Quantum Manipulations of Single Trapped-Ions Beyond the Lamb-Dicke Limit
2. Dynamics of single trapped ions driven by laser beams
2.3 Time evolutions of quantum states
The time evolution of the system governed by the above Hamiltonian can be solved by the time-evolution operator ˆT=exp(−itH/¯ˆ h), with ˆH taking ˆH0LDA, ˆHrLDA, ˆHLDAb , ˆHNLDAr , and HˆNLDAb , respectively. For an arbitrary initial state|ϕ(0), the evolving state at timetreads
|ϕ(t)=Tˆ|ϕ(0)=∑∞
n=0
1 n!
−it
¯ h
n
Hˆn|ϕ(0). (17) Typically, if the external vibrational state of the ion is initially in a Fock state |mand the internal atomic state is initially in the atomic ground sate|gor excited one|e, then the above dynamical evolutions can be summarized as follows (23; 26):
i) For the resonance or red-sideband excitationsK≤0:
⎧⎨
⎩
|m|g −→ |m|g,m<k,
|m|g −→cos(Ωm−k,kt)|m|g+ik−1e−iϑlsin(Ωm−k,kt)|m−k|e;m≥k,
|m|e −→cos(Ωm,kt)|m|e −(−i)k−1eiϑlsin(Ωm,kt)|m+k|g (18) ii) For the resonance or blue-sideband excitationsK≥0:
⎧⎨
⎩
|m|g −→cos(Ωm,kt)|m|g+ik−1e−iϑlsin(Ωm,kt)|m+k|e,
|m|e −→ |m|e,m<k,
|m|e −→cos(Ωm−k,kt)|m|e −(−i)k−1eiϑlsin(Ωm−k,kt)|m−k|g,m≥k.
(19)
The so-called effective Rabi frequency introduced above reads
Ωm,k=
⎧⎪
⎪⎨
⎪⎪
⎩
ΩLm,k=Ω2ηk
(m+k)!
m! , k=0, 1, ΩNm,k=Ω2ηke−η2/2
(m+k)!
m! ∑mj=0(j+k)!j!(m−j)!(iη)2jm! , k=0, 1, 2, 3, ... .
(20)
The above derivations show that the dynamics either within or beyond the LD approximation has the same form (see, Eqs. (18) and (19)), only the differences between them is represented by the specifical Rabi frequenciesΩLm,kandΩNm,k. Certainly, the dynamical evolutions without LD approximation are more closed to the practical situations of the physical processes.
Furthermore, comparing to dynamics with LD approximation (wherek=0, 1), the dynamics without LD approximation (where k =0, 1, 2, 3, ...) can describes various multi-phonon transitions.
Certainly, when the LD parameters are sufficiently small, i.e.,η1, the rate γ=Ωm,kL
Ωm,kN = eη
2/2
∑mj=0(j+k)!j!(m−j)!(iη)2jm!
∼1, (21)
and thus the dynamics within LD approximation works well. Whereas, if the LD parameter are sufficiently large, the quantum dynamics beyond the LD limit must be considered.
various engineered quantum states of trapped cold ions have been studied. The thermal, Fock, coherent, squeezed, and arbitrary quantum superposition states of motion of a harmonically bound ion have been investigated (4; 5; 6; 7). However, most of the related experiments are operated under the LD approximation. In this section we study the engineerings of various typical vibrational states of single trapped ions beyond the LD limit.
There are serval typical quantum states in quantum optics: coherence states, odd/even coherent states and squeezed states, etc.. They might show highly nonclassical properties, such as squeezing, anti-bunching and sub-Poissonian photon statistics. In the Fock space, coherence state|αcan be regarded as a displaced vacuum state, i.e.,
|α=D(α)|0=e−|α|2/2
∑∞ n=0
αn
√n!|n, (22)
where D(α) =exp(αaˆ†−α∗aˆ) is the so-called displace operator, with α being a complex parameter and describing the strength of displace. Similarly, the squeezed states |ψs are generated by applying the squeeze operator to a quantum state
|ψs=Sˆ(ξ)|ψ, (23) where ˆS=exp(ξ∗aˆ2/2−ξaˆ†2/2), andξis a complex parameter describing the strength of the squeezes. On the other hand, the odd/even coherent states are the superposed states of two coherent states with different phases
|αo=Co(|α − | −α), (24)
|αe=Ce(|α+| −α). (25) Above, Co= [2−2 exp(−2|α|2)]−1/2 andCe= [2+2 exp(−2|α|2)]−1/2 are the normalized coefficients.
According to the above definitions: Eqs. (22)-(25), one can obtain the squeezed coherent state:
|αs=Sˆ(ξ)|α=∑∞
n=0Gn(α,ξ)|n, (26)
squeezed vacuum state:
|0s=Sˆ(ξ)|α=0=∑∞
n=0Gn(0,ξ)|n, (27)
squeezed odd state:
|αo,s=Sˆ(ξ)|αo= ∑∞
n=0On(α,ξ)|n, (28)
and squeezed even state:
|αe,s=Sˆ(ξ)|αe= ∑∞
n=0En(α,ξ)|n. (29)
respectively. Above, Gn(α,ξ) = 1
cosh(r)n!
eiθsinh(r) 2 cosh(r)
n/2
exp[e−iθsinh(r)α2 2 cosh(r) − |α|2
2 ]Hn( α
2eiθsinh(r)cosh(r)). (30) withξ=rexp(iθ), andHn(x)being the hermitian polynomials. Finally,
On(α,ξ) =Co(Gn(α,ξ)−Gn(−α,ξ)) (31) and
En(α,ξ) =Ce(Gn(α,ξ) +Gn(−α,ξ)). (32) Based on the quantum dynamics of laser-ion interaction beyond the LD limit, in what follows we discuss how to use a series of laser pulses to generate the above vibrational quantum states of the trapped ions. Assuming the trapped ion is initialized in the state|0|g, to generate the desirable quantum states we use the following sequential laser pulses to drive the trapped ion:
1) Firstly, a pulse with frequencyωl=ωa, initial phaseϑ1, and durationt1is applied on the trapped ion to yield the evolution:
|0|g → |ψ1=cos(Ω0,0t1)|0|g+i−1e−iϑ1sin(Ω0,0t1)|0|e (33) 2) Secondly, a pulse with frequencyωl=ωa−ν, initial phaseϑ2, and durationt2is applied on the ion to generate:
|ψ1 → |ψ2=cos(Ω0,0t1)|0|g+iei(ϑ2−ϑ1)sin(Ω0,0t1)sin(Ω0,1t2)|1|g
+i−1e−iϑ1sin(Ω0,0t1)cos(Ω0,1t2)|0|e (34) 3) Thirdly, a pulse with frequencyωl=ωa−2ν, initial phaseϑ3, and durationt3is applied on the ion to generate:
|ψ2 → |ψ3=cos(Ω0,0t1)|0|g+iei(ϑ2−ϑ1)sin(Ω0,0t1)sin(Ω0,1t2)|1|g +ei(ϑ3−ϑ1)sin(Ω0,0t1)cos(Ω0,1t2)sin(Ω0,2t3)|2|g
+i−1e−iϑ1sin(Ω0,0t1)cos(Ω0,1t2)cos(Ω0,2t3)|0|e
(35)
4) Successively, after theNth pulse (with frequencyωl=ωa−(N−1)ν, initial phaseϑN, and durationtN) is applied on the ion, we have the following superposition state:
|ψN−1 → |ψN=N−1∑
n=0Cn|n|g+BN|0|e (36)
with
Cn=
⎧⎪
⎨
⎪⎩
cos(Ω0,0t1), n=0,
iei(ϑ2−ϑ1)sin(Ω0,0t1)sin(Ω0,1t2), n=1,
(−1)n−1inei(ϑn+1−ϑ1)sin(Ω0,0t1)sin(Ω0,ntn+1)∏nj=2cos(Ω0,j−1tj), n>1,
(37)
state|gand its external vibration is prepared on the superposition state|ψN=∑n=0 Cn|n. Because the durationstN and initial phasesϑN are experimentally controllable, we can set Cn=Gn(α,ξ),On(α,ξ), orEn(α,ξ)withn≤N−2 by selecting the propertNandϑN. As a consequence, we have the vibrational superposition state:
|ψGN=N−2∑
n=0Gn(α,ξ)|n+CN−1G |n−1, (39)
|ψON=N−2∑
n=0On(α,ξ)|n+CON−1|n−1, (40)
|ψEN=N−2∑
n=0En(α,ξ)|n+CEN−1|n−1, (41) and
|CN−1G |2=1−N−2∑
n=0|Gn(α,ξ)|2, (42)
|CON−1|2=1−N−2∑
n=0|On(α,ξ)|2, (43)
|CEN−1|2=1−N−2∑
n=0|En(α,ξ)|2. (44)
If the number of the laser pulses, i.e., N, are sufficiently large, the superposition states (39), (40), and (41) can well approach the desirable squeezed coherent state, squeezed odd and even coherent states, respectively. Specifically, whenξ=0, the usual coherent state, odd/even coherent states are generated.
Table 1 presents some numerical results for generating squeezed coherent states, where the typical values of parametersαandξ, e.g., α=2 andξ=0.5, are considered. When α=0, the squeezed coherent states correspond to the squeezed vacuum states. On the other hand, some results for generating squeezed odd/even coherent states are shown in table 2, and
α ξ N Ωt1(ϑ1) Ωt2(ϑ2) Ωt3(ϑ3) Ωt4(ϑ4) Ωt5(ϑ5) Ωt6(ϑ6) F
0 0.5 5 0.7080(0) 0(0) 53.9174(π) 0(0) 4065.1(π) 0.9983
0 0.8 5 1.0859(0) 0(0) 43.9507(π) 0(0) 4065.1(π) 0.9816
1 0.0 5 1.8966(0) 7.1614(3π/2) 46.0741(0) 343.5682(π/2) 4065.1(0) 0.9981
2 0.5 5 2.5666(0) 5.3267(3π/2) 43.7265(0) 371.8479(π/2) 4065.1(π) 0.9882
2 0.5 6 2.5666(0) 5.3267(3π/2) 43.7265(0) 371.8479(π/2) 1826.3(π) 3635.9(0) 0.9910
2 0.8 6 2.2975(0) 6.8280(3π/2) 43.9627(0) 59.8973(π/2) 1877.7(π) 3635.9(0) 0.9821
Table 1. Parameters for generating squeezed coherent states, with the typicalη=0.25
odd/even α ξ N Ωt1(ϑ1) Ωt2(ϑ2) Ωt3(ϑ3) Ωt4(ϑ4) Ωt5(ϑ5) Ωt6(ϑ6) F
odd 1 0.0 5 3.2413(0) 9.6943(3π/2) 0(0) 436.5962(π/2) 4065.1(0) 0.9927
odd 2 0.5 6 3.2413(0) 7.7328(3π/2) 0(0) 438.6774(π/2) 0(0) 3635.9(0) 0.9974
odd 2 0.8 6 3.2413(0) 9.9520(3π/2) 0(0) 85.3471(π/2) 0(0) 3635.9(0) 0.9763
even 1 0.0 5 1.3106(0) 0(0) 59.9889(0) 0(0) 4065.1(0) 0.9995
even 2 0.5 5 2.2687(0) 0(0) 61.3354(0) 0(0) 4065.1(0) 0.9870
even 2 0.8 5 1.8498(0) 0(0) 51.1086(0) 0(0) 4065.1(0) 0.9900
Table 2. Parameters for generating squeezed odd/even coherent states, with the typical η=0.25
where, ξ=0 corresponds to the usual odd/even coherent states. By properly setting the durationstNand phasesϑNof each laser pulse, the desirable squeezed states of the trapped ion could be generated with high fidelities (e.g., 99%), via few steps of laser pulse operations (e.g.,N=6). Here, the fidelitiesF=ψgenerated|ψexpectedare defined by the overlaps between the generated quantum states|ψgenerated(being|ψGN,|ψON, or|ψEN) and the expected states
|ψexpected.
Above, the frequencies, durations, phases, and powers of the laser pulses should be set sufficiently exact for realizing the high fidelity quantum state operations. In fact, it is not difficult to generate the desired laser pulse with current experimental technology. It has been experimentally demonstrated that, a Fock state up to |n=16 can be generated from the vibrational ground state|0, by sequentially applying laser pulses (with exact frequencies, durations, and phases, etc.) to the trapped ion (2; 4). This means that here the generation of the superposed Fock states, being coherent state, even/odd coherent states, or squeezed quantum states, etc., is also experimentally realizability.
On the other hand, the generation of the superposed Fock states is limited in practice by the existing decays of the vibrational and atomic states. Experimentally, the lifetime of the atomic excited states|ereaches 1s and the coherence superposition of vibrational states|0 and|1can be maintained for up to 1ms (32; 33). Therefore, roughly say, preparing the above superposed vibrational Fock states is experimentally possible, as the durations of quantum operation are sufficiently short, e.g., <0.1 ms. To realize a short operation time, the Rabi frequenciesΩ(describing the strength of the laser-ion interaction) should be set sufficiently large, e.g., whenΩ=105 kHz, the total durationt=t1+t2+t3≈0.04 ms for generating squeezed vacuum state (see, table 1).
Certainly, by increasing the Rabi frequencyΩ ∝√
Pvia enhancing the powerPof the applied laser beams, the operational durations can be further shorten. For example, if the Rabi frequencyΩincreases ten times, then the total durations presented above for generating the desired quantum states could be shortened ten times. Therefore, the desired quantum states could be realized more efficiently. Indeed, the power of the laser applied to drive the trapped cold ion is generally controllable, e.g., it can be adjusted in the range from a few microwatts to a few hundred milliwatts (32).