Mechanisms of local and global molecular quantum gates and their implementation prospects

We explore how the globality of quantum logic operations is ensured in the context of optimal control theory when qubits are encoded in vibrational eigenstates of different normal modes and specially shaped laser fields act as quantum logic operations. In a two-qubit model system, transition mechanisms for optimized laser fields generating single qubit flips, local NOT and global NOT and controlled-NOT (CNOT) gates are investigated and compared. We evaluate the participation of vibrational eigenstates beyond the qubit basis in the global gate mechanisms and how different features of CNOT and NOT gates relate to the characteristics of the vibrational manifold. When a non-qubit normal mode interacting via anharmonic resonances is introduced, neither the global gate mechanisms nor the optimized laser fields show a significant increase in complexity. Similar features of the global quantum gates in both model systems indicate a generality of the deduced principles. Finally, a primary concept for a realization of global quantum gates in an actual experiment referring to state-of-the-art techniques is presented. The possible reconstruction of optimized laser fields with sequences of simple Gaussian subpulses is demonstrated and some critical parameters are deduced.     

Vibrational molecular quantum computing: basis set independence and theoretical realization of the Deutsch-Jozsa algorithm

The phase of quantum gates is one key issue for the implementation of quantum algorithms. In this paper we first investigate the phase evolution of global molecular quantum gates, which are realized by optimally shaped femtosecond laser pulses. The specific laser fields are calculated using the multitarget optimal control algorithm, our modification of the optimal control theory relevant for application in quantum computing. As qubit system we use vibrational modes of polyatomic molecules, here the two IR-active modes of acetylene. Exemplarily, we present our results for a Pi gate, which shows a strong dependence on the phase, leading to a significant decrease in quantum yield. To correct for this unwanted behavior we include pressure on the quantum phase in our multitarget approach. In addition the accuracy of these phase corrected global quantum gates is enhanced. Furthermore we could show that in our molecular approach phase corrected quantum gates and basis set independence are directly linked. Basis set independence is also another property highly required for the performance of quantum algorithms. By realizing the Deutsch-Jozsa algorithm in our two qubit molecular model system, we demonstrate the good performance of our phase corrected and basis set independent quantum gates.     

Heat rectification in molecular junctions

Heat conduction through molecular chains connecting two reservoirs at different temperatures can be asymmetric for forward and reversed temperature biases. Based on analytically solvable models and on numerical simulations we show that molecules rectify heat when two conditions are satisfied simultaneously: the interactions governing the heat conduction are nonlinear, and the junction has some structural asymmetry. We consider several simplified models where a two-level system (TLS) simulates a highly anharmonic vibrational mode, and asymmetry is introduced either through different coupling of the molecule to the contacts, or by considering internal molecular asymmetry. In the first case, we present analytical results for the asymmetric heat current flowing through a single anharmonic mode using different forms for the TLS-reservoirs coupling. We also demonstrate numerically, studying a realistic molecular model, that a uniform anharmonic molecular chain connecting asymmetrically two thermal reservoirs rectifies heat. This effect is stronger for longer chains, where nonlinear interactions dominate the transfer process. When asymmetry is related to the internal level structure of the molecule, numerical simulations reveal a nontrivial rectification behavior. We could still explain this behavior in terms of an effective system-bath coupling. Our study suggests that heat rectification is a fundamental characteristic of asymmetric nonlinear thermal conductors. This phenomenon is important for heat control in nanodevices and for understanding of energy flow in biomolecules.     

Phase control in the vibrational qubit

In order to use molecular vibrations for quantum information processing one should be able to shape infrared laser pulses so that they can play the role of accurate quantum gates and drive the required vibrational transitions. In this paper we studied theoretically how the relative phase of the optimized transitions affects accuracy of the quantum gates in such a system. Optimal control theory and numerical propagation of laser-driven vibrational wave packets were employed. The dependencies observed for one-qubit gates NOT, pi-rotation, and Hadamard transform are qualitatively similar to each other. The results of the numerical tests agree well with the analytical predictions.     

Effect of diatomic molecular properties on binary laser pulse optimizations of quantum gate operations

The importance of the ro-vibrational state energies on the ability to produce high fidelity binary shaped laser pulses for quantum logic gates is investigated. The single frequency 2-qubit ACNOT(1) and double frequency 2-qubit NOT(2) quantum gates are used as test cases to examine this behaviour. A range of diatomics is sampled. The laser pulses are optimized using a genetic algorithm for binary (two amplitude and two phase parameter) variation on a discretized frequency spectrum. The resulting trends in the fidelities were attributed to the intrinsic molecular properties and not the choice of method: a discretized frequency spectrum with genetic algorithm optimization. This is verified by using other common laser pulse optimization methods (including iterative optimal control theory), which result in the same qualitative trends in fidelity. The results differ from other studies that used vibrational state energies only. Moreover, appropriate choice of diatomic (relative ro-vibrational state arrangement) is critical for producing high fidelity optimized quantum logic gates. It is also suggested that global phase alignment imposes a significant restriction on obtaining high fidelity regions within the parameter search space. Overall, this indicates a complexity in the ability to provide appropriate binary laser pulse control of diatomics for molecular quantum computing.     

Vibrational computing: simulation of a full adder by optimal control

Within the context of vibrational molecular quantum computing, we investigate the implementation of a full addition of two binary digits and a carry that provides the sum and the carry out. Four qubits are necessary and they are encoded into four different normal vibrational modes of a molecule. We choose the bromoacetyl chloride molecule because it possesses four bright infrared active modes. The ground and first excited states of each mode form the one-qubit computational basis set. Two approaches are proposed for the realization of the full addition. In the first one, we optimize a pulse that implements directly the entire addition by a single unitary transformation. In the second one, we decompose the full addition in elementary quantum gates, following a scheme proposed by Vedral et al. [Phys. Rev. A 54, 147 (1996)]. Four elementary quantum gates are necessary, two two-qubit CNOT gates (controlled NOT) and two three-qubit TOFFOLI gates (controlled-controlled NOT). All the logic operations consist in one-qubit flip. The logic implementation is therefore quasiclassical and the readout is based on a population analysis of the vibrational modes that does not take the phases into account. The fields are optimized by the multitarget extension of the optimal control theory involving all the transformations among the 2(4) qubit states. A single cycle of addition without considering the preparation or the measure or copy of the result can be carried out in a very competitive time, on a picosecond time scale.     

Car-Parrinello molecular dynamics study of anharmonic systems: a Mannich base in solution

A Car-Parrinello molecular dynamics study was performed for 4,5-dimethyl-2-(N,N-dimethylaminomethyl)phenol, a Mannich base, to investigate the vibrational properties in solution of its intramolecular hydrogen bond. The dynamic behavior of this hydrogen-bonded system was investigated using an explicit solvent model. Addition of a nonpolar solvent permitted inclusion of delicate environmental effects on the strongly anharmonic system which was studied from first principles. Molecular dynamics and a posteriori quantization of the O-H motion were applied to reproduce the vibrational features of the O-H stretching mode. Consistent application of Car-Parrinello dynamics based on the density functional theory with subsequent solution of the vibrational Schr?dinger equation for the O-H stretching motion offers an effective method for strongly anharmonic systems, and this is supported by the comparison of the results with experimental spectra. As a further element of the intramolecular hydrogen bond study, the effects of deuteration were taken into account and a successful application of the O-H stretching mode quantization technique to the liquid phase is demonstrated. This provides a valuable computational methodology for investigations incorporating nuclear quantum effects in the liquid phase and enzyme active centers and can be used to investigate numerous systems that are not readily susceptible to experimental analysis.     

Laser cooling of vibrational degrees of freedom of a molecular system

We consider the cooling of vibrational degrees of freedom in a photoinduced excited electronic state of a model molecular system. For the various parameters of the potential surfaces of the ground and excited electronic states and depending on the excitation frequency of a single-mode laser light, the average energy or average vibrational temperature of the excited state passes through a minimum. The amount of cooling is quantified in terms of the overlap integral between the ground and excited electronic states of the molecule. We have given an approach to calculate the Franck-Condon factor for a multimode displaced-distorted-rotated oscillator surface of the molecular system. This is subsequently used to study the effect of displacement, distortion, and Duschinsky rotation on the vibrational cooling in the excited state. The absorption spectra and also the average energy or the effective temperature of the excited electronic state are studied for the above model molecular system. Considering the non-Condon effect for the symmetry-forbidden transitions, we have discussed the absorption spectra and average temperature in the excited-state vibrational manifold.     

Quantum computing based on vibrational eigenstates: pulse area theorem analysis

In a recent paper [D. Babikov, J. Chem. Phys. 121, 7577 (2004)], quantum optimal control theory was applied to analyze the accuracy of quantum gates in a quantum computer based on molecular vibrational eigenstates. The effects of the anharmonicity parameter of the molecule, the target time of the pulse, and the penalty function on the accuracy of the qubit transformations were investigated. We demonstrate that the effects of all the molecular and laser-pulse parameters can be explained utilizing the analytical pulse area theorem, which originates from the standard two-level model. Moreover, by analyzing the difference between the optimal control theory results and those obtained using the pulse area theorem, it is shown that extremely high quantum gate fidelity can be achieved for a qubit system based on vibrational eigenstates.     

Vibrational frequencies in Car-Parrinello molecular dynamics

Car-Parrinello molecular dynamics (CPMD) are widely used to investigate the dynamical properties of molecular systems. An important issue in such applications is the dependence of dynamical quantities such as molecular vibrational frequencies upon the fictitious orbital mass μ. Although it is known that the correct Born-Oppenheimer dynamics are recovered at zero μ, it is not clear how these dynamical quantities are to be rigorously extracted from CPMD calculations. Our work addresses this issue for vibrational frequencies. We show that when the system is sufficiently close to the ground state the calculated ionic vibrational frequencies are ω(M) = ω(0M)[1 -C(μ/M)] for small μ/M, where ω(0M) is the Born-Oppenheimer ionic frequency, M the ionic mass, and C a constant that depends upon the ion-orbital coupling force constants. Our analysis also provides a quantitative understanding of the orbital oscillation amplitudes, leading to a relationship between the adiabaticity of a system and the ion-orbital coupling constants. In particular, we show that there is a significant systematic dependence of calculated vibrational frequencies upon how close the CPMD trajectory is to the Born-Oppenheimer surface. We verify our analytical results with numerical simulations for N(2), Sn(2), and H/Si(100)-(2×1).     

Selective excitation of molecular modes in a mixture by optimal control of electronically nonresonant femtosecond four-wave mixing spectroscopy

Femtosecond time-resolved coherent anti-Stokes Raman scattering (fs-CARS) gives access to ultrafast molecular dynamics. Due to the spectrally broad laser pulses, usually poorly resolved spectra result from this spectroscopy. However, it can be demonstrated that by shaping the femtosecond pulses a selective excitation of specific vibrational modes is possible. We demonstrate that using a feedback-controlled optimization technique, molecule-specific CARS spectra can be obtained from a mixture of different substances. A careful analysis of the experimental results points to a nontrivial control of the vibrational mode dynamics in the electronic ground state of the molecules as underlying mechanism.     

Manganese pentacarbonyl bromide as candidate for a molecular qubit system operated in the infrared regime

Our concept for a quantum computational system is based on qubits encoded in vibrational normal modes of polyatomic molecules. The quantum gates are implemented by shaped femtosecond laser pulses. We adopt this concept to the new species manganese pentacarbonyl bromide [MnBr(CO)5] and show that it is a promising candidate in the mid-infrared (IR) frequency range to connect theory and experiment. As direct reference for the ab initio calculations we evaluated experimentally the absorption bands of MnBr(CO)5 in the mid-IR as well as the related transition dipole moments. The two-dimensional potential-energy surface spanned by the two strongest IR active modes and the dipole vector surfaces are calculated with density-functional theory. The vibrational eigenstates representing the qubit system are determined. Laser pulses are optimized by multitarget optimal control theory to form a set of global quantum gates: NOT, CNOT, Pi, and Hadamard. For all of them simply structured pulses with low pulse energies around 1 microJ could be obtained. Exemplarily for the CNOT gate we investigated the possible transfer to experimental shaping, based on the mask function for pulse shaping in the frequency regime as well as decomposition into a train of subpulses.     

Bias-controlled selective excitation of vibrational modes in molecular junctions: a route towards mode-selective chemistry

We show that individual vibrational modes in single-molecule junctions with asymmetric molecule-lead coupling can be selectively excited by applying an external bias voltage. Thereby, a non-statistical distribution of vibrational energy can be generated, that is, a mode with a higher frequency can be stronger excited than a mode with a lower frequency. This is of particular interest in the context of mode-selective chemistry, where one aims to break specific (not necessarily the weakest) chemical bond in a molecule. Such mode-selective vibrational excitation is demonstrated for two generic model systems representing asymmetric molecular junctions and/or scanning tunneling microscopy experiments. To this end, we employ two complementary theoretical approaches, a nonequilibrium Green's function approach and a master equation approach. The comparison of both methods reveals good agreement in describing resonant electron transport through a single-molecule contact, where differences between the approaches highlight the role of non-resonant transport processes, in particular co-tunneling and off-resonant electron-hole pair creation processes.     

The workings of a molecular thermometer: the vibrational excitation of carbon tetrachloride by a solvent

An intriguing energy-transfer experiment was recently carried out in methanol/carbon tetrachloride solutions. It turned out to be possible to watch vibrational energy accumulating in three of carbon tetrachloride's modes following initial excitation of O-H and C-H stretches in methanol, in effect making those CCl(4) modes "molecular thermometers" reporting on methanol's relaxation. In this paper, we use the example of a CCl(4) molecule dissolved in liquid argon to examine, on a microscopic level, just how this kind of thermal activation occurs in liquid solutions. The fact that even the lowest CCl(4) mode has a relatively high frequency compared to the intermolecular vibrational band of the solvent means that the only solute-solvent dynamics relevant to the vibrational energy transfer will be extraordinarily local, so much so that it is only the force between the instantaneously most prominent Cl and solvent atoms that will significantly contribute to the vibrational friction. We use this observation, within the context of a classical instantaneous-pair Landau-Teller calculation, to show that energy flows into CCl(4) primarily via one component of the nominally degenerate, lowest frequency, E mode and does so fast enough to make CCl(4) an excellent choice for monitoring methanol relaxation. Remarkably, within this theory, the different symmetries and appearances of the different CCl(4) modes have little bearing on how well they take up energy from their surroundings--it is only how high their vibrational frequencies are relative to the solvent intermolecular vibrational band edge that substantially favors one mode over another.     

Ab initio centroid path integral molecular dynamics: application to vibrational dynamics of diatomic molecular systems

An ab initio centroid molecular dynamics (CMD) method is developed by combining the CMD method with the ab initio molecular orbital method. The ab initio CMD method is applied to vibrational dynamics of diatomic molecules, H2 and HF. For the H2 molecule, the temperature dependence of the peak frequency of the vibrational spectral density is investigated. The results are compared with those obtained by the ab initio classical molecular dynamics method and exact quantum mechanical treatment. It is shown that the vibrational frequency obtained from the ab initio CMD approaches the exact first excitation frequency as the temperature lowers. For the HF molecule, the position autocorrelation function is also analyzed in detail. The present CMD method is shown to well reproduce the exact quantum result for the information on the vibrational properties of the system.     

Infrared-infrared double resonance spectroscopy of the isomers of acetylene-HCN and cyanoacetylene-HCN in helium nanodroplets

Infrared-infrared double resonance spectroscopy is used to probe the vibrational dynamics of molecular complexes solvated in helium nanodroplets. We report results for the acetylene-HCN and cyanoacetylene-HCN binary complexes, each having two stable isomers. We find that vibrational excitation of an acetylene-HCN complex results in a population transfer to the other isomer. Photoinduced isomerization is found to be dependent on both the initially excited vibrational mode and the identity of the acetylene-HCN isomer. However, population transfer is not observed for the cyanoacetylene-HCN complexes. The results are rationalized in terms of the ab initio intermolecular potential energy surfaces for the two systems with particular emphasis on the long-range barriers to rearrangement.     

Identification of the atomic scale structures of the gold-thiol interfaces of molecular nanowires by inelastic tunneling spectroscopy

We examine theoretically the effects of the bonding geometries at the gold-thiol interfaces on the inelastic tunneling spectra of propanedithiolate (PDT) molecules bridging gold electrodes and show that inelastic tunneling spectroscopy combined with theory can be used to determine these bonding geometries experimentally. With the help of density functional theory, we calculate the relaxed geometries and vibrational modes of extended molecules each consisting of one or two PDT molecules connecting two gold nanoclusters. We formulate a perturbative theory of inelastic tunneling through molecules bridging metal contacts in terms of elastic transmission amplitudes, and use this theory to calculate the inelastic tunneling spectra of the gold-PDT-gold extended molecules. We consider PDT molecules with both trans and gauche conformations bound to the gold clusters at top, bridge, and hollow bonding sites. Comparing our results with the experimental data of Hihath et al. [Nano Lett. 8, 1673 (2008)], we identify the most frequently realized conformation in the experiment as that of trans molecules top-site bonded to both electrodes. We find the switching from the 42 meV vibrational mode to the 46 meV mode observed in the experiment to be due to the transition of trans molecules from mixed top-bridge to pure top-site bonding geometries. Our results also indicate that gauche molecular conformations and hollow site bonding did not contribute significantly to the experimental inelastic tunneling spectra. For pairs of PDT molecules connecting the gold electrodes in parallel we find total elastic conductances close to twice those of single molecules bridging the contacts with similar bonding conformations and small splittings of the vibrational mode energies for the modes that are the most sensitive to the molecule-electrode bonding geometries.     

Exploring the effect of anharmonicity of molecular vibrations on thermodynamic properties

Thermodynamic properties of selected small and medium size molecules were calculated using harmonic and anharmonic vibrational frequencies. Harmonic vibrational frequencies were obtained by normal mode analysis, whereas anharmonic ones were calculated using the vibrational self-consistent field (VSCF) method. The calculated and available experimental thermodynamic data for zero point energy, enthalpy, entropy, and heat capacity are compared. It is found that the anharmonicity and coupling of molecular vibrations can play a significant role in predicting accurate thermodynamic quantities. Limitations of the current VSCF method for low frequency modes have been partially removed by following normal mode displacements in internal, rather than Cartesian, coordinates.     

Comparative study of cluster Ag17Cu2 by instantaneous normal mode analysis and by isothermal Brownian-type molecular dynamics simulation

We perform isothermal Brownian-type molecular dynamics simulations to obtain the velocity autocorrelation function and its time Fourier-transformed power spectral density for the metallic cluster Ag(17)Cu(2). The temperature dependences of these dynamical quantities from T = 0 to 1500 K were examined and across this temperature range the cluster melting temperature T(m), which we define to be the principal maximum position of the specific heat is determined. The instantaneous normal mode analysis is then used to dissect the cluster dynamics by calculating the vibrational instantaneous normal mode density of states and hence its frequency integrated value I(j) which is an ensemble average of all vibrational projection operators for the jth atom in the cluster. In addition to comparing the results with simulation data, we look more closely at the entities I(j) of all atoms using the point group symmetry and diagnose their temperature variations. We find that I(j) exhibit features that may be used to deduce T(m), which turns out to agree very well with those inferred from the power spectral density and specific heat.