On ‘Solving’ a quantum many body problem by experiment Jörg Schmiedmayer

Vienna Center for Quantum Science and Technology, Atominstitut, TU-Wien T. Schweigler et al. arXiv:1505.03126 A. J. Steffens et al. nat. comm. (2015) arXiv:1406.3632 Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Quantum fields Correlation functions On the Green's functions of quantized fields J. Schwinger PNAS (1951)

! Solving a quantum many-body problem is equivalent to knowing all its correlation functions. ! In practice, an observer can only measure a finite number of correlations describing the propagation and scattering of excitations. ! To solve a problem one need to find degrees of freedom where only few (low order) correlation functions are relevant.

! If one finds the degrees of freedom (basis) where the correlation functions factorize, this is equivalent to diagonalization of the many body Hamiltonian. J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Outline 1d System Correlation functions

–  fields phase excitations

1000-10000 Rb atoms T = 10-100 nK ωR ≈ 2π x 2 - 3 kHz ωL ≈ 2π x 5 - 10 Hz µ, kBT non translation invariant correlation functions

with neglecting 4th order:

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Correlation functions excitations phase

T. Schweigler et al. arXiv:1505.03126

in experiment we measure the phase ϕ(z) directly -> look at phase correlators C(2)(z1, z2) = [ϕ(z1) with

Δϕ(z1, z2) = ϕ(z1) ϕ(z2)

ϕ(z2)]2 = [Δϕ(z1,z2)]2 Note: Δϕ is NOT restricted to 2π

using

-> phase correlators are related to the quasi particles 4th order C(4)(z1, z2,z3, z4) =

[ϕ(z1)

ϕ(z2)]2 [ϕ(z3) ϕ(z4)]2

-> quasi particle scattering J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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When do higher Correlation Functions factorize?

T.J.Schweigler et aal. arXiv:1505.03126 Schmiedmayer: On ‘Solving’ Quantum Many-Body Problem by Experiment

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Sine-Gordon physics

1d double-well with tunable coupling J experiment: T. Schweigler et al. theory: V. Kasper, S. Erne T Gasenzer, J. Berges

Quantum Sine-Gordon model:

phase coherence length that’s what we have seen so far … “uncoupled harmonic oscillators” phase (spin) healing length

Characteristic parameters anharmonic, non-gaussian, gapped, universality? J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Sine-Gordon physics

1d double-well with tunable coupling J Quantum Sine-Gordon model

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

experiment: T. Schweigler et al. theory: V. Kasper, S. Erne T Gasenzer, J. Berges

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Characterising the factorisation experiments probe the phase -> look at the ‘connected part’ of the phase correlation function

T. Schweigler et al. arXiv:1505.03126

Gaussian fluctuations Variance =0 =0 =0 Gaussian fluctuations =0

characterized by ‘Kurtosis’

=0 =0 J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Correlation functions fields phase

correlation functions for the fields:

T. Schweigler et al. arXiv:1505.03126

C(z1,z2) contains all orders of connected parts

for Gaussian fluctuations

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Observable and non-gauss measure T. Schweigler et al. arXiv:1505.03126

to study factorization of correlation functions we look at C(2)(z1, z2) = [ϕ(z1) ϕ(z2)]2 (4) C (z1, z2,z3, z4) = [ϕ(z1) ϕ(z2)]2 [ϕ(z3)

ϕ(z4

= [Δϕ(z1,z2)]2 = [Δϕ(z1,z2)]2 [Δϕ(z3,z4)]2 ,

)]2

C(4)(z1, z2, -15, 15)

Δϕ(z1, z2) = ϕ(z1) ϕ(z2) Δϕ is NOT restricted to 2π

no coupling scan4161 q=0

p

intermediate coupling scan3490 scan3426 q = 2.5 q=3.4

strong coupling scan4056a q = 15

20

full

experiment: T. Schweigler et al. theory: V. Kasper, S. Erne

20 0 -20 20

deviation

1

-20

Wick

Δϕ > 2π

Position z2 [µm]

0

0 -20 -20

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-1 J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Characterising non-Gaussian phase fluctuations Characterising the factorisation by the connected part:

T. Schweigler et al. arXiv:1505.03126

excess Kurtosis Experimental data, thermal state in a double well plasma-freq. = 0 Hz q=0

70 Hz q = 2.5

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

160 Hz q = 3.4

220 Hz q = 8.6

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Quantifying factorization of correlation functions full distribution function

T. Schweigler et al. arXiv:1505.03126

Kurtosis

•  the breakdown of factorization is evident in the full distribution functions of the phase by new peaks at multiples of 2π •  caused by the 2π periodic SG Hamiltonian -> 2π phase jumps, ‘kinks’ = SG solitons

•  SG Solitons are topological excitations •  Phase fluctuations around topologically different Vaccua J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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4-point phase correlators q

0

2.5

3.4

15

T. Schweigler et al. arXiv:1505.03126

full Wick factorization difference lower limit upper limit

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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6-point phase correlators q

0

2.5

3.4

15

T. Schweigler et al. arXiv:1505.03126

full Wick factorization difference lower limit upper limit

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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6-point phase correlators, connected part ωp

0 Hz

70 Hz

160 Hz

big

T. Schweigler et al. arXiv:1505.03126

full disconnected part

connected part lower limit upper limit

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Remove Solitons

Strongly coupled q = 8.6 ωp > 500 Hz T. Schweigler et al. arXiv:1505.03126

4-point correlator does not factorize:

without Solitons:

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Remove Solitons

intermediate coupling q = 3.4 ωp = 160 Hz T. Schweigler et al. arXiv:1505.03126

4-point correlator does not factorize:

without Solitons:

phase distribution: different sectors:

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

without solitons:

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What have we learned T. Schweigler et al. arXiv:1505.03126

•  high order (>10) correlation functions are accessible in experiment •  full distribution functions and the connected part of the higher order correlation functions contain genuine information about the quantum field theory •  quasi particles •  interaction of quasi particles •  vacuum states

•  gives insight in the effective theories describing the many body system •  for our resolution 6th order is sufficient -> necessary to take perturbation expansion up to 3rd order (3-3 scattering) J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Quench from J>0 to J=0 very preliminary

experiment: T. Schweigler et al.

Initial state q=3,4: non-Gaussian, dynamics Gaussian

collaboration with Berges & Gasenzer groups, Heidelberg J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Quench from J>0 to J=0 very preliminary

experiment: T. Schweigler et al.

Initial state q=3,4: non-Gaussian, dynamics Gaussian

collaboration with Berges & Gasenzer groups, Heidelberg J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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www.AtomChip.org

Quantum Field Tomography Theory: A. Steffens, et al, NJP 16 (2014) 123010. Experiment: Steffens et Many-Body al. nature (2015) arxiv:1406.3632 J. Schmiedmayer: .On ‘Solving’ a Quantum Problemcommunications by Experiment

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Reconstructing Quantum States •  Reconstruction of an unknown state based on data alone •  Generically, need d2 expectation values to reconstruct an unknown state in d dimensions •  full tomography tools for state identification inefficient, especially for continuous systems •  Brought down to O(rd log2(d)) for approx low-rank state with compressed sensing. •  Applicable for medium sized systems in conjunction with model selection. Approaches are based on using the right “data set” with the appropriate “sparsity structure” to capture quantum many-body systems. •  over permutation-invariant tomography •  matrix-product state tomography •  ....

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Quantum Field Tomography efficient tomographic state reconstruction for continuous quantum fields

continuous Matrix Product States (cMPS) naturally incorporate the locality present in realistic physical settings of locally interacting quantum field Phase correlation functions

with Extract low-order correlation functions

Reconstruct continuous matrix product states

Methods: Matrix pencils, prony methods Steffens, Friesdorf, Langen, Rauer, Schweigler, Huebener, Schmiedmayer, Riofrio, Eisert, Nature Comm (2015) Steffens, Riofrio, Huebener, Eisert, New J Phys 16, 123010 (2014) J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Quantum field tomography A. Steffens et al. Nature Comm (2015) arxiv:1406.3632

Use 2- and 4-point functions to reconstruct higher-order functions via continuous matrix product states with bond length 2 experimental 6-point function

reconstructed 6-point function

reconstruction of a quantum field with very weak assumptions Theory: A. Steffens, C. Riofrıo, R. Hubener, and J. Eisert, “Quantum field tomography,” NJP 16 (2014) 123010. J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Quantum field tomography A. Steffens et al. Nature Comm (2015) arxiv:1406.3632

Use 2- and 4-point functions to reconstruct higher-order functions via continuous matrix product states with bond length 2

reconstruction of the C-MPS wave functions gets worse with time C-MPS with bond length 2 have finite entanglement Question: Can one build a measure for entanglement growth after the quench? similar to data compressibility criteria?

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Outlook Non trivial (squeezed) initial states Relaxation in SG moel

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J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

Optimal Control of Splitting fast squeezing in a multi mode system

Vienna experiment: Preliminary

Optimal Controll applied to the problem of the fluctuation properties in splitting a BEC

Adiabatic splitting

J. Grond et al. PRA 79, 021603 R (2009) J. Grond et al. PRA 80, 053625 (2009)

–  Fancy splitting ramps inspired by OCT: t1+t2 = 17ms –  Leads to dramatic change of statistical distribution of interference

full cloud 140 µm long

t1

8ms

t2

tHold

10 ms

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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Squeezing T. Berrada, et al., Nat. Comm 4, 2077 (2013)

number'and'phase'distribu/on' (black:(measured,(blue:(binomial,(red:(detec4on(noise)(

RMS fluctuations of the number difference

Whereas Spin squeezing: Implies that ≈ 150 atoms are entangled! RMS fluctuations of the phase Whereas

Δn Δφ = 2.3 (7) J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

when correcting for measurement noise:

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Evolution of ξ2∼ -8dB 1d gas Tunnel Coupled ωp=14Hz

T. Berrada prelininary Could this oscillation come instead from the slow axial breathing? The fringe spacing oscillates at f = 18.2 ± 0.1 2.6 Hz.

0.6 z = (nL − nR )/N

0.05

0.4 0.2

0

⟨φ⟩ (rad)

−0.05

0 −0.1 −1

−0.2

−0.5

0 φ0 /π

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1

Figure 4.6: Phase portrait of the RFAmp = 0.65 double well Numerical resolution of the BJJ equations for ⇤ = 2400. Blue: Josephson oscillations, red: MQST (z0 = 0.1, 0.2, ..., 1), black: separatrix. The thick blue line is the trajectory for the initial conditions 0 = 0.5 rad and z = 0. Gray ellipse: number and phase spread in the initial state.

−0.4 −0.6

GPE 1D, φ0=0.25 rad, z0 = 0

−3

x 10

8

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60 t (ms)

80 pop imbalance

Separated

−1 0

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2 0

Figure 4.2: Set A: Oscillation of the mean phase A fit with a sine yields: frequency f = 15.6 ± 3.4 Hz; −2 amplitude: 0.5 ± 0.1 rad; o↵set: 0.3 ± 0.1 rad. −4 −6

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Figure 4.8: Carpets. Experimental carpets after integration along the longitudinal axis (central mF = 0 cloud only) and along the transvserse cloud. Each carpet is the result of averaging over ⇠ 50 repetitions. Note that the proportion of atoms in the three Zeman states is time-dependent.

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment102

101 0

80

t [ms]

t [ms]

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Figure 4.7: Simulated Josephson oscillation GPE 1D simulation in the RFAmp = 0.65 double well for the initial condition 0 = 0.25 rad and z = 0. f = 13.9 Hz, zmax = 0.007.

10 50 t [ms]

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0 0

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Figure 4.3: Set A: Oscillation of the fringe spacing and the axial width Left: fringe spacing (f = 18.2 ± 2.6 Hz); Right: FWHM in the axial direction (f = 23.9 ± 1.3 Hz)

Relaxation in coupled superfluids

number imbalance

number imbalance

phase

re-coupling starts SG model with a specific phase experiment: M. Pigeur theory: E. DelaTorre, E. Demler -> study phase locking

phase locking as a fix-point of the evolution time

phase

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

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What have we learned •  high order (>10) correlation functions are accessible in experiment •  Higher order correlation functions and the question if they factorize (full distribution functions) gives insight in the effective theories describing the many body system •  quasi particles •  interaction of quasi particles •  vacuum states •  Quantum field tomography opens up a way to extract information by using model cMPS wave-functions •  Experiments allow to probe how classical statistical properties emerge from microscopic quantum evolution through de-phasing of many body eigenstates.

J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

Schweigler et al. arXiv:1505.03126 Steffens, et al., Nature Comm (2015) Steffens, et al., NJP 16, 123010 (2014) LDQCM-15

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Papers Probing Quantum Fields by correlations Schweigler et al. arXiv:1505.03126 Quantum Field Tomography Steffens, et al., Nature Comm. (2015) Steffens, et al., NJP 16, 123010 (2014)

(exp) (theory)

Non equilibrium an relaxation in 1d systems Gring et al., Science 337, 1318 (2012) Kuhnert et al., PRL 110, 090405 (2013) Smith et al. NJP 15, 075011 (2013) Langen et al., Nature Physics 9, 460 (2013) Geiger et al. NJP 16 053034 (2014) Langen et al. Science 348, 207 (2015) Interferometer with trapped BEC Berrada, et al., Nature Comm. 4, 2077 (2013) Van Frank, et al., Nature Comm. 5, 4009 (2014)

(squeezing and entanglement)

Coolig a 1d quantum gas Rauer et al., arXiv:1505.04747 J. Schmiedmayer: On ‘Solving’ a Quantum Many-Body Problem by Experiment

Atom Chip Experiment S. Manz, T. Betz, R. Bücker, T. Berrada, S. vanFrank, M. Pigeur, A. Perrin, T Schumm, JF Schaff, R. Wu, M. Bonneau

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Atoms being loaded into a spiral on AtomChip

M. Kuhnert, M. Gring, B. Rauer, Th. Schweigler D. Smith, R. Geiger, T. Langen

Atom Chip Fabrication

D. Fischer, M. Trinker, M. Schamböck (ATI) S. Groth (HD), Israel Bar Joseph (WIS)

Theory Collaboration

I. Mazets, P. Grisins (ATI) J. Grond, U. Hohenester (Univ. Graz) E. Demler, T. Kitagawa + … (Harvard) T. Gasenzer, J. Berges, S. Erne, V. Kasper + … (Heidelberg) T. Calarco, S. Montanegro + … (Univ. Ulm) J. Eisert + …. (FU-Berlin) EU: SIQS, QIBEC, AQuS AT: FWF, CoQuS, Wittgenstein, Stadt Wien ERC AdG: QuantumRelax

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