Quantum Computation with Molecular Nanomagnets: Achievements, Challenges, and New Trends

Abstract

Molecular nanomagnets exhibit quanto-mechanical properties that can be nicely tailored at synthetic level: superposition and entanglement of quantum states can be created with molecular spins whose manipulation can be done in a timescale shorter than their decoherence time, if the molecular environment is controlled in a proper way. The challenge of quantum computation is to exploit the similarities between the coherent manipulation of molecular spins and algorithms used to process data and solve problems. In this chapter we shall firstly introduce basic concepts, stressing analogies between the physics and the chemistry of molecular nanomagnets and the science of computing. Then we shall review main achievements obtained in the first decade of this field and present challenges for the next future. In particular we shall focus on two emerging topics: quantum simulators and hybrid systems made by resonant cavities and molecular nanomagnets.

Appendices

Appendix 1: Quantum Description of the Spin Dynamics in a Resonant Cavity

In this section we provide further formalism to describe the interaction of single spin with a quantized electromagnetic field following the quantum approach [80, 87, 115]. We consider a cavity in which the field has a single harmonic mode of frequency ω. The intensity of the electromagnetic field determines the number n of photons in the cavity and we consider the situation for which few photons are present in the resonator. Let’s assume that the quality factor of the cavity Q is very high so that the photons lifetime is very long. Such a quantized electromagnetic field can be described as ${\mathrm{\mathcal{H}}}_c=\hslash \omega\;\left({a}^{\dagger }a+{\scriptscriptstyle \frac{1}{2}}\right)$, where a and a ^† are the creation and annihilation operators for photons, in analogy with a quantum one-dimensional oscillator [116]. The dipolar spin–photon interaction ℋ _cs  = − μ ⋅ B can be written as:

$$ {\mathcal{H}}_{cs}=\hslash {g}_c\left[\left(\mathbf{e}\cdot \mathbf{S}\right)a+\left({\mathbf{e}}^{*}\cdot \mathbf{S}\right){a}^{\dagger}\right]. $$

(12)

For this expression we make use of the Rotating Wave Approximation (RWA) that consists in neglecting fast-oscillating, non-energy-conserving terms which play a minor role in the dynamics of the system. The prefactor g _c is the coupling strength of the magnetic moment with the oscillating magnetic component of the electromagnetic field B ₁(t) The unitary vector e describes the polarization of B ₁(t), which can be conveniently chosen to obtain the circular polarization σ ₊ or σ ₋ with respect to the static field B ₀ along the z-axis. Being S _± = S _x  ± iS _y , we have thus

$$ {\mathrm{\mathcal{H}}}_{\sigma +}=\hslash {g}_c\left(a{S}_{+}+{a}^{\dagger }{S}_{-}\right), $$

(13)

for photons with helicity +ћ along z, and

$$ {\mathrm{\mathcal{H}}}_{\sigma -}=\hslash {g}_c\left(a{S}_{-}+{a}^{\dagger }{S}_{+}\right), $$

(14)

for photons with helicity −ћ along z. The Jaynes–Cummings model [117] considers the full Hamiltonian ℋ = ℋ _c  + ℋ _s  + ℋ _cs , i.e.:

$$ \mathrm{\mathcal{H}}=\hslash \omega \left({a}^{\dagger }a+\frac{1}{2}\right)+\hslash {\omega}_0{S}_z+\hslash {g}_c\left(a{S}_{\pm }+{a}^{\dagger }{S}_{\mp}\right). $$

(15)

being ℋ _s  = μ _B B ₀ S _z  = ℏω ₀ S _z the term describing the spin precession about B ₀. The interaction term ℋ _cs imposes the conservation of the z component of the total angular momentum since it has nonzero matrix element only between eigenstates of ℋ _c  + ℋ _s that are characterized by the same value of m _s  + n. This reproduces the selection rules Δm _s  = 1 for σ ₊ and Δm _s  = − 1 for σ ₋ expected for conventional perpendicular-mode EPR [118]. Since ${m}_s=\pm {\scriptscriptstyle \frac{1}{2}}$, we have only two possible values − 1/2 + n + 1 and + 1/2 + n, so the diagonalization of Eq. (15) can be carried out separately in each of the two-dimensional subspaces. It is convenient to make use of the dressed atom approach to describe the evolution of an isolated system composed by n photons and one spin [115]. Each subspace is represented by the photon plus spin states:

$$ \left|{\varphi}_a\right\rangle =\left|-\frac{1}{2},n+1\right\rangle \kern1em \left|{\varphi}_b\right\rangle =\left|+\frac{1}{2},n\right\rangle $$

(16)

related to the two allowed conditions, ground −1/2 spin state plus n + 1 photons and exited +1/2 spin state plus n photons. The correspondent eigenvalues

$$ {E}_a=\left(n+1\right)\hslash \omega -\left(\hslash {\omega}_0/2\right) $$

(17)

$$ {E}_b= n\mathit{\hslash \omega }+\left(\mathit{\hslash}{\omega}_0/2\right) $$

(18)

are separated by the detuning frequency ${\varDelta}_c={\scriptscriptstyle \frac{1}{\hslash }}\left({E}_a-{E}_b\right)=\omega -{\omega}_0$. At resonance (Δ _c  = 0), the unperturbed levels would be degenerate. The matrix elements of the interaction potential ${\mathrm{\mathcal{H}}}_{\sigma_{+}}$ result

$$ \left\langle {\varphi}_a\left|{\mathrm{\mathcal{H}}}_{\sigma_{+}}\right|{\varphi}_a\right\rangle =\left\langle {\varphi}_b\left|{\mathrm{\mathcal{H}}}_{\sigma_{+}}\right|{\varphi}_b\right\rangle =0 $$

(19)

$$ \left\langle {\varphi}_b\left|{\mathrm{\mathcal{H}}}_{\sigma_{+}}\right|{\varphi}_a\right\rangle =\hslash {g}_c\sqrt{n+1}. $$

(20)

showing that for a system with n photons, the coupling strength scales nonlinearly as $\sqrt{n+1}$. By defining the n-photon Rabi frequency as ${\varOmega}_n=2{g}_c\sqrt{n+1}$, the eigenvalues of Eq. (15) read

$$ {E}_{+}(n)=\hslash \left(n+\frac{1}{2}\right)\omega +\frac{\hslash }{2}\sqrt{\varDelta_c^2+{\varOmega}_n^2} $$

(21)

$$ {E}_{-}(n)=\hslash \left(n+\frac{1}{2}\right)\omega -\frac{\hslash }{2}\sqrt{\varDelta_c^2+{\varOmega}_n^2}. $$

(22)

They form two branches of hyperbola with the unperturbed energies as asymptotes (see Fig. 9a). With respect to the unperturbed states, the interaction potential determines the formation of an anticrossing centered on resonance. The minimum gap between E ₁ and E ₂ is ℏΩ _n for Δ _c  = 0. The corresponding eigenstates, expressed as function of the unperturbed basis, result

$$ \left|{\chi}_{+}(n)\right\rangle = \sin \kern0.5em \theta \left|-\frac{1}{2},n+1\right\rangle + \cos \kern0.5em \theta \left|+\frac{1}{2},n\right\rangle $$

(23)

$$ \left|{\chi}_{-}(n)\right\rangle = \cos \theta \left|-\frac{1}{2},n+1\right\rangle + \sin \theta \left|+\frac{1}{2},n\right\rangle $$

(24)

with mixing angle

$$ \tan \left(2{\theta}_n\right)=-\frac{\varOmega_n}{\varDelta_c}\kern1em 0\le 2{\theta}_n<\pi . $$

(25)

Each added photon creates a two-dimensional subspace, the complete manifold is a ladder of the two-level states shifted in energy by ћω.

Let’s now focus on the resonant case. For Δ _c  = 0 the mixing angle is θ _n  = π/4 and the perturbed states result

$$ \left|{\chi}_{+}(n)\right\rangle =\frac{1}{\sqrt{2}}\left[\left|-\frac{1}{2},n+1\right\rangle +\left|+\frac{1}{2},n\right\rangle \right] $$

(26)

$$ \left|{\chi}_{-}(n)\right\rangle =\frac{1}{\sqrt{2}}\left[\left|-\frac{1}{2},n+1\right\rangle -\left|+\frac{1}{2},n\right\rangle \right]. $$

(27)

The time evolution can be calculated by applying the unitary evolution operator to the perturbed dressed states and by recasting in the | − 1/2〉 or | + 1/2〉 unperturbed basis. The time evolution of the ground |Ψ ₋〉 state is

$$ \left|{\varPsi}_{-}(t)\right\rangle = \cos \left(\frac{\varOmega_nt}{2}\right)\left|-\frac{1}{2},n+1\right\rangle -i \sin \left(\frac{\varOmega_nt}{2}\right)\left|+\frac{1}{2},n\right\rangle $$

(28)

while the excited state evolves as

$$ \left|{\varPsi}_{+}(t)\right\rangle = \cos \left(\frac{\varOmega_nt}{2}\right)\left|-\frac{1}{2},n+1\right\rangle +i \sin \left(\frac{\varOmega_nt}{2}\right)\left|+\frac{1}{2},n\right\rangle $$

(29)

These expressions describe the dynamics of entangled spin and photon states which have a time evolution that recalls the beat signal of two coupled degenerate quantum oscillators. The eigenmodes are a symmetric and antisymmetric combination of the independent modes of the free oscillators. The cavity and the spin coherently exchange a photon, which is absorbed and then emitted following the spin flip.

The population of the | − 1/2, n + 1〉 and | + 1/2, n〉 states oscillates and for n ≫ 1 the transition probability can be written as

$$ {P}_{ba}(t)=\frac{\varOmega_n^2}{\varDelta_c^2+{\varOmega}_n^2}{ \sin}^2\left[\sqrt{\varDelta_c^2+{\varOmega}_n^2}\frac{t}{2}\right]. $$

(30)

This formula reproduces the classical result of Eq. (6) with Ω _n  = Ω _R.

Appendix 2: Planar Resonators

2.1 Fabrication of Microstrip and Coplanar Resonators

Planar transmission lines are commonly used in microwave technology as they provide a simple way to transmit electromagnetic waves on a printed board circuit realized by standard lithographic methods. Among many different geometries, microstrip and coplanar waveguides are the most frequent choices. Microstrip lines are constituted by a dielectric substrate having a metal strip on the top and a ground plane on the bottom side. Coplanar waveguides differ from microstrips for the presence of two ground planes placed beside the central strip on the top side. The ground conductor in the backside can also be removed. With these geometries, it is possible to match the impedance of the feeding coaxial lines (usually 50 Ω) with relative physical dimensions that spans from millimeter to micron size. By design, the transmission of quasi-transverse electromagnetic modes (TEM) can be achieved, while higher-order non-TEM modes can be appropriately suppressed [119].

Coplanar waveguides are the best choice for minimizing the irradiation of the microwave field outside the surface and to arrange ground electrodes close to the central signal line. A coplanar resonator of length l is realized when the central strip is interrupted in correspondence to two selected positions. These dielectric gaps are capacitors that electrically couple resonator and transmission line, acting like mirrors do in an optical cavity. Resonant conditions are met when input and reflected wave signals give constructive interference into the cavity. The value of the resonant frequency ω _c is determined by the length l of the resonator and by the speed of propagation of the electromagnetic wave in the coplanar waveguide. The latter is related to the effective dielectric constant ε _eff of the insulator. For a cavity resonating at half wavelength λ/2 [120], the resonance frequency is:

$$ {\omega}_c=\frac{2\pi c}{\sqrt{\varepsilon_{\mathrm{eff}}}}\frac{1}{2l} $$

(31)

As mentioned in the previous sections, the quality factor of the resonator must be maximized to reduce the decay rate of the cavity κ and to increase the photon lifetime. The Q-factor is defined as the ratio between the energy stored in the cavity and the power dissipated in a time interval 1/ω or, alternatively as the width of the resonance Δω _c since Q = ω _c /Δω _c . For a resonator coupled to the feedlines, the loaded quality factor must be considered

$$ \frac{1}{Q}=\frac{1}{Q_{\mathrm{ext}}}+\frac{1}{Q_{\mathrm{int}}}, $$

(32)

which is calculated by including the external quality factor (Q _ext) related to the coupling capacitances and the intrinsic Q _int, due to the internal losses of the resonators.

The capacitance of the input and output gaps controls the coupling with the transmission line and consequently the power flow κ _in and κ _out along the waveguide. The maximum transfer of microwave energy is obtained when the impedance of the resonator is matched to the feedline. This corresponds to the condition Q _ext = Q _int and the resonator is said to be critically coupled. For Q _ext < Q _int the resonator is undercoupled. This configuration corresponds to reduced transmission, thus lower signal-to-noise ratio, but maximum Q. In the experiments it is often reported because the low output signals can be restored by a low noise microwave amplifier inserted along the output line. Conversely, in the overcoupling regime (Q _ext > Q _int) high κ _in and κ _out are obtained, thus lower Q. This configuration has been used to get fast measurement rates of the cavity photon states [81].

Intrinsic losses often determine the loaded quality factor of the resonator. They are related to different dissipation mechanisms that finally determine the performances of the coplanar resonator. Losses depend on the geometry, material choice, temperature, frequency range, and applied magnetic field. Resonators are rather susceptible to their environment, so they are usually enclosed in metal boxes. Without applied magnetic field, three are the main dissipation mechanisms: resistive, dielectric, and radiative losses [121].

Resistive losses are due to energy dissipated by an electromagnetic wave traveling along a waveguide with finite conductance. Just considering resistive losses, the Q factor passes from ~10¹ to 10², typically obtained for resistive cavities, up to Q ~ 10⁷ for superconducting resonators [122]. Niobium is commonly employed for its relatively high critical temperature (T _c  ≃ 9.2 K) and critical field. Superconducting films of TiN, Al, Ta, Re, or YBCO are also reported. Spin systems usually require the application of static magnetic fields to split the degeneracy of the energy levels. For instance, X-band resonance of a spin 1/2 paramagnet requires about 340 mT. Trapping of magnetic flux can be minimized by aligning the field parallel to the resonator surface and experiments report limited degradation of Q up to 350 mT [123]. For higher field or other orientations the penetration of magnetic flux determines a decrease of the quality factor down to 10³ or lower values. Strategies for the reduction of the magnetic losses have been applied, for instance, by pinning the vortex motion by patterning of slots or microdots [124–126]. Magnetic hysteresis effects are also present and determine the dependence of the Q-factor on the magnetic history of the sample [127].

Dielectric losses are due to absorption of the electromagnetic power by the dielectric substrate. For a lossy material the complex dielectric constant ε = ε _r  + iε _i has a finite imaginary part ε _i and loss tangent (tan δ). The quality factor associated with the dielectric losses is Q _diel = 1/tan δ, thus it is desirable to choose insulating substrates with low loss tangent. Sapphire has very low losses with tan δ ~ 10^− 8 in high-purity crystals [128]. High resistivity silicon and thermally grown SiO₂ provide a valid alternative [129]. Fabrication strategies, like suspended resonators with grooves etched in the regions of high electric field, have been proposed for reducing the dielectric losses [130].

Radiative losses are an additional contribution due to the emission of electromagnetic radiation in the free space. The associated quality factor is Q _rad ~ (l/b)², where l and b are, respectively, the length and the distance between the ground electrodes in the top plane [131]. For a typical coplanar waveguide resonator Q _rad ~ 10⁶.

The temperature dependence of the Q-factor shows a sudden increase below T _c reaching a maximum value for T ≃ T _c /10 (T ≃ 1 K for Nb). At lower temperature, Q progressively decreases due to a further loss mechanism inducted by the two-level (spin) transitions. These losses, which dominate in the millikelvin range, are ubiquitously reported in lithographed resonators and they are independent by the materials used. They have been assigned to oxides or impurities located close to the active region of the resonator [132–135]

The fundamental resonance frequency of planar resonators is usually located in the 2–15 GHz range by appropriate choice of l. Higher-order harmonics provides further resonances, although the quality factor progressively deteriorates by increasing the mode number [136]. Tunable superconducting resonators have been realized by means of Josephson junctions demonstrating large tunable range and high quality factor [137–139], and the possibility to tune ω _c faster than photon lifetime [140].

2.2 Planar Resonators for Magnetic Resonance Experiments

Modern conventional three-dimensional EPR spectrometers report a spin sensitivity up to ~10⁹ spins Hz^−1/2 thanks to the high quality factor of cavity. The minimum detectable number of spins of an EPR cavity depends also on a set of different parameters, such as cavity volume and strength of the microwave field [118]. For small samples, such as thin films or nanostructures, an efficient way to improve the sensitivity of the EPR measurement is to increase the filling factor

$$ \eta =\frac{\int_{V_s}{\left|{B}_1\right|}^2dV}{\int_{V_c}{\left|{B}_1\right|}^2dV} $$

(33)

being V _c and V _s respectively, the e.m. mode and sample volume [141], by fabricating resonators that match the sample size and that can concentrate the microwave field in the sample space.

Planar resonating circuits show microwave fields confined in a small V _c , limited to about 100 μm above the surface, where the intensity of B ₁ can reach the 0.1 mT range with a limited input power (~100 μW). These devices have been proposed as EPR cavities [142, 143], also because they are suitable for low temperature experiments where microwave heating must be avoided. With the purpose to maximize the power to field conversation efficiency on the sample volume, several designs have been studied, including microstrips [144], planar microcoils [145, 146], and surface loop-gap microresonators [147]. These devices, investigated by means of both continuous-wave and pulsed EPR experiments, report an increase of the sensitivity up to ~10⁶ spins Hz^−1/2 [147]. Similar resonators were also used for ferromagnetic resonance measurements [148–150]. In addition, cross-shaped resonators were proposed for controlling the polarization of the microwave mode [151].

Continuous-wave EPR of different spin ensembles has been exploited for strong-coupling experiments with coplanar waveguide resonators [94–97, 152]. Superconducting resonators have also been studied for pulsed EPR [123, 153] or non-resonating frequency-sweeping EPR [108]. Optimized resonators made with parallel arrays of superconducting microstrip have been also developed for improving the homogeneity of B ₁ over a large region [123].