Taming Non-Equilibrium Systems with QuTiP, the Quantum Toolbox in Python
Nathan Shammah “Taming Non-Equilibrium Systems: from Theoretical Quantum Physics Lab Quantum Fluctuation to Decoherence” Cluster for Pioneering Research July 31st 2019 RIKEN, Saitama, Japan ICTP Trieste, Italy https://www.nature.com/articles/d41586-019-02046-0 QuTiP: The Quantum Physics Simulator The Quantum Toolbox in Python QuTiP
July 3rd, 2019: QuTiP 4.4.0 Released
`conda install qutip`
More info: qutip.org The steady growth of Python
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4 Python’s strengths A community-based programming language
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From reproducible data to reusable code.
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8 Cloud-based Notebooks on Open Quantum Systems The Quantum Toolbox in Python
Take a snapshot You can find an interactive notebook at
https://github.com/nathanshammah/
Repository: interactive-notebooks
You can run the notebooks live Create an open-source scientific project Use the tools of open source: https://github.com/nathanshammah/opensource Quantum Tech: Open Source Libraries More open-source is empowering broad research in the field
Library Year Creators Institution Language Description
Rob Johannson Simulation of open quantum systems; 2012 Python Paul Nation RIKEN quantum optics, cavity QED. Franco Nori Nikolas Tezak, Computer algebra package for QNet 2012 Michael Goerz Stanford Python quantum mechanics and photonic Hideo Mabuchi quantum networks Quantum optics and open quantum Sebastian Krämer U Innsbruck 2017 Julia systems framework inspired by et al. IQOQI the QO toolbox in Matlab and QuTiP
Damian S. Steiger Hardware-agnostic framework with ProjectQ 2016 Thomas Häner ETH Zurich Python compiler and simulator with Matthias Troyer emulation capabilities.
Google Fermionic potential calculations 2017 Ryan Babbush Python et al. (unofficial) for quantum chemistry
Studying many-body quantum The Simons C++ 2018 Giuseppe Carleo systems with artificial neural NetKet Foundation Python interface networks and ML techniques.
Nathan Killoran Photonic quantum computing with 2018 Xanadu Inc Python et al. continuous-variable optical circuits Checkout more open-source projects at https://qosf.github.io !11 QuTiP: The Quantum Physics Simulator The Quantum Toolbox in Python QuTiP
A toolbox to study the open quantum dynamics of realistic systems.
System ρ
Environment QuTiP: The Quantum Physics Simulator The Quantum Toolbox in Python Spin Lattices
Cavity QED
Condensed Matter QuTiP Quantum Quantum Optics Error Correction
qutip.org
Optomechanics Superconducting Circuits Ion Traps QuTiP in the lab: Rydberg atoms The Quantum Toolbox in Python
Experimental Research arXiv:1806.04682v1 [quant-ph] 12 Jun 2018 High-fidelity control and entanglement of Rydberg atom qubits H. Levine, […] and M.D. Lukin Phys. Rev. Lett. 121, 123603 (2018)
Numerical model for single atoms The numerical model is implemented using the Python package QuTiP [2]. It includes the following three effects: 1. A static but random Doppler shift in each iteration of the experiment […].
2. Off-resonant scattering from the intermediate state |r⟩. […]. This process is modeled by Lindblad operators.
3. Finite lifetime of the Rydberg state |r⟩. QuTiP in the lab: Superconducting qubits The Quantum Toolbox in Python
Experimental Research arXiv:1903.05672 [quant-ph] 13 Mar 2019 Phonon-mediated quantum state transfer and remote qubit entanglement A. Bienfait, […] and A.N. Cleland, Science 10, 1126 (2019) QuTiP: What research enables The Quantum Toolbox in Python
Qubit Dynamics Optomechanics Optimal Control
Superconducting cQED Circuits Superradiance
Spin Chains Ultrastrong Coupling Spin Squeezing
Non Markovian Spin-boson Models Heterodyne Detection Master Equations … and more QuTiP: The Quantum Toolbox in Python About QuTiP
QuTiP is a free and open source library for efficient simulation of a wide variety of quantum systems.
Field-specific intuitive framework Advanced mathematical techniques The mathematics of quantum mechanics • Noisy Dynamics: Master equation solvers • Complex number matrices • Correlation functions (quantum regression formula) • Non-commutative algebras • Modularity: Permutational invariant quantum solver • Operators and superoperators Hierarchical equations of motion • Intuitive Python classes: QObj Waveguide photon scattering Results Topological quantum circuits
Efficient numerical calculations • Fast complex-complex matrix-vector multiplication with SSE3 intrinsics. • Multiprocessing: Enhanced parallel performance with OPENMP
• Sparse Matrices: Fast CSR (Compressed Sparse Row) matrix class. • Up to 100x improvement: in CSR adjoint & transpose in Hermitian verification, Krönecker product More info at http://qutip.org/ in partial trace calculation QuTiP: The building blocks: states, operators and gates The Quantum Toolbox in Python
z >> from qutip import * H = >> H = sigmaz()/2
>> a = destroy(2) a,
p >> tensor(rho1, rho2)
⇢
Project Impact Authors Users Comp. Phys. Comm. 183, 1760–1772 (2012); ibid. 184, 1234 (2013). Code Distribution of 25k website visitors (2016)
>600 citations (Google Scholar)
(conda forge)
More info at http://qutip.org/ Rakuten Inc. IBM Q RIKEN / U. Michigan
Timeline: Previous Lead Developers Contributing Developers License: BSD Inspired by the Quantum • Neill Lambert (RIKEN) (Berkeley Software Distribution) Toolbox in MatLAB. • Denis Vasilyev (Leibniz) Style: PEP8 compliant • Kevin Fischer (Stanford) Libraries used: 2011-2012: QuTiP 1.0 • Jonathan Zoller (Ulm University) • Scipy • Ben Criger (RWTH Aachen) • Matplotlib Aug 2015: 100 citations • NumPy • … • SymPy Éric Giguère • Cython Aug 2016: 200 citations U. Sherbrooke • Louis Tessler (RIKEN) • Shahnawaz Ahmed (Chalmers) (Lead) Jan 2017: QuTiP 4.0 • Nathan Shammah (RIKEN) (Lead) • Jupyter notebooks July 2018: QuTiP 4.3 • Online documentation July 2019: QuTiP 4.4 • GitHub: 44 contributors, 4k commits • Independent testing QuTiP: Ad-hoc visualization tools The Quantum Toolbox in Python QuTiP: Ad-hoc visualization tools The Quantum Toolbox in Python
qutip.org/tutorials
Nathan Shammah, RIKEN – Open-Source Scientific Computing for Quantum Technology: QuTiP 29th Jan 2019 – Google AI Quantum, USA QuTiP: Ad-hoc visualization tools The Quantum Toolbox in Python
Bloch Sphere Wigner Function Quantum Circuits
State Tomography Surface Code 3D Wigner Function Topological Circuits*
*open pull request QuTiP: The QObj class The Quantum Toolbox in Python
J. R. Johansson, P. D. Nation, and F. Nori, Comp. Phys. Comm. 183, 1760–1772 (2012) QuTiP: An open-source Python framework for the dynamics of open quantum systems QuTiP: The QObj class The Quantum Toolbox in Python
• State and operators are declared as QObj >> q = Qobj([1], [0]) Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket
• Generate states and operators
• Algebra (bosonic) >> d = destroy(2) Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, AB − BA ≠ 0 isherm = False
ℰ(ρ) = AρB† � = B* ⊗ A >> q.dag() Quantum object: dims = [[1], [2]], ρ· = �ρ shape = (1, 2), type = bra QuTiP: Numerical Solvers The Quantum Toolbox in Python
The Result class stores the expectation values of the operators passed to the solver. result_ref = mesolve(H, rho0, times, c_ops, e_ops) plot_expectation_values(result_ref, y_labels = “E[a’a], …,) Simple Example: A driven, damped single mode cavity.
Solvers:
- `mesolve`: Lindblad master equations
- `mcsolve`: Monte-Carlo trajectory
- `Floquet_modes`: Floquet theory
- bloch_redfield: Bloch-Redfield master equation
- `ssesolve`: Stochastic Schrödinger equation
- `smesolve`: Stochastic master equations QuTiP lead team: Acknowledgements and funding
Shahnawaz Ahmed Alex Pitchford Eric Giguère Dr. Neill Lambert Prof. Franco Nori Chalmers, Sweden Aberystwyth University U. de Sherbrooke RIKEN, Japan RIKEN, Japan (RIKEN, Japan) United Kingdom Canada U. of Michigan (USA)
@NathanShammah medium.com/quantum-tech GitHub: nathanshammah Consulting: quantika.co
QuTiP 4.4.0: Major updates The Quantum Toolbox in Python
Faster, Flexible Quantum Trajectories (Monte Carlo, `mcsolve`) More General Stochastic Master Equations (`smesolve`)
50x faster Time-dependent jumps Several integration methods implemented
Quantum Information Processing User-defined Quantum Gates molmer_sorensen Quantum Hellinger distance
Bug Fixes More info: qutip.org QuTiP: A Growing Ecosystem What’s going on • 2018: Joined NumFOCUS, foundation for scientific code (NumPy)
• 2019: Participating to Google Summer of Code 2019:
Student Applications opened March 26th. Closed April 9th. 3 students working on summer projects. https://github.com/qutip/qutip/wiki//Google-Summer-of-Code-2019 2019: Applied to 1st Google Season of Docs 2019: Technical writers projects for documentation.
• 2018-2019: Reaching out to the sci-dev community.
EuroScipy 2018 PyData 2018 FOSDEM'19 (Quantum Computing) EuroScipy 2019 1st QuTiP developers workshop July 2018 November 2018 February 2, 2019, September 2019 February 19-21, 2019 Trento, Italy Warsaw, Poland Brussels, Belgium Bilbao, Spain RIKEN, Wako, Japan
• 2018-2019: A growing QuTiP ecosystem of satellite libraries:
piqs krotov matsubara qupulse QuTiP ecosystem: QuTiP library QuTiP-based quantum A qutip plugin for QuTiP-integrated Like AstroPy, Now a qutip module optimal control library non-Markovian dynamics hardware control but for Quantum QuTiP ongoing expansion (2019)
qutip.nisq qutip.lattice qutip.tiqs Model noise in quantum information Model lattices in QuTiP Translational invariant Lindblad dynamics processing (QIP) A qutip.lattice module implementing Driven-dissipative systems Hamiltonians for paradigmatic 1D Liouvillian spectrum QuTiP’s QIP module represents ideal quantum circuits. lattice models.
Student: Boxi Li (ETH Zurich) Student: Saumya Biswas (U. Oregon) Idea Exploit the symmetries of the dynamics Objectives: Objectives: Spin chains and bosonic lattices - Go beyond gates as instantaneous unitary Single particle picture, transformation Liouvillian spectrum - Noise model for realistic devices - SSH model (topological properties) - Noise model for dissipative dynamics Spin chains and bosonic lattices - Bose-Hubbard model ρ· = ℒρ First results: Complement existing libraries: - Added Mølmer-Sørensen gate - QuSpin - Allowed user-defined gates - Added optical pulses for gate shaping - pythontb [ℒ, S] = 0 Code: Boxi Li. Github: BoxiLi Code: Saumya Biswas Mentors: Alex Pitchford, Mentors: Clemens Gneiting, Neill Lambert, Eric Giguere, Shahnawaz Ahmed, Shahnawaz Ahmed, F. Minganti, et.al. Phys. Rev. A 98, 042118 (2018) Nathan Shammah Nathan Shammah https://gsoc2019-boxili.blogspot.com https://latticemodelfunctions.blogspot.com Code: Fabrizio Minganti. Github: fminga
Partly Deployed In Deployment In Development QuTiP recent additions (2018-2019)
qutip.scattering qutip.piqs qutip.nonmarkov Model nonlinear photon scattering Efficiently model local dissipation The environment has a memory in multiple waveguides Collective dissipation Non-Markovian dynamics vs. Local dissipation How photons scatter into the waveguide Hierarchical Equations of Motion vs. Coherent coupling when the system is driven (HEOM). with some excitation field
Matsubara RWA +
TLS TLS RC
Ohmic bath Qubit J(ω) Environment
A. Fruchtman, et al., Sci. Rep. 6, 28204 (2016) K.A. Fischer, et.al. (2017), arXiv: 1710.02875 N. Shammah et al., Phys Rev A 98, 063815 (2018) N. Lambert et al.i arXiv: 1903.05892 Matsubara Code: Ben Bartlett. Github: bencbartlett Code: Nathan Shammah and Shahnawaz Ahmed Code: Shahnawaz Ahmed and Neill Lambert RWA +
TLS TLS RC
Flat bath
Hierarchical Eq. of Motion Reaction Coordinate (HEOM) (RC) QuTiP recent additions (2018-2019)
qutip.scattering qutip.piqs qutip.nonmarkov Model nonlinear photon scattering Efficiently model local dissipation The environment has a memory in multiple waveguides Collective dissipation Non-Markovian dynamics vs. Local dissipation How photons scatter into the waveguide Hierarchical Equations of Motion vs. Coherent coupling when the system is driven (HEOM). with some excitation field
Matsubara RWA +
TLS TLS RC
Ohmic bath Qubit J(ω) Environment
A. Fruchtman, et al., Sci. Rep. 6, 28204 (2016) K.A. Fischer, et.al. (2017), arXiv: 1710.02875 N. Shammah et al., Phys Rev A 98, 063815 (2018) N. Lambert et al.i arXiv: 1903.05892 Matsubara Code: Ben Bartlett. Github: bencbartlett Code: Nathan Shammah and Shahnawaz Ahmed Code: Shahnawaz Ahmed and Neill Lambert RWA +
+ = import qutip.piqs TLS TLS RC Flat bath
Hierarchical Eq. of Motion Reaction Coordinate (HEOM) (RC) Open quantum systems: Lindblad master equation In general, the Liouvillian space grows exponentially as 4N
System Size 4N System [⇢] Hilbert space 1011 H ρ [⇢] Liouvillian space 109 L Environment 107 2N N 105 d i † 1 † 1 † ρ = − [H, ρ] + γij LiρL − L Ljρ − ρL Lj dt ℏ ∑ ( j 2 i \ 2 i ) i,j Space Dimension 103 Approximations: - Born d 101 Markov ρ ⃗ = ℒ ρ ⃗ Numerically Unfeasible - dt - Rotating wave Liouvillian 0 5 10 15 20 N Qubits Number qutip.piqs: Exploiting permutational symmetry About a recent QuTiP module simulating the dynamics of many qubits d N 1 1 iℏ ρ = [H, ρ] + γ L ρL† − L†L ρ − ρL†L Problem ∑ ( i i i i i i) Solution dt i 2 2 Permutational symmetry System The Su(2) Pauli operators can be written ρ using Su(4) generators in Liouvillian space … … obtaining an exponential size reduction.
Exponential Space Reduction Environment 1015 The Liouvillian space grows exponentially as 4N dim( TLS) System Size 12 D 10 dim( PIQS) D 11 [⇢] Hilbert space 10 H dim( TLS) [⇢] Liouvillian space 9 H 9 L 10 dim( ) 10 HPIQS 107 106 Space size 105 103 Space Dimension 103 101 100 NumericallyUnfeasible 0 25 50 75 100 0 5 10 15 20 N N Nathan Shammah, RIKEN – Open-Source Scientific Computing for Quantum Technology: QuTiP 29th Jan 2019 – Google AI Quantum, USA Numerical method in the Dicke space Homogeneous local dissipation can be included in the dynamical model
Exponential Space Reduction Block-Diagonal Density Matrix Dicke space visualization ⇢ = j,m,m ⇢j,m,m0 j, m j, m0 1015 0 | ih | P ⇢1,1,1 ⇢1,1,0 ⇢1,1, 1 0 dim( TLS) N m D j = 12 2 10 dim( PIQS) D dim( ) m TLS ⇢1,0,1 ⇢1,0,0 ⇢1,0, 1 0 H m0 9 N 10 dim( PIQS) j = 1 H 2 106 ⇢1, 1,1 ⇢1, 1,0 ⇢1, 1, 1 Space size 0 103 0 0 0 ⇢0,0,0 100 0 25 50 75 100 N m0 Number of Dicke States
B. A. Chase and J. M. Geremia, Phys. Rev. A 78, 052101 (2008) F. Damanet, D. Braun, and J. Martin, Rev. A 94, 033838 (2016) N. Shammah et al. Phys. Rev. A (2017) Dicke superradiance: light emission from an atomic cloud Probing the dynamics of many excited qubits / two-level systems
= sym Superradiance emerges from collective coupling Ideal case
N N S L D ⇢⇢˙˙ == i!i0[[!⇢0,JJxz]+, ⇢]+ (JJ [⇢⇢ ]++ J+J ⇢J ,n ⇢[⇢J]++J ) Jz,n [⇢] 2 L 2 L 2 L n =1 n=1 X X
sym = ) symmetricstates ) N+1 (
sym
EmissionIntensity t
tD
M. Gross & S. Haroche, Physics Reports 93, 301 (1982) Superradiance in the symmetric limit Standard study confined in a limited sector of the Hilbert space N N S L D ⇢⇢˙˙ == i!i0[[!⇢0,JJxz]+, ⇢]+ (JJ [⇢⇢ ]++ J+J ⇢J ,n ⇢[⇢J]++J ) Jz,n [⇢] 2 L 2 L 2 L n =1 n=1 = Dicke states: Collective spin statesX X states m Energy N+1
= j Cooperative Number
M. Gross & S. Haroche, Physics Reports 93, 301 (1982) The Dicke space generalizes the singlet-triplet system Local processes connect the singlet (dark) state
N N S L D ⇢˙ = i!0[⇢,Jz]+ J [⇢]+ J ,n [⇢]+ Jz,n [⇢] 2 L 2 L 2 L n=1 n=1 X X
N=2
N Shammah et al., Superradiance with local phase-breaking mechanisms, Phys. Rev. A 2017, arXiv 1704.07066 Analytical techniques Hierarchy Truncation: Higher-moment correlations are neglected
Mean-field approximation Factorization
Bosonic approximation: Holstein-Primakoff (not suitable for local processes)
I. Holstein-Primakoff
Bosonic approximation: Polariton Modes valid in the dilute regime
II. Dilute regime without cooperative number conservation
N Shammah et al., Superradiance with local phase-breaking mechanisms, Phys. Rev. A 2017, arXiv 1704.07066 Superradiance and phase-breaking effects Exploring inner states by including local incoherent processes
N N S L D = ⇢˙ = i!0[⇢,Jz]+ J [⇢]+ J ,n [⇢]+ Jz,n [⇢] 2 L 2 L 2 L n=1 n=1 X X
m Energy states j Cooperative Number N+1
=
N Shammah et al., Superradiance with local phase-breaking mechanisms, Phys. Rev. A 2017, arXiv 1704.07066 Superradiance and phase-breaking effects Exploring inner states by including local incoherent processes
N N S L D = ⇢˙ = i!0[⇢,Jz]+ J [⇢]+ J ,n [⇢]+ Jz,n [⇢] 2 L 2 L 2 L n=1 n=1 X X
m Energy states j Cooperative Number N+1
=
N Shammah et al., Superradiance with local phase-breaking mechanisms, Phys. Rev. A 2017, arXiv 1704.07066 Superradiance and phase-breaking effects Exploring inner states by including local incoherent processes
N N S L D = ⇢˙ = i!0[⇢,Jz]+ J [⇢]+ J ,n [⇢]+ Jz,n [⇢] 2 L 2 L 2 L n=1 n=1 X X
m Energy states j Cooperative Number N+1
This dissipative superradiant regime has been observed = In solid-state experiments with NV-centers and color centers C Bradac et al., Nature Communications, 2017 A Angerer et al., Nature Physics, 2018
N Shammah et al., Superradiance with local phase-breaking mechanisms, Phys. Rev. A 2017, arXiv 1704.07066 Exploiting permutational symmetry Generalization to all-to-all Hamiltonians: Validity and Limitations
N d i w + * # " ⇢ = [H, ⇢]+ J [⇢]+ Jz [⇢]+ J+ [⇢]+ J ,n [⇢]+ Jz,n [⇢]+ J+,n [⇢] + a† [⇢]+ a[⇢] dt 2 L 2 L 2 L 2 L 2 L 2 L 2 L 2 L ~ n=1 X ⇣ ⌘
TLS Pauli operators [Jx,n,Jy,n0 ]=iJz,n n,n0 •Unitary Dynamics
Lindblad [⇢]=2A⇢A† A†A⇢ ⇢A†A LA •Non-unitary Dynamics Superoperator •Collective Dissipation • Local Dissipation • Bosonic Dissipation Range of applicability
Hamiltonian Dissipation States Collective spin operators only Homogeneous local couplings. Limited to identical qubit states. Complete graph (fully connected). Constant edges weight: no lattice distance. Dicke state basis as a visualization tool Density matrices of collective quantum states
Forbidden 0 >0
Can also study: GHZ state CSS states Thermal states Literature overview Permutational symmetry in Lindblad dynamics N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, and F. Nori, PRA 98, 063815 (2018), arXiv:1805.05129 11
loc ⇢ ⇢ col L [ ] cav ⇢ L [ ] L [ ] i N w ⇢˙ H, ⇢ † h 2 J ⇢ 2 Jz ⇢ 2 J ⇢ n 1 2 J ,n ⇢ 2 Jz,n ⇢ 2 J ,n ⇢ 2 a ⇢ 2 a ⇢ ⇓ ⇑ ↓ ↑ = − [ ]+ − + + − + + = L [ ] + L [ ] + L [ ] L [ ] + L [ ] + L [ ] L [ ] + L [ ] 11 ∑ loc ⇢ ⇢ col L [ ] cav ⇢ L [ ] L [ ] i N w ⇢˙ H, ⇢ † h 2 J ⇢ 2 Jz ⇢ 2 J ⇢ n 1 2 J ,n ⇢ 2 Jz,n ⇢ 2 J ,n ⇢ 2 a ⇢ 2 a ⇢ ⇓ ⇑ ↓ ↑ = − [ ]+ − + + − + + Features Hamiltonian H Collective TLS col ⇢ Local TLS loc ⇢ Cavity cav ⇢ L [ ] + L [ ] + L [ ] ∑ = L [ ] + L [ ] + L [ ] L [ ] + L [ ] Features Hamiltonian H Collective TLS col ⇢ Local TLS loc ⇢ Cavity cav ⇢
TLS/TLS-cavity Emission DephasingL Pump[ ] Emission DephasingL [ Pump] EmissionL Pump[ ]
w
⇓ ⇑ ↓ ↑ TLS/TLS-cavity Emission DephasingL Pump[ ] Emission DephasingL [ Pump] EmissionL Pump[ ] Dicke space formalism [61, 62] General Superoperators [64] General
Superoperators & Dicke space [67] General
Multi-level systems [69, 74] General
w Optical bistability [58, 59] h!xJx 2 Collective quantum jumps [63] h!xJx h!zzJz
2 2 Bistability in the XY model [71] h!xJx h!0Jz− h!xy Jx Jy
⇓ ⇑ ↓ ↑ Two-axis spin squeezing* [61, 62] − ih⇤ J−2 J 2 +
2+ − Entanglement witness [82] h−!xxJ(x −h!0)Jz Dicke space formalism [61, 62] General − − Steady-state superradiance [20, 67] h!0Jz Ramsey spectroscopy [136] h!0Jz Superradiant emission [77] h!0Jz Superfluorescence/subradiance [56] h!0Jz Superoperators [64] General Spin synchronization [15] h!0Jz
† Superradiant lasing [137] h!0Jz
† † Non-classical light [60] h!x a a hg aJ a J
+ − State engineering [141] ( +hg J) +a J( a† + )
+ − Superoperators & Dicke space [67] General Lasing [20, 67] hg (J a + J a†) + − Photon anti-bunching [79] hg (J a + J a†)
+ − Super/subradiance [74, 75] hg (J a + J a†)
+ − ( + † ) Spaser [68, 70, 80] hgJx a a
( + †) Multi-level systems [69, 74] General Superradiant PT [78] hgJx a a PT, Lasing, Chaos [57] hg J a J a† ( hg+ J) a J a† ′ + − − + † Super/subradiant PT, squeezing [75] ( hgJ+ x a )a+ (h!xJ+x )
( + ) + Table II. Features studied in driven-dissipative open quantum systems comprising several TLSs in works in which permutational- invariant methods were applied. The works are grouped according to the general theory developed or according to the Hamilto- Optical bistability [58, 59] h!xJx nian studied, with !0, !x, !xx, !xy, ⇤, g,andg frequency parameters. For the master equation ⇢˙ i H, ⇢ ⇢ we show the relative interaction Hamiltonian, the rates relative′ to collective TLS processes, homogeneous local TLS processes, and cavity rates. PT stands for Phase Transition, and spaser stands for Surface Plasmon Amplification by Stimulated= [ ] + EmissionL[ ] of Radi- 1 1 1 1 ation. In Ref. [61, 62]thecollectiveandlocaldepolarizationchannelisconsidered,fixing 2 2 and 2 2 . † ∗ In Ref. [137]TLSadditionandsubtractionistreatedexploitingpermutationalsymmetry. ⇓ ⇑ ↓ ↑ 2 = = = = Collective quantum jumps [63] h!xJx h!zzJz
2 2 Bistability in the XY model [71] h!xJx h!0Jz− h!xy Jx Jy
Two-axis spin squeezing* [61, 62] − ih⇤ J−2 J 2 +
2+ − Entanglement witness [82] h−!xxJ(x −h!0)Jz
− − Steady-state superradiance [20, 67] h!0Jz Ramsey spectroscopy [136] h!0Jz Superradiant emission [77] h!0Jz Superfluorescence/subradiance [56] h!0Jz Spin synchronization [15] h!0Jz
† Superradiant lasing [137] h!0Jz
† † Non-classical light [60] h!x a a hg aJ a J
+ − State engineering [141] ( +hg J) +a J( a† + )
+ − Lasing [20, 67] hg (J a + J a†)
+ − Photon anti-bunching [79] hg (J a + J a†)
+ − Super/subradiance [74, 75] hg (J a + J a†)
+ − ( + † ) Spaser [68, 70, 80] hgJx a a
† Superradiant PT [78] hgJx(a + a )
PT, Lasing, Chaos [57] hg J a J a† ( hg+ J) a J a† ′ + − − + † Super/subradiant PT, squeezing [75] ( hgJ+ x a )a+ (h!xJ+x )
( + ) + Table II. Features studied in driven-dissipative open quantum systems comprising several TLSs in works in which permutational- invariant methods were applied. The works are grouped according to the general theory developed or according to the Hamilto- nian studied, with !0, !x, !xx, !xy, ⇤, g,andg frequency parameters. For the master equation ⇢˙ i H, ⇢ ⇢ we show the relative interaction Hamiltonian, the rates relative′ to collective TLS processes, homogeneous local TLS processes, and cavity rates. PT stands for Phase Transition, and spaser stands for Surface Plasmon Amplification by Stimulated= [ ] + EmissionL[ ] of Radi- 1 1 1 1 ation. In Ref. [61, 62]thecollectiveandlocaldepolarizationchannelisconsidered,fixing 2 2 and 2 2 . † ∗ In Ref. [137]TLSadditionandsubtractionistreatedexploitingpermutationalsymmetry. ⇓ ⇑ ↓ ↑ = = = = Dissipative open quantum systems simulation
d N 1 1 iℏ ρ = [H, ρ] + γ L ρL† − L†L ρ − ρL†L Open quantum systems Coherent Dynamics ∑ ij ( i j i j i j) dt i,j 2 2 with local and collective incoherent processes: Efficient numerical simulation using permutational invariance
N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, and F. Nori γij ≡ γδij γij ≡ γ ∀i, j Phys Rev A 98, 063815 (2018) arXiv:1805.05129 Dissipation Processes Local Collective
Open Quantum Systems
Local Emission Collective Emission Non-Equilibrium Phase Transitions
Quantum Optics, Spin Squeezing
Local Dephasing Collective Dephasing Ultrastrong Coupling of Light and Matter
Local Pumping Collective Pumping
!46 PIQS: Driven-dissipative two-level system ensembles The Permutational Invariant Quantum Solver d N iℏ ρ = [H, ρ] + γ ℒ [ρ] dt α ∑ Li i !xx =0
Ultrastrong Coupling (USC) Superradiant PT BoundaryBTC - spectrum, Time strong limit Crystals QuTiP jmat Resonance Fluorescence 15 L 5 !xx !x Non USC 10 ⌧ USC GS
i 5 a
† 0.01 a ) 0 !
h 0 Im(
5 ! )
t ( i 10 ( a S † 0.00 !xx = !x a
h 15 5 20 15.0 12.5 10.0 7.5 5.0 2.5 0.0 5 0 5 0 200 400 Re( ) !xx >!x t x GHZ and coherent spin states Noisy Heisenberg Model Dicke Superradiance Spin Squeezing 1 1 1 1 1 1 1 1 1 1 1 1 ρ = , , ρ = 1 , 1 1 , 1 ρ = , , CSS ρ = , , √2 √2 √2 √2 CSS √ √ √ √ CSS 20 0 20 √2 √2 √2 √2 √2 √2 √2 √2 CSS | − ρ⟩⟨= GHZ− |GHZ | 2 2⟩⟨ 2 2| ρ = GHZ GHZ | − ⟩⟨ − | | ⟩⟨ | max max 1.0 | ⟩⟨ | max N N max max | ⟩⟨ | max J 1 2 , 2 =0, =0.2 x | i # + ! h i N , 0 1.5 =0.2, =0 Jy , Jz | 2 i # + h i h i + CSS 0.5 ⇢ | i CSS
) h i
) | i t t 0, 0 ( m m (
m 1.0 m m | i i
m 0 2 0 i
GHZ ⇠ z 0
X 0.0 | i J h h 0.5 0.5 min min 0 0 0 0 m′ m′ m′ m′ m′ m′ N N N N N N N NN N NN N N N N N N N N 1 0.0 ρ = , , ρ = , 0 , 0 ρ = ρ =, , , , ρ = , 0 , 0 ρ0= , 20, 40 0.0 0.5 1.0 1.5 2.0 | 2 2 ⟩⟨ 2 2 | | 2 ⟩⟨ 2 | | 2 2| ⟩⟨2 2− 22⟩⟨| 2 − 2 | | 2 ⟩⟨ 2 | | 2 − 2 ⟩⟨ 2 − 2 | 0 5 10 t t N⇤t N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, and F. Nori, PRA 98, 063815 (2018) arXiv:1805.05129 m m m m m m + = import qutip.piqs m′ m′ m′ m′ m′ m′ !47 PIQS Example: Ultrastrong Coupling (USC) Regime Relaxation to the true steady state by deriving consistent operators: no light emission
Extra-cavity Light Field vs. Time Extra-cavity Light Spectrum vs. Frequency (a) (b) Non USC Non USC Hamiltonian USC USC GS 1.0 ⟩ a
† 0.01 a H = ! J + ! a a + gJ a + a ) ~ 0 z ~ cav † ~ x † ω −⟨ ( ) S t 0.5 ( H = ~!0Jz + ~!cava†a + ~gJx a + a†
⟩ a
† 0.00 a ⟨ 0.0 0 20 40 0.6 0.8 1.0 1.2 1.4 gt ω/ω0
Dissipation [⇢] Derive the “dressed” L light-matter jump operators for the Lindbladian
Case N=1: F. Beaudoin, J.M. Gambetta, A. Blais, PRA (2011) Case N>>1: N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, and F. Nori, PRA 98, 2018, arXiv:1805.05129 Steady-state light emission and transport Dissipation and ultrastrong coupling regime with transport: how to engineer light emission
Electron transport + Light-matter coupling: Multielectron Ground State Electroluminescence M. Cirio*, N. Shammah*, N. Lambert, S. De Liberato, and F. Nori Phys. Rev. Lett. 122, 190403 (2019) arXiv: 1811.08682
Features • Cooperative effect Robust with N Dr. Mauro Cirio Prof. Simone De Liberato • GSCAEP (China) U. of Southampton (UK) Possible Experiments • Quantum wells (Faist) • Quantum dots (Vandersypen, Petta) • Superconducting circuits • Hybrid devices (Wallraff)
Figure: Solid-state device. !49 QuTiP: A tool to explore quantum mechanics The Quantum Toolbox in Python
Quantum mechanics Tutorials at qutip.org/tutorials
Independent Expert
Beginner Hardness QuTiP
Proficiency • Over 60 Jupyter notebook tutorials • Over 20 quantum mechanics lectures QuTiP: Contributing to QuTiP – Tutorials The Quantum Toolbox in Python Recent addition: Cat vs coherent states in a Kerr resonator and the role of measurement
Dr. Fabrizio Minganti (RIKEN)
qutip.org/tutorials How to interact with the QuTiP project Multiple interaction channels Website: http://qutip.org/tutorials.html Tutorials
http://qutip.org/docs/latest/ Documentation User Guide Documentation API: docstring comments Documentation API: source code https://groups.google.com/forum/#!forum/qutip Help Group: Ask questions (installation, physics) Find already answered questions
https://github.com/qutip/qutip GitHub: Found a bug? - Search answer in already closed issue - Search an open (unsolved) issue on same topic - Unknown? Open an Issue Have a solution? Open a Pull Request (proposal for code modification of the official project source code) Thank you https://gitter.im/qutip Gitter Chat: @NathanShammah Join the conversation if you have other question GitHub: nathanshammah Extra Slides QuTiP: The Quantum Physics Simulator The Quantum Toolbox in Python: A toolbox to study the open quantum dynamics of realistic systems. QuTiP
Interactive Lectures @ ICTP, Leonardo Building Take a snapshot Tue 25th June - 11:45am, Seminar Room – Driven-dissipative models in quantum physics Wed 26th June - 11am, Seminar Room – Quantum Open Source & Introduction to QuTiP Thur 27th June - 9am, Computer Room – Hands-on session on QuTiP’s main features Mon 1st July - 9am, Computer Room – QuTiP stochastic solvers Tue 2nd July - 9am, Computer Room – PIQS and How to Build your Own Software (Wed 3rd July - 9am, Computer Room – Extra meeting: SISSA/ICTP projects)
https://github.com/nathanshammah/interactive-notebooks QuTiP: Some Statistics The Quantum Toolbox in Python
conda install qutip GitHub "Used by" allows to draw connection graphs on the quantum open source ecosystem:
- IBM’s qiskit is used by 122 other libraries.
- qutip by 60.
- Rigetti’s pyquil 53 (grove 8).
- ProjectQ 29.
- Google’s OpenFermion 22.
July 2018 downloads 43k - Dwave’s ocean-sdk 14. Feb 2019 100k QuTiP helps engineer an ecosystem of libraries. Apr 2019 downloads/year
Jun 2019 Solving a many-qubit dynamics: Quantum Trajectories Photon counting statistics: Monte Carlo solver with `mcsolve`
“quantum jump” d i 1 1 ρ = − [H, ρ] + γ LρL† − L†Lρ − ρL†L dt ℏ ( 2 2 ) continuous decay
A. Effective Hamiltonian: iℏ † shrinking the state H = H − L L eff sys 2
B. Quantum jump: L|ψ(t)⟩ |ψ(t + δt)⟩ = 1/2 1. Generale random number, r. ψ(t) L+L ψ(t) 2. Is r > norm(psi)? ⟨ ⟩ I. Yes: keep integrating (A) II. No: apply jump and renormalize Stochastic solvers Stochastic master equation: Continuous weak measurements with: `smesolve`
continuous weak measurements
Beam Splitter
Heterodyne Detection Signal Homodyne Detection Detector ωLO = ωsignal ωLO ≠ ωsignal (photodiode)
Local Oscillator
result = smesolve(H, rho0, times, [], c_ops, e_ops) Stochastic solvers Stochastic master equation: Continuous weak measurements with: `smesolve`
continuous weak measurements
Heterodyne Detection 1 1 dρ(t) = − i[H, ρ(t)]dt + γ�[a]ρ(t)dt + dW1(t) γℋ[a]ρ(t) + dW2(t) γℋ[−ia]ρ(t) 2 2 1 �[A]ρ = AρA† − (ρA†A + A†Aρ) Lindblad term 2 ℋ[A]ρ = Aρ + ρA† − Tr [Aρ + ρA†] ρ Nonlinear term
result = smesolve(H, rho0, times, [], c_ops, e_ops, method=‘heterodyne’)
method = ‘heterodyne’ method = ‘homodyne’ method = ‘photocurrent’