(1+1)-d U(1) Quantum link models from effective Hamiltonians of dipolar molecules
Jiayu Shen, Di Luo, Michael Highman, Bryan K. Clark, Brian DeMarco, Aida X. El-Khadra, Bryce Gadway
(( + ) -d U ( ) Quantum link models from effectiveHamiltonians of dipolar molecules
Jiayu Shen ∗†‡ , Di Luo ‡ , Michael Highman, Bryan K. Clark, Brian DeMarco, Aida X.El-Khadra, Bryce Gadway Department of Physics, University of Illinois, Urbana, IL 61801, USA
We study the promising idea of using dipolar molecular systems as analog quantum simulatorsfor quantum link models, which are discrete versions of lattice gauge theories. In a quantumlink model the link variables have a finite number of degrees of freedom and discrete values.We construct the effective Hamiltonian of a system of dipolar molecules with electric dipole-dipole interactions, where we use the tunable parameters of the system to match it to the targetHamiltonian describing a U ( ) quantum link model in 1 + ∗ Speaker. † E-mail: [email protected] ‡ These authors contributed equally to this work. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J a n LMs from dipolar molecules
Jiayu Shen
1. Introduction
The idea of quantum simulations is to study quantum field theories using quantum hardware[1]. Lattice gauge theories (LGT) are a particularly interesting target for quantum simulations.Both, digital [2] and analog [3] quantum simulations show promise for studies of the real-timeevolution of LGTs in a Hamiltonian formalism. Since, generally, the number of available quantumstates in an experimental system is finite, a truncation scheme (or other approximation) is neededto reduce the infinite degrees of freedom in a gauge theory to a finite number. Quantum linkmodels (QLMs) [4] provide a viable reformulation of LGTs for quantum simulations. Experimentsfor analog quantum simulations of QLMs using Hubbard models [5] and Rydberg atoms [6] havebeen proposed. In this work, we propose quantum simulation experiments of QLMs using dipolarmolecules at fixed positions. ( + ) -d U ( ) quantum link model We consider the massive lattice Schwinger model [7, 8], i.e., the ( + ) -d U ( ) LGT. Withstaggered fermions ψ x and in temporal gauge A =
0, the Hamiltonian takes the form H = − w ∑ x (cid:104) ψ † x U x , x + ψ x + + ψ † x + U † x , x + ψ x (cid:105) + m ∑ x ( − ) x ψ † x ψ x + g ∑ x E x , x + , (2.1)where w > m is the fermion mass, and g is the gauge coupling. Thegauge field A x , x + : = ( A ) x , x + is formulated in the compact form U x , x + = exp ( iagA x , x + ) , so thatthe electric field, which is proportional to the conjugate momentum of A x , x + , has a discrete butinfinite spectrum, E x , x + = − iag ∂∂ A x , x + ∈ Z [9]. In ( + ) dimensions staggered fermions have onlyone Dirac component. Each fermion site has two possible quantum states: occupied and empty.However, the degrees of freedom for the gauge links are infinite. The commutation relations ona link, [ E x , x + , U x , x + ] = U x , x + , [ E x , x + , U † x , x + ] = − U † x , x + , [ U x , x + , U † x , x + ] =
0, are reminiscent ofthe SU ( ) algebra [ S , S + ] = S + , [ S , S − ] = − S − , [ S + , S − ] = S except for the last commutator.Imposing [ U x , x + , U † x , x + ] = E x , x + , instead of [ U x , x + , U † x , x + ] =
0, changes this theory to a quantumlink model. A consequence of this modification is that the spectrum of the electric flux becomesfinite E x , x + = ∈ {− S , − S + , · · · , S − , S } for fixed S = / , , / , · · · , where different values of S correspond to different QLMs. In summary, QLMs are ideal for quantum simulations, becausetheir Hilbert spaces are finite. Here, we focus on S = / S = U ( ) QLMs. In particular,QLMs with S ≥ ( d + ) -dimensional quantum link model can be regarded asa D -theory of the d -dimensional Euclidean LGT through a dimensional reduction in a fictitiousEuclidean dimension [12].Analogous to its LGT counterpart, the Gauss’ law operator of a QLM evaluated at any x , (cid:101) G x = ψ † x ψ x − E x , x + + E x − , x + [( − ) x − ] must satisfy (cid:101) G x | phys (cid:105) = | phys (cid:105) denotes any physical state, | phys (cid:105) ∈ H phys . This Gauss’ law conditionis imposed by starting with a physical initial state | Ψ ( t = ) (cid:105) ∈ H phys , then the fact that (cid:101) G x is con-served, i.e. [ H , (cid:101) G x ] =
0, ensures that the time-evolved state | Ψ ( t ) (cid:105) = e − iHt | Ψ ( t = ) (cid:105) is guaranteedto stay in H phys . 1 LMs from dipolar molecules
Jiayu Shen
In the S = / C and P conjugates of one another. Oscillations betweenthese two directions of the string, referred to as string inversion [11, 6], occur in real-time dynamics.
3. Physics of dipolar molecules
Dipolar molecules are molecules composed of two atoms and are known as a physical platformof quantum magnetism [13]. Under the envisioned experimental conditions, they can be thoughtof as rods with fixed lengths. The rotation of a dipolar molecule is quantum mechanically char-acterized by an orbital angular momentum N of which the eigenstates are | N , m N (cid:105) as usual. Therotor Hamiltonian for the dipolar molecule is H rot = hB rot N where B rot is a rotational constant thatdepends on the molecule species. Here, we use only the N = N = N ≥ N = , | a (cid:105) : = | , (cid:105) , | b (cid:105) : = | , − (cid:105) , | c (cid:105) : = | , (cid:105) , | d (cid:105) : = | , (cid:105) .In the absence of an external field, a dipolar molecule is isotropic. However, a strong uniformmagnetic field can impose a specific direction for the basis of angular momentum eigenstates,namely, the z -axis of spherical harmonics or the quantization axis [14]. When the geometricalconfiguration of molecules already sets a direction, then the system itself becomes anisotropic andthe quantization axis has relative angles with respect to the geometric configuration.Two dipole molecules i and j at fixed positions have an electric dipole-dipole interaction V i j =( πε ) − [ d i · d j − ( d i · ˆ r i j )( d j · ˆ r i j )] / r i j where d i and d j are electric dipoles and r i j is the separationfrom i to j . V i j should be treated as a quantum operator whose matrix elements depend on thequantum states of the two molecules. Operators d i and d j act on two different molecules. Eachmolecule has a dipole selection rule that (cid:104) N (cid:48) , m (cid:48) N | d | N , m N (cid:105) (cid:54) = ∆ N = N (cid:48) − N = ± ∆ m N = m (cid:48) N − m N = , ± N = N = ∑ i , N , m N ε i , N , m N n i , N , m N where N , m N can take a , b , c and d .The molecular positions, external magnetic field, and laser beams are all experimentally ad-justable, yielding tunable parameters which can be used to map the molecular system to the targetQLM Hamiltonian.
4. Mapping the dipolar molecular system to the target QLM
In order to interpret the experimental system as the target theory, a mapping from the dipolarmolecular system to the QLM must be established. In our proposal, each dipolar molecule isidentified with either a site or a link in the QLM, where the accessible molecular states are mappedto one-body states on the sites or links of the QLM. The experimental system is simply a chainof molecules, where, in our case, we choose molecules that are characterized as hard-core bosons[15]. Hardcore bosons have strong repulsive hard-core potentials, so that each state can, at most,2
LMs from dipolar molecules
Jiayu Shen be occupied by one boson. This property enables their use as quantum spins on links, and, in 1 + | a (cid:105) and | c (cid:105) ), where | a (cid:105) is mapped to the occupiedstate and | c (cid:105) is mapped to the empty state. The | b (cid:105) and | d (cid:105) states are made off-resonant by adjustingtheir energies such that they are dynamically inaccessible (or highly suppressed) to the moleculesat the fermion sites.Each link has 2 S + S = / | b (cid:105) is mapped to the S = − / | d (cid:105) is mapped to the S = / | c (cid:105) is made off-resonant. | a (cid:105) is not directlyused in the mapping but it plays a role dynamically and can appear in intermediate states of virtualprocesses that will be discussed in a later section. For an S = | d (cid:105) is mapped to the S = − | b (cid:105) is mapped to the S = | c (cid:105) is mapped to the S = | a (cid:105) again is not useddirectly in the mapping but plays a role in virtual processes.
5. Engineering the effective Hamiltonian
The target QLMs contain one-body terms for the fermion mass ( m ∑ x ( − ) x ψ † x ψ x ) and the elec-tric flux energy ( g ∑ x E x , x + ). In addition, there is a three-body interaction, the fermion hoppingterm, − w ∑ x ( ψ † x U x , x + ψ x + + h . c . ) , which involves two neighboring sites and the link in betweenthem. In the dipolar molecular system the rotor Hamiltonian and the laser lights provide one-bodyterms. However, the molecular Hamiltonian contains only two-body interactions in the form ofdipole-dipole terms. In order to describe the three-body interactions in the QLM Hamiltonian,we must consider higher-order interactions in the molecular system. Here we use second-orderperturbation theory to construct the quasi-degenerate effective Hamiltonian [16] , which acts on aquasi-degenerate subspace of the Hilbert space.We denote the quasi-degenerate subspaces by α , β , γ ... where we are interested in the effectiveHamiltonian for one such subspace, say, α . We use m , l , n ... to denote the states in that subspace,which, in our case, are tensor product states of the individual molecular states, i.e., a sequence of | a (cid:105) , | b (cid:105) , | c (cid:105) and | d (cid:105) whose length equals the number of molecules. Since the states m are quasi-degenerate, their energies, while m dependent, vary by δ E , which is small. However, the energydifferences between two subspaces are large, E m α − E n β ∼ ∆ (cid:29) δ E (independent of the values of m and n ).In our construction of the QLM Hamiltonian, α should target H phys . Each state in α is aconfiguration of the molecular chain that obeys the Gauss’ law after mapping to the QLM states.In the Hamiltonian of the dipolar molecular system, H = H + V , we treat V = ∑ i < j V i j the dipole-dipole interactions, where i and j are molecular indices, as a perturbation. The Hamiltonian H = ∑ i ( H rot , i + H laser , i ) is exactly solvable. The eigenstates of H are tensor products of eachmolecule’s angular momentum eigenstate and serve as the basis of performing the perturbationtheory. Experimentally, H can be adjusted to the desired form by tuning the laser beams. At sec-ond order in perturbation theory, the matrix elements of the quasi-degenerate effective Hamiltonianfor the space H phys = α take the form (cid:104) m | H α eff | n (cid:105) = E m α δ m , n + (cid:104) m , α | V | n , α (cid:105) + ∑ l , γ (cid:54) = α (cid:104) m , α | V | l , γ (cid:105)(cid:104) l , γ | V | n , α (cid:105) (cid:20) E m α − E l γ + E n α − E l γ (cid:21) + · · · . (5.1)3 LMs from dipolar molecules
Jiayu Shen
The zeroth-order terms E m α δ m , n correspond to H projected to the subspace α . The first-orderterms (cid:104) m , α | V | n , α (cid:105) are matrix elements of the dipole-dipole interaction. Any two distinct states in α that satisfy Gauss’ law must differ in at least three molecules. However, because V is a two-bodyinteraction, only matrix elements of V between states that differ in two molecules are nonzero. Wecan therefore neglect the first-order terms in Eq. (5.1).The second-order term in Eq. (5.1) describes the effects on states in α from states outside α .The state | l , γ (cid:105) is an intermediate state that appears only in the virtual process | n , α (cid:105) → | l , γ (cid:105) →| m , α (cid:105) . The strength of this second-order perturbation is determined by the matrix elements of V and the energy differences in H . We can tune the matrix elements of V by varying the inter-molecular separations and adjust the energy differences in H by tuning the lasers. By virtue ofthese tunable, experimental parameters, we can adjust the effective Hamiltonian so that it becomesequivalent to − w ∑ x ( ψ † x U x , x + ψ x + + h . c . ) with a constant hopping parameter w . Self-interactionsalso arise at second-order in perturbation theory, because the virtual process described above caninvolve the same initial and final state, i.e., | n , α (cid:105) → | l , γ (cid:105) → | n , α (cid:105) . However, self-interactionsare diagonal and do not break Gauss’ law. In numerical tests of the dynamics, we find that theydo not significantly affect the important features. We can also use residual freedom in tuning theinter-molecular separations to further suppress the self-interactions. More details regarding theexperimental set-up and the tuning of the parameters are given in [17].
6. Numerical tests
We use the method of exact diagonalization (ED) to test how well the molecular systemdescribes the target theory, where we consider systems of up to two cells in this report. Westart the two systems in the same initial state and time-evolve them with the two Hamiltonians, | Ψ QLM ( t ) (cid:105) = e − iH QLM t | Ψ (cid:105) and | Ψ molecule ( t ) (cid:105) = e − iH molecule t | Ψ (cid:105) . We do not expect the two time evo-lutions to be identical even if we had perfect control over the experimental setup and parameters,because the effective Hamiltonian is based on second-order perturbation theory and includes termsonly through order V / ∆ . Our results for the time evolution of site and link density expectationvalues and comparisons between dynamics of the QLM and molecular systems for S = / S =
1, respectively, are plotted in Figure 1. More details of numerical tests can be found in [17].
7. Summary and outlook
We propose a new approach for analog quantum simulations of quantum link models that usesdipolar molecules. We show that it is possible to simulate S = / S = + LMs from dipolar molecules
Jiayu Shen position t i m e (a) Q L M D e n s i t i e s Site 1Link (1, 2)Site 2Link (2, 3)Site 3Link (3, 4)Site 4Link (4, 5)0 2 4 6 8 10 time M o l e c u l a r D e n s i t i e s (b) position t i m e (c) Q L M D e n s i t i e s Site 1Link (1, 2)Site 2Link (2, 3)Site 3Link (3, 4)Site 4Link (4, 5)0 2 4 6 8 10 time M o l e c u l a r D e n s i t i e s (d) Figure 1: (a) and (b): S = / m = . w in two unit cells with the open boundary condition.Time is in units of 1 / w . (c) and (d): S = m = . × √ w , g / = . × √ w in two unitcells with the open boundary condition. Time is in units of 1 / ( √ w ) . Time evolution of expectationvalues of fermion densities and electric fluxes in the dipolar molecular system after mapping toQLM states for (a) S = / S =
1. Filled circles denote fermion occupied states and hollowcircles denote fermion empty states. Red right-pointing arrows denote right-pointing electric fluxesand blue left-pointing arrows denote left-pointing electric fluxes. Full-sized symbols correspond toextrema of expectation values and non-full-sized symbols denote how close the expecation valuesare to the extrema. In (a), the direction of the electric flux string exhibits an oscillation over time,i.e., string inversion. In (c), the electric flux starts from a left-pointing string, but later on thestring breaks modulo finite size effects [18]. With the parameters we use, the broken state and therecovered state oscillates from one another. Comparisons between the QLM link/site densities anddipolar molecular link/site densities as functions of time for (b) S = / S =
1. The fermiondensity is rescaled to the interval [ , ] . The electric flux ranges from (b) − . . − Acknowledgments
We thank P. Draper, Y. Meurice, and J. R. Stryker for discussions. This work was supportedin part by the U.S. Department of Energy under award No. DE-SC0019213 and by the NationalScience Foundation Graduate Research Fellowship Program under Grant No. DGEâ ˘A ¸S17460475
LMs from dipolar molecules
Jiayu Shen (M.H.). J.S. is grateful for support from the UIUC Department of Physics and the Lattice 2019organizers which enabled him to attend this conference.
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