Symmetry reduction for tunneling defects due to strong couplings to phonons

Tunneling two-level systems are ubiquitous in amorphous solids, and form a major source of noise in systems such as nano-mechanical oscillators, single electron transistors, and superconducting qubits. Occurance of defect tunneling despite their coupling to phonons is viewed as a hallmark of weak defect-phonon coupling. This is since strong coupling to phonons results in significant phonon dressing and suppresses tunneling in two-level tunneling defects effectively. Here we determine the dynamics of a crystalline tunnelling defect strongly coupled to phonons incorporating the full 3D geometry in our description. We find that inversion symmetric tunnelling is not dressed by phonons whereas other tunnelling pathways are dressed by phonons and, thus, are suppressed by strong defect-phonon coupling. We provide the linear acoustic and dielectric response functions for a crystalline tunnelling defect for strong defect-phonon coupling. This allows direct experimental determination of the defect-phonon coupling. The singling out of inversion-symmetric tunneling states in single tunneling defects is complementary to their dominance of the low energy excitations in strongly disordered solids as a result of inter-defect interactions for large defect concentrations. This suggests that inversion symmetric two-level systems play a unique role in the low energy properties of disordered solids.


I. INTRODUCTION
Quantum tunnelling of substitutional defect ions in alkali halide crystals leads to particular low temperature properties 1 . Due to their misfit in size or shape such defect ions are confined to a potential energy landscape with a few degenerate off-center potential wells typically resulting in a finite electric and elastic moment for the defect. At defect concentrations above 100 ppm one typically faces a complicated many body problem 2,3 of interacting defects. Over a wide range of very large concentrations in the percent regime the system then shows glassy behaviour 4 , i.e. the low temperature universal properties of glasses, as revealed in the nearly linear specific heat and the particular characteristics related to phonon attenuation 5 . However, at concentrations lower than 100 ppm the situation can be described by isolated tunneling systems. Using a simplified two-states model allows to describe the thermal and dielectric properties 2, 6 . In contrast, the acoustic response shows a more complex behaviour reflecting the defect geometry and the according tensor character of the elastic moment 7 .
Relaxation of the defect as well as the dielectric and acoustic losses are caused by the coupling to phonons and are generally described within a spin boson type model assuming weak defect phonon coupling 7 . The defectphonon coupling strength is given by the acoustic (or elastic) moment γ of the defect and it is typically estimated from experimental results on either the acoustic response χ acu ∝ nγ 2 /v 2 , with the speed of sound v and the defect concentration n, or on the relaxation rate Γ ∝ ∆ 3 γ 2 /v 5 with the tunnel coupling ∆ of the defect. Thus, γ cannot be determined independently from ∆, n and v. Most experimental data is acquired in samples with defect concentrations n > ∼ 100 ppm, where many body effects complicate the situation. Accordingly, due to experimental uncertainties it remains unclear whether the weak coupling assumption holds. From a first-principle argumentation one actually expects rather strong defect-phonon couplings 8,9 . Treating the dynamics within a two-state spin phonon model allows to conclude that the dynamics at low tem-peratures is effectively always underdamped irrespective of defect-phonon coupling strength 10,11 due to the super-Ohmic spectrum of phonons. Thus, the question regarding to weak or strong defect-phonon coupling might seem hypocritical. We show, however, that a defect model incorporating the full 3D geometry results in acoustic and dielectric responses which qualitatively differ with the strength of the defect-phonon coupling. In detail, we find that strong defect-phonon coupling reduces the cubic symmetry of the defect effectively to inversion symmetry in the same way as observed for strong interdefect couplings 8,9 . Defect phonon coupling dresses, and thus suppresses, tunnelling except for tunnelling along paths reflecting inversion symmetry, which accordingly dominate at strong coupling. In turn, this provides a tool to differentiate experimentally between weak and strong defect-phonon coupling.
We organize the paper as follows: In the next section we briefly introduce substitutional tunnelling defects in more detail. Then we present our model for a [111] defect and its coupling to phonons. In the following section we introduce the Polaron transformation to decouple the defect from the phonons and then we calculate the dielectric and acoustic response. Finally, we discuss the experimental relevance of our results for [111] defect systems like KCl:Li and KCL:CN and end with a conclusion.

II. SUBSTITUTIONAL TUNNELLING DEFECTS
Typical defect systems are KCl doped with Li, OH or CN molecules. The potential energy landscape in which the defect ion moves is given by the host crystal and therefore reflects its symmetry which for most alkali halide crystals is cubic (e.g. the fcc-structure of potassium chloride). There are only three multi-well potentials which are consistent with this symmetry 12 : twelve wells at the edges of a cube, six wells in the middle of the surfaces and eight wells at the corners of a cube. Regarding polar molecules like CN and OH, the potential wells reflect equivalent molecular orientations whereas corre- At low temperatures thermally activated crossing over potential barriers is inhibited for the defect ions and quantum tunnelling remains which typically leads to a ground state splitting of about 1 Kelvin. Since the number density of such tunnelling states exceeds, even at low concentration, that of small-frequency phonon modes of the host crystal, the impurities govern the lowtemperature properties of the material. The dopants CN and Li in KCl favor 8 defect positions in [111]-directions and the dopant OH 6 positions in [100]-directions. We focus on defects with 8 minima in the [111]-direction (see Fig. 1b for a 2D illustration) and refer to these defect systems as [111]-defects in the following. Gomez et al. 12 have presented simplified impurity models whose energy spectra for the defects agree well with the Schottky peak observed by Pohl and co-workers for various impurity systems at low doping 1 .

III. THE [111] DEFECT
To model a [111] defect we restrict at low temperatures the Hilbert space to the 8 localized impurity positions with corresponding localized states For a lithium impurity, the off-center positions r form a cube of side-length d, whereas for cyanide impurities r indicates the orientation of the cigar-shaped polar molecule. The quantum operators of the defect may be written as direct products of pseudospin operators σ i α , where α = x, y, z label the usual Pauli matrices and α = 0 the identity operator. In terms of quantum states at r i = ±d/2 we have . Thus a lithium or cyanide impurity is described as the product of three two-state variables. We adopt the shorthand notation for its quantum operators where i = 1, 2, 3 label the crystal axes and Greek indices α = 0, x, y, z the pseudospin operators.
A single [111]-impurity exhibits eight identical potential minima with three different paths for tunnelling between them for the defect ion: (i) along the edges of the cube with corresponding tunnel coupling ∆ k , (ii) along a face diagonal with ∆ f and (iii) along a space diagonal with ∆ r . The corresponding Hamiltonian is Typically, edge tunnelling dominates the defect spectrum 12 since the edge length d is the shortest distance between potential minima. The path along a face diagonal is √ 2d and along a space diagonal √ 3d. Since tunnel couplings depend exponentially on the length of their respective tunnelling path 2 , one concludes that ∆ k ≫ ∆ f ≫ ∆ r .
Accordingly, the energy spectrum (shown in Fig. 1a) is expected to contain four almost equidistant energy levels with splitting ∆ k . The upper and lower ones are single quantum states, whereas the middle levels are threefold degenerate. Neglecting face and space diagonal tunnelling completely, the problem factorizes into three twolevel systems (TLS) with energy splitting ∆ k 1,6 . This justifies two-state approximations for thermal and dielectric properties but not for the more complex acoustic response due to the tensor character of the elastic moment 7 .
Strong defect-phonon coupling renormalizes the tunnel couplings by Debye-Waller (or Frank-Condon) factors. As one main result of our work we show that these Debye-Waller factors are different for the different tunnel couplings, and accordingly we observe for the renormalized tunnel couplings ∆ k ≫ ∆ f ≫ ∆ r .
Dipole moment Electic fields couple to the dipole of the defect with spatial unit vector e and cause transitions between adjacent levels as highlighted by the dashed arrows in the spectrum (see Fig. 1a). Elastic moment For a lattice distortion that varies sufficiently slowly in space, the interaction potential is given by the term that is linear in the elastic strain ǫ jl (r), Equation (3) is the lowest-order term of a multipole expansion with the elastic quadrupole operator 2,7 with, for example, Q xy = γA zz0 , the elastic coupling energy γ and the components e j of the spatial unit vector e (see Eq. (2)). Elastic perturbations induce two types of transitions as indicated by the solid arrows in Fig. 1a, namely between ground and second excited and between first and third excited states but also between the degenerate states of the first and between the second excited states as well.

IV. DEFECT-PHONON COUPLING
The elastic strain at the defect position r due to phonons 2,7 is determined as spatial derivative ǫ jl (r) = ∂ xj u l (r) of the displacement amplitude 15 u(r) = k,α iξ kα e ikr q kα , with polarisation vector ξ kα for phonon mode with frequency ω kα , wave vector k and polarisation α for longitudinal and transverse phonon branches and displacement operator q kα of mode k. Within our [111]-defect model the position of the defect ion r becomes a discrete quantum operator. We assume Debye phonons with linear dispersion up to the Debye frequency ω D which corresponds to a wavelength λ D ≫ a with lattice constant a. Since a > d and thus k j d ≪ 1 for all modes, we expand in a series in k j d up to first order only. We insert the resulting expression for the elastic strain ǫ jl (r) into Eq. 3 and find a defect-phonon coupling Hamiltonian with therein defining the defect-phonon coupling constants λ φ (k, α). In order to determine the effect of the phonons onto the dynamics of the defect we follow a standard system-bath approach 10,16 where we need the bath spectral functions: with equal α s and x s = 3 for φ = (s, xy), (s, xz), (s, yz) and α w,3 and x w,3 = 5 for φ = (w, 3) and α w,1 and x w,1 = 5 for φ = (w, 1, x), (w, 1, y) and (w, 1, z). Notice that all cross correlations vanish due to We assumed a Debye spectrum of phonons with linear dispersion up to the Debye frequency ω D with an exponential cut-off function. For the coupling strengths we The coupling strength α s can be related to material properties 7 , i.e.
with elastic moment γ α , speed of sound v α , geometric factors f α ∼ O(1) 7 for modes with polarisation α and mass density ρ of the host crystal. Similar relations could be derived for α w1 and α w3 but are not needed in the following.

The full Hamiltonian becomes
Notice that further contributions, formally similar as W w,3 and W w,1 with similar coupling strengths, i.e. weaker than W s , result when we include higher orders of the multipole expansion for the interaction of defect and lattice distortions 17 .

V. POLARON TRANSFORMATION
The dominant contribution to the coupling between tunnel defect and phonons is W s . In order to allow for strong defect-phonon coupling we employ a Polaron transformation with a subsequent lowest order Born-Markov approximation 11,18 . This treatment results in identical results 11,18 to using the non-interacting blip approximation (NIBA) as introduced by Leggett et al. 16 , and thus is able to treat the strong coupling case.
Defining shift operators with f jl,kα = λ s,jl (k, α)/ω 2 kα allows us to rewrite the Hamiltonian to Then we switch with the Polaron transformation which shifts all oscillator coordinates, to the shifted frame of reference resulting in with the shift of the zero point energy which we can safely ignore in the following.

A. System part
System operators A ijl only involving positions, i.e. i, j, l = 0 or z like A z0z , are unmodified by the transformation T . For the tunnelling operators in the Hamiltonian we find by using f jl = kα f jl,kα that space diagonal tunnelling, which inverts the impurity position, is unmodified. In contrast edge as well as face diagonal tunnelling is modified, for example, and all others accordingly. Thus, edge and face diagonal tunnelling become dressed. We now split the Hamiltonian into a phonon averaged part and a fluctuating part, i.e.
where H S − H ∆ reflects the residual defect-phonon coupling and the tunnelling of the defect is described by with the Debye-Waller factor W = cos f jl cos f jk phon for j = l = k. All cross-correlations vanish, i.e. cos f jl cos f jk phon = cos f jl phon cos f jk phon and we find (14) and W = exp(−(α s /π)ω 2 D ) at zero temperature. Strong coupling relates now to α s ω 2 D ≫ 1 leading to W ≪ 1. This results in a competition between the geometrical suppression of the face and space diagonal tunnelling with the phonon renormalization of edge and face diagonal tunnelling. If the geometrical suppression is stronger, we find the same Hamiltonian as for weak defect-phonon coupling and all results discussed in Ref. 7 will qualitatively prevail although quantitatively modified by the Debye-Waller factors. In contrast, when the phonon renormalization dominates, i.e. ∆ r ≫ ∆ k W , space diagonal (inversion symmetric) tunnelling will dominate with edge tunnelling a small correction and face diagonal tunnelling totally negligible leading to with a ground state energy E 1 = −(∆ r + 3∆ k W )/2, three-fold degenerate first excited state with energy E 2 = −(∆ r − ∆ k W )/2, three-fold degenerate second excited state with energy E 3 = −E 2 and third excited state with E 4 = −E 1 . We refer to ∆ r ≫ ∆ k W as the strong defectphonon coupling case in the following.Neither the dielectric nor the elastic moments are modified by the Polaron transformation. The spectrum (shown in Fig.2) exhibits now two fourfold quasi degenerate states with splitting ∆ r k W 2∆ ∆ r between which only dielectric transitions are allowed. The fourfold quasi degenerate states split into two states, one singlet and one triplet, between which acoustic transition are allowed.

B. Defect-phonon coupling
Compared with the defect-phonon coupling W s the residual coupling H S − H ∆ is suppressed by a Debye-Waller factor and we must discuss it at an equal footing with the subdominant defect-phonon couplings W w,1 and W w,3 . The Polaron transformation shifts the oscillators, leading to W w,1 (q kα ) = W w,1 (q kα + F kα ) and W w,3 (q kα ) = W w,3 (q kα + F kα ). Thus, new terms of the form of, for example, A z00 kα λ w,1,x (k, α)F kα , are generated. Since no phonon operator is involved anymore, the sum can be performed and we observe that all these terms vanish for symmetry reasons. Thus, W w,1 = W w,1 and W w,3 = W w,3 .

VI. DYNAMIC RESPONSE
We are interested in the response of the defect to applied dielectric and acoustic fields and thus must determine the linear response functions χ AB (t) = A(t)B − BA(t) and the influence of the phonons on them. We treat the coupling to phonons in an open quantum systems approach 10 . Therein we do not determine the full time dependent statistical operator R(t) of defect plus phonons but the reduced statistical operator of the defect ρ(t) = Tr B {R(t)} integrating out all phonon degrees of freedom. The time evolution is described by a time evolution superoperator U(t, t 0 )R(t 0 ) = R(t) which obeys the von-Neumann equation ∂ t U(t, t 0 ) = −i[H, U(t, t 0 )] = LU(t, t 0 ) thereby defining the Liouville operator L. In analogy, we can define the Liouvillians L 0 , L S and L W with the defect phonon coupling W = W s + W w,1 + W w,3 and the free Hamiltonian H − W = H 0 = H S + H B and also corresponding time evolution operators. The full time evolution can be expressed in form of a Dyson equation. Integrating out the phonon degrees of freedom results then in the integral equation for the effective time evolution superoperator U eff (t, t 0 ) which fulfils ρ(t) = U eff (t, t 0 )ρ(t 0 ). Herein, U 0 (t, t 0 ) = exp( L 0 (t − t 0 )) is the time evolution superoperator of the isolated defect. In order to obtain a simple representation for the memory kernel M(s, s ′ ) we use RESPET 14,19 which employs a lowest order Born-Markov approximation in the defect-phonon coupling, reasoning that after the Polaron transformation all remaining defect-phonon couplings are weak. This results in where the initial statistical operator is assumed to be factorized, i.e. R(t 0 ) = ρ(t 0 ) ⊗ ρ B (t 0 ) with the initial phonon statistical operator ρ B = exp(−H B /k B T ) and H B the Hamiltonian of the free phonons.
With the Laplace transformation defined as In order to obtain the time dependence we need to Laplace back transform U eff (z). Thereby, we focus solely on the damping rates resulting from M(z) which are given by its imaginary part 10,16,19 . Furthermore, we assume that the memory kernel only weakly influences the poles in U eff (z) and thus analyse M(z) at the poles of the unperturbed system 10,16,19 , i.e. U S (z). The effective time evolution superoperator U eff (t, t 0 ) then allows to determine the correlation functions where we define the action of A on an operator O as AO = 1 2 (AO + OA). The spectrum C ′′ AB (ω) is the imaginary part of the respective Laplace transform continued to the real axis. It is connected to the spectrum χ ′′ AB (ω) of the linear response function χ AB (t) = A(t)B−BA(t) via the fluctuation-dissipation-theorem Absorption or dielectric / acoustic loss are proportional to the spectra χ ′′ AB (ω) whereas change of dielectric constant or speed of sound are proportional to the real part of the susceptibility χ ′ AB (ω) which can be determined using the Kramers-Kronig relation.
is plotted versus temperature for several effective tunnel couplings reflecting cases from weak to strong defectphonon coupling.

A. Dielectric response
Due to the cubic symmetry dielectric responses for electric fields polarized in any direction are identical. We, thus, focus on the x direction and determine the correlation function of the dipole operator p x (neglecting damping at first) depending on the thermal occupations of the involved states. At strong coupling, i.e. ∆ r ≫ ∆ k W , it simplifies to C pxpx (z) = z/(z 2 − ∆ 2 r ) whereas at weak coupling, i.e. ∆ r ≪ ∆ k W , we obtain formally the same function with ∆ r → ∆ k . Since the tunnel couplings are not known, a priori, measuring the dielectric response provides the energy splitting but cannot differentiate between strong or weak defect-phonon coupling. However, for intermediate couplings the situation changes. With ∆ = (∆ r + ∆ k W )/2 and δ = ∆ r − ∆ k W the dielectric response becomes at low frequencieshω ≪ k B T, ∆, δ No significant contribution to the spectrum χ ′′ pxpx (T ) is observed. Fig.3 plots the χ ′ pxpx (T ) versus temperature. At weak defect-phonon coupling with ∆ r ≪ ∆ k W (black full line) one observes the expected tanh behaviour known for two-level systems. With increasing ∆ r (red dashed line) the plateau value at lowest frequencies diminishes but otherwise the qualitative behaviour is unchanged. Once ∆ r dominates at strong but not very strong defect-phonon coupling a hump at about T ≃ 0.2 evolves (dash-dotted blue line). In two-level systems such humps are a sign of relaxational behaviour. The according strong frequency dependence and contributions to the spectrum are, however, missing here 20 . At very strong defect-phonon coupling with ∆ r ≫ ∆ k W (dotted black line) we observe again the same tanh behaviour as at weak coupling.

B. Acoustic response
We focus on the response to the elastic operator Q xy and observe (neglecting damping at first) which exhibits a relaxational contribution (second term on r.h.s in Eq. (20)) and a resonant contribution (first term on r.h.s in Eq. (20)) which is governed for any defectphonon coupling by the edge tunnel coupling ∆ k W . At weak defect-phonon coupling this was observed before 7 . The remarkable result is that for any defect-phonon coupling the tunnel coupling 2∆ k W governs the acoustic response. Thus, at strong defect-phonon coupling, the dielectric response is governed by ∆ r whereas the resonant contribution of the acoustic response is governed by 2∆ k W . Next, we take damping into account and determine the acoustic response for small experimental frequencies hω ≪ k B T . As long as ∆ k W ≫hω, which holds for weak and intermediate but not necessarily at strong defectphonon coupling, we obtain with rates 21 and Bose factor n B (x) = 1/(exp(x/T ) − 1). The relaxational contribution (first and second term on the r.h.s. of Eq.(21)) yields also a corresponding contribution to the spectrum. line) one observes the tanh behaviour of the resonant plus the hump due to the relaxational contribution. The latter is suppressed with increasing frequency. We assumed in Fig.4 The actual rates Γ b/c are functions of 2∆ k W . Accordingly the frequency ω c above which the relaxational contributions vanish, i.e. ω ≫ ω c ≃ Γ b/c , changes with 2∆ k W . The striking difference between weak and strong defect-phonon coupling is that the onset temperature of the resonant response (tanh behaviour) shifts to lower temperatures with increasing coupling. At the same time the relaxational hump at low frequencies increases and its maximum shifts also to lower temperatures.

VII. EXPERIMENTAL RELEVANCE
Strong defect-phonon coupling suppresses edge tunnelling and space diagonal tunnelling dominates, i.e ∆ r ≫ ∆ k W . The dielectric response, however, is qualitatively not modified and thus it cannot be used to estimate the defect-phonon coupling strength. In contrast, the acoustic response shows strikingly different behaviour at weak and strong defect-phonon coupling. Specifically, with increasing defect-phonon coupling the relaxational part with strong frequency dependence becomes increasingly dominant over the resonant part. When comparing dielectric and acoustic response, it turns out that while at weak defect-phonon coupling both are governed by a single energy scale, i.e. ∆ k W , at strong defect-phonon coupling dielectric response is governed by ∆ r but acoustic response by ∆ k W .
Only for KCl doped with Li both, low frequency dielectric as well as acoustic, response measurements are reported at low defect concentrations. Tornow et al. report dielectric experiments at a concentration of 60 ppm 22 and Weiss et al. report for the same concentrations acoustic response 23 . In both cases resonant contributions are observed with roughly the same energy scale, i.e. ∆ ∼ 1.1K. Weiss et al. observed additionally relaxational contributions which points clearly to a scenario for weak defect-phonon coupling. Parameters for Li defects in KCL are for the defect-phonon coupling γ t = 0.04eV 24 and for KCL from table I the speed of sound v t = 1.7km/s and mass density ρ = 1.989g/cm 3 . The Debye temperature ishω D /k B = 230K for KCl 25 . Note that typically the longitudinal speed of sound is larger and thus its contribution in Eq.(10) is negligible due to the dependence on the fifth power. Accordingly, the sum in Eq.(10) over polarizations simplifies to the equal contribution of both transversal modes. The geometry factor 7 f t = 1/5. Thus, the Debye-Waller exponent is α s ω 2 D /π ≃ 7.8 · 10 −2 in line with the experimental observation of weak defect-phonon coupling.
The Debye-Waller exponent for CN defects (parameters from table I) in KCL is α s ω 2 D /π ≃ 1.9 which puts the system in the intermediate regime between strong and weak coupling and we expect mixed behaviour. Unfortunately, only acoustic response measurements at defect concentrations of 45 ppm (or higher) are reported 7,26 . They exhibit clearly relaxational as well as resonant contributions. An analysis in terms of weak defect-phonon coupling successfully described the data. Experimental data for dielectric response in these systems would be highly interesting in order to see whether they exhibit a tunnelling energy scale identical (weak coupling scenario) or different (strong coupling scenario) from the acoustic data.
Even larger defect-phonon couplings are observed in crystals doped with OH impurities which, however, form [100] defects with 6 potential minima. Since it posses two tunnelling path, of which the geometrically shorter path is, however, not inversion symmetric, [100] defects exhibit qualitatively the same physics as outlined in the presented theory. Ludwig et al. 27 observe an anomalous isotope effect when doping KCL and NaCl with OH or OD which they attribute to strong phonon renormalization. The [100] defect exhibits only two tunnel couplings. Ludwig et al. 27 find that both are roughly of the same size which points towards our proposed mechanism renormalizing one tunnel coupling to lower values while the geometrically suppressed tunnel coupling is unrenormalized.

VIII. CONCLUSIONS
We have discussed the dielectric and acoustic response of a [111] tunnelling defect, as for example, KCl doped with Li or CN impurities. We have shown that phonon dressing of tunnelling only suppresses tunnelling along paths which are not inversion symmetric. Since the [111] defect exhibits a geometrically subdominant tunnel path along a space diagonal (and thus an inversion symmetric path), this tunnelling dominates for strong defect phonon coupling when phonon dressing suppresses all other tunnelling paths. Thus, we have shown that assuming strong defect-phonon coupling dielectric and acoustic response are governed by different tunnel couplings in contrast to the weak coupling case where only one tunnel coupling dominates. This results in clear qualitative differences which allow easy experimental verification. Comparing our results with available experimental data we find that Li impurities are only weakly coupled to phonons but CN impurities are stronger coupled putting this case in an intermediate regime. We propose to carefully study dielectric response in these systems to fully characterize the defect-phonon coupling.
Furthermore our results might help to shed light on the microscopic nature of the tunnelling systems in disordered solids responsible for their universal low temperatures properties dominated by inversion symmetric tunnelling states 8,9 .