Realizing effective non-Hermitian time evolution with - PowerPoint PPT Presentation

Realizing effective non-Hermitian time evolution with superconducting circuits Conference on Quantum Measurement: Fundamentals, Twists, and Applications ICTP 2019 Kater Murch Department of Physics, Washington University in St. Louis Students: Mahdi Naghiloo, Maryam Abbasi Collaborators: Yogesh Joglekar (IUPUI)

A Hamiltonian must be Hermitian Hamiltonian described by Hermitian operator Real eigenvalues ☞ Unitary time evolution Complete set of eigenvectors ☞ Orthonormal basis

Or must it? � � Can have real eigenvalues with non-Hermitian Hamiltonian Cannonical example: � � Gain � � ! " " � � Time Parity � � ! ! " � � � � Loss � �

Non-Hermitian Hamiltonian Can have real eigenvalues with non-Hermitian Hamiltonian Cannonical example: Hamiltonian Gain ! � � ! Coupling Matrix representation " � � Loss Eigenvalues:

But who cares? Balanced gain and loss easily Gain ! � � achieved in classical systems ! Many experiments… Coupling " � � Loss

Non-Hermitian dissipation-experiments and systems Platforms for non-Hermitian physics: coupled mechanical/electrical oscillators J Schindler et al, JPA 45 , 444029 B. Peng et al Nature Phys. C. M. Bender et al A. J. Phys. 81 , 173 10 , 394 (2014). C. Dembowski et al PRL 86 , 5 (2001). Lasing: B. Peng (14), M. Brandstetter (14), propagating acoustical/optical waves M. Kim (14), Z. Wong (16), B. Peng (16), L. Feng (14), H. Hodaei (14) Asymmetric mode switching: J. Doppler (16), C. Shi et al Nat. Commun. 7 , 11110 Topological energy A. Regensburger et al Nature A. Guo et al Phys. Rev. Lett. transfer: Xu (2016) Enhanced sensing: W. Chen (2017), H. Hodae (17)

PT symmetry: 4 Key phenomena PT breaking transition from Re to Im eigenvalues (J> ɣ ) PT “un-broken” phase 1 (J< ɣ ) PT “broken” phase “Exceptional” point degeneracy (J= ɣ ) 2 Eigenvectors become degenerate 3 Enhanced sensitivity 4 Topological/non-reciprocal

PT symmetric quantum systems? Open systems Non-unitary dynamics Master equations Non-Hermitian Hamiltonian Mode selective loss (effective PT symmetry)

Effective non-Hermitian qubit ! $%& !"# ɣ f = 0.2 ! s -1 � ! " � *++ , � - .) / � ! ) " ɣ e = 8 ! s -1 � " � '() # ' '('1 '()0 � 2 � ' # Engineered decay rates Transmon circuit arXiv:1901.07968

Effective non-Hermitian qubit Sub-manifold evolution ! given by non-Hermitian coherent Hamiltonian ! drive " � Lindblad Master # equation: Aharonov et al PRL 96, Weisskopf & Wigner ‘30

Dynamics of non-Hermitian qubit Prepare in Post-select qubit Evolve under H eff qubit e-f manifold manifold ! ! ! ! " " " � � � # # # Ashida, Furukawa, Ueda Nat. Com. 2017

Effective PT symmetry In matrix representation: Overall PT symmetric (EP occurs at J = ɣ /4) loss Detuned drive: Time evolution under H eff is governed by the eigenvalue difference:

Effective non-Hermitian qubit overview !"# �� $ !"# �� $ ,. . /. / -. - / , - / / 0 , / 1 . � 2 �� #345$ 0 � 1 �� #234$ %&#'()* � +$ %&#'()* � +$ J< ɣ /4 J> ɣ /4 Exceptional "# is imaginary "# is real Point J J= ɣ /4 J=0

Exceptional point Degeneracy: “Diabolic point” “Exceptional Point” Hamiltonian: Hermitian non-Hermitian Eigenvalues: Degenerate Degenerate Degenerate Eigenvectors: Orthogonal J< ɣ /4 J> ɣ /4 Exceptional "# is imaginary "# is real Point J J= ɣ /4 J=0

Key phenomena PT breaking transition from Re to Im eigenvalues (J> ɣ ) PT “un-broken” phase 1 (J< ɣ ) PT “broken” phase “Exceptional” point degeneracy (J= ɣ ) 2 Eigenvectors become degenerate 3 Enhanced sensitivity 4 Topological/non-reciprocal

Probing the PT breaking transition 1 Measure time evolution under H eff (Rabi oscillations) Prepare |f ⟩ H eff Readout 0-2 ! s -. P(f) 30 9 J (rad/ ! s) �� !"#$%& � '( ?8" �� ( : 20 >#0=85 * 10 , ;5<#0=85 . 0 . - , + * ) 2 1 0 /0123456!#$78! !" "#$%& � '( Time ( ! s)

Distorted Rabi oscillations (calculation from Lindblad Master Equation) Measurement backaction P(f) from post-selection pushes toward |f ⟩ state. !"#$%& � '( Might seem surprising that we observe abrupt transition at J= ɣ /4 even though the qubit never decays in our data set.

Degenerate eigenstates at the EP 2 ● For J ⪼ ɣ eigenstates are |±x ⟩ ! ● Near the EP the eigenstates |- ⟩ coalesce toward |+y ⟩ ● At the EP, one eigenstate: |+y ⟩ |- ⟩ # |- ⟩ ● Past EP, eigenstates in Y-Z |+ ⟩ plane |+ ⟩ " |+ ⟩ J J= ɣ /4 J=0

Imaging eigenstates (stationary) 2 Prepare ( 훳 , ! ) H eff Readout 500 ns ! 훳 = $ /2 � ! " # +| & $% '()*+ � ,- !"# # !"! $!"# " ! % � � ./01234)5&)6758& � J J= ɣ /4 J=0

Imaging eigenstates (stationary) 2 Prepare ( 훳 , ! ) H eff Readout 500 ns ! = 0 훳 = $ /2 � � ! " # +| & $% '()*+ � ,- !"# ! !" "#$%& � '( !"! $!"# ! % � � &) * � � ./01234)5&)6758& � +,-$#!$./-0! � J J= ɣ /4 J=0

Non-orthogonal eigenstates 2 ! ! | | $ "# $ "# +| +| !"#$%&'" !"#$%& +| !"#$%&'()*+,-) !" .*+/0 � 12

Sensing advantages with dissipation 3 Degeneracy: “Diabolic point” “Exceptional Point” Hamiltonian: Hermitian non-Hermitian Eigenvalues: Degenerate Degenerate Degenerate Eigenvectors: Orthogonal -. Response 9 �� !"#$%& � '( ?8" �� ( : >#0=85 * , ;5<#0=85 . . - , + * ) (Perturbation) /0123456!#$78! !" "#$%& � '( Wiersig PRL 2014, Chen Nature 2017, Hodaie Nature 2017

Time evolution of H eff 3 P(f)/(P(f) + P(e)) Time ( ! s) ● Distorted ● Solve Lindblad ● Response to a trajectories due to Master Equation small perturbation H eff (can think of as ● J/ ɣ = 0.33 (EP ( % J/J = 0.7%) measurement back- at 0.25) action)

3 Cramér-Rao Bound For large data sets, the Cramer-Rao bound sets a limit on the mean squared deviation of some parameter d is the amount of data is an unbiased estimator of J I & is the Fisher information Quantum Fisher Information Bures distance: ' i ' f Evolve with H J ' i ' f Evolve with H J +d J Cramér 1946

Quantum Fisher Information 3 Rabi interferometry ! # "

Quantum Fisher Information 3 ! (calculation) � -.$ � & /0 $ ! J+dJ � -1$ � & /0 $ !"#$% � & ' ( # ! J " ! $%)*+, � &( QFI ∼ (dP f /dJ) 2

Measuring the QFI near the EP 3 Prepare |e ⟩ H eff Readout 500ns Steeper P(e)/(P(e)+ P(f) slope = higher QFI P(e) Low success rate = increased J (arb units) binomial EP error Kero Lau, Aash Clerk (Nat. Com. 2018)

Non-Hermitian qubit sensing summary 3 Improvement in the QFI about a perturbation to non-Hermitian Hamiltonian near EP. (In the post-selected qubit manifold) Post-selection introduces a cost due to the dissipation. Can be situations (technical noise) where there are still advantages. Also inspiration to look at non-lossy systems from a new angle. A. Jordan PRX 2014

Topological features of the EP 4 !"#$ !"# �� $ %"#$ +| | ,. &"#$ ! /. -. - / , +| +| / 0 1 ! � 2 !"#$ �� #345$ %&#'()* � +$ Adiabatically tune | | parameters to encircle # the EP Non-reciprocal behavior +| | !

Non-Hermitian qubit 30 Transition from imaginary to real eigenvalues 20 (J< ɣ ) PT “broken” (J> ɣ ) PT “un-broken” 10 phase phase +| � 0 2 1 0 � “Exceptional” point degeneracy (J= ɣ ) Eigenvectors not orthogonal and become degenerate � � Enhanced sensitivity � (prelim) +| | Topological/non-reciprocal features ! (theory) Decoherence! Dissipation: Lindblad vs non-Hermitian

Interplay of two types of dissipation !"# �� $ !"# �� $ . ,. /. / -. - - / , / / , / 0 . 1 0 � 2 � �� #345$ 1 �� #234$ %&#'()* � +$ %&#'()* � +$ " " " � $% � &#'()* � $% � &#%'()* � � !$ !# !" !"#"$ !"#"$ # ! # !" !" #" #"# #"% $"# $"% !"# #"# #"% $"# $"% !"# &'()*+ � ,- &'()*+ � ,- (additional decoherence)

Steady states from dissipation interplay Quantum state tomography after 4 ! s of evolution ���� !"./0( + ,* $ ! " # $ % & * + � ' ( ) -* ! !"#$#%& " # y bump: at the EP dissipation drives system to the single eigenstate.

Summary and thanks Non-Hermitian qubit: enhanced sensing, non-reciprocal/ topological features, interplay of dissipations. Naghiloo et al 2019 arXiv:1901.07968 Collaborators Yogesh Joglekar (IUPUI) Murch group at WUSTL 2019 Mahdi Maryam Naghiloo Abbasi