Generation of Mesoscopic Superpositions of Two Squeezed States of Motion for A Trapped Ion

1. Generation of Mesoscopic Superpositions of Two Squeezed States of Motion for A Trapped Ion Shih-Chuan Gou (郭西川) Department of Physics National Changhua University of Education 國立彰化師範大學物理系

2. Schemes for possible realization of quantum computer • Atom-cavity system • Ion trap • NMR • Quantum dots • Spintronics…

3. Reference: “Generation of mesoscopic superpositions of two squeezed states of motion for a trapped ion” , Phys. Rev. A 55, 3719 (1997). S.-C Gou, J. Steinbach, and P.L. Knight,

4. Working principle of the ion trap Penning trap: +magnetic field Paul trap:  +r.f. Combined trap:  + magnetic field+r.f. Linear and ring trap:…

5. Ion oscillations in a Penny trap

6. Realization of cavity QED in the ion trap homogeneous classical laser field : annihilation and creation operators of the harmonic oscillator

7. Quantized CM motion where the Lamb-Dicke parameter his defined as width of the ground-state wavepacket of the trapped ion = wavelength of driving laser Thus in the interaction picture, we have where

8. Choose D>0 blue sideband D=0 D<0  red sideband and (well-resolved sideband limit) (Lamb-Dicke limit) Thus to the leading order, we can engineer, for example, the l-photon-like interaction if we have an l-th red sideband excitation

9. Quantum state engineering in ion trap Theory: Squeezed states [Cirac, et. al. (1993)] Even and odd coherent states (Schrödinger cat states) [de Matos Filho and Vogel (1996)] Pair coherent states [Gou, Steinbach, Knight (1996)]  Experiment: D. Wineland’s group (NIST)

10. Squeezed states where displacement operator squeeze operator with squeezing factor Thus for two quadrature phase operators the minimum uncertainty product is reserved with

11. Even and odd squeezed states even squeezed states odd squeezed states Now since  where

12. Hamiltonian for a 2-level ion in 2-D trap y = -2x x = 0 = 2x =0 Superposed electric fields

13. The total Hamiltonian in the interaction picture

14. The evolution of the system can be described by a density matrix obeying the master equation accounts for the momentum transfer in the x-y plane due to spontaneous emission described by the angular distribution

15. For a highly anisotropic trap (x<<y), if y << x <<1 (Lamb-Dicke limit) and << j, then the master equation is reduced to

16. Steady-state solution of the master equation vibrational steady state (dark state)    Thus the eigenvalue  is determined by

17. The steady-state solutions depends on the parities of the initial state for initial state with even parity for initial state with odd parity for initial state with mixed parity

18. hx=0.02 hx=0.05 Number distribution P(n) of the vibrational steady state (grey bars) for various Lamb-Dicke parameters. The ion is initially prepared in the vacuum state. The number distribution of the even squeezed state, a=1, x=1 are shown in dark bars.

19. even squeezed state odd squeezed state Wigner distribution for even and odd squeezed states

21. Scheme of sideband cooling

22. Schrödinger’s cat If then what will you see when the chamber is open?

23. Δ=  Δ= 0 Δ= - For example, one may use the following π-pulse sequence to generate the number state n of vibration: g,0  e,1 g,2 e,2 … e,n g,n laser cooling laser off

24. Creation of entangled Schrödinger cat states with ions[(Monre, Meekhof, King and Wineland (1996)]

25. Various level schemes for the trapped ions

26. Measurement of quantum jump

27. Trapped ions as quantum computers [Cirac, Zoller, (1995)]

28. Vibrational mode as a quantum data bus (a) With the first laser pulse the state of ion 1 is mapped to the COM mode; (b) the state of ion 2 is changed conditional on the state of the COM mode.

29. (NIST, Ion Storage Group)

30. 10mm The scheme of the linear trap used in the Innsbruck group: A radio-frequency field (16 MHz, about 1000 Volts) is applied to the elongated electrodes (red) to provide the trapping in the radial direction. The ring-shaped electrodes at the two ends are responsible for the trapping in the axial direction, on which a static electric field of the order of +2000 Volts is applied. The ions (indicated by green dots ) oscillate in the radial and axial directions. However, since the trapping frequency in the radial direction (4 MHz) is much larger than that in the axial direction(700 kHz ), the ions arrange themselves in a linear string. The distance between the ions is typically only a few µm.

31. breathing mode center-of-mass motion Experimental demonstration of the motion of a string of 7 ions. (Figures by J.Eschner, F. Schmidt-Kaler, R. Blatt, Universität Innsbruck)

32. Perspectives of trapped ions Merits: • long decoherence times of the internal states of the ion • high efficiency to prepare, coherently control and detection of the states of the qubit using laser pulses Challenges: • fluctuations (intensity, frequency, phases…) of the driving lasers • collisions with background gas in the vacuum chamber • decoherence of the vibrational states that limits the number of operations • deviation between the laser focus and the position of the ion • difficulties to cool a string of ions to the ground state of motion