Geodesic paths for quantum many-body systems

Transcript

1 Geodesic paths for quantum many-body systems Michael Tomka, Tiago Souza, Steve Rosenberg, and Anatoli Polkovnikov Department of Physics Boston University Group: Condensed Matter Theory June 6, 2016 Workshop: Quantum Non-Equilibrium Phenomena

2 Adiabatic ground-state preparation Ĥ(λ(t)) Ψ gs (λ 0 ) Ψ gs (λ f ) λ(t) = λ 0 + (λ f λ 0 ) t t f Ψ i Ψ f

3 Adiabatic Quantum Computation Ĥ = (1 λ)ĥ 0 + λĥ f linear interoplation: λ(t) = 1 t f t Ground state of Ĥ 0 is easily accessible Ground state of Ĥ f encodes the solution to a hard computational problem E. Farhi, et al., Science 20, 472 (2001)

4 Superfluid - Mott Insulator Transition Superfluid Mott Insulator Phase coherence Macroscopic phase well defined in each potential well No phase coherence Macroscopic phase uncertain in each potential well Atom number uncertain in each potential well Atom number exactly known in each potential well M. Greiner, et al., Nature 415, (2002)

5 Outline Introduction Quantum metric tensor Geodesics Examples The Landau-Zener model XY spin chain in a transverse magnetic field Conclusion

6 Introduction Parameters of the Hamiltonian Hamiltonian λ(t) = (λ 1 (t),..., λ p (t)) T M H(λ(t)) Ĥ(λ(t)) ψ n (λ(t)) = E n (λ(t)) ψ n (λ(t)), n = 0, 1, 2,...

7 Introduction Ĥ(λ(t)) ψ 0 (λ 0 ) ψ 0 (λ f ) F [ λ(t f ) ] = ψ(t f ) ψ 0 (t f ) 2

8 Quantum metric tensor - a simplistic approach Consider two nearby ground states ψ 0 (λ) and ψ 0 (λ + dλ) and define a distance (λ, dλ) between λ and dλ in M 2 (λ, dλ) 1 ψ 0 (λ) ψ 0 (λ + dλ) 2 ψ 0 (λ + dλ) = ψ 0 (λ) + µ ψ 0 (λ) dλ µ µ ν ψ 0 (λ) dλ µ dλ ν (λ, dλ) 1 ψ 0 (λ) ψ 0 (λ + dλ) 2 = g µν dλ µ dλ ν ( = Re ψ 0 µ ν ψ 0 ψ 0 ) µ ψ 0 ψ 0 ν ψ 0 dλ µ dλ ν Quantum metric tensor ( g µν Re ψ 0 µ ν ψ 0 ψ 0 ) µ ψ 0 ψ 0 ν ψ 0 µ λ µ

9 Quantum metric tensor - a rigorous approach

10 Quantum metric tensor - a rigorous approach [ ψ0 ] : T [ ψ0 ]PH T [ ψ0 ]PH C u, v u v [ ψ0 ] X H projecting to a given vector u T [ ψ0 ]PH χ( u, v ) u v [ ψ0 ] = X 1 X 2 X 1 ψ 0 ψ 0 X 2 The non-degenerate hermitian metric χ pulls back to a non-degenerate hermitian metric χ only if pull back is injective where µ J.P. Provost, G. Valle, (Commun. Math. Phys. 76, 289 (1980)) χ µν ψ 0 µ ν ψ 0 ψ 0 µ ψ 0 ψ 0 ν ψ 0 λ µ and g µν = Re(χ µν ) F µν = 2 Im(χ µν )

11 Quantum metric tensor - Fidelity The quantum metric tensor is closely related to the fidelity F F(λ, λ + dλ) ψ 0 (λ) ψ 0 (λ + dλ) 2 The fidelity susceptibility χ F is defined by the expansion of F F(λ, λ + dλ) = 1 χ F +... and reads [ ψ 0 µ ν ψ 0 + ψ 0 ] ν µ ψ 0 χ F = ψ 0 µ ψ 0 ψ 0 ν ψ 0 dλ µ dλ ν 2 = g µν dλ µ dλ ν = 1 2 m 0 P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett. 99, (2007) ψ 0 µ Ĥ ψ m ψ m ν Ĥ ψ 0 + µ ν (E 0 E m ) 2 dλ µ dλ ν W.-L. You, Y.-W. Li, and S.-J. Gu, Phys. Rev. E 76, (2007) V. Mukherjee, A. Polkovnikov, and A. Dutta Phys. Rev. B 83, (2011)

12 Quantum metric tensor - Energy fluctuations ( ) gxx g g(x f, z f ) = xz, g µν = 1 2 g zx g zz m 0 ψ 0 µ Ĥ ψ m ψ m ν Ĥ ψ 0 + µ ν (E 0 E m ) 2 z = z f, x(t) = v x t + x 0, 0 t t f, x(t f ) = x f δe 2 ψ(t f ) Ĥ 2 ψ(t f ) ψ(t f ) Ĥ ψ(t f ) 2 ψ(t f ) = ψ 0 iv x δe 2 v 2 x m 0 m 0 ψ m x Ĥ ψ 0 (E m E 0 ) 2 + O(v2 x), ψ m x Ĥ ψ 0 2 (E m E 0 ) 2 = v 2 x g xx M. Kolodrubetz, V. Gritsev, and A. Polkovnikov, Phys. Rev. B 88, (2013)

13 Geodesics ds 2 = g µν dλ µ dλ ν = 1 ψ 0 (λ) ψ 0 (λ + dλ) 2 The quantum distance L between two ground-states ψ 0 (λ 0 ) and ψ 0 (λ f ) for a path λ(t), joining λ 0 = λ(0) and λ f = λ(t f ), is given by L(λ) = f i ds = A geodesics is defined by λf λ 0 gµν dλ µ dλ ν = δl = 0 tf 0 g µν λµ λν dt. and we can find the shortest path λ geo (t) connecting the two ground states at λ 0 and λ f.

14 Geodesics L = tf 0 g µν dλ µ dt dλ ν tf dt dt E = t dλ µ dλ ν f g µν 0 dt dt dt L 2 E, g µν dλ µ dt dλ ν dt = const. δe 2 where d 2 λ µ dt 2 + Γ µ dλ ν dλ ρ νρ dt dt = 0 Γ µ νρ = 1 ( gξν 2 gµξ λ ρ + g ξρ λ ν g ) νρ λ ξ g µν = (g µν ) 1 Γ µ νρ = Γ µ ρν

15 The Landau-Zener model ( ) Ĥ LZ (t) = x(t)ˆσ x + ɛ(t)ˆσ z ɛ(t) x(t) = x(t) ɛ(t) Eigenstates ψ 0,1 = 1 Ω ɛ + 1 x, 2 Ω(Ω ɛ) 2 Ω(Ω ɛ) Eigenenergies E 0,1 = Ω x 2 + ɛ 2 Find λ opt (t) = (x opt (t), ɛ opt (t)) T maximizing: F(t f ) = ψ(t f ) ψ 0 (t f ) 2 for ψ 0 (0) at λ i = (x i, ɛ) T ψ 0 (t f ) at λ f = (x f, ɛ) T

16 The Landau-Zener model (a) (b) (c) ϵ= xlin (t) xgeo(t) λ geo(t) x(t) t E 0 0 x ϵ λ i Δ x [ (1) x lin (t) = x i + (x f x i )t/t f = F(t f ) 1 exp λ f -log( F ) /t f π ɛ 2 (x f x i ) 1 t f (2) g xx = ɛ 2 4(x 2 +ɛ 2 ) 2 = x geo (t) = ɛ tan [α i + (α f α i ) t/t f ] α i,f = arctan(x i,f /ɛ) (3) λ(t) = Ω(t) ( ) sin θ(t) cos θ(t) (g µν ) = ( ) /4 ] = Ω geo (t) = Ω i θ geo (t) = θ i + (θ f θ i ) t/t f θ i,f = arctan(x i,f /ɛ i,f )

17 XY spin chain in a transverse magnetic field Ĥ XY = N [ 1 + g ˆσ j x ˆσ j+1 x g ] ˆσ y j 2 ˆσy j+1 + hˆσz j j=1 ( h cos(k) g sin(k) ) Ĥ XY = k g sin(k) [h cos(k)] PM FM PM

18 XY spin chain in a transverse magnetic field F = ψ(t f ) ψ 0 (h f ) 2 h(t) h 0 =1.1, h f =3.5, δ= t g hh = log(f(tf )) g=1, h 0=1.1, h f =3.5, N= h g 2 (h 2 1)(h 2 1+g 2 ) 3/2 δ = 1/t f M. Kolodrubetz, V. Gritsev, and A. Polkovnikov, Phys. Rev. B 88, (2013) P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett. 99, (2007)

19 XY spin chain in a transverse magnetic field γ sin φ γ cos φ H XY (h, γ, φ) = R z (φ)h XY R z(φ), R z (φ) = N l=1 exp( i φ 2 σz l ) g γγ = 1 16γ(1+γ) 2, g φφ = γ 8(γ+1), g γφ = 0

20 Bz By =0, g=1., B0= Summary Quantum metric ( tensor: g µν = Re ψ 0 µ ν ψ 0 ψ 0 ) µ ψ 0 ψ 0 ν ψ 0 Geodesics: d 2 λ µ dt 2 + Γ µ νρ dλν dt dλ ρ dλ dt = 0 g µ dλ ν µν dt dt = const. δe 2 ΔE Landau-Zener model and XY spin chain Used in experiments with superconducting quantum circuits (P. Roushan, et al.) Bx

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