Material comprising atoms excitable in rydberg states magnetized via inverse faraday effect.

Abstract

Material comprising atoms in Rydberg states magnetized via Inverse Faraday Effect having a magnitude of the effective magnetic field that scales substantially as the principal quantum number n raised to the power of 4 and method for magnetizing the material thereof, an atom-based quantum computing processor comprising a quantum platform populated with atomic quantum carriers configured to define qubits that are material as previously described and method for actuating the atom- based quantum computing processor thereof.

Description

[0001] MATERIAL COMPRISING ATOMS EXCITABLE IN RYDBERG STATES MAGNETIZED VIA INVERSE FARADAY EFFECT, ATOM-BASED QUANTUM COMPUTING PROCESSOR COMPRISING THE MATERIAL THEREOF, METHOD FOR MAGNETIZING SUCHMATERIAL AND METHOD FOR ACTUATING THE ATOM-BASED QUANTUM COMPUTING PROCESSOR THEREOF. The present invention relates to a material comprising atoms excitable in Rydberg states magnetized via Inverse Faraday Effect, an atom-based quantum computingprocessor comprising the material thereof, a method for magnetizing such materialand a method for actuating the atom-based quantum computing processor thereof. In some aspects, the present invention finds a preferred, though not exclusive, application in the field of quantum computing and particularly in ultrafast magnetic switching devices. It is known that Rydberg atoms are atoms characterized by a large orbital radius giving such excited states an interesting large angular moment suitable for severaldifferent technical applications. Excitation of the atoms can be achieved by light pulseand electric pulse to promote the atom to excited state. There is an increasing interest in developing improved ways of usage and optimization of such specific materials due to their surprising properties and innovative potentialities in quantum physics applications.Among them, Rydberg atoms applications have been explored in technical fields suchas atom-based quantum computing processors. Quantum computing processors are configured to perform operations on data adopting quantum-mechanical phenomena, such as superposition and entanglementand in this view, it is immediately apparent their difference from digital electroniccomputers based on transistors. Indeed, whereas digital computers elaborate data encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computing processors process quantum bits (qubits), which can be in a coherent superpositions of both states simultaneously.In this context with the terms “Inverse Faraday Effect” an opto-magneticphenomenon in which a static (dc) magnetization is dynamically induced in matterby a light field is identified.More specifically, the Inverse Faraday Effect (IFE) is in contrast to the conventional Faraday effect. In the latter, a linearly polarized light passing through a magnetized medium undergoes a rotation in its polarization axis. On the other hand, the IFE takes place when circularly polarized light, with a rotating polarization, induces amagnetization in the material. Further details will be disclosed in the following.In this context, the terms “uniform distribution” refer to a distribution where any point within a given volume has a same probability to be occupied by matter or atoms, determining in such a way a homogeneous material. Uniform distribution could be evaluated according to preferred directions, determining thus an anisotropic condition, or according all the directions, determining thus an isotropic condition. In this context, the terms “quantum platform” refer to a register which is an essential component used to store and manipulate quantum information. It comprises aplurality of qubits which due to superposition of states and entanglement can createcorrelations between qubits in order to perform quantum algorithms.In this context, with the terms “Rydberg states” are identified atoms having a low transition energy so that to produce a widely spatially extended electronic wavefunction thereby enabling manipulation of these states with external gates and laserpulses. Furthermore, with the terms “Rydberg atoms” are identified atoms having a high principal quantum number n. In this sense, thus, the terms “Rydberg states” can have a wider meaning than“Rydberg atoms” comprising, for instance, also dopant atoms in a semiconductor witha low principal quantum number n and a low transition energy.The Applicant during its continuous aim of producing technical improvements in the quantum physics technology field has felt the need to explore and find a technical solution that allowed a further efficient modification of the matter as a function of its specific composition in order to produce better magnetically responding platformsuitable to be applied in quantum physics applications such as atom-based quantumcomputing processes. In a first aspect, the invention in issue is addressed to a material comprising atomssuitable to be excited in Rydberg states having a low transition energy. The modalitiesfor exciting atoms could be e.g., the pulses of light or electronic control gates on topof semiconductor.Preferably, such atoms suitable to be excited in Rydberg states are magnetized via Inverse Faraday Effect. Preferably, such magnetization has a magnitude of the effective magnetic field thatscales substantially as the principal quantum number n of the atoms in Rydbergstates raised to the power of 4 (i.e. n4).Thanks to this technical solution it is possible to induce a really high magnetizationof the material that further shows ultra-fast magnetic switching properties.In other words, the IFE is applied to produce coherent orbital polarization of atoms with IFE light and thus obtaining the desired material magnetization. To put in a further perspective, the technical solution is addressed to a material comprising atoms excited in Rydberg states having a low transition energy and magnetized via Inverse Faraday Effect having a magnitude of the effective magnetic field that scales substantially as the principal quantum number n raised to the power of 4, inducing high magnetization of the material, long range coherence and alignment of orbital and spin moments of the quantum states allowing ultra-fast magnetic switching properties. In a second aspect the invention is addressed to an atom-based quantum computing processor. Preferably, the atom-based quantum computing processor comprises a quantumplatform populated with atomic quantum carriers configured to define qubits that arematerial having at least one of the features previously described. Preferably, the atom-based quantum computing processor comprises a quantum- state control hardware. Preferably, the quantum-state control hardware includes a photon system configured to cause individual atoms to move between quantum states switching reversibly from ground state to Rydberg states. Preferably, the quantum-state control hardware includes a vacuum system. Preferably, the quantum-state control hardware includes a magnetics system. Preferably, the quantum-state control hardware includes a circularly polarized light emitting system configured to induce Inverse Faraday Effect on the Rydberg states of the quantum platform. Preferably, the quantum-state control hardware includes a measurement systemconfigured to detect variations signal in the quantum platform due to superpositionstates. In this way it is possible to produce an atom-based quantum computing processorthat benefits of the technical advantages of having a quantum platform comprisingthe material having atoms in Rydberg states hugely and efficiently magnetized viaIFE. Such an innovative technical solution allows to increase significantly the magnetic switching and thus improving the efficiency and performances of the quantum computing processor with regards to the state of the art. More specifically, the IFE effect will induce long range coherence and alignment of orbital and spin moments of the quantum states. Furthermore, it would also facilitate the state initiation and control in the quantum processor. It is interesting to notice that moving individual atoms between quantum states switching reversibly from ground state to Rydberg states can occurs both directly and indirectly: directly upon applying light, indirectly using e.g. electric controls to excite Rydberg state. Furthermore, when intermediate states are achievable, the switching could happen either bypassing the intermediate state or bringing firstly the atom in the intermediate state and then to the Rydberg quantum state. In a third aspect the invention is addressed to a method for magnetizing a material comprising arranging a material comprising atoms suitable to be excited in Rydberg and magnetized via Inverse Faraday Effect. Preferably, the method for magnetizing a material comprises inducing a quantum state transition moving at least an atom of the material to a Rydberg state. Preferably, the method for magnetizing a material comprises magnetizing the at least one atom of material in the Rydberg states via Inverse Faraday Effect. In such an innovative way it is possible to obtain a new material which has new and surprising quantum magnetic features. In a fourth aspect the invention is addressed to a method for actuating an atom- based quantum computing processor. Preferably, the method for actuating an atom-based quantum computing processor comprises providing the atom-based quantum computing processor according to the features previously described. Preferably, the method for actuating an atom-based quantum computing processor comprises initializing the quantum computing process. Preferably, the method for actuating an atom-based quantum computing processor comprises activating the photon system configured causing at least one atom of the material to move to Rydberg states defining a first qubit. Preferably, the method for actuating an atom-based quantum computing processor comprises applying a circularly polarized light to the at least one atom in Rydberg state inducing a magnetization of the effective magnetic field having a magnitudethat scales substantially as the principal quantum number n raised to the power of 4(i.e. n4). Preferably, the method for actuating an atom-based quantum computing processor comprises actuating a superposition of the quantum states of the at least one atom in the Rydberg state magnetized defining a second qubit as a superposition state of the first qubit. Preferably, the method for actuating an atom-based quantum computing processor comprises activating a measurement system to read out the second qubit. Preferably, the method for actuating an atom-based quantum computing processor comprises completing the quantum computing process. In such a way it is possible to perform an innovative, fast and efficient way of adopting the new proposed material in atom-based quantum computing improving the computational performances. In at least one of the above-mentioned aspects, the present invention may also have at least one of the preferred features described below. Preferably, the Rydberg states are free Rydberg atoms with principal quantum number n>>1. Preferably, the Rydberg states are free Rydberg atoms with separation between atoms on the scale of optical trap sizes and sparse spacing of dopant atoms. Typical length scales involved in these devices is on a scale of fraction of microns and larger. Preferably, the free Rydberg atoms have principal quantum number n≥30. The Applicant found that the induced magnetic moment can be understood as the induced angular momentum of an orbital particle’s movement. Furthermore, the effect scales as the squared effective Bohr radius and the larger the effective radius of the bound state, the greater the induced moment becomes. More specifically, thanks to this solution the Applicant has been able to achieve a great amplification of the IFE adopting higher-lying Rydberg states in free atoms which exceeds significantly that for the hydrogen atoms (having n=1). Preferably, the Inverse Faraday Effect has an applied electric field of ε =104Vm-1, frequency ω = 1THz and an effective induced magnetic field Beff ≥10mT. In such a way it is possible to produce in an efficient way a stable material magnetization. According to a different embodiment, the Inverse Faraday Effect could be determinedas a function of the spectroscopic line of dopant atoms in quantum device. Theproposed method works in nonresonant condition where the applied frequency ofmicrowave or THz or electric modulation does not have to be in resonance with thetransition of the Rydberg state. Yet the estimates indicate the effect is most efficient when the applied agitation signal (light, electric modulation etc) is in resonance with the transition of the Rudberg state. Hence to maximize the efficiency of induced magnetization we suggest to apply the light or electric modulation close to resonance line of the targeted energy range of a Rydberg state.Preferably, the free Rydberg atoms are rubidium (Rb) or caesium (Cs).Due to their high principal quantum number n it is possible to obtain strong andadvantageous magnetization of IFE on their Rydberg states. Preferably, the Rydberg states are shallow dopants in a semiconductor. Preferably, the Rydberg atoms are shallow dopants in a semiconductor having principal quantum number n≥2. Thanks to this solution, the effect of the magnetization due to IFE is four orders of magnitude larger for shallow dopants in semiconductors, where the Bohr radius, which when multiplied with n^2 sets the length scale for all Rydberg states with principal quantum number n, exceeds by roughly two orders of magnitude that for the hydrogen atom. Preferably, the dopants in shallow semiconductor have a uniform distribution involume and an average inter-dopant atoms distance d ≤ 10-7 – 10-6 m.In such a way the distance is greater than the Bohr radius of the dopants by over anorder of magnitude and wave functions of the dopants do not overlap. Allowing andefficient achievement of the desired effect. In this sense, dopants are deposited either in the bulk material homogeneously and / or on the surface preferably using electrostatic gates to better control Rydberg states.Preferably, the shallow dopants in a semiconductor are phosphorus (P) dopants insilicon (Si) or phosphorus (P) dopants in germanium (Ge).Thanks to these innovative solutions, when silicon is exposed, for instance, to TeraHertz light it behaves a s a transparent material neglecting any or major dissipative effects conferring the advantage that not only dopants on the surface of the silicon sample will respond to the THz light, but also dopants within the volume increasing the response efficiency. Furthermore, germanium has a higher dielectric constant than silicon leading to a larger Bohr radius and thus to enhanced magnetization response.According to one embodiment, the dopants concentration is below or on the order of 1014cm-2, preferably comprised between 1010cm-2and 1014cm-2and more preferably comprised between 1013cm-2and 1014cm-2for surface doping. Preferably, the dopants concentration is below or on the order of 1018cm-3, preferably comprised between 1016cm-3and 1018cm-3for bulk doping. In such as way it is possible to balance the required amount of dopants concentration with the advantageous production and usage of the magnetization IFE induced in the Rydberg states. Preferably, the photon system is a TeraHertz pumping ion system. In such a way it is possible to efficiently induce a transition of the material to aRydberg state in order to apply the invention in issue.Furthermore, when applied to silicon atoms such radiation does not interact withthem avoiding dissipating effects and thus optimizing the excitation process.Preferably, the TeraHertz pumping ion system works substantially at 0.1 – 10 THz,more preferably substantially at 1 THz. The Applicant found that such a value is an effective solution in order to perform the invention in issue. Preferably, the inducing a quantum state transition moving at least an atom of the material to a Rydberg state is performed via TeraHertz ion pumping. As previously discussed, in such a way it is possible to efficiently induce a transition of the material to a Rydberg state in order to apply the invention in issue. Preferably, the method for actuating an atom-based quantum computing processor comprises providing one or more optical trapping units configured to generate a plurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms of the material. Preferably, the method for actuating an atom-based quantum computing processor comprises providing one or more entanglement units configured to quantum mechanically entangle at least a subset of the at least one atom in the one or more superposition states. Preferably, the method for actuating an atom-based quantum computing processor comprises providing one or more readout optical units as measurement system to read out the second qubit. In such a way it is possible to efficiently produce a preferred quantum computing process. The characteristics and advantages of the invention will become clearer from the detailed description of an embodiment illustrated, by way of non-limiting example, with reference to the appended drawings in which:-Figure 1 is a schematic representation of induced orbital angular momentumgenerating magnetic moment according to one embodiment of the invention in issue; -Figure 2 is a schematic view of a splitting of degenerate energy levels due to a second-order interaction with the applied circularly polarized field ℰ^ / ^where the energy splitting ∆E is attributed to the effective magnetic field Beff; -Figure 3 is a plot reporting the calculated values for the effective induced magnetic field for rubidium and caesium over a range of principal quantum numbers n. The inverse Faraday effect (IFE) is an opto-magnetic phenomenon in which a static (dc) magnetization is dynamically induced in matter by a light field. It is often stated as an induction of dc magnetization as a result of illumination by circularly polarized light: ^^^⃗ ^^ ∼ ^^⃗ (^) × ^^⃗∗(^)  (1)Here ^ ^^⃗ (ω) is the complex-valued electric field of light vector at frequency ω, which isassumed to be monochromatic hereafter.As previously discussed, the inverse Faraday effect (IFE) is in contrast to theconventional Faraday effect. In the latter, a linearly polarized light passing through a magnetized medium undergoes a rotation in its polarization axis. On the other hand, the IFE takes place when circularly polarized light, with a rotating polarization,induces a magnetization, de-noted as ^^^⃗ dc. One particularly intriguing application ofthe IFE in magnetic systems is ultrafast magnetic switching. At its core, the strength of the effect is linked to the size of an electron’s bound state. In simpler terms, the induced magnetic moment can be understood as the induced angular momentum of an orbital particle’s movement. ^^^⃗ ^^ ∼ ^^⃗ (^) × ^^⃗∗(^) . (2)Further, the effect scales as the squared effective Bohr radius ^^: ^∼ ^ ^^^ ^  . (3)Here ^⃗ is the operator of the electron centre of mass and we estimate the average ofthe electron angular momentum scale as ^^. Thus, the larger the effective radius of the bound state, the greater the induced moment becomes. This insight led us to hypothesize a great amplification of the IFE as we go to higher-lying Rydberg states in free atoms. The effect is four orders of magnitude larger for shallow dopants in semiconductors, where the Bohr radius, which when multiplied with n^2 sets the length scale for all Rydberg states with principal quantum number n, exceeds byroughly one to two orders of magnitude that for the conventional hydrogen atom.Figure 1 illustrates excited atomic wave functions, characteristic of either the dopant states in a semiconductor or Rydberg atoms. The incoming circularly polarized beam induces an emergent ferromagnetic state with oriented magnetic moments ^^arising from the driven orbital angular momentum. The IFE is characterized by a nonlinear optical response which can be described by^^(0) = ^^^^(0; ^, −^)^^(^)^^(−^), where the indices α, b, and c denote Cartesiancoordinates and the Einstein summation convention is implied.In the present document, firstly, the IFE is discussed as heuristic classical model.Expanded details of the development of the quantum theory are reported in the followings. Classically, the inverse Faraday effect can be understood as the magnetic moment arising from the orbital angular momentum of a charged particle per unit mass as where q is the electric charge of this particle and m* its effective mass. This expression illustrates that the magnitude of the induced magnetic moment is proportional to the square of the orbital radius |^(⃗^)|^. In the context of Rydberg atoms and shallow semiconductor dopants discussed above, the large effective Bohr radius they possess therefore implies the possibility of inducing a large magnetization due to the Inverse Faraday Effect. The magnetic moment in Eq. (4) can be obtained by considering free electrons under the Drude-Lorentz model. In this model, the equation of motion for the electrons is given by For circularly polarized light of frequency ω, the external force takes the form of with the amplitude ℰ and complex-valued vector ^^⃗ (^) = ℰ[^ + ^^]of the externalmonochromatic electric field, and unit vectors ^ and ^ in Cartesian coordinates. Thesolution in the frequency domain for the equation of motion reads as This results in a magnetic moment induced by the IFE as given by Eq. (4) as To model a bound state, we consider the resonance coherent light source, when the damping term dominates in the denominator. The induced magnetic moment then takes the form (9)With a damping dominated term, 1 / ^ is proportional to the average radius ^^^ as^(^) ∼ 1 / (^^ + ^^^ − ^ ^^) in an undriven system. We identify this with the Bohr radius^^. We then interpret the magnetic moment then as Here we see the functional dependence of the induced magnetization on ^^, ω, and ℰ. This scaling dependence will also be seen in the quantum calculation. The quantum mechanical description of the IFE can be treated as an effect arising from a second-order process in the dipole interaction. The Hamiltonian for this systemis given by ^ = ^^ + ^(^) with the potential Where ^(^) = −^ ^⃗ ⋅ ^^⃗ (^). The applied field ^^⃗ is taken to be circularly polarized andpropagating in the ^̂ direction, which is defined to be the direction perpendicular tothe target sample surface. Defining ^±= ^ℰ^ ∓ ^ℰ^^, we have^ = −^(^^ℰ^ + ^^ℰ^), where L and R denote left- and right-hand circular polarizations.An effective Hamiltonian can be defined to describe the second-order dipoleinteraction process yielding the IFE. The effective Hamiltonian can be expressed asan operator whose matrix elements are given bywhere ^(^) = ^^^^^ / ℏ^(^)^^^^^^ / ℏ, |^^ and |^^ are eigen-states of the ground multipletwith energies ^^ and ^^, while the summation is taken over the excited states |^^ withenergies ^^. These matrix elements for the zeroth harmonic generation components for the dipole interaction can be computed as where ℏ^^^ = ^^ − ^^ and ^± and ℰ^ / ^ defined as above. Details on the derivation ofthe expression Eq. (13) can be found in Appendix A. The analysis of the effect in the present discussion is valid for long times, where the characteristic time scale offluctuations in the applied light source ^(^) is on the order of nanoseconds orpicoseconds. For the ultrafast regime of femtoseconds, the analysis needs to be augmented by considering the shape of the incident pulse. Details regarding modifications to the effective Hamiltonian necessary for this regime can be found in D. Popova, A. Bringer, and S. Blügel, Theory of the in-verse Faraday effect in view of ultrafast magnetization experiments, Phys. Rev. B 84, 214421 (2011). The action of the effective Hamiltonian is to break the degeneracy in the energy levels between the initial and final states as illustrated in Fig. 2. Given that the external field breaks time-reversal symmetry, the effective Hamiltonian can be interpreted as an effective magnetic field with an effective magnetic moment. The matrix element Eq. (13) can therefore be re-expressed in terms of this effective magnetic field instead of the physical electric field, reading as where ^^ = ^ℏ / (2^∗) is the effective Bohr magneton and the quantity in the bracketshas units of magnetic field strength. The effective Hamiltonian now takes the form ofa Zeeman term ^eff = ^^ℬeff. We emphasize here that there is no real magnetic fieldpresent, however the atoms respond as if they possessed a unit Bohr magneton magnetic moment immersed in a magnetic field of strength ℬeff. It is possible now to apply this formalism to calculate the magnitude of the IFE for Rydberg atoms and shallow dopants in semiconductors. Since the interest inRydberg-like states, the states |^^, |^^, and |^^ are taken to be hydrogenic states definedin terms of their principal quantum number n, orbital angular momentum number ℓ, and magnetic quantum number m, such that the states may be labelled by the tuple|^^ =|^, ℓ, ^^. Their specific form is given by Eq. (A17). The magnetization consideredis an orbital angular momentum effect and spin states are not considered in this analysis, leaving it for further more detailed descriptions. Rydberg atoms are isolated atoms with a valence electron occupying a state with n≫ 1, which results in a highly extended radial wave function and a large electric dipolemoment. The large radial extent of the Rydberg wave function means that the outer electron experiences a very weak (shallow) potential. The shallowness of this potential allows Rydberg atoms to be highly sensitive to external electric fields which makes them an attractive platform also for metrological technologies. Rydberg atom- based platforms can be sensitive to electric fields on the order of 10−1V m−1. Rydbergatoms can be adopted as precision radio frequency sensors, which are highly tunableto specific frequencies and are robust to noise. Moreover, these states are consideredto be highly sensitive to magnetic fields as well.A microscopic description of the IFE effect for Rydberg states is hereby provided. Thedependence of the IFE on Rydberg quantum numbers can be estimated from quasi- classical considerations. A particle of charge q in a bound state of a central potentialwith coupling constant k obeys ^^ / (2^^) = ^^^ / ^^. A dependency on theprincipal quantum number can be obtained from the Bohr-Sommerfeld quantizationcondition ^ ⋅ ^ = ^ℏ or ^ = ^ℏ / ^. Inserting this condition into the energy balanceexpression yields ^ = ^^ℏ^ / (2^^^^). The radius of the bound particle therefore scalesas n2. It is therefore expected that Rydberg states yield a strong IFE as a consequence of their large n values. In the followings the magnitude of the effective magnetic field induced by the IFE for the typical Rydberg atoms rubidium (Z = 37, ^^= 235 pm) and caesium (Z = 55, ^^= 260 pm) is calculated. In these calculations spin-orbit coupling is ignored. Since the Rydberg states have wave functions with wide spatial extent, the coupling between the valence electron and the atomic nucleus is very weak, making the omission of spin-orbit coupling a reasonable assumption. To estimate the size of the effect, the expression Eq. (14) is used over Rydberg wavefunctions with parameters of the applied beam energy of ℐ = 10 W / cm^, correspondingto applied electric field of |ℰ| = 10^ V / m and ω = 1 THz. We calculate the value of theIFE for states with the quantum numbers |^, 1, ±1^. The calculated values for theeffective induced magnetic field for rubidium and caesium over a range of principalquantum numbers n is plotted in figure 3. More specifically, in figure 3 it is shown the magnitude of the induced effective magnetic field ℬeffdue to the inverse Faraday effect for Si:P (green), rubidium (red circles), and caesium (blue diamonds) evaluated for a range of principal quantumnumbers ^ with an applied 1 THz beam with intensity 10 W / cm^. The data points forSi:P are for |ℓ, ^^ (+), |2, −1^ (×), and |2, −2^ (△). The curves are fits to thenumerical data and are 4th order polynomials, given in Eqs. (C1), (C2), and (C3) forSi:P, and Eqs.(B1) and (B2) for the Rydberg atoms. The curves for Si:P terminate at^ = 2 and 3 as the IFE is only finite for states where ^ ≥ ℓ + 1. The numerical valuesfor ℬeff are shown in Table 4 in Appendix C, and Tables 2 and 3 in Appendix B.Numerical values are given in Tables II and III in Appendix B. It is found that themagnitude of the IFE in Rydberg systems scales as n4. This scaling behaviour is expected and reasonable since the spatial extent of a Rydberg wave function scales as n2and the IFE scales as the radius squared. The size of the IFE for both rubidiumand caesium at n ∼ 30 is Beff∼ 10 mT. For the numerical values in Fig. 3 and TablesII and III, the IFE scales as The above calculation shows that for Rydberg atoms, the IFE obeys ℬ^^ / ^eff^≈ const.This scaling can be used to extrapolate estimates for other species of Rydberg atoms. The IFE for the hydrogen atom was reported with an effect scaling linearly with theaverage radial extent ^^^ of the wave function, or as n2, in contrast with our resultsabove. However, in regards of the hydrogen atom analysis the effect was onlycalculated up to n = 3. It is indeed relevant to note that for n ≤ 4 it is possible toconfirm that n2does accurately describe the scaling of the hydrogenic IFE, however for larger values of n this scaling no longer holds and n4is the appropriate scaling behaviour, as disclosed here. A Rydberg system (or state) is generally an isolated atom excited to a state with avery high principal quantum number n ≫ 1. However, due to the shallow potential ofcertain dopants in semiconductors, a Rydberg state can be achieved already at lown, such as n = 2. This means that that only a low transition energy is required toproduce a widely spatially extended electronic wave function. TeraHertz (THz) pumping of doped silicon reveal that there exists a large cross- section for two photon absorption (2PA) of shallow dopants to an excited Rydberg- like state. These Rydberg states possess a very large effective Bohr radius. The shallow potential contributing to this large radius arises from the high dielectric constant of silicon and the low effective electron mass. The effective mass ofelectrons in silicon is ^∗ = 0.3^^ of the electron mass and the Bohr radius is ^^ == 3.17nm, which is significantly larger than a typical atomic Bohr radius such ashydrogen, where ^^ = 0.05 nm, or even the Rydberg atoms considered in the previoussection which had ^^ = 0.235 nm and 0.260 nm for rubidium and caesium respectively.The shallow potential of these dopant atoms means that their low energy excited states possess a very large spatial extent with a large dipole moment. Our goal is to exploit this property to induce a large orbital magnetization via the IFE.For concreteness, in this paper the case of phosphorus doped silicon, Si:P will beconsidered. The phosphorus atoms (Z = 15, ^^ = 3.17 nm) form the shallow Rydberg-like states. While focus on silicon in the present disclosure, it is possible to notehowever that compared to silicon, which has a dielectric constant of ^Si = 11.8 ^^,germanium has a higher dielectric constant of ^Ge = 16.0 ^^, which denotes thatdopants deposited on germanium reasonably have an even more pronounced IFEthan those for silicon as the larger dielectric constant leads to a larger Bohr radius.We emphasize that the electron in the excited Rydberg state is still bound to thedopant atom. The donor electron is not excited into the continuum conduction band and remains in a bound state with the dopant atom. In the context of the present work, the doped semiconductor serves as a solid state platform for demonstrating the IFE for individual atomic systems in contrast to the free or trapped atoms of the previous section. With the expression Eq. (14), it is possible to evaluate the magnitude of the IFE for shallow dopants in semiconductors. A series of excited Rydberg states are accessible from THz pumping of phosphorus donors in silicon. These states are used to form the basis states for the matrix elements in Eq. (13). To obtain numerical estimates of the magnitude of the effective magnetic field, it can be assumed typical experimentalvalues of 1 THz for the frequency of the applied beam, 108 W cm−2 for the appliedbeam intensity or 10^ V / m in terms of electric field strength. This electric fieldstrength is an advantageous terahertz spectroscopy range in such experiments. Furthermore, Silicon is transparent to terahertz light, so it is not needed to consider any dissipative effects. This transparency also implies that not only dopants on the surface of the silicon sample will respond to the THz light, but dopants within the volume will as well. The diagonal matrix elements of the effective Hamiltonian for aselection of basis states |^^ =|^, ℓ, ^^ yielding values for the effective magnetic field isgiven in Table I. We find that for matrix elements with finite amplitude, the effective magnetic field has magnitude of ^(10^T). The total energy splitting induced by theeffective Hamiltonian for the relevant states is ^(10 meV). Off-diagonal matrixelements are those with initial and final states differing in the principal quantumnumber n, but with same angular state ℓ, ^.In terms of relevant parameters, the magnitude of the effective magnetic field scales as A magnetization per dopant atom can be obtained from the relation where ^ is the volume magnetic susceptibility and µ0 is the permeability of freespace. For silicon, ^Si / ^^ ≈ −2.97 A^s^ / kg / m. This leads to an induced magnetizationfrom the IFE as ℳeff ∼ 103 A / m. For further reference it is also noted that the value forgermanium, ^ / ^ ≈ ^ ^Ge ^ −6.35 A s / kg / m. As ℳ ∝ ^ and ^Ge > ^Si, we again see that the IFEfor germanium is advantageously greater than for that of silicon.The calculation presented here is for the induced magnetization per dopant atom.Here it is assumed that each dopant can be analysed independently from each other,so for a bulk material we take as an assumption that the density of the dopants is such that the wave functions of the dopants do not overlap. A typical concentration of dopants in Si:P is 10^^cm^^and some of the functional elements like implanted ions can have concentration as low as 10^^cm^^. Assuming a uniform distribution ofdopants in the volume, this implies an inter-dopant distance of ^ ∼ 10–^m > ^^. Sincethis distance is greater than the Bohr radius of the dopants by over an order of magnitude, we expect that samples of this concentration are compatible with our approximations and will allow for the effect described and claimed. Hereafter, Table I is reported expressing a diagonal matrix elements of the effective Hamiltonian showing values for the magnitude of the effective magnetic field ℬefff and the corresponding magnetization ℳefffor Si:P.Table I. Diagonal matrix elements of the effective HamiltonianIt is, thus, considered that in such a disclosure, among several technical aspects, theinverse Faraday effect (IFE) has been discussed in the bulk materials. Building upon this understanding, new IFE applications to shallow semiconductor dopants and Rydberg states are described and claimed. It has furthermore demonstrated thecondition of inducing large orbital magnetization IFE in these examples of Rydbergsystems.The first example was that of an isolated atom in a large n Rydberg state, specificallyrubidium and caesium atoms in states up to n = 32, where we find that the IFE caninduce an effective magnetic field on the order of 10 mT, and we find that the effect scales as n4, implying a very large enhancement for higher n states. The second example studied was that of shallow semi-conductor dopants. We find that for phosphorus doped silicon, Si:P, the IFE can produce an induced effectivemagnetic field for the dopant states of ℬeff ∼ ^(10^T), or a corresponding orbitalmagnetization of ℳeff ∼ ^(10^A / m). This process was shown schematically in Fig. 1. Terahertz pumping of silicon is practical and efficient technical solution. It is furthermore considered that germanium is considered an innovative embodiment of the invention in issue due to the comparatively higher susceptibility of germanium increasing the magnitude of the effect.For the above cases analysis laser time scales on the order of nanoseconds witheffectively stationary magnetization dynamics. In a broader context, Rydberg states are hereby described as platforms for atom- based quantum computing, owing to their advantageous long range entanglement properties.As previously discussed, shallow dopants in semiconductors exhibit characteristicssimilar to solid-state analogues of Rydberg states being adapted for quantumcomputing with ultrafast switching of magnetic moments via IFE.In the following further details are discussed in the Appendixes attached heretoA. Derivation of the effective HamiltonianIn this appendix we review the derivation of the effective Hamiltonian employed in the main text to describe the inverse Faraday effect. Our derivation follows theconstruction originally developed in Ref. (Pershan, Ziel, and Malmstrom 1966). Thewave function for the dopants obeys the evolution equation where we consider a Hamiltonian of the form ^ = ^^ + ^(^) with ^^ the unperturbedHamiltonian and ^(^) is an interaction with an external field to be consideredperturbatively. Spin-orbit coupling is not considered in the following. In the interaction representation, the evolution equation takes the form of where ^(^) = ^^^^^ / ℏ^(^) and ^(^) = ^^^^^ / ℏ^(^)^^^^^^ / ℏ. The formal solution to theevolution equation can be obtained as We d an asgenerator of transitions between states |^^ and |^^ due to the interaction ^(^) by The term of interest here is the process second-order in ^(^). The first-order term is neglected as we assume that there are no relevant absorption processes, assuming the optical frequency is non-resonant with any atomic frequency, so we have^^^^(^′)^^^ = 0. This leads to the effective Hamiltonian being defined in terms of thematrix elements of the second-order interaction as Here the states |^^ and |^^ are eigenstates of the ground multiplet and |^^ are excitedintermediate states. The integral can be evaluated by using the approximation that the interaction is slowly varying over the time scale considered. Assuming variationsin the interaction ^(^) occur on a characteristic times scale of ^, the integral in Eq.(A5)can be approximated with This approximation is valid in the case where the characteristic time scale is in theregime where ^|^ ± ^^^| ≫ 1, such that ^(^) can be considered constant over theintegration time, with an adiabatic ramp-up initial condition ^(−∞) = 0. With theunderstanding of this approximation, we suppress the time dependence of ^ and ^∗in the following. This approximation also assumes that the transitions are off-resonant, |^^^ ± ^|   ≮   ^^^. Here we define the notation ℏ^^^ ≡ ^^ − ^^ where ^^ is theenergy of state ^. Using this approximation we can write the effective Hamiltonian as The second pair of terms on the right-hand side of Eq.(A9) are proportional to ^±^^^^and therefore represent second harmonic generation (SHG) processes. These terms will not be considered in the following as our focus in this paper is only in the zeroth harmonic generation (dc) effects. The interaction we consider is the electric dipole interaction, with the interaction taking the form of ^= −^(^^ℰ^ + ^^ℰ^) , ^ ∗ = −^(^^ℰ∗^ + ^^ℰ∗^) . (^10)The rectified zeroth-order harmonic mode can now be expanded asExpansion of the matrix elements of ^ and ^∗ yields contributions to the effectiveHamiltonian which are proportional to ℰ^^ / ^ = ℰ ∗^ / ^ ℰ^ / ^. For transitions involving ^^ = 0, the energies ^^ and ^^ are5 degenerate, so these second two terms do not contribute. For transitions where ^^ ≠0, ^^^ ∼ 10^^^ (for ^ ∼ 1 THz), so these terms again do not significantly contributecompared to the leading order ℰ^^ − ℰ^^term. The final expression we utilize for the effective Hamiltonian matrix elements used in the main text [Eq. (13)] is then Here ^ is obtained fr ( )^^^^ / ℏ^^^^^ / ℏeff om ^eff ^ = ^ eff^such that . As defined in Eq. (A4), the full dynamical expression for the transitionamplitude is given 15 We may appropriately use the expression Eq. (A15) by assuming a time scale suchthat ^^|^^^eff|^^^^^^ / (^^^^) ≈ ^^^|^eff|^^ where ^ is a time scale.The set of states |^^, |^^, |^^ in Eq. (A15) are eigenstates of the unperturbedHamiltonian ^^. The Rydberg behavior of the excited atomic systems and the semiconductor dopant states means that they can be approximated with hydrogenic20 wave functions with atomic number ^ and Bohr radius ^^: The energies of the eigenstates are given by (^18) where ^ is the dielectric constant and ^ = ^^ / (4^^ℏ^) is the fine-structure constant.The matrix elements are calculated considering wave functions of the form Eq. (A17)in spherical coordinates with ^ . Source code for thenumerical calculations presented in this paper can be foundat (“https: / / github.com / patrickjwong / RydbergIFE / ,” n.d.).B. Numerical data for Rydberg atomsPresented in this appendix are the numerical values of the induced effective magneticfield for Rydberg states plotted in Fig. 2 for rubidium (Table 2) and caesium (Table3). We calculate the effective magnetic field from the matrix element ^^|^eff|^^ with|^^ =|^, 1, ±1^. The intermediary states we sum over are |^^ ∈ {|^, 0,0^, |^, 2, ±2^}. Thefitted functions for the effective magnetic field in Fig. 2 areℬ (^) = −4.21189 × 10^^ + 1.6737 × 10^^^ ^^ ^ ^^ ^Rb − 2.3217 × 10 ^ + 9.33475 × 10 ^ ++1.44179 × 10^^^^ (B1)andℬ (^) ^^ ^^ ^^ ^ ^^ ^Cs = −7.08553 × 10 + 1.90833 × 10 ^ − 1.97067 × 10 ^ + 7.15958 × 10 ^ +7.77412 × 10^^^^ (B2)which have ^^at dominant order for large ^.Table 2. Magnitude of the induced effective magnetic field by the IFE for the |^, 1, ±1^state of Rb.

[0002] 30 ±12.632 ±16.3Table 3. Magnitude of the induced effective magnetic field by the IFE for the |^, 1, ±1^state of Cs. ^ℬeff [mT]8 ±2.52 × 10−210 ±7.12 × 10−212 ±1.58 × 10−914 ±3.05 × 10−916 ±5.33 × 10−918 ±8.69 × 10−920 ±1.3422 ±1.9724 ±2.8226 ±3.9028 ±5.2830 ±6.9632 ±9.02C. Numerical data for shallow dopantsIn this appendix we give the numerical data of the IFE calculations for the shallowsilicon dopants presented in Sec. IV and plotted in Fig. 2. The intermediate stateswhich are summed over are those which obey the selection rules: ^^′, ℓ ± 1, ^ ±1^^±^^, ℓ, ^^, and we take ^′ ∈ {^, ^ ± 1}. Terms not obeying these rules vanish, eitherbecause their overlap is zero, or because summing over intermediary states with ±^ results in values which are equal in magnitude and opposite in sign, and thereforecancel. Transitions with ^^ > 1 are finite, but of subleading order and so are not takeninto account in our calculations. The diagonal matrix elements of the effectiveHamiltonian including all intermediate virtual states considered is shown in Table 4.As with the effective magnetic field generated by the Rydberg atoms, the shallow dopants also exhibit scaling with ^^. The IFE for the dopants was computed for statesof the form |^, 1, ±1^, |^, 2, ±1^, and |^, 2, ±2^. Curves fitted to the numerical data forthese states as plotted in Fig. 2 areℬ (^) = 148.914 − 14 ^ ^ ^|^,^,±^^ 8.675^ + 67.1985^ − 4.06475^ + 0.144196^ , (C1)ℬ (^) = 142. ^ ^ ^|^,^,±^^ 125 − 163.611^ + 64.4927^ − 3.31341^ + 0.105551^ , (C2)andℬ ^ ^ ^|^,^,±^^(^) = 385.522 − 388.863^ + 141.487^ − 7.67950^ + 0.242497^ . (C3)Table 4. Effective magnetic field values of Si:P for the diagonal matrix elements|^, 1, ±1^, |^, 2, ±1^, and |^, 2, ±2^. 2∓81 — —3 ∓221 ∓146 ∓2924 ∓415 ∓342 ∓6845 ∓671 ∓593 ∓11806 ∓967 ∓899 ∓18007 ∓1350 ∓1270 ∓25208 ∓1750 ∓1680 ∓33609 ∓2260 ∓2190 ∓438010 ∓2770 ∓2700 ∓540011 ∓3350 ∓3280 ∓656012 ∓4010 ∓3930 ∓785013 ∓4730 ∓4640 ∓927014 ∓5640 ∓5460 ∓10930

Claims

CLAIMS1. Material comprising atoms suitable to be excited in Rydberg states having a lowtransition energy and magnetized via Inverse Faraday Effect having a magnitude of the effective magnetic field that scales substantially as the principal quantum number n raised to the power of 4.

2. Material according to claim 1, wherein the Rydberg states are free Rydberg atomswith principal quantum number n>>1.

3. Material according to the previous claim, wherein the free Rydberg atoms haveprincipal quantum number n≥30.

4. Material according to any of the previous claims, wherein the Inverse FaradayEffect has an applied electric field of ε =104Vm-1, frequency ω = 1THz and an effective induced magnetic field Beff ≥10mT.

5. Material according to any of the previous claims, wherein the free Rydberg atomsare Rb or Cs.

6. Material according to claim 1, wherein the Rydberg states are shallow dopants ina semiconductor.

7. Material according to the previous claim, wherein the Rydberg atoms have principalquantum number n≥2.

8. Material according to claim 6 or 7, wherein the dopants in shallow semiconductorhave a uniform distribution in volume and an average inter-dopant atoms distance d ≤ 10-7m9. Material according to any of claims 6 to 8, wherein the shallow dopants in asemiconductor are P dopants in Si or P dopants in Ge.

10. Material according to the previous claim, wherein the dopants concentration iscomprised between 1010cm-2and 1014cm-2preferably for surface doping.

11. Atom-based quantum computing processor comprising:- A quantum platform populated with atomic quantum carriers configured todefine qubits that are material according to any of claims 1 to 10, -A quantum-state control hardware including^ A photon system configured to cause individual atoms to movebetween quantum states switching reversibly from ground state to Rydberg states, ^a vacuum system,^ a magnetics system,^ A circularly polarized light emitting system configured to induceInverse Faraday Effect on the Rydberg states of the quantum platform, ^a measurement system configured to detect variations signal inthe quantum platform due to superposition states.

12. Quantum computing processor according to the previous claim, wherein thephoton system is a TeraHertz pumping ion system.

13. Quantum computing processor according to the previous claim, wherein theTeraHertz pumping ion system works substantially at 1 THz.

14. Method for magnetizing a material comprising:- Arranging a material comprising atoms suitable to be excited in Rydbergstates according to any of claims 1 to 10, -Inducing a quantum state transition moving at least an atom of the materialto a Rydberg state, -Magnetizing the at least one atom of material in the Rydberg states viaInverse Faraday Effect.

15. Method according to the previous claim, wherein the inducing a quantum statetransition moving at least an atom of the material to a Rydberg state is performed via TeraHertz ion pumping.

16. Method for actuating an atom-based quantum computing processorcomprising: -Providing the atom-based quantum computing processor according to anyof claims 11 to 13, -Initializing the quantum computing process,- Activating the photon system configured causing at least one atom of thematerial to move to Rydberg states defining a first qubit and, -Applying a circularly polarized light to the at least one atom in Rydberg stateinducing a magnetization having a magnitude of the effective magnetic field that scales substantially as the principal quantum number n raised to the power of 4, -Actuating a superposition of the quantum states of the at least one atom inthe Rydberg state magnetized defining a second qubit as a superposition state of the first qubit, -Activating a measurement system to read out the second qubit, and- Completing the quantum computing process.thod according to the previous claim, comprising:- Providing one or more optical trapping units configured to generate aplurality of spatially distinct optical trapping sites, the plurality of optical trapping sites configured to trap a plurality of atoms of the material,- Providing one or more entanglement units configured to quantummechanically entangle at least a subset of the at least one atom in the one or more superposition states,- Providing one or more readout optical units as measurement system to readout the second qubit.

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