Exploring Quantum Simulations with Superconducting Circuits

Transcription

1 Exploring Quantum Simulations with Superconducting Circuits Andreas Wallraff (ETH Zurich) Team: J.-C. Besse, R. Buijs, M. Collodo, S. Garcia, S. Gasparinetti, J. Heinsoo, S. Krinner, P. Kurpiers, M. Oppliger, M. Pechal, A. Potocnik, Y. Salathe, M. Stammeier, A. Stockklauser, T. Thiele, T. Walter (ETH Zurich)

2 The ETH Zurich Quantum Device Lab incl. undergrad and summer students

3 Acknowledgements Former group members now Faculty/PostDoc/PhD/Industry A. Abdumalikov (Gorba AG) M. Allan (Leiden) M. Baur (ABB) J. Basset (U. Paris Sud) S. Berger (AWK Group) R. Bianchetti (ABB) D. Bozyigit (MIT) C. Eichler (Princeton) A. Fedorov (UQ Brisbane) A. Fragner (Yale) S. Filipp (IBM Zurich) J. Fink (IST Austria) T. Frey (Bosch) M. Goppl (Sensirion) J. Govenius (Aalto) L. Huthmacher (Cambridge) D.-D. Jarausch (Cambridge) K. Juliusson (CEA Saclay) C. Lang (Radionor) P. Leek (Oxford) P. Maurer (Stanford) J. Mlynek (Siemens) M. Mondal (Johns Hopkins) G. Puebla (IBM Zurich) A. Safavi-Naeini (Stanford) L. Steffen (AWK Group) A. van Loo (Oxford) S. Zeytinoğlu (ETH Zurich) Collaborations with (groups of): A. Blais (Sherbrooke) C. Bruder (Basel) M. da Silva (Raytheon) L. DiCarlo (TU Delft) K. Ensslin (ETH Zurich) J. Faist (ETH Zurich) J. Gambetta (IBM Yorktown) K. Hammerer (Hannover) T. Ihn (ETH Zurich) F. Merkt (ETH Zurich) L. Novotny (ETH Zurich) T. J. Osborne (Hannover) B. Sanders (Calgary) S. Schmidt (ETH Zurich) R. Schoelkopf (Yale) C. Schoenenberger (Basel) E. Solano (UPV/EHU) W. Wegscheider (ETH Zurich)

4 Classical and Quantum Electronic Circuit Elements basic circuit elements: charge on a capacitor: quantum superposition states of: current or magnetic flux in an inductor: charge q flux φ commutation relation (c.f. x, p):

5 Constructing Linear Quantum Electronic Circuits basic circuit elements: harmonic LC oscillator: energy: electronic photon classical physics: quantum mechanics: Review: M. H. Devoret, A. Wallraff and J. M. Martinis, condmat/ (2004)

6 Constructing Non-Linear Quantum Electronic Circuits circuit elements: anharmonic oscillator: non-linear energy level spectrum: Josephson junction: a non-dissipative nonlinear element (inductor) electronic artificial atom Review: M. H. Devoret, A. Wallraff and J. M. Martinis, condmat/ (2004)

7 How to Operate Circuits in the Quantum Regime? control circuit quantum circuit read-out circuit recipe: avoid dissipation work at low temperatures isolate quantum circuit from environment Review: M. H. Devoret, A. Wallraff and J. M. Martinis, condmat/ (2004)

8 Cavity QED with Superconducting Circuits coherent quantum mechanics with individual photons and qubits basic approach: What is this good for? Isolating qubits from their environment Maintain addressability of qubits Reading out the state of qubits Coupling qubits to each other Converting stationary qubits to flying qubits

9 Cavity QED with Superconducting Circuits coherent interaction of photons with quantum two-level systems... J. M. Raimond et al., Rev. Mod. Phys. 73, 565 (2001) S. Haroche & J. Raimond, OUP Oxford (2006) J. Ye., H. J. Kimble, H. Katori, Science 320, 1734 (2008) Properties: strong coupling in solid state sys. easy to fabricate and integrate Research directions: quantum optics quantum information hybrid quantum systems A. Blais, et al., PRA 69, (2004) A. Wallraff et al., Nature (London) 431, 162 (2004) R. J. Schoelkopf, S. M. Girvin, Nature (London) 451, 664 (2008)

10 Realization device fabrication: L. Frunzio et al., IEEE Trans. on Appl. Supercon. 15, 860 (2005)

11 J. Mlynek et al., Quantum Device Lab, ETH Zurich (2012)

12 Sample Mount ~ 2 cm M. Peterer et al., Quantum Device Lab, ETH Zurich (2012)

13 Cryostate for temperatures down to 0.02 K Microwave control & measurement equipment ~ 20 cm

14 How to do the Measurement prevent leakage of thermal photons (cold attenuators and circulators) average power to be detected (ω r /2π = 6 GHz, κ/2π = 1 MHz) efficient with cryogenic low noise HEMT amplifier T N = 6 K

15 Resonant Vacuum Rabi Mode Splitting... with one photon (n=1): very strong coupling: forming a 'molecule' of a qubit and a photon first demonstration in a solid: A. Wallraff et al., Nature (London) 431, 162 (2004) this data: J. Fink et al., Nature (London) 454, 315 (2008) R. J. Schoelkopf, S. M. Girvin, Nature (London) 451, 664 (2008)

16 Resonant Vacuum Rabi Mode Splitting... with one photon (n=1): very strong coupling: vacuum Rabi oscillations forming a 'molecule' of a qubit and a photon first demonstration in a solid: A. Wallraff et al., Nature (London) 431, 162 (2004) this data: J. Fink et al., Nature (London) 454, 315 (2008) R. J. Schoelkopf, S. M. Girvin, Nature (London) 451, 664 (2008)

17 Quantum Optics with Supercond. Circuits Strong Coherent Coupling Chiorescu et al., Nature 431, 159 (2004) Wallraff et al., Nature 431, 162 (2004) Schuster et al., Nature 445, 515 (2007) Root n Nonlinearities Fink et al., Nature 454, 315 (2008) Deppe et al., Nat. Phys. 4, 686 (2008) Bishop et al., Nat. Phys. 5, 105 (2009) Parametric Amplification & Squeezing Castellanos-Beltran et al., Nat. Phys. 4, 928 (2008) Abdo et al., PRX 3, (2013) Microwave Fock and Cat States Hofheinz et al., Nature 454, 310 (2008) Hofheinz et al., Nature 459, 546 (2009) Kirchmair et al., Nature 495, 205 (2013) Vlastakis et al., Science 342, 607 (2013) Wang et al., Science 352, 1087 (2016) Waveguide QED Qubit Interactions in Free Space Astafiev et al., Science 327, 840 (2010) van Loo et al., Science 342, 1494 (2013)

18 Hybrid Systems with Superconducting Circuits Spin Ensembles: e.g. NV centers D. Schuster et al., PRL 105, (2010) Y. Kubo et al., PRL 105, (2010) Polar Molecules, Rydberg, BEC P. Rabl et al, PRL 97, (2006) A. Andre et al, Nat. Phys. 2, 636 (2006) D. Petrosyan et al, PRL 100, (2008) J. Verdu et al, PRL 103, (2009) CNT, Gate Defined 2DEG, or nanowire Quantum Dots M. Delbecq et al., PRL 107, (2011) T. Frey et al., PRL 108, (2012) K. Petersson et al., Nature 490, 380 (2013) Nano-Mechanics J. Teufel et al., Nature 475, 359 (2011) X. Zhou et al., Nat. Phys. 9, 179(2013) Rydberg Atoms S. Hoganet al., PRL 108, (2012) x z vz and many more

19 Quantum Computing with Superconducting Circuits Teleportation L. Steffen et al., Nature 500, 319 (2013) M.. Baur et al., PRL 108, (2012) Circuit QED Architecture A. Blais et al., PRA 69, (2004) A. Wallraff et al., Nature 431, 162 (2004) M. Sillanpaa et al., Nature 449, 438 (2007) H. Majer et al., Nature 449, 443 (2007) M. Mariantoni et al., Science 334, 61 (2011) R. Barends et al., Nature 508, 500 (2014) Error Correction M. Reed et al., Nature 481, 382 (2012) Corcoles et al., Nat. Com. 6, 6979 (2015) Ristè et al., Nat. Com. 6, 6983 (2015) Kelly et al., Nature 519, (2015) Deutsch & Grover Algorithm, Toffoli Gate L. DiCarlo et al., Nature 460, 240 (2009) L. DiCarlo et al., Nature 467, 574 (2010) M. Reed et al., Nature 481, 382 (2012)

20 Quantum Simulation describes physical system of interest (physics, chemistry, biology, ) encodes hard classical problem interesting toy model use universal quantum computer still to be realized! simulating a quantum system OR difficult on classical computer! map! quantum simulator sufficient controllability, flexibility! Feynman, Int. Journal of Th. Phys. 21, 467 (1982) Llloyd, Science 273, 5278 (1996)

21 Systems for Quantum Simulation Ultracold gases Trapped ions more established Optical photons Bloch et al., Nat. Phys. 8, 267 (2012) Nuclear magnetic resonance Blatt & Roos, Nat. Phys. 8, 277 (2012) under development Solid state quantum devices Aspuru-Guzik & Walther, Nat. Phys. 8, 258 (2012) Vandersypen & Chuang, RMP 76, 1037 (2004) Georgescu et al., RMP 86, 153 (2014)

22 Quantum Simulation with Superconducting Circuits Digital simulation of exchange, Heisenberg, Ising spin models two-mode fermionic Hubbard models Salathe et al., PRX 5, (2015) Quantum simulation of correlated systems with variational Ansatz based on MPS Barends et al., Nat. Com. 6, 7654 (2015) Eichleret al., Phys. Rev. X 5, (2015) O Malley et al., arxiv: (2015) Analog simulations with cavity and/or qubit arrays Houck et al., Nat Phys. 8, 292 (2012) Raftery et al., Phys. Rev. X 4, (2014)

23 Heisenberg Model Simulated with Supercon. Circuits Digital simulation of Heisenberg spin model Salathe et al., PRX 5, (2015)

24 Quantum Properties of Propagating Microwaves Single photon sources and their anti-bunching Preparation and characterization of qubitpropagating photon entanglement Bozyigit et al., Nat. Phys 7, 154 (2011) Lang et al., PRL 107, (2011) Full state tomography and Wigner functions of propagating photons Eichler et al., PRL 106, (2011) Eichler et al., PRL 109, (2012) Eichler et al., PRA 86, (2012) Hong-Ou-Mandel: Two-photon interference incl. msrmnt of coherences at microwave freq. Lang et al., Nat. Phys. 9, 345 (2013)

25 On-Demand Pulsed Single Photon Source Step 2: Map qubit state to resonator by 1/2 vacuum Rabi oscillation Step 3: Measure at the output using linear amplifier and signal processing hardware Step 1: Prepare qubit state by Rabi oscillation D. Bozyigit et al., Nat. Phys. 7, 154 (2011), S. Deleglise et al. Nature 455, (2008) M. Hofheinz et al., Nature 454, 310 (2008), A. Houck et al., Nature 449, 328 (2007)

26 Single Sided Cavity and Beam Splitter

27 Detection Scheme using Beam Splitters, Amplifiers... beamsplitter: bosonic field operators signal mode output mode linear amplifier: noise mode C. Caves, PRD 26, 1817 (1982).

28 ... and Quadrature Detectors measuring field amplitude instead of photon number: measured observable signal optical analogue IQ mixer M. P. da Silva et al., PRA 82, (2010). homodyne detection scheme

29 The Signal in One Channel of the Setup signal mode, vacuum mode Beam splitter Linear amplifier Mixer effective noise mode analogous for second channel!

30 Field Quadrature and Photon Number Measurements Measure quadratures at channel b: Measure cross-power between channel e&f:

31 Single-Channel Power vs. Two-Channel Cross Power Single channel power: system noise uncorrelated from signal signal photons... vs. cross power: added noise photons system noise is Gaussian with vanishing mean in vacuum Noise in 2 detection channels uncorrelated similar for higher order moments: whereas

32 G (2) Measurement of Pulsed Single Photon Source First measurement of intensity/intensity correlation function of a microwave frequency single photon source D. Bozyigit et al., Nat. Phys. 7, 154 (2011) M. P. da Silva et al., PRA 82, (2010) C. Eichler et al., PRA 86, (2012)

33 Photon Blockade: A Single Photon Turnstile Level diagram: Vacuum Rabi mode splitting: Drive: Photon blockade: first photon enters cavity second is blocked mediated photon/photon interactions Effective two-level system (polariton) Mollow-type triplet: C. Lang et al., PRL 107, (2011)

34 Polariton Mollow Triplet Measurement (cross-)power spectrum: observations: Rayleigh scattering resonance fluorescence C. Lang et al., PRL 106, (2011)

35 G (2) of Continuously Pumped Single Photon Source resonant photon blockade: anti-bunching and sub-poissonian photon statistics thermal (bunching) & coherent (poissonian) fields C. Lang et al., PRL 106, (2011) M. P. da Silva et al., PRA 82, (2010) C. Eichler et al., PRA 86, (2012)

36 Microwave Photon Field Detection amplitude detector = ADC & digital filtering linear amplifier complex amplitude: adds vacuum/thermal noise signal Eichler et al., PRA 86, (2012) M. P. da Silva et al., PRA 82, (2010) noise C. M. Caves, PRD 26, 1817 (1982) noise added by HEMT amplifier : signal photon G G...~50 noise photons

37 Full Tomography of a Single Propagating Mode 1) prepare in vacuum : vacuum noise G photon source noise mode record histogram of measurement results normal distribution with variance introduces thermal noise with mean photon number

38 Coherent State Histograms 2) prepare in coherent state : MW generator 3) rotate phase : MW generator Question: What can we learn about state when?

39 Single Photon Source Histogram store 2D histogram from measurement results: corresponding phase space distribution signal mode in vacuum Q - function of noise mode : convolution with P function of signal subtracted histograms to visualize difference signal mode in single photon Fock state separate noise signal from systematically! C. Eichler, et al., PRL 106, (2011), PRA 86, (2012)

40 Statistical Analysis of Histograms systematic mode separation: histogram moments: 1. calculate histogram moments 2. algebraic inversion vacuum Fock state reminder: C. Eichler, et al., PRL 106, (2011), PRA 86, (2012)

41 State Dependent Moments of Probability Distribution moments for different prepared states: Fock state mean photon number mean amplitude superposition anti bunching coherent state coherent state C. Eichler, et al., PRL 106, (2011), PRA 86, (2012)

42 Reconstructed Wigner Function of Itinerant Photon Wigner function reconstructed from measured moments: with Fock state superposition C. Eichler, et al., PRL 106, (2011), PRA 86, (2012)

43 Reconstruct Density Matrices and Wigner Functions for propagating multi-photon Fock states and their superpositions: Density matrices Wigner functions measured using near-quantum-limited parametric amplifier C. Eichler, et al., PRL 106, (2011) C. Eichler et al., PRA 86, (2012)

44 Two Single Photon Sources and Beam Splitter

45 Two Single Photon Sources and Beam Splitter

46 (2) Hong-Ou-Mandel g (τ) for Microwave Photons Observations: Photon-Pair anti-bunching For τ > 0: Broadening of satellite peaks Triple-peak structure of satellite peaks Full recovery of double-peak at Exp.: C. Lang, C Eichler et al., Nat. Phys. 9, 345 (2013) Theo.: M. Woolley et al., New J. Phys. 15, (2013)

47 Learning More: Two-Channel Tomography Idea: Measure 4D histogram and evaluate relevant photon statistics analogous to 1-channel case 1 average photon in each channel 2-photon bunching quantum superposition! no simultaneous click in both channels C. Lang, C Eichler et al., Nat. Phys. 9, 345 (2013)

48 Density Matrix Displaying Two-Mode Entanglement Density matrix reconstruction: moments linear map Fock space density matrix use maximum likelihood procedure final state: F = 0.93 F = 0.86 C. Lang, C Eichler et al., Nat. Phys. 9, 345 (2013)

49 Dissipative Bose Hubbard Dimer Unconventional photon blockade effect Liew & Savona, PRL (2010) Bamba et al., PRA 83, (2011) emission hopping Simulate quantum many body physics with light Greentree et al. Nat. Phys., (2006) Angelakis et al, PRA 76, (2007) Hartmann et al., Nat Phys (2006) Schmidt et al., PRB (2010) photon-photon interaction Realize nondegenerate parametric amplifier Eichler et al., Phys. Rev. Lett. 113, (2014) Josephson oscillations Gerace et al. Nat. Phys., (2009) Abbarchi et al. Nat. Phys., (2013)

50 Why parametric amplifiers? QUANTUM LIMITED AMPLIFICATION FOR MW photon correlations Qubit readout Mallet et al. Nat. Phys., (2009) Vijay et al, PRL 106, (2011) Sank et al., arxiv (2014) Quantum feedback Mallet et al., PRL. 106, (2011) Eichler et al., PRL & PRA (2012) G Pechal et al., PRX 4, (2014) Riste et al., PRL 109, (2012) Mlynek et al., Nat. Com. 5, 5186 (2014) Vijay et al., Nature 490, 77 (2012) Steffen et al., Nature 500, (2013) 2-5% best commercial amps Detection efficiency: -> 100% parametric amps Squeezing Microwave CV Entanglement Quantum dots, spectroscopy/ Mechanical devices, Castellanos-Beltran et al., Nat. Phys. 4, 929 (2008) Hybrid systems Rydberg atoms, Eichler et al., PRL. 107, (2011) Flurin et al., PRL 109, (2012). ANY CRYOGENIC SETUP Menzel et., PRL109, (2012). OPERATING at MW FREQUENCY!

51 Parametric amplifier: Basic working principle coherent signal Spectral density pump Pump nonlinear medium signal + idler photon pairs idler 4-wave mixing: occupy same mode degenerate paramp Conversion stimulated by: 1) vacuum fluctuations -> spontaneous parametric down-conversion (SPDC) 2) input (quantum) signals ->parametric amplification Yurke et al., PRL 60, 764 (1988) Castellanos-Beltran et al., Nat. Phys. 4, 929 (2008) Eichler and Wallraff, EPJ 1, (2014)

52 Degenerate vs. nondegenerate paramp Degenerate paramp idler gain pump signal gain desirable for: single mode squeezing fast readout quantum feedback Non-degenerate paramp desirable for: conventional phase-preserving amplification multiplexed readout photon correlation measurements

53 JPD amplifier: implementation finger capacitor array of SQUIDs Features: Arrays with M SQUIDs to control nonlinearity: asymmetric SQUIDs -> homogeneous coupling to external flux Eichler et al., Phys. Rev. Lett. 113, (2014)

54 JPD amplifier: implementation Features: Arrays with M SQUIDs to control nonlinearity: lumped element -> large bandwidth possible asymmetric SQUIDs -> homogeneous coupling to external flux Eichler et al., Phys. Rev. Lett. 113, (2014)

55 Linear spectroscopy of JPD device Extract From fit to input-output model Eichler et al., Phys. Rev. Lett. 113, (2014)

56 Frequency Tuneability: Linear Response Eichler et al., Phys. Rev. Lett. 113, (2014)

57 JPD amplifier: non-degenerate operation Idler gain Signal gain ~ 0.5 GHz Increase pump power Eichler et al., Phys. Rev. Lett. 113, (2014)

58 JPD amplifier: degenerate operation Idler gain Signal gain Eichler et al., Phys. Rev. Lett. 113, (2014)

59 Entanglement of signal and idler noise vacuum noise + pump Entangled signal + idler photons + pump How does the entanglement become evident? idler signal Vacuum fluctuations detuning from pump phase between LO and pump

60 Entanglement of signal and idler fields anti-squeezing Vacuum level squeezing squeezing detuning from pump phase between LO and pump Eichler et al., Phys. Rev. Lett. 113, (2014)

61 Summary JPD G Very good properties in terms of dynamic range, bandwidth, noise, tunability, Two-mode squeezing of more than -12 db Degenerate and Non-degenerate operation possible Broadly applicable in cryogenic systems Eichler et al., Phys. Rev. Lett. 113, (2014)

62 Quantum Simulation describes physical system of interest (physics, chemistry, biology, ) encodes hard classical problem interesting toy model use universal quantum computer large scale one still to be realized! simulating a quantum system OR difficult on classical computer! map! quantum simulator sufficient controllability, flexibility! Feynman, Int. Journal of Th. Phys. 21, 467 (1982) Llloyd, Science 273, 5278 (1996)

63 Find Ground State of Hamiltonian Typically: Quantum system Cooling or Annealing Ground state

64 Alternative Paradigm for Quantum Simulation Ground state well described by Variational Ansatz Here, specific class of states: - Matrix product states (MPS) - Projected entangled pair states Verstraete, Murg & Cirac Advances in Physics (2008) Not necessarily identical! Use to simulate Controllable quantum system Barrett et al., PRL 110, (2013) Eichleret al., Phys. Rev. X 5, (2015)

65 Variational Quantum Simulation using Cavity QED 1) Create variational (cmps) states using CQED 2) Evaluate Hamiltonian of interest using correlation measurements of. 3) Vary state using external control Proposal: Barrett et al., PRL 110, (2013) Realization: Eichleret al., Phys. Rev. X 5, (2015)

66 (Discrete) Matrix Product States 1D lattice model (d levels per site): Most general state has coefficients Idea: Reduce states space to relevant one. Properties: Satisfy area law for the entanglement entropy Represent exact ground state of specific models Generalization to higher dimensional lattices Close relation to DMRG Continuous limit Connection with cavity QED and input/output theory with D-dim. matrices -> matrix elements Reviews: Verstraete et al., Adv. Phys. (2008) Schollwoeck, Ann. Phys (2012) Orus, Ann. Phys. (2014) cmps and cavity QED: Verstraete et al., PRL 104, (2010) Barrett et al., PRL 110, (2013) Osborne et al., PRL 105, (2010)

67 (Continuous) Matrix Product States Definition: field operator with Equivalence to discrete MPS by choosing (for bosons): act on auxiliary system and taking the continuum limit. Features: Also possible for fermionic fields Generalization to vector fields with multiple components Verstraete and Cirac, PRL 104, (2010)

68 Cavity QED output as cmps generated by auxiliary system through with Compare with non-hermitian representation of cavity QED Hamiltonian Cavity acts like auxiliary system to generate MPS in the 1D continuum. Assumptions made here: Cavity input field is in the vacuum Internal decay negligible Osborne et al., PRL 105, (2010) Barrett et al., PRL 110, (2013)

69 What we Simulate Here: Gas of interacting bosons in 1D continuum Described by the Lieb-Liniger model kinetic energy only one parameter in the model! repulsive interaction chemical potential Lieb & Liniger Phys. Rev. 130, 1605 (1963) Paredes et al., Nature 429, 277 (2004)

70 What is needed? 1) Tunable cavity QED system 2) Efficient & programmable measurement apparatus 3) Simulation of the Lieb-Liniger model 4) Future perspectives

71 Cavity QED with Superconducting Circuits cavity Cavity Atom Transmission line resonator Superconducting qubit atom Small mode volume (1D) Large dipole moments Strong coupling A. Blais, et al., PRA 69, (2004) A. Wallraff et al., Nature (London) 431, 162 (2004) R. J. Schoelkopf, S. M. Girvin, Nature (London) 451, 664 (2008)

72 Circuit QED Approach to Creating cmps cavity = piece of a coax cable Radiation field stored inside: sum of harmonic oscillators Energy atom transmon = Josephson Junction = Nonlinear inductor anharmonic system f e g qubit Y. Nakamuraet al., Nature 398, 786 (1999) J. Koch, et al., PRA 76, (2007) Change transition energy on ns timescales!

73 Circuit QED Device with Tunable Coupling Transmon Out In asymmetric resonator coupling control using global field + local flux line tunable frequency and tunable coupling Local flux line Mediate effective photon-photon interactions with tunable strength Eichleret al., Phys. Rev. X 5, (2015) Srinivasan et al., PRL 106, , (2011)

74 Measurement Apparatus Variational parameters: At GHz frequencies? Lieb-Liniger Hamiltonian Field operator Radiation field Cavity output field Measure photon correlation functions!

75 Experiments with Propagating Quantum Microwaves Time-correlations functions for continuous single photon source Lang et al., PRL 107, (2011) Bozyigit et al., Nat. Phys 7, 154 (2011) Eichler et al., PRL 106, (2011) since then experiments on: Squeezing Wigner Tomography Qubit-Photon-Entanglement Hong-Ou-Mandel interference Photon shaping Superradiance See ETH Qudev lab publications improved detection efficiency with quantum limited amplifiers linear amplifier adds vacuum/thermal noise Quantum limited amplifiers: (Special requirements in terms of dynamic range, bandwidth, phase-insensitivity) Eichler et al., PRL (2014) c.f.: Castellanos-Beltran et al., Nat. Phys. 4, 929 (2008) Bergeal et al., Nature 465, 64 (2010) reduce g 2 measurement time by ~10000

76 Time-resolved Correlation Measurements Drive nonlinear cavity mode vs. two variational parameters: Drive rate Effective anharmonicity: Eichleret al., Phys. Rev. X 5, (2015)

77 Simulate Lieb-Liniger Hamiltonian How to measure? Barrett et al., PRL 110, (2013)

78 Correlations vs. Variational Parameters Measured correlations vs. variational parameters: Chemical Potential: Kinetic energy: Interaction energy: Energy in variational space Eichleret al., Phys. Rev. X 5, (2015)

79 Measured Energy Landscape Energy landscape vs. variational parameters: Local minimum in variational space Ground state depends on interaction strength v Parametric amplifier reduces g 2 measurement time by ~10000 variational ground state Eichleret al., Phys. Rev. X 5, (2015)

80 Properties of the Simulated Ground State Ground state energy vs. interaction strength Experimentally obtained Tonks-Giradeau limit Exact numerical result We can do more than that: We can probe any quantity of interest! at particle density Eichleret al., Phys. Rev. X 5, (2015)

81 Properties of the Simulated Ground State Experimentally obtained first order correlation function: Conversion of temporal into spatial coordinates Decrease of correlation length with increasing interaction strength No Bose condensation in 1D (Mermin Wagner) Numerical result with small bond dimension D similar to experimental data

82 Properties of the Simulated Ground State Experimentally obtained particle-particle correlations: crossover from weakly to strongly interacting Bose gas (Tonks-Giradeau) qualitative agreement already for a simulation with two variational parameters quantitative agreement requires additional variational parameters in experiments

83 Continuous Matrix Product States: Summary Variational approach System suitable for creating cmps experimentally Efficient correlation measurements High level of control/tunability Some potential for scaling Many-body physics Atomic, solid state, bio-physics Quantum information Quantum field theories Discrete Lattice models Vector fields Fermionic systems Extension to higher dim s Eichleret al., Phys. Rev. X 5, (2015)

84 The ETH Zurich Quantum Device Lab incl. undergrad and summer students

85 Want to work with us? Looking for Excellent Postdoc and PhD Candidates!