Protocol for Optically Pumping AlH + to a Pure Quantum State
Panpan Huang, Schuyler Kain, Antonio de Oliveira-Filho, Brian C. Odom
PProtocol for Optically Pumping AlH + to a Pure Quantum State Panpan Huang, Schuyler Kain, and Brian C. Odom
Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
Antonio de Oliveira-Filho
Departamento de Química, Laboratório Computacional de Espectroscopia e Cinética,Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto,Universidade de São Paulo, Ribeirão Preto-SP 14040-901, Brazil (Dated: September 9, 2020)We propose an optical pumping scheme to prepare trapped
AlH + molecules in a pure state, thestretched hyperfine state | F = , m F = (cid:105) of the rovibronic ground manifold | X Σ + , v = 0 , N = 0 (cid:105) .Our scheme utilizes linearly-polarized and circularly-polarized fields of a broadband pulsed laser tocool the rotational degree of freedom and drive the population to the hyperfine state, respectively.We simulate the population dynamics by solving a representative system of rate equations thataccounts for the laser fields, blackbody radiation, and spontaneous emission. In order to modelthe hyperfine structure, new hyperfine constants of the A Π excited state were computed using aRASSCF wavefunction. We find that adding an infrared laser to drive the – vibrational transitionwithin the X Σ + manifold accelerates the cooling process. The results show that under optimumconditions, the population in the target state of the rovibronic ground manifold can reach 63 % after68 µs (330 ms) and 95 % after 25 ms (1.2 s) with (without) the infrared laser. I. INTRODUCTION
In recent years, molecules have played increasingly im-portant roles in many active areas research such as pre-cision measurement[1–4] and cold chemistry[5–7]. Also,the rich internal degrees of freedom and long-rangedipole-dipole interaction between polar molecules offerpossibilities of developing a toolkit for quantum infor-mation processing and quantum simulation[8–12]. Cen-tral to these applications is the development of qubits,which can be represented by the pure states of atoms ormolecules[13]. The long hold times, environmental isola-tion, and quantum control demonstrated for atomic ionsin radiofrequency traps have made these platforms pop-ular for work on atomic qubits. As for atomic qubits,one critical feature of molecular qubits is the ability torapidly reset them into pure quantum states. In thiswork, we explore the use of optical pumping to preparetrapped molecular ions in pure states.Our group has previously shown that rotational coolingof diatomic molecules can be achieved using a spectrally-filtered femtosecond laser (SFFL) with species that haverelatively large rotational constants and fairly diagonalFrank-Condon factors (FCFs)[14]. One such exampleis aluminum monohydride cation,
AlH + , for which wedemonstrated an increase in the rotational ground statepopulation from a few percent to ∼ % within a second[15]. The cooling of the ions to a single rotational Zee-man state was also theoretically investigated using theapproach of optimal control theory [16]. However, theoperation of cooling AlH + to a specific hyperfine statehas not yet been addressed. Such hyperfine cooling hasbeen demonstrated on the molecular ion, HD + , with thetransfer taking a few tens of seconds and the target popu-lation reaching 19 %[17, 18]. Also, a quantum-logic tech-nique has demonstrated the ability to project a single trapped molecular ion into a pure state[13].This manuscript proposes an efficient method to trans-fer the AlH + population to a single stretched hyper-fine state of the rovibronic ground manifold and is or-ganized as follows. In Section II, we review the theoryand describe our method of performing optically-drivenand laser-enhanced rotational cooling. We then presentour design to optically pump the system to the singlestretched hyperfine state. The simulation details are de-scribed in Section III while Section IV presents and dis-cusses our results. We conclude in Section V. II. THEORY AND METHODS
In the electronic ground state of
AlH + , X Σ + , the an-gular momenta are well-described by Hund’s case (b),with good quantum numbers { Λ , N, S, J, ± } . J ( (cid:126)J = (cid:126)N + (cid:126)S ) is the quantum number of the total angu-lar momentum exclusive of nuclear spin; N is the ro-tational quantum number; and S is the electron-spinquantum number. The A Π electronic state of AlH + is better-described using the Hund’s case (a) basis of { Λ , S, Σ , J, Ω , ± } . Here, Λ , Σ , and Ω are projectionsalong the molecular axis of the electronic orbital angu-lar momentum, electron-spin, and their sum ( Λ + Σ ),respectively. The eigenvalues of the parity operator, P ,are represented as ± .Though N is not a good quantum number in A Π , forconvenience, we still use N to denote different J valuesin the A Π state (see Figure 1) and to label transitionbranches. It should be noted that the rotational statesof both the X and A manifolds of AlH + exhibit doublets.In the X manifold, the doubling is a result of the interac-tion of the electron-spin and molecular rotation whereasthe doubling in the A Π manifold is produced by the a r X i v : . [ phy s i c s . a t m - c l u s ] S e p FIG. 1: (a) Rotational cooling mechanisms of
AlH + and (b) the energy level structure of AlH + . In (a), the rotational coolinglaser that drives the P , P Q and O P branches from | X Σ + , v = 0 (cid:105) to | A Π / , v = 0 (cid:105) are shown. Though the rotationalangular momentum is not a good quantum number in A Π , N (cid:48) is used here as a convenient label for different J values. Thedashed green arrows represent the electronic spontaneous emission with vibrational excitation. The dashed orange arrowsrepresent the rovibrational relaxation within the X state. The solid orange arrow represents the rovibrational transition drivenby the V10 laser. The dashed red line is the rovibronic ground state. In (b), the states are arranged vertically to reflect theirapproximate energies; the colors of the arrows represent the relative transition energies. States that would be dark to thecooling laser if it were not for the transverse component of the magnetic field are represented by dashed black lines. The dottedarrows do not connect any particular final m J -states; they indicate that the initial dark-state populations are transferred to amixture of m J states, thus returning them to the cooling cycle. The target states of rotational cooling, namely the rovibronicground states, are represented by dashed red lines.FIG. 2: (a) A schematic of the experimental setup and (b) the hyperfine structure of the rovibrational ground state of X Σ + and A Π / . In (a), quadrupole rods of a linear Paul trap and two SFFLs are shown. The blue arrow represents thelinearly-polarized SFFL (RC laser) that performs rotational cooling. The light-purple arrow represents the σ + -polarized SFFL(HC laser) that drives the Q (0 . -branch transition, optically pumping the population into the stretched hyperfine state ofthe ground rovibronic manifold. The orange arrow represents the infrared laser (V10 laser) that drives the – vibrationaltransition. In (b), the hyperfine structure and the set of transitions driven by the σ + -polarized SFFL are shown. States of agiven energy level are non-degenerate due to Zeeman shifts (not drawn; ∆ E Z (cid:28) ∆ E F ). The population in the rovibrationalground state of X Σ + is driven towards a single dark state, | X Σ + , v = 0 , N = 0 , F (cid:48)(cid:48) = 3 . , m F (cid:48)(cid:48) = 3 . (cid:105) , represented by thedashed red line. The solid purple arrows represent transitions driven by the HC laser. The dashed purple arrows representspontaneous emission channels to the target state. coupling between electronic orbital angular momentumand molecular rotation (known as Λ -doubling). For bothelectronic states, v is the vibrational quantum number.We invoke the convention that v (cid:48) ( v (cid:48)(cid:48) ) denotes the vibra-tional quantum number of the upper (lower) state of aspectroscopic transition. A. Rotational Cooling
Our group has previously demonstrated broadband ro-tational cooling of
AlH + using a linearly-polarized SFFLwith 100 fs pulses centered at 360 nm. The pulses aregenerated from a frequency-doubled femtosecond laser(Spectra-Physics Mai Tai) with 80 MHz repetition rate,so the pulse spectrum is divided into a frequency combwith 80 MHz spacing between neighboring teeth. Itslarge bandwidth pumps many rotational transitions si-multaneously. We use spectral filtering in the Fourierplane to selectively excite rotational cooling transitions.Within the A Π – X Σ + set of transitions, the v (cid:48) = 1 – v (cid:48)(cid:48) = 1 band lies ∼
150 cm − above the 0 – 0 band. Theneighboring 1 – 0 and 0 – 1 bands are also nearly a vibra-tional constant away from the 0 – 0 band . Thus, we ig-nored the 1 – 1 and 1 – 0 bands because they are outsidethe passband of our spectrally-filtered SFFLs. However,the 0 – 1 band was included because it provides a criticalrelaxation pathway. Informed by these choices, we onlyincluded the | X Σ + , v = 0 , (cid:105) and | A Σ + , v = 0 (cid:105) statesin our population dynamics model.As shown in Figure 1, the rotational cooling processhas two parts. The first part is a fast cycle in whichlinearly-polarized 360 nm pulses of the SFFL drive theelectronic transition connecting | X Σ + , v (cid:48)(cid:48) = 0 (cid:105) and | A Π / , v (cid:48) = 0 (cid:105) . Following excitation to the A state,electronic spontaneous emission without vibrational ex-citation occurs with a lifetime of ∼ | A Π / , v (cid:48) = 0 (cid:105) → | X Σ + , v (cid:48)(cid:48) = 0 (cid:105) is ∼ | v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 0 , (cid:105) , withina few microseconds. However, the parity is flipped fordipole transitions such as | A Π (cid:105) – | X Σ + (cid:105) . As a result,the population in | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 1 , −(cid:105) cannot betransferred to the rovibronic ground state, | X Σ + , v (cid:48)(cid:48) =0 , N (cid:48)(cid:48) = 0 , + (cid:105) via this fast cycle because each cycle in-volves a pair of transitions that together conserve the par-ity. The population in | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 1 , −(cid:105) canstill transfer to | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 0 , + (cid:105) , but must doso in an odd number of transitions to flip the parity. Theshortest parity-flipping process happens in three transi-tions: first, the population in | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 1 , −(cid:105) is excited by the SFFL to the A state; then there isspontaneous decay to an intermediate state with neg-ative parity, | X Σ + , v (cid:48)(cid:48) = 1 , N (cid:48)(cid:48) = 1 , −(cid:105) ; last, thepopulation undergoes a vibrational relaxation to reach | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 0 , + (cid:105) . This parity-flipping pro-cess, which constitutes the second part of the coolingprocess, is relatively slow because the vibrational life-time for the | X Σ + , v = 1 (cid:105) – | X Σ + , v = 0 (cid:105) transition is140 ms.In our laboratory, AlH + molecular ions are heldin a linear Paul trap and sympathetically cooled tosub-Kelvin translational temperatures using co-trappedDoppler-cooled Ba + atomic ions. To remove the darkstates of the barium ion during the Doppler cooling pro-cess, a 2 G magnetic field is applied. After translationalcooling, the linearly-polarized SFFL (with polarizationof 45 ◦ relative to the direction of the magnetic field) isturned on to rotationally cool the molecules into theirrovibronic ground state. Rotational cooling of the negative-parity populationsis rate-limited by the vibrational-decay timescale. Wepropose to address this bottleneck through the additionof a 6.7 µm continuous-wave laser (V10) which drivesthe | X Σ + , v (cid:48)(cid:48) = 1 , N (cid:48)(cid:48) = 1 , −(cid:105) – | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) =2 , + (cid:105) transition to accelerate the parity-flipping process.This technique has not yet been applied to rotationalcooling using broadband lasers. We simulate the rota-tional cooling process and show that it is accelerated bythe additional laser. The simulation results are summa-rized in Figure 3 and Table V. B. Hyperfine Cooling
AlH + has one unpaired electron ( S = ), and nucleiwith nuclear spins I Al = and I H = . We define thehyperfine states of the electronic ground state of AlH + in terms of a set of total angular momentum quantumnumbers, { F } . We define the members of the set as fol-lows: F = J + I Al , and F = F + I H . In its rovibronicground state ( X , v = 0 , N = 0 ), AlH + thus has four hy-perfine states: F = (cid:8) , (cid:9) for F = 2 and F = (cid:8) , (cid:9) for F = 3 .As we are interested in pumping our system to a sin-gle hyperfine state, we also added a circularly-polarized360 nm beam propagating along the direction of the 10G magnetic field as shown in Figure 2(a). By taking ad-vantage of the selection rules for dipole transitions drivenby σ + -polarized light, we can optically pump the systemto the stretched hyperfine state in which the total angu-lar momentum ( F ) has the largest projection ( m F ) alongthe quantization axis. The schematic plot of our setup isshown in Figure 2(a). When we apply the σ + -polarizedlaser, the selection rule are as follows: ∆ F = 0 , ± m F = 1 As can be seen in Figure 2(b), if we set the σ + -polarized laser to drive the Q (0 . -branch transition , | A Π / , v (cid:48) = 0 , N (cid:48) = 0 (cid:105) – | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 0 (cid:105) ,most of the population in the rovibronic ground state of X Σ + will be further optically pumped to the stretched This notation describes the branch in terms of both N and J : ∆ N ∆ J ul ( J (cid:48)(cid:48) ) . Note that N is not a good quantum number in A Π , and simply serves as a convenient label. If ∆ N = ∆ J ,then the notation uses one letter to mark the type of branch. u and l denote the spin orientations of the upper and lower statesof a transition. In our case, the upper state could be either A Π / or A Π / . These states, corresponding to | Ω = Λ − (cid:105) or | Ω = Λ + (cid:105) , are denoted as u = 1 and u = 2 , respectively.Analogously, the lower-state, part of the X Σ + manifold, has S = 1 / . Our convention is that l = 1 and l = 2 represent | J = N + (cid:105) and | J = N − (cid:105) , respectively. state of maximal F . This stretched hyperfine state is adark state; it cannot absorb any more σ + -polarized pho-tons because there is no higher m F -state available withinthe | A Π / , v = 0 , N = 0 (cid:105) manifold. Thus the popula-tion will accumulate in the stretched state over time. Theaddition of the 6.7 µm continuous-wave laser also accel-erates this hyperfine cooling process since it is dependentupon the rate of rotational cooling. The simulation re-sults are shown in Figure 4 and Table VI. III. SIMULATION DETAILS
Our population dynamics were modeled by the follow-ing system of rate equations: ∂ρ i ∂t = − (cid:88) j (cid:54) = i B ij ( I BBR + I laser ) ρ i − (cid:88) ji A ji ρ i (1)where ρ i is the population fraction in state i . The sys-tem of equations includes the rovibronic and hyperfinestates of interest. The initial population was assumed tobe thermal with a temperature of 300 K. I BBR and I laser are the energy densities of the blackbody radiation andlaser. A and B are the spontaneous emission and stim-ulated emission Einstein coefficients, respectively. TheEinstein coefficients can be expressed using the followingequations: A ul = 2 π (cid:101) ν q e (cid:15) m e c g l g u f lu B ul = q e (cid:15) m e hc (cid:101) ν g l g u f lu B lu = q e (cid:15) m e hc (cid:101) ν f lu (2)where (cid:101) ν is the wavenumber energy of the transition; q e and m e are the charge and the rest-mass of an elec-tron; (cid:15) is the vacuum permittivity; c is the speed oflight; g l and g u are the degeneracies of the lower andthe upper states, respectively; and f lu is the oscillatorstrength of the transition. In order to determine the Ein-stein coefficients using Equation (2), we utilized West-ern’s PGOPHER [21] software to compute transition ener-gies and oscillator strengths for
AlH + . In the absence ofthe magnetic field, for J – J + 1 rovibronic transitionswith linearly-polarized light, the two stretched states, | J m J (cid:105) = {| ( J + 1) ± ( J + 1) (cid:105)} , are inaccessible by elec-tric dipole transitions and constitute dark states as shownin Figure 1(b). A 10 G magnetic field was applied toaddress this problem while driving the P-branch rota-tional cooling transition with a linearly-polarized beam.By adding the 10 G magnetic field at an angle of 45 ◦ with respect to the polarization direction of the rota-tional cooling laser (see Figure 2(a)), bright states are mixed with the dark state. The brightened state evolvesat the Larmor frequency (10 s − ), which is sufficientlyfast compared to the Rabi frequency of the rotationalcooling laser (10 s − ) to destabilize the dark populationand expose it to the cooling laser. PGOPHER required anumber of empirical parameters to describe the statesof
AlH + . Table I and II present the values we used todescribe the X Σ + and A Π states.We used Le Roy’s LEVEL [22] to calculate vibrationally-averaged permanent and transition electric-dipole mo-ments from potential-energy, permanent and transitionelectric-dipole moment functions of a prior work[23].These results are presented in Table III.
Dalton [24, 25] quantum computational software wasused to compute the hyperfine and nuclear electric-quadrupolar coupling constants of the X Σ + and A Π states at fixed geometry ( R = 1 . Å). We chose thepcJ-1 basis set[26] because the pcJ- n family was opti-mized for calculating indirect nuclear spin-spin couplingconstants and has tight functions that are well-suitedfor describing the electron density near the nucleus .The computations invoked a restricted active space self-consistent field (RASSCF)[27] wave function with 2,439determinants for the X Σ + state and 1,947 determinantsfor the A Π state. The wave function was defined by5 inactive orbitals in the RAS1 space, a full-valence (5-orbital) RAS2 space, and single and double excitationsfrom the RAS2 space into the RAS3 space for the 30 re-maining orbitals. Dalton outputs the values of the hyper-fine tensor components ( A xx , A yy , A zz ) and the Fermicontact term ( A iso ). However, PGOPHER takes as inputsof hyperfine constants the Frosch–Foley coefficients ( a , b , c , d ). The conversion formulas are listed below: c = 32 A zz d = A xx − A yy b = A iso − c a = d + c (3)We used the electric-field gradients ( q xx ≡ ∂ V x /∂x , q yy ≡ ∂ V y /∂y , q zz ≡ ∂ V z /∂z in MHz ) and the nu-clear electric-quadrupole moment ( Q in barn) computedby Dalton to calculate the nuclear electric-quadrupolarcoupling constants, eQq and eQq (in cm − ). The for-mulas are given by: We performed test computations at the same level of theory for
HCl + and OH and found that the calculated hyperfine couplingconstants for these two molecules were within 10–15 % of pub-lished experimental values. We feel the relative agreement justi-fies our choices. TABLE I: Molecular constants for the X Σ + state of AlH + ‡ Constant[19] v = 0 v = 1 B v . . D v × . . H v × − . − . L v × − . M v × . N v × − . γ v × .
665 5 . γ Dv × − . − . origin ‡ in cm − TABLE II: Molecular constants for the A Π state of AlH + ‡ Constant v = 0 B v . [20] A v [20] p v × . [19] q v × . [19] D v × − . [20]origin [19] ‡ in cm − TABLE III: Permanent and transition dipole moments ( (cid:104) i | ˆ µ | j (cid:105) ) †‡ State i State j X Σ + , v = 0 X Σ + , v = 1 A Π , v = 0X Σ + , v = 0 − . Σ + , v = 1 0 . − . Π , v = 0 1 . − . − . † The signs of the dipole moments reflect the choice of coordinatesystem in which the lighter atom was placed at +ˆ z . ‡ in debye TABLE IV: Hyperfine constants of
AlH + ‡ Constant X Σ + A Π Al H Al Ha . × − . × − . × − . × − b . × − . × − . × − − . × − c . × − . × − − . × − . × − d . × − . × − eQq − .
341 211 × − .
126 36 × − .
203 58 × − .
342 31 × − eQq − .
896 52 × − − .
633 15 × − ‡ in cm − eQq = 7 . × − cm − ( q zz MHz )( Q barn ) eQq = 7 . × − cm − ( q xx − q yy MHz )( Q barn ) (4)The input hyperfine coupling constants for PGOPHER are listed in Table IV.At room temperature, 99 % of the AlH + populationis in the lowest vibrational state, v = 0 , within the X Σ + manifold. In turn, within this vibrational groundstate, thermal distribution produces significant popula-tions among the first ten J -levels, J = { . , . . . , . } , and less than 4 % in J > . . Therefore, we included the low-est ten J -states ( J = { . , ..., . } ) of each vibrationalstate (i.e. | X Σ + , v = 0 , (cid:105) and | A Π , v = 0 (cid:105) ) in therate equation simulation. This manifold of states wasable to resolve the vibronic relaxation from | A Π , v = 0 (cid:105) to | X Σ + , v = 1 (cid:105) and the parity-flipping process via theintermediate states.Our femtosecond laser was given a frequency-domainrepresentation in the simulation. The spectrum was de-scribed by 80 MHz-spaced comb teeth within a Gaussianenvelope of ∼ P , O P and the P Q branches using the linearly-polarized SFFL.To subsequently cool the hyperfine manifold, we addedthe Q (0 . branch and drove it with the σ + -polarizedSFFL. Informed by our previous experimental work, wesplit the 200 mW laser power equally between the linear-and the σ + -polarized beams, each having a focused 400µm-diameter spot at the center of the ensemble of AlH + molecules.The typical optical transition linewidth for AlH + is ∼
20 MHz, which is smaller than the 80 MHz comb-teethspacing of the femtosecond laser spectrum. As a result,it was possible for an SFFL comb-line to fall outside ofthe transition linewidth of some of the transitions we de-sired to drive. We solved this issue in the simulation byintroducing a Doppler-broadened linewidth contributioncorresponding to a ∼ AlH + ion cloud. Doing so ensured that there wasat least one comb-line within the linewidth of every de-sired cooling transition. One can accomplish this form ofbroadening in the experiment by raising the translationaltemperature of the AlH + ions in a couple ways. One canexcite the secular motion of the AlH + with an AC fieldor introduce additional micro-motion by shifting the en-tire ion cloud away from the geometric center of the Paultrap using a DC field. After internal cooling is finished,one can then turn off the source of translational heating,allowing the AlH + to be sympathetic cooled once againby the laser-cooled Ba + atoms.We simulated the laser-enhanced parity-flipping pro-cess as well. For such cases, we represented the infraredlaser (V10) that drives the | X Σ + , v (cid:48)(cid:48) = 1 , N (cid:48)(cid:48) = 1 , −(cid:105) – | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 2 , + (cid:105) transition to match thespecifications of a commercial Fabry-Perot quantum-cascade laser from Thorlabs. This laser outputs ∼ ∼
15 cm − (FWHM) and canbe tuned to lase ∼ v (cid:48) = 1 – line in the electronic ground state of AlH + . IV. RESULTS AND DISCUSSION
Figure 3 and Table V present rotational cooling ratesfor two schemes. In the first scheme, we apply thelinearly-polarized rotational cooling laser (RC). In thesecond scheme, we apply the rotational cooling laser(RC) as well as the infrared laser (V10) that drives the | X Σ + , v (cid:48)(cid:48) = 1 , N (cid:48)(cid:48) = 1 , −(cid:105) – | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) =2 , + (cid:105) transition. From Figure 3(a), it can be seen thatwithout the V10 laser, the population in the rovibra-tional ground state ( v , N ) = (0,0) increases to 45 % within a few microseconds through the fast rotationalcooling cycle. Afterwards, the population in (0,0) con-tinues to increase but with a slower rate as shown inFigure 3(b). This behavior can be attributed to twotime scales. At shorter times, population accumulates in the | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 1 , −(cid:105) state after it under-goes a parity flip via the | X Σ + , v (cid:48)(cid:48) = 1 , N (cid:48)(cid:48) = 1 , −(cid:105) state. At longer times, vibrational relaxation (140 msdecay constant) begins to dominate the process. Theaddition of the V10 laser mitigates the effect of vibra-tional relaxation between | X Σ + , v (cid:48)(cid:48) = 1 , N (cid:48)(cid:48) = 1 , −(cid:105) and | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 2 , + (cid:105) . The time it takesfor the population in the rovibronic ground state, ρ R0 ,to grows to 63 % reduces from 60 ms to 8.7 µs. Thetrend continues as ρ R0 reaches 95 % in only 160 µs, alsosignificantly shorter than the 360 ms required in the ab-sence of the V10 laser. However, the V10 laser addressesmost but not all of the populations that accumulate inthe | X Σ + , v (cid:48)(cid:48) = 1 (cid:105) manifold. The populations in thehigher rotational states of | X Σ + , v (cid:48)(cid:48) = 1 (cid:105) either relaxand re-enter the cooling cycle or undertake sequencesof rotational and/or vibrational relaxations to reach therovibronic ground state directly. Thus, as can be seenin Figure 3(d), after reaching ∼
95 %, the growth rateof the rovibronic ground state population slows and be-comes asymptotic.Figure 4 and Table VI present our simulation resultsfor two hyperfine-cooling schemes. In the first scheme,we apply the rotational cooling laser (RC) and the σ + -hyperfine-cooling laser (HC). The second scheme addsthe infrared laser (V10). As can be seen in Figure 4(a),in the absence of the V10 laser, the population in thestretched hyperfine state increases to ∼ during thefirst tens of microseconds via the fast rotational coolingcycle and the hyperfine optical pump. Since the hyperfinecooling process takes additional cycles to transport thepopulation to the stretched state, the time scale is longerwhen compared to the case during which only the RClaser is applied. The transfer rate eventually slows dueto the relatively long vibrational relaxation time scale.After one second, the hyperfine population reaches morethan 90 % . From Figure 4(d), we can see that when theV10 laser is added, the impact of vibrational relaxationis reduced. The time it takes for the stretched hyperfine-state population in the rovibronic ground state, ρ H0 , toreach 63 % is shortened from 330 ms to 67 µs. If we leavethe lasers on, ρ H0 can reach 95 % in 25 ms, a near 50-foldreduction from the 1200 ms duration in the absence of theV10 laser. At longer times, the growth rate of ρ H0 slowsdown due to the more complicated relaxation dynamicsof the higher rotational states in | X Σ + , v (cid:48)(cid:48) = 1 (cid:105) .A final point that deserves consideration is the effectof the polarization purity—that is, just how well one pre-pares the polarization of each laser field. While we do notexpect the polarization to have significant effect on thetimescale of the cooling dynamics, it will set an asymp-totic limit on the population transferred to the targetstate. μs )0.00.51.0 P opu l a ti on fr ac ti on (a) RC |X Σ + , v = 0, N = 0, + ⟩ | X Σ + , v = 0, N = 1, − ⟩ |X Σ + , v = 1, N = 1, − ⟩ P opu l a ti on fr ac ti on (b) RC0 10 20 30 40Time ( μs )0.00.51.0 P opu l a ti on fr ac ti on (c) RC+V10 0 100 200 300 400Time (ms)0.00.51.0 P opu l a ti on fr ac ti on (d) RC+V10 FIG. 3: Simulated population dynamics of the rotational cooling: The plots in the top row show rotational cooling performedby application of the linearly-polarized SFFL (RC) for (a) 40 µs and (b) 400 ms, respectively. The plots in the bottom rowshow rotational cooling performed by RC while enhanced by the V10 laser for (c) 40 µs and (d) 400 ms, respectively. The V10laser drives the | X Σ + , v (cid:48)(cid:48) = 1 , N (cid:48)(cid:48) = 1 , − (cid:105) – | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 2 , + (cid:105) transition. μs )0.00.51.0 P opu l a ti on fr ac ti on (a) RC+HC | El , v , J , N , F , F , m F ⟩| X Σ + , 0, 0.5, 0, 3, 3.5, 3.5⟩ | X Σ + , 0, 1.5, 1, 4, 4.5, 4.5⟩ | X Σ + , 1, 1.5, 1, 4, 4.5, 4.5⟩ P opu l a ti on fr ac ti on (b) RC+HC 0 1 2 3 4Time (s)0.00.51.0 P opu l a ti on fr ac ti on (c) RC+HC0 10 20 30 40Time ( μs )0.00.51.0 P opu l a ti on fr ac ti on (d) RC+HC+V10 0 100 200 300 400Time (ms)0.00.51.0 P opu l a ti on fr ac ti on (e) RC+HC+V10 0 1 2 3 4Time (s)0.00.51.0 P opu l a ti on fr ac ti on (f) RC+HC+V10 FIG. 4: Simulated population dynamics of the single hyperfine-state preparation: The plots in the top row show hyperfinecooling performed by application of the linearly-polarized SFFL (RC) and the σ + -polarized SFFL (HC) for (a) 40 µs, (b) 400ms, and (c) 4 s, respectively. The plots in the bottom row show hyperfine cooling performed by RC and HC while enhancedby the V10 laser for (d) 40 µs, (e) 400 ms, and (f) 4 s, respectively. The V10 laser drives the | X Σ + , v (cid:48)(cid:48) = 1 , N (cid:48)(cid:48) = 1 , −(cid:105) – | X Σ + , v (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) = 2 , + (cid:105) transition. TABLE V: Population rise times for the rovibronicground state Laser fields
63 % 95 % RC ms msRC, V10 . µs µs TABLE VI: Population rise times for | F = , m F = (cid:105) in the rovibronic ground stateLaser fields
63 % 95 %
RC, HC ms msRC, HC, V10 µs ms V. CONCLUSIONS
In order for the fields of molecular quantum comput-ing and simulations to mature, simple, efficient, and pre-cise qubit state preparation will be critical. We have de-scribed an improvement to our previous work, in whichthe previously rate-limiting step of a parity-flip is spedup by the addition of a new infrared laser. We furtherdescribed an extension to our rotational cooling setupthat should enable optical pumping to a single hyperfinestate. In support of this work, it was necessary to com-pute hyperfine matrix terms for the A Π state of AlH + .Simulations show that we should be able to drive 95 % of an ensemble of AlH + molecules to a single quantumstate within 25 ms. Conflicts of interest
There are no conflicts to declare.
Acknowledgements
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