Model for calorimetric measurements in an open quantum system Brecht - PowerPoint PPT Presentation

Model for calorimetric measurements in an open quantum system Brecht Donvil 1 Paolo Muratore-Ginnaneschi 1 Jukka Pekola 2 1 Univeristy of Helsinki 2 Aalto University 20.12.2017

Introduction Motivation: proposal for experimental setup to perform a calorimeteric measurement of work on a driven qubit 1 T e Q W Drive E Calorimeter T p T e Qubit t Problem: How to model evolution of small quantum system while continuously measuring a macroscopic property of the bath Two cases: • Weak coupling, based on the Lindblad equation 2 • Strong coupling: path integral formalism and filtering 1 Pekola et al (New. J. Phys. 2013) 2 A. Kupiainen et al PRE (2016), B. D. et al, PRA (2018) and B. D. et al, PRA (2019)

Qubit-Calorimeter Qubit H I H ep � ω q T e T p H d ( t ) Drive Calorimeter Phonon bath Figure: Experimental setup � ( σ + + σ − ) a † H I = g k a l k , l

Time scales ◮ τ ee = O (10 0 )ns: Landau quasi-particle relaxation rate to Fermi–Dirac equilibrium in a metallic wire ◮ τ ep = O (10 4 ): Electron-phonon interactions ◮ τ R = 2 − 5 × O (10 5 )ns: Transmon qubit relaxation times 3 ◮ τ eq ≈ g − 2 : Fermi’s golden rule estimate of characteristic qubit-calorimeter time scale 3 Wang et al., Appl. Phys. Let., (2015)

Time scales ◮ τ ee = O (10 0 )ns: Landau quasi-particle relaxation rate to Fermi–Dirac equilibrium in a metallic wire ◮ τ ep = O (10 4 ): Electron-phonon interactions ◮ τ R = 2 − 5 × O (10 5 )ns: Transmon qubit relaxation times 3 ◮ τ eq ≈ g − 2 : Fermi’s golden rule estimate of characteristic qubit-calorimeter time scale Time scale separations τ ee ≪ τ eq ≪ τ ep ≪ τ R 3 Wang et al., Appl. Phys. Let., (2015)

Time scales ◮ τ ee = O (10 0 )ns: Landau quasi-particle relaxation rate to Fermi–Dirac equilibrium in a metallic wire ◮ τ ep = O (10 4 ): Electron-phonon interactions ◮ τ R = 2 − 5 × O (10 5 )ns: Transmon qubit relaxation times 3 ◮ τ eq ≈ g − 2 : Fermi’s golden rule estimate of characteristic qubit-calorimeter time scale Time scale separations τ ee ≪ τ eq ≪ τ ep ≪ τ R We assume that the qubit is interacting with the calorimeter at a well-defined temperature 3 Wang et al., Appl. Phys. Let., (2015)

Weak coupling: Stochastic Jump Process The evolution of a closed quantum system is described by the Schr¨ odinger equation ψ ( t + d t ) − ψ ( t ) = d ψ ( t ) = − iH ψ d t For an open system the Schr¨ odinger equation is modified

Weak coupling: Stochastic Jump Process The evolution of a closed quantum system is described by the Schr¨ odinger equation ψ ( t + d t ) − ψ ( t ) = d ψ ( t ) = − iH ψ d t For an open system the Schr¨ odinger equation is modified ◮ Dissipative terms are added to the Hamiltonian H ψ ( t ) d t → G ( ψ ( t )) d t

Weak coupling: Stochastic Jump Process The evolution of a closed quantum system is described by the Schr¨ odinger equation ψ ( t + d t ) − ψ ( t ) = d ψ ( t ) = − iH ψ d t For an open system the Schr¨ odinger equation is modified ◮ Dissipative terms are added to the Hamiltonian H ψ ( t ) d t → G ( ψ ( t )) d t ◮ Addition of jump terms ( | + � − ψ ( t )) d N ↑ , d N ↑ = 0 , 1 , ( |−� − ψ ( t )) d N ↓ , d N ↓ = 0 , 1 , E ψ ( d N ↑ / ↓ ) = γ ↑ / ↓ � σ ± ψ � 2 d t

Temperature Process Using the Sommerfeld expansion we find the dependence of the temperature on the change in internal energy E of the calorimeter e = d E d T 2 N γ . The qubit-electron interaction gives d E = � ω ( d N ↓ − d N ↑ )

Temperature Process Using the Sommerfeld expansion we find the dependence of the temperature on the change in internal energy E of the calorimeter e = d E d T 2 N γ . The final result is the set of coupled equations  d ψ ( t ) = − iG ( ψ ( t )) d t    � � � �  + | + � − ψ ( t ) d N ↑ + |−� − ψ ( t ) d N ↓    d T e = � ω  N γ ( d N ↓ − d N ↑ )

Temperature Process Using the Sommerfeld expansion we find the dependence of the temperature on the change in internal energy E of the calorimeter e = d E d T 2 N γ . The final result is the set of coupled equations  d ψ ( t ) = − iG ( ψ ( t )) d t    � � � �  + | + � − ψ ( t ) d N ↑ + |−� − ψ ( t ) d N ↓    d T e = � ω  N γ ( d N ↓ − d N ↑ )

Add the substrate A Fr¨ ohlich electron-phonon interaction leads to extra terms  d ψ ( t ) = − iG ( ψ ( t )) d t   � � � �   + | + � − ψ ( t ) d N ↑ + |−� − ψ ( t ) d N ↓  √ 10Σ Vk B T 3  Σ V ( T 5 p − T 5  e ) d t 4 + d T e = � ω d w t 5  p N γ ( d N ↓ − d N ↑ ) + N γ N γ 4 Kaganov, Lifshitz and Tanatarov (1956) 5 Pekola and Karimi (2018) 4 Kaganov, Lifshitz and Tanatarov (1956) 5 Pekola and Karimi (2018)

Add the substrate A Fr¨ ohlich electron-phonon interaction leads to extra terms  d ψ ( t ) = − iG ( ψ ( t )) d t   � � � �   + | + � − ψ ( t ) d N ↑ + |−� − ψ ( t ) d N ↓  √ 10Σ Vk B T 3  Σ V ( T 5 p − T 5  e ) d t 4 + d T e = � ω d w t 5  p N γ ( d N ↓ − d N ↑ ) + N γ N γ 0,28 0,24 T e (K) 0,20 0,16 0,12 x10 4 0,0 0,5 1,0 1,5 2,0 2,5 3,0 Periods of driving 4 Kaganov, Lifshitz and Tanatarov (1956) 5 Pekola and Karimi (2018) 4 Kaganov, Lifshitz and Tanatarov (1956) 5 Pekola and Karimi (2018)

Effective temperature process Performing multi-timescale analysis eliminates the jumps process and adds a correction to the drift and noise e = 1 � � � d T 2 Σ V ( T 5 p − T 5 e ) d t + J ( T 2 � 10Σ Vk B T 3 S ( T 2 e ) d t + p d w t + e ) d w t . γ J ( T 2 e ) = Average heat dissipated by the qubit in a thermal state T e + O ( ǫ ) 0,42 0,39 0,36 T S (K) 0,33 18 15 a (10 6 s -1 ) 0,30 12 9 0,27 6 0,24 3 0 0,00 0,02 0,04 0,06 0,08 0,10 0,21 0,00 0,02 0,04 0,06 0,08 0,10 g 2

Strong Coupling: Central fermion (Work in Progress) Issues with the strong coupling spin-fermion models: ◮ Quadratic coupling ◮ Performing spin path integrals requires ”Non Interacting Blip Approximation” (NIBA) 6 or other approximations 6 Leggett et al., Rev. Mod. Phys. (1987)

Strong Coupling: Central fermion (Work in Progress) Issues with the strong coupling spin-fermion models: ◮ Quadratic coupling ◮ Performing spin path integrals requires ”Non Interacting Blip Approximation” (NIBA) 6 or other approximations To avoid making non-trivial approximations, we consider the ”central-fermion”-model � � g k ( b † ω k b † ω 0 c † c k c + c † b k ) + H = + k b k � �� � k k central fermion This Hamiltonian can be used to model a quantum dot 6 Leggett et al., Rev. Mod. Phys. (1987)

Central fermion Our goal is to find a set of equations which describes the evolution of the density matrix of the central fermion ρ ( t ) and the energy of the bath E ( t ) � d ρ ( t ) = L t ρ ( t ) d E ( t ) = tr ( E ( t ) ρ ( t )) We calculate both operators separately

Central fermion Our goal is to find a set of equations which describes the evolution of the density matrix of the central fermion ρ ( t ) and the energy of the bath E ( t ) � d ρ ( t ) = L t ρ ( t ) d E ( t ) = tr ( E ( t ) ρ ( t )) We calculate both operators separately ◮ L t we find by exactly integrating the bath and qubit dynamics

Central fermion Our goal is to find a set of equations which describes the evolution of the density matrix of the central fermion ρ ( t ) and the energy of the bath E ( t ) � d ρ ( t ) = L t ρ ( t ) d E ( t ) = tr ( E ( t ) ρ ( t )) We calculate both operators separately ◮ E ( t ) is obtained from performing the partial trace tr B ( � k ω k b † k b k ρ T ( t ))

Fermionic path integral The dynamics of the full central fermion-bath system can be represented in terms of a path integral � d [ x ′ , X ′ , x , X ]Φ TOT ( x ′ , X ′ | x , X , t ) ρ 0 ( x , X ) ρ TOT ( t ) = � D [ x t , X t ] e iS T [ x t , X t ] Φ( x ′ | x , t ) =

Fermionic path integral The dynamics of the full central fermion-bath system can be represented in terms of a path integral � d [ x ′ , X ′ , x , X ]Φ TOT ( x ′ , X ′ | x , X , t ) ρ 0 ( x , X ) ρ TOT ( t ) = � D [ x t , X t ] e iS T [ x t , X t ] Φ( x ′ | x , t ) = For linear system-bath coupling , similar to the Caldeira-Leggett model, the bath fields X t can be integrated over � d ( x , x )Φ( x ′ | x , t ) ρ 0 ( x ) ρ ( t ) = with � D [ x t ] e iS [ x t ] Φ( x ′ | x , t ) = The action S [ x t ] is now time non-local

Central fermion: dynamics Solving the qubit path-integral gives an expression for the propagator 1 Φ( x ′ | x , t ) = N ( t ) e x ′ K ( t ) x Differentiating the propagator leads to a master equation for the qubit state 7 ρ ( t ) = L t ρ ( t ) ˙ =Ω[ c † c , ρ ( t )] + f ( t )( c † c ρ ( t ) + ρ ( t ) c † c ) + g ( t ) c ρ ( t ) c † + h ( t ) c † ρ ( t ) c + k ( t ) ρ ( t ) 7 Tu and Zang, PRB, 78 (2008)

Energy of the electron bath The energy of the electron bath is given by the operator � k ω k b † k b k . Using similar path integral techniques as before, we find A ( t ) such that � ω k b † E ( t ) = tr ( k b k ρ T ( t )) = tr S ( A ( t ) ρ ( t )) k Thus, we have the set of equations � ∂ t ρ ( t ) = ¯ L t ρ t ∂ t E ( t ) = tr(( ∂ t + L † t ) A ( t ) ρ ( t ))