Quantum quantum information and thermodynamics with ions • Introduction to ion trapping and cooling Trapped ions as qubits for quantum computing and simulation • Qubit architectures for scalable entanglement • Quantum thermodynamics introduction • Heat transport, Fluctuation theorems, • • Phase transitions, Heat engines Outlook • www.quantenbit.de F. Schmidt-Kaler Mainz, Germany: 40 Ca +
Ion Gallery Innsbruck, Austria: 40 Ca + coherent breathing motion of a 7-ion linear crystal Oxford, England: 40 Ca + Boulder, USA: Hg + Aarhus, Denmark: 40 Ca + (red) and 24 Mg + (blue)
Why using ions? • Ions in Paul traps were the first sample with which laser cooling was demonstrated and quite some Nobel prizes involve laser cooling… • A single laser cooled ion still represents one of the best understood objects for fundamental investigations of the interaction between matter and radiation • Experiments with single ions spurred the development of similar methods with neutral atoms and solid state physics • Particular advantages of ions are that they are - confined to a very small spatial region ( d x< l ) - controlled and measured at will for experimental times of days - strong, long-range coupling • Ideal test ground for fundamental experiments • Further applications for - precision measurements - quantum computing - thermodynamics with small systems - quantum phase transitions - cavity QED - optical clocks - quantum sensors - exotic matter
Introduction to ion trapping Modern segmented micro Paul trap • Paul trap • Ion crystals • Eigenmodes of a linear ion crystal • Non-harmonic contributions Traditional Paul trap
Dynamic confinement in a Paul trap
Invention of the Paul trap Wolfgang Paul (Nobel prize 1989)
Binding in three dimensions Electrical quadrupole potential trap size: Binding force for charge Q leads to a harmonic binding: Ion confinement requires a focusing force in 3 dimensions, but Laplace equation requires such that at least one of the coefficients is negative, e.g. binding in x- and y-direction but anti-binding in z-direction ! no static trapping in 3 dimensions
Dynamical trapping: Paul‘s idea time depending potential with leads to the equation of motion for a particle with charge Q and mass m takes the standard form of the Mathieu equation (linear differential equ. with time depending cofficients) with substitutions radial and axial trap radius
Theodor Hänsch‘s video celebrating Wolfgang Paul invention
Regions of stability time-periodic diff. equation leads to Floquet Ansatz If the exponent µ is purely real, the motion is bound, if µ has some imaginary part x is exponantially growing and the motion is unstable. The parameters a and q determine if the motion is stable or not. Find solution analytically (complicated) or numerically: a =0, q =0.5 a =0, q =0.1 a =0, q =0.9 a =0, q =0.7 a =0, q =0.3 a =0, q =1.0 a =0, q =0.8 a =0, q =0.6 a =0, q =0.2 a =0, q =0.4 6 10 19 unstable excursion excursion excursion excursion excursion excursion excursion excursion excursion excursion -3 10 19 time time time time time time time time time time
3-Dim. Paul trap stability diagram for a << q << 1 exist approximate solutions The 3D harmonic motion with frequency w i can be interpreted, approximated, as being caused by a pseudo-potential Y leads to a quantized harmonic oscillator Pseudo potential approximation: RMP 75, 281 (2003), NJP 14, 093023 (2012), PRL 109, 263003 (2012)
non-ideal surfaces Real 3-Dim. Paul traps r ring ~ 1.2mm ideal 3 dim. Paul trap with equi-potental surfaces formed by copper electrodes RMP 82, 2609 (2010) numerical calculation of equipotental lines similar potential near the center
2-Dim. Paul mass filter stability diagram time depending potential with y dynamical confinement in the x- y-plane x with substitutions radial trap radius
Innsbruck design of linear ion trap 1.0mm 5mm Blade design w w 5 MHz 0 . 7 2 MHz radial axial trap depth eV F. Schmidt-Kaler, et al., Appl. Phys. B 77, 789 (2003).
Ion crystals: Equilibrium positions and eigenmodes
Equilibrium positions in the axial potential z-axis trap potential mutual ion repulsion find equilibrium positions x 0 : ions oscillate with q(t) arround condition for equilibrium: dimensionless positions with length scale D. James, Appl. Phys. B 66, 181 (1998)
Equilibrium positions in the axial potential set of N equations for u m force of the Coulomb force trap potential Coulomb force of all ions from left side of all ions from left side numerical solution (Mathematica), e.g. N=5 ions equilibrium positions -1.74 -0.82 0 +0.82 +1.74
Eigenmodes and Eigenfrequencies describes small excursions Lagrangian of the axial ion motion: arround equilibrium positions N N m=1 m,n=1 D. James, Appl. Phys. N N B 66, 181 (1998) m=1 m,n=1 N with and linearized Coulomb interaction leads to Eigenmodes, but the C. Marquet, et al., next term in Tailor expansion leads to mode coupling, which is Appl. Phys. B 76, 199 however typically very small. (2003)
Eigenmodes and Eigenfrequencies numerical solution (Mathematica), e.g. N=4 ions Matrix, to diagonize Eigenvectors pictorial Eigenvalues for the radial modes: Market et al., Appl. Phys. B76, (2003) 199 depends on N does not
Common mode excitations Center of mass mode position breathing mode H. C. Nägerl, Optics time Express / Vol. 3, No. 2 / 89 (1998).
Breathing mode excitation H. C. Nägerl, Optics Express / Vol. 3, No. 2 / 89 (1998).
1D, 2D, 3D ion crystals Wineland et al., J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998) Depends on a =( w ax / w rad ) 2 • Enzer et al., PRL85, Depends on the number of ions a crit = cN b • 2466 (2000) 1D Generate a planar Zig-Zag when w ax < w y rad << w x • rad • Tune radial frequencies in y and x direction Planar crystal 2D equilibrium positions d x~50nm ±0.25% Kaufmann et al, PRL 3D 109, 263003 (2012)
Ion crystal beyond harmonic approximations Marquet, Schmidt- Kaler, James, Appl. E kin U pot,harm. U Coulomb Phys. B 76, 199 (2003) Z 0 wavepaket size l z ion distance g,l ion frequencies D n,m,p coupling matrix
Non-linear couplings in ion crystal Self-interaction Cross Kerr coupling Resonant inter-mode coupling …. remind yourself of non-linear optics: frequency doubling, Kerr effect, self-phase modulation , …. Lemmer, Cormick, Schmiegelow, Schmidt-Kaler, Plenio, PRL 114, 073001 (2015)
Ding, et al, PRL119, 193602 (2017) Non-linear couplings in ion crystal Cross Kerr coupling Frequ. of mode a depends on occupation in mode b two phonons in mode b generate one phonon in mode a Resonant inter-mode coupling
Basics: Harmonic oscillator Why? The trap confinement is leads to three independend harmonic oscillators ! here only for the linear direction of the linear trap no micro-motion treat the oscillator quantum mechanically and introduce a+ and a and get Hamiltonian Eigenstates |n> with:
Harmonic oscillator wavefunctions Eigen functions with orthonormal Hermite polynoms and energies:
Two – level atom Why? Is an idealization which is a good approximation to real physical system in many cases two level system is connected with spin ½ algebra using the Pauli matrices D. Leibfried, C. Monroe, R. Blatt, D. Wineland, Rev. Mod. Phys. 75, 281 (2003)
Two – level atom Why? Is an idealization which is a good approximation to real pyhsical system in many cases together with the harmonic oscillator leading to the ladder of eigenstates |g,n>, |e,n>: n 1 , e n , e n 1 , e 1 , n g n , g n 1 , g levels not coupled
Laser coupling 2-level-atom harmonic trap dressed system „molecular Franck Condon“ picture dressed system n 1 , e n , e n 1 , e „energy ladder“ picture 1 , n g n , g n 1 , g
Laser coupling dipole interaction, Laser radiation with frequency w l , and intensity |E| 2 Rabi frequency: Laser with the laser interaction (running laser wave) has a spatial dependence: momentum kick, recoil:
Laser coupling in the rotating wave approximation using and defining the Lamb Dicke parameter h : if the laser direction is at an angle f to the vibration mode direction: Raman transition: projection of D k=k 1 -k 2 single photon transition x-axis x-axis
Lamb Dicke Regime carrier: laser is tuned to blue sideband: the resonances: red sideband:
Wavefunctions in momentum space kick by the laser: kicked wave function is non- orthogonal to the other wave functions
Experimental example carrier e , 1 e , 0 g , 1 g , 0 sideband carrier and sideband Rabi oscillations with Rabi frequencies and
„Weak confinement“ n + 1, e n , e n 1 , e n 1 , g n , g n 1 , g weak confinement: Sidebands are not resolved on that transition. Simultaneous excitation of several vibrational states
Two-level system g/2p = 15MHz dynamics spont. decay rate g Rabi frequency W W/2p = 100MHz incoherent: W < g coherent: W > g W/2p = 50MHz W/2p = 10MHz Solution of optical Bloch equations W/2p = 5MHz Steady state population of |e>:
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