Holographic Storage of Ultrafast Photonic Qubit in Molecules

Herein, it is demonstrated that ultrashort spatially structured beams can sculpt a sample of gas‐phase molecules like a 4D material to produce a spatial pattern of aligned molecules whose shape and temporal evolution allow to restore the spatial light information on a time‐delayed reading pulse. To do so, the spatial phase and amplitude information of ultrashort light beams are encoded into rotational coherences of molecules by exploiting the interplay between spin angular momentum and orbital angular momentum. The field‐free molecular alignment resulting from the interaction leads to an inhomogeneous spatial structuring of the sample allowing to transfer the encoded information into a time‐delayed probe beam. The demonstration is conducted in CO2$\left(\text{CO}\right)_{2}$ molecules. Besides applications in terms of THz bandwidth buffer memory, the strategy features interesting prospects for establishing versatile optical processing of orbital angular momentum (OAM) fields, for studying various molecular processes, or for designing new photonic devices enabling to impart superpositions of OAM modes to light beams.


Introduction
The ability to establish fine control of matter with light or, conversely, manipulating light on demand with matter has opened up fascinating applications ranging from coherent and quantum control, [1,2] to laser structuring process, [3] via pulse shaping technologies, [4] laser filamentation, [5] fiber communication, [6] or data storage [7] to name a few.10][11] Such fields are characterized by a helical phase distribution lφ, where φ is the azimuthal angle in the transverse plane of the beam, and l, the topological charge, corresponds to the number of interwined helical phase over one propagation wavelength. [8]ight-matter interaction with such fields has been extensively investigated.The matter can be used as an interface to store [12][13][14] or manipulate [15] structured light beams, while the interaction of OAM beams with matter has found far-reaching applications covering innovative photonic functionalities, [16] particle manipulation, [17,18] quantum information technology, [19][20][21] optical communication, [22][23][24][25] laser micromachining, [26] or high-resolution imaging, [27] to cite only a few.The interweaving between chirality of light and matter [28][29][30] as well as the interaction of matter with multiplexed light embedding both spin angular momentum (SAM) and OAM, [31,32] constitute a new emerging paradigm of great importance.In this frame, we have recently reported [33] the use of gas-phase molecules as a quantum interface to store pure OAM states carried by ultrashort laser pulses.Through a coupling scheme exploiting the interplay between SAM and OAM, [34] the helical phase information of light beams was encoded into the rotational coherences of linear molecules.The underlying mechanism at the origin of this storage was explained by the spatial structuring of the molecular sample induced by the field.The polarization vector distribution of the writing field was shown to induce an inhomogeneous spatial distribution of molecular axes orientation in the transverse plane of the beam.This last was found to follow the same azimuthal pattern as the one of neutral axes of q-plates, [15] commonly used for the production of OAM beams.The periodical revival of this "molecular q-plate" associated with the quantum beatings of the rotational wavepacket was exploited to restore the encoded helical phase structure on-demand.Although the strategy has proven its suitability for memorizing a spatial phase pattern, the ability of the method to store a spatial amplitude information, in addition to a spatially varying phase, remains to be demonstrated.Such a property turns out to be particularly relevant for encoding a coherent superposition of different OAM, namely an OAM-based photonic qubit, carried by an ultrashort laser pulse.A coherent superposition of OAM bears both spatial phase and amplitude modulation requiring, for its storage, more elaborate molecular structuring than that required with a pure OAM.While for pure OAM, the molecules have to be confined into the transverse plane of the beam along a specific direction depending on the azimuthal angle, the storage mechanism of the present work can be interpreted by the production of a spatial pattern of aligned molecules in 3 dimensions (which evolves over time).This specific and refined structuring allows to restore the overall information on a time-delayed reading pulse.The experiment is conducted in CO 2 molecules at room temperature with the storage of a superposition of l ¼ AE1 OAM states, also called photonic gear. [35]The choice of this molecule was driven by its large polarizability anisotropy but the method is applicable to all molecules except spherical top molecules.The restored state is analyzed for different relative amplitude or phase between the two components and our observations reveal a great fidelity in the preservation of the mode superposition validating the ability of the storage process.The experimental observations are supported by a theoretical model to capture the underlying effects.

Principle
The photon state of the input field to be stored (called "input beam" hereafter) can be written as: with a 2 1 þ a 2 À1 ¼ 1 (a 1 and a À1 being real and positive) and jli, the OAM mode of topological charge l.This state corresponds to a coherent superposition of two OAM modes with a phase shift β.In practice, the storage in atomic states of such structured light beams is obtained using an auxiliary beam, ideally a plane wave, analogously to holography imaging.Excitation schemes are usually based on a resonant Λ-configuration, through the absorption (resp.emission) of one photon of the structured beam (resp.auxiliary beam) in a stimulated Raman process, resonant with an excited state.The spatial phase and amplitude information, stored through the coherences induced between the two lower states, can be restored at will with a reading beam.Such a strategy, implying a resonance with an excited state, remains limited to the storage of spatial information embedded into narrowband laser beams. [13,14]The coupling scheme of the present work relies for its part on a nonresonant Λ excitation scheme (see Figure 1) well-suited for the storage of spatial information carried by broadband laser pulses.We point out that this configuration, offering THz bandwidth buffer memory, is standard for the production of rotational coherences and field-free molecular alignment in molecular systems. [36]As depicted in Figure 1, an ultrashort pump pulse consisting of the field carrying the OAM superposition jΨi in and the auxiliary field (jl ¼ 0i) interact with the molecular sample.This interaction stores the spatial information (phase and amplitude) of the input beam by producing a rotational wavepacket in the vibronic ground state of the molecules.The encoded spatial structure can be restored later on with a time-delayed probe pulse by exploiting the quantum beating of the wavepacket.We recall that with short pulses (of duration much shorter than the rotational period of the molecules), the interaction prepares a broad rotational wavepacket in the ground state of the molecule by impulsive stimulated Raman transitions.The periodical rephasing of the rotational wavepacket then manifests itself by the occurrence of molecular alignment [36] at well-defined times after the pulse turn-off.The induced anisotropy is exploited for the restoration of the spatial information.As explained in Ref. [33], the sequence of photons implied in the Raman process for such a nonresonant and broadband excitation is not well determined when using linear polarization.Besides, the sequence of photons depicted in Figure 1, the pump excitation can also be achieved with either one of the two beams alone (input or auxiliary), or by exchanging the input and auxiliary beam.This results in the occurrence of unwanted quantum excitation channels detrimental to the information to be encoded: the channel is only driven by the auxiliary beam stores j0i while the one driven by the structured field is a superposition of j0i, j2i, and j À 2i.The relevant information to be stored can nevertheless be preserved from this uncontrolled superposition by using circular polarizations of opposite helicity for the two fields implied in the Raman transition as depicted in Figure 1.Such a strategy enables the selection of one specific quantum channel by spin constraint.In our experiment, the SAM and OAM are conserved on their own since the weak focusing geometry prevents any spin-to-orbital conversion [37] and since dipole electric transition are allowed with no significant role played by quadrupole transitions. [38]In the scheme of Figure 1, a left circularly polarized (LCP) probe field is used to read the medium at the occurrence of field-free molecular alignment producing a field of opposite helicity, i.e., right circularly polarized (RCP), due to the sample anisotropy.This "signal" field is isolated by a circular analyzer in our experiment.Using the convention from the point of view of the receiver for the SAM, the corresponding Raman coupling of the reading process implies ΔJ ¼ 0, AE 2 and ΔM ¼ þ2 transitions, where J and M correspond to the total angular momentum and magnetic quantum numbers, respectively.The sequence of pump pulses helicity in the global parametric process is then constrained by SAM conservation imposing a sequence of RCP-LCP for the pump Raman excitation, as depicted in Figure 1.Depending on their polarization, the spin constraint, therefore, assigns the role played by both input and auxiliary beams in the excitation, preventing unwanted quantum channels.This strategy has been successfully applied to the storage of pure OAM modes [33] embedding spatial phase modulation and the purpose of the present work is to study a possible operation with a coherent superposition of OAM states bearing both spatial phase and amplitude structures.As already mentioned the storage relies on the production of a spatial pattern of field-free molecules aligned in 3 dimensions and the assessment of how this pattern can restore the superposition mode along with its relative phase holds significant importance.

Theoretical Model
To assess the ability of the method to operate with fields carrying an OAM superposition state, we write the time-dependent Hamiltonian interaction describing the coupling of the molecule with the pump field.This last (propagating along z) is written in circular basis as , where e R (resp.e L ) corresponds to the right (resp.left)-handed circular polarization in the (x, y) plane, namely, e R ¼ 1 ffiffi 2 p ð1, À iÞ and e L ¼ 1 ffiffi 2 p ð1, iÞ.According to excitation scheme of Figure 1, E L corresponds to the auxiliary field while E R is the phase and amplitude structured light beam to be stored that can be written as: (2) where ε 0 ðtÞ and ω 0 denote the pulse envelope and the central angular frequency of the pulse.a s ðφÞ and ϕ s ðφÞ stand for the spatial amplitude and phase of the beam to be encoded that are related to the parameters a 1 , a À1 , and β of Equation ( 1) (see Section 2, Supporting Information).The spatial amplitude a s ðφÞ (real and positive) includes a possible global amplitude factor between the components E R and E L .We point out that, for the sake of clarity, the radial dependence has been omitted in these expressions since the relevant information in our case is encoded into the azimuthal coordinate φ (a radial dependance would not change the main conclusion).Within the high-frequency approximation, one can show (see Section 3, Supporting Information) that the Hamiltonian interaction can be written as: where θ and Φ are the Euler angles of the molecule with θ the angle between the molecular axis and the fixed-frame quantum axis z (corresponding to the propagation axis for all fields) and Φ the azimuthal angle (with respect to the fixed-frame x-axis) of the projection of the molecular axis onto the x À y plane.Δα ¼ α k À α ⊥ is the difference between the parallel and perpendicular components of the polarizability tensor.The first two terms of the Hamiltonian, independent of Φ, are associated to transitions ΔM ¼ 0 and correspond to the excitation channels only driven by the input field or the auxiliary field as explained before.It can be noted that these two terms do not induce any preferential direction of the molecules in the transverse plane and therefore do not contribute to produce any anisotropy for the reading beam.Only the last term of the interaction Hamiltonian relies on the ΔM ¼ AE2 transitions of interest here that contribute to the measured signal in the experiment (see Figure 1).This last term tends to preferentially confine the molecules in the (x, y) plane (corresponding to θ ¼ π=2) at an angle Φ ¼ ϕ s ðφÞ 2 that depends on the azimuthal coordinate φ inside the beam.This coupling term varies linearly with a s ðφÞ the spatial amplitude of the input field so that the induced alignment (in the transverse plane) is expected to follow the same trend at least below the saturation regime. [39]The direction and amplitude of molecular alignment therefore depends on the coordinate φ of the molecule inside the beam.A probe beam E pr ðtÞ interacting with such a structured medium will experience a spatially dependent modification of its ellipticity at the origin of the OAM superposition carried by the signal field, namely the field of opposite helicity isolated by the circular analyzer (Figure 1).By calculating the dipole moment resulting from the interaction of the probe pulse with the molecular sample (see Section 3, Supporting Information), the signal field writes as: We consider a probe pulse time-delayed by τ of the form E pr ðtÞ ¼ ε 0 ðt À τÞe Àiω 0 ðtÀτÞ .For a molecular alignment along θ ≈ π=2 and Φ ≈ ϕ s ðφÞ 2 , varying with the amplitude a s ðφÞ as discussed above, the signal field becomes: which is a replica of the RCP pump field encoded into the molecular sample with the same amplitude and phase distribution.It should therefore be possible to store and restore any OAM or superposition of OAM in this way.It can be shown (see Section 3, Supporting Information) that a possible phase and/or amplitude radial distribution in the beam would also be identically restored.

Experimental Set-Up and Results
The experimental set-up to verify this prediction is shown in Figure 2. Briefly, the laser system produces pulses of 35 fs duration centered around 800 nm at 1 kHz repetition rate and of maximum energy 5 mJ.The beam is split into three parts so as to produce the input and auxiliary components of the pump beam as well as the probe beam.The production of the pump beam is depicted in Figure 2b.The OAM superposition is produced with a q plate [15] enabling to generate OAM beams via SAM-OAM exchange.The chosen q plate (half-waveplate, q= 1/2) in conjunction with a Berek compensator (BC 3 ) and a polarizer encodes the superposition of OAM modes jΨi in of Equation ( 1) at once into the beam (see Section 1, Supporting Information).This structured beam is then recombined with a part of the incident beam, the auxiliary beam j0i ("Aux" in Figure 2b), passing through a delay line and whose polarization is rotated by 90°with a half waveplate.Both components, temporally synchronized, are recombined with a beam splitter (BS), and circularly polarized with an opposite handedness by a quarter waveplate so as to produce the overall pump field.The probe beam for its part is frequency doubled with a Beta barium borate (BBO) crystal, circularly polarized and time-delayed with a delay-line (Figure 2a).Pump and probe beams, recombined by an optical wedge, propagate along the same axis and are focused by a lens (f ¼ 30 cm) in a static cell filled with 1 bar of CO 2 .At the exit of the cell, the probe is filtered from the pump with a dichroїc filter.
A circular analyzer (BC 2 with polarizer) isolates the signal field produced during the interaction which can be measured either by a CCD camera to get its spatial profile, or by a photomultiplier, for having the spatially integrated signal.The spatial phase profile is obtained by measuring the interference pattern with a reference plane wave (called "Ref" in Figure 2a) of same polarization state.
The first experiments have been conducted with a superposition of balanced j AE 1i OAM states, namely, jΨi in ¼ j1iþe iβ jÀ1i ffiffi 2 p in Equation (1).It corresponds to a field amplitude varying as cosðφ À β=2Þ, where the dephasing β can easily be tuned by modifying the neutral axis orientation of the Berek compensator before the q-plate (see Section 1, Supporting Information).Typical results are depicted in Figure 3 where all signals (shown in left column) have been measured at a pump-probe delay of τ ¼ 21.18 ps.Such a delay corresponds to a revival peaks of molecular alignment but similar results can be obtained at any other delays and even between two revivals as already discussed. [33]The first line (Figure 3a,d) corresponds to the case β ≈ 0. The spatial distribution of the restored signal field, displayed in Figure 3a, shows a petal-like pattern which is characteristic of a superposition of two conjugate (AEl) OAM modes.The number of petals and their orientation are in perfect agreement with the expectation 2l petals around φ ¼ 0, π.It reproduces with a good fidelity the spatial distribution of the input beam encoded into the molecular sample and is shown in inset of the right panel.Besides, the intensity distribution, another important ingredient of the restoration is the spatial phase, experimentally characterized by the interference pattern with a reference beam.This last, displayed as a color scale in Figure 3a, well reproduces the expected π-phase shift between the two lobes of the field pattern.We also proceeded to the decomposition of the signal field into the first OAM eigenstates.As observed in Figure 3d, the dominant contributions are concentrated within the l ¼ AE1 modes of similar amplitude in agreement with the spatial shaping of the field applied by the combination of the Berek compensator (tuned as a half-waveplate) and the q-plate (see Section 1, Supporting Information).Others configurations with different relative phases β (β ≈ π in Figure 3b,e and β ≈ π=2 in Figure 3c,f ) have also been tested.Again the signal field read out by the probe beam well reproduces the expectation of two petals of equal intensity rotated by an angle β=2 and bearing a relative π-phase.
In a next step, the case of unbalanced superpositions has been tested (Figure 4).Such a configuration is experimentally produced by modifying the phase induced by the Berek compensator so as to produce an elliptical polarization before the q-plate (see Section 1, Supporting Information).The associated intensity pattern must then feature two petals superimposed with a residual background contribution since: for a 1 6 ¼ a À1 and a 2 1 þ a 2 À1 ¼ 1, one has a 1 a À1 < 1=2.The previous equation therefore indicates that no angle φ cancels the intensity unlike balanced qubit, providing therefore an intensity pattern with two lobes surrounded by a "figure-eight halo."The experiment described below was conducted with a Berek compensator set to produce a superposition with a 1 ¼ 2a À1 .The intensity pattern measured for the input and signal beams (at τ ¼ 21.18 ps) are depicted in Figure 4a,b.As evident, these last display the anticipated figure-eight halo.To compare the input and restored beams, we have computed their integrated angular distribution in Figure 4c.The figure reveals a high degree of similarities between the two distributions which is a strong indicator of the quality of the restoration.The relative contribution l ¼ AE1 (a 1 and a À1 ) in the input beam can further be retrieved through the contrast of the angular distribution from Equation (7).The result, shown as a red line in Figure 4d, is found to be in good agreement with the modulation a 1 ¼ 2a À1 applied in the input beam and with the complete decomposition of the restored beam.
A superposition of AEl OAM states corresponds to the generalized Hermite-Laguerre-Gaussian beams (GHLG beams) which lie between LG and HG beams. [40]GHLG modes can be described through a Bloch sphere representation, analogous  to the Poincaré sphere (used for the description of polarization states), with the pole corresponding to pure LG eigenstates.A GHLG state jΨi is then defined by jΨi ¼ cosðθ p =2Þjli þ e iβ sinðθ p =2Þj À li, where β describes the relative phase between OAM modes and θ p their weighted contribution.The evolution of light field modes explored in this article and in our previous work [33] are depicted in Figure 5.As it can be shown, our strategy enables the complete coverage of the Poincaré sphere (along the equator for the balanced superpositions and from the equator to the poles with unbalanced superpositions).
The above results demonstrate the storage of OAM modes superposition into rotational coherences can be interpreted by the field-induced spatial structuring of the molecular sample.A typical example is depicted in Figure 6 for the case of (a 1 ¼ a À1 with β ¼ 0).The associated input field can be written as E R ðt, φÞ ¼ ε 0 ðtÞ cosðφÞe Àiω 0 t (if a global amplitude factor 1= ffiffi ffi 2 p is considered) and its superposition with the auxiliary beam of opposite helicities provides a field with an azimuthal distribution of amplitude and ellipticity, as depicted in the middle part of Figure 6a (see also Section 4, Supporting Information).The interaction of such spatially structured vector fields with atoms, and matter in general, has demonstrated significant potential both in classical and quantum applications. [41]Here, the light-matter interaction tends to align the molecules along the axis of the field but with a degree of alignment in the (x, y) plane that decreases as the ellipticity increases. [42]As a result, the molecules are predominantly aligned along the x-axis in the region Àπ=2 < φ < π=2 (right part of the interaction area) but with a degree of alignment which decreases with jφj.In particular for φ ¼ π=2, the pump field is perfectly circularly polarized so that the axis of molecular alignment points toward z, the propagation axis.As a function of the time, the molecules will be alternately confined along z or delocalized in the transverse plane without any preferential direction.There is therefore no net anisotropy in the x À y plane for φ ¼ π=2.The same situation occurs for the left part of the interaction area but with molecules preferentially aligned along the y-axis.This result is confirmed by the Hamiltonian interaction given in Equation ( 4).For the input field E R ðt, φÞ ¼ ε 0 ðtÞ cosðφÞe Àiω 0 t under consideration, we have a s ðφÞ ¼ jcosðφÞj and ϕ s ¼ 0 (resp.π) in the right (resp.left) part of the interaction area in Figure 6a.As already explained, the molecules will be confined along the angle Φ ¼ ϕ s 2 and therefore along the x (resp.y)-axis in the right (resp.left) part of the beam profile with a coupling term that varies as a s ðφÞ ¼ jcosðφÞj.This spatial structuring of the sample leads to an azimuthal variation of its anisotropy in the (x, y) plane, this last being maximal for φ ¼ ½0, π and zero for φ ¼ AEπ=2.A circularly polarized probe beam interacting with such a sample will experience a spatially dependent modification of its ellipticity enabling to restore the OAM mode or OAM mode superposition after a circular analyzer.It produces for the case jΨi in ¼ j1iþjÀ1i ffiffi 2 p a signal with two petals located at φ ¼ 0 and π that are phase-shifted by π as a result of the orthogonal direction of the molecules in the right and left part of the interaction area (Figure 6b).From the birefringence of the molecular sample, it can readily be shown (see Section 4, Supporting Information) that the signal field is of the form E sig ∝ cosðφÞ that corresponds to the photon state jΨi in ¼ ðj1i þ j À 1iÞ= ffiffi ffi 2 p stored in the molecular sample.

Conclusion
We have shown the storage of OAM modes superpositions carried by ultrashort laser pulses into rotational coherences of a molecular system.The approach, exploiting the field-free molecular alignment and the interplay between SAM and OAM, features the ability to store both the spatial phase and amplitude of light beams.The storage mechanism can be interpreted by the production of a time-varying spatial pattern of   The two empty circles at the poles have been reported in our previous work. [33].
(a) (b) : the superposition of jΨi in with the auxiliary beam (j0i) of opposite helicities provides a global field with an azimuthal distribution of amplitude and polarization inducing a spatially varying molecular alignment.A schematic representation of the axis of alignment with its amplitude is depicted on the right part.b) The interaction of a circularly polarized probe field with the molecular sample at the occurrence of molecular alignment produces a field of opposite handedness (isolated by a circular analyzer CA) embeddingjΨi in .
aligned molecules in three dimensions embedding the spatial structure of the beam to be stored.Once produced, this pattern can restore the spatial information into a reading beam.The storage time relying on molecular alignment is of the order of 100 ps in the condition of the present work but can be increased up to several nanoseconds [43] if the gas density is reduced.Furthermore, while the storage time is a crucial parameter for long-distance communication, such a coherent buffer memory can be attractive for ultrafast local quantum processing where other properties as the bandwidth and multimode ability can be of more interest. [44]In comparison to other quantum memory protocols such as controlled-reversible inhomogeneousbroadening [45] or atomic frequency comb [46] with multimode ability, the current approach presents the capability for a truly on-demand readout of the spatial information.This is made possible through the field-free permanent molecular alignment providing a residual alignment between the revivals. [33]esides applications as THz-bandwidth storage medium, the strategy exploiting molecules as light-matter interface features interesting prospects for various applications.The present strategy could be for instance applied for the design of q-plates enabling to generate beams with a superposition of various l values.Standard q plates, used for embedding pure OAM states, usually rely on liquid crystal films carefully orientated with a specific pattern in the plane perpendicular to the plate surface.A 3D design, as prescribed in the present work, would allow an extension of the method to impart OAM superpositions.The rich structure of states also constitutes a specific attribute of molecules compared to atoms.This specificity could be exploited to store complex waveform as space-time wavepackets embedding time-varying OAM [47] or multiplexed light embedding both SAM and OAM. [31,32]Finally, the set of strategies developed in this last decades to control the rotational coherences and the angular distribution of molecules in space [48][49][50][51] could be of great interest for the manipulation and tunability of OAM beams.A controllable laser-induced modification of the molecular pattern and therefore of the state stored by the molecular sample constitutes the basic building block of quantum logic gates and can be exploited to produce complex optical waveforms with a spatially (and possibly time) varying phase and amplitude structure.

Figure 3 .
Figure 3. Storage of a balanced superposition l ¼ AE1 with in a,d) β ≈ 0, b,e) β ≈ π, and c,f ) β ≈ π=2.Left column: spatial distribution of the restored field with its spatial phase as a color scale.Right column: decomposition of the signal field into first OAM eigenstates, the inset depicts the spatial distribution of the input beam stored in the molecular sample.

Figure 4 .
Figure 4. Superposition of unbalanced l ¼ AE1 OAM states.a) Intensity pattern of the input pump beam to be stored, b) intensity pattern of the measured signal field, c) dashed blue (resp.solid red) angular distribution of the integrated signals depicted in b) (resp.a), and d) decomposition of the signal field into the first OAM eigenstates (blue square) compared to the encoded ratio a 1 /a À1 (red line) evaluated from the contrast measured in c).

Figure 5 .
Figure 5. Evolution of light field modes through a Bloch sphere description.The red, green, blue, and yellow full circles correspond to the structured beams stored in Figure3a-c, and 4, respectively.The two empty circles at the poles have been reported in our previous work.[33] .