* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download OPENNESS OF MANY-ELECTRON QUANTUM SYSTEMS FROM
Double-slit experiment wikipedia , lookup
Quantum fiction wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Particle in a box wikipedia , lookup
Renormalization group wikipedia , lookup
Quantum computing wikipedia , lookup
Coupled cluster wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Coherent states wikipedia , lookup
Probability amplitude wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Dirac equation wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Wave–particle duality wikipedia , lookup
Quantum decoherence wikipedia , lookup
Wave function wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Quantum key distribution wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum teleportation wikipedia , lookup
Franck–Condon principle wikipedia , lookup
EPR paradox wikipedia , lookup
Hydrogen atom wikipedia , lookup
Rotational–vibrational spectroscopy wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
History of quantum field theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Canonical quantization wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum group wikipedia , lookup
Quantum entanglement wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Ensemble interpretation wikipedia , lookup
OPENNESS OF MANY-ELECTRON QUANTUM SYSTEMS FROM THE GENERALIZED PAULI EXCLUSION PRINCIPLE 1 ROMIT CHAKRABORTY THE MAZZIOTTI GROUP THE UNIVERSITY OF CHICAGO THE PAULI EXCLUSION PRINCIPLE • No two fermions can occupy the same quantum state (Pauli, 1925) • Fermion occupation numbers must lie between 0 and 1 0 ≤ ni ≤ 1 • Comes from the skew-symmetry of the N-fermion wave function (Dirac, Heisenberg, 1926) 1A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963) Wolfgang Pauli 2 numbers. Coleman showed that the Pauli exclusion principle applied to the natural occupation numbers imposes necessary and sufficient ensemble N -representability conditions on the 1-RDM, that the eigenvalues of the 1-RDM (1) must lie between 0 and 1 [6]. While the Pauli conditions of the 1-RDM are complete nberg [4] showed ensemble N -representability conditions, additional consymmetry of the ditions on the 1-RDM are required to ensure that it arises • A general from N-fermion pure state outermatrix product of the Nthe integration of is anexpressible N -fermion by purethe density eneral N -fermion fermion wave function ermion ensemble N D(1, 2, .., N ; 1̄, 2̄, .., N̄ ) = Ψ(1, 2, .., N )Ψ∗(1̄, 2̄, .., N̄) (4) , .., N )Ψ∗i (1̄, 2̄, .., N̄) where the N D can be spectrally resolved in terms of the single N -fermion wave function Ψ(1̄, 2̄, .., N̄ ). These (2) additional 1-RDM constraints are known as pure N at sum to unity, representability conditions or generalized Pauli condictions, and each tions [6–11]. The pure N -representability conditions coordinates of a of the 1-RDM depend only on its natural occupation ensemble density numbers [6], and hence, we will use the terms N ns save one yields representability of the 1-RDM and N -representability of (1-RDM) the 1-RDM spectrum, interchangeably. Smith showed that pairwise degeneracy of occupation numbers are suffi)d2d3..dN. (3) cient to ensure pure N -representability of the 1-RDM [7]. Furthermore, he showed that such degeneracy occurs nat1-RDM must be urally in even-N quantum systems with time-reversal positive semidefsymmetry. In 1972 Borland and Dennis reported pure obey additional 1A. J. Coleman, Rev. Mod. N -representability conditions for active space of three Phys. 35, 668 (1963) ed otherwise, the upation numbers en 0 and 1 DEFINITIONS 3 3 arXiv:1404.5223v1 [physi numbers. Coleman showed that the atomic Pauli exclusion prin-[2], by Pauli in 1925 to explain transitions ed otherwise, the tulated ues of the 1-R ciple appliedplays to thea natural numbers imposes principle key roleoccupation in predicting electronic conupation numbers this necessaryof and sufficient N -representability con-the numbers. Cole en 0 and 1 figurations atoms and ensemble molecules. Stated otherwise, ditions on the says 1-RDM, eigenvalues of the 1-RDM ciple applied t Pauli principle thatthat thethe fermion occupation numbers (1) lie betweensystem 0 and 1must [6]. lie between 0 and 1 necessary and λi must of a quantum While the Pauli conditions of the 1-RDM are complete ditions on the nberg [4] showed 0 ≤ λi conditions, ≤ 1. ensemble N -representability additional con-(1) must lie betw symmetry of the ditions on the 1-RDM are required to ensure that it arises the P Subsequent work Dirac and Heisenberg [4]matrix showed of the While • A general N-fermion pure by state by outer product Nfrom the integration of is anexpressible N[3] -fermion purethe density eneral N -fermion ensemble N -r fermionthat wave function this principle arises from the antisymmetry of the ermion ensemble ditions on the fermion wave function. N ∗ D(1, 2, .., N ; 1̄, 2̄, .., N̄ ) = Ψ(1, 2, .., N )Ψ (1̄, 2̄, .., N̄) from the integ As discussed by von Neumann [5], a general N -fermion (4) , .., N )Ψ• ∗i (A 1̄, general 2̄, .., N̄) quantum state expressible byexpressible anresolved N -fermion ensemble N N-fermion quantum system is by N-fermion ensemble where the Dis can be spectrally in an terms of N matrix density density matrix D(1, 2, .., N the single N -fermion wave function Ψ(1̄, 2̄, .., N̄ ). These (2) ! are known as pure N additional 1-RDM constraints N at sum to unity, D(1, 2, .., N ; 1̄, 2̄, .., N̄) = N )Ψ∗icondi(1̄, 2̄, .., N̄) where the N D i Ψi (1, 2, ..,Pauli representability conditions or w generalized ctions, and each tions [6–11]. The pure N i-representability conditions the single N -f coordinates of a (2) of the 1-RDM depend only on its natural occupation additional 1-R ensemble density where wi are non-negative weights that sum to unity, numbers [6], and hence, we will use the terms N representabilit ns save one yields Ψ (1, 2, .., N ) are N -fermion wave functions, and each i representability of the 1-RDM and N -representability of tions [6–11]. (1-RDM) number denotes the spatial and spin coordinates of a the 1-RDM spectrum, interchangeably. Smith showed of the 1-RDM fermion. Integration of the N -fermionnumbers ensemble density that pairwise degeneracy of occupation are suffi)d2d3..dN. (3) numbers [6], cient over to ensure pure N -representability of thesave 1-RDM [7]. matrix the coordinates of all fermions one yields representabili Furthermore, hereduced showed that such matrix degeneracy occurs natthe one-fermion density (1-RDM) 1-RDM must be the 1-RDM s urally in even-N quantum systems with time-reversal " positive semidefthat pairwise symmetry. Borland Dennis reported pure(3) 1 N D(1, 2, .., Nand ; 1̄, 2, .., N )d2d3..dN. D(1; 1̄) = In 1972 obey additional 1A. J. Coleman, Rev. Mod. N cient to4ensure -representability conditions for active space of three Phys. 35, 668 (1963) DEFINITIONS 3 arXiv:1404.5223v1 [p arXiv:1404.5223v1 [physi numbers. Coleman showed that the atomic Pauli exclusion prin1-RDM kn ed otherwise, the tulated appliedare to th Pauli in 1925 explain transitions Pauli by principle says thatto the fermion occupation numbers[2], ciple ciple applied to the natural occupation numbers imposes ues of and the suffi 1-R upation numbers this playssystem a key role predicting con- necessary λiprinciple of a quantum mustinlie between 0electronic and 1 necessary and sufficient ensemble N -representability conen 0 and 1 ditions on the Cole 1-R numbers. figurations of atoms and molecules. Stated otherwise, the ditions on the 1-RDM, that the of the 1-RDM 0≤λ 1. (1) i ≤eigenvalues must lie applied between t0 ciple Pauli principle says that the fermion occupation numbers (1) must lie between 0 and 1 [6]. While the Pauli necessary and work by Dirac [3] and Heisenberg [4] showed λi Subsequent of a quantum system must lie between 0 and 1 While Pauli conditions of the are complete nberg [4] showed ensemble -repre ditionsNon the that thisthe principle arises from the 1-RDM antisymmetry of the ensemble N -representability conditions, additional consymmetry of the thebetw 1-R (1) ditions fermion wave function.0 ≤ λi ≤ 1. mustonlie ditions on the 1-RDM are required to ensure that it arises from the integratio As discussed bystate von Neumann [5], abygeneral N -fermion • A general N-fermion pure is expressible the outer product of Nthe P workisbyexpressible Dirac andanHeisenberg [4]matrix showed the While from the integration of an N[3] -fermion pure density eneral N -fermion Subsequent quantum state by N -fermion ensemble fermion wave function N -r this principle arises from the antisymmetry of the Nensemble ermion ensemble that density matrix D(1, 2, .., N ; 1̄, ditions on the N ∗ fermion wave function. ! D(1, 2, .., N ; 1̄, 2̄, .., N̄ ) = Ψ(1, 2, .., N )Ψ (1̄, 2̄,∗.., N̄) N fromthe theNinteg wi[5], Ψi (1, .., N )ΨN 1̄, 2̄,(4) .., N̄ ) where 2, .., N ;by 1̄, 2̄, .., N̄ )= AsD(1, discussed von Neumann a 2, general i (-fermion D ca ∗ , .., N )Ψ• i (A 1̄, general 2̄, .., N̄) N i N-fermion quantum system is by N-fermion ensemble quantum state expressible byexpressible anresolved N -fermion ensemble where the Dis can be spectrally in an terms of the single N -ferm (2) density density matrix N the single N -fermion wave function Ψ(1̄, 2̄, .., N̄ ). These matrix additional D(1,1-RDM 2, .., N (2) where wi are non-negative weights that sum to unity, additional 1-RDM constraints are known as pure N representability co !wave functions, and each at sum to unity, N Ψi (1, 2, .., N ) are N -fermion ∗ Ψi (1, 2, ..,Pauli N )Ψicondi(1̄, 2̄, .., N̄) D(1, 2, .., N ; 1̄, 2̄,conditions .., N̄) = or w representability generalized tions [6–11]. N where the Th D number denotes the spatial andi spin coordinates of a ctions, and each of the 1-RDM de tions [6–11]. The pure N -representability conditions fermion. Integration of the iN -fermion ensemble density coordinates of a the single N -f (2) numbers [6], and of the 1-RDM depend only on its natural occupation matrix over the coordinates of all fermions save one yields save one ensemble• density additional 1-R Integration of the N-fermion wave function over all co-ordinates representability of numbers [6], and hence, we will use the terms N where w are non-negative weights that sum to unity, i the one-fermion reduceddensity densitymatrix matrixor(1-RDM) ns save oneyields yieldsthe one-electron reduced the 1-RDM representabilit representability 1-RDM wave and Nfunctions, -representability Ψi (1, 2, .., N ) are -fermion and of each the 1-RDM spect " ofNthe (1-RDM) tions [6–11]. the 1 1-RDM interchangeably. Smith showed N number denotes the spatial coordinates of a that pairwise dege D(1, 2, .., Nand ; 1̄, 2,spin .., N )d2d3..dN . (3) D(1; 1̄) =spectrum, of tothe 1-RDM cient ensure pur that pairwise degeneracy of occupation numbers are suffi)d2d3..dN. (3) fermion. Integration of the N -fermion ensemble density Furthermore, numbers he [6],sh cient to ensure pure N -representability thesave 1-RDM [7]. matrix coordinates ofmatrix, all fermions one Like over the Nthe -fermion density theof1-RDM mustyields be urally in even-N representabili Furthermore, showed that such degeneracy occurs natHermitian,he(ii) normalized, and (iii) positive semidefone-fermion reduced density matrix (1-RDM) 1-RDM must be the(i) symmetry. In 197 urally even-N quantum withobey time-reversal the 1-RDM s inite. in However, 1-RDMsystems must also additional " the positive semidefN -representability symmetry. In 1972 Borland and Dennis reported pure that pairwise 1 N obey additional D(1, 2, .., N ; 1̄, 2, .., N )d2d3..dN. (3) D(1; 1̄) = fermions in5 six or 1A. J. Coleman, Rev. Mod. N -representability conditions for active space of three Phys. 35, 668 (1963) cient to ensure 3 of numerical calcu DEFINITIONS arXiv:1404.5223v1 [p arXiv:1404.5223v1 [physi numbers. Coleman showed that the atomic Pauli exclusion prin-[2], ciple 1-RDM aretokn ed otherwise, the tulated applied t Pauli in 1925 tothe explain transitions Pauliby principle says that fermion occupation numbers ciple applied to the natural occupation numbers imposes ues of the upation numbers thisλprinciple and 1-R suffi plays system a key role in lie predicting con- necessary must between 0electronic and 1 i of a quantum necessary and sufficient ensemble N -representability conen 0 and 1 ditions on theCole 1-R numbers. figurations of atoms and molecules. Stated otherwise, the ditions on the 1-RDM, that eigenvalues of the 1-RDM 0 ≤the λi ≤ 1. (1) must lieapplied betweent ciple Pauli principle says that the fermion occupation numbers (1) must lie between 0 and 1 [6]. While the Paul necessary and work by Dirac [3]lie andbetween Heisenberg [4] showed λi ofSubsequent a quantum system must 0 and 1 While theprinciple Pauli conditions of the are complete nberg [4] showed ensemble ditions N on-repr the that this arises from the1-RDM antisymmetry of the ensemble N -representability conditions, additional consymmetry of the 1-R 0 ≤ λi ≤ 1. (1) ditions fermion wave function. muston liethe betw ditions on the 1-RDM are required to ensure that it arises from the integrat As discussed by vonisNeumann [5],by a general N -fermion • A general N-fermion pure state expressible the outer product of Nthe P work by Dirac and Heisenberg showed the While from the integration of an N[3] -fermion pure density[4] matrix eneral N -fermion Subsequent quantum state is expressible by an N -fermion ensemble fermion wave function ensemble N -r this principle N ermion ensemble thatdensity matrix arises from the antisymmetry of the D(1, 2, .., N ; 1̄ ditions on the N ∗ fermion wave function. ! D(1, 2, .., N ; 1̄, 2̄, .., N̄ ) = Ψ(1, 2, .., N )Ψ (1̄, 2̄, .., N̄) ∗ N fromthe the Ninteg w[5], 2, .., N )Ψ (1̄, 2̄, 2, .., Nby ; 1̄, von 2̄, .., Neumann N̄ ) = i Ψi (1, As D(1, discussed a general Ni -fermion (4).., N̄ ) where D c ∗ , .., N )Ψ• i (A 1̄, general 2̄, .., N̄) N i N-fermion quantum system is by N-fermion ensemble quantum state expressible byexpressible anresolved N -fermion ensemble where the Dis can be spectrally in an terms of the single N -ferm (2) density density matrix N the single N -fermion wave function Ψ(1̄, 2̄, .., N̄ ). These matrix additional D(1, 1-RDM 2, .., N (2) where wi are non-negative weights that sum to unity, additional 1-RDM constraints are known as pure N representability c ! wave functions, and each at sum to unity, N Ψi (1, 2, .., N ) are N -fermion ∗ N )Ψicondi(1̄, 2̄, .., N̄)tions D(1, 2, .., N ; 1̄, 2̄,conditions .., N̄) = or w representability generalized [6–11]. i Ψi (1, 2, ..,Pauli where the NTD number denotes the spatial and spin coordinates of a ctions, and each of the 1-RDM d tions [6–11]. The pure N -representability conditions fermion. Integration of thei N -fermion ensemble density coordinates of a the single N -f (2) numbers [6], an ofmatrix the 1-RDM depend only on its natural occupation over the coordinates of all fermions save one yields save one ensemble• density additional 1-R Integration of the N-fermion wave function over all co-ordinates representability o numbers [6], and hence, we will use the terms N where w are non-negative weights that sum to unity, i the one-fermion reduced densitymatrix matrix (1-RDM) ns save oneyields yieldsthe one-electron reduced density or the 1-RDM representabilit representability 1-RDM wave and Nfunctions, -representability Ψi (1, 2, .., N ) are"ofNthe -fermion and of each the 1-RDM spec (1-RDM) tions [6–11]. the 11-RDM Smith showed N spatial number denotes and spin of a that pairwise deg D(1,interchangeably. 2, .., N ; 1̄, 2, .., N coordinates )d2d3..dN . (3) D(1; 1̄) spectrum, = the of the 1-RDM cient to ensure pu that pairwise degeneracy of occupation numbers are suffi)d2d3..dN. (3) fermion. Integration of the N -fermion ensemble density Furthermore, he s numbers [6], cient to ensure pure Ndensity -representability of 1-RDM thesave 1-RDM [7].be matrix the coordinates all fermions one Likeover the N -fermion matrix, the must • Eigenfunctions of the 1-RDM are theof natural orbitals while its yields eigenvalues are urally in even-N representabili Furthermore, he(ii) showed that such degeneracy occurs nat(i) Hermitian, normalized, (iii) positive semidefnatural occupation numbers one-fermion reduced densityand matrix (1-RDM) 1-RDM must be the symmetry. In 19 urally even-N quantum systems withobey time-reversal the 1-RDM s inite. inHowever, the 1-RDM must also additional " positive semidefN -representabilit symmetry. In 1972 Borland and Dennis reported pure that pairwise 1 N obey additional D(1, 2, .., N ; 1̄, 2, .., N )d2d3..dN. (3) D(1; 1̄) = fermions in 6 six or 1A. J. Coleman, Rev. Mod. N -representability conditions for active space of three Phys. 35, 668 (1963) cient to ensure 3 of numerical calc DEFINITIONS N-REPRESENTABILITY Ensemble 1-RDM is derivable from the Nfermion ensemble density matrix The Pauli Exclusion Principle is necessary and sufficient for ensemble N-representability of the 1-RDM 0 ≤ ni ≤ 1 (Coleman, 1963) 1A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963) 7 N-REPRESENTABILITY Ensemble Pure 1-RDM is derivable from the Nfermion ensemble density matrix 1-RDM arises from the pure density matrix The Pauli Exclusion Principle is necessary and sufficient for ensemble N-representability of the 1-RDM Generalized Pauli Conditions are pure N-representability conditions on the 1-RDM 0 ≤ ni ≤ 1 (Coleman, 1963) 1A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963) More stringent and complicated than the Pauli condition Defines a convex polytope in the space of natural occupations 8 GENERALIZED PAULI CONSTRAINTS Constraints Feasible Set ni ≥ ni+1,∀ i ∈ {1..(r −1)} Pauli 0 ≤ ni ≤ 1 r ∑n = N i i=1 defines convex set 1E(N,r ) 2R. 3A. E. Borland and K. Dennis, J. Phys. B 5, 7 (1972) A. Klyachko, J. Phys. Conf. Ser. 36, 72 (2006) 1E (3,6) 9 GENERALIZED PAULI CONSTRAINTS Constraints Feasible Set ni ≥ ni+1,∀ i ∈ {1..(r −1)} Pauli 0 ≤ ni ≤ 1 r ∑n = N i i=1 defines convex set 1E(N,r ) Generalized Pauli r 0 ≤ ni ≤ 1, ∑ ni = N, i=1 r ij i j i=1 defines convex set 1P(N,r ) 3A. 5 E. Borland and K. Dennis, J. Phys. B 5, 7 (1972) A. Klyachko, J. Phys. Conf. Ser. 36, 72 (2006) (3,6) ni ≥ ni+1,∀ i ∈ {1..(r −1)} ∑A n ≥ b 2R. 1E 1P (3,6) Feasible sets in the Borland Dennis setting 10 FEASIBLE SETS Occupation numbers 1.0000 1.0000 0.9000 0.9996 0.6000 0.9996 0.3000 0.0004 0.1000 0.0004 0.1000 0.0000 Both sum to 3 Both obey the Pauli principle Which one of these sets come from the wave function? 11 GENERALIZED PAULI EXCLUSION Occupation numbers 1.0000 1.0000 0.9000 0.9996 0.6000 0.9996 0.3000 0.0004 0.1000 0.0004 0.1000 0.0000 Spectrum of occupations in Li 12 OPEN QUANTUM SYSTEMS • Generalized Pauli constraints are necessary for conditions for a pure quantum state A closed system 13 OPEN QUANTUM SYSTEMS • Generalized Pauli constraints are necessary for conditions for pure quantum states An open system • Violation of Generalized Pauli conditions provide a sufficient condition for the openness of a many-electron quantum system • These conditions can be used to study the interaction of a quantum system with its environment 14 SUFFICIENT CONDITIONS FOR OPENNESS 6R. • Spectra can be represented by an N-fermion wavefunction (pure) if and only if they lie inside the pure set 1P(N,r) • Spectra can be represented by an N-fermion density matrix if and only if they lie inside the ensemble set 1E(N,r) • Spectra in the outside the pure set but inside the ensemble set (1E(N,r)\1P(N,r)) cannot be represented by an N-fermion wavefunction • Violation of Generalized Pauli Conditions are sufficient to certify openness of a many-electron quantum system form sole knowledge of the 1-RDM Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) 15 DEGREE OF OPENNESS 6R. • The nature and extent of spectral deviation from Generalized Pauli condtions can be used to quantify the degree of openness in an interacting quantum system • We use a euclidean metric (pure distance) to quantify deviation from the facets of the pure set (polytope) defined by the Generalized Pauli conditions Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) 16 DEGREE OF OPENNESS 6R. • The nature and extent of spectral deviation from Generalized Pauli condtions can be used to quantify the degree of openness in an interacting quantum system • We use a euclidean metric (pure distance) to quantify deviation from the facets of the pure set (polytope) defined by the Generalized Pauli conditions • Pure distance is written as the Sequential Quadratic Program: Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) 17 DEGREE OF OPENNESS • The nature and extent of spectral deviation from Generalized Pauli condtions can be used to quantify the degree of openness in an interacting quantum system • We use a euclidean metric (pure distance) to quantify deviation from the facets of the pure set (polytope) defined by the Generalized Pauli conditions • Pure distance is written as the Sequential Quadratic Program: ! ! Pure Distance = min min σ n −p ! j p r such that ∑ pi = N i=1 pi ≥ pi+1∀ i ∈ [1, r −1] 0 ≤ pi ≤ 1∀ i ∈ [1, r] r ∑A n = b ji 6R. i=1 i j Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) 18 DEGREE OF OPENNESS • The nature and extent of spectral deviation from Generalized Pauli conditions can be used to quantify the degree of openness in an interacting quantum system • We use a euclidean metric (pure distance) to quantify deviation from the facets of the pure set (polytope) defined by the Generalized Pauli conditions • Pure distance is written as the Sequential Quadratic Program: ! ! Pure Distance = min min σ n −p ! j p r such that ∑ pi = N i=1 pi ≥ pi+1∀ i ∈ [1, r −1] 0 ≤ pi ≤ 1∀ i ∈ [1, r] r ∑A n = b ji 6R. i=1 i j Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) Ø σ is positive (negative) when the spectrum is inside (outside) the pure set Ø Minimum distance to the facets of the ensemble set 1E(N,r) and Slater point calculated for comparison Ø Pure ≤ Ensemble ≤ Slater by definition Ø Pinned (quasi-pinned) if constraints are saturated (close to being saturated) 19 ting antenThe computations presented reveal methods that the functional entanglement within chromophores. Two here advanced in ules, called subsystems achieve their efficiencies by a quantum mechanism 1 sity-matrix group [17] and including two-electron reducedThe renormalization light. similar to that of the full system the roles of entanglement and environmental noise. We assess each subsystem’s mplexes to 12,13 as 8, 19],2 have recently shown that networks of conjugated bonds based on entanglement by a global entanglement measure ency. For 14!19 of these chrothe and squared Euclideanare distance. a classical acene sheets [20], chlorophyll associatedThe withstudy polyradical mophore subsystems gives more information about the role of y transfers adequately described without a strongly correlated each chromophore in the energy transfermany-electron in the whole FMO ion until it 20!31 This more in-depth understanding can provide complex. • Environmental interactions in photosynthetic energy transfer perimental insight to other antennae complexes and ultimately m coherent • The Fenna-Mathews-Olson complex (FMO) be applied to create synthetic solar cells that rival the efficiency of nature. Matthews! We consider a single monomer of the FMO complex with M acteria has Figurechromophore 1. Populations ofsites chromophores and the reaction center for (A) the subsystem 123 and (B) the full FMO system 1234 with the 1%3 Hamiltonian electronic of the FMO complex transfers energy more efficiently from chromophore 1 to the reaction center (sink) than the full FMO syste es and over M þ ! ! þ þ ! ^elements pω σl þ σl Þ ð1Þ ¼ j σdensity j, l ðσ Fkl are theH of the matrix pv D in thej basis set σ ofj wave stronger than any classical correlation.” Par j σj þ j ¼ 1 j6 ¼ l harvesting functions {Ψk}, and L^ is the Lindblad operator, which accounts system become entangled when the total dens y we mean for interactions +of the!M chromophores with the environment. At system cannot be expressed as a product of the where σ (σ ) creates (annihilates) a single excitation on chroj j ple efficient t = 0 we initialize the density matrix with a single excitation for the parts.12,13,34 Global entanglement12,13,3 the mophore j. The of each chromophore is pω by the squared Euclidean distance be on either site site 1 or energy 6. Because the Hamiltonian does notj, whilemeasured tion center. (exciton) . coupling constant between the pair of chromophores j and l is pv j,l matrix F and the nearest classical density matr mix singly excited wave functions with other wave functions, the set nae’s chro• {Ψ Dynamics in the FMO complex The time evolution of the system’s density matrix is governed k} only includes M wave functions formed from sequentially antennae by thea single quantum Liouville equation considering excitation on each of the M chromophores. σðFÞ ¼ ∑ jFij % ξij j2 where each or correlated Each theofseven i, j is theofsum three chromophores operators that in The chromophores. Lindblad operator32,30 d dephasing,i dissipation, a chromo- account for ^in D' ^and to the(left), reaction center D electron ¼ ! ½H, þ L ðDÞloss ally treated as a single a two-state model and yet the ð2Þ monomer dt p On the basis of M wave functions, in which ea (sink) or chromoted from chlorophyll molecules with many strongly correlated electrons has a single excitation on one of the M ch where is the density matrix ^diss ðDÞ ^sink ðDÞ ^D þ L ð4Þ ¼L deph ðDÞ þ L re 3, which L^ðDÞ measure corresponds to the sum of the sq ch of the chromophores by a correlated N -electron model by Lipkin, diagonal elements of F, or the sum of the re we show where D ¼ Fkl jΨk æÆΨl j ð3Þ concurrences. The squared Euclidean distance there exist k, l tration by K. ^Naftchi-Ardebili, The University of Chicago, 2011. Used 20 chrom if the excitations (excitons) on the ð5Þ ncies close P.LRebentrost, deph ðDÞ ¼ α ∑ 2ÆkjDjkæjkæÆkj % fjkæÆkj, Dg 5M. Mohseni, S. Lloyd, and A. Aspuru-Guzik, J. Chem. Phys. 129, 174106 (2008) entangled. Both the squared Euclidean distance k PHOTOSYNTHETIC ENERGY TRANSFER ∑ ∑ ∑ TRAJECTORY OF THE FMO • The three chromophore subsystem has similar efficiency to the full 7-chromophore network • We are able to visualize the time dependent trajectory in the space of natural orbital occupations Closed Trajectory in population space with femtosecond resolution. Points in green lie inside the pure set 6R. Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) 21 CHARACTERIZED TRAJECTORY IN THE FMO • Environmental noise increases the size of the set of 1-RDMs accessible to a quantum system facilitating in the transfer of excitation to the reaction center Open Trajectory in population space Points in green lie inside the pure set and points in red are outside 6R. Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) 22 Table 2 reports not only the minimum distance to a facet of the pure 1-RDM set (pure distance) but also the harmonic, geometric, and arithmetic means of the Euclidean distances to each facet of the set. While the harmonic mean is similar to the minimum distance, the geometric and arithmetic means provide information about the distance to the majority of fac- seven excited sta ence of electron seven excited sta RDMs of these st The pure distanc states 3 and 6 are and 6 have the g • Assuming single excitations, the 7x7 matrix of site and coupling energies, on diagonalization, gives seven stationary state wavefunctions have the highest Table 3. Minimum distances to the pure, ensemble, and Slater boundaexcited states. Int ries for the ground and excited states of the seven-site FMO complex are • We use theshown. euclidean distance metric to study the nature of pinning in the FMO state 2, and state and 6 have the Euclidean distances degenerate pairs. 21 FMO State Energy (cm ) Pure Ensemble Slater in the FMO comp plexity of excited Ground 0 0.00 0.0000 0.0000 0.0000 Excited 1 223.74 0.0000 0.0005 0.8876 quasi-pinned, or b STATIONARY STATES OF THE FMO 2 101.97 0.0000 0.0048 3 120.96 0.0020 0.0085 4 268.37 0.0000 0.0141 5 307.13 0.0000 0.0001 6 332.00 0.0003 0.0003 7 513.32 0.0000 0.0003 Ø Ground state corresponds to a Slater determinant 0.8528 0.9958 0.9246 0.9320 0.9784 0.8797 Because the ground state in this model is a Slater determinant, its Slater, ensemble, and pure distances vanish. In contrast, the Slater distances of all seven excited states are nonzero (! 0:85), reflecting the presence of electron correlation. The ensemble distances of all seven 8R. Chakraborty, D. A. Mazziotti Int. J. Quantum Chem. 116, 784-790 (2016) excited states are also nonzero. The pure distances, however, for all the Conclusions The role of the g chemistry of excit tions to three-, fo ecules. While th boundary of the excited-state 2-RD 23 ensemble N-repre Table 2 reports not only the minimum distance to a facet of the pure 1-RDM 1-RDM set (pure (pure distance) but but also the the harmonic, the the pure pure 1-RDM set set (pure distance) distance) but also also the harmonic, harmonic, geometric, and arithmetic means of the Euclidean distances to geometric, geometric, and and arithmetic arithmetic means means of of the Euclidean distances to each facet of of the set. set. While the the harmonic mean is similar to each each facet facet of the the set. While While the harmonic harmonic mean is similar to the minimum distance, the geometric and arithmetic means the the minimum minimum distance, distance, the the geometric geometric and arithmetic means provide information information about about the the distance distance to to the the majority provide provide information about the distance majority of of facfac- seven excited sta ence of electron ence of electron seven excited sta seven excited sta RDMs of these st RDMs of these st The pure distance The pure distanc states states 33 and and 66 are are and and 66 have have the the gg • Assuming single excitations, the 7x7 matrix of site and coupling energies, on have diagonalization, gives sevendistances stationary state wavefunctions have the the highest highest Table 3. 3. Minimum Minimum to the the pure, ensemble, and Slater Table pure, Table 3. Minimum distances distances to to the pure, ensemble, ensemble, and Slater boundaboundaexcited ries for for the the ground ground and and excited excited states states of of the the seven-site seven-site FMO ries excited states. states. Int Int ries for the ground and excited states of the seven-site FMO complex complex are are • We use theshown. euclidean distance metric to study the nature of pinning in the FMO shown. state shown. state 2, 2, and and state state and and 66 have have the the h Euclidean Euclidean distances Euclidean distances degenerate degenerate pairs. pairs. 21 21 21 FMO State Energy (cm ) Pure Ensemble Slater FMO State Energy Pure in FMO State Energy (cm (cm )) Pure Ensemble Slater in the the FMO FMO comp comp plexity Ground 0.00 0.0000 Ground 000 0.00 0.0000 0.0000 0.0000 plexity of of excited excited Ground 0.00 0.0000 0.0000 0.0000 Excited 1 223.74 0.0000 0.0005 0.8876 Excited 11 223.74 0.0000 Excited 223.74 0.0000 0.0005 0.8876 quasi-pinned, quasi-pinned, or or bb STATIONARY STATES OF THE FMO 222 33 44 55 66 77 101.97 101.97 101.97 120.96 120.96 268.37 268.37 307.13 307.13 332.00 332.00 513.32 513.32 0.0000 0.0000 0.0000 0.0020 0.0020 0.0000 0.0000 0.0000 0.0000 0.0003 0.0003 0.0000 0.0000 0.0048 0.0048 0.0085 0.0141 0.0001 0.0003 0.0003 0.8528 0.8528 0.9958 0.9246 0.9320 0.9784 0.8797 Ø Ground state corresponds to a Slater determinant Ø Excited states are correlated Because Because the the ground ground state state in in this this model model is is aa Slater determinant, its Slater, Slater, ensemble, ensemble, and and pure pure distances distances vanish. vanish. In contrast, the Slater distances distances of of all all seven seven excited excited states states are are nonzero nonzero (! 0:85), reflecting the presence of of electron electron correlation. correlation. The The ensemble ensemble distances of all seven 8R. Chakraborty, D.presence A. Mazziotti Int. J. Quantum Chem. 116, 784-790 (2016) excited excited states states are are also also nonzero. nonzero. The The pure pure distances, distances, however, for all the Conclusions Conclusions The role role of of the the gg chemistry of excit tions to to three-, three-, fo fo ecules. While While th th boundary of the excited-state 2-RD 2-RD excited-state 24 ensemble N-repre N-repre ensemble Table Table 2 2 reports reports not not only only the the minimum minimum distance distance to to aa facet facet of of the pure 1-RDM set (pure distance) distance) but but also also the the harmonic, harmonic, the pure 1-RDM set (pure distance) but also the harmonic, geometric, and arithmetic means means of the the Euclidean distances distances to geometric, and arithmetic means of of the Euclidean Euclidean distances to to each facet of the set. While the harmonic mean is similar to each facet of the set. While the the harmonic harmonic mean mean is is similar similar to to the geometric and and arithmetic arithmetic means means the minimum minimum distance, distance, the the geometric geometric and arithmetic means provide distance to to the the majority majority of of facfacprovide information information about about the the distance distance to the majority of fac- seven seven excited excited ss ence of electro electr ence ence of of electr seven excited seven seven excited excited RDMs of these these RDMs RDMs of of these The pure pure distan dista The The pure dista states 333and and666 states states and and 666 have have the th and and have the • Assuming single excitations, the 7x7 matrix of site and coupling energies, on have the the high high have have the high Table Minimum the pure, pure, ensemble, and and Slater Slater boundaboundaTable to ensemble, diagonalization, sevendistances stationary state wavefunctions Table 3. 3.gives Minimum distances to the the pure, ensemble, and Slater boundaexcited states. states. excited ries states of of the the seven-site seven-site FMO FMO complex complex are are ries excited states. ries for for the the ground ground and and excited excited states states of the seven-site FMO complex are shown. shown. state 2,2, and andsta st state • We use theshown. euclidean distance metric to study the nature of pinning in the FMO state 2, and sta and 666 have have th th and and have th Euclidean distances Euclidean distances Euclidean distances degenerate pai pa degenerate degenerate pai 21 21 FMO State Pure Ensemble Slater FMO Energy Pure Ensemble Slater in the the FMO FMO co co in FMO State Energy (cm (cm21))) Pure Ensemble Slater in the FMO co plexity of of excit exci plexity Ground 00 0.0000 0.0000 0.0000 Ground 0.00 0.0000 0.0000 0.0000 plexity of excit Ground 0.00 0.0000 0.0000 0.0000 Excited 11 0.0000 0.0005 0.8876 Excited 223.74 0.0000 0.0005 0.8876 Excited 223.74 0.0000 0.0005 0.8876 quasi-pinned,oo quasi-pinned, STATIONARY STATES OF THE FMO 22 33 4 5 6 7 101.97 101.97 120.96 268.37 307.13 332.00 513.32 0.0000 0.0000 0.0000 0.0020 0.0020 0.0000 0.0000 0.0000 0.0000 0.0003 0.0003 0.0000 0.0000 0.0048 0.0048 0.0048 0.0085 0.0085 0.0141 0.0141 0.0001 0.0001 0.0003 0.0003 0.0003 0.0003 0.8528 0.8528 0.8528 0.9958 0.9958 0.9246 0.9246 0.9320 0.9320 0.9784 0.9784 0.8797 0.8797 Because state in this is aa Slater Because the this model isdeterminant Slater determinant, determinant, its its the ground Ø Ground state corresponds to model a Slater Slater, distances Slater, ensemble, distances vanish. vanish. In In contrast, contrast, the the Slater Slater ensemble, Ø Excited statesand arepure correlated distances of allensemble seven excited states distances states are are nonzero nonzero (! (! 0:85), 0:85), reflecting reflecting the the Ø Nonzero distances presence of of electron correlation. presence correlation. The The ensemble ensemble distances distances of of all all seven seven states are also nonzero. The excited states excited The pure pure distances, distances, however, however, for for all all the the statesInt.except excited statesChem. 3 and are to 8R. Chakraborty, D. excited states and are zero zero to arbitrary arbitrary digits digits of of A. Mazziotti J. Quantum 116,66784-790 (2016) precision. precision. Conclusion Conclusions The The role role of of the th chemistry chemistryof ofex e tions tions to to three-, threeecules. ecules. While While boundary boundary of of t excited-state excited-state 22 ensemble ensembleN-rep N-re 25 RDMs RDMs to to lie lie ini the pure 1-RDM set (pure distance) but also the harmonic, geometric, and arithmetic means of the Euclidean distances to each facet of the set. While the harmonic mean is similar to the minimum distance, the geometric and arithmetic means provide information about the distance to the majority of fac- ence of electron seven excited sta RDMs of these s The pure distanc states 3 and 6 are and 6 have the g have the highes • Assuming single excitations, 7x7 matrix site and coupling energies, on Table 3. Minimumthe distances to theofpure, ensemble, and Slater boundadiagonalization, gives seven stationary state wavefunctions excited states. In ries for the ground and excited states of the seven-site FMO complex are shown. state 2, and state • We use the euclidean distance metric to study the nature of pinning in the FMO and 6 have the Euclidean distances degenerate pairs. 21 FMO State Energy (cm ) Pure Ensemble Slater in the FMO comp plexity of excited Ground 0 0.00 0.0000 0.0000 0.0000 Excited 1 223.74 0.0000 0.0005 0.8876 quasi-pinned, or b STATIONARY STATES OF THE FMO 2 3 4 5 6 7 101.97 120.96 268.37 307.13 332.00 513.32 0.0000 0.0020 0.0000 0.0000 0.0003 0.0000 0.0048 0.0085 0.0141 0.0001 0.0003 0.0003 0.8528 0.9958 0.9246 0.9320 0.9784 0.8797 ground state in this a Slater determinant, its Ø Because Groundthe state corresponds to amodel Slaterisdeterminant Ø Slater, Excited states are ensemble, andcorrelated pure distances vanish. In contrast, the Slater Ø distances Nonzeroofensemble distances all seven excited states are nonzero (! 0:85), reflecting the Ø presence Quasi-degenerate states areThe quasi-pinned of electron correlation. ensemble distances of all seven excited states are also nonzero. The pure distances, however, for all the 8R. Chakraborty, D. A.excited states states 3 and are zero to arbitrary digits of Mazziotti Int. J.except Quantum Chem. 116, 6784-790 (2016) Conclusions The role of the g chemistry of excit tions to three-, fo ecules. While th boundary of the excited-state 2-RD ensemble N-repre 26 RDMs to lie insi the pure 1-RDM set (pure distance) but also the harmonic, geometric, and arithmetic means of the Euclidean distances to each facet of the set. While the harmonic mean is similar to the minimum distance, the geometric and arithmetic means provide information about the distance to the majority of fac- ence of electron seven excited sta RDMs of these s The pure distanc states 3 and 6 are and 6 have the g have the highes • Assuming single excitations, 7x7 matrix site and coupling energies, on Table 3. Minimumthe distances to theofpure, ensemble, and Slater boundadiagonalization, gives seven stationary state wavefunctions excited states. In ries for the ground and excited states of the seven-site FMO complex are shown. state 2, and state • We use the euclidean distance metric to study the nature of pinning in the FMO and 6 have the Euclidean distances degenerate pairs. 21 FMO State Energy (cm ) Pure Ensemble Slater in the FMO comp plexity of excited Ground 0 0.00 0.0000 0.0000 0.0000 Excited 1 223.74 0.0000 0.0005 0.8876 quasi-pinned, or b STATIONARY STATES OF THE FMO 2 3 4 5 6 7 101.97 120.96 268.37 307.13 332.00 513.32 0.0000 0.0020 0.0000 0.0000 0.0003 0.0000 0.0048 0.0085 0.0141 0.0001 0.0003 0.0003 0.8528 0.9958 0.9246 0.9320 0.9784 0.8797 ground state in this a Slater determinant, its Ø Because Groundthe state corresponds to amodel Slaterisdeterminant Ø Slater, Excited states are ensemble, andcorrelated pure distances vanish. In contrast, the Slater Ø distances Nonzeroofensemble distances all seven excited states are nonzero (! 0:85), reflecting the Ø presence Quasi-degenerate states areThe quasi-pinned of electron correlation. ensemble distances of all seven excited states are also nonzero. The pure distances, however, for all the 8R. Chakraborty, D. A.excited states states 3 and are zero to arbitrary digits of Mazziotti Int. J.except Quantum Chem. 116, 6784-790 (2016) Conclusions The role of the g chemistry of excit tions to three-, fo ecules. While th boundary of the excited-state 2-RD ensemble N-repre 27 RDMs to lie insi ENVIRONMENTAL EFFECTS • We use the pure distance as a metric to study the interaction of a quantum system with its environment • Pure distance is positive or negative based whether the spectrum of occupations is inside or outside the pure set respectively Ø Pure distance is ≥ 0 at all times for the closed system Ø Becomes positive when sites 1 and 2 become maximally entangled 28 ENVIRONMENTAL EFFECTS • Pure distance as a metric to study the interaction of a quantum system with its environment 6R. Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) 29 ENVIRONMENTAL EFFECTS • Pure distance as a metric to study the interaction of a quantum system with its environment • Pure distance is positive or negative based whether the spectrum of occupations is inside or outside the pure set respectively 6R. Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) 30 ENVIRONMENTAL EFFECTS • Pure distance as a metric to study the interaction of a quantum system with its environment • Pure distance is positive or negative based whether the spectrum of occupations is inside or outside the pure set respectively Ø Pure distance is ≥ 0 at all times for the closed system Ø Becomes positive when sites 1 and 2 become maximally entangled 31 ENVIRONMENTAL EFFECTS • We use the pure distance as a metric to study the interaction of a quantum system with its environment • Pure distance is positive or negative based whether the spectrum of occupations is inside or outside the pure set respectively Ø Pure distance is ≥ 0 at all times for the closed Ø Pure distance is ≤ 0 at most times system Ø Spectrum enters the pure set when sites 1 and Ø Becomes positive when sites 1 and 2 become 2 become maximally entangled 32 maximally entangled ANTISYMMETRY AND ENERGY MINIMIZATION • Environmental noise increases the size of the set of 1-RDMs accessible to a quantum system facilitating in the transfer of excitation to the reaction center • Information about the interaction of a many-electron quantum system with its environment is encoded in the 1-RDM which scales polynomially with system size Closed 6R. Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) Open 33 OPENNESS OF QUANTUM SYSTEMS FROM GENERALIZED PAULI CONDITIONS • Generalized Pauli constraints ensure orbital occupations arise from pure quantum states • Give sufficient conditions for openness of a many-electron quantum system from sole knowledge of the 1-RDM • Can be used to quantify the degree of openness in many-electron quantum systems • Highlight interplay between antisymmetry and energy minimization photosynthetic energy transfer • Emphasize the coaction of entanglement and decoherence in quantum information transport 6R. 7R. Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) Chakraborty, D. A. Mazziotti Int. J. Quantum Chem. 2015 34 REFERENCES 1. A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963) 2. R. E. Borland and K. Dennis, J. Phys. B 5, 7 (1972) 3. A. A. Klyachko, J. Phys. Conf. Ser. 36, 72 (2006) 4. C. Schilling, M Christandl, D. Gross, Phys, Rev. A. (2013) 5. M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, J. Chem. Phys. 129, 174106 (2008) 6. R. Chakraborty, D. A. Mazziotti, Phys. Rev. A, 89, 042505 (2014) 7. R. Chakraborty, D. A. Mazziotti Phys. Rev. A, 91, 010101(R) (2015) 8. R. Chakraborty, D. A. Mazziotti Int. J. Quantum Chem., 115, 1305-1310 (2015) 9. R. Chakraborty, D. A. Mazziotti Int. J. Quantum Chem. 116, 784-790 (2016) 10. R. Chakraborty, D. A. Mazziotti, 2016, Manuscript being prepared 35 THANKS! The Mazziotti Group 36