Robust Preparation of Many-body Ground States in Jaynes-Cummings Lattices
Kang Cai, Prabin Parajuli, Guilu Long, Chee Wei Wong, Lin Tian
RRobust Preparation of Many-body Ground States in Jaynes-Cummings Lattices
Kang Cai, Prabin Parajuli, Guilu Long,
2, 3, 4, 5
Chee Wei Wong, and Lin Tian ∗ School of Natural Sciences, University of California, Merced, California 95343, USA Beijing Academy of Quantum Information Sciences, Beijing 100193, China State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China Beijing National Research Center for Information Science and Technologyand School of Information Tsinghua University, Beijing 100084, China Frontier Science Center for Quantum Information, Beijing 100084, China Electrical & Computer Engineering, and Center for Quantum Science & Engineering,University of California, Los Angeles, California 90095, USA
Strongly-correlated polaritons in Jaynes-Cummings (JC) lattices can exhibit quantum phase transitions be-tween the Mott-insulating and the superfluid phases at integer fillings. Here we present an approach for therobust preparation of many-body ground states of polaritons in a finite-sized JC lattice by optimized nonlinearramping. In the deep Mott-insulating and deep superfluid regimes, polaritons can be pumped into a JC latticeand be prepared in the ground state with high accuracy via engineered pulse sequences. Using such states asinitial state and employing optimized nonlinear ramping, we demonstrate that many-body ground states in theintermediate regimes of the parameter space can be generated with high fidelity. We exploit a Landau-Zener-typeof estimation on this finite-sized system and derive an optimal ramping index for selected ramping trajectories,which greatly improves the fidelity of the prepared states. With numerical simulation of the ramping process,we further show that by choosing an appropriate trajectory, the fidelity can remain close to unity in almost theentire parameter space. This method is general and can be applied to many other systems.
I. INTRODUCTION
The Jaynes-Cummings (JC) model is a prototype for study-ing light-matter interaction, where a quantum two-level sys-tem is coupled to a cavity mode [1]. This model has beenutilized to study cavity or circuit quantum electrodynamics(QED) in a wide range of systems, from individual particlesin the atomic scale to collective modes in mesoscopic de-vices [2–4]. More recently, advances in device fabrication andquantum technology enabled the exploration of novel many-body physics in arrays of JC models (i.e., JC lattices), whichcan be realized with optical cavities coupled to defects insemiconductors [5–9] and superconducting circuit QED sys-tems [10–15]. The light-matter coupling in the JC models inthe lattice induces intrinsic nonlinearity in the energy spec-trum, which can be mapped to an onsite repulsive interactionbetween polariton excitations. The competition between thisonsite interaction and the polariton hopping between neigh-boring sites gives rise to rich many-body physics for strongly-correlated polaritons in JC lattices, such as quantum or dissi-pative phase transitions and novel transport phenomena [16–18]. One effect of particular interest is the quantum phasetransition between the Mott-insulating (MI) and the superfluid(SF) phases for polaritons at integer fillings of JC lattices, fea-tured by the occurrence of off-diagonal long-range orders incorrelation functions. It was shown that this effect can be ob-served in coupled cavity arrays [5–7] and multi-connected JClattices [13–15].The prerequisite to observe the MI-SF phase transition is topump polariton excitations into a JC lattice and prepare theminto appropriate ground states. However, preparing many-body ground states is a challenging task in engineered systems ∗ [email protected] such as quantum simulators [19–21] and adiabatic quantumcomputers [22, 23]. A number of approaches have been stud-ied to tackle this problem, including adiabatic quantum evolu-tion [24–26], quantum shortcut method with counter-diabaticinteractions [27, 28], quantum phase estimation (QPE) viaquantum Fourier transformation [29, 30], variational quan-tum eigensolver [31, 32], full quantum eigensolver [33], andengineered dissipative environment approach [34, 35]. De-spite these efforts, it is still hard to generate desired many-body states with high fidelity in the noisy intermediate-scalequantum (NISQ) era [36], in particular, for systems workingwith excitations such as the JC lattices. The barriers to gener-ating desired many-body states efficiently and accurately in-clude the lack of a priori knowledge of the energy spectrum,the difficulty in engineering complicated counter-diabatic in-teractions, the rapid decrease of energy gaps and quick in-crease of the size of parametrized quantum circuits with thesize of quantum simulators, and the limited decoherence timesin NISQ devices. Furthermore, many-body states in strongly-correlated systems can be highly entangled, unknown, andhence, often impossible to be generated with pre-programedquantum logic gates.Here we study the robust generation of many-body groundstates in a finite-sized JC lattice at unit filling using optimizednonlinear ramping. By adjusting its parameters to have nohopping between adjacent unit cells, or in the opposite limit,diminishing light-matter coupling, the JC lattice can be tunedto the deep MI or deep SF regimes, where the JC lattice iscomposed of uncoupled subsystems of individual unit cellsor collective cavity modes. We show that polaritons can bepumped into the ground states of these limiting cases withhigh accuracy via engineered pulse sequences by using theapproach developed in [37]. The number of pulses is O ( N ) ,linear to the number of polaritons. With such states as initialstate, we employ optimized nonlinear ramping to generate the a r X i v : . [ qu a n t - ph ] J u l many-body ground states in the intermediate regimes of theparameter space. In previous works [38–40], it was shownthat nonlinear ramping can reduce diabatic transitions to ex-cited states or the production of domain walls when a thermo-dynamic system evolves across the quantum critical point of aquantum phase transition due to the scaling of the many-bodysystem. In this work, we apply nonlinear adiabatic rampingto a finite-sized system, where the energy gap between theground and the excited states remains finite. By exploitinga Landau-Zener-type of estimation [41, 42] and the spectralfeature along a given trajectory, we derive an optimal rampingindex for the trajectory, which can significantly improve thefidelity of the prepared state. Our result agrees well with thedata from numerical simulation of the ramping process. Fur-thermore, we show that by selecting an appropriate trajectoryfor a given set of target parameters combined with the optimalramping index, the fidelity can remain close to unity in almostthe entire parameter space.Finite-sized JC lattices have been implemented in recent ex-periments [18, 43, 44] and the adiabatic evolution studied hereis within reach of current technology [45–47]. Using prac-tical parameters from recent experiments, we estimate thathigh fidelity can be achieved for the final states on a timescale much shorter than the observed decoherence times ofthese devices. The method of optimized nonlinear rampingfor finite-sized systems is general and can be applied to manyother problems. This work can hence guide future works onhigh-fidelity preparation of many-body states and observationof many-body correlations in quantum simulators.This paper is organized as follows. In Sec II, we introducethe Hamiltonian of a one-dimensional (1D) JC lattice and theMI-SF phase transition in this model. In Sec. III, we show thatpolaritons can be pumped into the JC lattice and prepared inthe ground states of the deep MI and SF regimes. We then em-ploy optimized nonlinear ramping to prepare the many-bodyground states in the intermediate regimes of the parameterspace in Sec. IV. We derive the optimal ramping index forgiven trajectories and compare this result with data from nu-merical simulation. In Sec. V, we discuss the improvementof the final-state fidelity by choosing an appropriate rampingtrajectory and the total evolution time for practical devices.Conclusions are given in Sec. VI. II. QUANTUM PHASE TRANSITION IN JC LATTICE
Consider the JC lattice depicted in Fig. 1(a). Here each unitcell contains a qubit coupled to a cavity mode with couplingstrength g , and adjacent unit cells are connected via photonhopping with hopping rate J . The total Hamiltonian of thismodel is H t = H + H int (¯ h = H = ω c ∑ j a † j a j + ω z ∑ j σ jz + + g ∑ j (cid:16) a † j σ j − + σ j + a j (cid:17) (1)is the Hamiltonian of uncoupled JC models with ω c the fre-quency of cavity modes, ω z the level splitting of the qubits, a j ( a † j ) the annihilation (creation) operator of the j -th cavity, and (b) J ∆ MI SF ! ! ! ! ! J J J J a a a a a (a) g g g g g
FIG. 1. (a) Schematic of a 1D JC lattice. Circles (rectangles) repre-sent qubits (cavity modes) with light-matter coupling g and hoppingrate J . (b) Single-particle density matrix ρ ( , ) vs hopping rate J and detuning ∆ for a finite-sized lattice with N = L =
6. Here we let g ≡ σ j ± , σ jz the Pauli operators of the j -th qubit, and H int = − J ∑ j (cid:16) a † j a j + + a † j + a j (cid:17) (2)is the photon hopping between neighboring unit cells. Let {| n , s (cid:105)} be the basis set of an individual JC model with thecavity in the Fock state n and the qubit in the state s = ↑ , ↓ . Theeigenstates of the JC model include the ground state | g (cid:105) = | , ↓(cid:105) with no excitation and the polariton doublets | n , ±(cid:105) withinteger number n excitations, where | n , + (cid:105) = cos ( θ / ) | n , ↓(cid:105) + sin ( θ / ) | n − , ↑(cid:105) , (3) | n , −(cid:105) = sin ( θ / ) | n , ↓(cid:105) − cos ( θ / ) | n − , ↑(cid:105) (4)with θ = (cid:112) [ − ∆ / χ ( n )] / χ ( n ) = (cid:112) ∆ + ng , and ∆ = ω c − ω z the detuning between the cavity mode and thequbit. The corresponding eigenenergies are E g = E n , ± = ( n − / ) ∆ ± χ ( n ) /
2. When the coupling g is nonzero,the energy spacings between the eigenstates become unequalwith ( E n + , − − E n , − ) > ( E n , − − E n − , − ) . Specifically, E , − > E , − for n =
1, which indicates that the energy to add two po-laritons to the JC model is more than twice the energy to adda single polariton. The extra energy to add a second polaritoncan be viewed as an effective onsite interaction or nonlinearityof the polaritons, which is at the root of many interesting phe-nomena in JC models or lattices, such as the photon blockadeeffect [5–7] and electron-phonon-like effect [48].In the limit of J =
0, the JC lattice is composed of isolatedJC models. The ground state at unit filling, where the numberof polaritons N is equal to the number of lattice sites L , is | G (cid:105) J = = ∏ j | , −(cid:105) j (5)with one polariton excitation occupying the state | , −(cid:105) at eachsite, which is in the deep MI regime. States with more thanone excitations at the same site are energetically unfavorabledue to the onsite interaction. In the opposite limit of g = J , the cavity modes are decoupled fromthe qubits. The hopping Hamiltonian (2), now the dominantterm, can be transformed to the momentum space under theperiodic boundary condition with H int = − J ∑ k cos ( k ) a † k a k ,where a k = ∑ j a j e ik · j / √ N ( a † k ) is the annihilation (creation)operator of a collective cavity mode at quasimomentum k = π m / N with integer number m ∈ [ − ( N − ) , N ] . At ∆ < | G (cid:105) g = = √ N ! (cid:16) a † k = (cid:17) N ∏ j | , ↓(cid:105) j (6)with all polaritons occupying the mode a k = , which is a non-local state in the deep SF regime.With the mean-field approximation [5–7] and numericalmethods [8, 9, 13–15], it was shown that quantum phase tran-sitions between the MI and SF phases due to the competitionbetween the onsite interaction and the photon hopping can oc-cur in the intermediate regimes of the parameter space of a JClattice. For a finite-sized lattice with N = L =
6, we numer-ically calculate the many-body ground states using the exactdiagonalization method. The spatial correlation in the many-body ground state | G (cid:105) can be characterized by the normalizedsingle-particle density matrix defined as [49, 50] ρ ( i , j ) = (cid:104) G | a † i a j | G (cid:105) / (cid:104) G | a † i a i | G (cid:105) , (7)which reveals the off-diagonal long-range order of the state.The single-particle density matrix decreases algebraicallywith the spatial separation | i − j | in the SF phase and decreasesexponentially in the MI phase [13–15]. For a finite | i − j | , ρ ( i , j ) of the SF phase is much larger than that of the MIphase. In Fig. 1(b), we plot our numerical result of ρ ( , ) as functions of the hopping rate J and the detuning ∆ , with g ≡ ρ ( , ) increaseswith J for an arbitrary detuning. In the deep MI regime with J = ρ ( , ) = ρ ( , ) can approach unity. This resultclearly indicates the occurrence of the MI-SF phase transition.Meanwhile, in this finite-sized system, the energy separationbetween the ground and the excited states decreases as the pa-rameters approach the intermediate regimes, but maintains afinite energy gap. III. STATE INITIALIZATION
In this section, we present the methods to pump N = L po-laritons to the JC lattice in the limiting cases of J = (b) | N, ↓" | N − , ↓ , ↓" | , ↓" | , ↓"| , ↑"| , ↑"| , ↑"| N − , ↑ N, ↑" (a) | g ! | g ! | g ! N − | g ! N − | g ! N | , −" | , −" | , −" N − | , −" N − | , −" N FIG. 2. Engineered pulses for state initialization at (a) J = g =
0. In (a), vertical arrows are Rabi flips between states | g (cid:105) and | , −(cid:105) of JC models. In (b), vertical (slant) arrows are the operations C l ( Q l ) with l ∈ [ , N ] on the auxiliary qubit and mode a k = . g =
0, respectively, by applying engineered pulses. The po-laritons are pumped into the many-body ground states at thecorresponding parameters. These states will be used as theinitial state in the ramping process studied in Sec. IV.
A. Deep MI regime At J =
0, the ground state is given by (5) with each unitcell in the polariton state | , −(cid:105) . Because the unit cellsare decoupled, we can perform a Rabi rotation between thestates | g (cid:105) and | , −(cid:105) on individual JC models, as illustratedin Fig. 2(a). The driving Hamiltonian can have the form H d ( t ) = ∑ j [ ε exp i ω L t σ − j + h . c . ] with ε the driving amplitudeand ω L = E , − − E g the driving frequency. The correspond-ing Rabi frequency can be derived as Ω d = | ε cos ( θ / ) | fol-lowing Eq. (4). The duration of the Rabi flip from the initialstate | g (cid:105) to the final state | , −(cid:105) is τ d = π / Ω d . To pre-vent the driving pulse from inducing unwanted transitions tohigher states such as | , + (cid:105) , it requires that | ε | (cid:28) g . B. Deep SF regime
In the opposite limit of g = J , the ground stateis given by (6) with all polaritons occupying the collective(nonlocal) mode a k = . To generate this state, we introduce anauxiliary qubit with Pauli operators σ ± , σ z and Hamiltonian H d ( t ) = ω σ z + ε ( t ) e i ω L t σ − + g d ( t ) ∑ j a † j σ − + h . c ., (8)which includes the qubit energy term with frequency ω , adriving on the qubit with amplitude ε ( t ) and frequency ω L ,and a time-dependent coupling between the qubit and mode a k = with coupling strength g d ( t ) . With ω L , ω = ω c − J in resonance with mode a k = , we have H d ( t ) = ε ( t ) σ − + √ Ng d ( t ) a † k = σ − + h . c . in the rotating frame. The first termof H d generates a Rabi rotation on the auxiliary qubit, and thesecond term is a coupling between the qubit and mode a k = .Both terms can be tuned on and off within nanoseconds, as hasbeen demonstrated in recent experiments on superconductingtransmon qubits [43, 44].With the qubits in the JC lattice decoupled from the cavities,the initial state of the coupled system of mode a k = and theauxiliary qubit is | , ↓(cid:105) . To generate the state (6), we utilizethe approach in [37] to design a pulse sequence by switching ε ( t ) and g d ( t ) on and off alternately. The unitary operator forthis pulse sequence is U = Q N C N Q N − C N − · · · Q C Q C , (9)where the unitary operator C l ( l ∈ [ , N ] ) incurs a Rabi flipon the auxiliary qubit by applying a driving pulse with fi-nite amplitude ε ( g d =
0) for a duration of τ cl = π / | ε | , andthe unitary operator Q l causes the exchange of excitationsbetween the auxiliary qubit and mode a k = by turning onthe coupling g d ( ε =
0) for a duration of τ ql = π / √ Nl | g d | .Following this pulse sequence, the state evolves as | , ↓(cid:105) →| , ↑(cid:105) → | , ↓(cid:105) · · · → | N , ↓(cid:105) , as shown in Fig. 2(b). The totalduration of this pulse sequence is τ d = ∑ l (cid:0) τ cl + τ ql (cid:1) . As-suming that the magnitudes of ε and g d are fixed, we find τ d = N π / | ε | + ∑ l π / √ Nl | g d | , which increases with thenumber of polaritons as τ d = O ( N ) . Meanwhile, it requiresthat | ε | , √ N | g d | (cid:28) ω L to achieve high fidelity for the finalstate. Note that at g = a k (cid:54) = , which are separated from mode a k = only by afrequency difference of π J / N . But couplings between theauxiliary qubit and modes a k (cid:54) = during the pulse sequence areprohibited by the symmetry of the Hamiltonian H d . IV. OPTIMIZED NONLINEAR RAMPING
Many-body ground states in the intermediate regimes of theparameter space cannot be calculated analytically, and we can-not design quantum logic operations to generate such states, incontrast to the ground states in the deep MI or SF regimes. Inthis section, we employ optimized nonlinear ramping to reachsuch states via adiabatic evolution.
A. Ramping trajectory and optimal index
In this scheme, a parameter p has the time dependence: p ( t ) = p ( ) [ − ( t / T ) r ] + p ( T ) ( t / T ) r , (10)where p = g , J , ∆ is a tunable parameter of the JC lattice, p ( ) is the initial value of the parameter at time t = p ( T ) is thetarget value at the final time T , and r is the ramping index.For r =
1, it is the linear ramping studied in adiabatic quan-tum computing [22, 23]; and r (cid:54) = t , e.g., ∆ ( t ) − ∆ ( ) ∆ ( T ) − ∆ ( ) ≡ J ( t ) − J ( ) J ( T ) − J ( ) . (11)Hence the evolution trajectory in the parameter space is in-dependent of the ramping index r for given initial and targetvalues of the parameters. On the other hand, the ramping in-dex can strongly affect the sweeping rate of the Hamiltonianalong a trajectory. Using (10), we obtain the time derivative p (cid:48) = d p ( t ) / dt as a function of p : p (cid:48) ( p ) = r [ p − p ( )] ( r − ) / r T [ p ( T ) − p ( )] − / r . (12)The sweeping rate of the Hamiltonian at a given set of pa-rameters { p } can be written as (cid:104) dH / dt (cid:105) = ∑ p (cid:104) ∂ H / ∂ p (cid:105) p (cid:48) ( p ) ,which depends on r via p (cid:48) ( p ) , with (cid:104) . (cid:105) the operator average atthe ground state of parameters { p } .Let | ψ ( T ) (cid:105) be the wave function of the final state of the evo-lution at time T . The fidelity of the final state can be definedas F = |(cid:104) ψ ( T ) | G T (cid:105)| with | G T (cid:105) the many-body ground stateat the target parameters { p ( T ) } = { g ( T ) , J ( T ) , ∆ ( T ) } . Dur-ing a continuous evolution, the probability of diabatic tran-sitions can be approximated by the Landau-Zener formula ∼ e − π E / H (cid:48) gp [41, 42], where the energy gap E gp is definedas the minimum of the energy separation between the groundand the excited states along the evolution trajectory, and H (cid:48) gp denotes sweeping rate of the Hamiltonian at the parameters ofthe energy gap. To reach the desired state with high fidelity,the adiabatic criterion, a commonly expressed as H (cid:48) gp (cid:28) E ,needs to be satisfied so that the diabatic transitions are negligi-ble. For a given trajectory, we can optimize the ramping index r to minimize H (cid:48) gp so as to suppress the diabatic transitions inthe most vulnerable region of the evolution and greatly im-prove the fidelity of the final state. With (11) and (12), wefind that H (cid:48) gp ∝ r (cid:8) [ p ( T ) − p ( )] / (cid:2) p gp − p ( ) (cid:3)(cid:9) / r with p gp the value of parameter p at the energy gap. At the optimalramping index r min , ∂ H (cid:48) gp / ∂ r =
0. This leads to r min = log (cid:20) p ( T ) − p ( ) p gp − p ( ) (cid:21) , (13)which only depends on the initial and final values of the tra-jectory and the position of the energy gap.Below we will conduct numerical simulation on two trajec-tories to calculate the fidelity of the final state and comparethe above result with the numerical data. B. Trajectory I: from deep MI regime
We first consider a trajectory following (10) with g ( t ) ≡ ∆ ( t ) ≡ J ( ) =
0, and J ( T ) ∈ [ , . ] , where the photon hop-ping rate is continuously increased from zero to a finite value.The initial state is (5) in the deep MI phase. Using the ex-act diagonalization method, we calculate the eigenstates and (a) E g p (b) J ′ g p (d) r F
0 0.5 0.1 0.2 0.3 0.4 0.6 0.8 0.4 0.2 (c) F J ( T ) FIG. 3. (a) Energy spectrum of the lowest excited states vs hop-ping rate J ( T ) . Solid (dotted) curve is for the symmetric (asym-metric) state. Here g ≡ ∆ ≡
0, and the ground-state energy isset to zero. (b) Time derivative J (cid:48) gp vs ramping index r at T = π / g , π / g , π / g from top to bottom, and J ( T ) = .
5. (c) Fidelity F vs J ( T ) for r = / / T = π / g . (d) Fidelity F vs r at T = π / g , π / g , π / g from bottom to top, and J ( T ) = . eigenenergies of the JC lattice along this trajectory. The en-ergy spectrum of the lowest excited states is plotted as a func-tion of the hopping rate J ( T ) in Fig. 3(a). The solid curvecorresponds to the energy of a state that is symmetric with re-gard to all lattice sites, and the dotted curves are for asymmet-ric states. As both the initial state and the Hamiltonian H ( t ) are symmetric with regard to lattice sites, the wave function | ψ ( t ) (cid:105) at an arbitrary time t must remain symmetric. Hencediabatic transitions can only happen between the ground stateand this symmetric state, and the energy gap related to theadiabatic criterion is also determined by the energy separationbetween these two states. From our numerical result, we findthat the energy gap occurs at J gp = .
122 with E gp = . H (cid:48) gp ∝ J (cid:48) gp with J (cid:48) gp the time derivative of the hopping rate atthe gap position. Using (12), we have J (cid:48) gp = rJ ( r − ) / r gp T [ J ( T )] − / r , (14)which is plotted as a function of r in Fig. 3(b). The deriva-tive J (cid:48) gp contains a local minimum at the optimal ramping in-dex r min = log (cid:2) J ( T ) / J gp (cid:3) . With the above parameters and J ( T ) = . r min = .
41, which indicates that the best fidelityfor the final state can be achieved between linear and quadraticramping. Using (14), we obtain J (cid:48) gp = .
01 at a total evolutiontime T = π / g . With the gap energy E gp = .
31, the adi-abatic criterion is well satisfied. Meanwhile, J (cid:48) gp , and hencethe probability of diabatic transitions, increase with the tar-get value J ( T ) . We numerically simulate the ramping process
0 0.5 0.1 0.2 0.3 0.4 0 2 4 -2 -4 1 0 0.2 0.4 0.6 0.8 (c) J ( T ) ∆ ( T ) (a) ∆ ( T ) (b) ∆ ( T )
0 0.5 0.1 0.2 0.3 0.4 0 2 4 -2 -4 1 0 0.2 0.4 0.6 0.8 (d) J ( T ) ∆ ( T ) FIG. 4. Fidelity F vs target parameters J ( T ) and ∆ ( T ) for (a) r = /
3, (b) r = /
2, (c) r =
1, and (d) r =
2. Here g ( t ) ≡ J ( ) = ∆ ( ) = T = π / g . along this trajectory and calculate the fidelity of the final state.In Fig. 3(c), the fidelity vs J ( T ) is plotted for several values oframping index r at T = π / g . The fidelity decreases quicklyas J ( T ) becomes larger due to the increase of J (cid:48) gp . It can alsobe seen that for J ( T ) sufficiently larger than the value at theenergy gap, the fidelity is much higher for r = , r = / , /
2. As shown in Fig. 3(d), maximal fidelity isachieved when r ∈ ( , ) , which agrees well with our result ofthe optimal ramping index.Using numerical simulation, we obtain the fidelity of the fi-nal state for a wide range of target parameters following thetrajectories of g ( t ) ≡ J ( ) = ∆ ( ) =
0, and finite values of J ( T ) , ∆ ( T ) with the MI initial state (5). The fidelity is pre-sented in Fig. 4(a-d) for r = / , / , ,
2, respectively, at T = π / g . It can be seen that the fidelity decreases as thetarget parameters move further towards the SF phase. In par-ticular, the fidelity exhibits a sharp drop when the parametersenter the SF phase crossing the energy gap. Meanwhile, thefidelity demonstrates strong dependence on the ramping in-dex in the intermediate regimes of the parameter space, whichagrees with our analytical prediction above. C. Trajectory II: from deep SF regime
Now we consider a trajectory that starts from the deep SFphase with g ( ) = g ( T ) = J ( ) = . J ( T ) ∈ [ , . ] ,and ∆ ( t ) ≡
0, and follows the time dependence in (10) withinitial state (6). In Fig. 5(a), we plot the energy spectrum ofthe lowest excited states vs the hopping rate J ( T ) . The solidcurve is for a symmetric state and the dotted curves are forasymmetric states with regard to all lattice sites. As discussedin Sec. IV B, the ground state can only make diabatic tran-sitions to the symmetric state, and the wave function of thequantum state will remain symmetric during the entire evo- (a) E g p
0 2 0.5 1 1.5 0.4 0.6 0.8 0.2 0 (d) r J ′ g p (b)
0 0.5 0.1 0.2 0.3 0.4 0.6 0.8 1 (c)
F F J ( T ) FIG. 5. (a) Energy spectrum of the lowest excited states vs hop-ping rate J ( T ) . Solid (dotted) curve is for the symmetric (asym-metric) state. Here g ( ) = g ( T ) = J ( ) = . ∆ ≡
0, andthe ground-state energy is set to zero. (b) Time derivative J (cid:48) gp vsramping index r at T = π / g , π / g , π / g from top to bottom, and J ( T ) =
0. (c) Fidelity F vs J ( T ) for r = / / T = π / g . (d) Fidelity F vs r at T = π / g , π / g , π / g from bottom to top, and J ( T ) = lution. From our numerical result, the energy gap occurs at J gp = .
101 with E gp = .
25 along this trajectory.The sweeping rate of the Hamiltonian H (cid:48) gp = g (cid:48) gp I g − J (cid:48) gp I J with g (cid:48) gp the time derivative of the coupling g at the gapposition, I g = (cid:104) ∑ j ( a † j σ j − + σ j + a j ) (cid:105) , and I J = (cid:104) ∑ j ( a † j a j + + a † j + a j ) (cid:105) . Using (11), we have g (cid:48) gp = g ( T ) T rb / r , (15) J (cid:48) gp = J ( T ) − J ( ) T rb / r (16)with b = g ( T ) / g gp = [ J ( T ) − J ( )] / (cid:2) J gp − J ( ) (cid:3) . Hence H (cid:48) gp depends on the ramping index as rb / r , and the optimal ramp-ing index can be derived as r min = log ( b ) . With the aboveparameters and J ( T ) =
0, we have r min = . J ( T ) , the tra-jectory in the parameter space will be different, which re-sults in different gap positions g gp , J gp and different r min . Wenumerically simulate the ramping process and obtain the fi-delity of the final state vs the target hopping rate, as plotted inFig. 5(c). The fidelity decreases as J ( T ) decreases away from J ( ) , because | J (cid:48) gp | increases with the difference | J ( T ) − J ( ) | .As J ( T ) →
0, the fidelity for r = , / r = , /
3. In Fig. 5(d), we plot the fidelity vs theramping rate for J ( T ) =
0, which indicates that the best fi-delity can be achieved for r ∈ ( / , ) at T = π / g , π / g and for r ∈ ( / , / ) at T = π / g . This result confirms ourestimation that the optimal ramping index of this trajectory J ( T )
0 0.5 0.1 0.2 0.3 0.4 0 2 4 -2 -4 1 0 0.2 0.4 0.6 0.8 (c) ∆ ( T ) (a) ∆ ( T ) (b) ∆ ( T ) J ( T )
0 0.5 0.1 0.2 0.3 0.4 0 2 4 -2 -4 1 0 0.2 0.4 0.6 0.8 (d) ∆ ( T ) FIG. 6. Fidelity F vs target parameters J ( T ) and ∆ ( T ) for (a) r = /
3, (b) r = /
2, (c) r =
1, and (d) r =
2. Here g ( ) = ∆ ( ) = g ( T ) = J ( ) = . T = π / g . will shift to a smaller value with r min ≤ r min could be owing to the small sep-aration between the gap position and the target value, whichwill affect the accuracy of the Landau-Zener formula in adia-batic processes [41, 42].We also obtain the fidelity of the final state for a widerange of target parameters following the trajectory (10) with g ( ) = ∆ ( ) = g ( T ) = J ( ) = .
5, finite values of J ( T ) , ∆ ( T ) , and the SF initial state (6), as given in Fig. 6 for r = / , / , ,
2, respectively, at T = π / g . The numericalresult shows that the fidelity decreases quickly as the targetparameters enter the MI phase and strongly depends on theramping index in certain regimes of the parameter space. V. DISCUSSIONS
In Sec. IV, we showed that the fidelity of the final state inintermediate regimes of the parameter space can be greatlyimproved by choosing an optimal ramping index for a chosentrajectory and by increasing the total ramping time T . Here weshow that for a given set of target parameters, the choice of thetrajectory can have dramatic impact too. When the target pa-rameters are in the MI phase, it is better to start from an initialstate in the deep MI regime such as (5) so that the adiabaticevolution does not need to cross the energy gap to reach thetarget parameters and diabatic transitions remain negligible.Similarly, when the target parameters are in the SF phase, wecan choose the initial state to be in the deep SF regime suchas (6). For illustration, in Fig. 7, we plot the maximal fidelityfrom the two trajectories described in Sec. IV B [Fig. 4(c) for r =
1] and in Sec. IV C [Fig. 6(c) for r = J ( T ) ∆ ( T ) FIG. 7. Maximal fidelity vs target parameters J ( T ) and ∆ ( T ) fromthe two trajectories in Fig. 4(c) and Fig. 6(c), respectively. space. Further improvement could be achieved by optimizingthe trajectory to the target parameters.An obvious approach to improve the fidelity of adiabaticprocesses is to increase the ramping time T , which can reducethe time derivative p (cid:48) gp and the sweeping rate of the Hamilto-nian. This can be seen from the numerical result in Fig. 3(d)and Fig. 5(d). For quantum devices in the NISQ era, however,the decoherence times of the qubits and the cavity modes ina JC lattice set limits on the total evolution time. The many-body ground state studied here works with finite number ofpolariton excitations. In the presence of decoherence, the ex-citations can decay in a timescale comparable to the decoher-ence times. The ramping time needs to be much shorter thanthe decoherence times. In experiments, superconducting res-onator cavities with frequency ω c / π ∼ can be readily realized, which have a decay time of3 µ sec, and superconducting qubits can have a decoherencetime of 100 µ sec [47]. With typical coupling strengths of g / π , J / π ∼ MHz , the evolution time T = π / g ∼ . ∼ VI. CONCLUSIONS
To conclude, we studied an optimized nonlinear rampingscheme to prepare the many-body ground states of polari-tons in a finite-sized JC lattice. The polaritons are initializedinto the ground states in the deep MI and SF regimes via en-gineered pulse sequence and are subsequently prepared intothe many-body ground states in the intermediate regimes ofthe parameter space by adiabatic ramping. Using a Landau-Zener-type of estimation, we derived the optimal ramping in-dex for given ramping trajectories that can ensure minimalsweeping rate of the Hamiltonian at the energy gap. Our nu-merical simulation confirmed that the fidelity of the final statescan be significantly improved by using the optimal rampingindex. We further showed that by choosing an appropriatetrajectory, the fidelity can remain close to unity in almost theentire parameter space. We want to emphasize that this ap-proach is general and can be applied to other finite-sized sys-tems. This work can shed lights on high-fidelity preparationof many-body states in engineered quantum systems, such asquantum simulators, and advance the implementation of quan-tum simulation with NISQ devices.
VII. ACKNOWLEDGEMENTS
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