Feb 22, 2017 - we propose a realistic platform to probe fluctuation theorems in the quantum regime. .... In what follows we drive the qubit transition with a.
1 2 1 2 Alexia Au↵` , Au↵` Jean-Michel G´ rard , and Jean-Philippe 1 evesAlexia 2, eJean-Michel Au↵` ves G´2e,rard , Jean-Phi and1Poiz Jea 1eerard Alexia Au↵` e ves , Jean-Michel G´ , and Jean-Philippe Poizat Alexia e ves , Jean-Michel G´ e rard and 1 1 Pure emitter dephasing : a resource for advanced solid-state 1 CEA/CNRS/UJF Joint team ” Nanophysics semiconductors CEA/CNRS/UJF Joint teamand ” Nanophysics and ”,Institut semiconduN 1
arXiv:1702.06811v1 [quant-ph] 22 Feb 2017
CEA/CNRS/UJF CEA/CNRS/UJF Joint team ” Nanophysics and” semiconductors ”,Institut N´eel-CN Joint team Nanophysics and semiconductors BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France an BP 166, 25, des rue des Martyrs, 38042 Grenoble quantum fluctuation theorems in engineered reservoirs BPProbing 166, 25, rue des 38042 Grenoble Cedex 9, France and BP Martyrs, 166, 25, rue Martyrs, 38042 Grenoble Cedex Ce 9, 2 1 2 2 2 Pure emitter dephasing : a resource for advanced solid Alexia Au↵` e ves , Jean-Michel G´ e rard , and Jean-Philippe CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors” 2 CEA/CNRS/UJF Joint team ” Nanophysics and CEA/CNRS/UJF Joint ” resource Nanophysics and semiconductors”, 1 2 3 4 CEA/CNRS/UJF Joint Nanophysics and semi⇤ C. Elouard, N. K.dephasing Bernardes, A. R. team R. Carvalho, M. F. Santos,team and A.”Auff` eves1, ∗ 1 emitter : a for advanced solid⇤ CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Inst CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France PurePure emitter dephasing : a resource for advanced solid-state emitter dephasing : a resource for advanced solid-state single ph CEA/INAC/SP2M, 17 rue des Martyrs, 38054 G 1 CEA/INAC/SP2M, 17 rue Martyrs, 38054 France 1 Grenoble, CNRS and Universit´e CEA/INAC/SP2M, Grenoble Alpes, des Institut N´eel, F-38042 Grenoble, 17Au↵` rue Martyrs,G´ Grenob Alexia evesdes , France Jean-Michel e38054 rard2 , and Jean-P 2Pure emitter dephasing : a resource for advanced solid-stat Departamento de F´ısica, Universidade Federal de Minas Gerais, BPdephasing 166, 25, rue Martyrs, 38042 Grenoble Cedex 9, Fran 1 resource (Dated: December 9,December 2016) Pure :aDecember ades for solid-state sing (Dated: 9,single 2016) Pure emitter dephasing :Joint aadvanced resource for advanced s CEA/CNRS/UJF team ” December Nanophysics and semiconducto (Dated: 9,resource 2016) (Dated: 9, 2016) Pureemitter emitter : Postal resource for advanced solid-state pho Belodephasing Horizonte, Caixa 702, 30161-970, Brazil 2 Pure emitter dephasing : a resource for advanced solid-state single photon sou 1 2 Pure emitter dephasing : a for advanced solidBP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 3 CEA/CNRS/UJF Joint team ” Nanophysics and semiconduc Alexia Au↵` ves Jean-Michel rard , and Jean-Ph 2Queensland 1 1 University, 2Australia CentreAu↵` for Quantum 4111,G´ 1, and 2 Alexia eInstituto ves1Dynamics, ,deAu↵` Jean-Michel G´ e,Nathan, rard Jean-Philippe Poizat eGriffith ves , eJean-Michel G´ erard ,eG´ and 4 Alexia Au↵` e2ves ,deJean-Michel erard ,” and Jean-Philipp CEA/CNRS/UJF Joint teamJean-Philippe Nanophysics and se 1 Alexia 1 2 1 2 F´ ısica, Universidade Federal do Rio Janeiro, 1 2 1and CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, FrJ Alexia Au↵` eves, Jean-Michel , Jean-Michel G´ erard ,17Jean-Philippe and Jean-Philippe Poiza 1 1 Alexia e ves , Jean-Michel G´ e rard , 2Au↵` 1 CEA/CNRS/UJF Joint team ” Nanophysics and semiconductor 1ves1 Alexia Au↵` e ves G´ e rard , and Poizat Alexia Au↵` e , Jean-Michel G´ e rard , and Jean-Philippe Poizat CEA/INAC/SP2M, rue des Martyrs, 38054 Gre CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,In CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´ e el-CN CEA/CNRS/UJF Joint team ”Au↵` Nanophysics ”,Insti Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil and semiconductors 1 42.50.Pq 2 solid-stat 11 numbers: 1a numbers: 42.50.Gy; 42.50.Pq ; 42.65.Hw PACS 42.50.Ct; 42.50.Gy; ; 42.65.Hw Pure emitter dephasing : resource for advanced 1 42.50.Ct; Alexia e ves , Jean-Michel G´ e rard , and Jean-Ph CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N9 PACS PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw CEA/CNRS/UJF Joint team ” Nanophysics and semicon CEA/CNRS/UJF Joint team ” 25, Nanophysics semiconductors ”,Institut N´ eel-CNR PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ;and 42.65.Hw (Dated: February 21, 2017) CEA/CNRS/UJF Joint team ”BP Nanophysics and semiconductors ”,Institut N´ eel-CNRS, (Dated: December 9, 2016) BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 166, rue des Martyrs, 38042 Grenoble Cedex 9, Fra 1 25, BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and Fluctuation Theorems are central in stochastic thermodynamics, as they allow for quantifying the BP 166, rue des Martyrs, 38042 Grenoble Cedex 9, Franc CEA/CNRS/UJF Joint team ” rue Nanophysics and semiconductor 166, 25, rue Martyrs, 38042 Grenoble Cedex 9, and 2 BP 166,des 25, ruedesdes Martyrs, 38042 Grenoble Cedex 9,and France and BP 166, 25, des 38042 Grenoble BP 166,2 BP 25, rue Martyrs, 38042 Grenoble Cedex 9,”Martyrs, France andFrance Joint team Nanophysics semicond irreversibility of single they CEA/CNRS/UJF have been experimentally checked in the classical 2 emitter 2 trajectories. 22 :Although CEA/CNRS/UJF Joint team ”for Nanophysics and sem 2 2 Nanophysics Pure dephasing a resource for advanced solid-state single photon sou CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”, Pure emitter dephasing : a resource advanced solid-state CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”, CEA/CNRS/UJF Joint team ” and semiconductors”, CEA/CNRS/UJF Joint team ” Nanophysics and semiconduc CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”, CEA/CNRS/UJF Joint team ” Nanophysics a BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex regime, a practical demonstration in the framework of quantum open systems still to come. CEA/INAC/SP2M, 17is rue des Martyrs, 38054 Grenoble, 1numbers: 2 Here PACS 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw ⇤ ⇤ ⇤ Alexia Au↵` e ves , Jean-Michel G´ e rard , and Jean-Philipp ⇤ 2 CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Greno weCEA/INAC/SP2M, propose a realisticCEA/INAC/SP2M, platform to probe fluctuation theorems in regime. It is 17 based on 9, CEA/INAC/SP2M, 17therue desJoint Martyrs, 38054 Grenoble, France CEA/INAC/SP2M, des Martyrs, CEA/CNRS/UJF team ” rue Nanophysics and38054 sem 17 des Martyrs, Grenoble, France (Dated: December 2016) 17 rue des Martyrs, 38054 Grenoble, Fra PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ;quantum 42.65.Hw 1 rue(Dated: 21 38054 1 2 (Dated: February 10,detection 2017) February 21, 2017) an effective 1two-level system coupled to an engineered reservoir, that enables the of the Alexia Au↵` e ves , Jean-Michel G´ e rard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,In Alexia Au↵` e ves , Jean-Michel G´ e rard , and Jean-Philippe (Dated: December 9, 2016) (Dated: December 9, 2016) (Dated: December 9, 2016) Pure emitter dephasing :December a resource for advanced soli CEA/INAC/SP2M, 17 ruea9, des Martyrs, 38054 Gren 1 (Dated: 2017) 1teamWhen (Dated: 2016) photons emitted and absorbed by the system. the systemJoint is21, coherently measurable CEA/CNRS/UJF Joint ”February Nanophysics and semiconductors ”,Institut N´eel-CNRS, ”,Ins CEA/CNRS/UJF team ”driven, Nanophysics and semiconductors 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, Fr quantum component inBP the BP entropy production is evidenced. We quantify the error due to photon (Dated: December 9, 2016) 166, 25, numbers: rue des Martyrs, 38042 Grenoble Cedex 9,38042 FranceGrenoble and 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw BP 166, 25, rueefficiently des Martyrs, Cedex 9, Fran PACS and numbers: 42.50.Ct; 42.50.Gy; ; 42.65.Hw PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.50.Pq 42.65.Hw 2 PACS detection inefficiency, show that the missing information can be corrected, based 2 0 1 and 2 semicond 0G´ M 2team CEA/CNRS/UJF Joint ” Nanophysics and M CEA/CNRS/UJF Joint ”42.50.Ct; Nanophysics semiconductors”, PACS numbers: 42.50.Gy; 42.50.Pq 42.65.Hw Alexia Au↵` e;team ves ,quantum Jean-Michel eL[⇢] rard , and Jean PACS numbers: 42.50.Gy; 42.50.Pq ; 42.65.Hw CEA/CNRS/UJF Joint team ” Nanophysics and semicondu solely on the detected events. Our 42.50.Ct; findings provide new insights into how the character of ⇤ 1 Pure emitter dephasing : a resource for advanced solid-state single photon CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, F PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw CEA/CNRS/UJF Joint team ” Nanophysics and semiconduc a physical system impacts its thermodynamic evolution. CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; December 42.65.Hw9, 2016) (Dated: December 9, 2016) (Dated: BP 166, 25, rueDecember des Martyrs, 38042 Grenoble Cede (Dated: 9, 2016) 2 Alexia Au↵`eves1 , Jean-Michel G´erard2 , and Jean-Philippe Poizat1 I.
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team ” Nanophysics and s cl Joint a) CEA/CNRS/UJF 1 INTRODUCTION b) CEA/CNRS/UJF Joint team ” 42.50.Ct; Nanophysics N´eel-CNRS, Qcl [ 42.50.Gy; ] and W [ semiconductors ] L[⇢] PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ;CEA/INAC/SP2M, 42.65.Hw d (t) PACS numbers: 42.50.Pq ; 42.65.Hw 17 rue ”,Institut des H Martyrs, 38054 Gr
BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and O and O 9, 2016) (Dated: December 2 42.50.Ct; The existence PACS of some numbers: preferred direction of time42.50.Gy; is Joint team 42.50.Pq ; 42.65.Hw CEA/CNRS/UJF ” Nanophysics semiconductors”, ⇤ a fundamental concept of physics, captured by the secCEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France L[⇢] ond Law of thermodynamics [1, 2]. The irreversibility (Dated: [December ] 42.50.Gy; P1 Q[9, ]2016) PACSposnumbers: cl 42.50.Ct; 42.50.Pq ; 42.65.Hw of physical phenomena is manifested by a strictly Hd (t) M M itive entropy production, classically identified with the S S numbers: 42.50.Ct; 42.50.Gy; change of entropy of PACS the closed system considered. In 42.50.Pq ; 42.65.Hw c) the textbook situation of a system S exchanging work W 1 1 1 1 ! with some external operator and heat Qcl with a thermal Hd (t) Hd (t) ⌦ L[⇢] PR R reservoir R of temperature T , entropy production equals 1 O the entropy change of the system ∆Ssys and the reservoir P P1 ∆Sres = −Qcl /T , and reads d 1 M
S
R
Q [ ]
P
P
∆i S = (W − ∆F )/T.
(1)
! ⌦ P1
S
P
P
0
M [ ]
[ ]
! ⌦
R
R H (t)
! ⌦
! ⌦
S
! ! ! ✓/⇡ ⌦ ⌦P P1 Hd (t) 1 ⌦ S ! FIG. 1.(t) a) Classical to test FT on a system S exchangR setup ! H ⌦ ⌦✓ + 1 ◆ d with a thermalP! P ing heat bath R and work with an external 1 1 z ✓/⇡ Pe (t) TrQq [γ] ⇢qand (t) work ✓/⇡ ⌦amounts operator O! . The stochastic of = heat !
∆F is the change of the system’s free energy, satisfying ∆Ssys = (W + Qcl − ∆F )/T . cl ⌦ Focusing on the transformations of microscopic sys⌦classical W [γ] are reconstructed by recording the ! trajectory 2 ✓/⇡ tems, stochastic thermodynamics has extended the conγ of the system. b) S is a quantum system⌦ driven through a cepts of thermodynamics to single stochastic realizations P through Hamiltonian (t) and interacting with a heat bath 1 HPd✓ ◆ z + ✓ P1S can ◆randomly 1perturb or “trajectories” of the system in its phase space [3– 1✓/⇡ the Lindbladian L[ρ]. Monitoring ✓/⇡ its h(⌫0 + (t)) z +1 Pe (t) = Tr (t)Tr q= evolution and corresponds to an additional dissipative Tr [⇢m (t)(H =◆ 5]. In this framework, heat and work exchanges bem 0 + V )] chanPeq(t)⇢✓/⇡ ⇢2qq (t) = ✓ q H 2 2 nel. c) Proposed strategy: Monitoring a✓/⇡ reservoir zengineered come stochastic quantities defined for single trajectories +1 to simulate a thermal ! Pbath. e (t) = Trq ⇢q (t) ! γ, which can give rise to a negative entropy production ⌦ ✓/⇡ ◆ ⌦ 2 ✓ ◆ ✓ ∆i s[γ] < 0. The fluctuations of ∆i s[γ] verify the cen◆ ✓ −∆i s[γ]/kB ! h(⌫ + (t)) + 1 + 1 ◆ ✓ z + † 0 z tral Fluctuation Theorem (FT) [6] he iγ = 1, z z (t)) Tr [⇢qm (t)g (b + b )] zh(⌫ 1q∆F Hqe(t) = Trm [⇢mq(t)(H )]= = Tr [0(t) += 1]= mz ∆ s[γ] = (Wzq[γ] ⇢ − )/TP . z+ W [γ] is m the stochastic 0P+ + (t) Tr ⇢q1(t) iV ⌦ e (t) q[⇢ H (t) = verifies Tr (t)(H =(t) while the Second Law of thermodynamics remains valid Pe (t) =m Tr (t) qfor q= qe 0 + VP)] z z+ 1] q mq⇢ qe (t) Trqqeγ,2q2 ⇢q (t)q[ JE: 2 e2 e2 trajectory work exchange the yielding 2 0 + (t)) on average. A famous example of such central FT is h(⌫ ◆[ z + Hq (t) ✓/⇡ =heTr [⇢mB T(t)(H +zV B)]T = ✓ provided by Jarzynski’s equality (JE) [5, 7] characteriz−W m[γ]/k iγ = e0−∆F/k (2)1 + 2 z ing isothermal transformations of systems driven out of ✓/⇡ e [⇢ (t)gHm q(t) †q Tr= Peb=(t) Tr†q ⇢mq+ (t) V )] = h⌦b† b + hgm Pe q [⇢q (t)(H (b + )] equilibrium. Eq.(1) remains valid at the trajectory level,(t) = m classical m mregime, h(⌫ + (t)) InTr the the stochastic work exchanges h(⌫ + 0 2 (t)) h(⌫0 + h(⌫ 0 =VTr [⇢m (t)gm (b +[ b )]+ 1] (t)) 0+ Hqtrajectory (t) = Tr [⇢m= (t)(H + )] m = 0 [⇢ z+ such that entropy produced along a single Hqmγ(t) Tr(t) (t)(H +[⇢the V )] = [ Hinferred =Hfrom Tr (t)(H V )] = = Tr [⇢ (t)(H + V q(t) m m 0 m m z)]+=1][ z + 1] W [γ] are knowledge of the trajectory, m m 0 ◆ ✓q0(t) 2 1✓/⇡ 2 0z + 2 2 0 be recorded without perturbing the system † in 0 P0ewhich (t) = can Trm (t) (t) =0 Trm0[⇢m (t)g q m m z m (b + b )] q ⇢q m q z ◆ ✓ arbitrary Using q m m principle 0with anzm z this † 2 precision0(Fig.1a). q m + 1 † † h⌦Tr [⇢ z(t)b E (t) = b] z m m m protocol, JE has been experimentally probed in various Hm (t)e= Trq [⇢q (t)(H +q V )] = h⌦b b + hgPm P†(t) (t)(bTr + b )⇢ z q m (t)+h(⌫ ∗ alexia.auffeves@neel.cnrs.fr e be += q0e (t)(b †h⌦b Hm (t) =(t) Tr [⇢Tr (t)(H V(b )]+= hg+ b† )0 + (t)) [ qb(t) qe qH mP setups, e.g. Brownian [8, orq= electronic = (t)g )](t)(H (t) =+mqTr V†9])] mq[⇢ mm = Trmq[⇢ (t)g + b† )] 2 m [⇢ m 0m ewith qparticles m (b †
✓/⇡ ! ⌦
✓/⇡ ✓/⇡ ! ! ⌦ ⌦ ✓/⇡ ◆ ✓ ✓ ◆ ✓1 ◆ +✓ 1 + 1 ◆ + + 1 P (t) = (t)P ⇢(t)(t) Tr = ⇢Tr(t) ⇢ (t) P Tr (t) =⇢ Tr P2=(t) 2✓ ◆2 2 ✓/⇡ ✓/⇡ ✓/⇡ + 1 P (t) = Tr ⇢ (t) 2 h(⌫ +h(⌫ (t))+ (t)) h(⌫ + h(⌫(t)) +[ ( = VTr)] = ✓ ◆+ H (t) H = Tr [⇢ (t)(H + [ + 1] H (t) [⇢ (t)(H V )] = ◆ ✓ (t) = Tr [⇢ H (t)(H V )]+ [= + 1] (t) =+Tr [⇢= +✓ V )] ◆ + 12(t)(H 2 1 2 P (t) = Tr P ⇢(t)(t) + 12 =PTr ⇢ (t) h(⌫ + (t)) 2 (t) = Tr ⇢ (t) 2=b Tr Hq (t) = Trm [⇢m(t) (t)(H V(t)g )] =(b(t) [ 2z(b++1] = Tr0m+ [⇢m + )] m [⇢2 b ) m (t)gm ◆ h(⌫0 + m (t))✓ 2 † Hq (t) = Trm [⇢m +Tr V )] =q (t)(Hm + V [)]z=+ 1] + 1b + hgm Pe (t)(b + H(t)(H m (t) 0= q [⇢ zh⌦b h ˙ (t) † † ⇢2 (t) † ˙ P (t) = Tr e q q ˙ E (t) = h⌦Tr [⇢ (t)b b] (t) = Tr (b + b )] m [⇢m (t)gm m= †= m†2(b = w˙ b=h(⌫ Em+† (t) †U (t) Tr [⇢ (t)g + )]+ m m m m † E (t) = h⌦Tr [⇢ (t)b b] † 0 b )] =mTr (t)g (b + b )] m m 2+ = Tr [⇢ (b Hm (t) = Trq [⇢(t) + V )][⇢ == h⌦b b(t) + hg P (t)(b +=m bh⌦b )(t)g m(t) m m q (t)(H m e+ m m Hm Tr [⇢ (t)(H V )] b + hg P h(⌫ + (t)) m q q m m H (t) = Tr [⇢ (t)(H V )] = † e (t)(b + 0(t)m= Tr q m 0 h(⌫ ++1] †[[⇢m (t)g (b ++h⌦b b+)](t)) m m Hq (t) = Trm [⇢m +[⇢V (t)(H )] =H (t) + H(t)(H (t) =0Tr + V=)]Tr = h⌦b b+ P b††2b) + h z0 hg [⇢ (t)(H V(t)) )](t)(b = h(⌫
✓/⇡
PACS numbers: 42.50.Ct; 42.50.Gy; ; Jean-Michel 42.65.Hw (Dated: February ⇤ CEA/CNRS/UJF Joint ” Nanophysics and 10, semiconductors”, Alexia Au↵`e42.50.Pq ves ,team G´erard , 2017) and Jean-Philip
! Cedex CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France9, F BP 166, 25, rueP/(⌦.~! des Martyrs, 38042 Grenoble /2)
0 ⇤ (Dated: February 2017) 1 ⌦ 10, 1 Joint 2 ! 1 CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ✓/⇡ 2 Au↵` (Dated: February 10, 2017) 1rard 1 2Nanophysics CEA/CNRS/UJF team ” Nanophysics and semiconductors ”, Alexia e ves , Jean-Michel G´ e rard , and Jean-Philippe Poizat PACS PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw CEA/CNRS/UJF Joint team ” and semicon Alexia Au↵` e ves , Jean-Michel G´ e , and Jean-Philippe Poiza numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw D 1
1 ”,Institut N´ February 22, 2017) 1 CEA/CNRS/UJF Joint team ”team Nanophysics and semiconductors e”,Institut el-CNRS, BP 166,Joint 25,(Dated: rue des Martyrs, 38042 Grenoble Cedex 9, N´ Fe 2Grenoble, CEA/CNRS/UJF ”42.50.Pq Nanophysics and semiconductors PACS numbers: 42.50.Ct; 42.50.Gy; ; 42.65.Hw CEA/INAC/SP2M, 17 rue des Martyrs, 38054 ⌦ BP 166, 25, rue des Martyrs, |gi 38042 Grenoble Cedex 9, France and 2 PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw BP 166, 25, rue des Martyrs, 38042 Grenoble France and CEA/CNRS/UJF Joint team ”and Nanophysics semico 2 (Dated: February 22,Cedex 2017)9,and CEA/CNRS/UJF Joint team ⌦⌧ ” Nanophysics semiconductors”, 2 /⇡ H a) w systems [10]. |mi CEA/CNRS/UJF Joint teamrue ” Nanophysics and semiconductors”, PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; des 42.65.Hw CEA/INAC/SP2M, 17 des Martyrs, 38054 CEA/INAC/SP2M, 17 rue Martyrs, 38054 Grenoble, TFrance = 0K⇤ Grenoble CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France⇤ ! (Dated: February 10, 2017) 2February 10, 2017) ⌦⌧0w(Dated: /⇡ P/(⌦.~! /2) The recent developments of quantum and nano10,D 2017) 2 ! PACS 42.50.Gy;(Dated: 42.50.Pq ; 42.65.Hw |ei February technologies have urged to explorenumbers: the validity42.50.Ct; of FTs ⌦ ! ⌦ PACSsystems. numbers:In 42.50.Ct; 42.50.Gy; 42.50.Pq ; ⌦ 42.65.Hw for out-of-equilibrium quantum the partic ✓ ◆ |gi Hd (t) !1 + 1 PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw ular case of Jarzynski’s protocol, the system coupled to 42.50.Gy; z D PACS numbers: 42.50.Ct; 42.50.Pq ; 42.65.Hw Pe (t) = Trq 1 ⇢q (t) |gi !1 a thermal reservoir can now be coherently driven. HowH! 2 ever in the quantum world, monitoring a system per⌦ ⌦ b) turbs its evolution (Fig.1b), posing severe difficulties to Te↵ > 0K !|ei ! ✓/⇡⌦⌧ ⌦ /⇡ D measure and even to define work exchanges. In a se-
2 !1 ⌦ h(⌫0 + (t ries of pioneering papers [11–14] it was shown that JE is |ei ! ! Trm [⇢2m (t)(HD 0 2+ V )] = still valid and can be probed, provided that the system’s Hd (t)H ✓ q⌦(t) ! + =◆ 2 !1 ! ✓/⇡ z + 1 ⌦ ✓ ◆ (resp. reservoir’s) internal energy change ∆U [γ] (resp. P (t) = Tr ⇢ (t) ! e q q + 1 z ⌦ 2 −Qcl [γ]) can be known from projective energy measure|gi ✓/⇡ Pe (t) =⌦Trq ⇢q (t) ⌦ ! 2 ments performed in the beginning and at the end of the ⌦ ! transformation. The work is then inferred from the First ✓ ◆ + = Trm [⇢m (t)gm (b + b† )] ✓/⇡ ⌦ ✓/⇡ (t) + 1 D ⌦ (t) z 2 h(⌫ + (t)) Law: ∆U [γ] = W [γ] + Qcl [γ]. Known as the two-points P (t) = Tr ⇢ H d (t) ✓ 0environment e (t)(H 2. 0a) based Trm [⇢FIG. +qEngineered V)]q= 1] on a three-level ◆[ z + q (t) m measurement scheme, this seminal extension ofHJE in=the 2 ✓/⇡ !2lifetime due to sponta-h(⌫0 + (t ✓/⇡ + 1 short |ei has az2! atom. The state |mi very quantum world has triggered numerous investigations, Pneous Trq ⇢in ✓/⇡ e (t) = q (t) ⌦ H (t) = Tr [⇢ V )] emission vacuum characterized by the rate Γ. = The 0+ ✓/⇡q 2 m m (t)(H motivated by two major challenges. The first challenge 2 transition |ei to |gi (resp. |mi to |ei) is associated with emis✓◆ ◆ ✓/⇡ † of a photon ✓mof(t) is practical: How to measure energy changes induced by ! † H = Tr [⇢ (t)(H + V )] = h⌦b b + hgm sion energy ω (resp. ω ) detected by D (resp. q◆ + 12 m ! 1 (t)) (t) = Trm [⇢m (t)g +✓h(⌫ b11)]⌦ 0qz+H ! m (b + d (t) a microscopic system on a large reservoir? OneHpossible P2 ).m =transition ⇢ (t) (t)(H +qV! )]z⌦ [ + 1] The |ei − |gi+is1driven by the Hamiltonian ✓ ◆ ⌦ e (t) q= q (t) = TrmD[⇢ 0Tr z z 22 Petempera(t) = TrHqPd (t). ⇢q=(t) ✓ h(⌫ ◆⌦ strategy relies on finite size reservoirs whose 1✓effective therEquivalent description of an Trq ⇢⌦ z+ ◆ (t)) e (t)b) q (t) 01+in terms + z 2(t) † (t) Tr2 Hq (t)[15– = TrmP [⇢e (t) (t)(H +⇢= V =qe↵of⇢ [Tr +[⇢ 1]ofz(t)g T >aq◆ 0K(t) mal presence weak classical Rabi freture is sensitive to heat exchanges with the system + 1 mbath. 0qthe zdrive eIn ✓)] =PTr = q (t) m m m (b + b )] 2 q |mi Pe†z(t) = Tr ⇢q2(t) + 21transition quency Ω resonant with the →†|ei, level |mi can 17], such that both emission and absorption events can Tr+q V ⇢)]q (t) Hm (t) = TrqP [⇢eq(t) (t)(H = eliminated. h⌦b b + hg +inb= ) h⌦Tr 2minco! m m Pe (t)(b be= adiabatically This an effective E [⇢m (t)b† b] be detected using fine calorimetry. But such reservoirs m (t) † result 2 2 ⌦ h(⌫ + (t)) ! (t) = Tr [⇢ (t)g (b + b )] 0 m m m herent rate Γ = 4Ω /Γ associated with transition |gi → ✓/⇡ + ✓/⇡ are by essence non Markovian, therefore affectingHthe dy+⌦coupled (t)) ◆✓/⇡ + V )] =h(⌫ [to zan+effective 1] |ei 0✓ q (t) = Tr(dotted m [⇢m (t)(H 0 simulating arrow), a qubit H (t) = Tr [⇢ (t)(H + V )] = [ + 1] 2z + 1 z namics of the system and the resulting work distribution q m m 0 h(⌫ +†(t)) ✓/⇡ h(⌫ + (t) = Tr (t)g (b +0Tbeff )] m [⇢ m0 2(t)) thermal bath ofm temperature = ~ω 1 − /Γ + ). † ! h(⌫ + (t)) Hq (t)(t)(H = Trm [⇢+ V )] = = [/kzB 0log(Γ ++ 1] m (t)(H 0+ P (t) = Tr ⇢ (t) [18]. H (t) Tr [⇢ (t)(H + V )] = h⌦b b+ e q q † Hq (t) = Trm [⇢ V )] = [ 1] m q q m mHq (t)E= 0 z 2 ⌦ Tr [⇢ (t)(H + V )] = [ + 1]hgm P (t) = h⌦Tr [⇢ (t)b b] + (t)) 2 h(⌫ + (t)) m m m h(⌫ m m 0 z 0 0 Another strategy involves the convenient 2 ✓ ◆ ˙ Hq (t) =platTrm [⇢m (t)(H + V )] = [ + 1] 2(t)b† ) 0 = Tr Hq (t) [⇢m†✓ (t)(H + V )] = [ z h+ mh⌦b 1 0mU Hm (t) semi= Trq [⇢q (t)(H + V )]= hg (t)(b ˙◆ez= 2+z0K forms provided by superconducting circuits and Te↵bz> =✓w˙ = 2 E˙ ◆ +† 1†P m Pe(t) (t)m= = Tr ⇢ (t) q q Tr b )] ✓ 2 † ◆ + 1 Pe(t) (t)==Tr Tr ⇢m(t)g (t)m†m conducting quantum photonics. In both cases, the ra(b2(b +†+ b )] q[⇢[⇢ q(t)g mm m Hm (t) = Trq [⇢q (t)(H + V )] = h⌦b b + hg Pe (t)(b +(t) b ) z 2 m (t) = Tr [⇢ (t)g (b + b )] ✓/⇡ diation produced by quantum emitters (superconductm m to Pean(t)engineered =mh(⌫ Tr ⇢qz(t)) + 1 [⇢+al-1](t)b† b] 0 h ˙(TLS) (t) m coupled system thermal q+ ˙ ˙ E[26, (t) = h⌦Tr Hq (t)ready (t)(H V†ofq)]us= [bath U= = Tr = ˙(t) =by= E+ m [⇢P mew 0 m mzincom2 ing qubits or quantum dots) is efficiently funneled into m Tr ⇢ (t) † q introduced some 27] to perform † )] 2 [⇢ 2 b◆)]2 (t) =[⇢ Trm + bm (t) = Tr [⇢ (b + ✓ m (t)g m (b †(t)g m m well-designed waveguides, and is thus recorded with (t)high = Tr (t)g (b + b )] ✓/⇡ † m mquantum m computing Em (t) = h⌦Tr h(⌫Tr (t)) herent [28]. great of˙ bthis m [⇢ m (t)b 0b]+ A 1interest = [⇢ + )] zm+ w =(bemission Hq (t) VTr )](t) = for [(t)g +˙m 1] efficiency. This recent experimental ability has lead to = Tr m [⇢m (t)(H 0+ zof † m † † the † Em scheme is that it allows detection the P (t) = ⇢ (t) e+ qh⌦b qbh(⌫ H (t) = Tr (t)(H V )] = + hg P (t)(b + b ) H (t) = Tr [⇢ (t)(H + V )] = h⌦b b + hg P (t)(b + b ) + (t)) 2 m q m m e m q q m m e 0 † † † the development of bright single photon sources [19, 2 ✓ +[ bzwhile Hm =20] TrqTr [⇢and V 0)]+ = of h⌦b hg Pe (t)(b )+ 1]◆ E(t)(H (t) =+h⌦Tr [⇢ qm m inmthe H(t) = [⇢the )] =b + b] absorption photons reservoir, still m m (t)b q (t) m (t)(H ˙ m mV w ˙ = E and to the monitoring of quantum trajectories of super† 2 h(⌫ ˙ + 1 keeping its(t) Markovian character. Eventually, both the 0 + (t) z = Tr [⇢ (t)g (b + b )] h (t) m† m[⇢ (t)(H m˙ † )] = H (t) = Tr + V conducting quantum bits [21–23]. However, q m m 0 P (t) = Tr ⇢ (t) Hm (t) standard = Trq [⇢q (t)(H + V )] = h⌦b b + hg P (t)(b + b ) U = = w ˙ = E˙ m ˙ e and the quantum qm qetrajectory heatmexchanges followed (t)) h (t) 0 +◆by † ✓ h(⌫ 2[ b† )+ † ˙ ˙ 2 2m schemes based on photo-counters do not allow H formthe (t)H= Tr [⇢=qTr (t)(H + Vreconstructed. )]+ + hg P (t)(b + Trcan [⇢ (bE b† h⌦b )]V )] b= U(t) = = w(t)g ˙fully =m (t)(H + q= m + the system be Note that a†1equite m[⇢ m m= q (t) m m 0 z † † z h(⌫ +h⌦b 2= (t) h⌦Tr [⇢+ (t)b b]⇢†[29] 0access (t) = Tr [⇢ (t)(H + )](t)(b Pe (t)( †(t)b recording of photons absorbed system, a majorm H Hm (t)by=theTr + VE )] = h⌦b b= hg PVb] +(t)) bb[2)+ hg m m m m qh⌦Tr qproposed m m P (t) Tr q [⇢q (t)(H m E (t) = similar strategy ism in to= quantum e(t)(H qb] q†e(t) E (t) =˙[⇢ h⌦Tr [⇢ (t)b m m m h (t) mTr m[⇢ H (t) = + V )] = + 1] q m m 0 z drawback for quantum thermodynamics purposes. U˙(t) = = Tr w˙ (t)g = Harmonic E˙(b [⇢ )] 2 2 trajectories ofma= forced m + boscillator. m m † 2 H (t) = Tr [⇢ (t)(H + V )] = h⌦b b + hgm Pe (t)(b + b† ) The second challenge is of fundamental nature, and † m q q m E (t) = h⌦Tr [⇢ (t)b b] ˙m m m m † validate ˙mh(⌫ = consists in identifying quantum signatures in H quantum (t) [⇢w (t)g b† )] We first and the platform +(bby+ (t)) (t)(H b += hgTr + b†0)E m m (t) = Trq [⇢q m m Pm e (t)(b w˙ +˙introduce =V )]E˙=m h⌦b † FTs. For instance, the coupling to a thermal bath erasesHq (t) (t) = Tr [⇢ (t)(H +[⇢ Vm )] = (b [ z+ checking in the simple case where the external Eh˙h(t) h⌦Tr (t)b b] + b†drive m m= 0˙[⇢ hJE m ˙(t) ˙== † (t) (t) = Tr (t)g )] ˙ ˙ U = w ˙ = E † m m m m 2 U = w ˙ = E quantum coherences during the system’s evolution. This m does U not coherences the+system, and heat ˙ =create E (t) =˙ in h⌦Tr b] Tr (b b [⇢ )]h(⌫ †= ˙m m m m (t)b = w ˙[⇢ m m (t)g m 0+ 2==any w(t) ˙2 [⇢ E †Em † (t)) E h⌦Tr (t)b b] is an irreversible process, expected to induce some m m Hm (t)mgen=(t) Tr [⇢ + V )] = h⌦b b + hg P (t)(b H= (t) =˙ Tr [⇢ (t)(H + V )] = [ z+ exchanges are perfectly detected. We turn + to bthe) 2 †then qq q (t)(H m m e m m 0 Em (t) = h⌦Trmdrive [⇢m (t)b b] we provide h (t) 2 uinely quantum entropy production [24, 25]. Furthercase of a coherence-inducing for which † U˙ =Em (t) ==h⌦Tr w˙ =m [⇢E˙mm(t)b b] more, the incoherent energy exchange between the sysa full thermodynamic 2 H (t) = Tr˙ q˙˙[⇢qanalysis. (t)(Hm Genuinely + V )] =quantum h⌦b† b + hgm P m tem and the bath (heat) is carried by quantized excitasignatures w in quantities are evidenced, =E E ˙ w˙=thermodynamic † h (t) m m † ˙m (b † (t) = [⇢ (t)g +P+eb(t)(b )] + ˙ w(t)(H †m ˙Tr ˙h⌦b m m tions. Thus any information depending on the recording U = = w ˙ = E H (t) = Tr [⇢ (t)(H + V )] = b + hg b(t)(b ) as well as how to measure them. Finally, we investigate ˙ = E h (t) m q q m m m H (t) = Tr [⇢ + V )] = h⌦b b hg P m q q m m e † ˙ [⇢=m (t)bfinite ˙ †mthe ˙2 of these excitations may be significantly hindered by the E (t) = h⌦Tr b] U = w ˙ = E h (t) the influence of the detectors efficiency on m m ˙ ˙ (t) ˙ w˙ =hofU(t) ˙(t) = Trm (t)g )] = = E˙ m 2mThe m (b + b heat eventual inefficiency of the detection scheme. So far, an h expression E ˙w˙m[⇢ the measured FTs. non-detected ˙ = m U= =w = w ˙ ˙=m=E U ˙ = E 2 2 experimental observation of such genuinely quantum efresults expression of JE that includes an † 2 in a modified Eproduction. m (t) = h⌦Tr m [⇢m (t)b b] fects has remained elusive. extra term in the entropy † We show that Erelated (t) =to (t)b b] † b and m(t)(H m Here we propose a strategy that addresses bothH the (t)this the record can P (t)(b w˙ h⌦Tr =+measurement E˙[⇢ = term Trq is[⇢ V )]m= h⌦b + † hg m m m m e hq˙from (t)mthe E (t) = h⌦Tr [⇢ (t)b b] ˙ practical and fundamental issues raised above (Fig.1c and be computed available data. ˙ m m ˙ ˙ ˙ w˙m= Em † w˙mw= E U = w˙ = = ˙ = Em E Fig.2). Our proposal is based on an effective two-level Hm (t) = Trq [⇢q2(t)(Hm + V )] = h⌦b b + hgm Pe (t)(b w
w˙ =
E˙ m
h ˙ (t)
h ˙ (t) U˙ = = w˙ = E˙ m U˙ = = w˙ = 2E˙ m 2E˙ h=˙ (t) h⌦Trm [⇢m (t)b† b] m (t) w˙E=
U˙ = m = w˙ = E˙ Em (t) =2 h⌦Trm [⇢m (t)bm† b]
3 II.
ENGINEERED THERMAL BATH
We consider a system whose three levels are denoted by |mi, |ei and |gi of respective energy Em > Ee > Eg (see Fig.2a). This three-level system is coupled to an electro-magnetic reservoir at zero temperature, such that |mi decays to state |ei (resp. |ei decays to |gi) with a spontaneous emission rate Γ (resp. Γ− ). We assume that |mi is a metastable level, such that Γ Γ− . We denote ω1 = (Ee − Eg )/~ and ω2 = (Em − Ee )/~. In addition, the atom is weakly driven by a laser resonant with transition ω1 +ω2 and of Rabi frequency Ω, satisfying Ω Γ. Due to its short lifetime, level |mi can be adiabatically eliminated, resulting in an effective incoherent transition rate Γ+ = 4Ω2 /Γ from state |gi to state |ei [26]. The states |ei and |gi define our effective qubit of interest (Fig.2b). Without external drive, this qubit relaxes towards some effective thermal equilibrium characterized + by the temperature Teff , satisfying e−~ω1 /kB Teff = ΓΓ− . In what follows we drive the qubit transition with a Hamiltonian Hd (t), such that ω1 may depend on time: the rates Γ± can be adjusted accordingly to keep the effective temperature constant. These adjustments are discussed in Ref.[26] and the Methods Section at the end of this manuscript. The dynamics of the effective qubit density matrix ρ is ruled by the master equation: i ρ˙ = − [H0 + Hd (t), ρ] + Γ+ L[σ † ]ρ + Γ− L[σ]ρ, ~
(3)
where H0 = (~ω1 /2)σz with σz = |eihe| − |gihg|. We have introduced the dissipation super-operator L[X]ρ = XρX † − 12 {X † X, ρ}, with σ = |gihe|. Assuming that a photo-counter D1 (resp. D2 ) detects the photons emitted at frequency ω1 (resp. ω2 ), one can formulate the evolution of the qubit state conditioned to the measurement records of the detectors in terms of quantum jumps [30]. Assuming an initial known pure state |Ψ0 i of the qubit, and discretizing the time between ti and tf such as tn = ti + ndt (with n ∈ J0, N K and tN = tf ), the evolution of the system features a stochastic trajectory γ of pure states |ψγ (tn )i. The trajectory is generated by applying a sequence of operators MK(tn ) , where K(tn ) stands for the stochastic measurement outcome at time tn .
Namely, detecting a photon on detector D1 (resp. D2 ) at the operator M1 = p time tn corresponds topapplying Γ− dtσ (resp. M2 = Γ+ dtσ † ) on the qubit state |ψγ (tn )i: remarkably in this scheme, the absorption of a photon from the effective heat bath (Fig.2b) is detectable and actually corresponds to the emission of a photon of frequency ω2 (Fig.2a). If no photon is detected, the nojump operator M0 = 1 − idtH(t) − 21 M1† M1 − 12 M2† M2 is applied. The qubit state is then renormalized. Each of these possible evolutions occurs at time tn with probabilities † MK(tn ) |ψγ (tn )i, pK(tn ) = hψγ (tn )|MK(t n)
III.
∆U [γ] = Uγ (tf ) − Uγ (ti ) = W [γ] + Qcl [γ] + Qq [γ],(9)
(4)
STOCHASTIC THERMODYNAMIC QUANTITIES
We now define the thermodynamic quantities associated with the system’s quantum trajectory |ψγ (tn )i. Following [25], the internal energy of the qubit at time tn is defined as Uγ (tn ) = hψγ (tn )|H(tn )|ψγ (tn )i, with H(t) = H0 + Hd (t) the total Hamiltonian of the system. The variation of this quantity along trajectory γ splits into three terms: there is a work increment δWγ (tn ) due to the Hamiltonian time-dependence and given by: δWγ (tn ) = hψγ (tn )|dHd (tn )|ψγ (tn )i,
K(tn ) = 0. (5)
There is also a classical heat Qcl [γ] corresponding to the energy exchanged with the effective thermal bath under the form of emitted or absorbed excitations of energy ~ω1 (tn ), where ω1 (tn ) is the qubit effective transition frequency defined as ω1 (t) = (he|H(t)|ei − hg|H(t)|gi)/~. This increment reads −~ω1 (tn ), K(tn ) = 1 δQcl,γ (tn ) = (6) ~ω1 (tn ), K(tn ) = 2. Finally, a third contribution must be introduced whenever the system’s dynamics creates coherences in the {|ei; |gi} basis. Defined as quantum heat in [25], its increment when a jump takes place reads: ~ω1 (tn )|hg|ψγ (tn )i|2 , Kγ (tn ) = 1 δQq,γ (tn ) = (7) −~ω1 (tn )|he|ψγ (tn )i|2 , Kγ (tn ) = 2, while if no jump takes place, it reads
1 δQq,γ (tn ) = − hψγ (tn )|{M1† M1 − p1 (tn ), H(tn )}|ψγ (tn )i 2 1 − hψγ (tn )|{M2† M2 − p2 (tn ), H(tn )}|ψγ (tn )i, 2
The first Law expressed for the trajectory γ is then given by
K ∈ {0, 1, 2}.
Kγ (tn ) = 0.
(8)
where O[γ] denotes the integrated quantity over the whole trajectory.
4
IV.
transformation and, in our example, amounts to a large exchange of photons between the system and the bath before the energy of the system significantly changes. Finally, entropy production drastically increases when & γ. In this case, very few photons are exchanged and the system is driven far from equilibrium.
JARZYNSKI EQUALITY IN THE QUANTUM REGIME
Jarzynski’s protocol generally consists in preparing the qubit in the thermal equilibrium state ρi = Zi−1 eH(ti )/kB Teff , then performing a strong measurement of the qubit energy state to project it on an initial pure state |ii of internal energy Ui . The qubit is driven out of equilibrium until time tf when a final strong energy measurement is performed, projecting the qubit onto the final state |f i of internal energy Uf . Note that the final density matrix is in general different from the final thermal equilibrium state Zf−1 eH(tf )/kB Teff . We have introduced the initial and final partition functions Zi and Zf . The two strong measurements allow inferring the qubit internal energy change ∆U [γ] = Uf − Ui , while the bath monitoring allow recording the classical heat exchange Qcl . As stated in the introduction (the demonstration is provided in Section V), the entropy produced in a single trajectory γ equals ∆i s[γ] = (∆U [γ] − Qcl [γ] − ∆F )/Teff [25, 31]. Inserting Eq.(9), it becomes
∆i s[γ] =
Qq [γ] 1 (W [γ] − ∆F ) + . Teff Teff
(10)
While fully compatible with former expressions for entropy production, Eq.(10) reveals two components respectively involving the amounts of work and quantum heat exchanged. The second component represents a genuinely quantum contribution due to the presence of quantum coherences in the qubit bare energy basis, that quantitatively relates entropy production and energetic fluctuations. In the following, we show that this quantum contribution can be measured in our setup. We first validate our platform by considering as driving Hamiltonian Hd (t) = (~ω1 /2)σz (t − ti ) for ti ≤ t ≤ tf . The work and heat exchanged in this case take particularly simple forms, the work being due to time-varying energy levels, and the heat to stochastic population changes induced by the bath. More precisely, we have δWγ (tn ) = hσz (tn )i~dt and δQcl,γ (tn ) = (hσz (tn+1 )i − hσz (tn )i)~ω1 (tn ), while the quantum heat increment and the quantum component of entropy production are zero. Note that this transformation boils down to the famous Landauer’s protocol [9, 32]. In Fig.3a and b, we have plotted the quantity he−∆i s[γ]/kB iγ and mean entropy production h∆i s[γ]iγ for different values of the coupling constant and temperatures. First note that, as expected, JE is verified for the thermodynamic quantities previously defined. Also note that the entropy production vanishes for Γ− . This is also expected since, in this limit, the drive is quasi-static and the qubit is always in equilibrium with the heat bath. This situation corresponds to a reversible
We now turn to a case with no classical counterpart by considering a drive Hd (t) = (~g/2)(σeiω1 t + σ † e−iω1 t ) for ti < t < tf and Hd (ti ) = Hd (tf ) = 0. This can be implemented by coupling the atom with a classical light field resonant with the transition ω1 . It gives rise to Rabi oscillations, i.e. to the reversible exchange of energy (work) between the qubit and the field under the form of the coherent and periodic emission and absorption of photons at the Rabi frequency g. Note that this scenario deviates from the previous one as, here, work exchanges induce coherences and population changes in the qubit bare energy basis. According to our definitions, the coupling of this coherently driven qubit to the bath will now give rise to both classical and quantum heat exchanges, as well as a quantum component in the entropy production. We first check that JE is recovered from the detection events in this quantum regime, as it appears in Fig3c. This shows that our generalized thermodynamic quantities are properly defined. A more thorough analysis can be drawn from Fig.3d, where we plot the average entropy production as a function of the ratio g/Γ− . There, one can identify three different situations, two of which corresponding to reversible transformations (h∆i siγ → 0). For a very weak drive g Γ− , the qubit is always in a thermal equilibrium state. This regime corresponds to reversible quasi-static isothermal transformations. In the other extreme, i.e. when g Γ− , the drive is strong enough to induce almost unperturbed Rabi oscillations. In this case, the transformation is too fast to allow for a stochastic event to take place and heat to be exchanged, i.e. it is adiabatic in the thermodynamic sense and once again reversible. The intermediate scenario where g ∼ Γ− gives rise to a maximal average entropy production. Note that, in this situation, the thermodynamic time arrow has a completely different nature from the classical case studied before. Here, entropy is mostly produced by the frequent interruptions of the qubit Rabi oscillations caused by the stochastic exchanges of photons with the bath (a.k.a. heat) (Inset of Fig3d). During such quantum jumps, the quantum coherences induced by the driving process are erased, giving rise to the exchange of quantum heat, and to entropy production of quantum nature. The deep connection between quantum heat and entropy production is captured in Eq.(10) where they are quantitatively related, and in Fig3(e to h), where both histograms concentrate around zero in the reversible regime, and take finite values in the irreversible regime. Such a connection between quantum irreversibility and energy fluctuations has already been evidenced in [25] in a different context, where stochasticity is caused
5 a)
c)
1.1 1.05
1.5
1
1
0.95
0.5
0.9 − 2 10
b)
1.5 1
1 0 -1 0
0.5 0 10− 2
1 0 -1 0
d) 10 5 10
0.05
1 0 -1 0
10− 1
1
101
0
-1
1
0
0.74 1 0 -1 0
0 10− 1
0.05 1
2
100
101
0
0
20
40
0
20
40
h)
2
5 10
10− 2
100
f) 0.1
5
100
0.8 0.6 0.4 0.2 0
1
0 10− 2
100
g)
e) 0.1
2
-1
0
1
0.8 0.6 0.4 0.2 0
FIG. 3. a),c) Evolution of the parameter he−∆i s[γ]/kB iγ and b),d) of the mean entropy production h∆i s[γ]iγ . We have considered the case of Landauer’s drive Hd (t) = (~ω1 /2)(t − ti )σz for a),b) and of a coherent drive Hd (t) = (~g/2)(σeiω1 t + σ † e−iω1 t ) for c),d). Both quantities are studied as a function of the respective driving strengths /Γ− and g/Γ− . Insets: evolution of hσz i = hψγ (t)|σz |ψγ (t)i along a single trajectory γ, in two different regimes of the studied transformation. In b) two different temperatures are used: ~ω1 /kB Teff = 3 (red ‘+’) and ~ω1 /kB Teff = 0.3 (blue ‘x’). In d), ~ω1 /kB Teff = 3. Distributions of the quantum heat increment e),f) and of the entropy produced along a single trajectory γ g),h) in the case of a coherent drive for g/Γ− = 0.01 (e,g) and g/Γ− = 1 (f,h). Parameters: Number of trajectories Ntraj = 105 for a),b), Ntraj = 2 × 106 for c),d), Ntraj = 103 for e) to h).
by quantum measurement instead of a thermal bath. In both cases, quantum heat exchanges and entropy production appear as two thermodynamic signatures of the same phenomenon, i.e. coherence erasure.
V.
FINITE DETECTION EFFICIENCY.
Up to now, we have assumed that the detection of heat is 100% efficient, i.e. that all the photons emitted by the three level atom can be detected. In practice, state-of-the art devices only allow for a fraction η of these photons to be indeed detected [33, 34]. A more realistic scenario is now investigated by considering that the photo-detectors have a finite efficiency η. As a consequence, between two detected photons, there exist several different possible (or fictitious) trajectories of pure states, which cannot be distinguished by the measurement record: Each of these trajectories corresponds to a particular sequence of undetected emissions and absorptions. This effect induces a decrease of the purity of the qubit state which has to be described by a stochastic density matrix ργ , rather than a wavevector. In the limit where η = 0 (no detector), the evolution of this density matrix is captured by Eq.(3), such that ργ (t) = ρ(t). For 0 < η < 1, the evolution of ργ (t) is still conditioned on the stochastic measurement outcome of the detector Kγ (tn ), that now corresponds to applying a set of super(η) operators {EK [ργ ]} (K ∈ {0, 1, 2}). Detecting one photon in detector D1 (resp. D2 ) between time tn and tn +dt
(η)
corresponds to applying the super-operator E1 [ργ ] = (η) ηM1 ργ M1† (resp. E2 [ργ ] = ηM2 ργ M2† ), which oc(η) curs with probability p1 (t) = ηTr{M1† M1 ργ (t)} (resp. (η) p2 (t) = ηTr{M2† M2 ργ (t)}). When no photon is de(η) (η) tected, which occurs with probability p0 = 1 − p1 − (η) p2 , a super-operator decreasing the state purity is applied: (η)
E0 [ργ ]= 1 − idt[H(t), ργ (t)] ηdt (Γ− σ † σ, ργ (t) + Γ+ σσ † , ργ (t) ) − 2 +(1 − η) Γ− dtL[σ] + Γ+ dtL[σ † ] ργ (t). (11) (η)
Note that E0 is a linear interpolation between (1) E0 [ρ] = M0 ργ M0† applied in the perfect efficiency (0) quantum jump formalism and E0 [ρ] which is the Lindbladian (right-hand terms in Eq.(3)). After applying (η) the super-operator EK , the density matrix has to be (η) divided by pK in order to be renormalized.
Generalized Jarzynski equality.
We now extend the definitions of thermodynamic quantities into the finite efficiency regime. For the sake of simplicity, we restrict the study to a driving Hamiltonian of the form Hd (t) = (~ω1 /2)(t − ti )σz , such that
6 no quantum heat is exchanged. With the definitions proposed above, the increment of measured classical heat reads: −~ω1 (tn ), K(tn ) = 1 δQηcl (tn ) = . (12) ~ω1 (tn ), K(tn ) = 2. Despite its apparent similarity with the definition in the perfect efficiency regime, this energy flow does not capture the entire classical heat flow dissipated in the heat bath, as it does not take into account the energy carried by the undetected photons. Therefore, the measured entropy production ∆i sη [γ] = (∆U −Qηcl [γ]−∆F )/Teff does not check Jarzynski equality as evidenced in Fig.4a) and b). The violation of JE is all the larger as the efficiency η is smaller, and JE is recovered in the limit η → 1, showing that our definitions for the finite-efficiency case still hold in the perfect efficiency limit. Remarkably, however, it is possible to formulate another equality taking into account the finite efficiency, converging towards the standard JE when η → 1: η
he−∆i s
[γ]/kB −ση [γ]
iγ = 1.
(13)
This equality is reminiscent of JE, but it includes a trajectory-dependent correction term ση [γ]. We now show that ση [γ] can be computed from the measurement record only. The entropy creation along one trajectory γ is usually defined as ∆i s[γ] = kB log(Pd [γ]/Pr [γ r ]), where Pd [γ] is the probability of a trajectory γ in the direct protocol defined by Hd (t): −→ N Y (η) Pd [γ] = pi Tr |f ihf | EKγ (tn ) |iihi| , (14) n=1
We have introduced pi = Zi−1 e−Ui /kB Teff the probability of drawing the initial pure state |iihi| of internal energy Ui in the initial thermal distribution. The arrow indicates the order in which the sequence of super-operators is applied. Pr [γ r ] is the probability of the time-reversed trajectory γ r corresponding to γ. It is generated by the time-reversed drive Hd (tf − t) and the reversed sequence (η),r of super-operators EKγ (tn ) [25, 35, 36], such that: ←− N Y (η),r Pr [γ r ] = pf Tr |iihi| EKγ (tn ) |f ihf | , (15) n=1
where pf = Zf−1 e−Uf /kB Teff the probability of drawing the final pure state |f ihf | of internal energy Uf in the final thermal distribution. When η = 1, the time-reversed operators at time tn reduce to (1),r
E1
[ργ ](tn ) = e−~ω1 (tn )/kB Teff M1† ργ M1
(1),r E2 [ργ ](tn ) (1),r E0 [ργ ](tn )
= =
e~ω1 (tn )/kB Teff M2† ργ M2 M0† ργ M0 .
(16) (17) (18)
By inserting these expressions in Eq.(15), we express the ratio Pr [γ r ]/Pd [γ] = e−(∆U [γ]−∆F )/kB Teff eQcl [γ]/kB Teff , which is JE. When η < 1, the ratio Pr [γ r ]/Pd [γ] is modified. It (η) is useful to decompose E0 into three super-operators conserving purity: (η)
E0
(η)
(η)
(η)
= E00 + E01 + E02 ,
(19)
with (η)
(1)
(η)
(1)
E01 [ργ ] = (1 − η)E1 [ργ ],
E02 [ργ ] = (1 − η)E2 [ργ ], (η)
(1)
E00 [ργ ] = E0 [ργ ].
(20)
From this decomposition, we generate a set F[γ] of fictitious trajectories γF of pure states, compatible with γ: a trajectory γF ∈ F[γ] is built by applying a sequence of super-operators {EKγF (tn ) }n , with KγF (tn ) = Kγ (tn ) if Kγ (tn ) belongs to {1, 2} (i.e. when a photon has been detected), and KγF (tn ) ∈ {00, 01, 02} when Kγ (tn ) is zero (no photon detected). F[γ] thus contains 3N0 [γ] fictitious trajectories, where N0 [γ] is the number of time-steps during which no photon is detected. The perfect-efficiency fictitious trajectories are interesting because they can be time-reversed using the rules Eq.(16)-(18), P while their probability distribution checks Pd [γ] = Eventually, the following γF ∈F [γ] Pd [γF ]. equality is derived: X Pr [γ r ] = 1 γ
=
XX γ
= =
F [γ]
XX γ
Pr [γFr ]
X
Pd [γF ]e−(∆U [γF ]−∆F −Qcl [γF ])/kB Teff
F [γ]
η
Pd [γ]e−(∆U [γR ]−∆F −Qcl [γ])/kB Teff
γ
×
X
F [γ]
η
Pd [γF |γ]e(Qcl [γ]−Qcl [γF ])/kB Teff . (21)
We have introduced the conditional probability Pd [γF |γ] = Pd [γF ]/Pd [γ] of the fictitious trajectory γF ∈ F[γ], given the detected trajectory γ. In order to find Eq.(13), we now define: ση [γ] = − log
X
F [γ]
η
P (γF |γ)e(Qcl [γF ]−Qcl [γ])/kB Teff . (22)
ση [γ] can be numerically computed, based solely on the measurement outcomes. The computation involves the simulation of a sample of the trajectories γF ∈ F[γ] for each trajectory γ. The corrected expression
7 a)
0.8
1.3
0.1
0.2
1 0.5
0
1
0
5
0
10
d)
1.6
0.8 0.6
1.2
0.4
1
0.2
10− 1
100
0
5
10
0
5
10
f)
1.4
10− 2
0.2
0.4
1.1 0.9 0
0.70
0.6
1.2
b)
e)
c) 1.4
0
0.71
0.2 0.1 0
5
0
10
FIG. 4. Test of Jarzynski equality at finite efficiency, for a drive Hd (t) = ~ω1 (t − ti )/2σz . Purple ‘x’: Measured parameter η he−∆i s [γ]/kt extB iγ . The magnitude of the deviation (positive for this transformation) increases monotonically when η decreases. 4 Blue ‘+’: The plotted parameter included the correction term σ[γ]. We have used Ntraj = 10p trajectories γ and 104 ficitious trajectories γF for each trajectory. The error bars stand for the standard deviation divided by Ntraj . b) Jarzynski’s parameter without correction for η = 0.1 as a function of the temperature. The error bars are smaller than the marker size. c)f) Distributions of the entropy production during the transformation for η = 1 in c),e) and η = 0.3 in d),f). c) and d) correspond to high temperature (βω1 = 7 10−3 ) and therefore a large mean number of photons exchanged per trajectory hN [γ]iγ = N − hN0 [γ]iγ . The inserts show the same distribution with rescaled axes to evidence the similarity of shape. e) and f) correspond to a lower temperature (βω1 = 1) and a few photon exchanged. Parameters: For a) ~ω1 /kB Teff = 0.1, γtf = 0.5, /γ = 600. For b) and c): γtf = 1, /γ = 9, Ntraj = 105 .
he−∆i s[γ]/kB −ση [γ] iγ is plotted in Fig.4a): JE is verified, showing that the experimental demonstration of a FT in a realistic setup is within reach. Our results explicit a new quantum-classical border in the thermodynamic framework, which is illustrated in Fig.4(b to f) where the deviation from JE is studied as a function of the reservoir temperature. In the quasi-static limit characterized by Γ− (high temperature case), many photons are exchanged and the system is allowed to thermalize before any significant work is done: the transformation is reversible. We see from Fig.4c) and d) that in this limit, the effect of detection inefficiency in the entropy production is merely to widen the spread of the distribution but conserving its Gaussian shape around zero. The statistical properties of the distribution are not affected, even if many photons are missed, and JE is verified (Fig.4b). This corresponds to a semi-classical situation where, even though the exchange of heat is quantized, the overall thermodynamic behavior of the system is equivalent to a classical ensemble. In this limit, there is no significant information loss due to the undetected photons. On the other hand, as soon as work and heat are exchanged at similar rates ∼ Γ− , the system is
brought out of equilibrium and the transformation is irreversible. In this limit, few photons are exchanged before a non negligible amount of work is done on the system. Each missed photon represents, then, a significant information loss, such that the measured distribution of entropy production is severely affected (see Fig.4e) and f)). Such information loss is quantifiable by the violation of the non-corrected JE, and is a direct consequence of the quantization of heat exchanges.
VI.
CONCLUSION
We presented a realistic setup based on a driven three-level atom allowing to simulate a qubit in equilibrium with an engineered reservoir, giving full access to the distribution of the heat dissipated by the qubit. We exploited this scheme to characterize irreversibility in the case of a coherently driven qubit, coupled to a thermal bath: we evidenced genuinely quantum contributions to the entropy production, and showed these contributions do have some energetic counterpart. In a second part, we took into account finite detection efficiency and evidenced deviations from Jarzynski
8 equality that can be tested and corrected. We derived a modified equality, involving only quantities computed from the finite-efficiency measurement record. This work opens avenues for the experimental verification of fluctuation theorems in quantum open systems. Owing to the versatility of the scheme, various reservoirs could be simulated, including non-thermal ones. Furthermore, quantitative relations between entropy production and energy fluctuations are the basis for energetic bounds for classical computation [32, 37, 38]. In this work we study situations where equivalent relations can be derived in the quantum regime, providing new tools to explore energetic bounds for quantum information processing.
ACKNOWLEDGMENTS
MFS acknowledges CNPq (Project 305384/2015-5). NKB thanks the support from the Brazilian agency CAPES. This work was supported by the Fondation Nanosciences of Grenoble under the Chair of Excellence ”EPOCA”, by the ANR under the grant 13-JCJCINCAL and by the COST network MP1209 ”Thermodynamics in the quantum regime”.
By identification, we find the conditions: p Ω(t) = Γ0 (¯ n[ω1 (t)] + 1)¯ n[ω1 (t)] Γ− (t) = Γ0 (¯ n[ω1 (t)] + 1),
Eq.(23) can be fulfilled by tuning accordingly the intensity of the laser drive. Eq.(24) requires to tune the incoherent rate of transition from |ei to |gi. This task can be performed by embedding the three level atom in a quasi-resonant optical cavity and tuning the frequency between state |ei and |gi e.g. with a non-resonant laser (Stark effect) according to a protocol ω10 (t) designed to fulfill Eq.(24). Note that the implemented protocol ω10 (t) is in general different from the target protocol ω1 (t) which defines the constraints Eq.(23)-(24) on the qubit dynamics.
B.
Simulating a heat bath with an engineered environment
The rates Γ± induce the same dynamics as a heat bath, if and only if n ¯ [ω1 (t)] P˙e = −Γ0 (2¯ n[ω1 (t)] + 1) Pe (t) − . 2¯ n[ω1 (t)] + 1 We have introduced the population of the excited qubit’s state Pe = (Tr[ρ(t)|eihe|], the mean number of thermal photons in the effective bath n ¯ [ω] = (e~ω/kB Teff − 1)−1 and Γ0 the spontaneous emission rate of the qubit. The dynamics induced by the engineered environment is: Γ+ P˙e = −(Γ+ + Γ− ) Pe (t) − . Γ+ + Γ −
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Expression of the quantum heat
For a general driving Hamiltonian Hd (t) = δ(t)σz + µ(t)σ + µ∗ (t)σ † , the quantum heat δQq (tn ) exchanged when a jump takes place reads (for η = 1):
APPENDIX A.
(23) (24)
δQq =
~ω1 (t)Pg (t) − 2Re µ(t)hσ(t)i, Kγ (tn ) = 1 −~ω1 (t)Pe (t) − 2Re µ(t)hσ(t)i, Kγ (tn ) = 2
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δQq (tn ) = −(Γ− − Γ+ )dtRe(hσi)(~µ(tn )Pg (tn ) − ~µ∗ (tn )Pe (tn )) −~ω1 (tn )(Γ− − Γ+ )dtPe (tn )Pg (tn ), Kγ (tn ) = 0 (25) Note that δQq (tn ) is zero when the qubit state has no coherence in |ei − |gi basis, such that hσi = hσ † i = 0, (Pe (tn ), Pg (tn )) = (1, 0) (resp. (0, 1)) when Kγ (tn ) = 1 (resp. Kγ (tn ) = 2): the qubit has zero probability to absorb a photon at time t when Pe (tn ) = 1, and zero probability to emit a photon when Pg (tn ) = 1.
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