Eth Zurich,¨ Trapped Ion Quantum Information Group Semester Project Spectroscopy of high energy Rydberg states of Magnesium Theo Dressler Prof. Jonathan Home Dr. Matt Grau Christoph Fischer April 15, 2019 Abstract The Rydberg energy states of magnesium atoms for different an- gular momentum quantum states are measured. Transitions in a mag- nesium gas cloud are driven using lasers of wavelength 285 nm for a transition into the first excited state, and 375 nm to drive a transition into a Rydberg state. Unexpected states at energies in between the theoretically predicted energy states of different angular momentum are discovered in combination with the expected transitions. Addition- ally in contradiction to the possible transitions, following the dipole selection rules, angular momentum states are reached which should not be possible: from a p-state to another p-state. This experiment determines the energies of the realized states of Magnesium atoms. 1 Contents 1 Introduction 3 2 Quantum Defect and Excitation Theory 4 3 Experimental Set-up 6 4 Measurement Method 9 5 Spectroscopy Results 12 5.1 Possible Transition . 17 5.2 Transition Characteristics . 19 5.3 Diversity of Spectral Lines . 20 1 5.4 n P1 Correspondence . 21 5.5 High Rydberg states close to ionization . 23 6 Discussion and Determination of Quantum defect parame- ters 24 2 1 Introduction Magnesium ions can be used to perform quantum state manipulations in a controlled way. The rough procedure is described in the following. With trapped ions quantum states can be generated and quantum information processed. This requires ionization and trapping with lasers. Different prop- erties of ground or excited states can be used to generate certain quantum states. Having full experimental control over the system, qubits are used to perform calculations. A qubit consists of two orthogonal quantum eigen- states. These superposition states enable information processing. [1] The Rydberg atoms allow to perform a similar procedure. Rydberg atoms have properties similar to ions. In high Rydberg states the electron is further departed from the atom core most of the time. The atom now has strong dipole properties. With the dipole interaction of the atoms, one can work similarly to ions. Rydberg atoms allow analytical treatment similar to the hydrogen atom, adding a quantum defect which affects the energy levels of the atom. Being able to apply the properties of a neutral atom as well as these of ions, allows to combine an extended as well as a narrow range of interactions. These make Rydberg atoms attractive for quantum entangle- ments. [2] With Rydberg atoms a so called dipole blockade is used to inhibit transitions into certain collective quantum states of an atom cluster. This results in an extended control other the quantum system in form of a limitation of possible states. Rydberg states have longer lifetimes which makes it easier to work with these quantum states. A realization of quantum information processing using Rydberg state atoms promises an more facile treatment due to a more secure control over the system. [3]. A Rydberg atom is characterized by a high energy quantum state of an atom, as the valence electron is excited into a state of high principal quan- tum number n. Rydberg atoms can hence be treated in a classical approach with quantum mechanical properties. Rydberg atoms are therefore related to the Bohr atom model given a high quantum number. Classically an electron orbits the atom core at a large distance. From the quantum mechanical per- spective this means that the electron is more likely to be found at a position further away from the atom core. Due to the larger distance the effect of the electron-electron repulsion as well as an exact coulomb attraction of all core protons loses its effect. Rydberg atoms can hence be treated equally to the analytically solvable hydrogen atom. [4] In order to further analyse the properties and experimental possibilities of Rydberg atoms, this spectroscopy experiment with Magnesium atoms was realized. Laser excited magnesium gas into higher quantum states. Split up 3 into different values of the angular momentum quantum number, these states could be sectioned. The energy of different angular momentum was mapped. It allowed to determine the quantum defect parameters for a range of Ryd- berg states of the atoms. Prime intention was to explore possible transitions in lower as well as higher Rydberg states closer to the ionization limit. 2 Quantum Defect and Excitation Theory Rydberg atoms were first introduced in transitions of the hydrogen atom. Explicitly from the second excited state to higher energy states. The visi- ble spectral lines allowed to identify these transitions. Their frequency was described using the Balmer formula which was later generalized for any tran- sition from an s to an p state. The formula for the frequency of a transition connecting two states of quantum number m and n resulted: Ry Ry hν = 2 − 2 (m − δs) (n − δp) Ry = 13:605 eV is the so called Rydberg constant and the δ parameters describe the deviation due to an s and p state: a quantum defect. To an- alytically determine the Rydberg constant one can exploit the classical cor- respondence of Rydberg atoms: An electron on a circular orbit, held by the e4m coulomb force, combined with energy conservation yields in Ry = 2 2 2 . 320π ~ With e; m the electron mass and charge [4]. In the case of heavier atoms such as magnesium this Rydberg constant needs to be adopted slightly to obtain the correct result because of electron reduced mass. The wave function of the hydrogen atom can be solved analytically starting from the Schroedinger equation and a Coulomb potential. Due to the spher- 1 ical symmetry and potential of the form V (r) ∼ r this yields an analytically solvable solution. In light of the solution of the hydrogen atom a likewise ap- proach for the Rydberg atom is possible. The energy eigenvalues, frequency with respect to ionization, are known from above approximations (m ! inf): Ry En = − 2 . Again, the potential is spherically symmetric considering (n−δl) the valence electron. The change in nucleus from hydrogen to magnesium leads to a lower potential for for small radii. Hence, the kinetic energy of the valence electron is higher in comparison to hydrogen. Since the spheri- cal component does not differ from the hydrogen atom, the consequence is a change in the radial component, reflected by shortened wavelength and a phase shift of the wavefunction. This phase shift τ, dependent on the Ry- τ dberg atom, can be described by the quantum defect parameters δl = π . The resulting wavefunction matches the wavefunction of the hydrogen atom, 4 assessed with additional factors of quantum defect: [5] Y (Θ; Φ)[a rl+1 cos πδ − b r−l sin πδ ] Ψ (Θ; Φ; r) = l;m l l l l n;l;m r Yl;m(Θ; Φ) are the solutions of the spherical Laplace operator, the well known spherical harmonics. n is the principal quantum number, l the angular mo- mentum number and m the angular momentum in direction of the z-axis. The energy eigenstates are given by the Rydberg-Ritz formula [6]: −RMg En;l = 2 (n − δl) The defect parameters δl in the most general form are also dependent on n. As the goal of this experiment is to find approximated parameters for a broad range of Rydberg states this defect parameter can be mathematically β γ expended: δl = α+ (n−α)2 + (n−α)4 . The parameters α, β and γ determine the energy states of the magnesium atoms. For any angular momentum quantum number the parameters can be calculated and taken from literature [6]. For optical transitions the 'selection rules' need to be considered. Transitions ∗ are possible if hΨn;l;mjΨn0;l0;m0 i 6= 0. Z ∗ 3 Ψn;l;m(r; Θ; Φ)Ψn0;l0;m0 (r; Θ; Φ)r dr cos Θ sin ΘdΘdΦ 6= 0 This problem is solved using 'Parity inversion'. The Parity operator P^, per- forms an inversion of the coordinate directions. Applying it on the integral, returns the properties of the integral. Odd parity means that the integral is zero whereas even parity leads to a non zero integral. The clue is to only consider the angular part of the wavefunction, Yl;l(Θ; Φ). The parity on this state has the eigenvalue (−1)l, an uneven parity. All other angular momen- tum states are created by application of the angular momentum operator ^ l−. This operator has even parity. Since the parity inversion transforms the azimuthal angle Θ via Θ ! π − Θ the sin has uneven, whereas the cosin has even parity. The resulting condition reflects (−1)l(−1)l0 (−1)1 = 1 to be stated as follows: l0 = l ± 1 When performing an optical transition to a quantum state of principal quan- tum number n , the angular momentum quantum number needs to be changed accordingly. [7] With the discrete wavefunction of the Rydberg atom and the selection rule as 5 derived above, optical transitions can be specified. In a gas at room temper- ature most atoms are in the ground state. This state of principal quantum number n = 0 is specified with angular momentum l = 0. Departing from this ground state jn = 0; l = 0i ^=jgi, the principal quantum number can be excited into any state, assuring that the angular momentum quantum number is increased by one. The resulting transition into the first excited state has the form jgi ! jn = 1; l = 1i. An excitation into Rydberg states of high principal quantum number can then be leveled into two different ex- cited states, following the selection rules: jn; l = 0i and jn; l = 2i.