Energy-level structure of ion cloud and crystal in a linear Paul trap

Ion cloud and crystal are recognized as two distinct states in ion traps, but there is no theory that can describe the energy-level structures of these two states from the perspective of quantum theory. In this paper, we construct a model describing ions in a linear Paul trap to investigate the energy-level structure with the linear response function, which is often used in the field of condensed matter to compute the energy-level of electrons or plasmas. We support a method to calculate the energy-level of both the crystal and cloudy states of trapped ions in a linear Paul trap. Furthermore, we present the energy-level diagram of two trapped ions and give a quantum interpretation for the fluorescence hysteresis loops that were observed in our laboratory. This is a fundamental theory that will help us to understand the crystallization and cooling processes of ions in a linear Paul trap.


Introduction
Over the past several decades, experiments to control the level of trapped ions have been so widely developed that the experimental techniques for the confinement, cooling and coherent manipulation of ions have been extensively exploited for quantum information processing purposes. During the laser cooling process, the ions undergo a series of phase transitions [1,2]. These phase transitions are due to the competition between the repulsive Coulomb interaction and the kinetic energy of each ion, and they can be controlled by varying the detuning frequencies of a cooling laser. In the ion trap, the phenomena concerning the 'ordered' state, such as the phase transition of the 'ordered' state (zigzag transition) and the energy-level of the 'ordered' state (crystal state) [3][4][5][6][7], are mentioned more frequently than the 'disordered' (cloudy-crystal transition and cloudy state) [8][9][10][11].
However, the 'disordered' state has proven to be as important as the 'ordered' state, especially in the phase transition process. The 'disordered' state always seems to be an ion cloud. In early times, descriptions of the ion 'cloud' in the Penning trap and Paul trap were reported by Bollinger and Flory respectively [12,13]. They showed that the temperature of the ion cloud is associated with its size. A further interpretation was made by G Z Li, who studied the density of ion cloud by using the Thomas-Fermi theory and gave a theoretic interpretation [14]. However, the description of the ion 'cloud' in a linear Paul trap still has no final conclusion.
In some situations, the theory of molecular dynamics can be used to simulate the actions of an ion cloud. However, it is still necessary to find a quantum theoretical description for two reasons. The first is because the classical theory will fail in the descriptions of phase transition parameters and the energy-levels of the ions, especially under conditions such that the kinetic energy of the ions is comparable to ω  , where ω is the secular frequency. In the case of this low kinetic energy, the quantum effects become significant and the classical theory description leads to erroneous results. In fact, Wineland explained that the energy of ions in a cloudy state is very low and can be comparable to ω  [12], which forces the adopting of the quantum theoretical description. The other reason to employ a quantum theoretical description is that it is thought to be more exact than the classical theoretical description for trapped ions. It can reveal the real values for the state and energy of trapped ions in different states, and can also be used to predict new phenomena.
This paper explores a proper model that enables us to deal with the manipulation of both the crystal and cloudy states, using the original Hamiltonian. To eliminate the difficulty brought about by using a singular point of Coulomb potential, a proper field operator is chosen and the Coulomb potential is transferred to the second quantization formulation. Based on the calculations, the Hamiltonian is reduced further and to show this reduction can keep both the long-range and short-range actions of ions in the linear Paul trap; therefore, it can describe the ions' behaviour no matter how near or how far they are. This is different from phonon models of ions, which can be used only in crystal states of low kinetic energy, or when the oscillation amplitude of ions in equilibrium is very small. If the kinetic energy of trapped ions is very large, the ions may be very close together and the phonon model will be invalid. An advantage of the model presented here, is that it can be used to describe the competition between the Coulomb interaction and the kinetic energy per ion, which leads to the phase transition between the cloudy state and crystal state according to this formulation.
Our model can show the energy variation during the process of phase transition and as well as the ion temperature in the Paul trap. By using the linear response function, it will be shown that the crystal state is stable, which corresponds to collective excitation, while the cloudy state is unstable, which corresponds to individual excitation. By using this model, we explain the hysteresis-loop-like behaviour formed by fluorescence intensity of trapped ions, which was proposed by Diedrich et al [15]. Furthermore, we show that the results arising from our formulation are in accordance with our experiments. This paper falls into three parts. In the first part, we will introduce our ion-trap experiment and show the phenomena that we encounter in our experiments. In the next, we will construct a Hamiltonian model and show the way to calculate the energy-level of trapped ions with the linear response function. In this part, we will focus on the rationality of this model. In the last part, we will exhibit the comparison between our theory and the experiment.

The outline of experiment
In this experiment, a Paul trap was used to capture + Ca 40 . The set-up can be seen in figure 1. RF voltage was applied to the two pairs of blades and DC voltage was applied to the caps. The cooling laser used is a 397 nm, and the re-pumping laser is an 866 nm laser. The 397 nm laser was used for the initial cooling down of the ions in the trap. The trapped ions were then in the crystal state and formed a line along the axis. The frequency of the repumping laser was kept constant, but the 397 nm laser was detuned to heat the ions in the trap. Two directions are typically used for detuning, namely red-detuned and blue-detuned lasers. Red-detuning is usually used to cool down the ions, but when the detuning frequency difference is large, it may also heat the ions because of the broad kinetic energy range of the ions and the frequency of the cooling laser.
In the heating process, the ions became increasingly warmer. This was mainly reflected in their increasingly larger fluorescent images in EMCCD. This wonʼt last long if the cooling laser is detuned along one direction. It will experience a phase transition when the detuning frequency of the cooling laser, which is 397 nm in wavelength, is over 20 MHz for bluedetuning or 200 MHz for red-detuning in our laboratory. In the process of phase transition, the fluorescent image of the ions altered significantly to a large cloud. Sometimes, it is hard to cool the ion cloud to be a crystal because cooling and warming are not inverse processes in ion traps. To cool a big ion cloud to be a Coulomb crystal, two methods are commonly adopted. Either the 397 nm laser is red-detuned (in this process, the 866 nm laser should be continuously sweeping over one or more hours to assist the cooling when the compensation for micromotion is good) or the ion cloud is moved by reducing the voltage intensity of the caps (in this process, some ions in the trap escape, reducing the ion cloud, and then the phase transition happens). In fact, there is not an easy way to control the ion cloud in our experiment; therefore, to clearly define what the ion cloud is and how to control the process of phase transition is very important for us.
One puzzling phenomenon we observed in the laboratory is the hysteresis-loop-like behaviour of ion fluorescence, which was first detailed by Diedrich et al [15]. Diedrich et al found that during the process of a phase transition between the 'crystalline' and 'gaseous' state, the fluorescence intensity changes with a hysteresis-loop-like as a function of the cooling laser power, which had been never found before. However, they still did not give an explanation of the hysteresis-loop-like in theory. We observed this phenomenon in our laboratory, and it is shown in the left of figure 2. To interpret this phenomenon, physicists historically proposed some theories [16][17][18][19][20][21][22][23]. However, most of these theories are based on Newtonian mechanics and can not reveal ions' energy varying in the process of phase transition. As we know, there are still no perfect 'quantum' theoretical explanations of the hysteresis loop mentioned above and present an explanation from the quantum perspective. Moreover, further hysteresis-loop-like behaviour that has not previously been reported is shown in the right of figure 2. In that process, the ions in the trap seemed to attract each other despite the Coulomb repulsion between them. This paper will interpret this interaction. The phase transition between the cloudy state and crystal state in our laboratory. The fluorescent light is collected by EMCCD and imaging in a computer through special software processing. We observe two kinds of phase transition in the laboratory, the normal phase transition, shown in picture I, and abnormal phase transition, shown in II. The normal phase transition is only driven by cooling laser, but for the abnormal phase transition, we should use micro-amplification RF drive voltage under the compensating pole to assist the transition.

The Hamiltonian model
To construct a proper Hamiltonian, we first consider the behaviour of the trapped ions. In the linear Paul trap, there are two main kinds of potential: the Coulomb potential, which comes from each pair of ions, and the trapped potential, which comes from both the axial DC voltage and radial RF voltage. The RF voltage can cause the micromotion of the trapped ions, and the amplitude of micromotion is proportional to the distance of the ion from the trap centre [1,2]. Usually, this micromotion cannot be overlooked, but in low temperatures, if the number of ions is fewer than 10 (this number is the best condition to observe the phase transition between the cloudy state and crystal state in our laboratory), the ions are crowded in the centre of the trap and their micromotion is close to zero and can be ignored. Under these conditions, the potential from the RF voltage is the pseudopotential of RF voltage, which is treated as a constant potential. Furthermore, the inner states of ions, spin states, are not considered in this paper. Therefore, the total Hamiltonian is where, ω x and ω r are the secular frequencies in the axial and radial directions respectively, and m p is the mass of the ions.Ĥ contains three parts, the kinetic part, the harmonic potential and the repulsive Coulomb potential. The Coulomb potential is always difficult to calculate for its singularities points at = r r i j , therefore, it is transferred to the second quantization to eliminate these singularities. We define the field operator as  (3) contains two parts H 0 and H 1 . H 0 is the Hamiltonian with no transition-term, and H 1 is the transition-terms Hamiltonian. For strong trapped potential in axial-direction, 〈 〉 ≫ 〈 〉 H H 0 1 , meaning ≈ H H 0 . In this paper, we use the bold-face and lowercase letters to denote vectors and capital letters to denote matrices for simplification. In equation (3), the Hamiltonian after the second quantization is written as the bold-face H to be distinguished fromĤ before the second quantization. The other undefined expressions in equation (3)  The functions γ and κ can be calculated as where, '+' corresponds to bosons and '−' to fermions. Θ here is a dimensionless function. Its concrete expressions can be seen in the appendix. x , corresponding to strong axial confinement, α → 0. In this condition, the terms that associate with conditions It means that in very strong axial potential, there is no transition term in the axial direction. It is because the energy gap along this direction is so big that the ions are not easily excited by Coulomb repulsion. In strong axial confinement, the ions need a considerable amount of energy to jump to the others energy level. This energy cannot be gotten from the Coulomb potential scattering because this scattering is weak.
For the other extreme, namely ω → 0 x , corresponding to weak axial confinement, α → ∞, rd rd . It means that there is no coupling in the radial direction, consequently no excitation in the radial direction. This is because in the weak axial confinement, the ions repel each other, making the distance of each pair of ion so long that the overlapping of wave function in radial direction can be overlooked, and therefore, the ions cannot feel the force from the others in radial direction.
For different cases, we divide the transition-term H 1 into the axial transition H x 1 , the radial transition H rd 1 and the coupling term In linear Paul trap, the secular frequency ω x is usually much less than ω rd , namely α ≫ 1. In this condition, the terms about the excitation in the axial direction are much bigger than those in the radial direction, namely, 〈 1 . Additionally, the coupling term is also much less than H x 1 . H x 1 can be further reduced. As can be seen in figure 3,  . The abscissa of the right picture is n2 and ordinate is − n n 1 2. We can see that few data have considerable value, which corresponds to the light line in right contour figure, and the other are near to zero, which can be overlooked. These considerable terms correspond to The relations between the terms in equation (9)  when the quantum numbers | | m , | | n are small and γ ω ≪ α  when quantum numbers | | m , | | n approach infinity.
(1) for a few ions in traps. H T 1 and H R 1 can be treated as the perturbation. They scatter any two states | 〉 m x and | 〉 n x into other two | ′〉 m x and | ′〉 n x , but keep the parities of the differences of these two pairs of quantum numbers constant. Namely x x x x . In low temperature, the ions will tend to stay in low energy level. Therefore, it is useful for us to find out the minimum value of γ. We find, if we let m and n keep constant but d vary in According to the characteristics of transition terms H R 1 and H T 1 , the green line in the picture is the transition path. The ion tends to stay in low energy level at low temperature, which is corresponding to point A or C in figure 4. According to figure 4, the ions in point A are more stable than those in point ′ C , because the ions here need much more energy to jump to the next point ′ A than ions in point C to the next point ′ C . On the contrary, point B is unstable, because B . lays in band of quasi continuous energy. Therefore, ions in B will be quickly excited by thermal fluctuation. In fact, we will see, B forms the energy band of cloudy state and A constructs the energy level of crystal state.

Method: linear response function
To exactly calculate the energy of ions, we take the linear response function, which is usually applied in condense matter field for electrons or plasmas, to calculate the energy-level of ions [24]. The linear response function is defined on Green function, The θ t ( ) is Heaviside step function here. It equals 1 when ⩾ t 0 and 0 when < t 0, if the Hamiltonian of system does not explicitly depend on time. In equation (11) where, η → 0.
0 . This singularity ω 0 is the eigen frequency in this system.
The more explicated introduction for linear response function can be seen in the references [25,26]

Energy level of ions
Let us return to the expressions of H T x 1 and H R x 1 . The task for us is to find out the appropriate operator that satisfies equation (14). We show the appropriate operator is a linear combination ofQ md . The commutation relation aboutQ H        Based on the conclusions above, we show the operator is analogous to A in equation (14) and it can be written as  can be calculated out according to equation (21). The singularities of its real part are the eigen frequences. To get the singularity, we let η → 0, where, is the matrix. The eigen vector is denoted by g and the corresponding eigen value is ε d . Therefore, equation (22) can be simplified into Namely, are both real. This will contradict equation (21).

Comparison with experiments
We consider a situation in which there are only two ions in the Paul trap. This scenario was chosen in part because we observed the evidence of phase transition between the ion cloud and ion crystal in our laboratory. The other reason for using the two ions as the comparison example is that the possibility that the ion cloud is the electron gas or an order state can be ruled out if the ion cloud is formed by two ions.
Based on equation (22), the energy levels in figure 5 was determined. The energy of the cloudy state formed a quasi continuous area, which corresponds to the filled area in figure 5. The value of the cloudy state is E d m, x , which corresponds to individual excitation, and the state has infinite lifetime. The line hanging on the bottom of the cloudy state area corresponds to collective excitation, which represents the energy of the crystal state. Its elementary excitations are the phonons, which have a long lifetime and are hard to decay.
To prove the rationality of our theory, the crystal energy that arises from our theory is first observed. As can be seen in figure 5, the minimum energy of the ion trap system appears in the crystal state. Its energy is calculated to be = × − E J 6.30957 10 , where l is half of the distance between the equilibrium positions of these two ions. The agreement between the two values is evidence that the model in this paper is effective in calculating for the crystal state energy.
However, for the cloudy state, we have no other theoretical comparison; the results of the model can only be compared with the experimental results. According to our theory, in the cloudy state, the transition index d x0 is proportional to the axial width of the movement areas of the ions in the trap. Therefore, we can evaluate the axial width of the fluorescence image of the cloudy state from EMCCD when the phase transition between the cloudy state and crystal states happens. The axial width is approximately μ = ∼ l 76.3 98.6 m t at ω = 0.2 MHz x in theory. The limit superior of l t is the limit inferior of the valid area of the phonon model (introduced in the following text), and the limit inferior of l t is the point at which the energy of the crystal state equals that of the cloudy state. The experimental data of fluorescence images from EMCCD are shown in figure 6. To create strong radial confinement, the power of RF should be increased, but the increased power of RF causes the increased micromotion of ions in our laboratory. The radial RF power applied is 5 W, which corresponds to 1710 V in our laboratory. Under this high voltage, the axial width of fluorescence image became unstable and varied in every measurement. However, the width of the cloudy state measured in the experiments fell into the predicted area.
After the phase transition, the kinetic energy of the ions became smaller; hence, the distance between these two ions in trap was smaller than the width of the cloudy state.  crystal, and their movements are treated as micro-oscillations around their equilibrium positions [27][28][29][30]. The question is how the phonon model can remain valid in calculating ion energy. We define the valid area for the phonon model as the 10% deviation between the phonon model and our model. We show the valid area varying with ω x in the right graph of figure 6. The valid areas for the breathing mode and the centre-of-mass mode are the same in theory. According to our theory, in weak axial confinement, the phonon model is exact in crystal energy level calculation at low temperature. However, at high temperature or strong axial trap, the phonon model will be invalid and the model in our paper provides a good substitution for the phonon model in these situations. In the following discussion, two phenomena observed in the experiments, shown in figures 2 and 7 are described. We will use the model proposed in this paper to give their interpretations.
The transition process shown in picture I of figure 2, is described in the route map on the right side of the energy-levels in figure 8. The fluorescent images A B C , , ,... in figure 8 are drawn according to the characteristics of the quantum states that they correspond.
The assembled ions are initially in state A, which has close to the lowest energy the ions are able to reach (in fact, the ions cannot reach this low energy solely through the Doppler cooling process). The cooling laser is then red-detuned with a large frequency difference. The ions are heated by RF, and the state transfers to B. The fluorescent image of the ions is much bigger at this stage. The size reflects the increased movement area that the ion can reach, which   transition. In this process, the Coulomb potential converts to the kinetic energy but the total energy is unchanged. Because of the large amount of kinetic energy that ions get in state C, the temperature of the ion system increases.
From the cloudy state C, if we want the ions to be crystal again, we slowly detune back. In the back-scan process, the ions slowly cool down but they do not transfer back to B and then to A. They transfer to stage D first during the back-scan process. In the C-to-D process, the kinetic energy of the ions in the radial direction is transferred to the axial direction to overcome 'attractive' potential, which is essentially the Coulomb potential. The energy of these ions in state D is almost lowest in cloudy states, depending on the balance of the cooling laser and heating RF. The ions' fluorescence image in D becomes very long and narrow in shape. The ions then become unstable and quickly decay into state E, the crystal state. In this process, the kinetic energy is used to overcome the 'attractive' potential and the temperature of the ion system drops. The state E can be further cooled into A. Then, the normal hysteresis-loop-like forms.
The abnormal hysteresis-loop-like shown in II of figure 2 is the other evidence for our theory. 5 V RF is added under the compensating pole to excite the ions in the trap. When the RF frequency is near the radial secular frequency, the state ′ B will be created. The cooling laser is then blue-detuned. The ions in the trap will quickly transfer to ′ E through the stage ′ C . The ions in ′ C are very unstable, rapidly decaying into D, then further transferring to A. In the process ′ → ′ → B C D, the ions exhibit a strong attractive force, which is contrary to our understanding because the Coulomb force among ions is a repulsive force. Why could the Coulomb potential support the 'attractive' force here? To answer this question, let us review a classical mode: the electrical potential of two electrified rings with the same radius R in a plane. The electric potential of this system might reach a maximum when the distance between these two rings equals R 2 , and the electric potential will drop as the distance decreases. The 'attractive' force between two ions in a trap is similar to that of these two rings. The movement areas of the ions are the size of rings. If one ion, P1, enters the movement area of the other, P2, ion P1 will not feel the force of P2 because the average force of P2 on P1 in a very short time approximately equals zero. Then, the electric potential will quickly drop when the movement areas of the ions overlap.

Conclusion
This paper exhibited the drawbacks of previous work on the description of the cloudy state and proposed a new model that can calculate the energy-levels of both the cloudy state and crystal state. In accordance with the linear response function, we presented a method to calculate the energy-level of any number of trapped ions in the Paul trap and exhibited the energy level diagram of two trapped ions as an example. We showed the imaginary part of the function ρ 〈〈ˆ| 〉〉 ω H md 0 is nonzero for the cloudy state, meaning the ions in cloudy state have finite lifetime. In contradiction to this, the crystal state is much more stable and the corresponding ρ 〈〈ˆ| 〉〉 ω Im H ( ) md 0 equals zero. We use this to interpret the phenomena that we encountered in our laboratory. We show that these quantum interpretations for the fluorescent hysteresis loops that are formed by two trapped ions in a Paul trap conform to the phenomena that we observed in the laboratory. This theory is fundamental and it can be used in the areas of ion cooling, ion state manipulation and ion transport in further studies. Quantum algebra is a simple way to calculate exactly inner products, especially in the condition that we cannot give a concrete expression of a wave function in coordinate or momentum representation. For the radial direction here, the formulas of wave functions are more complicated than the axial direction. It is hard to get the normalizing factors of them. In this condition, we must employ quantum algebra to calculate the inner products in radial direction. Considering the symmetry in the radial direction, we should choose the polar coordinate plane expression to calculate inner products. Under the polar coordinate, the basis can be written as