Dmitry Abanin (Harvard) Eugene Demler (Harvard)

1. Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012

2. Entanglement Entropy: Definition -Many-body system in a pure state -Divide into two parts, -Reduced density matrix for left part (effectively mixed state) -Entanglement entropy: -Characterizes the degree of entanglement in

3. Entanglement entropy across different fields -Many-body quantum systems: scaling laws, a universal way to characterize quantum phases -Guide for numerical simulations of 1D quantum systems (e.g., spin chains) -Topological entanglement entropy: measure of topological order -Black hole entropy, Quantum field theories

4. Scaling law for entanglement entropy c -- central charge Wilczek et al’94 Vidal et al’ 03 Cardy, Calabrese’04 -1D system, ? -Gapped systems: -1D Fermi gas -Any critical system (conformal field theory): IMPLICATIONS: -Measure of the phase transition location and central charge -Independent of the nature of the order parameter

5. Topological entanglement entropy Topological order -no symmetry breaking or order parameter -degeneracy of the ground state on a torus -anyonic excitations -gapless edge states (in some cases) Physical realizations: -Fractional quantum Hall states -Z2 spin liquids (simulations) -Kitaev model and its variations DIFFICULT TO DETECT

6. Topological entanglement entropy -Three finite regions, A, B, C -Define topological entanglement entropy: (Kitaev, Preskill’06; Levin, Wen ’06) -In a topologically non-trivial phase, -A unique way to detect top. order -Proved useful in numerical studies invariant characterizing the kind of top. order Isakov, Melko, Hasting’11 Grover, Vishwanath’11…

7. Existing proposals to measure entanglement entropy experimentally Klich, Levitov’06 Song, Rachel, Le Hur et al ’10, ‘12 Hsu, Grosfield, Fradkin ’09 Song, Rachel, Le Hur ‘10 -Free fermions in 1D (e.g., quantum point contact) -Relate entanglement entropy to particle number fluctuations in left region in the ground state (Physical reason: particle number fluctuations in a Fermi gas grow as log(l)) -Limited to the case of free particles -Breaks down when interactions are introduced (e.g., for a Luttinger liquid)

8. Is it possible to measure entanglement in a generic interacting many-body system? (such that the measurement complexity would not grow exponentially with system size) Challenging – nonlocal quantity, requires knowledge of exponentially many degrees of freedom..

9. Proposed solution: entangle (a specially designed) composite many-body system with a qubit Will show that Entanglement Entropy can be measured by studying just the dynamics of the qubit

10. Renyi Entanglement Entropy -Many-body system in a pure state -Reduced density matrix -n-thRenyi entropy: PROPERTIES: -Universal scaling laws -Analytic continuation n1 gives von Neumann entropy -Knowing all Renyi entropies  reconstruct full entanglement spectrum (of ) -As useful as the von Neumann entropy

11. System of interest -Finite many-body system -short-range interactions and hopping (e.g., Hubbard model) -Ground state separated from excited states by a gap Gapped phase: Correlation length Gapless phase Fermi velocity

12. Useful fact: relation of entanglement and overlap of a composite many-body system -Consider two identical copies of the many-body system 2 Different ways of connecting 4 sub-systems: Way 1: Way 2: -Overlap gives second Renyi entropy: Ground state Ground state

13. Derivation Schmidt decomposition of a ground state for a single system Orthogonal sets of vectors in L and R sub-systems

14. Derivation Schmidt decomposition of a ground state for a single system Orthogonal sets of vectors in L and R sub-systems Represent ground states of the composite system using Schmidt decomposition:

15. Derivation Schmidt decomposition of a ground state for a single system Orthogonal sets of vectors in L and R sub-systems Represent ground states of the composite system using Schmidt decomposition: Zanadri, Zolka, Faoro ‘00, Horodecki, Ekert’02; Cardy’11, others

16. Main idea of the present proposal -Quantum switch coupled to composite system (a two-level system) -Controls connection of 4 sub-systems depending on its state Ground state Ground state Abanin, Demler, arXiv:1204.2819, Phys. Rev. Lett., in press

17. Spectrum of the composite system Switch has no own dynamics (for now); Two decoupled sectors Eigenstates of a single system Energy eigenfunction

18. Introduce switch dynamics -Turn on -Require: (not too restrictive: gap is finite) -For our composite many-body system, such a term couples two ground states -Effective low-energy Hamiltonian Renormalized tunneling:

19. Rabi oscillations: a way to measure the Renyi entanglement entropy Slowdown of the Rabi oscillations due to the coupling to many-body system Bare Rabi frequency (switch uncoupled from many-body system) Rabi frequency is renormalized: Gives the second Renyi entropy Abanin, Demler, arXiv:1204.2819, Phys. Rev. Lett., in press

20. Generalization for n>2 Renyi entropies -n copies of the many-body system -Two ways to connect them Ground state Ground state Overlap gives n-th Renyi entropy

21. Proposed setup for measuring n>2 Renyi entropies -Quantum switch controls the way in which 2n sub-system are connected -Renormalization of the Rabi frequency  overlap  n-thRenyi entropy

22. A possible design of the quantum switch in cold atomic systems -quantum well -polar molecule: *forbids tunneling of blue particles -particle that constitutes many-body system tunneling

23. A possible design of the quantum switch in cold atomic systems -Doubly degenerate ground state that controls connection of the composite many-body system -Q-switch dynamics can be induced by tuning the barriers between four wells -Study Rabi oscillations by monitoring the population of the wells

24. Generalization to the 2D case S/A A -2 copies of the system, engineer “double” connections across the boundary

25. Generalization to the 2D case -Couple to an “extended” qubit living along the boundary -Depending on the qubit state, tunneling either within or between layers is blocked -Measure n=2 Renyi entropy, and detect top. order

26. Details: Abanin, Demler, arXiv:1204.2819, Phys. Rev. Lett., in press (see also: Daley, Pichler, Zoller, arXiv:1205.1521) Summary -A method to measure entanglement entropy in a generic many-body systems -Difficulty of measurement does not grow with the system size APPLICATIONS -Test scaling laws; detect location of critical points without measuring order parameter -Extensions to 2D – detect topological order? MESSAGE: ENTANGLEMENT ENTROPY IS MEASURABLE

27. In collaboration with: Michael Knap (Graz) Yusuke Nishida (Los Alamos) AdiletImambekov (Rice) Eugene Demler (Harvard) PART 2: Time-dependent impurity in cold Fermi gas: orthogonality catastrophe and beyond

28. Strongly imbalanced mixtures of cold atoms Many groups: Salomon, Sengstock, Esslinger, Inguscio, I. Bloch, Ketterle, Zwierlein, Hulet.. -Minority (impurity) atoms can be localized by strong optical lattice -A controlled setting to study impurity dynamics -Fermi-Fermi and Fermi-Bose mixtures realized

29. Probing impurity physics: cold atomic vs. solid state systems Cold atoms: -Wide tunabilityvia Feshbach resonance: strong interactions regime -Fast control: quench-type experiments possible -Rich atomic physics toolbox: direct, time-domain measurements Solid state systems -Limited tunability -Many-body time scales too fast; dynamics beyond linear response out of reach -No time-domain experiments Energy-domain only (X-ray absorption)

30. Orthogonality catastrophe and X-ray absorption spectra in solids -Response of Fermi gas to a suddenly introduced impurity Without impurity With impurity -Relevant overlap: -- scattering phase shift at Fermi energy -Manifestation: a power-law edge singularity in the X-ray absorption spectrum Nozieres, DeDominicis; Anderson ‘69

31. Preview: Universal OC in cold atoms (very small energies) -No universality at short times/large energies (band structure,scattering parameters unknown,…) Previously: (very long times)

32. Preview: Universal OC in cold atoms (very small energies) -No universality at short times/large energies (band structure,scattering parameters unknown,…) -This work: exact solution for (all times and energies); Previously: (very long times)

33. Preview: Universal OC in cold atoms (very small energies) -No universality at short times/large energies (band structure,scattering parameters unknown,…) -This work: exact solution for (all times and energies); -Universal, determined only by impurity scattering length -Time domain: new important oscillating contribution to overlap -Energy domain: cusp singularities in with a new exponent at energy above absorption threshold Previously: (very long times)

34. Setup -Pseudospin can be manipulated optically *flip *create coherent superpositions, e.g., -Study orthogonality catastrophe in frequency and time domain -Fermi gas+single localized impurity -Two pseudospin states of impurity, and • -state scatters fermions -state does not -Scattering length

35. Ramsey interferometry –probe of OC in the time domain -Entangle impurity pseudospin and Fermi gas; -Utilize optical control over pseudospin study Fermi gas dynamics -Ramsey protocol 1) pi/2 pulse 2) Evolution 3) pi/2 pulse, measure

36. RF spectroscopy of impurity atom: OC in the energy domain Atom in a Fermi sea – OC completely changes absorption function New cusp singularity Free atom

37. Origin of singularities in the RF spectra: an intuitive picture Threshold singularity Singularity at Ef -Certain sets of excited states are important -Edge singularity (standard): multiple low-energy e-h pairs -Singularity at : extra electron -- band bottom to Fermi surface + multiple low-energy e-h pairs

38. Functional determinant approach to orthogonality catastrophe Combescout, Nozieres ‘71; Klich’03, Muzykantskii’03; Abanin, Levitov’04; Ivanov’09; Gutman, Mirlin’09-12….. -Solution in the long-time limit is known(Nozieres- DeDominicis’69); based on solving singular integral equation OUR GOAL: full solution at all times -Approach 1: write down an integral equation with exact Greens functions; solve numerically (possible, but difficult) -Approach 2: reduce to calculating functional determinants (easy)

39. Functional determinant approach to orthogonality catastrophe Desired response function Many-body trace Fermi distribution function Time-dep. scattering operator -Long-time behavior: analytical solution possible Muzykantskii, Adamov’03, Abanin, Levitov’04,… -Arbitrary times (this work): evaluate the determinant numerically; certain features (prefactors, new cusp singularity) found analytically Represent as a determinant in single-particle space

40. Results: overlap, a<0 -No impurity bound state -Leading power-law decay -Sub-leading oscillating contribution due to van Hove singularity at band bottom

41. Universal RF spectra for a<0 -Impurity potential does not create a bound state -Single threshold

42. Universal RF spectra for a<0 -Single threshold -New non-analytic Feature at

43. Cusp singularity at Fermi energy Zoom -Origin: combined dynamics of hole at band bottom+e-h pairs -Becomes more pronounced for strong scattering -Smeared on the energy scale -At the unitarity, evolves into true power-law singularity with universal exponent ¼! Knap, Nishida, Imambekov, DA, Demler, to be published

44. Universal RF spectra for a>0 -Impurity potential creates a bound state -Double threshold (bound state filled or empty) -Non-analytic feature at distance from first threshold -Characteristic three-peak shape

45. Summary -New regimes and manifestation of orthogonality catastrophe in cold atoms -Exact solutions for Fermi gas response and RF spectra obtained; New singularity found -Spin-echo sequences  probe more complicated dynamics of Fermi gas -Extensions to multi-component cold atomic gases  simulate quantum transport and more… Knap, Nishida, Imambekov, Abanin, Demler, to be published

46. Results: overlap a>0; bound state Strong oscillations (bound state either filled or empty) a<0; no bound state Weak oscillations from van Hove singularity at band bottom

47. Functional determinant approach to orthogonality catastrophe w/o impurity with impurity Density matrix Trace is over the full many-body state; dimensionality -number of single-particle states Represent

48. A useful relation Then Trace over many-body space (dimensionality )  Determinant in the single-particle space (dimensionality ) -Holds for an arbitrary number of exponential operators -Derivation: step1 – prove for a single exponential (easy) step2 – for two or more exponentials, use Baker-Hausdorf formula reduce to step 1 Consider quadratic many-body operators

49. Single impurity problems in condensed matter physics -Edge singularities in the X-ray absorption spectra (asympt. exact solution of non- Equilibrium many-body problem) -Kondo effect: entangled state of impurity spin and fermions Influential area, both for methods (renormalization group) and for strongly correlated materials Rich many-body physics

50. Results: overlap no bound state -Power-law decay -Weak oscillations from van Hove singularity at band bottom