Repositorio Institucional de Documentos

End-Of-Degree Thesis Generation of Spin Squeezing in hybrid materials Author: SEBASTIÁN ROCA JERAT Supervisors: DAVID ZUECO LÁINEZ FERNANDO LUIS VITALLA Department of Condensed Matter Physics Faculty of Science University of Zaragoza “Aunque no podemos adivinar el tiempo que será, sí que tenemos al menos el derecho de imaginar el que queremos que sea.” Eduardo Galeano, “Derecho al delirio” Me gustaría agradecer y dedicar este trabajo a todas las personas que ayudaron a apartar piedras del camino durante estos cuatro años y seguro de los que quedan por recorrer. Espero que mis escuetas palabras no sean consideradas como un menosprecio a la inmensa ayuda y apoyo que me han dado, pero seguro que mi torpeza con las palabras resulta peor. Agradecer particularmente a David y Fernando su infinita paciencia durante todo el trabajo y el interés en guiarme e introducirme en el vasto mundo de la investigación. I Contents Introduction 1 1 Theoretical background 3 1.1 Light-Matter interaction ................................ 3 1.2 Coupling term ...................................... 4 1.3 Dicke model and symmetries .............................. 5 2 Symmetry broken system 6 2.1 Break in parity conservation .............................. 6 2.2 Effective model close to the anticrossing ....................... 9 2.3 Transmission ...................................... 11 3 Experimental setup 15 3.1 Coplanar waveguide resonators (CPW) and nanoconstrictions ........... 15 3.2 Measurements and data analysis ........................... 17 3.3 Feasible potential improvements ............................ 21 Conclusions & outlook 22 References 23 A Qubit Hamiltonian diagonalization 25 B Dynamics calculations 26 B.1 Equation of motion for mean values .......................... 26 B.2 Dynamics for hai,b2and b†b........................... 26 B.3 RWA and MF approximations ............................. 28 B.4 Solutions with physical meaning ............................ 29 II Introduction The development of quantum simulators, as well as quantum technologies in general, based on atomic or molecular qubits are subject to understanding and controlling coherent interactions between the atoms. These interactions can be further exploited by driving our atomic system with an electromagnetic field; light-matter coupling has been largely studied and its control deeply consolidated in the laboratories. We may use its quantum spin to define the qubit degrees of freedom. Here, we will do so and we will study its interaction with the electromagnetic field in the same framework for both subsystems. For this, we quantize the radiation and thus the interaction is reduced to an exchange between photons and excited states in the spin ensemble. In order to keep the radiation tunable and localized in space, cavities are often used. There, both the electromagnetic modes and the material sample are confined inside it. The basic scenario in such systems consists in the excitation and deexcitation of one spin 1/2, a two level system that realizes each qubit, via annihilation and creation, respectively, of one photon. Those states are superpositions of light and matter, i.e. cavity and spin excitations. They are known as polaritons. They are one of the most basic objects of cavity Quantum Electrodynamics (QED); the theory that studies this kind of systems. Polaritons are also present in systems with more than one two-level system, where the spin excitation is shared by all atoms in the ensemble. However, the above description deals only with cases where the total number of excitations within the system is conserved: for each excitation created (either one photon or one spin flip in the atomic ensemble) another one is consumed and vice versa. In a more general situation symmetry does not need to hold. This is the main focus of this thesis, specifically, systems where not only the number of excitations is not conserved but neither does its parity. We will be interested in processes where one single photon can create two excitations in the spin ensemble, creating a coherent interaction between one photon and two collective spin states. This kind of process was theorized recently in the one photon two atoms case [1] but has to be seen yet in a laboratory. Figure 1: The coherent interaction between one photon and two spin states (a) has been shown to produce optimal spin squeezing [2]. Spin squeezed states can be represented as we do in (b): by diminishing the uncertainty in the x-direction of the spin momentum, paying the cost by the Heisenberg principle of increasing the uncertainty in the y-direction, we are able to increase the precision in the measurement (∆θns for the non-squeezed state vs ∆θsin the squeezed one) of spin momentum (z-direction) precession originated by the magnetic field applied in the y-direction. One of the most appealing applications of this interaction lies in quantum sensing [3]. It has 1 been shown that this process creates optimal spin squeezing [2]. To introduce the concept of spin squeezed states, let’s suppose that we have one spin polarized sample along the z-direction and that we apply to it a magnetic field along the y-direction. Then, by the Larmor effect, the spin momentum of the sample will rotate in the zx-plane, as it is sketched in Fig.1. If we now want to measure the Larmor frequency, or the time that this momentum takes to rotate a certain angle, we have to deal with the intrinsic uncertainty, arising from Heisenberg’s uncertainty principle, in all three spin components. This uncertainty introduces an imprecision in our measurement. Due to Heisenberg’s uncertainty principle, we know that uncertainties on the three spin components are bound and that they are related in such a way that they cannot be eliminated. However, we can reduce the uncertainty in one direction at the expense of increasing it in another one. For our particular case, we could reduce the uncertainty in the x-direction in order to gain precision in our measurements giving that the resulting increase in the y-direction uncertainty does not suppose a negative consequence. Such states in which the uncertainty has been "transferred" from one component to the other two are referred to as "spin squeezed" states. As the Larmor frequency is proportional to the magnetic field, one could use these states as quantum sensors to measure very subtle magnetic field changes. For this, it is necessary to find methods able to generate and optimize spin squeezing. Besides the theoretical exploration of the one photon-two spin excitations we are also interested in its experimental implementation. After identifying the critical parameters in our models and the necessary conditions for this interaction to occur, we would review the most suitable circuits in which we could reach this situation and check if it is feasible. We will study in detail the current technological available possibilities, measuring and computing the reachable values of these parameters and finally discussing if they are enough or if improvements are needed. 2 1 Theoretical background 1.1 Light-Matter interaction For our purposes it is more than sufficient start with the Dirac equation: (i¯h/ ∂−mc)ψ= 0 (1) where / ∂≡γµ∂µ, being γµthe gamma matrices. As important remark, the Dirac equation was the first equation that jointed special relativity and quantum mechanics, being for many people the most beautiful equation in physics. Among the several achievements of this theory, maybe the most important one was to theoretically account for the the spin term which was first introduced by Pauli phenomenologically in order to explain the experimental results obtained in the well-known Stern-Gerlach experiment. This is particularly relevant for this thesis, that is devoted to the coupling of spins and light. In the non-relativistic limit, v << c, the Dirac equation reduces to the Pauli equation for an electron with mass me: HPauli =1 2me (p+eA)2−eφ −e¯h 2me σ·B(2) where σare the Pauli matrices1,φis the Coulomb electrostatic potential, Ais the potential vector and B=∇ × Ais the magnetic field. Hamiltonian (2) is the microscopic foundation of this thesis. It covers the interaction between light and matter in the limit stated. Our next step is to quantize the light2. With this aim we introduce the gauge transformations: A0 µ=Aµ−1 e∂µχ(3) where we have written the potential vector in its covariant notation, Aµ≡(φ, A), as well as the partial differential, ∂µ≡(1 c∂t,∇)and χis some arbitrary scalar function. Equations (1) and (2) are said to be gauge invariant because they are invariant to transformations of this kind; i.e. the phase space is enlarged due to this degeneracy in the gauge choice. Fixing the gauge consists in choose certain restrictions to Aµ. For our purpose, that is, to quantize the electromagnetic field, is convenient to choose the Coulomb gauge, which imposes ∇ · A= 0 (transversality condition). Now the field only has transversal component and satisfies ∂µ∂µA= 0 and we can express the potential vector as a Fourier series: A(r, t) = X k ck(t)uk(r)(4) where ck(t)are the time dependent coeficients and uk(r)orthogonal functions for the propagation mode with wave vector kand angular frequency ωk=|k|c. If we introduce this expression in the classic electromagnetic Hamiltonian Hrad =1 2Z(0|E|2+µ0|B|2)dV (5) 1These matrices are σx= 0 1 1 0!;σy= 0−i i0!;σz= 1 0 0−1! 2The formal quantization of the electromagnetic field is too vast to be presented here in detail, so we restrict ourselves to this modest summary. The complete procedure can be seen in [4,5] 3 and take into account the Maxwell equations E=∂tAand B=∇ × A, we end with the Hamiltonian for a collection of coupled harmonic oscillators: Hrad =X k 1 2(P2 k+ω2 kQ2 k)(6) where the canonical variables Qk≡(ck+c∗ k)and Pk≡ −iωk(ck−c∗ k)have been introduced. The final step is to make the change from field vectors to operators. The states that will be representate the radiation will be denoted as |n0, n1, . . . , nk, . . .iwhere nistands for the number of excitations (photons) the electromagnetic mode ihas with energy ¯hωi. In doing so, we also introduce the annihilation, a, and creation, a†operators: ak=1 √2¯hωk (ωkQk+iPk); a† k=1 √2¯hωk (ωkQk−iPk)(7) which obeys the commutation relation [a, a†] = Iand their action over the radiation states reads ak|nki=√nk|nk−1iand a† k|nki=√nk+ 1 |nk+ 1i. As their names indicate, the operator akdestroys one excitation in the mode kwhereas a† kcreates it. The quantized electromagnetic field Hamiltonian finally reads: Hrad =X k (a† kak+ 1/2)¯hωk(8) The operator a† kakjust accounts for the number of photons which exist in the mode k:a† kak|nki= nk|nki. Note that the energy in the vacuum state is non-zero; indeed h0|Hrad |0i=Pk1 2¯hωk. Furthermore, the expression for the quantized magnetic field is [see last term in Eq.(2)]: B=X k Brms(ak+a† k)(9) being Brms the root-mean-square magnetic field fluctuations. Thus, from this point on, for simplicity’s sake, we are going to work with atoms with two relevant energetic states. These systems are refered as two-level systems (TLS) or qubits and are isomorphic to a 1/2 spin particle. If the energy difference between the two states is labelled as ¯h∆, the Hamiltonian reads: H=¯h∆ 2σz+X k a† kak¯hωk−Hint (10) In fact, we will focus to magnetic molecules where the electrons are symmetrically distributed, so the electric dipole is zero and therefore we will only have the Zeeman coupling term in the interaction, which is the last term in Eq.(2). 1.2 Coupling term Now we are going to consider that the magnetic molecules are inside a cavity, which prevents the electromagnetic modes to dissipate, with some material sample placed inside. The modes allowed can be restricted by the geometry of the cavities. Here we are going to consider only one resonant mode (single mode approximation), ωc, so the sum in Eq.(10) can be omitted. The theory that studies the interaction between light and matter inside such a cavity is known as Cavity QED. The interaction intensity between the sample and the radiation is usually denoted through the parameter g, giving an interaction energy proportional to ¯hg. We can deduce an 4 expression for this parameter, following the previous discussion, supposing that the interaction occurs via Zeeman coupling: Hint =geµB ¯hS·B(11) being ge≈2the electron g-factor, µB≡e¯h 2methe Bohr magneton and S=¯h 2σis the spin operator. Projecting this Hint onto the TLS basis and introducing the magnetic field Bjust quantized (Eq.(9)): Hint = ¯hgσx(a+a†)(12) Where, denoting as |0iand |1ithe ground and excited states from our TLS, respectively, the interaction parameter has been expressed as: g≡geµB|h0|S·Brms |1i| (13) It indicates how likely is the spin to be led by the light from its initial state to the final one and it is clear that stronger the magnetic field, the stronger the coupling is. Eq.(12) is easy to understand: while the σxoperator switches the spin state |0i ←→ |1i,(a+a†)does similarly for light states |n−1i ←→ |ni ←→ |n+ 1i. In fact, we can express σxas σx= (σ++σ−), being σ± the raising/lowering operators in the spin space. 1.3 Dicke model and symmetries The most basic model in cavity QED is the Rabi model [6] in which we have only one TLS inside the cavity. The Hamiltoninan describing such a system is [cf. Eqs. (10) and (12)]3: HRabi =∆ 2σz+ωca†a+gσx(a+a†)(14) Figure 2: Schematic representation of cavity QED models: Rabi (a) and Dicke (b). The difference is the number of TLS considered inside the cavity. Moreover, in the figure of the left we represent the typical parameters present in this theory. The coupling intensity between light and matter is parametrized by g; the dissipation losses of the cavity by κand the atomic decoherence ratio by γ. We sketch in figure 2the typical parameters involved in cavity QED. Generally, our system is an open system; i.e. it interacts with its environment which yields dissipation. The cavity has some dissipation ratio κaccounting for photon leakage whereas the spin sample presents an intrinsic decoherence ratio γ[7] due to the fact that the atom can deexcitate in a non radiative process, without emitting any photon to the cavity, e.g. coupling to phonons. This will be 3We are going to use from now on natural units where ¯h= 1 5 Hence, we will be interested in calculating hai. But, as we will see and it is depicted in detail in App.B, we will also need b2and b†b. The equation of motion for these values is, also derived in App.B:d dt hθi= +iD[ˆ HT, θ]E+γDD† θ[b]E+κDD† θ[a]E(33) where DD† θ[A]E=DA†θAE−1 2D{A†A, θ}E(34) giving us the following closed system of equations (see App.B):                d dt hai=−iωc+κ 2hai−igeff b2−iακ eiωt +e−iωt d dt b2=−i2ωqb2−i2geff Da2b†b+ 1E−γb2 d dt Db†bE=i2geff Da†b2E−Dab†2E−γDb†bE (35a) (35b) (35c) To obtain the solution of these equations we start getting rid of the oscillating terms applying the Rotating Wave Approximation (RWA): hAi=D˜ AEe−iωt =⇒d dt hAi=d dt D˜ AEe−iωt −iω D˜ AEe−iωt (36) Besides, we use mean field decoupling to get a closed set of equations. hABi=hAihBi(37) After the application of these approximations (see App.B) the stationary averages of (35) are the solution of:                  h˜ai=−i2 κhακ +geff D˜ b2E+ (ωc−ω)h˜aii D˜ b2E=−i2 γgeff h˜ai2D˜ b†bE+ 1+1 2(2ωq−ω)D˜ b2E D˜ b†bE=−i2 γh˜aiD˜ b†2E−h.c. (38a) (38b) (38c) This solution is: hai=−iακ κ 2+i(ωc−ω) + 2g2 eff (2hb†bi+1) γ+i(2ωq−ω) (39) Db†bE=8g2 eff|hai|2 γ2+ (2ωq−ω)2−16g2 eff|hai|2(40) which yields a third order equation for hai. This set has 3 solutions. Only one is physically meaningful (see App.B.4) 12 Figure 6: We show the b†bdependence on the effective coupling geff and the external driving intensity λ=ακ. The inset show that b†bturns out to be proportional to λ. The parameters used were γ/ωc=κ/ωc= 0.5in both graphs and geff /ωc= 0.02 in the inset. In Fig.6we plot b†bat resonance ω=ωc= 2ωq. We observe a pronounced peak in the spin mean occupation number when γ∼4geff |hai|2which increases its height as we increase the driving intensity, denoted as λ=ακ. As it is expected, the effective coupling needs to be sufficiently large to produce some interaction between the sample and the radiation yielding a finite b†b. We also obtain that, for a sufficient large coupling geff ,b†bis proportional to this intensity, scaling linearly with λ. This is actually an important result that will be useful in further discussions. If we put together Eq.(32) and Eq.(39) we get the following expression for the transmission: t=−2πκ i(ωc−ω) + κ 2+2g2 eff (2hb†bi+1) γ+i(2ωq−ω) (41) As we discussed in the first section, in the Rabi oscillation case, the coupling between the states of our interest will be represented in the transmission spectrum as two well-resolved peaks. These peaks could merge into one if the coupling is extremely small or the dissipation terms, γand κ, too big. These two peaks will be symmetric respect to the resonant condition, here: ω=ωc= 2ωq. We can check this by numerically solving Eq.(41), as we show in Figure 7. 13 Figure 7: (a) Normalized transmission peaks obtained for ωc/ωq= 2,κ/ωc=γ/ωc= 0.01 and geff/ωc= 0.002 as we vary the driving frequency ωand the mean occupation number b†b. Each curve has been rescaled for a better comparison between them. (b) We can see how the double peak merges into a single one as we increase the decoherence rate γ. We find that the critical value is around Ω≈2 3γ. The parameters used were geff /ωc= 1/600,κ/ωc= 10−3and b†b= 0. (c) Splitting width, Ωin (a), as function of p2hb†bi+ 1. In Fig.7(a), we see that initially, for null mean occupation number, b†b= 0, we only obtain one transmission peak. This would tell us that the coupling is too weak to induce by itself the transition |n, m + 2i ←→ |n+ 1, mi. However, as we increase b†b, for example increasing the driving intensity as we saw in Fig.6, we obtain a progressive splitting of the two peaks, Ω. This parameter appears subtly in Eq.(41) as Ω 22 ∝2g2 eff 2Db†bE+ 1(42) This relation is confirmed numerically in Fig.7(c). Besides, the decoherence rate γalso sets if we obtain two well differentiated peaks or only one as it can be seen in Fig.7(b). We find that increasing γmodifies the transmission spectrum until we reach γ≈3 2Ωwhen the double peak merges into a single one. The challenge now consists in attaining a Ωlarge enough through the effective coupling as well as the largest b†bpossible, although both are bounded [cf. Eq.(39) & Eq.(40)], to surpass the previous condition. Furthermore we have to consider that we also need the largest possible splitting because of the finite sensitivity of actual devices. 14 3 Experimental setup The theory described in the previous two sections allows estimating the coupling between sample and cavity that one needs to attain in to observe the anti-crossing associated with the creation of 2 excitations in the spin ensemble by a single photon. This condition comes from Eq.(25), where the collective coupling, g√N, needs to be comparable to the qubits’ transition frequency to make this phenomenon observable. This effective term is what determines the peak splitting in the transmission spectrum (See Eq.(41) and Eq.(42)), which characterizes the coherent coupling between the two states we seek. We can already see that this constitutes a very demanding condition: in addition of reaching a collective coupling comparable to ωq, one also needs a strong single spin coupling to the cavity. Keeping this in mind we should aim for the optimal architecture of the cavity to achieve the highest possible coupling. The experimental platform that was considered throughout Sec.2is a coplanar waveguide resonator (CPW). This device has been shown to be really efficient in this endeavour as well as being a feasible architecture for quantum computation[17–20]. In this section we will explore in detail the CPW as well possible improvements that can be brought about by introducing constrictions; we will explore the reachable parameters with the current accessible technology to later discuss if the anticrossing could be detected and finally we will discuss what further feasible improvements could be done. 3.1 Coplanar waveguide resonators (CPW) and nanoconstrictions As we introduced in Sec.1, an electromagnetic resonator consists in an optical cavity, in our case a section of coplanar waveguide coupled to external feed lines (input and output) through capacitors. These capacitors act as “mirrors” that trap only the radiation modes whose wavelenghts match the dimensions of the resonator, being the cavity’s geometry what fixes the resonant frequencies allowed. The photons inside these cavities range in the region of 1-10 GHz with long life times [19]. The central line consists of a niobium superconducting line, refrigerated below the critical temperature, 9K, which allows to suppress losses due to resistivity increasing the quality factor. This factor, Q, accounts for the ratio between the resonant frequency and the dissipation losses. In addition, this line is surrounded by two Nb planes that prevent the central line to be perturbed by external electromagnetic noise. All these conductor components are on top of a dielectric substrate, usually sapphire (Al2O3). 15 Figure 8: Graphical representation of a coplanar waveguide resonator (CPW). In the left figure we present the front view where the geometrical parameters both from the resonator and the sample are depicted as well as the direction of the electrical current. In the right one we show a side view with the first harmonic mode of the magnetic field. Reprinted from [20]. As it is shown in Fig.8, the current through the central line will create a magnetic field around it that will interact with the sample. This magnetic field will be maximum at the center of the line and larger the closer is the sample to the line in the vertical direction. The coupling of the sample to the resonator is intrinsically dependant of this field intensity through the Zeeman effect as it was shown in Sec.1. The couplings reported in the literature [19,20] for single spins 1/2 with this set-up are found to be around 1−10 Hz for central line widths of 15 µm. Since typical photon resonance frequencies are of the order of a few GHz, this results to be too weak for our proposal. One way of enhancing the magnetic field and so, the single spin coupling, consists in narrowing as much as possible the central line. By confining in this way the supercurrent, the energy density and therefore the interaction between the spins and the cavity is enhanced. One way of effectively achieve this consists in the introduction of constrictions in the central line, narrowing them down to 10 nm and thereby improving the single spin coupling up to two orders of magnitude [18,20]. In Fig.9we show the physical implementation of one nanoconstriction in a CPW via nanolitography. Galium ions, Ga+, are used to locally mill down the transmission line, with resolutions of a few nm. In order to avoid unwanted electrical discharges due to the presence of ions, a Platinum bridge is built between the line and the ground wire during the deposition. This process is divided in two phases: first, the regions labeled as 2-3 and 1-4 in Fig.9.C are made with less resolution but faster and then, the most important region, labeled as 1-2, employs the most of the time for accuracy’s sake. Once the constriction has been made, the Pt bridge is removed. The dimensions of the constriction that was used to measure the data we will analyze next are shown in the image D of the same figure. Note that we go from the original 15 µm central line down to about 150 nm, 100 times narrower, being this possibly further exploited as we will explore later. About the resonator’s original parameters, namely the resonant frequencies and the quality factor, it is shown that the introduction of the constriction does not suppose a significant perturbation [18,20] so the benefits outweigh the drawbacks. 16 Figure 9: Circuit used to obtain the data analyzed. (A): general overview of the resonator where the curves in the superconducting line gives the total length desired to obtain the precise resonant frequencies we want. The capacitor gaps are at each extreme of the line. (B), (C), (D): central point of the line where the constriction is made in order to profit the maximum intensity magnetic field. (E), (F): the platinum bridge that avoids discharges during the fabrication of the circuit and is removed afterwards. 3.2 Measurements and data analysis Now that the device which was used to obtain the data has been presented, we proceed to present the measurements performed and its pertinent analysis. The microwave transmission through the device was measured using a network analyzer (Fig.10), which ports 1 and 2 were conected to the feed lines of the superconducting resonator. One port for the signal input (an electromagnetic wave ranging from 0.1 GHz to 3 GHz) and the other port to detect the transmission through the cavity. Figure 10: Network analyzer employed to measure the transmission. About the sample deposited on the resonator, we worked in this case with organic free rad17 ical molecules which act as two level systems (or a 1/2 spin) from 2,2-diphenyl-1-picrylhydrazyl, known as DPPH. The procecure to deposit the sample in the nanoconstriction is really accurate and the technique employed was dip-pen nanolithography. In Fig.11 we show the best result of various deposition tries to obtain the maxixum number of molecules near the constriction. Being this zone so small in comparison with the drop it is not trivial to place it. Moreover, the electrostatic interactions between the circuit and the sample make the latter assemble not always in the same place. In our case the number of spins deposited was N= 3.1×109. Figure 11: Sample deposition by dip-pen nanolithography. The circuit transmission resonances were measured for different magnetic fields, which bring the spins to resonance with the cavity. Lorentzian fits of these resonances provide the quality factor, Q= f0/κ, and the resonance frequency f0as a function of magnetic field. From this we can obtain the cavity dissipation κ as a function of magnetic field. This is what is shown in Fig.12 where the Lorentzian fit used was Eq.(43) [20] and the frequency that appears in the x axis is the magnetic field frequency, Ω/2π= −gµBBz/h, where the giromagnetic ratio g= 2.02745,µBis the Bohr magneton, Bzthe magnetic field and hthe Planck constant. Figure 12: Fits using Eq.(43) of the data measured for the first harmonic (a) and the third (b). It can be seen how the higher magnetic field in the third harmonic produces a higher dissipation in the cavity, about one order of magnitude more. 18 κ≈κ0+G2γ (ωn−Ω)2+γ2(43) We present the data collected for the first and third modes of the resonator. As we explained before, the geometry of the cavity only allows certain electromagnetic modes to propagate. However, the spacial distribution of each mode is different and therefore also will be its interaction with the sample. In Fig.13 we can see why these modes were considered and not the second harmonic which presents a magnetic field node where the sample is placed, near the constriction. Having a node in this zone implies no field intensity and so no interaction with the sample and no coupling. The first and third harmonics, on the contrary, present a maximum in its intensity there. Parameters Resonant mode 1 Resonant mode 3 ωq/2π1.397 GHz 4.193 GHz κ/2π0.434 MHz 4.021 MHz γ/2π4.372 MHz 5.071 MHz g/2π116.28 Hz 191.12 Hz Table 1: Parameters fitted from experimental data. The single spin coupling is stronger in the third harmonic as it is also greater the resonant frequency. The T−1 2decoherence time, γ, remains similar due to the only change between modes corresponds to the cavity parameters, not to the sample’s. We can see this in the cavity’s dissipation term, κ, which indeed is increased with the mode. The parameters obtained from the fits correspond to the decoherence rate T−1 2of the sample, γ, the collective coupling of all the spins to the cavity, G, and the resonant frequency which in this case is the same for the cavity, ωc, and the qubit, ωq. Recall that we are interested in the resonance condition ωc= 2ωqbut here we are obtaining the achievable magnitudes for the single spin coupling in order to estimate if with this technology we could perform an experiment to observe that phenomenon. In the Table 1we show the values obtained for these parameters. The value of single spin coupling, g, was computed from the collective coupling through G=gpNeff , where Neff accounts for the effective ensemble size due to thermal polarization. As we are at non-zero temperature, there will be spins that are in their excited state by thermal excitation and they will neither contribute to the transmission nor to the coupling measured. The experiment was realized at T= 4.2Kso the effective ensemble size: Neff = Sz(T) S N(44) Sz(T) S=e−β0 Z(−1) + e−β1 Z(+1) = e−β1−e−β0 e−β1+e−β0=e−βωq−1 e−βωq+ 1 (45) It is evident that the lower the temperature, the closer to unity the quotient in Eq.(45) becomes; i.e., in the limit T→0all the spins in the ensemble will be in their ground state and therefore all of them could absorb photons from the cavity. Hence, we can now obtain the fundamental parameter that will give us the information about if we could be able to observe the interaction we seek. This parameter was introduced in section 2as the effective coupling, geff , in the resonance condition, ωc= 2ωq, and which is 19 related with the single spin coupling through Eq.(25). Taking into account that in our sample (See Fig.11) there were N= 3.1×109spins, we present in Table 2the values for the effective size and coupling for each resonant mode. Figure 13: Magnetic distribution modes through the cavity. Although the first and third harmonic modes present a maximum field intensity where the constriction is made and the sample placed, the second harmonic does present a node, being the coupling between the sample and the circuit non-existing in this scenario. Reprinted from [20]. Parameters Resonant mode 1 Resonant mode 3 Neff 2.39 ·1077.19 ·107 geff /2π0.49 µHz 0.72 µHz Table 2: Effective ensemble’s size and coupling for each resonant mode. The circuits employed although being efficient lack of single spin coupling strength to provide an effective coupling non negligible. As we can see, the values obtained for geff in both modes are absolutely negligible as compared to the photon dissipation and spin decoherence rates, and they are also well below the frequency resolution of the experiments. We recall that observing the phenomenom described in the previous section would require a collective coupling, g√N, comparable to the resonant frequency, ωq, to not end with a tiny quotient in Eq.(25) as it is squared. According to our calculations (see Eq.(41) and Eq.(42)), this geff needs to be comparable to the dissipation terms γand κin order to have the double peak displayed in Fig.7. Although the double peak splitting can be enhanced by increasing the mean bosonic number b†bin addition to the effective coupling, this occupation does not suppose any difference; with the maximum driving we could introduce, i.e. the number of photons (about the same of spins in the sample to avoid distortion effects [20]) which in essence equals the electromagnetic wave intensity and this is proportional to the square of its amplitude, we cannot populate enough the sample due to the extremely weak coupling and so the weak interaction between the spins and the light. Therefore, it seems that we cannot aim to observe the desired anti-crossing with these circuits. Besides the excellent quality of the parameters obtained with these kind of resonators, we need to explore which modifications in the resonator architecture could bring higher coupling intensities that could close the gap between the effective couplings obtained in Table 2and the ones we need, approximately of the order of MHz, the same as γ[cf. Section 2]. Fortunately, we have the tools required to explore higher single spin couplings that could potentially give inspiring results. 20 3.3 Feasible potential improvements Introducing nanoconstrictions in the resonators provided an enhancement in the reachable single spin couplings in that region due to the increase in the magnetic field intensity by forcing the same energy to flow in a narrower space. The data analyzed before were obtained for a 150 nm wide nanoconstriction (see Fig.9) but circuits with 30 nm wide and 10 µm long constrictions can be built, constituting the state of the art of this technology, and providing up to 50 kHz of intensity coupling for each S=1/2 spin. Taking into account these dimensions and that the DPPH has a density of ρ= 1.4g/cm3and a molecular weight of M= 394.32g/mol we end with N≈1.92 ×107spins. We can estimate the new effective coupling with these new parameters. If we cool the system down to, let’s say T= 100 mK, to obtain the optimal effective number of spins, Neff , in Eq.(44) and we work with a resonator of frequency ωc≈3GHz, so we tune the qubit transition frequency to be about ωq≈1.5GHz to be in the resonant condition ωc= 2ωq, we end with an effective ensemble size of Neff ≈1.18×107and an effective coupling of geff ≈4Hz. This results in an increment of 7 orders of magnitude respect to the values obtained in Table 2. Although it remains too small, it shows that increasing the single spin coupling, g, maintaining the spin density is crucial due to the cubic dependence in Eq.(25). So far, we have been dealing with the lowest spin possible, S = 1/2, but working with single magnetic molecules which posses higher spins could be as well as improving the resonator’s architecture a great option. Increasing the number of states that can interact with the magnetic field is reported to increase the single spin coupling, g, [17,20] at least in a factor √S. Thus, working with the improved nanoconstrictions just presented combined with using samples of S > 1/2 could lead to a further enhancement in the effective coupling we seek. Examples of this kind of molecules could be the cluster [Gd(H2O)P5W30O110]12−[21], or GdW30, whose ground state spin is 7/2 due to its Gd3+ ion trapped inside; the magnetic cluster compound Mn6O4Br4(Et2dbm)6 [22] which contains 6 Mn3+ ions that add up to a total spin S = 12 in an octahedral simmetry; or, finally, the highest spin ground state reported [23], S = 91, of a giant Ni20Gd20 cage. In figure 14 we present the molecular structure of the first two single magnetic molecules. Figure 14: SMM that could suppose an improvement in the effective coupling measured to observe the one photon-two atoms anti-crossing. Left: Mn6octahedral core. Right: GdW30 cluster. 21 Finally, for b†b: d dt Db†bE=iD[HT, b†b]E+γDD† b†b[b]E(75) D[HT, b†b]E=geff D[ab†2+a†b2, b†b]E=geff Da[b†2, b†b]E+Da†[b2, b†b]E (76) Like before, we compute each commutator separately: [b†2, b†b] = (b†b†b†b−b†bb†b†) = b†b†b†b−b†(1 + b†b)b†= =b†2[b†, b]−b†2=−2b†2(77) [b2, b†b] = (bbb†b−b†bbb) = b(1 + b†b)b−b†bbb = = [b, b†]b2+b2= 2b2(78) DD† b†b[b]E=Db†b†bbE−1 2Db†bb†bE−1 2Db†bb†bE= =Db†b†bbE−Db†bb†bE=Db†[b†, b]bE=−Db†bE(79) Thus, d dt Db†bE=i2geff Da†b2E−Dab†2E−γDb†bE(80) With this we obtain the equation system that appears in the main text:                d dt hai=−iωc+κ 2hai−igeff b2−iακ eiωt +e−iωt d dt b2=−(i2ωq+γ)b2−i2geff Da2b†b+ 1E d dt Db†bE=i2geff Da†b2E−Dab†2E−γDb†bE (81a) (81b) (81c) B.3 RWA and MF approximations Now, in order to be able to solve these equations analytically, we resort to a couple of approximations: mean field theory, which allows us to consider that the mean values of each operator is independient from each other, so hABi=hAihBi(82) and the rotating wave approximation. In the latter, we place ourselves in the interaction picture and suppose that the the cavity operators rotate with the same frequency as the driving, ω, and the spins with ω0. If we place ourselves in this rotating frame:    hai=h˜aie−iωt hbi=D˜ bEe−iω0t (83a) (83b) and we proceed to apply these approximations to the system. Starting with hai: d dt hai=d dt h˜aie−iωt −iω h˜aie−iωt =−iωc+κ 2h˜aie−iωt −igeff D˜ b2E−iακ(eiωt +e−iωt)(84) 28 If we now extract e−iωt as common factor and we apply that ω= 2ω0, which is the resonant condition that interests us: d dt h˜ai=−i(ωc−ω) + κ 2h˜ai−igeff D˜ b2E−iακ(ei2ωt + 1) (85) For b2: d dt b2=d dt D˜ b2Eei2ω0t−i2ω0D˜ b2Ee−i2ω0t(86) d dt b2=−(i2ωq+γ)D˜ b2Ee−i2ω0t−i2geff h˜aie−iωt 2D˜ b†bE+ 1(87) And, again, applying that ω= 2ω0: d dt D˜ b2E=−(i(2ωq−ω) + γ)D˜ b2E−i2geff h˜ai2D˜ b†bE+ 1(88) Finally, for D˜ b†bE, applying MF: d dt Db†bE=d dt D˜ b†bE=−γD˜ b†bE−i2geff h˜aie−iωt D˜ b†2Eei2ωt −h.c.(89) d dt D˜ b†bE=−γD˜ b†bE−i2geff h˜aiD˜ b†2E−h.c.(90) Which gives us the following system of equations in this frame:                d dt h˜ai=−i(ωc−ω) + κ 2h˜ai−igeff D˜ b2E−iακ ei2ωt + 1 d dt D˜ b2E=−(i(2ωq−ω) + γ)D˜ b2E−i2geff h˜ai2D˜ b†bE+ 1 d dt D˜ b†bE=−γD˜ b†bE−i2geff h˜aiD˜ b†2E−h.c. (91a) (91b) (91c) Finally, in order to obtain the stationary solutions that will be used for the transmission calculation, we neglect oscillating terms and suppose the stationary state t→0and d/dt →0                  h˜ai=−i2 κhακ +geff D˜ b2E+ (ωc−ω)h˜aii D˜ b2E=−i2 γgeff h˜ai2D˜ b†bE+ 1+1 2(2ωq−ω)D˜ b2E D˜ b†bE=−i2 γh˜aiD˜ b†2E−h.c. (92a) (92b) (92c) B.4 Solutions with physical meaning We proceed to solve this closed system and for that we simplify the notation defining A ≡ 2ακ κ;Gκ≡2geff κ;Gγ≡2geff γ(93) X≡ih˜ai;Y≡D˜ b2E;Z≡D˜ b†bE(94) 29 so the equations read:        X= (A+GκY) Y=−XGγ(2Z+ 1) Z=−Gγ(XY ∗+X∗Y) (95a) (95b) (95c) Introducing the expression for Y in Z and we do so also in X we obtain the following system of two equations for Z and X:          X=A 1 + GκGγ(2Z+ 1) Z=2G2 γ|X|2 1−4G2 γ|X|2 (96a) (96b) We know that Z has to be a real number since the mean bosonic occupation b†bhas to be so. By looking directly the expression for X we note that it has to be real either which leads to the conclusion that haiis purely imaginary due to how is defined X. In addition, if we solve this system we end with a third order equation for X, which gives three solutions. However, only one should be the “real”, the one with physical meaning. If we continue examining each variable as before, we obtain that if X∈ R then Y∈ R. Moreover, Z has to be positive (we cannot have a negative number of spins in their excited state) so X and Y must have contrary signs. Now only remains to assign which sign correspond to X and Y: if Y > 0,Xwould be also positive due to the fact that, without loss of generality, Aand Gκare both positive. So Y has to be negative and we obtain the last restriction for X: Y=X−A Gκ <0 =⇒X < A(97) which makes sense if we think about it as that the annihilation operator cannot be greater than the ratio of photons we introduce through the driving and the photon losses by the dissipation in the cavity. With all these considerations we can solve for X and we obtain: F(X)≡ −4GγX3+ 4G2 γAX2+ (1 + GγGκ)X−A (98) for which as we can see in Fig.15 only has one solution that obeys all the restrictions imposed before. 30 Figure 15: Graphical solution for Eq.(98) for Gγ=Gκ=A= 1. The vertical dashed line indicates the value of A. 31