Abstract
Singlequantum level operations are important tools to manipulate a quantum state. Annihilation or creation of single particles translates a quantum state to another by adding or subtracting a particle, depending on how many are already in the given state. The operations are probabilistic and the success rate has yet been low in their experimental realization. Here we experimentally demonstrate (near) deterministic addition and subtraction of a bosonic particle, in particular a phonon of ionic motion in a harmonic potential. We realize the operations by coupling phonons to an auxiliary twolevel system and applying transitionless adiabatic passage. We show handy repetition of the operations on various initial states and demonstrate by the reconstruction of the density matrices that the operations preserve coherences. We observe the transformation of a classical state to a highly nonclassical one and a Gaussian state to a nonGaussian one by applying a sequence of operations deterministically.
Introduction
In quantum mechanics, bosonic creation and annihilation operators bear the following operator relations
where stands for a Fock state of n bosons. The proportionality factors and appear due to the symmetric indistinguishable nature of bosons^{1,2}. Thus, the addition or subtraction in quantum domain involves the modification of the probability amplitude of state due to the excitation ndependent factor. The bosonic annihilation and creation operation have been proposed to be a building block to generate an arbitrary quantum state^{3,4}, to distill entanglement or nonlocality^{5} and to transform a classical state to a nonclassical state^{6}. In recent times, there have been seminal works to realize the bosonic operations at the singleboson level for the test of foundations and applications of quantum mechanics^{7,8,9,10,11,12,13}. However, such bosonic operations are not trace preserving and inherently probabilistic. The probability of success has so far been extremely low in their implementation for photonic fields. A higher probability may be obtained at the expense of lowering the performance fidelity^{14}. Owing to this, a repetition of these operations has been limited.
The conventional addition and subtraction of a particle can be written as
The addition takes the n particle state to the (n+1) state, whereas the subtraction operation brings the n state to the (n−1) state without incurring additional factors. Seeing the form of the operations in equation (2), we immediately recognize that is a deterministic process while may not be, as it is not possible to subtract a particle from vacuum 0〉. When the vacuum component of the initial state is small, the subtraction can be done neardeterministically. In recent times, there have been theoretical proposals of the operations (equation (2)) for the generation of an arbitrary quantum state^{15}, the measurement of vacuum^{16}, the transformation to a nonclassical state^{17} and the amplification of a quantum state^{18}. In particular, such arithmetic operations form an important component of a qubit gate operation for ions in a harmonic potential^{19}. The operations (equation (2)) were also suggested as the elements of a phase operator^{20}. Beyond the quantumstate engineering, the subtraction can be used for the subDoppler cooling in the trapped ion system^{21}.
In this study, we experimentally demonstrate deterministic addition and neardeterministic subtraction of a bosonic particle, in particular a phonon of a ^{171}Yb^{+} ion trapped in a harmonic potential. We realize the operations by coupling phonons to an auxiliary twolevel system, so called the hybrid scheme of discrete and continuous variable^{22}, and applying a transitionless quantumdriving scheme. We perform the operations on superpositions of Fock states and coherent states, and we demonstrate that our singlephonon operations are (near) deterministic and preserve coherence. By applying a sequence of operations deterministically, we show that classical states are turned into nonclassical ones manifesting highly subPoissonian statistics and negativity in the Wigner function.
Results
Experimental implementations
We implement the and operations of equation (2) on a vibrational mode of frequency ω_{X} for a single trapped ^{171}Yb^{+} ion in a threedimensional harmonic potential^{23} through its interaction with the twolevel system of atomic energy levels. The harmonic potential is generated by an oscillating electric field in the radial axis with trap frequency ω_{X}=(2π)2.8 MHz. The twolevel system is represented by two hyperfine states and of the S_{1/2} manifold with the transition frequency ω_{HF}=(2π)12.6428 GHz. As shown in Fig. 1a, the antiJaynes–Cummings (aJC) interaction or bluesideband transition, , is realized by the stimulated Raman laser beams with beatnote frequency (ω_{R1}−ω_{R2})=(ω_{HF}+ω_{X})+Δ. Here, Ω is the Rabi frequency of the twolevel system, the Lamb–Dicke parameter, Δk the net wave vector of the Raman laser beams and M the mass of ^{171}Yb^{+} ion. The aJC coupling produces the transition between and with the oscillation frequency of , where the factor comes from the fundamental property of and operators in equation (1). Therefore, the application of the simple aJC interaction does not transfer to in an nindependent manner at a fixed duration of time.
The full population transfer independent of the initial motion state, the uniform bluesideband transition, , can be obtained by the application of the stimulated Raman adiabatic passage^{24,25,26}. The adiabatic scheme provides a robust transfer against the variations in the transition rate either from the intensity change of the control Raman beams or from the property of the transitions^{25}. Therefore, a properly designed adiabatic passage would allow a decent state transfer for a wide range of phonon number states through the aJC interaction, despite the dependence. In the adiabatic passage, typically ηΩ slowly increases at the beginning and decreases at the end, that is, Ω(t)=Ω_{0} sin(πt/T), whereas the detuning Δ changes according to Δ(t)=Δ_{0} cos(πt/T), where T is the total transfer time, across the resonance. However, for the applicability of the scheme to a wide range of initial phonon numbers with high fidelity, we should set Δ_{0} as high as , where n_{M} is the largest phonon number for the transfer and should fulfill the adiabatic condition, . In our experimental conditions, the reasonable duration T for such an adiabatic transfer is around 21 times that of πpulse duration for the bluesideband transition of the ground state π/ηΩ_{0}, when we include up to the maximum phonon number n_{M}=6.
In recent times, the transitionless quantumdriving scheme^{27,28,29,30} has been developed to speed up the adiabatic control. When the transfer is nonadiabatic, it introduces an additional term in the Hamiltonian of the instantaneous basis, which is , where β is in the order of π/ηΩ_{0}/2T, the ratio between the πpulse duration and the total operation time. By adding a counterdiabatic term in the control, we can suppress the nonadiabatic excitation with a reasonable speed up over the adiabatic passage. For the aJC interaction, the optimal values of Δ_{0} and β are dependent on the phonon number n for the given Ω_{0}. In our experiment, we optimize Δ_{0} and β for the case of the geometric average of the minimum and maximum phonon number, . By doing this, we are able to reduce the total duration of the operation from 21 to 7 times π/ηΩ_{0} without sacrificing the fidelity of the rapid adiabatic passage for the same range of phonons n_{M}=6.
To rigorously achieve the uniform bluesideband operation , it is important not only to increase the phonon number but also to preserve the relative phases between component states of the quantum state. For our previous realization^{23}, the different extra phases were accumulated depending on the phonon number of the initial state, which prevented from keeping the initial phase coherences (see ‘AC Stark shift compensation’ in Methods). In this study, we have developed a sequence of operations compensating the phononnumberdependent phases based on the spinecho principle. As shown in Fig. 1b, we invert the sign of Ω and reverse the control of Δ in the middle of the sequence of the operation, which symmetrizes the whole operation and produces the accumulated phases of opposite signs before and after the inversion and reverse. Therefore, the total phases are cancelled out at the end of the operation (see ‘AC Stark shift compensation’ in Methods).
Conventional phasecoherent addition
We implement the coherent addition operation in equation (2) by applying the uniform bluesideband transfer followed by the πpulse of carrier transition as shown in Fig. 2a. By this operation, we add single phonons deterministically, independent of the initial number of phonons. To manifest the coherence property of the addition operations, we prepare an initial state , apply the operations up to three times and measure the density matrices of the resulting phonon states. As shown in Fig. 2b, the coherences represented by the offdiagonal terms of the density matrix clearly remain after the multiple addition processes up to three times.
We also consider an initial coherent state of the amplitude α=0.81, which is achieved by displacing the vacuum. The density matrix is reconstructed (see Fig. 2c,d) by using the iterative maximumlikelihood algorithm^{31} on the phonon number distributions for eight different angles (see ‘Reconstruction of the phonon density matrix’ in Methods). The phonon number distributions are obtained by observing the time evolutions of the standard bluesideband transitions, similar to the direct reconstruction scheme of the phonon density matrix^{32}. One immediate consequence of adding single phonons on the coherent state is the production of subPoissonian phonon statistics, because the addition increases the average phonon number but not the shape of the distribution nor the variance. We observe that the ratios between the variance and the average phonon number reduce from 1 to 0.43, 0.39 and 0.2, after 1–3 singlephonon additions, respectively. Applying the first addition operation, we detect negativity in the Wigner function as shown in the second column of Fig. 2d. It is important to note that the addition operation, which converts a coherent state to a highly nonGaussian and nonclassical state, is deterministic unlike the case of operation^{9,10,11}. There is a technical limit in the number of additions we can apply, owing to the validity of the adiabatic approximation and the heating process of phonons^{33}. Under this limitation, we could perform the operations three times without the significant loss of fidelity. As shown in Fig. 2d, the experimental results and the theoretical predictions for the Wigner functions are in agreement, except the shift of the orientation, which would come from the phase offset between the preparation and the analysing pulses. This is significant in comparison with the photonic realization of bosonic operations of singlephoton creation and annihilation^{11}. We detect negativity in the Wigner function as shown in the second column of Fig. 2, where we obtain the Wigner function of the state from the reconstructed density matrix.
Conventional subtraction operation
The subtraction operation in equation (2) is realized by reversing the sequence of the addition operation, that is, the application of the πpulse of carrier transition and the uniform bluesideband transfer, followed by the fluorescence detection as shown in Fig. 3a. The zero phonon state is eliminated after the subtraction, which is implemented by the conditional measurement in our experimental scheme. After the detection sequence, we only collect the data with no fluorescence, which has the success rate given by the probability of the nonzero phonon states. We examine the performance of the subtraction operation with an initial phonon superposition state . As shown in Fig. 3b, the subtraction operation reduces the phonon excitation by one quanta. After the second application of the subtraction, the offdiagonal terms of the density matrix are significantly reduced, which shows the current limit in experiments due to heating of the system. We also prepare a coherent state and apply the subtraction twice as shown in Fig. 3c, which shows qualitatively that the subtraction operation works for any initial quantum state. The subtraction operation can squeeze a coherent state, which is different from annihilation that has the coherence state as its eigenstate. However, our experimental precision is not high enough to observe the squeezing effect.
Commutation relation of addition and subtraction operations
We study experimentally how the quantum states are changed depending on the order of the addition and subtraction for an initial coherent state . If we add and then subtract , the state after the sequence is the same as the original one, as there is no amplitude modification. For the case of subtractionthenaddition the final state does not have vacuum component, because a vacuum component is annihilated by and cannot be restored by . Figure 4b shows the experimental result of , which is basically identical to the initial state of Fig. 4a. Figure 4c shows the result after the operation of , where there is no significant vacuum component in the density matrix. The vacuum component is not perfectly absent because of the detection error during the projective measurement based on the atomic fluorescence and heating of the system. The fluorescence detection duration is comparable to the motional coherence time of our system, which makes the offdiagonal part of the final state suppressed significantly (see ‘Error analysis’ in Methods). Our experimental result is well in line with the noncommuting relation of the Susskind–Glogower’s phase operators, that is, (ref. 20).
Discussion
Owing to the capability of deterministically generating a nonclassical and nonGaussian states, the conventional addition and subtraction operations provide an efficient scheme for quantum state engineering^{34,35}. The current work can also be a stepping stone to realize the ion qubit gate operation as proposed by Schneider et al.^{19} without having to worry about long operation times and loss of coherences. This scheme may be further applied to various quantum optics setups such as cavity quantum electrodynamics and optomechanics. We note a theoretical scheme that the nonGaussian information of a phonon state would be transferred to that of photon through phonon–photon coupling^{36,37}. It has also been discussed that the conventional arithmetic addition and subtraction can be used to measure the vacuum state without disturbing the state^{16} and therefore it can be directly applied to construct the Qfunction of a quantum state of motion. We have noted that an arithmetic subtraction was performed for a photonic system^{38}.
Methods
AC Stark shift compensation
The AC Stark shift in the adiabatic operations mainly comes from the offresonant coupling to the carrier transition, the transition between S_{1/2}P_{1/2} states of ^{171}Yb^{+} ion and the other radial motional mode (ω_{Y}≈ω_{X}+(2π)0.4 MHz). The dominant AC stark shift comes from the carrier transition of frequency kHz with Ω_{0}=(2π)38.5 kHz and the Lamb–Dicke parameter η=0.089. The amount of the shift brought by the Y mode is given by , that is, ∼20 times smaller than that from the carrier coupling. The AC stark shift between qubit states from the Raman laser beams due to S_{1/2}P_{1/2} transition is kHz, where g_{R1} and g_{R2} are the coupling strengths of Raman 1 and Raman 2 beams, respectively, and the Δ_{R}=(2π) 18 THz is the detuning from the level of ^{2}P_{1/2}.
We consider total AC stark shift as the form of , where Δ_{total} is the detuning effectively including all the possible origin of AC stark shifts discussed above. We obtain and Δ_{total} by fitting the several points of with the equation . The actual frequency of bluesideband is measured by observing the resonant excitation. Including the AC stark shift, the actual waveform of Ω(t) that we apply on our arbitrary waveform generator is as follows:
Here, Δ(t′)=Δ_{0} cos(πt/T) and we note that the imaginary part in original form of Ω(t)=Ω_{0}[sin(πt/T)+iβ] is changed to sinwave, which has the phase difference. Here, φ(t) is calculated as follows:
Reconstruction of the phonon density matrix
We use an iterative algorithm in ref. 31 for the reconstruction of a quantum state. It consists of a maximumlikelihood estimation solved by the expectationmaximization algorithm followed by a unitary transformation of the eigenbasis of the density matrix ρ.
Based on the measured phonon distribution f_{n} by N measurements after a displacement of the quantum state, we aim to get real probabilities that are as close to the observed frequencies f_{n} as possible, subject to the maximumlikelihood functional
from which we reconstruct ρ. This likelihood functional can be interpreted as a linear and positive problem in the classical signal processing:
where r_{i} is the eigenvalues of ρ and h_{in} is a positive kernel. We can solve this linear and positive problem with the expectationmaximization algorithm^{39,40}:
which is initially set to a positive vector r . This is repeated for different displacements.
The second part of the reconstruction scheme aims at getting the eigenbasis diagonalizing the density matrix. This part consists of two steps: reconstruction of the eigenvectors of ρ in a fixed basis and rotation of the basis using a unitary transformation
with the infinitesimal form and is a small positive real number. G=i[ρ, R] is chosen as a Hermitian generator of the unitary transformation, where R is a semipositive definite Hermitian operator .
Starting from some positive initial density matrix ρ, we continue repetition of first finding eigenvalues r_{i} using the expectationmaximization iterative algorithm (equation (9)) and then finding eigenvectors φ_{i} by unitarily transforming the old ones. The likelihood of the estimate p_{n} is increased and we finally reach to determine the density matrix
Error analysis
Dominant error comes from the phonon heating process caused by the electricfield noise from the trapped electrodes^{41}. Heating decreases the Fock state preparation fidelity and affects the adiabatic blue sideband process. Its time evolution is known to be described by^{41,42}:
where γ is the coupling strength between the ion motion and the thermal reservoir and is the average phonon number for the thermal reservoir. In our experimental setup, the heating rate is 150 Hz. It can be reduced by using a large trap, cleaning the electrodes^{43} (equivalent to reducing γ) or cooling the trap (equivalent to reducing ) (ref. 44).
Error may be caused by fitting, as some noise in the bluesideband curves may be recognized as a high phonon population. Although the Fock state preparation error will be involved in the transition probability, fortunately the error for a small n is relatively insignificant and the population mainly resides in small nvalues for the initial states we prepared in the experiments.
There are some other small errors that we did not calculate or simulate, including the fluctuation of the trap frequency ∼0.1 kHz coming from ∼2% intensity fluctuation of Raman laser and ∼1 kHz trap frequency fluctuation coming from sudden changes of the ion position. We see how much our experimental results are modified when comparing with the simulations^{23}.
Additional information
How to cite this article: Um, M. et al. Phonon arithmetic in a trapped ion system. Nat. Commun. 7:11410 doi: 10.1038/ncomms11410 (2016).
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Acknowledgements
We thank Suyong Lee for the helpful discussion. This work was supported by the National Basic Research Program of China under grants number 2011CBA00300 (2011CBA00301) and the National Natural Science Foundation of China 11374178, 11574002 and 11504197. K.K. acknowledges the first recruitment programme of global youth experts of China. M.S.K. was supported by the UK EPSRC (EP/K034480/1) and the Royal Society. H.N. was supported by the NPRP grant 87511157 from the Qatar National Research Fund.
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M.U., J.Z., D.L., Y.L. and S.A. developed the experimental system. M.U. led the data taking and analysis. J.Z. developed the protocols for the control and measurement on phonons. J.N.Z., H.N. and M.S.K. provided theory and motivation. K.K. supervised the project. All authors contributed to writing of the manuscript.
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Um, M., Zhang, J., Lv, D. et al. Phonon arithmetic in a trapped ion system. Nat Commun 7, 11410 (2016). https://doi.org/10.1038/ncomms11410
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