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Crystal structures of Calcium ions have been prepared in a linear Paul trap and their collective motion excited with resonant rf-fields. The trapped ions are laser-cooled and images of the fluorescing ions are obtained with a CCD camera and show high spatial resolution. Crystals with up to 15 ions arrange in a linear string and their eigenmodes can subsequently be selectively excited. The collective motion of the string can then be observed via the CCD images.
©1998 Optical Society of America
Single or few ions in a Paul trap are ideally suited to study a quantum system with very small perturbations by the environment. Ion storage has therefore long been applied to ultra-high precision spectroscopy and the development of frequency standards1. More recently, single trapped ions have been used to demonstrate and test some of the intriguing features of quantum mechanics2,3. In particular, both the internal electronic state and the motional state of a trapped ion can be modified using laser light or resonant radio-frequency fields. Decoherence of internal superposition states is nearly negligible and coherence of the motion can be maintained for substantial times. With almost perfect control of the quantum state of a single ion2, the attention has turned to systems of few ions with well controlled interactions between them4. A first step in this direction is the cooling and control of the motion of ion strings. Although cooling to the ground state of motion was only demonstrated for one5 and, most recently, two ions6 it is possible to cool strings to thermal states with relatively low mean quantum number (n ≃ 50). Starting with these states one can excite coherent states of motion with a resonant radio-frequency interaction. The thermal component of the final state is insignificant compared to the coherent motion in the sense that the mean quantum number of the final state can be much larger (n ≃ 10000 or more) than the one of the initial thermal distribution. Controlling the motion of strings of ions will be a crucial ingredient for the implementation of quantum gates along the lines proposed by Cirac and Zoller7.
The paper is organized as follows: In section 1 we briefly present the theoretical background for normal mode excitation. In section 2 the experimental setup for trapping strings of 40Ca+ ions is described and in section 3 experimental results on the coherent excitation of normal modes in the string are presented and discussed.
1. Linear strings and normal modes
In a linear ion trap, ions can be confined and optically cooled4 such that they form ordered structures. A few cooled ions arrange linearly along the trap axis and the distance between the ions is determined by the equilibrium of the Coulomb repulsion and the axially confining potential. The actual positions of the ions can be numerically evaluated, and if the trap potential is harmonic, the positions can be described by a single parameter only, i.e. the axial frequency ωz 4,8. Small displacements of the ions from their equilibrium positions then cannot be described in terms of individual ion motion since the Coulomb interaction between the charged particles leads to coupled motion. Hence, the motion of the string of ions must described in terms of normal modes and the entire chain of particles moves with distinct frequencies7,8. As an example, consider two ions confined in a linear ion trap: The first mode corresponds to an oscillation where all of the ions are moving back and forth as if they were rigidly joined together. This oscillation is usually referred to as the center of mass mode of the string 8. The second normal mode corresponds to an oscillation where the ions move towards each other; this is the so-called breathing mode. More generally, for a string with N ions an oscillation where each ion is vibrating with an amplitude proportional to its equilibrium distance from the trap center is called a breathing mode. Explicit calculation of the normal modes (i. e. the eigenmodes) and the respective eigenfrequencies of an ion string yield the following simple results8,9: i) for a string consisting of N ions in one dimension there are N normal modes and normal frequencies, ii) the center of mass mode has a frequency which is equal to the frequency of a single ion, iii) higher order frequencies are nearly independent of the ion number N and are given by (1, 1.732, 2.4, 3.05(2), 3.67(2), 4.28(2), 4.88(2), …) ωz, where the numbers in brackets indicate the maximum frequency change for N up to 10 ions, iv) the relative amplitudes of the normal modes have to be evaluated numerically (at least for strings with more than 3 ions, see equation (28) in Ref.8). Note that the relative amplitudes are important if one tries to excite the modes with a laser wave (equation (63) in Ref.8). For radio-frequency excitation however, one has to take into account the coupling strengths of the rf wave to the different modes. In general, the fact that for higher modes than the center of mass mode the ions do not oscillate in phase requires field gradients across the ion string.
The predicted motion of a string consisting of N=5 ions is shown in Fig. 1: Ion positions are indicated with a black dot, the different mode frequencies are given on the vertical axis. The red lines represent the relative amplitude of the individual normal modes and the arrows (at the tip of the respective red lines) indicate the relative sign of the motion. The lowest mode frequency represents the center of mass mode and the second mode shows a breathing behavior. Higher order modes show a more complicated motional structure, e.g. the third normal mode is excited when the inner three ions are moving 180 degrees out of phase with the outer ions (higher order breathing mode).
Figure 1. Normal modes: Frequencies and amplitudes for 5 ions. Note that in the upper part the amplitudes for the center of mass mode have been reduced to 30% for clarity.
2. Experimental setup
2.1 Trap design
The linear trap for confining strings of 40Ca+ ions4 consists of a UHV vessel containing 4 parallel steel rods (diameter: 0.6 mm, distance from the trap axis to electrode surface 1.2 mm) which serve for (dynamic) radial x-y confinement and two ring electrodes (inner diameter: 4 mm) for (static) longitudinal z confinement (see Fig. 2). Pairs of additional electrodes for DC bias fields (micromotion compensation) are spaced 20 mm below and to the side of the center axis parallel to the quadrupole electrodes (pair spacing 15 mm). Pairs of Helmholtz coils in all 3 dimensions outside the vacuum are used to generate a well defined magnetic field.
We drive the trap at a frequency of Ω/2π = 18 MHz (given by the resonance frequency of the helical resonator amplifier) with a voltage amplitude of typically 800 V (400 to 1200 V are also used). This results in a radial secular frequency of about 1.4 MHz. On the DC ring electrodes we usually apply 100 to 200 V to achieve longitudinal center of mass oscillation frequencies of 160 to 200 kHz. In general, we apply dc voltages of up to 600 to 700 V to the compensation electrodes. These amplitudes are necessary because the compensation electrodes are rather far away from the trap electrodes. We believe that most of the compensation is needed because of imperfections in the alignment of the electrodes. Patch charges etc. do not seem to play an important role because we would expect that their effects vary from load to load which is not observed.
The ions are cooled in all 3 dimensions by laser light from the (x, y, z) = (0,1,-1) and (-1,0,-1) directions (see Fig. 2). The fluorescence light at 397 nm is collected in opposite directions onto a photomultiplier tube (Hamamatsu) and an intensified CCD camera (Princeton Instruments). For the CCD camera we use f/1.4 optics (Nikon) with a 20-fold magnification which allow for a resolution of better than 4 μm.
2.2 Laser systems
The Doppler cooling and detection light at 397 nm (see Fig. 3) is produced by a frequency doubled Ti:Sapphire ring laser which is radio-frequency locked to an external cavity (finesse ≃ 150) resulting in 50 mW of power with a rms linewidth of 250 kHz. Since this light is produced on a separate optical table (to isolate the experiment from vibrations due to the Ar+ ion pump laser) it is fibre-coupled to the ion trap. While loading the trap we generally use 4 mW of power in each of the two beams which is focused down to 100 μm. Afterwards we reduce the power to a level of 50 to 400 μW in order to reduce stray light without reducing the signal from the ions. The repumping light at 866 nm is produced by a grating stabilized laser diode (SDL) with a usable power of up to 15 mW. For loading we use about 2 mW of power in each beam. Again, the power is reduced when single ions are present. This laser diode is also radio-frequency locked to an external cavity (finesse ≃ 1000) and has a rms linewidth of better than 100 kHz. Both lasers are superposed and are shone onto the ions from the directions indicated above.
2.3 Ion storage
We see crystallization of ions for a wide range of parameters of the trap AC and DC voltages and laser parameters4. Crystallization from a large cloud produces 3-dimensional structures with up to 60 ions arranged in different shells (for this, the laser beam waists should be widened to assure cooling of the entire crystal). Smaller clouds result in 2-dimensional zigzag structures. Depending on the ratio between the strength of the radial and longitudinal confinement we get strings of ions if the number of ions is about 15 (AC amplitude ≃ 800 V, DC amplitude ≃ 150 V). Loading directly from the atomic beam preferably produces smaller strings with up to 5 ions. The positions of the ions agree well with a simple theoretical calculation4 and the mean distance between the ions is of the order of 10 to 20 μm depending on the strength of the longitudinal potential and the number of ions stored. The ions’ fluorescence can be well resolved on a CCD camera.
So far we have not measured the temperature of the ions directly. This will be done with the help of a sideband spectrum on the narrow S 1/2 – D 5/2 quadrupole transition10. However, excitation spectra on the 866 nm transition show a line whose width is close to the natural linewidth. From this we infer a temperature close to the Doppler limit. In z-direction this translates into an average occupation number for a thermal state of < n > = 50 for the center of mass mode.
3. Normal mode oscillations
After loading the trap with a string of ions4 normal modes can be excited by either applying additional AC voltages to the ring or to the compensation electrodes. For what follows, we have used the compensation electrodes. It is important to note that the normal mode excitation can be seen by an increase of the spot width on the CCD camera long before there is a dip in the fluorescence collected by the photomultiplier. This agrees with the fact that the integrated counts on the CCD camera stay constant for small excitation. In general, the measured frequency of the first breathing mode agrees within 1% with the expected frequency at √3 times the center of mass frequency. We have measured this for N = 2, 3, 5, 7 and 9 ions. Fig. 3 shows the excitation of the breathing mode (around 276.0 kHz) for 3 ions as a function of excitation frequency. The center ion does not change its width at all.
As can be seen in Fig. 4 for high excitation the classical turning points of the outer vibrating ions become already visible. From the crude estimate that the ions vibrational amplitude can be as large as 10μm one can infer an average occupation number of about 100000 for a coherent state. In this regime, the initial thermal occupation is completely negligible.
Fig. 5 shows the excitation of the center of mass mode (158.5 kHz) for various excitation amplitudes and the excitation of the breathing mode (276.0 kHz) for 5 ions. In order to excite the breathing mode we need to apply voltages which are typically about 300 times higher than the ones needed for excitation of the center of mass mode (3 V compared to 0.01 V). We did not observe excitations for higher order modes with the AC voltages available in our setup. We attribute this to the fact that our exciting field is nearly spatially constant across the ions so that the higher modes which need field gradients across the ions are not very well excited. We have also found that the amplitude of the excitation sensitively depends on the laser detunings (i.e. at 397 nm and 866 nm) which determine the laser cooling rate. Also, as we tune across the normal mode resonance we have seen prominent hysteresis effects. This we also attribute to the damping mechanism due to laser cooling whose efficiency in turn depends on the motion of the ions.
Figure 5. No excitation, slight and strong excitation of the center of mass mode (158.5 kHz), breathing mode (276.0 kHz) for 5 ions.
4. Conclusion and outlook
We have shown how strings of 40Ca+ ions in a linear Paul trap can be laser cooled and we observed crystallization in form of a linear string. Using a CCD camera, we have been able to resolve individual ions. The strings were excited by rf-fields resonant with the frequencies of the center of mass mode and the breathing mode. The results agree with the theoretical value of √3 for the frequency spacing in a harmonic trap. Together with the agreement of experimental and theoretical distances between the ions4 we conclude that the realized trap potential is harmonic and the agreement between theory and experiment is encouraging for a future application of the 40Ca+ strings as a quantum register. With additional cooling schemes (sideband laser cooling) it should be possible to reach the vibrational ground state5,10,11 and then the ion strings can be applied to a quantum gate operation.
Appendix
After submission of the paper we have taken movies of ions oscillating in the center of mass mode (107 kHz) and in the breathing mode (185 kHz). The trap DC voltages are different from the ones used in the main text. This results in lower oscillation frequencies.
For each frame of the movie the CCD camera was gated for 250 ns with a repetition rate of about 7 kHz phaselocked to the exciting radio frequency field. About 420000 pictures have been accumulated corresponding to an overall exposure time of 105 μs. The time between successive frames is 500 ns. In the case of the breathing mode it was necessary to apply very high excitation amplitudes (10 V, 1000 times higher than for the center of mass mode, see main text) so that to some extend the center of mass mode has also been excited.
Acknowledgments
This work is supported by the Fonds zur Förderung der wissenschaftlichen Forschung (FWF) under contract number P11467-PHY and in parts by the TMR networks “Quantum Information” (ERB-FMRX-CT96-0087) and “Quantum Structures” (ERB-FMRX-CT96-0077).
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