Coupled quantized mechanical oscillators
The harmonic oscillator is one of the simplest physical systems but also one of the most fundamental. It is ubiquitous in nature, often serving as an approximation for a more complicated system or as a building block in larger models. Realizations of harmonic oscillators in the quantum regime include electromagnetic fields in a cavity haroche06 (); miller05 (); houck07 () and the mechanical modes of a trapped atom leibfried03 () or macroscopic solid oconnell10 (). Quantized interaction between two motional modes of an individual trapped ion has been achieved by coupling through optical fields monroe01 (), and entangled motion of two ions in separate locations has been accomplished indirectly through their internal states jost09 (). However, direct controllable coupling between quantized mechanical oscillators held in separate locations has not been realized previously. Here we implement such coupling through the mutual Coulomb interaction of two ions held in trapping potentials separated by 40 m (similar work is reported in a related paper harlander11 ()). By tuning the confining wells into resonance, energy is exchanged between the ions at the quantum level, establishing that direct coherent motional coupling is possible for separately trapped ions. The system demonstrates a building block for quantum information processing and quantum simulation. More broadly, this work is a natural precursor to experiments in hybrid quantum systems, such as coupling a trapped ion to a quantized macroscopic mechanical or electrical oscillator heinzen90 (); wineland98 (); tian04 (); hensinger05 (); tian05 ().
The direct coupling of atomic ions in separate potential wells is a key feature of proposals to implement quantum simulation schmied08 (); chiaverini08 (); schmied09 (), and it could allow logic operations to be performed in a multi-zone quantum information processor wineland98 (); cirac00 (); kielpinski02 () without the requirement of bringing the ion qubits into the same trapping potential. Moreover, the coupling could prove useful for metrology and sensing. For example, it could extend the capabilities of quantum-logic spectroscopy heinzen90 (); schmidt05 (); rosenband08 () to ions that cannot be trapped within the same potential well as the measurement ion, such as oppositely charged ions or even anti-matter particles heinzen90 (); wineland98 (). Coupling could be obtained either through mutually shared electrodes heinzen90 (); daniilidis09 () or directly through the Coulomb interaction wineland98 (); cirac00 (); tan02 (); ciaramicoli03 ().
The Coulomb interaction potential for two trapped ions, and , with charges and in potential wells separated by a distance is given by
Here and are the displacements of the ions from the external potential minima and is the permittivity of free space. The first term is constant and does not affect the dynamics. The second term represents a steady force between the ions that displaces them slightly; if necessary, it can be counteracted with additional potentials applied to nearby electrodes. The terms proportional to and represent static changes in the trap frequencies that could also be compensated with potentials applied to nearby electrodes. The term proportional to represents the lowest-order coupling between the ions’ motions. For small deviations, and , from equilibrium, the coupling is
and , , , and represent the harmonic oscillator lowering (, ) and raising (, ) operators, and are respectively the ion masses and motional frequencies, is Planck’s constant divided by , and we have neglected fast-rotating terms. Minimizing the distance, , between ions is crucial, because for fixed , the coupling rate scales as .
When (the resonance condition), we find
At time , the operators have changed roles up to a phase factor, and the oscillators have completely swapped their energies, regardless of their initial states. At , the energies have returned to their initial values in each ion. The mean occupation, , of ion as a function of time exhibits oscillations with period .
The trap can produce two potential minima at a height m above the surface and separated by m along the axis. Each potential well confines a single ion with an axial (parallel to ) motional frequency of MHz and a barrier between wells of meV. Pseudopotential confinement in radial directions (normal to ) is accomplished with a peak potential of V at 170 MHz applied to the radio-frequency electrodes, yielding radial frequencies of MHz. By applying static potentials to the d.c. electrodes, we can independently vary the separation between the ions and the curvatures of the two trapping wells. In this way, the ion axial motional frequencies can be brought into or out of resonance, allowing a tunable interaction. For MHz, we predict that , where we have included a 2% correction in owing to the metallic electrodes beneath the ions (Methods Summary).
With currently achieved size scales in ion traps, the Coulomb interaction is relatively weak, so low ion heating rates and stable trapping potentials are essential. Heating rates can be suppressed by operating at cryogenic temperatures deslauriers06 (); labaziewicz08_2 (), so that the direct Coulomb coupling rate can exceed the heating rate. In this work, the trap electrodes and surrounding vacuum enclosure are cooled to 4.2 K with a liquid helium bath cryostat. With similar versions of this trap at MHz, heating rates expressed as (where denotes the quantum number of motional Fock state and is its expectation value) were observed to be as low as 70 quanta per second, consistent with the results of ref. 26. However, for the experiments described here ( MHz) the heating rate was quanta per second and varied between the two wells. We observed in this trap, in agreement with previous reports deslauriers06 (); labaziewicz08_2 (); epstein07 (), so large values of are beneficial. The use of , the lightest of the commonly trapped atomic ions, is an advantage here, because for given d.c. trapping potentials the heating rate should remain unchanged while . Cryogenic operation decreases the background gas pressure to negligible levels, such that ion loss rates due to collisions with background gas are smaller than one per day.
A signature of coupling between the ions is the splitting between the two axial normal mode frequencies. As the trap potential is tuned into the resonance condition, this splitting, , reaches a theoretical minimum kHz. A plot of the mode frequencies will therefore show an avoided crossing. We measure the mode frequencies by applying a nearly resonant oscillating potential pulse to one of the trap electrodes. We then illuminate both ions with laser radiation resonant with the cycling transition at 313 nm. A decrease in the resulting fluorescence indicates that a mode of the ions’ motion has been resonantly excited. For pulse lengths , we resolve the two modes (Fig. 2a).
We sweep the trapping wells through resonance by varying the static potentials that are applied to the trap electrodes. A plot of the resulting mode frequencies, determined as above, is given in Fig. 2b, c, showing a minimum of kHz, in agreement with theory.
To demonstrate coupling at the level of a few motional quanta, we observe the exchange of energy between the two ions as follows. The ion motional frequencies are initially detuned by kHz, which is much greater than , effectively decoupling the ions’ motions. The ions are then simultaneously illuminated with a laser detuned by MHz from the cycling transition, cooling them into a thermal state at the Doppler limit with mean occupation . Subsequently, ion is cooled to by several cycles of stimulated Raman cooling with the and hyperfine states monroe95 (). The Raman beams are counter-propagating and oriented at 45° relative to the axis. At this point the potentials are brought into resonance ( MHz) within an interval (9 s) short in comparison with but long in comparison with the axial oscillation period. They remain on resonance for a time , allowing energy to transfer between the ions. After a time , the potentials are adiabatically returned to their off-resonance values and we determine the mean quantum number, , in ion by observing asymmetry between the red and blue motional sidebands of the -to- hyperfine Raman transition monroe95 ().
As seen in Fig. 3, energy exchanges between the ions during an interval s.
The 5% disagreement between the measured and predicted (162 s) values for is probably due to uncertainty in the ion separation, (even a 1-m uncertainty would account for the disagreement). The first maximum of corresponds to the cooling limit of ion ( quanta). The underlying linear growth in corresponds to heating of the ions at a rate of quanta per second (Methods Summary).
As a final experiment, we demonstrate energy exchange at approximately the single-quantum level. Ideally, the experiment takes the following form. The ions are tuned to the resonance condition throughout and are initially Doppler cooled. Ion is Raman-cooled, sympathetically cooling ion and thereby preparing the state . To create a single motional quantum, we drive ion with a blue-sideband Raman pulse (of duration s, which is much less than ), creating the state . The system oscillates between and with period . After a time , we drive ion with another blue-sideband pulse, conditionally flipping the spin from to , dependent on the presence of a motional quantum in ion . The final internal state probability will be given by . In practice, contrast in the oscillations (Fig. 4) is significantly reduced by incomplete cooling, motional decoherence and decoherence due to imperfect Raman sideband pulses.
We estimate that ion is cooled initially to . Although we were unable to measure the initial temperature of ion directly, comparison of the contrast and temporal behavior of our exchange data (Fig. 4) with simulations indicates that . Motional decoherence results from heating and from trap frequency instability over the time required to acquire the data. Raman sideband pulses suffer from variations in laser intensity and fluctuations in the sideband coupling caused by thermal spread in the and motional states (Debye-Waller factors wineland98 ()). For MHz, we observe oscillations with period s. The 2% disagreement with the prediction (447 s) of equations 2 and 3 is probably due to uncertainty in the ion separation and the difficulty of maintaining the exact resonance condition.
Significant improvements in coupling fidelity seem to be in reach with current technology. It should be possible to improve Raman laser intensity stability, and Debye-Waller factors from the and motion can be eliminated by proper choice of beam directions monroe95 (). We believe that our motional frequency instability can be alleviated by ensuring that the trap surface is free of charged contaminants. Faster exchange can be achieved by scaling down trap dimensions, but this puts a premium on reduced motional heating. Although the work presented here uses the axial mode to couple the ions, it may prove advantageous to use the radial modes in future experiments because of the lower heating rates associated with their higher frequencies. As an example, radial mode frequencies of MHz are routinely achieved in the apparatus, compared with axial frequencies MHz.
These results could lead to several possible applications in quantum state engineering and spectroscopy. For example, from the motional state the state of the ions at time is the Bell state . Transferring this state onto the ions’ internal states with sideband pulses would create an entangled spin state, even between ions of dissimilar species. This could be used as an entangled-pair factory in the scheme of refs 10,18, with the advantage over previous schemes jost09 () that the ions are already in separate wells, ready for distribution to separate locations. The coupling could be used to read out the state of one ion species with another, an ability useful for error correction protocols and for quantum logic spectroscopy schmidt05 (). When the harmonic wells are not in resonance, the spin state of one ion can be read out without destroying the state of the other, so schemes for weak or quantum non-demolition measurements that use either Kerr-type nonlinearities or quantum logic may become feasible haroche06 (). Hybrid quantum systems, incorporating similar interactions to couple a trapped ion to other quantum devices, could serve as a means of transferring quantum information between different qubit implementations in a future quantum network. For example, a trapped ion could act as a quantum transformer between a superconducting qubit tian05 () and a photonic qubit cirac97 (); moehring07 (). The sympathetic cooling through exchange might also be used to cool neutral molecules idziaszek10 ().
The shielding factor, , represents the ratio of the exchange rates with and without the presence of the trapping electrodes. To a good approximation, we ensure that at the motional frequencies all trap electrodes are held at ground. Therefore, assuming that gaps between the electrodes are negligible, the shielding factor can be calculated with the method of images. The result is
which reaches a maximum of at .
The evolution of under the influence of equation 1, including heating effects and assuming that both ions begin in a thermal state with mean quantum numbers and , can be predicted with a Langevin equation heinzen90 (). On resonance, the evolution is
where represents the mean of and for uncorrelated noise sources.
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Acknowledgements This work was supported by IARPA, DARPA, ONR and the NIST Quantum Information Program. We thank M. Biercuk, A. VanDevender, J. Amini, and R. B. Blakestad for their help in assembling parts of the experiment, and we thank U. Warring and R. Simmonds for comments. This paper, a submission of NIST, is not subject to US copyright.
Author Contributions K.R.B. and C.O. participated in the design of the experiment and built the experimental apparatus. K.R.B. collected data, analysed results and wrote the manuscript. Y.C. fabricated the ion trap chip and collected data. A.C.W. maintained laser systems and collected data. D.L. participated in the design of the experiment, collected data and maintained laser systems. D.J.W. participated in the design and analysis of the experiment. All authors discussed the results and the text of the manuscript.
Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of this article at www.nature.com/nature. Correspondence and requests for materials should be addressed to K.R.B. (email@example.com).