t ρ ph = i [ ρ ph , ˜ H ph ] + (cid:88) (cid:126) k γ ↓ ,(cid:126) k (cid:32) b (cid:126) k ρ ph b † (cid:126) k − (cid:26) b † (cid:126) k b (cid:126) k , ρ ph (cid:27)(cid:33) + (cid:88) (cid:126) k γ ↑ ,(cid:126) k (cid:32) b † (cid:126) k ρ ph b (cid:126) k − (cid:26) b (cid:126) k b † (cid:126) k , ρ ph (cid:27)(cid:33) . (41)The rates γ ↓ ,(cid:126) k , γ ↑ ,(cid:126) k describe the incoherent annihilation, gener-ation of a phonon at wave vector (cid:126) k and will not be specifiedhere. Their ratio γ ↓ ,(cid:126) k γ ↑ ,(cid:126) k = exp (cid:18) c | (cid:126) k | T (cid:19) , however, fulfills detailedbalance.Solving the Heisenberg equations of motion ∂ t n (cid:126) k ≡ ∂ t Tr (cid:18) b † (cid:126) k b (cid:126) k ρ ph (cid:19) for the stationary state, ∂ t n (cid:126) k ! = n (cid:126) k = γ ↑ ,(cid:126) k γ ↓ ,(cid:126) k − γ ↑ ,(cid:126) k + A ˜ ω (cid:126) k + ( γ ↓ ,(cid:126) k − γ ↑ ,(cid:126) k ) (42) ⇒ n ( E ) = n B ( E ) + A ( E − ν d ) + δγ ( E ) , (43)where we assumed in the second step that the dissipationrates are isotropic and depend only on energy, i.e., δγ ( E ) = γ ↓ ,(cid:126) k − γ ↑ ,(cid:126) k with E = c | (cid:126) k | , and we inserted the Bose-Einstein dis-tribution n B ( E ). An illustration of the nonequilibriium phonondistribution function in the presence of phonon driving is dis-played in Fig. 9(b).Replacing the Bose distribution in Eq. (14) with thenonequilibrium phonon distribution from Eq. (43) and pullingout one factor of n B ( E ) from the second part of the equationyields the nonequilibrium magnetic dissipation rate γ noneq ( E ) = γ ( E ) (cid:32) + ν d T A ( E − ν d ) + δγ ( ν d ) (cid:33) . (44)Here we have used the notation γ ( E ) for the equilibriumdissipation rates without phonon driving and approximated n B ( E ) ≈ T ν d for T (cid:29) ν d in the vicinity of the Lorentzian peak.
100 200 300 400 500 600 700-5-4-3-2-10 10 -13 frequency ⌫ d acoustic phonon drive amplitude A -16 E n ( E ) ⌫ d [arb. units] [Hz] [arb. units] ! [Hz] ( a ) ( b )( c ) A = 0kHz ( ! )
40 60 A = 45kHz A = 64kHz T = 100mK = 2 ⇡ ⇥ H d = 3MHz = 2Hz Figure 9: Manipulating the magnetic dissipation rates in aLiHo x Y − x F crystal yields experimental control over the degree towhich hole burning (or in general quantum e ff ects) can be observed.Changing the temperature populates or depopulates all low energylattice degrees of freedom at once. In contrast, monochromatic driv-ing of the lattice, illustrated in (a), can be used to populate onlyphonon modes in a narrow frequency regime. This is shown inthe nonequilibrium phonon distribution n ( E ) in (b), resulting froma monochromatic drive at frequency ν d = π ×
262 and with variabledrive amplitude A . The drive populates phonon modes around en-ergy E ∼ ν d , placing a Lorentzian with width δγ ( ν d ) (inverse phononlifetime) and height A δγ ( ν d ) on top of the common Bose-Einstein distri-bution, cf. Eq. (43). The additional weight in the phonon distributionincreases the dissipative magnetic transition rates γ ( E ) at energies E ∼ ν d close to the drive frequency and leads to dissipation ratesdescribed by Eq. (44). Λ -schemes with energy di ff erences matching ν d will thus experience much stronger dissipation and their contribu-tion to hole burning is suppressed. We term this phenomenon anti-hole burning. Its manifestation in the magnetic susceptibility for aLiHo x Y − x F sample of n =
12 magnetic moments is shown in (c).The parameters in (c) are taken from Fig. 6 (a) and the system is sub-ject to an additional phonon drive at frequency ν d = π × δω ∼ ff ect on the resonance at δω ≈ δω ∼ ω p = ω d + δω = ν d and is thus strongly influenced by the phonon drive. The controlledmanipulation of magnetic dissipation rates via a monochromatic lat-tice modulations yields an additional playground for nonequilibriumphenomena in LiHo x Y − x F and provides a verification mechanismof the hole burning phenomenology via anti-hole burning. Within this simple model, one finds that driving lattice vi-brations with a frequency ν d and strength A modifies the mag-netic dissipation rate by an additional Lorentzian, peaked at E = ν d and with maximum ∼ A ν d γ ( ν d ) T δγ ( ν d ) and width δγ ( ν d ).In order to account for the modified dissipation rates in themagnetic susceptibility, one has to replace γ in Eq. (34) by γ → γ noneq ( E ), where E = E lm is the energy of the corre-sponding transition | l (cid:105) ↔ | m (cid:105) . While the complete evalua-tion of χ ( ω ) becomes complicated with this substitution andcan only be performed numerically, we can devise a simple5rule of thumb for the modifications due to the phonon drive:since a Fano signal appears only for near resonant transitions E lm = ω p , d , anti-hole burning will be most pronounced at ω p = ν d , i.e. when the phonons are driven close to the probefrequency of the oscillating magnetic field. This behavior isdemonstrated via the numerical evaluation of χ (cid:48) ( δω ) in thepresence of phonon driving in Fig. 9(c).Probing anti-hole burning via acoustically driving latticevibrations should be accessible for most state of the art ex-periments on LiHo x Y − x F and should be able to either con-firm or invalidate our present toy model approach. In the casethat our predictions survive the experimental reality, the addi-tion of acoustic driving represents a rather simple additionalcontrol mechanism for the low energy physics in disorderedLiHo x Y − x F magnets. VI. CONCLUSIONS
In this work, we present a numerical analysis of thelevel structure and magnetic susceptibility of strongly drivenLiHo x Y − x F samples. It is inspired by experiments that ob-served spectral hole burning in the susceptibility as the defin-ing characteristic of the antiglass state .We demonstrate that this spectral hole burning, i.e., Fanoresonances in the magnetic susceptibility in LiHo x Y − x F canbe explained on the basis of small spin clusters ( n = ,
3) andthat it can be seen as a consequence of quantum corrections tothe common Ising approximation. The Fano resonances per-sist also when extending the system to the many-body regime,in our numerical study represented by LiHo x Y − x F samplesof n =
12 spins and a dilution of x = .
04. In the presenceof more and more spins, hole burning is caused by interfer-ence between many-body quantum spin levels and can be ob-served at surprisingly low energies and driving frequencies.The crucial requirement for its observation at low frequenciesis, however, dissipation rates, i.e., phonon lifetimes, which areof the order of the driving frequencies or even smaller. Thisrequires strong isolation of the system from its environmentas also reported in experiments . We also propose an experi-ment that would test our hypothesis through the excitation ofthe phonon degrees of freedom crucial for the observation ofhole burning. Indeed, we expect that exciting phonons at theappropriate frequency provides an accessible mean of controlof hole-burning.The explanation of the Fano resonances, originating fromalmost isolated, many-body Λ -schemes, which are in turncaused by quantum corrections to the classical Ising approx-imation in LiHo x Y − x F without transverse field paves theway further studies on the emergence of quantum e ff ects instrongly diluted and strongly frustrated LiHo x Y − x F samplesand the role these corrections play for the low energy phasediagram, i.e., for a glass or anti-glass phase.Possible further directions include the e ff ect of a transversefield, whose common e ff ect on LiHo x Y − x F samples is to in-troduce or increase quantum e ff ects and to investigate re-finements to the model from hyperfine e ff ects in the presenceof large transverse fields . Acknowledgments
This work was partially supported by the Department ofEnergy under Grant No. de-sc0019166. M. B. acknowl-edges support from the Alexander von Humboldt foundation.T. F. R. acknowledges support from US Department of En-ergy Basic Energy Sciences Award DE-SC0014866. We thankGabriel Aeppli and Markus M¨uller for fruitful discussions.
Appendix A: Crystal field Hamiltonian
The actual form of the crystal field Hamiltonian H cf for agiven electron configuration depends on the symmetries ofthe crystal (space group C h – I / a for LiHo x Y − x F ) and theground state manifold of the ion ( I ). It is commonly ex-pressed in terms of the so-called Stevens operators O α n .The crystal field Hamiltonian is H cf = (cid:88) n ,α B α n O α n (A1)and for LiHo x Y − x F only n = , , , B α n . In terms of the angular momentum operators J ± , J z and the total angular momentum J , the list of relevantStevens operators is O = J z − J , (A2) O = J − J (cid:16) + J z (cid:17) + J z (cid:16) + J z (cid:17) , (A3) O C = (cid:16) J + + J − (cid:17) , (A4) O S = i (cid:16) J + − J − (cid:17) , (A5) O = − J + J (cid:16) J z + (cid:17) − J (cid:16) J z + J z + (cid:17) + J z (cid:16) J z + J z + (cid:17) , (A6) O C = (cid:110)(cid:16) J + + J − (cid:17) , (cid:16) J z − J − (cid:17)(cid:111) , (A7) O S = i (cid:110)(cid:16) J + − J − (cid:17) , (cid:16) J z − J − (cid:17)(cid:111) . (A8)The numerical values for the parameters B α n are taken from in-elastic neutron scattering experiments on LiHoF . The exacteigenstates of H cf for this data indeed show up to numericalprecision a degenerate ground state doublet and a single ex-cited state at ∆ E = . ∆ E = . Appendix B: Strongly two-mode driven two-level systems
In this section we consider a two-mode driven two level sys-tem and show that in the limit of strong drive amplitudes thesystem performs Rabi oscillations with a strongly suppressed,e ff ective Rabi frequency. The two-level system is describedby the Hamiltonian H ( t ) =
12 ( Ω cos( ω t ) + Ω cos(( ω + δ ) t )) σ z + ∆ σ x , (B1)6with a hierarchy of scales Ω > Ω (cid:29) ω (cid:29) ∆ , δ . The commonrotating wave approximation is not applicable since both Rabifrequencies are much larger than any other energy scale.We follow the approach outlined in Ref. andtransform the Hamiltonian into a rotating frame˜ H ( t ) = U † ( t ) H ( t ) U ( t ) − iU † ( t ) ∂ t U ( t ) with U ( t ) = exp (cid:16) − i σ z (cid:104) Ω ω sin( ω t ) + Ω ω + δ ) sin(( ω + δ ) t ) (cid:105)(cid:17) . This yields˜ H ( t ) = ∆ (cid:34) exp (cid:32) i Ω sin( ω t ) ω + i Ω sin(( ω + δ ) t ) ω + δ (cid:33) σ + + H.c. (cid:35) . (B2)The Jacobi-Anger expansion e iz sin θ = (cid:80) ∞ n = −∞ J n ( z ) e in θ , withthe Bessel functions of the first kind J n ( z ), of this term yields˜ H ( t ) = ∆ σ + (cid:88) n , m J n − m (cid:32) Ω ω (cid:33) J m (cid:32) Ω ω + δ (cid:33) e i ( n ω + m δ ) t + H.c. = (cid:88) n , m (cid:32) ∆ m , n e i ( n ω + m δ ) t σ + + H.c. (cid:33) . (B3)For strong driving, the e ff ective ’Rabi-frequencies’ ∆ m , n aremuch smaller than the original frequency | ∆ m , n | (cid:28) | ∆ | since | J n ( x ) | ∼ (cid:112) π x . This enables a rotating wave type approxi-mation in the Floquet frame. The two-dimensional FloquetHamiltonian corresponding to Eq. (B3) is H m , n , m (cid:48) , n (cid:48) = δ m , m (cid:48) δ n , n (cid:48) ( ω n + δ m ) + ∆ m − m (cid:48) , n − n (cid:48) σ + + ( − n − n (cid:48) σ − ) . (B4)Changes in n are strongly suppressed by the large frequency ω compared to the Rabi-frequencies. We therefore only consider n = n (cid:48) =
0. Multiplication with the unitary U x = √ ( σ z + σ x )yields a long-range hopping model in Floquet space H m , , m (cid:48) , = δ m , m (cid:48) δ m + ∆ m − m (cid:48) , σ z . (B5)Translating this model back to our original spin model,the detuning δ ∼ mHz and the Rabi frequency ∆ m − m (cid:48) , ∼ (cid:15)ω d π √ Ω d Ω p ∼ − (cid:15) where (cid:15) ∼ mHz is a quantum level split-ting. This yields incredibly slow Rabi oscillations which donot interfere with the susceptibility at the probe frequency. Appendix C: Lattice induced dissipation in the Born-Markovapproximation
This section provides a short review over the derivation ofphonon induced dissipation rates as shown in Eq. (14), whichwere obtained from tracing out the phonon bath in the so-called Born-Markov approximation. We consider a generalHamiltonian of the form H tot = H mag + H D + H mag-ph . Here, H mag = (cid:80) α E α | α (cid:105)(cid:104) α | is the Hamiltonian for the magnetic de-grees of freedom, e.g., from Eq. (3), expressed in its eigen-basis and H D is the Debye-phonon Hamiltonian (12). Thephonon and magnetic degrees of freedom are coupled via H mag-ph as shown in Eq. (13). The time evolution of the to-tal density matrix ρ tot , which describes the coupled system ofmagnetic and phonon modes is given by the von Neumannequation ∂ t ρ tot ( t ) = i [ ρ tot ( t ) , H tot ] . (C1)It is common to switch to a Dirac representation of the densitymatrix, ˜ ρ tot ( t ) ≡ e − it ( H D + H mag ) ρ tot ( t ) e it ( H D + H mag ) and the phonon-magnet coupling ˜ H mag-ph ( t ) ≡ e it ( H D + H mag ) H mag-ph e − it ( H D + H mag ) .This yields the equation of motion ∂ t ˜ ρ tot ( t ) = i [ ˜ ρ tot ( t ) , ˜ H mag-ph ( t )] . (C2)It is formally solved by˜ ρ tot ( t ) − ρ (0) = (cid:90) t i [ ˜ ρ tot ( t (cid:48) ) , ˜ H mag-ph ( t (cid:48) )] dt (cid:48) , (C3)which we insert into (C2) and find ∂ t ˜ ρ tot ( t ) = i [ ρ tot (0) , ˜ H mag-ph ( t )] − (cid:90) t (cid:104) [ ˜ ρ tot ( t (cid:48) ) , ˜ H mag-ph ( t (cid:48) )] , H mag-ph ( t ) (cid:105) dt (cid:48) . (C4)The density matrix of the magnetic degrees of freedom isobtained from ˜ ρ tot by taking the partial trace over the phonondegrees of freedom, i.e., ˜ ρ mag ( t ) = Tr ph ( ˜ ρ tot( t ) ). Assuming thatthe initial density matrix is a direct product of the magneticand phonon Hilbert spaces and that it commutes with the H D and H mag , one finds the formally exact expression ∂ t ˜ ρ mag ( t ) = − Tr ph (cid:32)(cid:90) t (cid:104) [ ˜ ρ tot ( t (cid:48) ) , ˜ H mag-ph ( t (cid:48) )] , H mag-ph ( t ) (cid:105) dt (cid:48) (cid:33) . (C5)Within the Born-Markov approximation only terms up tosecond order in the magnetic-phonon coupling g are takeninto account and one assumes that the phonon system relaxestowards its equilibrium on time scales much faster than g − ,i.e., the phonon system always remains in its thermal equi-librium state. As a consequence of both approximations thedensity matrix can be written as an instantaneous product,˜ ρ tot ( t (cid:48) ) → ˜ ρ mag ( t ) ⊗ ρ ph (0). 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