Enantiomer superpositions from matter-wave interference of chiral molecules
Benjamin A. Stickler, Mira Diekmann, Robert Berger, Daqing Wang
EEnantiomer superpositions from matter-wave interference of chiral molecules
Benjamin A. Stickler,
1, 2, ∗ Mira Diekmann, Robert Berger, and Daqing Wang † Faculty of Physics, University of Duisburg-Essen, 47048 Duisburg, Germany QOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom Fachbereich Chemie, Philipps-Universität Marburg, 35032 Marburg, Germany Experimentalphysik I, Universität Kassel, 34132 Kassel, Germany
Molecular matter-wave interferometry enables novel strategies for manipulating the internal me-chanical motion of complex molecules. Here, we show how chiral molecules can be prepared in aquantum superposition of two enantiomers by far-field matter-wave diffraction and how the resultingtunnelling dynamics can be observed. We determine the impact of ro-vibrational phase averagingand propose a setup for sensing enantiomer-dependent forces, parity-violating weak interactions,and environment-induced superselection of handedness, as suggested to resolve Hund’s paradox.Using ab-initio tunnelling calculations, we identify [4]-helicene derivatives as promising candidatesto implement the proposal with state-of-the-art techniques. This work opens the door for quantumsensing and metrology with chiral molecules.
Introduction—
Controlling the quantum dynamics ofmolecules enables exploiting their mechanical degreesof freedom for metrology [1–3], for quantum informa-tion processing [4–8], and for testing the quantum su-perposition principle [9]. State-of-the-art experimentsrange from cooling diatomic molecules into the deepro-vibrational quantum regime [10–19], to coherentlycontrolling the rotational and vibrational dynamics ofmolecules [20, 21], to centre-of-mass interference of mas-sive molecules [22]. Simultaneously addressing the inter-nal and external quantum dynamics of large moleculesis complicated due to the vast number of interactingdegrees of freedom. Here, we show how single chiralmolecules can be brought into a quantum superpositionof oppositely-handed configurations.Understanding the properties of chiral molecules andtheir interaction with environments is an active interdis-ciplinary field of research [23, 24]. In the absence of parityviolating interactions [25], their vibrational groundstateis a quantum superposition of two oppositely handedconfigurations, coherently tunnelling between the twomirror-images [26], see Fig. 1. Preparing a molecularbeam in a quantum superposition of enantiomers andobserving their coherent tunnelling dynamics will openthe door for interferometric measurements of handedness-dependent interactions, for enantiomeric state manipula-tion schemes [27], for observing the tunnelling shifts dueto parity-violating weak interactions [28–32], and for ob-serving the environment-induced decay of tunnelling, asproposed to resolve Hund’s paradox [26, 33–35].Matter-wave interference is a versatile tool for test-ing the quantum superposition principle with largemolecules [22, 36] and for measuring molecular propertiesin the gas phase [37–39]. In addition, far-field matter-wave interferometry combined with spatial filtering hasbeen proposed for sorting conformers [40] and enan-tiomers [41, 42] from a racemic mixture. In this article, ∗ [email protected] † [email protected] we achieve three important goals: First, we demonstratethat far-field diffraction from a standing-wave gratingtogether with a handedness-dependent phase shift froman optical helicity grating can be used to prepare enan-tiomeric superposition states of individual thermally ro-tating molecules. Second, we show how interferometricsensing and the environmental decay of tunnelling canbe observed. Third, we identify helix-shaped [4]-helicenederivatives as suitable candidates, exhibiting tunnellingfrequencies in a wide range and featuring large opticalrotations. Our work presents a promising platform for in-vestigating chiral molecules, complementing conventionalspectroscopic techniques [3, 23]. Ro-vibrational dynamics—
To illustrate the interfero-metric preparation of enantiomer superpositions, we con-sider a freely rotating chiral molecule of mass M withthree orientational degrees of freedom Ω = ( α, β, γ ) , | L | R Diffraction I n t en s i t y Position Time P opu l a t i on V ( q ) q Tunnelling Interferometric sensing - ℓ ℓ DetectionOptical Gratings ScreenMoleculeSource InteractionCollimator zx v z Figure 1. Proposed setup to generate quantum superpositionsof enantiomers and exploit their tunnelling dynamics for in-terferometric sensing. A racemic beam of chiral molecules isdiffracted from optical gratings and filtered at the detectionscreen. Left inset: During the transit, the molecules contin-uously tunnel between left- and right-handed molecular con-figurations. Middle inset: Adjusting the grating phases pre-pares superpositions of enantiomer states. Right inset: Theresulting tunnelling dynamics can be observed and exploitedin subsequent beam experiments. a r X i v : . [ qu a n t - ph ] F e b e.g. Euler angles in the z - y (cid:48) - z (cid:48)(cid:48) convention. The vibra-tional dynamics are characterized by the coordinate q ,parametrizing the one-dimensional tunnelling path witheffective mass µ and double-well potential V ( q ) . The lat-ter gives rise to tunnelling between oppositely handedmolecular configurations localized at q = ± (cid:96) .The vibrational motion can strongly interact withthe rotations via the centrifugal coupling and theCoriolis effect [43]. The former is due to theconfiguration-dependence of the inertia tensor I(Ω , q ) =R(Ω)I ( q )R T (Ω) , where I ( q ) is the body-fixed inertiatensor and R(Ω) is the rotation matrix specifying theorientation of the molecule. The Coriolis coupling is pro-portional to the product of vibrational velocity ˙ q andangular velocity ω , related to the time-derivative of theorientation by ˙R(Ω) = ω × R(Ω) .In total, the ro-vibrational Lagrangian is of the form L = 12 ω · I(Ω , q ) ω + κ ˙ q ω · n (Ω) + µ q − V ( q ) , (1)where we defined the Coriolis coupling parameter κ andthe body-fixed axis of Coriolis coupling n = n (Ω) . Thefirst term describes the rotational kinetic energy and cen-trifugal coupling, the second term describes the Cori-olis effect, and the final two terms describe the body-fixed elastic motion of the molecule. Depending on themolecule, the tunnelling mass µ , as well as the Corio-lis coupling parameters κ and n can depend on q , seeApp. A.The canonical vibrational and angular momentum co-ordinates follow from the derivatives of (1) with respectto ˙ q and ω , respectively, as p = µ ˙ q + κ ω · n and J = I(Ω , q ) ω + κ ˙ q n . Thus the mechanical deformationcontributes to the total angular momentum along n andmechanical rotations around n contribute to the elas-tic momentum. A straight-forward calculation yields theHamilton function H = 12 (cid:18) J − κµ p n (cid:19) · Λ − (Ω , q ) (cid:18) J − κµ p n (cid:19) + p µ + V ( q ) , (2)with the effective tensor of inertia Λ(Ω , q ) = I(Ω , q ) − κ n ⊗ n /µ . The first term on the right-hand side describesfor the rotational motion, including its centrifugal andCoriolis coupling to the vibrational dynamics Enantiomer tunnelling—
The classical Hamiltonian (2)can be quantized by promoting all phase space variablesto operators, which are denoted by sans-serif characters.The two lowest lying vibration states in the double-wellpotential V ( q ) have tunnelling splitting (cid:126) ∆ . As shownbelow, interferometric preparation works best if ∆ / π ison the Hz to kHz level, while the spacing to the nexthigher vibration states is approximately THz – a sit-uation realistically achievable with [4]-helicene deriva-tives. Finally, typical rotation rates are on the orderof ω rot / π (cid:39) GHz . This large frequency spacing between the two low-est and all higher vibration states constrains the vi-brational dynamics to the left- and right-handed enan-tiomeric states | L (cid:105) and | R (cid:105) , whose even and odd super-positions are the vibrational ground- and first excitedstates, see Fig. 1. There are thus three distinct contri-butions from the rotational Hamiltonian in the first lineof (2): (i) The centrifugal coupling is approximately di-agonal in the enantiomer basis because the tunnellingcontribution is suppressed by |(cid:104) L | R (cid:105)| (cid:28) . (ii) The Cori-olis coupling is negligible in comparison to the centrifu-gal coupling since the molecular rotation rate clearly ex-ceeds the tunnelling splitting, ∆ /ω rot (cid:28) . (iii) Thepurely elastic contribution effectively modifies the tun-nelling splitting from ∆ to ∆ ∗ .We explicitly derive the ro-vibrational Hamiltonian (2)and ∆ ∗ by performing the two-level approximation for ahalf- and full helix in Apps. B and C. The helix modelcan be used to estimate the influence of ro-vibrationalcouplings for helix-shaped [ n ] -helicene molecules.In summary, the Hamiltonian in the vibrational two-level approximation reads H (cid:39) − (cid:126) ∆ ∗ ( | L (cid:105)(cid:104) R | + h . c . )+ H L ⊗ | L (cid:105)(cid:104) L | + H R ⊗ | R (cid:105)(cid:104) R | (3)Here, the free rotational Hamiltonians H L = J · Λ − L J / ,and likewise for R , describes the motion of an asymmetricrotor with inertia tensor Λ L/R = Λ(Ω , ± (cid:96) ) . These tensorshave identical eigenvalues but distinct eigenvectors. Theyare thus related to each other by a rotation of the body-fixed frame.The Hamiltonians H L/R share the same eigen-energies E η , where η = ( jn ) labels the n -th rotation eigenstatewith total angular momentum j . The free rotationalHamiltonian is isotropic and thus its eigenstates andeigenenergies are independent of the space-fixed angu-lar momentum quantum number m . However, the eigen-states | η ; L/R (cid:105) are related by a rotation of the body-fixed frame. The relation between left- and right-handedrotation states is given in App. D for the half- and fullhelix. The rotational eigenstates | η ; L/R (cid:105) at fixed molec-ular configuration must not be confused with the config-uration states | L/R (cid:105) , which cannot be interconverted bya rotation.The rotations can be eliminated adiabatically by per-forming a rotating-wave approximation since the rotationrates significantly exceed the tunnelling splitting. Theresulting Hamiltonian describes combined ro-vibrationaltunnelling between states of identical rotational energybut opposite handedness, H (cid:39) − (cid:126) (cid:88) η ∆ η e iγ η | ηL (cid:105)(cid:104) ηR | + h . c ., (4)where | ηL (cid:105) = | jn ; L (cid:105)| L (cid:105) and likewise for R . Thus eachrotation state η yields a distinct tunnelling frequency,which depends on the overlap of rotation states, ∆ η e iγ η = / - . - . - . - . . . . n=0 n=1 n=2 - . - . - . - . . . . . - . - . - . . . . . I n t e n s i t y | L | R
75 50 25 0 25 50 75
Position on screen ( m ) I n t e n s i t y | L | R P o p u l a t i o n P L P R Time (ms) P o p u l a t i o n P L P R (a) (b)(d) (c)(e) Figure 2. Enantio-selectivity for [4]-helicene derivatives: (a) Numerically computed difference of left- and right-handed moleculepopulations after spatially filtering the zeroth (left), first (middle) and second (right) diffraction orders as a function of thegrating phases φ and χ [prefactors of Eqs. (7) and (8)]. Blue and red refer to an excess of right- and left-handed enantiomers,respectively, indicating that high enantio-selectivity can be achieved in all diffraction orders. (b) Diffraction pattern for ( φ, χ ) = (0 . π, . π ) . Red and blue shaded areas depict the numerically computed diffraction pattern for the | L (cid:105) and | R (cid:105) states. The solid lines denote the position and amplitude of Eq. (10). (c) Evolution of the enantiomer populations P L/R ( t ) fromthe grating to the screen (hatched background) and after spatial filtering (blank background). The evolved states are selectedby filtering the first diffraction order, as indicated by the grey-shaded area in (b). The influence of rotations is calculated withthe full-helix model at a rotational temperature of K. Plots (d) and (e) show the same as (b) and (c) but for selecting thezeroth diffraction order at ( φ, χ ) = (0 . π, . π ) . ∆ ∗ (cid:104) η ; L | η ; R (cid:105) . Physically, this describes that the enan-tiomeric states and the molecular rotations become en-tangled by the free dynamics with the Hamiltonian (3).The eigenstates of (4) with energies ∓ (cid:126) ∆ η are givenby | ψ ησ (cid:105) = ( | ηL (cid:105) − σe − iγ η | ηR (cid:105) ) / √ with σ = ∓ . Theresulting ro-vibrational tunnelling dynamics of the purestates | ηL (cid:105) and | ηR (cid:105) follow as | Ψ ηL (cid:105) = cos(∆ η t ) | ηL (cid:105) + ie − iγ η sin(∆ η t ) | ηR (cid:105) , (5a) | Ψ ηR (cid:105) = cos(∆ η t ) | ηR (cid:105) + ie iγ η sin(∆ η t ) | ηL (cid:105) . (5b)In what follows, we will show how these tunnelling dy-namics can be combined with centre-of-mass molecule in-terferometry to prepare enantiomer superposition states. Matter-wave diffraction—
A beam of molecules freelypropagates with constant forward velocity v z from thesource to the optical grating, traverses the light field,and continues freely to the interference screen at distance d further downstream, see Fig. 1. At the source, themolecular centre-of-mass state is uncorrelated with theinternal degrees of freedom and described by Gaussiansmeared point sources [40]. The ro-vibrational degreesof freedom are in a thermal state of temperature of 4 Kand with the Hamiltonian (4). The centre-of-mass statedisperses during the passage to the grating, while theinternal state does not change.The centre-of-mass and ro-vibrational states becomeentangled through diffraction at two optical gratings.The gratings are each generated by a single laser pulse,which is split into two beams and recombined: thefirst pulse generates a standing-wave grating, imprintingthe spatial phase modulation φ L/Rη ( x ) on the centre-of- mass state. The second pulse realises a helicity grating[41, 42], introducing the enantiomer-specific phase mod-ulation χ L/Rη ( x ) . The combined action of both pulsescan be described by a grating transformation operator,which is diagonal in the transverse centre-of-mass posi-tion as well as in the ro-vibrational states, t = (cid:88) η (cid:0) t Lη ( x ) ⊗ | ηL (cid:105)(cid:104) ηL | + t Rη ( x ) ⊗ | ηR (cid:105)(cid:104) ηR | (cid:1) . (6)The grating transformation operators can be calcu-lated in the eikonal approximation [40, 41, 44]. Formolecules with negligible absorption this yields t Lη ( x ) =exp[ iφ Lη ( x )+ iχ Lη ( x )] , where x must lie within the gratingof width w , and likewise for R . For transverse positions x outside the grating, t Lη ( x ) = 0 , because the molecularbeam is blocked by a collimation aperture, see Fig. 1. Thephase due to the pulsed standing-wave grating, generatedby overlapping two co-propagating laser beams at angle θ with wavelength λ and the same linear polarization,is given by φ Lη ( x ) = (cid:114) π α Lη τ E (cid:126) (cid:104) (cid:16) π sin θ xλ (cid:17)(cid:105) . (7)Here, E is the amplitude of a Gaussian pulse of duration τ , E ( t ) = E exp( − t /τ ) and α Lη is the expectation valueof the polarisability tensor of state | η ; L (cid:105) in direction ofthe laser polarisation. For large and cold molecules, theinfluence of the polarizability anisotropy on interferenceis typically small [44], so that we can replace α Lη by therotationally averaged polarizability α , which is indepen-dent of the molecule handedness [45].The helicity grating is generated by superposing twolaser beams at angle θ but with orthogonal linear polar-izations and electric field strength E hel . This yields theeikonal phase [41] χ Lη ( x ) = (cid:114) π G Lη τ E cos θ (cid:126) c sin (cid:16) π sin θ xλ + ϑ (cid:17) , (8)and similar for R . The transverse spatial offset betweenthe standing-wave and the helicity gratings is quanti-fied by ϑ . Again, we approximate the | η ; L (cid:105) -expectationvalues G Lη of the electric-magnetic dipole polarizabilitypseudotensor [45] by its rotational average G L . Since itdepends on the molecular handedness, G L = − G R , thehelicity grating imprints phases of opposite sign on left-and right-handed states. Thus, the centre-of-mass andro-vibrational states become entangled at the gratings,enabling enantiomer selection further downstream.The diffraction pattern follows from freely propagat-ing the total ro-vibrational and centre-of-mass statefrom the grating to the screen and tracing out the ro-vibrational degrees of freedom. Denoting the combinedro-vibrational and centre-of-mass unitary time evolutionoperators from the source to the grating by U and fromthe grating to the screen by U , the final diffractionpattern for L -enantiomers and the centre-of-mass sourcestate | x (cid:105) reads S L ( x, x ) = 1 Z (cid:88) ησ e − β(cid:15) ησ |(cid:104) ηL | (cid:104) x | U tU | ψ ησ (cid:105)| x (cid:105)| , (9)where Z is the ro-vibrational partition function and sim-ilar for R . The trace over the ro-vibration states isweighted with the Boltzmann factor of energy (cid:15) ησ = E η + σ (cid:126) ∆ η and temperature /k B β . Finally, one mustaverage (9) over an Gaussian ensemble of centre-of-masspoint sources. Enantiomer superpositions—
The resulting enantiomerstate at the screen can be approximated analyticallyin the far field. For this purpose, we consider thatthe centre-of-mass state before the grating is given bythe transverse plane wave | p x = 0 (cid:105) , which then prop-agates with constant forward velocity v z towards thescreen at distance d . Starting out in the ro-vibrationalstate | ψ ησ (cid:105) , the total state at the screen is ( | Ψ ηL (cid:105)| Φ L (cid:105) − σe − iγ η | Ψ ηR (cid:105)| Φ R (cid:105) ) / √ . Here, the ro-vibrational statesare given by Eq. (5) with t = d/v z the time from thegrating to the screen. Similarly, the centre-of-mass states | Φ L/R (cid:105) = U (cm)2 t L/R ( x ) | p x = 0 (cid:105) have been diffracted withthe L - and R -grating transformations (6), respectively,where U (cm)2 describes the free centre-of-mass time evo-lution from the grating to the screen.In the far-field regime, mv z w / (cid:126) d (cid:28) , individualdiffraction orders of the spatial wave functions | Φ L/R (cid:105) are well separated by π (cid:126) d sin θ/M v z λ . Tuning the rela-tive grating offset to ϑ = π/ , the amplitude of the n -th diffraction order is given by c Ln ∝ i n J n (cid:20)(cid:114) π ατ E (cid:126) (cid:18) G L cα E E cos θ (cid:19)(cid:21) , (10)and likewise for R , where J n ( · ) denotes the n -th orderBessel function and we dropped the normalization, seeApp. E. Spatially filtering the n -th diffraction order fromthe full interference pattern would thus prepare the ro-vibrational superposition state c Ln | Ψ ηL (cid:105) − σe − iγ η c Rn | Ψ ηR (cid:105) from the pure initial state | ψ ησ (cid:105) .The amplitudes (10) depend on the molecular hand-edness through the sign of the electric-magnetic dipolecoupling G L = − G R . Thus, the relative weight of theleft- and right-state in the filtered wave packet can beset via the ratio of laser intensities E /E as well asthe distance d between the grating and the screen. Forinstance, tuning the intensity ratio such that c Rn (cid:29) c Ln prepares the state ρ (cid:39) (cid:88) η p η | Ψ ηR (cid:105)(cid:104) Ψ ηR | , (11)with statistical weights p η = (cid:80) σ exp( − βε ησ ) /Z (cid:39) exp( − βE η ) /Z . The spatial filtering thus post-selectsstates of a definite handedness at the grating. The re-sulting interference patterns for a set of experimentallyfeasible parameters are illustrated in Fig. 2(b) and (d),showing remarkable agreement between the exact expres-sion (9) and the analytic approximation (10). Note thatarea of the interferograms from left- and right-handedmolecules are equal, while they distribute differentlyacross the diffraction orders. The enantiomer-excess inFig. 2(a) demonstrates that matter-wave diffraction canbe used for creating enantiomer superpositions.The enantiomeric composition of the molecular beamat a variable distance behind the screen can be mea-sured with state-of-the-art photoionization and velocity-map imaging techniques [46]. In the absence of decoher-ence, the resulting probability of observing the moleculein its L configuration at the distance v z t behind the in-terference grating is given by P R ( t ) = (cid:80) η p η cos (∆ η t ) .It decays with time and approaches / on a timescaledetermined by the ro-vibrational coupling. This is calcu-lated with the full-helix model and shown in Fig. 2(c,e). Interferometric sensing—
Having prepared moleculesin a superposition of enantiomer states, the setup enablesinterferometric sensing and metrology of handedness-dependent external forces. For instance, post-selectingthe R -enantiomer at the detection screen, the state atdistance v z t after the grating is given by Eq. (11). Ifthe molecules interact with the potential V int ( r ⊥ , z ) = V L ( r ⊥ , z ) | L (cid:105)(cid:104) L | + V R ( r ⊥ , z ) | R (cid:105)(cid:104) R | for a short period τ int (cid:28) / ∆ η , where z denotes the centre-of-mass posi-tion of the molecule along the flight direction and r ⊥ isits transverse position, the L - and R -enantiomers accu-mulate the relative phase ϕ (cid:39) (cid:126) (cid:90) ∞−∞ dt (cid:48) [ V R ( r ⊥ , v z t (cid:48) ) − V L ( r ⊥ , v z t (cid:48) )] . (12)The probability of measuring the R configuration afteranother free time evolution over the distance v z t is thenmodified by the presence of the potential according to P R ( t, t ) = 1 − cos ( ϕ/ − P R (2 t )] , as detailed in App. F,where P R (2 t ) is the probability without the phase shift.The modification becomes observable for phase differ-ences on the order of π/ , amounting to energy differ-ences of − Joule and force differences of − N, for τ int (cid:39) µ s and v z (cid:39) m/s.Interferometric sensing with a beam of chiral moleculescan be used to measure enantiomer-dependent forces dueto nearby surfaces, other chiral molecules, or opticalfields with unprecedented accuracy. For instance, let-ting the molecules fly briefly at distance d to a chiralsurface induces the phase difference ϕ (cid:39) (cid:80) i R i [1 / ω i c/d )] τ int / π (cid:126) ε d [47], where R i and ω i are therotatory strengths and frequencies of the electronic tran-sitions. For a separation of d (cid:39) nm and consideringthe dominant electronic transitions of [4]-helicene [48]this yields ϕ (cid:39) π/ . In a similar fashion, this setupcan be exploited for interferometric metrology of chi-ral molecules. For instance, the electric-magnetic dipolecross-polarizability tensor can be measured by illuminat-ing the molecular beam with pulses of chiral light [49],which leads to differential interaction with the L/R enan-tiomers.In addition, imprinting the relative phase ϕ = π per-forms motional decoupling, reversing rotational phase av-eraging of the tunnelling dynamics. Sensing of parity violation—
Detecting parity-violating weak interaction through enantiomer tunnellingwas proposed in Ref. [28]. Here, we calculate how themodified tunnelling dynamics influence the measurementsignal. The parity violating contribution to the molecularvibration state adds H pv = (cid:126) ω pv (cid:88) η ( | ηL (cid:105)(cid:104) ηL | − | ηR (cid:105)(cid:104) ηR | ) , (13)to the Hamiltonian (2) [28, 31]. This modifies the tun-nelling frequency, Ω η = ω +∆ η , and introduces a phaseoffset, replacing Eqs. (5) by | Ψ (cid:48) ηL (cid:105) = (cid:20) cos(Ω η t ) − i ω pv Ω η sin(Ω η t ) (cid:21) | ηL (cid:105) + i ∆ η Ω η sin(Ω η t ) e − iγ η | ηR (cid:105) (14a) | Ψ (cid:48) ηR (cid:105) = (cid:20) cos(Ω η t ) + i ω pv Ω η sin(Ω η t ) (cid:21) | ηR (cid:105) + i ∆ η Ω η sin(Ω η t ) e iγ η | ηL (cid:105) (14b)The resulting probability P (cid:48) R ( t ) , after post-selecting the R -enantiomer, oscillates with reduced amplitude aroundthe shifted mean of / (cid:80) η p η ω / η , see Fig. 3(b).In naturally occurring molecules, parity violation dueto the electro-weak interaction slightly lifts the degen-eracy of the L - and R -enantiomers on the sub-mHz (a) P L P R P o p u l a t i o n (b) (c) Figure 3. (a) Interferometric sensing of enantio-selectiveforces: Imprinting relative phases ϕ ≤ π/ after the screen(vertical dashed line) strongly affects the subsequent tun-nelling dynamics, as indicated by the shaded region. (b)Measuring parity violation: The presence of parity violat-ing interactions shifts the mean values of the probabilitiesfor observing left- and right-handed enantiomers, as indi-cated by the dashed lines. (c) Environmental super-selectionof handed enantiomers: The environment-induced decay ofenantiomer tunnelling can be probed in a dedicated collisionchamber starting right after the screen. The solid line showsthe free tunnelling dynamics (evacuated collision chamber),while dashed and dotted show the influence of N moleculesat pressures × − mbar and × − mbar, respectively. level [25, 28, 50]. While such small splittings are notobservable with this setup, molecules containing heavyelements can reach splittings of up to ∼ Hz [51].Assuming ω pv / π = 10 Hz and Ω η / π (cid:39) Hz, themeans of P (cid:48) L/R ( t ) are separated by (cid:39) , as shownin Fig. 3(b), which can be resolved with state-of-the-artphoto-electron circular dichroism measurements [46]. Environmental decay of tunnelling—
In addition to in-terferometric sensing, the proposed setup also enablesobserving the tunnelling decay under environmental in-teractions, which has been proposed to resolve Hund’sparadox [26, 33]. In the presence of a background gas,the free motion of the molecules is interrupted by randomcollisions with individual gas particles. These scatter-ing processes exert linear and angular momentum kicksto the molecules, thereby localizing their position andorientation [52, 53]. The resulting decay of rotationalcoherences implies decoherence of the enantiomer super-positions, since the rotational and elastic degrees of free-dom are entangled, see Eq. (5). The decoherence rateof enantiomer superpositions can thus be estimated bythe total rate of collisions with gas particles of mass m and density n g , Γ LR (cid:39) n g (cid:104) pσ tot ( p ) (cid:105) /m . Here, the aver-age is taken over the Maxwell-Boltzmann distribution ofgas momenta p at temperature T and σ tot ( p ) is the totalscattering cross section of van der Waals scattering.The total scattering cross section can be calculated inthe eikonal approximation via the optical theorem forhighly-retarded van der Waals scattering with the rota-tionally averaged potential V ( r ) = − C/r , where thevan der Waals constant depends on the gas polarizability α g as C = 23 (cid:126) cα g α/ ε π [54, 55]. A straight-forwardcalculation based on Ref. [52] yields Γ LR (cid:39) n g Γ (cid:18) (cid:19) Γ (cid:18) (cid:19) (cid:114) πk B T m (cid:18) √ mC (cid:126) √ k B T (cid:19) / , (15)adding an exponential decay to the chiral tunnelling dy-namics of the reduced state (11), see App. G.The resulting tunnelling dynamics for various pressuresof N background gas are shown in Fig. 3, where we as-sumed that the region after spatial filtering is homoge-neously filled with the gas. This demonstrates that theenvironmental suppression of chiral tunnelling is observ-able under realistic experimental conditions. Identifying suitable molecules—
Interferometric prepa-ration of enantiomer superposition requires (i) strongchiro-optical response for achieving sufficiently largephase shifts at the interference grating and (ii) ground-state splittings in the range of Hz to kHz, wheretunnelling dynamics can be realistically observed duringthe transit from the screen to the detector. Tunnellingsplitting calculations in this regime are computationallyexpensive because of the large size of the molecules andthe required numerical accuracy. We perform extensivecalculations with density functional theory and the in-stanton approach (see App. H) to identify [4]-helicenederivatives as a promising class of molecules.The optical properties of [4]-helicene render it suitablefor interferometric enantio-separation, given that α =3 . × − Cm / V and G L/R = ± . × − mA s / kg at λ = 1 µ m. The high vibrational frequency ∼ justifies the two-level approximation at 4 K. The elec-tronic barrier for inversion is ∼
50 THz , rendering theinstanton approach [56, 57] appropriate for calculatingthe tunneling splitting. These calculations yield ∆ / π ∼ pHz for [4]-helicene, which is too small for the pro-posed experiment. The tunnelling splitting of helicenescan be significantly increased by chemical substitutions,which only affects mildly their optical properties [58, 59].Calculations for 1H-naphtho[2,1-g]indole (naphthindole)yield ∆ / π ∼ −
200 MHz , while α and G L/R remainon the same order of magnitude. These two compoundsbracket the desired frequency range and thus provide anexcellent starting point for future investigations. We sug-gest replacing the side groups to slightly change the tun-nelling barrier, while using additional substituents or iso-topes in the periphery of the molecule to alter the effec- tive tunneling mass.
Proposed experiment—
The required molecular beamcan be generated in a seeded supersonic jet [60] or froma buffer gas cell [61], yielding ro-vibration temperaturesof a few Kelvin. The laser field strengths required fordiffraction can be achieved by a pulsed Nd:YAG laseroperating at nm, exhibiting a pulse energy of 1 J, apulse duration of 10 ns, and a beam diameter of 0.5 mm.The laser wavelength lies in the transparent region of [4]-helicene derivatives to minimize photon absorption. Inaddition, molecules, which absorbed a photon during thegrating transit, can be filtered out at the screen becauseof the ensuing momentum kick [40]. The beam crossingangle θ = 0 . implies a grating period of . µ m, yield-ing a rather long total interferometer length of m at v z (cid:39) m/s. The resulting interference pattern andenantiomer tunnelling dynamics are illustrated in Figs. 2and 3, see App. I for further details. The enantiomericcomposition of the final beam can be measured with highresolution using photo-electron circular dichroism mea-surement [46] or Coulomb explosion [62]. In both detec-tion methods, the molecules can be ionized using a laserspot of approximate size µ m, which when extendedover the vertical direction of the molecular beam im-plies a longitudinal velocity spread of . m/s. Averagingthe tunnelling signal over this velocity spread shows thatenantiomer tunnelling is observable for molecules withtunnelling frequencies smaller than kHz, thus comple-menting existing spectroscopic techniques [3, 23, 63, 64].This interferometric scheme can reach enantiomeric puri-ties beyond 90% at relatively small helicity-grating phaseshifts on the order of π/ . Conclusion—
In conclusion, we demonstrated that far-field matter-wave diffraction can be used to prepare enan-tiomer superpositions with feasible experimental param-eters. This will open the door to the observation of enan-tiomer tunnelling and its exploitation for sensing andmetrology, including the determination of chiral Casimir-Polder forces and potentially parity-violating effects in-side the molecule. We identify [4]-helicene derivatives assuitable molecular candidates. Extended ab-initio cal-culations will help to further narrow this down to a spe-cific compound. Future work could investigate how inter-ferometric preparation of enantiomer superpositions canbe achieved in near-field interferometers [2] or trappedmolecules, potentially alleviating the requirements on themolecular properties. In addition, the presented setupcan in principle also be used to observe and manipulatethe tunnelling dynamics between distinct conformationsof achiral molecules.
Acknowledgements—
We thank Klaus Hornberger,Melanie Schnell, and Kilian Singer for stimulating dis-cussions and feedback on the manuscript. This work issupported by Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) through CRC 1319. [1] S. Patra, M. Germann, J.-P. Karr, M. Haidar, L. Hilico,V. I. Korobov, F. M. J. Cozijn, K. S. E. Eikema,W. Ubachs, and J. C. J. Koelemeij, Proton-electron massratio from laser spectroscopy of HD + at the part-per-trillion level, Science , 1238 (2020).[2] K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter,and M. Arndt, Colloquium: Quantum interference ofclusters and molecules, Rev. Mod. Phys. , 157 (2012).[3] B. Darquie, C. Stoeffler, A. Shelkovnikov, C. Daussy,A. Amy-Klein, C. Chardonnet, S. Zrig, L. Guy, J. Cras-sous, P. Soulard, et al. , Progress toward the first obser-vation of parity violation in chiral molecules by high-resolution laser spectroscopy, Chirality , 870 (2010).[4] D. DeMille, Quantum computation with trapped polarmolecules, Phys. Rev. Lett. , 067901 (2002).[5] C. M. Tesch and R. de Vivie-Riedle, Quantum compu-tation with vibrationally excited molecules, Phys. Rev.Lett. , 157901 (2002).[6] P. Rabl, D. DeMille, J. M. Doyle, M. D. Lukin, R. J.Schoelkopf, and P. Zoller, Hybrid quantum processors:Molecular ensembles as quantum memory for solid statecircuits, Phys. Rev. Lett. , 033003 (2006).[7] P. Yu, L. W. Cheuk, I. Kozyryev, and J. M. Doyle, Ascalable quantum computing platform using symmetric-top molecules, New J. Phys. , 093049 (2019).[8] V. V. Albert, J. P. Covey, and J. Preskill, Robust encod-ing of a qubit in a molecule, Phys. Rev. X , 031050(2020).[9] M. Arndt and K. Hornberger, Testing the limits ofquantum mechanical superpositions, Nat. Phys. , 271(2014).[10] P. F. Staanum, K. Højbjerre, P. S. Skyt, A. K. Hansen,and M. Drewsen, Rotational laser cooling of vibrationallyand translationally cold molecular ions, Nat. Phys. , 271(2010).[11] T. Schneider, B. Roth, H. Duncker, I. Ernsting, andS. Schiller, All-optical preparation of molecular ions inthe rovibrational ground state, Nat. Phys. , 275 (2010).[12] C.-Y. Lien, C. M. Seck, Y.-W. Lin, J. H. Nguyen, D. A.Tabor, and B. C. Odom, Broadband optical cooling ofmolecular rotors from room temperature to the groundstate, Nat. Commun. , 4783 (2014).[13] J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Stei-necker, and D. DeMille, Magneto-optical trapping of adiatomic molecule, Nature , 286 (2014).[14] S. Truppe, H. J. Williams, M. Hambach, L. Caldwell,N. J. Fitch, E. A. Hinds, B. E. Sauer, and M. R. Tarbutt,Molecules cooled below the Doppler limit, Nat. Phys. ,1173 (2017).[15] L. Anderegg, B. L. Augenbraun, Y. Bao, S. Burchesky,L. W. Cheuk, W. Ketterle, and J. M. Doyle, Laser coolingof optically trapped molecules, Nat. Phys. , 890 (2018).[16] S. Ding, Y. Wu, I. A. Finneran, J. J. Burau, andJ. Ye, Sub-doppler cooling and compressed trapping of yomolecules at µ K temperatures, Phys. Rev. X , 021049(2020).[17] F. Wolf, Y. Wan, J. C. Heip, F. Gebert, C. Shi, andP. O. Schmidt, Non-destructive state detection for quan-tum logic spectroscopy of molecular ions, Nature ,457 (2016).[18] C.-W. Chou, C. Kurz, D. B. Hume, P. N. Plessow, D. R. Leibrandt, and D. Leibfried, Preparation and coherentmanipulation of pure quantum states of a single molecu-lar ion, Nature , 203 (2017).[19] M. Sinhal, Z. Meir, K. Najafian, G. Hegi, andS. Willitsch, Quantum-nondemolition state detection andspectroscopy of single trapped molecules, Science ,1213 (2020).[20] D. Brinks, F. D. Stefani, F. Kulzer, R. Hildner,T. H. Taminiau, Y. Avlasevich, K. Müllen, and N. F.Van Hulst, Visualizing and controlling vibrational wavepackets of single molecules, Nature , 905 (2010).[21] C. P. Koch, M. Lemeshko, and D. Sugny, Quantum con-trol of molecular rotation, Rev. Mod. Phys. , 035005(2019).[22] Y. Y. Fein, A. Shayeghi, L. Mairhofer, F. Kiałka,P. Rieser, P. Geyer, S. Gerlich, and M. Arndt, Quantumsuperposition of molecules beyond 25 kDa, Nat. Phys. , 1242 (2019).[23] M. Quack, J. Stohner, and M. Willeke, High-resolutionspectroscopic studies and theory of parity violation inchiral molecules, Annu. Rev. Phys. Chem. , 741 (2008).[24] Y. Saito and H. Hyuga, Colloquium: Homochirality:Symmetry breaking in systems driven far from equilib-rium, Rev. Mod. Phys. , 603 (2013).[25] M. Quack, Structure and dynamics of chiral molecules,Angew. Chem. Int. Ed. , 571 (1989).[26] P. Mezey, New Developments in Molecular Chirality (Springer Netherlands, 2012).[27] J. A. Cina and R. A. Harris, On the preparation andmeasurement of superpositions of chiral amplitudes, J.Chem. Phys. , 2531 (1994).[28] R. A. Harris and L. Stodolsky, Quantum beats in opticalactivity and weak interactions, Phys. Lett. B , 313(1978).[29] R. Berger, M. Gottselig, M. Quack, and M. Willeke,Parity violation dominates the dynamics of chiralityin dichlorodisulfane, Angew. Chem. Int. Ed. , 4195(2001).[30] M. Quack, How important is parity violation for molec-ular and biomolecular chirality?, Angew. Chem. Int. Ed. , 4618 (2002).[31] A. MacDermott and R. Hegstrom, A proposed experi-ment to measure the parity-violating energy differencebetween enantiomers from the optical rotation of chi-ral ammonia-like “cat” molecules, Chem. Phys. , 55(2004).[32] K. Gaul, M. G. Kozlov, T. A. Isaev, and R. Berger, Chiralmolecules as sensitive probes for direct detection of P -odd cosmic fields, Phys. Rev. Lett. , 123004 (2020).[33] F. Hund, Zur Deutung der Molekelspektren. III.: Be-merkungen über das Schwingungs- und Rotationsspek-trum bei Molekeln mit mehr als zwei Kernen, Z. Phys. , 805 (1927).[34] J. Trost and K. Hornberger, Hund’s paradox and the col-lisional stabilization of chiral molecules, Phys. Rev. Lett. , 023202 (2009).[35] F. T. Ghahramani and A. Shafiee, Emergence of molecu-lar chirality by vibrational Raman scattering, Phys. Rev.A , 032504 (2013).[36] S. Nimmrichter, K. Hornberger, P. Haslinger, andM. Arndt, Testing spontaneous localization theories with matter-wave interferometry, Phys. Rev. A , 043621(2011).[37] S. Eibenberger, X. Cheng, J. P. Cotter, and M. Arndt,Absolute absorption cross sections from photon recoilin a matter-wave interferometer, Phys. Rev. Lett. ,250402 (2014).[38] L. Mairhofer, S. Eibenberger, J. P. Cotter, M. Romirer,A. Shayeghi, and M. Arndt, Quantum-assisted metrologyof neutral vitamins in the gas phase, Angew. Chem. Int.Ed. , 10947 (2017).[39] Y. Y. Fein, A. Shayeghi, L. Mairhofer, F. Kiałka,P. Rieser, P. Geyer, S. Gerlich, and M. Arndt, Quantum-assisted measurement of atomic diamagnetism, Phys.Rev. X , 011014 (2020).[40] C. Brand, B. A. Stickler, C. Knobloch, A. Shayeghi,K. Hornberger, and M. Arndt, Conformer selection bymatter-wave interference, Phys. Rev. Lett. , 173002(2018).[41] R. P. Cameron, S. M. Barnett, and A. M. Yao, Discrim-inatory optical force for chiral molecules, New J. Phys. , 013020 (2014).[42] R. P. Cameron, A. M. Yao, and S. M. Barnett, Diffractiongratings for chiral molecules and their applications, J.Phys. Chem. A , 3472 (2014).[43] P. R. Bunker and P. Jensen, Molecular symmetry andspectroscopy (NRC Research Press, 2006).[44] B. A. Stickler and K. Hornberger, Molecular rotationsin matter-wave interferometry, Phys. Rev. A , 023619(2015).[45] L. D. Barron, Molecular light scattering and optical ac-tivity (Cambridge University Press, 2009).[46] A. Kastner, C. Lux, T. Ring, S. Züllighoven, C. Sarpe,A. Senftleben, and T. Baumert, Enantiomeric excess sen-sitivity to below one percent by using femtosecond pho-toelectron circular dichroism, ChemPhysChem , 1119(2016).[47] P. Barcellona, R. Passante, L. Rizzuto, and S. Y. Buh-mann, Dynamical casimir-polder interaction between achiral molecule and a surface, Phys. Rev. A , 032508(2016).[48] F. Furche, R. Ahlrichs, C. Wachsmann, E. Weber,A. Sobanski, F. Vögtle, and S. Grimme, Circular dichro-ism of helicenes investigated by time-dependent densityfunctional theory, J. Am. Chem. Soc. , 1717 (2000).[49] A. Canaguier-Durand, J. A. Hutchison, C. Genet, andT. W. Ebbesen, Mechanical separation of chiral dipolesby chiral light, New J. Phys. , 123037 (2013).[50] M. Quack, On the measurement of the parity violat-ing energy difference between enantiomers, Chem. Phys.Lett. , 147 (1986).[51] R. Berger and J. Stohner, Parity violation, WIREsComp. Mol. Sci. , e1396 (2019).[52] B. A. Stickler, B. Papendell, and K. Hornberger, Spatio-orientational decoherence of nanoparticles, Phys. Rev. A , 033828 (2016).[53] B. Papendell, B. A. Stickler, and K. Hornberger, Quan-tum angular momentum diffusion of rigid bodies, New J.Phys. , 122001 (2017).[54] S. Y. Buhmann, Dipsersion Forces I - MacroscopicQuantum Electrodynamics and Ground-State Casimir,Casimir-Polder and van der Waals Forces (Springer -Berlin, 2012).[55] D. Craig and T. Thirunamachandran,
Molecular Quan-tum Electrodynamics: An Introduction to Radiation- molecule Interactions (Dover Publications, 1998).[56] J. O. Richardson and S. C. Althorpe, Ring-polymer in-stanton method for calculating tunneling splittings, J.Chem. Phys. , 054109 (2011).[57] N. Sahu, J. O. Richardson, and R. Berger, Instanton cal-culations of tunneling splittings in chiral molecules, J.Comput. Chem. , 210 (2020).[58] R. H. Janke, G. Haufe, E.-U. Würthwein, and J. H.Borkent, Racemization barriers of helicenes: A compu-tational study, J. Am. Chem. Soc. , 6031 (1996).[59] P. Ravat, R. Hinkelmann, D. Steinebrunner, A. Presci-mone, I. Bodoky, and M. Juricek, Configurational stabil-ity of [5]helicenes, Org. Lett. , 3707 (2017).[60] T. E. Wall, Preparation of cold molecules for high-precision measurements, J. Phys. B , 243001 (2016).[61] N. R. Hutzler, H.-I. Lu, and J. M. Doyle, The buffer gasbeam: An intense, cold, and slow source for atoms andmolecules, Chem. Rev. , 4803 (2012).[62] M. Pitzer, M. Kunitski, A. S. Johnson, T. Jahnke,H. Sann, F. Sturm, L. P. H. Schmidt, H. Schmidt-Böcking, R. Dörner, J. Stohner, J. Kiedrowski,M. Reggelin, S. Marquardt, A. Schießer, R. Berger, andM. S. Schöffler, Direct determination of absolute molec-ular stereochemistry in gas phase by coulomb explosionimaging, Science , 1096 (2013).[63] D. Patterson, M. Schnell, and J. M. Doyle, Enantiomer-specific detection of chiral molecules via microwave spec-troscopy, Nature , 475 (2013).[64] S. R. Domingos, A. Cnossen, W. J. Buma, W. R. Browne,B. L. Feringa, and M. Schnell, Cold snapshot of a molec-ular rotary motor captured by high-resolution rotationalspectroscopy, Angew. Chem. Int. Ed. , 11209 (2017).[65] A. R. Edmonds, Angular Momentum in Quantum Me-chanics (Princeton University Press, Princeton, 1996).[66] M. S. Newman, The synthesis of 4,5-dimethylchrysene,J. Am. Chem. Soc. , 2295 (1940).[67] M. S. Newman and W. B. Wheatley, Opti-cal activity of the 4,5-phenanthrene type: 4-(1-methylbenzo[c]phenanthryl)-acetic acid and 1-methylbenzo[c]phenanthrene, J. Am. Chem. Soc. ,1913 (1948).[68] M. S. Newman and D. Lednicer, The synthesis and res-olution of hexahelicene, J. Am. Chem. Soc. , 4765(1956).[69] R. N. Armstrong, H. L. Ammon, and J. N. Darnow,Molecular geometry and conformational stability of 4,5-dimethyl- and 3,4,5,6-tetramethylphenanthrene: a lookat the structural basis of the buttressing effect, J. Am.Chem. Soc. , 2077 (1987).[70] G. Winnewisser and K. Yamada, Millimetre, submillime-tre and infrared spectra of disulphane (hssh) and its iso-topic species, Vib. Spectrosc. , 263 (1991).[71] M. Quack and M. Willeke, Theory of stereomutationdynamics and parity violation in hydrogen thioperoxideisotopomers , , HSO , , H, Helv. Chim. Acta , 1641(2003).[72] M. Gottselig, M. Quack, and M. Willeke, Mode-selectivestereomutation tunneling as compared to parity violationin hydrogen diselenide isotopomers , , H Se , Isr. J.Chem. , 353 (2003).[73] M. Gottselig, M. Quack, J. Stohner, and M. Willeke,Mode-selective stereomutation tunneling and parity vi-olation in HOClH + and H Te isotopomers, Int. J. ofMass Spectrom. , 373 (2004). [74] D. D. Fitts and J. G. Kirkwood, The theoretical opti-cal rotation of phenanthro[3,4-c]phenanthrene, J. Am.Chem. Soc. , 4940 (1955).[75] A. Brown, C. M. Kemp, and S. F. Mason, Electronicabsorption, polarised excitation, and circular dichro-ism spectra of [5]-helicene (dibenzo[c,g]phenanthrene), J.Chem. Soc. A , 751 (1971).[76] A. Brown, C. Kemp, and S. Mason, π -SCF and INDOcalculations and the absorption, polarized excitationand circular dichroism spectra of [4]-helicenes, Molecu-lar Physics , 787 (1971).[77] W. S. Brickell, A. Brown, C. M. Kemp, and S. F. Mason, π -electron absorption and circular dichroism spectra of[6]- and [7]-helicene, J. Chem. Soc. A , 756 (1971).[78] R. H. Martin, The helicenes, Angew. Chem. Int. Ed. ,649 (1974).[79] S. Grimme and S. Peyerimhoff, Theoretical study of thestructures and racemization barriers of [n] helicenes (n=3–6, 8), Chem. Phys. , 411 (1996).[80] V. Buss and K. Kolster, Electronic structure calculationson helicenes. concerning the chirality of helically twistedaromatic systems, Chem. Phys. , 309 (1996).[81] Y. Nakai, T. Mori, and Y. Inoue, Theoretical and experi-mental studies on circular dichroism of carbo[n]helicenes,J. Phys. Chem. A , 7372 (2012).[82] M. Gingras, G. Félix, and R. Peresutti, One hundredyears of helicene chemistry. part 2: stereoselective syn-theses and chiral separations of carbohelicenes, Chem.Soc. Rev. , 1007 (2013).[83] R. Martin and M. Marchant, Thermal racemisation of[6], [7], [8] and [9] helicene, Tetrahedron Lett. , 3707(1972).[84] R. Martin and M. Marchant, Thermal racemisation ofhepta-, octa-, and nonahelicene: Kinetic results, reac-tion path and experimental proofs that the racemisa-tion of hexa- and heptahelicene does not involve an in-tramolecular double diels-alder reaction, Tetrahedron ,347 (1974).[85] H. Lindner, Atomisierungsenergien gespannter kon-jugierter kohlenwasserstoffe i: Razemisierungsenergienvon helicenen, Tetrahedron , 281 (1975).[86] M. S. Newman and M. Wolf, A newsynthesis of benzo(c)phenanthrene: 1,12- dimethylbenzo(c)phenanthrene, J. Am. Chem. Soc. , 3225 (1952).[87] J. Barroso, J. L. Cabellos, S. Pan, F. Murillo, X. Zarate,M. A. Fernandez-Herrera, and G. Merino, Revisiting theracemization mechanism of helicenes, Chem. Commun. , 188 (2018).[88] G. Gamow, Zur Quantentheorie des Atomkernes, Z.Phys. , 204 (1928).[89] R. Berger, Do heavy nuclei see light at the end of thetunnel?, Angew. Chem. Int. Ed. , 398 (2004).[90] S. G. Balasubramani, G. P. Chen, S. Coriani, M. Dieden-hofen, M. S. Frank, Y. J. Franzke, F. Furche, R. Grot-jahn, M. E. Harding, E. Hättig, et al. , Turbomole:Modular program suite for ab initio quantum-chemicaland condensed-matter simulations, J. Chem. Phys. ,184107 (2020).[91] R. D. Amos, Electric and magnetic properties of CO, HF,HCI, and CH F, Chem. Phys. Lett. , 23 (1982).[92] H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby,M. Schütz, P. Celani, W. Györffy, D. Kats, T. Korona,R. Lindh, et al. , Molpro, version 2019.2 (2019).[93] A. Garg, Tunnel splittings for one-dimensional potentialwells revisited, Am. J. Phys. , 430 (2000).[94] S. K. Mohanty, K. D. Rao, and P. K. Gupta, Opticaltrap with spatially varying polarization: application incontrolled orientation of birefringent microscopic parti-cle(s), Appl. Phys. B , 631 (2005).[95] G. Cipparrone, I. Ricardez-Vargas, P. Pagliusi, andC. Provenzano, Polarization gradient: exploring an orig-inal route for optical trapping and manipulation, Opt.Express , 6008 (2010).[96] H. S. Chung, B. S. Zhao, S. H. Lee, S. Hwang, K. Cho,S.-H. Shim, S.-M. Lim, W. K. Kang, and D. S. Chung,Molecular lens applied to benzene and carbon disulfidemolecular beams, J. Chem. Phys. , 8293 (2001).[97] R. Fulton, A. I. Bishop, and P. F. Barker, Optical starkdecelerator for molecules, Phys. Rev. Lett. , 243004(2004).[98] D. Spelsberg and W. Meyer, Static dipole polarizabili-ties of N , O , F , and H O, J. Chem. Phys. , 1282(1994).
Appendix A: Derivation of the ro-vibrational Hamiltonian
We will show how the ro-vibrational Lagrangian L = 12 N (cid:88) n =1 m n ˙ r n − V ( r , . . . , r N ) , (A1)describing the dynamics of N atoms with masses m n and centre-of-mass positions r n interacting via the molecularpotential V ( r , . . . , r N ) , can be simplified in the limit that only a single vibrational degree of freedom q matters [43].In this case, the atomic positions are fully determined by the value of q and by the molecule orientation Ω , r n (Ω , q ) = R(Ω) r (0) n ( q ) , (A2)where R(Ω) is the matrix rotating between body- and space-fixed frames and r (0) n ( q ) are the atom positions in thebody-fixed frame. The resulting velocities are related to the angular velocity vector ω and to ˙ q by ˙ r n (Ω , q ) = ω × r n (Ω , q ) + ˙ q R(Ω)[ ∂ q r (0) n ( q )] . (A3)0Assuming that there are no external forces and torques, V ( r , . . . , r N ) = V ( q ) , and plugging Eqs. (A2) and (A3)into the Lagrangian (A1) yields the Lagrangian (1) with the effective mass µ ( q ) = N (cid:88) n =1 m n [ ∂ q r (0) n ( q )] , (A4a)the inertia tensor I(Ω , q ) = R(Ω) (cid:34) N (cid:88) n =1 m n (cid:16) [ r (0) n ( q )] − r (0) n ( q ) ⊗ r (0) n ( q ) (cid:17)(cid:35) R T (Ω) , (A4b)and the Coriolis-coupling constants κ ( q ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n =1 m n r (0) n ( q ) × [ ∂ q r (0) n ( q )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A4c) n (Ω , q ) = 1 κ ( q ) R(Ω) (cid:34) N (cid:88) n =1 m n r (0) n ( q ) × [ ∂ q r (0) n ( q )] (cid:35) . (A4d)In the following, we will drop the arguments for brevity.The Lagrangian (1) yields the canonical momenta p = ∂L/∂ ˙ q = µ ˙ q + κ ω · n and J = ∂L/∂ ω = I ω + κ ˙ q n . Solvingthe resulting relations for the velocities yields ω = Λ − (cid:18) J − κµ p n (cid:19) (A5a)and ˙ q = pµ − κµ ω · n . (A5b)Here, we defined the effective inertia tensor Λ = I − κ µ n ⊗ n . (A6)With these relations one can carry out the Legendre transformation, H = ˙ qp + ω · J − L = 12 ˙ qp + 12 ω · J , (A7)yielding the Hamiltonian H = 12 (cid:18) J − κµ p n (cid:19) · Λ − (cid:18) J − κµ p n (cid:19) + p µ + V ( q ) . (A8)Both ro-vibrational canonical momenta p and J contain contributions due to mechanical rotations and deformations.As a consequence, the limit of no mechanical rotation is described by the condition that J = κp n /µ , see Eq. (A5a),rather than J = 0 . Appendix B: Half- and full-helix Hamiltonian
Half helix—
To quantify the effect of rotations on the tunnelling dynamics for [4]-helicene derivatives, we perform acontinuum approximation, which models the latter as homogeneous helix wires of total mass M , radius r , and smallopening q (cid:28) r . Denoting the body-fixed helix axis by n and its body-fixed opening axis as n , the helix is in thecenter-of-mass frame described by ξ ( ϕ ) = r ( n cos ϕ + n sin ϕ ) − r n /π + ϕq n /π , where the angle ϕ ∈ [ − π/ , π/ parametrizes the helix, see Fig. 4. The opening q determines whether the helix is left-handed ( q < ) or right-handed( q > ) handed and n = n × n .1 X Z m=0,1 qq n n n n n n (a) (b) (c) Figure 4. (a) Skeletal formula of the [4]-helicene derivatives considered herein. The parent [4]-helicene is obtained when X andZ, which mark the fjord positions, correspond to C–H groups and m = 1 , so that four fused six-membered rings are obtained.For m = 0 , the ring containing Z becomes a five-membered ring, as in the naphthindole (X=C–H, Z=N, m = 0 ). Variation ofX, Z and m has a strong impact on the barrier for stereomutation and the corresponding tunneling splittings. (b) Illustrationof a half-helix body used for modelling the ro-vibrational coupling. (c) Same as in (b) but for a full helix. The molecule’s angular velocity ω quantifies the rotation of the body-fixed frame { n , n , n } as ˙ n k = ω × n k .Together with the elastic deformations of the helix along the helix axis n , this determines the velocity of eachinfinitesimal wire segment, ˙ ξ ( ϕ ) = ϕ ˙ q n /π + ω × ξ ( ϕ ) . Note that the chosen parametrization of the helix implies J vib ∝ M r ˙ q n (cid:54) = 0 , not fulfilling the Eckart equation since there is no unique vibrational equilibrium [43]. As aconsequence, the canoncical momentum of elastic deformations is not equal to the kinetic momentum.Integrating the kinetic-energy density over the helix volume and substracting the double-well potential V ( q ) yieldsthe Lagrangian L = M (cid:90) π/ − π/ dϕπ ˙ ξ ( ϕ ) − V ( q ) = µ q + 12 ω · I( q ) ω + 2 M rπ ˙ q ω · n − V ( q ) , (B1)with µ = M/ . Comparing this with Eq. (1) shows that κ = 2 M rπ . The opening-dependent inertia tensor is I =
M r (cid:20) + n ⊗ n − qπ r ( n ⊗ n + n ⊗ n ) − π ( n ⊗ n + n ⊗ n ) + q r ( n ⊗ n + n ⊗ n ) (cid:21) . (B2)The resulting Hamiltonian is of the form (2). It can be further simplified by neglecting terms proportional to q /r and using that the effective inertia tensor fulfills Λ n = M r (cid:18) − π (cid:19) n , (B3)yielding H = 12 J · Λ − J + p µ ∗ + V ( q ) − pπ µ ∗ r J · n , (B4)where we defined the effective mass µ ∗ /µ = 1 − /π . Full helix—
The full helix is parametrized by ξ ( ϕ ) = r ( n cos ϕ + n sin ϕ ) + ϕq n / π with ϕ ∈ [ − π, π ] , see Fig. 4.All other conventions are as for the half helix. Again, integrating the kinetic-energy density over the helix volumeyields L = µ q + 12 ω · I(Ω , q ) ω + M r π ˙ q ω · n − V ( q ) , (B5)with µ = M/ and κ = M r/ π . The inertia tensor reads I =
M r (cid:20) + n ⊗ n − qπr ( n ⊗ n + n ⊗ n ) + q r ( n ⊗ n + n ⊗ n ) (cid:21) . (B6)The Hamiltonian (2) can be simplified by neglecting terms of order q /r , H = 12 J · Λ − J + p µ ∗ + V ( q ) − pπµ ∗ r J · n , (B7)with µ ∗ /µ = 1 − /π . In Figs. 2 and 3, we used the full-helix model to describe the ro-vibrational phase averagingdynamics of [4]-helicene derivatives.2 Appendix C: Vibrational two-level approximation
Canonical quantization of the Hamiltonian (2) replaces the phase-space coordinates by operators (denoted by sans-serif characters) obeying the canonical commutation relations. For instance, this implies [ p , J ] = 0 .The tunnelling dynamics can be approximately described by the two lowest-lying vibration states, which are wellseparated from all higher vibration states (see main text). The two vibration states can be expressed as even and oddsuperpositions of the left- and right-handed states | L (cid:105) and | R (cid:105) , which are well approximated as harmonic oscillatorgroundstates of mass µ , frequency ω , and centred at ± (cid:96) , (cid:104) q | L (cid:105) = (cid:16) µωπ (cid:126) (cid:17) / exp (cid:104) − µω (cid:126) ( x + (cid:96) ) (cid:105) , (C1)and likewise for R localized at (cid:96) .Describing the double-well potential by V ( q ) = µω (cid:0) q − (cid:96) (cid:1) / (cid:96) , with barrier height µω (cid:96) / , the tunnellingsplitting of a non-rotating molecule can be calculated as ∆ = − (cid:126) (cid:28) L (cid:12)(cid:12)(cid:12)(cid:12) p µ + V ( q ) (cid:12)(cid:12)(cid:12)(cid:12) R (cid:29) = ω (cid:18) ζ − ζ − (cid:19) e − ζ . (C2)Here, we introduced the dimensionless parameter ζ = µω(cid:96) / (cid:126) . Since the tunnelling splitting ∆ is defined for non-rotating molecules it depends on the effective mass µ rather than µ ∗ . Half helix—
For a rotating half helix, the effective mass µ is replaced by µ ∗ = µ (1 − /π ) , so that the correspondingtunnelling splitting becomes ∆ ∗ = ∆ − (cid:126) µ ∗ π (cid:104) L | p | R (cid:105) = ∆ + 48 ωπ µµ ∗ (cid:18) ζ − (cid:19) e − ζ . (C3)For values ζ > / , the tunnelling splitting is larger if rotational degrees of freedom are included than in the purevibration Hilbert space, ∆ ∗ > ∆ .The resulting half-helix Hamiltonian reads H (cid:39) (cid:18) − (cid:126) ∆ ∗ + iγ J · n + 12 J · Λ − LR J (cid:19) ⊗ | L (cid:105)(cid:104) R | + h . c . + H L ⊗ | L (cid:105)(cid:104) L | + H R ⊗ | R (cid:105)(cid:104) R | , (C4)Here γ denotes the Coriolis coupling and Λ LR = Λ(0) / (cid:104) L | R (cid:105) quantifies rotation-induced tunnelling. The centrifugalcoupling frequency follows as γ = 4 iπ µ ∗ r (cid:104) L | p | R (cid:105) = 4 µω(cid:96)π µ ∗ r e − ζ , (C5)and the overlap (cid:104) L | R (cid:105) = exp( − ζ ) . Full helix—
A similar calculation for the full helix shows that ∆ ∗ = ∆ + 3 ωπ µµ ∗ (cid:18) ζ − (cid:19) e − ζ , (C6)and γ = µω(cid:96)πµ ∗ r e − ζ , (C7)with µ ∗ /µ = 1 − /π .3 Appendix D: Rotation states of left- and right-handed enantiomers
Half helix—
The left- and right-handed effective tensors of inertia Λ L/R have identical eigenvalues but distincteigenvectors. Up to linear order of (cid:96)/r , the eigenvalues are λ = M r (1 − /π ) / , λ = M r (1 − /π ) / , and λ = M r (1 − /π ) with corresponding eigenvectors e L = n , e L (cid:39) n − (cid:96)π r n , e L (cid:39) n + 4 (cid:96)π r n , (D1a) e R = n , e R (cid:39) n + 4 (cid:96)π r n , e R (cid:39) n − (cid:96)π r n . (D1b)The principal axes of the left- and right-handed enantiomers are thus related by a rotation by the angle (cid:96)/π r aroundthe body-fixed axis n .The resulting rotational Hamiltonians describe asymmetric rigid rotors with moments of inertia λ < λ < λ .Again, their eigenvalues are identical but their eigenstates are related by the rotation | η ; R (cid:105) = exp (cid:18) − i (cid:126) (cid:96)π r J · n (cid:19) | η ; L (cid:105) . (D2)In order to calculate the eigenstates, we numerically diagonalize the Hamiltonian H L by expanding its eigenstatesin the symmetric-top basis | jkm (cid:105) , which are simultaneous eigenstates of the total angular momentum J | jkm (cid:105) = (cid:126) j ( j + 1) | jkm (cid:105) , the body-fixed angular momentum J · n | jkm (cid:105) = (cid:126) k | jkm (cid:105) , and the space-fixed angular momentumcomponent J · e x | jkm (cid:105) = (cid:126) m | jkm (cid:105) . Here, m, k = − j, . . . , j .The rotational Hamiltonian H = L / λ + L / λ + L / λ , with L i = J · n i the body-fixed angular momentumoperators projected on the eigenvectors (D1), commutes with L = L + L + L and with J · e x so that j and m are good quantum numbers. The remaining j + 1 eigenstates for fixed j and m are labeled by η = ( jnm ) with n = − j, . . . , j , so that | η ; L (cid:105) = j (cid:88) k = − j c ηk | jkm (cid:105) . (D3)The resulting eigenvalue problem for the coefficients c ηk is found by using (cid:104) jkm | L | jkm (cid:105) = (cid:104) jkm | L | jkm (cid:105) = (cid:126) j ( j + 1) − k ] , (D4a) (cid:104) jkm | L | jk + 2 m (cid:105) = (cid:104) jk + 2 m | L | jkm (cid:105) = (cid:126) (cid:112) ( j − k )( j − k − j + k + 1)( j + k + 2) , (D4b) (cid:104) jkm | L | jk + 2 m (cid:105) = (cid:104) jk + 2 m | L | jkm (cid:105) = − (cid:126) (cid:112) ( j − k )( j − k − j + k + 1)( j + k + 2) , (D4c)and using that all other matrix elements are zero. Given the coefficients c ηk , the overlaps follow from Eq. (D2), (cid:104) η ; L | η ; R (cid:105) = j (cid:88) k = − j | c ηk | exp (cid:18) − ik (cid:96)π r (cid:19) . (D5) Full helix—
In a similar fashion, the eigenvalues of the effective inertia tensor of the left- and right-handed full helixare λ = M r (1 − /π ) / , λ = M r / , and λ = M r with corresponding eigenvectors e L = n , e L (cid:39) n − (cid:96)πr n , e L (cid:39) n − (cid:96)πr n , (D6) e R = n , e R (cid:39) n + (cid:96)πr n , e R (cid:39) n + (cid:96)πr n . (D7)The left- and right-eigenvectors are thus related by a rotation by the angle (cid:96)/πr around the body-fixed axis n , sothat the eigenstates of the rotational Hamiltonians fulfill | η ; R (cid:105) = exp (cid:18) − i (cid:126) (cid:96)πr J · n (cid:19) | η ; L (cid:105) , (D8)4and (cid:104) η ; L | η ; R (cid:105) = j (cid:88) k = − j | c ηk | exp (cid:18) − ik (cid:96)πR (cid:19) . (D9)Finally, we remark that the Hamiltonian H L/R is spatially isotropic and thus its eigenenergies E η are independentof the space-fixed angular momentum quantum number m , implying that the orientational diagonal elements of thero-vibrational state (11) fulfill (cid:104) Ω | ρ | Ω (cid:105) = tr rot ( ρ ) / π . Here, tr rot ( · ) denotes the partial trace over the rotationalHilbert space. This follows from the completeness of the Wigner D -matrices [65] j (cid:88) m = − j (cid:104) Ω | jkm (cid:105)(cid:104) jk (cid:48) m | Ω (cid:105) = 2 j + 18 π δ kk (cid:48) . (D10)This relation expresses that all orientations are equally probable. Appendix E: Far-field diffraction
Writing the phase (7) as φ cos (2 π sin θx/λ ) and the phase (8) as χ L/R sin(4 π sin θx/λ ) , the grating transformation(6) for fixed handedness takes on the form t L ( x ) = ∞ (cid:88) n = −∞ i n J n (cid:18) φ χ L (cid:19) exp (cid:18) i πn sin θxλ (cid:19) , (E1)and likewise for R . Here, we used that ϑ = π/ . The resulting far-field diffraction pattern follows from propagatingthe spatial wave-function | x (cid:105) first to the grating with the unitary U , and then to the detection screen at distance d behind the grating, (cid:104) x | Φ L (cid:105) ∼ ∞ (cid:88) n = −∞ i n J n (cid:18) φ χ L (cid:19) (cid:90) w/ − w/ dx (cid:48) (cid:104) x (cid:48) | U | x (cid:105) exp (cid:20) − i mv z x (cid:48) (cid:126) d (cid:18) x − n π sin θ (cid:126) dmv z λ (cid:19)(cid:21) . (E2)The spacing of neighboring diffraction orders is π sin θ (cid:126) d/mv z λ . Appendix F: Interferometric sensing and motional decoupling
Filtering the | R (cid:105) -state at the grating and imprinting the relative phase ϕ at time t yields | Ψ (cid:48) ηR (cid:105) = cos(∆ η t ) | ηR (cid:105) + ie iγ η e iϕ sin(∆ η t ) | ηL (cid:105) , (F1)which after a free evolution for time t (cid:48) gives | Ψ (cid:48)(cid:48) ηR (cid:105) = a η ( t, t (cid:48) ; ϕ ) | ηR (cid:105) + ie iγ η b η ( t, t (cid:48) ; ϕ ) | ηL (cid:105) with the coefficients a η ( t, t (cid:48) ; ϕ ) = cos(∆ η t ) cos(∆ η t (cid:48) ) − e iϕ sin(∆ η t ) sin(∆ η t (cid:48) ) , (F2a) b η ( t, t (cid:48) ; ϕ ) = cos(∆ η t ) sin(∆ η t (cid:48) ) + e iϕ sin(∆ η t ) cos(∆ η t (cid:48) ) . (F2b)The probability P R ( t, t (cid:48) ) = (cid:80) η p η | a η ( t, t (cid:48) ; ϕ ) | at fixed times t and t (cid:48) strongly depends on the relative phase ϕ andcan thus be used to measure it. In particular, a straight forward calculation yields P R ( t, t (cid:48) ) = cos (cid:16) ϕ (cid:17) P R ( t + t (cid:48) ) + sin (cid:16) ϕ (cid:17) P R ( t − t (cid:48) ) , (F3)where P R ( t ) denotes the probability of measuring the R -enantiomer in the absence of the phase shift. Thus, for t = t (cid:48) ,the unperturbed signal is modulated with cos ( ϕ/ . In addition, applying ϕ = π implements motional decoupling byundoing the rotational phase averaging. For t (cid:48) = t , this yields the initial condition a η ( t, t ; π ) = 1 and b η ( t, t ; π ) = 0 ,as illustrated in Fig. 5(a).5 θ E E θ E E xy (b) z (c)(a) P o p u l a t i o n P L P R W / cm )0.00.10.20.30.4 / Figure 5. (a) Motional decoupling implemented by applying a relative phase shift φ = π at the time indicated by the dashedgreen line, which effectively reverses the rotational phase averaging of the tunnelling signal. (b) Optical gratings created bysuperposing two linearly polarized laser fields each of electric field amplitude E and copropagating in the transverse x − y plane.Left: for the standing-wave grating, the two laser beams have the same polarization. Right: for the helicity grating, the twobeams have orthogonal polarizations. (c) Phase shift of the helicity grating as a function of peak optical intensity in each laserbeam considering Gaussian-shaped pulse envelope and beam profile. Appendix G: Decoherence dynamics
The ro-vibrational dynamics in presence of decoherence with the rotation-state-independent rate Γ LR is describedby the Lindblad master equation ∂ t ρ = − i (cid:126) [ H, ρ ] + Γ LR (cid:88) η (cid:88) h = L,R | ηh (cid:105)(cid:104) ηh | ρ | ηh (cid:105)(cid:104) ηh | − ρ , (G1)with the tunnelling Hamiltonian (4). The rate Γ LR is given by (15) if the molecule is inside the collision chamber,and zero before and after it. Since this Hamiltonian induces no rotational coherences and since the initial rotationstate is thermal with weights p η , we can write the ro-vibration state at all times as a mixture of vibration states, ρ = (cid:80) η p η ρ η .The dynamics of each vibration state ρ η with tr( ρ η ) = 1 can be calculated by recasting the master equation (G1) foreach η in the form ∂ t c η = i A η c η , with the vector of matrix elements c η = ( (cid:104) L | ρ η | L (cid:105) , (cid:104) R | ρ η | R (cid:105) , (cid:104) L | ρ η | R (cid:105) , (cid:104) R | ρ η | L (cid:105) ) T and the non-Hermitian matrix A η = − ∆ η e − iγ η ∆ η e iγ η η e − iγ η − ∆ η e iγ η − ∆ η e iγ η ∆ η e iγ η i Γ LR η e − iγ η − ∆ η e − iγ η i Γ LR . (G2)The solution to this equation is given by c η ( t ) = exp( i A η t ) c η (0) .Filtering R -states at the grating, c η (0) = (0 , , , T , and denoting by T the time from the grating to the screen,the state at the screen is c η ( T ) = sin (∆ η T )cos (∆ η T ) ie iγ η sin(2∆ η T ) / − ie − iγ η sin(2∆ η T ) / , (G3)since there is no notable decoherence in the interferometer. After traversing the collision chamber for time τ , theprobability of measuring the R state can be calculated as P ηR ( τ ) = 12 + 12 e − Γ LR τ/ (cid:20) cos(2 ω η τ ) cos(2∆ η T ) − ∆ η ω η sin(2 ω η τ ) sin(2∆ η T ) + Γ LR ω η sin(2 ω η τ ) cos(2∆ η T ) (cid:21) , (G4)with ω η = (cid:113) ∆ η − (Γ LR / . The final signal is obtained by summing over all rotation states with weights p η , P R ( τ ) = (cid:88) η p η P ηR ( τ ) . (G5)6Equation (G4) contains two relevant limiting cases: (i) For vanishing decoherence Γ LR (cid:28) ∆ η , the probability ofmeasuring the R -chiral state reduces to P ηR ( τ ) (cid:39) / − Γ LR τ /
2) cos[2∆ η ( T + τ )] / . (ii) For long tunnellingtimes ∆ η (cid:29) Γ LR and ∆ η T (cid:28) , the state stays pure and there is no influence of decoherence, P ηR (cid:39) . Appendix H: Choice of molecules and computational methods
The possibility that molecular chirality is induced in overcrowded structures of angularly fused benzene rings wasnoted for 4,5-dimethyl phenanthrene in the 1940s [66, 67]. It was demonstrated by synthesis and enantioresolutionof [6]-helicene with subsequent measurement of optical rotation [68], but it took four decades to demonstrate itexperimentally for the original 4,5-dimethyl phenanthrene [69], due to fast racemisation. For the present work, fastracemisation and comparatively low barrier of stereomutation is, however, desirable to obtain tunneling splittings inthe range of
100 Hz to
50 kHz .Tunneling splittings of
150 kHz have been observed experimentally for the axially chiral inorganic compounddisulfane (H S ) [70] and splittings about an order or magnitude smaller have been predicted for TSOT, H Se andH Te [57, 71–73], but the optical rotation for this compound class is expected to be small. Helicenes and derivativesthereof, in contrast, are well-known for comparatively large optical rotation [68, 74–82], which is crucial to realise thegrating transformation in the present work. Optical rotation increases with increasing n in [ n ]-helicenes [78] whereasactivation energies for racemisation become lower for smaller helicenes but reach a plateau for n > [58, 78, 79, 83–85].In [4]-helicene (also denoted as 3,4-benzphenanthrene or benzo[c]phenanthrene; see Fig. 4), the barrier for racemi-sation is reported to be too low to allow separation of enantiomers at room temperature [86]. Lindner was the firstto study the racemisation mechanism of helicenes computationally and found a planar transition structure for [4]-helicene with an activation barrier of about
50 THz , which was confirmed also in later studies [79, 87], which was themotivation for choosing [4]-helicenes and derivatives thereof as a starting point in the present work.Tunability of the activation barrier for stereomutation by substitution is well known for [4-6]-helicenes [58, 59, 86]and Ref. [59] found a correlation between the torsional twist in the equilibrium structure induced by substituents in thefjord region of [5]-helicenes and the corresponding activation barrier for stereomutation, by which a quick assessmentof expected barrier heights is possible. Additional fine-tuning of the tunneling splittings is possible by isotopicsubstitution or by replacement of hydrogen with fluorine on the periphery because of the exponential dependenceon the square root of the tunneling mass [88, 89]. Our primary target is thus to identify [4]-helicene derivativesthat can bracket the desired tunneling splitting, while maintaining the comparatively large optical rotation that ischaracteristic for the helicene family. Chemical tuning will allow to determine subsequently ideal compounds.The equilibrium structures of [4]-helicene as determined in Ref. [81] on the density functional theory (DFT) levelB97-D with the D2 dispersion correction and a basis set of triple zeta quality (TZVP) were taken as input structures.Equilibrium and transition structures were in the present work then determined on the same level of theory with theprogram package Turbomole [90].The isotropic part of the optical rotatory dispersion tensor β ( ω ) at real and imaginary frequencies and the frequencydependent electric dipole polarisability tensor α ( ω ) were calculated with the same program package on the time-dependent DFT level with the B3LYP hybrid density functional and a triple-zeta basis set (TZVPP) basis set andthe recommended basis sets for the resolution of the identity (RI) of the Coulomb term (RI-J). β ( ω ) is related to thegyration tensor G (cid:48) ( ω ) [45] by G (cid:48) ( ω ) = − ω β ( ω ) (H1)and can possess in contrast to G (cid:48) ( ω ) a finite static limit [91]. The proper units that result when G (cid:48) and β are reportedin atomic units, are obtained by the collection of the following constants [ G (cid:48) ] = ea e (cid:126) m − E − = e a (cid:126) − (H2) [ β ] = ea e (cid:126) m − (cid:126) E − = e a m e (cid:126) − , (H3)with e being the electric charge of the proton, a being the Bohr radius, and E h the Hartree energy.For the equilibrium structure of [4]-helicene, we estimate the static-limit isotropic optical rotatory dispersion β iso ≈ e a m e (cid:126) − , which is only a factor 5 to 6 smaller than the corresponding value in [6]-helicene and thus consistentwith the ratio computed in Ref. [81] for the frequency dependent optical rotation of the two compounds at the sodiumD-line.For the equilibrium structure of the naphthindole (see Fig. 4), we obtain β iso ≈ . e a m e (cid:126) − , which is nearlya factor of 3 smaller than the value for [4]-helicene. The lowest harmonic vibrational frequency remains almostunchanged (about ), but the electronic barrier for stereomutation drops to only .7Tunnelling splittings were estimated with the instanton code as implemented in the program package Molpro [92] onthe DFT level with the B3LYP functional and the 3-21G basis set for all atoms. For further details on the instantonapproach we refer to Ref. [93]. The computed action S for the instanton path of [4]-helicene is ∼ (cid:126) using 160 beadsand β = 20000 (cid:126) . This gives a tunneling splitting of ∼ pHz. Using the B97-D functional with the D4 dispersioncorrection and a correlation consistent double zeta basis set (cc-pVDZ) for all atoms, instanton paths with up to 65beads were calculated. The resulting action is ∼ (cid:126) . As this term enters exponentially in the calculation of thetunneling splitting with ∆ E ± ∝ exp( − S/ (cid:126) ) , it essentially determines the magnitude of the tunneling splitting. Thisgives a difference of 1-2 orders of magnitude in the tunneling splitting for the different methods, which is sufficientfor our estimates.We thus considered several other derivatives and turned to the naphthindole sketched in Fig. 4. Although the in-stanton approach becomes less accurate for shallow tunneling situations [91], it can still provide an order of magnitudefor the tunneling splitting. With the B3LYP functional we calculated 320 beads with β = 15000 (cid:126) for naphthindole,which gave an action ∼ (cid:126) and a splitting ∼ MHz. At 160 beads the action has already converged to (14 . ± . (cid:126) for β values between (cid:126) and (cid:126) , giving a sufficient accurate estimate of the splitting. With the better B97-Dfunctional using 65 beads and the same β the action is (cid:126) . Though the action has not converged in these calcula-tions, we can estimate of the splitting in the range of 1 to 200 MHz, so that the two compounds selected here clearlybracket the desired range of 100 Hz to 50 kHz used in the present estimates. Appendix I: Simulation parameters
For the simulations presented in Figs. 2 and 3 , we consider a molecular beam generated by seeding [4]-helicenes intoa supersonic jet of Xenon, yielding a mean velocity v z (cid:39) m/s and a ro-vibrational temperature of approximately K. The beam is first filtered by a source skimmer with a diameter of µ m, before propagating freely for a distanceof m to a collimation slit with an opening of µ m. After passing the slit, the molecules are illuminated by two pulsedoptical gratings. The gratings are realized by superposing two laser beams of the same intensity and co-propagatingat an angle θ = 1 perpendicular to the molecular beam, as illustrated in Fig. 5. For the first grating, the two laserbeams have their polarization aligned parallel to each other, see Fig. 5(b). The interference leads to an intensitygradient in the transverse direction x and gives rise to an electric-dipole interaction potential in the form of V Lη ( x, t ) = − α Lη E ( t ) [1 + cos ( ζx )] . (I1)Integrating this potential over the temporal profile of the laser pulse yields the phase shift (7). The right-handedenantiomer experiences the same phase shift since α Lη = α Rη .The second optical grating, acting directly after the first, so that the motion of the molecules between the gratings isnegligible, is created in a similar fashion, but with the polarization of the two beams perpendicular to each other. Here,the superposition of the two laser beams creates an optical helicity-density gradient which interacts with the moleculesvia their electric-magnetic dipole polarizability pseudotensor G L/Rη [41, 42]. The resulting interaction potential canbe written as ˜ V Lη ( x, t ) = − G Lη c E ( t ) cos θ sin ( ζx + ϑ ) , (I2)yielding the phase shift (8). The right-handed enantiomer experiences the opposite phase shift since G Lη = − G Rη . Aftertraversing the gratings, the molecule propagates for a distance of m, where a screen with a slit of µ m-openingblocks all but a single diffraction peak. The requirement on the total length of the interferometer can be alleviatedby using a slower molecule beam, e.g., from a buffer gas cell [61].The two optical gratings can be generated by a pulsed Nd:YAG laser operating at 1064 nm with pulse duration of10 ns. Both gratings can be implemented by splitting the laser beam with a beam splitter and superposing them inthe interaction region using a Mach-Zehnder configuration [94, 95]. Considering a collimated Gaussian beam profilewith beam diameter w x,y = 0 . mm, pulse energy of 250 mJ is necessary in each beam to reach a phase shift of π/ for the helicity grating. Generating the same phase shift with the standing-wave grating, requires only . µ J in eachbeam. The phase shifts in Eq. 7 and Eq. 8 assume a flat intensity profiles in the interaction region of µ m, whichcan be achieved by using beam shaping optics or by adapting an active resonator design of the Nd:YAG laser. Therelation between the phase shift of the helicity grating versus the peak optical intensity of each laser beam is plottedin Fig. 5(c). To reach a phase shift of π/ , a peak intensity of × W / cm is required in each beam. The laserwavelength 1064 nm is far detuned from the electronic (<400 nm ) and vibrational (>3 µ m) absorption bands of helicenederivatives. In addition, the absorption of a photon in the grating transfers a momentum kick of magnitude π (cid:126) /λ tothe molecule in the propagation direction of the laser beam, so that the corresponding molecules will be filtered out8at the screen. The intensity required for the helicity grating is one order of magnitude lower than those employed insensitive dipole force measurements using aromatic hydrocarbons and Nd:YAG laser at the same wavelength [96, 97].To simulate the tunnelling dynamics of [4]-helicene derivatives , we use the full-helix model with µ = 24 u, ω/ π =3 THz, r = 2 . Å and (cid:96) = 0 . Å , yielding that Coriolis coupling and the overlaps are negligibly small, while thetunnelling splitting is on the order of ∆ ∗ / π (cid:39) Hz. To calculate the decoherence dynamics illustrated in Fig. 3,we considered nitrogen molecules with mass of 28 u and rotation averaged-polarizability α g / πε = 1 . Å3