Many interacting fermions in a 1D harmonic trap: a quantum chemist s perspective

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1 Many interacting fermions in a 1D harmonic trap: a quantum chemist s perspective Tomasz Grining, Michał Tomza, Michał Lesiuk, Pietro Massignan, Maciej Lewenstein, Robert Moszynski Quantum Chemistry Laboratory, Faculty of Chemistry University of Warsaw Quantum Technologies Conference VI,

2 Motivation Selim Jochim s experiments transition between two-, few- and many-body physics pursuit of high accuracy predictions 1 S. Jochim et al. Science, (2013) and S. Joachim et al. Phys. Rev. Lett. 111, (2013) Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

3 Table of contents 1 Theory full configuration interaction coupled cluster 2 Results convergence with the size of one-particle basis convergence with the excitation level in the wavefunction comparison with the experiment approaching the thermodynamic limit with QCh methods 3 Summary Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

4 The Hamiltonian The model Hamiltonian describing N structureless spin-1/2 fermions in a one-dimensional harmonic trap reads Ĥ = 1 2m N i=1 2 x 2 i mω2 N i=1 x 2 i + g δ(x i x j ) (1) i<j where m is the mass of the atom, ω is the frequency of the trap and g is the interaction strength. Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

5 Methods algebraic approximation finite (n b ) number of single-particle functions of the form: ϕ n (x i ) = 1 2 n n! ( ) mω 1/4 e mωx2 i ( ) 2 H n mωxi π (2) full configuration interaction method also known as the exact diagonalization, coupled cluster methods Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

6 Full Configuration Interaction Wave function of the form: Ψ = (1 + Ĉ)Φ = Φ + c α i ρ i e ρ i α i Φ (N!) 2 cα 1...α N ρ 1...ρ N e ρ 1...ρ N α 1...α N Φ (3) were Φ is the Slater determinant, e ρ 1...ρ k α 1...α k are the k-tuple excitation operators and coefficients c α 1...α k ρ 1...ρ k are variationally optimized. E = Ψ ĤΨ Ψ Ψ (4) Cutting the number of excitations resulting in the loss of size-consistency. Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

7 Coupled Cluster Full CC equivalent to full CI, as the wave function is of the form: Ψ = e ˆT Φ, where e ˆT = 1 + Ĉ (5) Possible cutoff in included excitations (preserving size-consistency): ˆT = ˆT 1 + ˆT 2 CCSD, ˆT = ˆT 1 + ˆT 2 + ˆT 3 CCSDT, ˆT = ˆT 1 + ˆT 2 + ˆT 3 + ˆT 4 CCSDTQ. Energy expression (nonvariational): Equations for ˆT : E = Φ e ˆT Ĥe ˆT Φ (6) 0 = e ρ k...ρ k α k...α k Φ e ˆT Ĥe ˆT Φ, k = 1,..., N (7) Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

8 Extrapolation to the infinite basis set typical electronic Schrödinger equation: L 3, where L highest angular momentum present in the basis set this Hamiltonian, rigorously proven two-body behaviour ( 1 E E nb = const + g ) ( 1 + O nb π many-body case: three-terms extrapolation formula E E nb = n b n 3 2 b ). (8) A nb + B n b (9) Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

9 Convergence of the FCI results with the basis set size (E nb -E CBS ) / (E nb -E CBS ) / (a) N=2+2 (b) N=3+3 g=1 g=2 g=5 g= /n 1/2 b Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

10 Convergence of the CC results with the level of excitation E X corr /EFCI (%) corr E X corr /EFCI (%) corr CCSD CCSD(T) CCSDT CCSDTQ (a) N=2+1 (c) N=3+2 (b) N=2+2 (d) N= g g Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

11 Comparison with the previous state-of-the-art n b =20 n b = Fit PT 8.0 E / /g M. Lewenstein et al., Phys. Rev. A (2013) Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

12 Comparison with Jochim s experiments Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

13 Comparison with Jochim s experiments E-E F / E(N) / (a) (b) Exp. FCI g -0.6, Exp. g=-0.6, FCI g=2.8 g=1.14 g=0.36 g=-0.4, FCI N Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

14 What is possible with the coupled cluster methods CCSD CCSD(T) E-E F / N= g Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

15 Approaching the thermodynamic limit, rescaled energies (FCI) 3+3 (FCI) 3+3 (CC) (CC) (CC) E N = E N /E (0) N, γ = πg/ N Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

16 Thermodynamic limit, rescaled many-body contributions - γ=-π/ γ=π/ γ=π/2 Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

17 Thermodynamic limit, BCS pairing gap N = E N (E N+1 + E N 1 )/2, = 8E F γ ( ) π 2 2π 3 exp γ Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18

18 Summary 1. Successful implementation of the quantum chemical methods in quantum gas physics. 2. Pushing the boundary of what is possible in the numerical calculations for 1D Fermi gases. 3. Importance of the extrapolation to the infinite set of one-particle functions. 4. Accurate predictions for the past and future experiments. 5. Unprecedented description of the transition to the thermodynamical limit. Tomasz Grining (QChL Warsaw) Fermions in 1D trap: chemist s view Quantum Tech., / 18