Microscopic structure of entanglement in the many-body environment of a qubit

Transcription

1 Microscopic structure of entanglement in the many-body environment of a qubit Serge Florens, [Ne el Institute - CNRS/UJF Grenoble] displacements fpol. fanti. fsh : Microscopic structure of entanglement in the many-body environment of a qubit ω

2 The team Soumya Bera (Néel, Grenoble) Ahsan Nazir (Imperial, London) Harold Baranger (Due, USA) Alex Chin (Cavendish, Cambridge) Nicolas Roch (ENS, Paris) : Microscopic structure of entanglement in the many-body environment of a qubit 2

3 Outline Outline Motivation : from cavity-qed to photonic Kondo effect Dissipative quantum mechanics in a nutshell Physical picture of environmental entanglement : quantum superpostions of polarons and antipolarons Many-body ground state Ansatz : coherent state expansion Wigner tomography & entanglement spectroscopy from NRG Perspectives S. Bera et al., arxiv : : Microscopic structure of entanglement in the many-body environment of a qubit 3

4 From cavity-qed to photonic Kondo From cavity-qed to photonic Kondo : Microscopic structure of entanglement in the many-body environment of a qubit 4

5 From cavity-qed to photonic Kondo Cavity-QED Pioneering experiment : Haroche, 2012 Nobel Prize Coupling of light and matter to manipulate and measure quantum states Original experiment very complex (15 years to build!) simple conceptually (single cavity mode+two-level system) Rabi model : H = ω 0 a a + σ x + g(a + a)σ z + H out : Microscopic structure of entanglement in the many-body environment of a qubit 5

6 From cavity-qed to photonic Kondo Circuit QED Recent progresses : on-chip cavity QED Bereley, Paris, Yale, Zurich... Atom superconducting qubits (non-linear element) Macroscopic two-level system with large dipole moment Microwave circuits : design more complex electromagnetic environment New route to explore strong correlations? : Microscopic structure of entanglement in the many-body environment of a qubit 6

7 From cavity-qed to photonic Kondo Generation and measurement of arbitrary quantum states Setup : Single electromagnetic mode (cavity) Qubit-coupling used to generate complex photonic states Measurement : quantum tomography Nice example : Ψ = 0 + e iφ Hofheinz et al., Nature (2009) : Microscopic structure of entanglement in the many-body environment of a qubit 7

8 From cavity-qed to photonic Kondo Generation and measurement of arbitrary quantum states Setup : Single electromagnetic mode (cavity) Qubit-coupling used to generate complex photonic states Measurement : quantum tomography Other example : Ψ = [ ] + [ 1 2 ] Qubit-conditioned photon moments Eichler et al., PRL (2012) : Microscopic structure of entanglement in the many-body environment of a qubit 8

9 From cavity-qed to photonic Kondo Towards many-body photonic problems Setup : Need many electromagnetic modes Taylored environment (Josephson junction arrays) Goldstein, Devoret, Houzet, Glazman, PRL (2013) Model : H = i,j 2e 2 n i ( C 1 ) ij n j E ij J cos(φ i φ j ) Limit E J E C : harmonic environment (free bosonic bath!) : Microscopic structure of entanglement in the many-body environment of a qubit 9

10 From cavity-qed to photonic Kondo Towards many-body photonic problems Quantum dot (qubit) : local defect with E L/R J E C Quantized 2e charge on the island (Cooper pair box) Tuned to charge degeneracy N N + 2 Effective two-level system : Pseudospin ˆσ z = [ ˆQ (N + 1)]/2 Bosonic impurity model! Goldstein, Devoret, Houzet, Glazman, PRL (2013) : Microscopic structure of entanglement in the many-body environment of a qubit 10

11 From cavity-qed to photonic Kondo Standard model for dissipative qubit Effective theory : spin boson hamiltonian Leggett et al. RMP (1987) H = 2 σ x σ z g 2 (a + a ) + Spectral density : J(ω) g 2 δ(ω ω ) Ohmic bath : J(ω) = 2παω Dissipation strength : α = 2E Cg /E J 1 Tunneling amplitude : = E L/R J ω a a : Microscopic structure of entanglement in the many-body environment of a qubit 11

12 From cavity-qed to photonic Kondo Aim of the tal Physics at play : The qubit generates a non-linearity among photons Strong similarity with the electronic Kondo problem (even more than just an analogy...) Questions to be answered : What ind of correlations are created in the environment due to its coupling to the qubit? What controls the coherence of the qubit? : Microscopic structure of entanglement in the many-body environment of a qubit 12

13 Dissipative quantum mechanics in a nutshell Dissipative quantum mechanics in a nutshell : Microscopic structure of entanglement in the many-body environment of a qubit 13

14 Dissipative quantum mechanics in a nutshell Wea dissipation regime : α < 0.4 Spin dynamics : underdamped Rabi oscillations Bosonic NRG solves the model [Bulla et al. PRL (2003)] Spin-spin dynamical correlation functions for arbitrary dissipation strength [Florens et al. PRB (2011)] χ z (ω)/ω α = 0.01 α = 0.1 α = ω/ Pea at renormalized scale R < Non-lorentzian lineshape for α > 0.1 : Microscopic structure of entanglement in the many-body environment of a qubit 14

15 Dissipative quantum mechanics in a nutshell Strong dissipation regime : α > 0.4 Spin dynamics : overdamped Rabi oscillations Linewidth Γ > R : incoherent qubit Boring? No! : universal (Kondo) regime for α 1 strongly correlated many-body photonic state χ z (ω)/ω α = 0.3 α = 0.4 α = 0.5 S(ω) 6*10 4 4*10 4 =0.1 = = = α = 0.9 S(ω)/S(0) *10 4 ω/ r ω/ 0*10 4 0*10 3 2*10 3 4*10 3 6*10 3 8* *10 3 ω : Microscopic structure of entanglement in the many-body environment of a qubit 15

16 Dissipative quantum mechanics in a nutshell Exact mapping to Kondo model Spin boson Hamiltonian : H = 2 σ g x σ z 2 (a + a ) + ω a a g Unitary transformation : U γ = exp{ γσ z 2ω (a a )} U γ HU γ = 2 σ+ e γ P g (a ω a ) g +h.c.+(γ 1)σ z 2 (a +a )+ ω a a Bosonization (ohmic bath only) : [Guinea, Haim, Muramatsu PRB (1985)] U γ HU γ = σ + c c + h.c. J = +(1 α)ω c σ z [c c c c ] J z 1 α + σ ɛ c σ c σ : Microscopic structure of entanglement in the many-body environment of a qubit 16

17 Dissipative quantum mechanics in a nutshell Summary : phase diagram delocalized coherent dynamics incoherent dynamics localized Ferro J AntiFerro α 0 J z That s not all, fols! Question : what ind of correlations/entanglement does the qubit generate into its environment? : Microscopic structure of entanglement in the many-body environment of a qubit 17

18 Energetics : polarons vs. antipolarons Energetics : polarons vs. antipolarons : Microscopic structure of entanglement in the many-body environment of a qubit 18

19 Energetics : polarons vs. antipolarons Classical limit Spin boson Hamiltonian : H = 2 σ x σ z g 2 (a + a ) + ω a a Oscillators subject to a spin-dependent potential = 0 limit : frozen potential doubly-degenerate ground state Ψ,f cl. = f cl. Ψ, f cl. = f cl. where f cl. = g /(2ω ) and ± f e ± P f (a a ) 0 Zero tunneling localized coherent states! : Microscopic structure of entanglement in the many-body environment of a qubit 19

20 Energetics : polarons vs. antipolarons Bare polaron theory for 0 Ground state Ansatz : antisymmetric combination (polaron) GS +f cl. f cl.... but no quantum fluctuations among the oscillators! Testing on Rabi model : consider single mode ω 1 below Exact solution (blue dots) vs. bare polaron (blue dashed line) Failure at increasing /ω 1 : problematic for many modes! : Microscopic structure of entanglement in the many-body environment of a qubit 20

21 Energetics : polarons vs. antipolarons Variational polaron theory : Silbey-Harris state Ground state Ansatz : polaron with optimized displacement GS +f pol. f pol. f pol. = g /2 ω + R Renormalized tunnel rate R reduces the displacement Quantum fluctuations within the well Testing on Rabi model : consider single mode ω 1 below Exact solution (blue dots) vs. SH polaron (red dot-dashed line) Energy improves, but something is still missing : Microscopic structure of entanglement in the many-body environment of a qubit 21

22 Energetics : polarons vs. antipolarons Energetics : competition localization vs. tunneling Elastic cost of displacement f : δe = ω [f g /2ω ] 2 Favors positive displacement f = +g /2ω Applies for high frequency modes : adiabatic response Tunnel process : tends to delocalize Polaron state couples to a displaced state f with matrix element : Ψ, e f K + Ψ,g /2ω = e 1 2 P (f +g /2ω ) 2 Favors negative displacement f = g /2ω Energy gain = (bare!) Small ω modes can tae advantage : anti-adiabatic response Strong energetic constraints : These reduce the phase space volume of allowed displacements : Microscopic structure of entanglement in the many-body environment of a qubit 22

23 Energetics : polarons vs. antipolarons Physical picture [A] : one polaron (adiabatic) [B] : one polaron (SH) [D] X (A. U.) [C] : polaron + anti-polaron X (A. U.) X (A. U.) Idea : quantum mechanics allows for state superposition The best compromise is to allow positive and negative displacements : Microscopic structure of entanglement in the many-body environment of a qubit 23

24 Energetics : polarons vs. antipolarons Checing the wavefunctions (one mode, = 4ω 1 ) 0.4 g Ω g Ω Φ x 0.0 Φ x x x 0.4 g Ω g Ω Φ x 0.0 Φ x x x Exact wavefunction (dots) vs SH single polaron state (dashed line) : Microscopic structure of entanglement in the many-body environment of a qubit 24

25 Energetics : polarons vs. antipolarons Polaron+antipolaron state (one mode, = 4ω 1 ) GS = [ + f pol. 1 + p + f anti. 1 ] [ f pol. 1 + p f anti. 1 ] 0.4 g Ω g Ω Φ x 0.0 Φ x x x 0.4 g Ω g Ω Φ x 0.0 Φ x x x Trial wavefunction and energy (solid line) are nearly perfect! : Microscopic structure of entanglement in the many-body environment of a qubit 25

26 Many-body ground state Ansatz : coherent state expansion Many-body ground state Ansatz : coherent state expansion : Microscopic structure of entanglement in the many-body environment of a qubit 26

27 Many-body ground state Ansatz : coherent state expansion How do we build the many-body wavefunction? Insight from two modes : X X X 1 Left panel : exact wavefunction Ψ (X 1, X 2 ) compatible with Ψ = f pol. 1 f pol. 2 + p f anti. 1 f anti. 2 Right panel : hypothetical product state (no entanglement) Ψ = { f pol. 1 + p f anti. 1 } { f pol. 2 + p f anti. 2 } : Microscopic structure of entanglement in the many-body environment of a qubit 27

28 Many-body ground state Ansatz : coherent state expansion Many-modes : two-polaron variational Ansatz Proposed wavefunction : built with f e P f (a a ) 0 GS 2pol. = [ +f pol. + p +f anti. ] [ f pol. + p f anti. ] 0.4 α = 0.5 /ω c = 0.01 displacements f pol. f anti. f SH Adiabatic/anti-adiabatic crossover captured! ω : Microscopic structure of entanglement in the many-body environment of a qubit 28

29 Many-body ground state Ansatz : coherent state expansion Many modes : ground state properties Ground state coherence σ x : σ x 10-2 NRG N=4 N=3 N=2 N=1 (Silbey-Harris) Coupling Strength (α) The one-polaron (SH) breas down at large dissipation Antipolarons (entanglement) helps in preserving σ x Add more polarons : convergent expansion! : Microscopic structure of entanglement in the many-body environment of a qubit 29

30 Many-body ground state Ansatz : coherent state expansion General multi-polaron (coherent state) expansion displacements f (n) N pols [ GS = p n + f (n) f (n) ] f (1) f (2) α =0.2 f (3) f (4) f (5) f (6) n=1 f (7) f (8) p 3 p 4 p 5 p 6 p 7 p 8 ω p 2 p 1 p 1 p 1 p 1 p 1 p 1 p 1 α = ω 0.0 p p 3 p 4 p 5 p 6 p 7 p 8 p 1 p 1 p 1 p 1 p 1 p 1 p 1 Adiabatic/antiadiabatic crossover obeyed α =0.8 ω 0.00 p p 3 p 4 p 5 p 6 p 7 p 8 p 1 p 1 p 1 p 1 p 1 p 1 p 1... but new displacements with double ins for n > 6 Approach to the scaling limit α 1 : data collapse : Microscopic structure of entanglement in the many-body environment of a qubit 30

31 Many-body ground state Ansatz : coherent state expansion Nature of the strong dissipation state (a) (c) 10-1 p n /p 1 p n /p (b) α p 2 /p 1 p 3 /p 1 p 4 /p 1 p 5 /p (d) /ω c delocalized localized The displacements vanish for 0 : bare polarons The strong dissipation state at α 1 is non-trivial α : Microscopic structure of entanglement in the many-body environment of a qubit 31

32 Many-body ground state Ansatz : coherent state expansion z Including a bias Tuning the charge asymmetry : Leggett et al. RMP (1987) 10-2 H = ɛσ z + 2 σ x σ z finite magnetization σ z 0 σ z Bethe ansatz 6 polarons ansatz ɛ σz g 2 (a + a ) + ω a a BA Mpols Bias /ω c = α Perfect agreement with (fermionic) Bethe Ansatz Deviations at α > 0.9 : more polarons needed : Microscopic structure of entanglement in the many-body environment of a qubit 32

33 Many-body Wigner tomography How do we chec the form of the wavefunction? Many-body Wigner tomography : Microscopic structure of entanglement in the many-body environment of a qubit 33

34 Many-body Wigner tomography Wigner spectroscopy of the polaron component Definition : W () (X ) = d 2 λ π 2 ex ( λ λ) GS [ e λa λa ] GS From the two-polaron Ansatz : W () (X ) = 1 pol. f e 2(X ) 2 + (small terms) π NRG computation : use moments GS [a ]m [a ] n GS W () (X) NRG N=1 W () (X) NRG N=8 N=3 N=2 N=1 The SH state captures the complete density matrix reduced to a given but arbitrary high energy oscillator X X : Microscopic structure of entanglement in the many-body environment of a qubit 34

35 Many-body Wigner tomography Wigner spectroscopy of the antipolaron component NRG N= Definition : W () σ + (X ) = From the two polaron Ansatz : W () σ (X ) p + π NRG computation : α = 0.8 here W () (X) X d 2 λ π 2 ex ( λ λ) GS [ e λa λa σ + ] GS [e 2 (X + f pol. NRG N=8 N=3 N=2 N=1 f anti. 2 ) 2 + e 2 (X f pol. f anti. 2 ) 2] These correlations are only captured by the antipolaron part of the wavefunction : Microscopic structure of entanglement in the many-body environment of a qubit 35

36 Many-body Wigner tomography Entanglement properties Entropy spectroscopy : S spin+ = Tr spin+ [ρ spin+ log ρ spin+ ] S spin = Tr spin [ρ spin log ρ spin ] S Spin + - S Spin α=0.2 α=0.4 α=0.6 α= ω Negative part in S spin+ S spin : Fully accounted by SH S Spin + S Spin α =0.3 α =0.5 α = α =0.5, /ω c =0.01 Origin : entanglement of spin with modes NRG N=8 N=3 N=2 N= ω : Microscopic structure of entanglement in the many-body environment of a qubit 36

37 Many-body Wigner tomography Entanglement properties Entropy spectroscopy : S spin+ = Tr spin+ [ρ spin+ log ρ spin+ ] S spin = Tr spin [ρ spin log ρ spin ] S Spin + - S Spin α=0.2 α=0.4 α=0.6 α= ω Positive part in S spin+ S spin : S Spin + S Spin α =0.3 α =0.5 α = α =0.5, /ω c =0.01 NRG N=8 N=3 N=2 N= ω Positive pea is not accounted by SH Origin : entanglement of modes with modes (antipolarons) Kondo lineshape (wide entanglement) at α 1 Massively entangled photonic Kondo cloud : Microscopic structure of entanglement in the many-body environment of a qubit 37

38 Conclusion Conclusion Quantum tunneling of a qubit subsystem can drive strong correlations in its environment Environmental entanglement builds from superposition of polarons and antipolarons These ideas can be rationalized quantitatively from a coherent state expansion of the many-body ground state Strong phase space constraints mae the expansion quicly convergent : Microscopic structure of entanglement in the many-body environment of a qubit 38

39 Conclusion Open questions : Quantum certification : can we really prove convergence of the polaron expansion for the complete many-modes wavefunction (for all many-body density matrices)? Can we reliably simulate quantum quenches and non-linear photon transport experiments using a coherent-state based time-dependent variational framewor? [Related wors in quantum chemistry, e.g. Burghardt et al. J. Chem. Phys. (2003)] Can future superconducting circuit experiments give evidence for the wide-entanglement that characterizes the strongly dissipative (Kondo-lie) photonic states? Can we learn something on the structure of the wavefunction for strongly correlated fermions via the Kondo analogy? : Microscopic structure of entanglement in the many-body environment of a qubit 39