Quantum Adiabatic Doping for Atomic Fermi-Hubbard Quantum Simulations
QQuantum Adiabatic Doping for Atomic Fermi-Hubbard Quantum Simulations
Jue Nan,
1, 2
Jian Lin, Yuchen Luo, Bo Zhao,
1, 3, ∗ and Xiaopeng Li
2, 4, † Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China. State Key Laboratory of Surface Physics, Institute of Nanoelectronics and Quantum Computing,and Department of Physics, Fudan University, Shanghai 200433, China Shanghai Branch,CAS Center for Excellence and SynergeticInnovation Center in Quantum Information and Quantum Physics,University of Science and Technology of China, Shanghai 201315, China. Shanghai Qi Zhi Institute, Xuhui District, Shanghai, 200032, China.
There have been considerable research efforts devoted to quantum simulations of Fermi-Hubbardmodel with ultracold atoms loaded in optical lattices. In such experiments, the antiferromagneticallyordered quantum state has been achieved at half filling in recent years. The atomic lattice awayfrom half filling is expected to host d-wave superconductivity, but its low temperature phases havenot been reached. In a recent work [1], we proposed an approach of incommensurate quantumadiabatic doping, using quantum adiabatic evolution of an incommensurate lattice for preparationof the highly correlated many-body ground state of the doped Fermi-Hubbard model starting froma unit-filling band insulator. Its feasibility has been demonstrated with numerical simulations ofthe adiabatic preparation for certain incommensurate particle-doping fractions, where the majorproblem to circumvent is the atomic localization in the incommensurate lattice. Here we carry out asystematic study of the quantum adiabatic doping for a wide range of doping fractions from particle-doping to hole-doping, including both commensurate and incommensurate cases. We find that thereis still a localization-like slowing-down problem at commensurate fillings, and that it becomes lessharmful in the hole-doped regime. With interactions, the adiabatic preparation is found to be moreefficient for that interaction effect destabilizes localization. For both free and interacting cases, wefind the adiabatic doping has better performance in the hole-doped regime than the particle-dopedregime. We also study adiabatic doping starting from the half-filling Mott insulator, which is foundto be more efficient for certain filling fractions.
I. INTRODUCTION
Ultracold atoms in optical lattices provide a fascinat-ing platform for quantum simulations of correlated many-body physics [2–5]. Since the atomic tunneling and in-teractions are both controllable in these systems, theyhave widely been used to study quantum many-bodyphases and quantum phase transitions. One main themeof quantum simulation with optical lattices is to inves-tigate the low-temperature phase diagram of the Fermi-Hubbard model [6–8], and help uncover the fundamentalmechanism of high-temperature superconductivity [9].Whether and how the d-wave superconductivity arisesin the doped region including both hole- and parti-cle doped cases, in the repulsive Fermi-Hubbard modelhas been attracting continuous research efforts [9–12],but this remains an open question with no consensusreached [11, 12], one reason being that the numerical sim-ulations on classical computers meet fundamental chal-lenges of exponentially growing Hilbert space of the quan-tum many-body system. This makes quantum simula-tions of the doped Fermi-Hubbard model very much de-manded as it is has potential to address the importantquestion of existence of d-wave superconductivity in the ∗ [email protected] † xiaopeng [email protected] model. With the development of ultracold atom experi-ments, the low-temperature antiferromagnetic phase hasnow been reached at half filling [13–15]. The doping ofantiferromagnet with hole has been realized by reducingthe density of the trapped gases [15]. However, it is diffi-cult to cool down the system to a sufficiently low temper-ature to enter the d-wave superconducting phase. How tooptimally perform doping for the atomic Fermi-Hubbardoptical lattice system while maintaining low entropy ofthe system demands more theoretical study.A plausible method to maintain the quantum simula-tor at low entropy is to perform the quantum adiabaticdoping. The adiabatic quantum state preparation hasbeen widely applied in the quantum state engineering andquantum simulation [1, 16–21]. For the fermions confinedin the optical lattice, a different filling factor is achievedby adiabatically converting the lattice with one spatialperiod to a lattice with a different period. Going throughan adiabatic evolution from insulating states to the dopedregime, the system remains at the ground state of the in-stantaneous Hamiltonian. This protocol has been stud-ied to prepare Fermi-Hubbard antiferromagnet insulat-ing state[18, 20], and also doped ground state [1] withincommensurate fillings. For the incommensurate case,it has been found that the major difficulty in carrying outthe quantum adiabatic doping is from fermion localiza-tion [1]. The physics of fermion localization occurring inthe intermediate dynamics prevents efficient state prepa-ration, causing a problem of localization slowing down. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n As a generic solution of the localization slowing down[22–25], the atomic interaction is introduced and foundto improve the preparation efficiency.Since the focus of Ref. 1 is to show the feasibility ofthe quantum adiabatic doping for incommensurate fill-ing, only one filling factor at the particle-doped regimewas studied. It is worth more systematic study how thequantum adiabatic doping behaves from different fillings,and in particular whether it remains efficient in the hole-doped regime should be addressed.In this work we carry out a systematic study on thequantum adiabatic doping in a one-dimensional opticallattice for a broad range of filling factors, with numericalsimulations based on the time-dependent density matrixrenormalization group (DMRG) method. We find thatthe localization slowing down is a generic problem forboth commensurate and incommensurate fillings. For theincommensurate case, the localization problem is morefundamental because the localization persists in the ther-modynamic limit. This problem also causes slowing downfor the commensurate case considering a finite-size sys-tem with a localization length significantly smaller thanthe system size, although it scales linearly with the sys-tem size in the thermodynamic limit. For the particledoping, we show that the adiabatic preparation efficiencycan be enhanced by introducing atomic interaction forboth commensurate and incommensurate fillings. Forthe hole doping, we find that the localization is muchweaker, which we attribute to the large particle tunnelingof the final lattice. The quantum adiabatic hole dopingis consequently more efficient than the particle doping,and the efficiency can be further improved by includ-ing strong atomic interaction. Besides starting from theband insulator, we also consider the adiabatic particledoping starting from the Mott insulator at half filling.Our numerical simulation shows that the quantum adia-batic doping starting form the Mott insulator has betterperformance for certain fillings. We expect these numer-ical results on the quantum adiabatic doping for a one-dimensional optical lattice would also shed light on thetwo-dimensional lattice.
II. THEORETICAL SETUP
The atomic quantum simulator of the Fermi-Hubbardmodel consists of two-component Fermionic atoms con-fined in the periodical optical lattice. The system is de-scribed by the Hamiltonian H = (cid:90) d d x ψ † σ (cid:16) − (cid:126) ∇ M + V ( x ) − µ (cid:17) ψ σ + gψ †↑ ψ †↓ ψ ↓ ψ ↑ , (1)where ψ σ = ↑ , ↓ ( x ) is the quantum field operator for cor-responding pseudospin (hyperfine state) ↑ and ↓ com-ponents, M the atomic mass, µ the chemical potential, g the interaction strength between the two components, and V ( x ) the optical lattice potential. We neglect theharmonic potential in our calculation for simplicity. Inthis work we first consider a one-dimensional( d = 1) bandinsulator as the initial state of the adiabatic evolution.In optical lattice experiments, band insulators with lowentropy have been achieved [20]. The initial lattice po-tential reads as V I ( x ) = V cos(2 πx/λ ) . (2)with λ the lattice constant and V the strength. Thenwe adiabatically convert the lattice to another one witha different period V F ( x ) = V (cid:48) cos(2 πx/λ (cid:48) ) . (3)During the time evolution, the potential has a time de-pendent form V ( x, t ) = (1 − s ( t/T )) V I ( x ) + s ( t/T ) V F ( x ) , (4)which is standard in the context of adiabatic algorithmwith s ( t/T ) giving the path of the evolution and T thetotal evolution time [26]. In the calculation, we focus atthe parameter choice of V = V (cid:48) , and examine the conse-quence of varying the overall lattice strength. Assumingthat the atomic loss during the evolution is negligible,which holds when the evolution time T is smaller thanthe lifetime of the cold atom system, the filling factor ofthe final state is f = λ (cid:48) /λ . By controlling the the ratiobetween the lattice constants, a generic filling factor isaccessible with the quantum adiabatic doping.With the incommensurate potential in Eq. (4), thequantum dynamics during the adiabatic evolution can-not be described by a valid tight-binding model. Wethus have to take into account the continuous degrees offreedom of the lattice. In the numerical simulation, wediscretize the space as x → j × a , with a the length of agrid, and j the discrete index. This leads to a Hamilto-nian H = (cid:88) j (cid:110) − t (cid:104) c † jσ c j +1 ,σ + H.c. (cid:105) + V j c † jσ c jσ + U n j ↑ n j ↓ (cid:111) (5)with parameters t = (cid:126) / (2 M a ), V j = V ( ja ) and U = g/a . In the following calculation, we divide each spatialperiod of the initial lattice into 20 grids if not specified.The linear path s ( t/T ) = t/T is adopted for the adia-batic evolution. To evaluate the performance of the adi-abatic doping, we calculate the wave function overlapOverlap = |(cid:104) Ψ g | Ψ T (cid:105)| (6)and the excitation energy∆ E = (cid:104) Ψ( T ) | H F | Ψ( T ) (cid:105) − (cid:104) Ψ g | H F | Ψ g (cid:105) , (7)where | Ψ g (cid:105) is the ground state of the final Hamiltonian H F , and | Ψ( T ) (cid:105) is the final state of the quantum adi-abatic evolution. To confirm our finding is generic forfinite-size system, we consider more than one system sizesin following numerical study.
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200 400 600 8000.00.20.40.60.81.0 (a) (b)(c) (d)
V/E R =1V/E R =2V/E R =4V/E R =8 T[ ħ /E R ] T[ ħ /E R ] FIG. 1. Quantum adiabatic particle doping of free fermionsin one-dimensional optical lattice with rational filling factors.The lines show the dependence of the final state wave func-tion overlap on total evolution time T with varying strengthsof lattice potential. The filling factor after the lattice conver-sion takes the rational values f = 3 / f = 2 / L = 36 and 60, with the results shown in (a),(c)and (b),(d), respectively. The wave function overlap system-atically increases with the adiabatic time T . For weak latticepotential, the overlap approaches to 1 with sufficiently longevolution time and the doped final state can be efficiently pre-pared. For stronger potential, the overlap remains nearly zeroand the adiabatic doping proposal fails for free fermions. III. PARTICLE DOPINGA. Free fermion
We first study the adiabatic particle doping of freefermions. This corresponds to setting the filling factor f larger than 1 /
2. We simulate the adiabatic evolu-tion with rational filling factors f = 2 / , / T is shown inFig. 1. The initial state is a one-dimensional band insula-tor without interaction in optical lattice with L periods.We consider two different choices of system size, L = 36,and 60. For current and following calculations, we use pe-riodical boundary condition for non-interacting fermions,while the open boundary condition is adopted for theDMRG simulation of interacting fermions for numericalimplementation convenience. For weak lattice potential,say V = 1 , E R ( E R is the single photon recoil energy),the overlap increases with T and quickly approaches to1 as we increase the total evolution time. This implies the adiabatic preparation of the final state is efficient forthe weak lattice confinement. However, with strong lat-tice potential ( V = 8 E R , for example), the final stateoverlap essentially remains at zero for all evolution timewe have simulated, which means the adiabatic doping isinefficient.For the adiabatic doping with incommensurate lattice,the localization in the intermediate dynamics leads toslowing down of the adiabatic state evolution. This hasbeen shown by the increase of inverse participation ra-tio(IPR) with the lattice potential strength [1]. In thecommensurate case, similarly, the breakdown of adiabaticstate preparation under strong lattice confinement indi-cates a localization-like problem also occurs during evo-lution. To investigate the slowing down problem in thecommensurate lattice, we calculate the normalized par-ticipation ratio(NPR) [27]. The NPR of a single-particleeigenstate φ m ( x ) is defined asNPR ( m ) = L (cid:88) j | φ m ( j ) | − , (8)where L is the system size, and j labels the discretespatial coordinate(see Eq. (5)). This quantity remainsfinite for spatial extended states but vanishes for local-ized states. For a one-dimensional localized system withlength L , it goes like L − . We calculate the single-particle NPR of L lowest eigenstates and average themto get (cid:104) NPR (cid:105) = 1 L L (cid:88) m =1 NPR ( m ) . (9)The averaged NPR is multiplied by L to compensate the L − scaling: M ( L ) = L (cid:104) NPR (cid:105) . (10)Therefore, for localized one-dimensional system, M ( L ) isexpected to be nearly independent of L . In Fig. 2 we showthe dependence of the quantity log M ( L ) on L followingthe evolution path. The filling factor is set to be f = 3 / V = V (cid:48) = 8 E R . It isevident that M ( L ) increases quickly with L at each pointof the path. While this is consistent with the well-knownfact that commensurate lattice models do not have lo-calization in the thermodynamic limit, having a (cid:104) NPR (cid:105) significantly smaller than 1 in a finite size system impliesthe coupling between different modes is severely sup-pressed owing to the locality of the Hamiltonian, whichthen causes the slowing down of the quantum adiabaticevolution.
B. Interacting fermion
The localization makes the minimal energy gap be-tween the ground state and the first excited state expo-nentially small, and therefore causes slowing down of the log (L) FIG. 2. The dependence of the quantity log M ( L )(Eq. (10)) on the system size L for the adiabatic particledoping of one-dimensional free fermions. Here s ∈ [0 ,
1] isthe path parameter of adiabatic evolution (Eq. (4)). The sys-tem size L takes values from 15 to 597 with a rational fillingfactor f = 2 /
3. In the calculation, the potential strength is V = V (cid:48) = 8 E R , for which the intermediate regime of evo-lution is strongly localized. It is evident that this quantityincreases quickly with L at each time instant s . For a one-dimensional localized system, M ( L ) is expected to be nearlyindependent of L due to the L − scaling of (cid:104) NPR (cid:105) . The in-crease of log M ( L ) implies that the localization becomes un-stable as we increase the system size and tends to disappearin the thermodynamic limit. Nonetheless, the significant de-viation of M(L) from L already implies the coupling betweenwave functions is rather weak, which affects the quantum adi-abatic doping. adiabatic state preparation. In the study of many-bodylocalization, it has been established that interaction ef-fect tends to destabilize localization [22–25, 28–30], whichwould then improve the efficiency of the quantum adia-batic doping. This has been shown to be efficient for anincommensurate lattice with strong interaction in Ref. 1.In this work, we consider introducing a time-dependentinteraction H U = g ( t ) (cid:90) dxψ †↑ ψ †↓ ψ ↓ ψ ↑ (11)with the time sequence shown in the inset of Fig. 3. Thesystem is initially prepared in a one-dimensional nonin-teracting band insulator with 14 periods. The interactionis turned on slowly and ramped to a constant. Then theinitial optical lattice is adiabatically converted to anotherspatial period, after which the interaction is turned offslowly. The size of the final lattice is L (cid:48) = 21, which cor-responds to a rational filling factor f = 2 /
3. We simulatethe lattice conversion process using density matrix renor-malization group (DMRG) method. The total evolutiontime is T = 200 (cid:126) /E R . We use the second-order Trotteri- O v e r l a p Δ E / N [ E R ] g(t)/g(0)s(t/T) g/ λ [E R ] FIG. 3. Quantum adiabatic particle doping of one-dimensional interacting fermions with rational filling. Wesimulate the adiabatic evolution with DMRG, taking theHamiltonian in Eq. (5). The inset shows the time sequence ofthe evolution. The interaction is adiabatically tuned on for aone-dimensional noninteracting band insulator, held constantto implement the lattice conversion, and then adiabaticallytuned off. We perform the calculation of the lattice conver-sion process in the regime t ∈ [0 , T ]. In the DMRG calcu-lation, we take second-order Trotterization of the evolutionoperator, with the evolution time T = 200 (cid:126) /E R divided into26000 Trotter steps. The filling factor is chosen as f = 2 / L = 14 and L (cid:48) = 21. The strength of the potential is V = V (cid:48) = 8 E R . As we increase the interaction strength from g = 0 to g = 0 . λE R , the final state wave function overlapincreases and the single-particle excitation energy ∆ E/N ( N is particle number) becomes smaller. The localization prob-lem is reduced and the performance of the adiabatic dopingis improved by introducing interaction of proper strength. zation [31] of the evolution operator with 26000 evolutionsteps. The wave function overlap and the single-particleexcitation energy at t = T are shown as functions of theinteraction strength g in Fig. 3. Here the overall potentialstrength is V = V (cid:48) = 8 E R , for which the non-interactingquantum adiabatic doping is inefficient. As we increasethe interaction strength g from 0 to 0 . λE R with λ the spatial period of the initial lattice), the wavefunction overlap is improved from 0 .
128 to 0 . E/N ( N is the numberof particles) becomes much suppressed. We thus find thatin the commensurate case, the slowing down problem canstill be solved by introducing atomic interaction. Andramatically enhancement of the preparation efficiencyis achieved with a proper interaction strength. However,when we further increase the interaction strength, we findthat the overlap decreases and ∆ E/N increases slightly.In the dynamical simulation, the largest bond dimensionof the matrix product state(MPS) is χ = 120, with whichnumerical convergence is reached (see Ref. 1)We also consider performing quantum adiabatic dop-ing starting from a Mott insulating state. In experi-ments, such an initial state is accessible since the low-temperature antiferromagnetic order has been observedat half filling in two-dimensional optical lattices [15]. Tocompare the performances of starting from the two dif-ferent initial states of Mott and band insulators, we sim-ulate the adiabatic lattice conversion starting from inter-acting band insulator with L periods and Mott insula-tor with 2 L periods. We choose five different fillings inthe particle-doped regime, both rational and irrational.Without loss of generality, the irrational filling factor isset to be golden ratio, which is approximated by the Fi-bonacci sequence as 8 /
13 and 13 /
21 in our finite-sizecalculation. Here we consider several different systemsizes, L = 8 , , ,
13. For the Mott insulator, each pe-riod of the initial lattice is divided into 10 grids, whilethat for band insulator is divided into 20 grids. This pa-rameter choice is chosen such that the final states of twoprocedures have the same spatial periods and discretegrids. The total evolution time is T = 200 (cid:126) /E R for thesmaller system of L = 8 , T = 200 , , (cid:126) /E R with the corresponding numbers of Trotter steps 26000,38000, and 50000 for the larger one of L = 12 ,
13. Dur-ing the evolution, the interaction strength is held con-stant of g = 0 . λE R and the overall potential strengthtakes the value V = V (cid:48) = 8 E R . The final state wavefunction overlap is shown in Fig. 4 with the solid anddashed lines representing the results for band insulatorand Mott insulator, respectively. The overlap is largerfor band insulator at f = 3 / √ − / f = 2 / /
4, the adiabatic doping startingfrom Mott insulator has better performance. The ad-vantage is especially evident for f = 3 /
4. As shown inFig. 4(c), the overlap reaches 0 .
93 at T = 300 (cid:126) /E R for Mott insulator while that for band insulator is 0 . f = 4 /
5, the overlaps of the two protocols are bothsmaller than 1 / . T = 400 (cid:126) /E R . We expect the overall performancecan be further improved by increasing the total evolu-tion time. IV. HOLE DOPINGA. Free fermion
In this section we study the adiabatic hole doping,which corresponds to the filling factor smaller than 1 / L periods. Both therational and irrational fillings are considered. The ratio-nal filling factors are chosen as f = 1 / /
4, while forthe irrational case, we take f = (cid:2) − √ (cid:3) /
2, which is ap-proximated by Fibonacci sequence as f = F n − /F n . Wechoose four different values for the system size, L = 36 , L = 34 ,
55 for the irrationalcase. The dependence of the final state wave function O v e r l a p BI MI3/5 Golden Ratio 2/3 3/4 4/5 f O v e r l a p BIBI200 250 300 350 4000.00.20.40.60.81.0 O v e r l a p BIMI 200 250 300 350 4000.00.20.40.60.81.0 BIMI (a)(b)(c) (d)
MIMI T[ ħ /E R ] T[ ħ /E R ]T=200 ħ /E R :T=300 ħ /E R :T=400 ħ /E R : FIG. 4. Performance of quantum adiabatic doping of one-dimensional interacting fermions starting from band insulatorand Mott insulator. We simulate the adiabatic evolution usingDMRG with several different system sizes, L = 8 , , , L with the lattice constant λ/
2. During thelattice conversion, the interaction strength is held constant of g = 0 . λE R . The final state wave function overlap of thesetwo different protocols are compared in (a) for L = 8 , T = 200 (cid:126) /E R and (b) for L = 12 ,
13 with T = 200 , , (cid:126) /E R . We choose 5 different final statefillings, both rational and irrational. In (c) and (d), we showthe dependence of wave function overlap on the adiabatic time T with f = 3 / /
5, respectively. For f = 2 / /
4, the adiabatic doping tarting from the Mott insulator hasbetter performance. The comparison is most dramatic for f = 3 /
4, as shown in (c). For f = 4 /
5, the wave functionoverlaps of both procedures are smaller than 1 / T , the protocol startingfrom a Mott insulator is more efficient than that from a bandinsulator. overlap on the total evolution time is shown in Fig. 5.The overlap systemically increases with evolution timefor all potential strengths we consider. It should be no-ticed that a wave function overlap larger than 1 / L = 55 ,
60 with V = 8 E R and smaller system of L = 34 ,
36 with V = 16 E R as weincrease the evolution time to T = 800 (cid:126) /E R . The adi-abatic doping remains efficient in the hole-doped regimeeven for moderate lattice potentials, for example with V = 8 E R , E R , in contrast to the particle-doping, whichis severely subjected to localization slowing down prob-lem. For sufficiently strong lattice potential, the slowingdown problem still occurs in the quantum adiabatic evo-lution of the hole-doped case.To investigate the localization in the hole-dopedregime, we calculate the inverse participation ratio(IPR),which tends to vanish in the extended system and re-mains finite in the localized system. The IPR is aver-aged over the L lowest lying single-particle eigenstatesfollowing the evolution path. The results are shown inFig. 6. The irrational filling is set to be f = (cid:2) − √ (cid:3) / /
144 with L = 55. Theaveraged IPR systematically increases with the potentialstrength, which means the breakdown of adiabatic prepa-ration corresponds to the atom localization. The resultsare compared to that in the particle-doped regime with f = (cid:2) √ − (cid:3) / /
89) and V = 8 E R .For the most part of Hamiltonian evolution path, the IPRof hole doping is smaller than that of particle doping,especially for s > . B. Interacting fermion
We further consider adiabatic hole doping of interact-ing fermions, and simulate the evolution process startingfrom a one-dimensional interacting band insulator usingDMRG. We choose f = 2 / L = 12 for the rationalfilling and f = (cid:2) − √ (cid:3) / L/L (cid:48) = 13 /
34 in the calculation.The dependence of the final state wave function overlapand single particle excitation energy on the interactionstrength is shown in Fig. 7 (solid lines) with an overall po-tential strength V = 8 E R . The DMRG calculation sharesthe same time sequence and parameter choice with thatin the particle-doped regime. Two choices of total evo-lution time, T = 50 (cid:126) /E R and 200 (cid:126) /E R , are consideredwith 7000 and 26000 evolution steps, respectively. Thepreparation efficiency is improved by introducing strongatomic interaction. In our numerical results, we find sys-tematic increase of wave function overlap (Eq. (6)) andreduction of excitation energy with increasing interactionstrength. By increasing interaction strength g from 0 to0 . λE R ), the wave function overlap ultimatelyreaches 0 .
96 for f = 13 /
34 and 0 .
965 for f = 2 /
5. Wecompare the results with that of particle doping (dashedline) for the same interaction and confinement potentialstrengths. As shown in Fig. 7 (c), the wave functionoverlap for T = 50 (cid:126) /E R in the hole-doped regime is evenlarge than that of particle doping for T = 200 (cid:126) /E R . Thehole doping is evidently more efficient than the parti-cle doping in the interacting regime, as is true for non-interacting case as well.
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200 400 600 8000.00.20.40.60.81.0 (a) (b)(d)(e) (f)
V/E R =4 V/E R =8 V/E R =16 V/E R =24 V/E R =32V/E R =4 V/E R =8f<1/2:f>1/2: T[ ħ /E R ]T[ ħ /E R ] FIG. 5. Quantum adiabatic hole doping of free fermionsstarting from one-dimensional band insulator. The solid anddashed lines show the dependence of final state wave func-tion overlap on total evolution time T for hole doping andparticle doping, respectively. For hole doping, the filling fac-tor takes rational values 1 / / L = 36 in (a),(c) and L = 60 in (b), (d). It also takes irrational value (cid:2) − √ (cid:3) /
2, which is approximate by the Fibonacci sequenceas f = 34 /
89 with L = 34 in (e) and 55 /
144 with L = 55in (f). The non-interacting hole doping is efficient for weaklattice confinement, say V = 4 E R . That is, the wave functionoverlap quickly approaches to 1 for rational fillings we con-sidered, and also the smaller system of L = 34 with irrationalfilling. Under strong lattice confinement with V = 32 E R , theoverlap remains small and the quantum adiabatic doping failsto prepare the final state. For the same strength, the adia-batic doping is more efficient in the hole-doped regime. Forthe fillings in hole-doped regime we consider, a overlap largerthan 1 / T = 800 (cid:126) /E R for L = 34 ,
36 with V = 16 E R and L = 55 ,
60 with V = 8 E R , while particle dop-ing is inefficient with V = 8 E R (shown by dashed lines withthe corresponding filling factors 1 − f ). s < I P R > f=55/89:f=55/144: V/E R =8V/E R =8V/E R =16V/E R =32 FIG. 6. Averaged inverse participation ratio(IPR) along theevolution path with V = 8 , , E R in the hole-doped regimeand V = 8 E R in the particle-doped regime. The filling factoris set to be f = (cid:2) − √ (cid:3) / (cid:2) √ − (cid:3) / F n − /F n and F n − /F n − with the initial lattice size L = 55, respectively. The aver-aged IPR increases with potential strength, which implies thelocalization becomes stronger. Compared to particle doping,the IPR for hole doping is smaller for most part of the evo-lution path, which explains the high preparation efficiency inthis regime. V. CONCLUSION
To conclude, the doped quantum phase of atomicFermi-Hubbard model with low thermal entropy can beprepared by adiabatically converting two optical latticeswith different spatial periods. In this adiabatic dopingproposal, an arbitrary filling fraction can be achieved bychoosing the lattice constant ratio of the initial and finallattices. In this work, we consider the quantum adiabaticdoping starting from band insulator in one-dimensionallattice and systemically study the proposal for a broadrange of filing fractions from particle-doping to hole-doping, including both rational and irrational cases. It isfound that the atom localization, which is a fundamentalproblem in incommensurate lattice, also prevents efficientadiabatic doping for commensurate filling at strong lat-tice confinement. Through DMRG simulation, we showthat the localization is suppressed by introducing strongatomic interaction and the state preparation efficiencyis consequently improved. Compared to particle dop-ing, the adiabatic hole doping is more efficient in bothfree and interacting regime. We also consider the adi-abatic doping starting from the Mott insulator state athalf filling. From the results of DMRG calculation, wefind that this protocol has significantly higher prepara-tion efficiency than the adiabatic doping starting fromband insulator for a certain range of filling factors.Although the numerical simulation of the quantum adi-abatic doping process is restricted to one dimension inthis work due to numerical cost, we anticipate a bet- ter performance in two dimensions for the localization O v e r l a p T=200 ħ /E R T=50 ħ /E R g/ λ [E R ] O v e r l a p T=200 ħ /E R T=50 ħ /E R Δ E / N [ E R ] Δ E / N [ E R ] (a) (b)(c) (d) T=200 ħ /E R T=50 ħ /E R T=200 ħ /E R f>1/2:f<1/2: g/ λ [E R ]T=200 ħ /E R T=50 ħ /E R T=200 ħ /E R f>1/2:f<1/2: FIG. 7. Performance of adiabatic hole doping of inter-acting fermions in one-dimensional lattice for both rational( f = 2 /
5) and irrational fillings( (cid:2) − √ (cid:3) / L = 12 and (c),(d) for irrational filling, which isapproximated by L/L (cid:48) = 13 /
34. Here we consider the totaladiabatic time T = 50 (cid:126) /E R and 200 (cid:126) /E R . The overall poten-tial strength is V = 8 E R for both particle and hole doping.It is found that the preparation efficiency is improved by in-troducing strong interaction. As we increase the interactionstrength, the wave function increases and approaches to 1 for T = 200 (cid:126) /E R , and the excitation energy is reduced accord-ingly. Compared to particle doping (shown by dashed lineswith filling factor f = 13 / physics is largely expected to be weaker in higher di-mensions. Our systematic study of different fillings andinteraction effects should also shed light on the quantumadiabatic doping in two dimensions. Acknowledgement.
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