Jarzynski equality for superconducting optical cavities: an alternative path to determine Helmholtz free energy
Josiane Oliveira Rezende de Paula, J. G. Peixoto de Faria, J. G. G. de Oliveira Jr., Ricardo de Carvalho Falcão, Adélcio C. Oliveira
aa r X i v : . [ qu a n t - ph ] J a n Jarzynski equality for superconducting optical cavities:an alternative path to determine Helmholtz free energy
Josiane Oliveira Rezende de Paula
Escola Estadual Cˆonego Luiz Vieira da Silva, Ouro Braco, MG 36420-000, Brazil
J. G. Peixoto de Faria
Departamento de Matem´atica, Centro Federal de Educa¸c˜ao Tecnol´ogica de Minas Gerais,Belo Horizonte, MG, 30510-000, Brazil
J. G. G. de Oliveira Jr.
Departamento de Ciˆencias Exatas e Tecnol´ogicas, Universidade Estadual de Santa Cruz,45.662–900, Ilh´eus – BA – Brazil
Ricardo de Carvalho Falc˜ao
Departamento de Estat´ıstica, F´ısica e Matem´atica, Universidade Federal de S˜ao Jo˜ao DelRei, C.P. 131,Ouro Branco, MG, 36420-000, Brazil
Ad´elcio C. Oliveira
Departamento de Estat´ıstica, F´ısica e Matem´atica, Universidade Federal de S˜ao Jo˜ao DelRei, C.P. 131,Ouro Branco, MG, 36420-000, Brazil
Abstract
A superconducting cavity model was proposed as a way to experimentally in-vestigate the work performed in a quantum system. We found a simple mathe-matical relation between the free energy variation and visibility measurement inquantum cavity context. If we consider the difference of Hamiltonian at time t and t λ (protocol time) as a quantum work, then the Jarzynski equality is validand the visibility can be used to determine the work done on the cavity. Keywords: quantum work, quantum heat, quantum Jarzynski equality, cavityquantum electrodynamics
1. Introduction
Fluctuation theorems have been developed to describe systems far from equi-librium, that is the case of Jarzynski equality (JE) [1, 2, 3, 4] and Crooks relation[5] and Bochkov-Kuzovlev [6]. The classical JE is a relation between the free en-ergy difference of two equilibrium states (∆ F ) and the work ( W ) averaged over Preprint submitted to Physica A January 7, 2019 ll possible paths of a nonequilibrium process linking them. Mathematically theJE is e − β ∆ F = h e − βW i . (1)The JE was developed assuming that the system is isolated from the reservoirwhile the protocol is performed. Morgado and Pinto [7] have obtained JE fora massive Brownian particle connected to internal and external springs, theirresult does not depend on the decoupling of system and bath along the protocoltime, a Brownian particle was also experimentally investigated in context of JE[8]. Minh and Adib [9] have used path integral formalism and demonstratedthat the validity of JE in the context of Brownian particle subject a class ofharmonic potential.Experimentally some important results were achieved, Liphardt and collab-orators [10] demonstrated the validity of JE by mechanically stretching a singlemolecule of RNA reversibly and irreversibly between two conformations. Toy-abe and collaborators [2] have investigated experimentally the JE for a dimericparticle comprising polystyrene beads by attaching it to a glass surface of achamber filled with a buffer solution, they have found a discrepancy smallerthan 3% between the observed result and what was expected with JE. Douarcheand collaborators [11] have experimentally checked the Jarzynski equality andthe Crooks [5] relation on the thermal fluctuations of a macroscopic mechanicaloscillator in contact with a heat reservoir and found a good agreement with JEand crooks relation. Hoang et al. [12] have performed an experimental testof JE and Hummer-Szabo relation [13] using an optically levitated nanosphere.These, among many others experimental investigation consolidates the JE inthe classical domainIn this work, we use the quantum analog of Jarzynski equality (JE) to pro-pose a way to obtain experimentally the work performed in a quantum system.The Quantum version of (JE) is a controversy area, the first attempts to deriveJarzynski equalities for quantum systems failed [6, 15, 14] leading to mislead-ingly believed that the equality was not valid for quantum systems. In some ofthis earlier derivations of Jarzynski equation for quantum systems a work oper-ator was defined [6, 14, 15, 16, 17], but this work definition is not, in general, aquantum observable [18] this is due to the fact that work characterize a processrather than an instantaneous state of the system. This earlier attempts had ledto quantum corrections to the classical Jarzynski result and the classical resultwas recovered only when the Hamiltonian in a time t commutes with itself in atime t ′ [16].Recently the discussion has been changed to how to define operational waysof measuring work since Jarzynski’s equality has already been obtained for closedquantum systems [25, 26, 18, 19, 22] for open systems [27, 5, 23] even for sys-tems with strong couplings [24]. Most of these proposals are linked to thequestion of measuring energy in two moments, which from a quantum pointof view introduce several questions since a quantum systems have a dynamicalbehavior that is affected by the measurements, thus since one performs energymeasurements the system state changes, this problem is circumvented if one use2on-demolition measurements, in reference [28] they show that POVM (positiveoperator valued measure) can be used to sample the work probability distribu-tion. Experimentally, some advances have been achieved, An and collaborators[29] have investigated experimentally the JE in the quantum domain. Theyhave used Y b + ion trapped in a harmonic potential and perform projectivemeasurements to obtain phonon distributions of the initial thermal state, theyhave concluded that JE still valid, a similar result was obtained by measuringa single-molecule [30].In this work, we study a transition between two equilibrium states of aquantum system, namely a quantum harmonic oscillator coupled to a thermalbath. This model can be implemented with a cavity quantum electrodynamics(CQED) [31]. The protocol can be executed by injecting a coherent field inthe cavity. The CQED experimental setup was widely used to explore quantummechanical foundations with many interesting results (see [32, 33, 34, 35] andreferences therein). Even for a more realistic model [36], that consider environ-ment action, the quantum nature of the electromagnetic field was demonstrated.Experimentally, the initial state was prepared in a pure state [34], usually ina vacuum. We consider a thermal state, as the initial state, and the work isgiven by the difference of cavity’s Hamiltonian ∆ H = H ( τ ) − H (0) where τ isa time bigger than the protocol time. The thermal state is not a guarantee ofa “classical state” [38], but surprisingly, for the work as defined above, the JEis valid in all quantum domain [28, 18, 20, 21]. Cerisola and collaborators [39]have shown that JE is valid in quantum domain for a more general measurementclass named “a quantum work meter”. Assuming that JE is valid, we show thatthe free energy variation and also the mean h e − βW i can be simply inferred bya measurement of fringes visibility in the context of CQED.
2. Quantum Jarzynski Equality
We consider a Quantum analog of the model studied by [40]. It consistsof N non-interacting harmonic oscillators all initially in thermal equilibrium attemperature T , then the partition function is Z (0) = N Y n =1 Z n (0) (2)with Z n (0) = exp (cid:0) − β ¯ hω (cid:1) − exp ( − β ¯ hω ) . (3)After the action of the protocol, the equivalent quantum Hamiltonian of n thoscillator for t ′ < t is ˆ H n ( t ′ ) = ˆ p n m + 12 mω ˆ x n + l ˆ x n L ( t ′ ) . (4)3hen its eigenvectors are the same of harmonic oscillator and the energies are E n ( t ′ ) = (cid:18) j + 12 (cid:19) ¯ hω − l L ( t ′ ) mω , (5)where j are positive integers. Thus the partition function reads Z ( t ′ ) = N Y n =1 Z n ( t ′ ) , (6)with Z n ( t ′ ) = exp (cid:20) βl L ( t ′ )2 mω (cid:21) " exp (cid:0) − β ¯ hω (cid:1) − exp ( − β ¯ hω ) , and Helmholtz free energy to the n -th oscillator of the system is F n = ¯ hω − l L ( t ′ )2 mω + 1 β ln [1 − exp ( − β ¯ hω )] . (7)Again, the protocol changes L parameter from L to L . Thus,∆ F n = l mω (cid:0) L − L (cid:1) . (8)It is easy to see that the variation of the Helmholtz free energy to the systemwill be ∆ F = N X n =1 ∆ F n . (9)Since we are dealing with N non-interacting harmonic oscillators, without lossof generality, we can restrict our analysis to a single oscillator of system. Wewill do this from now on. The Hamiltonian ( ˆ H ( t ′ )) to a single oscillator of system isˆ H n ( t ′ ) = ˆ p n ω (cid:20) ˆ x n + 2 l ˆ x n L ( t ′ ) ω (cid:21) . (10)where we set m = ¯ h = 1. We can also write the Hamiltonian in terms of creationand annihilation operators, and it will be useful in the next sections, it is givenby ˆ H n ( t ′ ) = ω (cid:18) ˆ a n † ˆ a n + 12 (cid:19) + ˜ L ( t ′ ) (cid:0) ˆ a + ˆ a † (cid:1) . (11)where ˜ L ( t ′ ) = q ¯ h mω L ( t ′ ). 4e assume that the system environment coupling is not relevant during theprotocol time, if we assume that the necessary work [40] to change L L is the same as the system energy variation ∆ E , then at t = 0 we find an energy E (0) n with a probability given by P (0) n = exp (cid:16) − βE (0) n (cid:17) Z . (12)After a time t f the system is in a state ˆ U ( t f ) (cid:12)(cid:12) ψ n (cid:11) and the transition probabilityis w mn = (cid:12)(cid:12)(cid:12)D ψ ( f ) m (cid:12)(cid:12)(cid:12) ˆ U ( t f ) (cid:12)(cid:12)(cid:12) ψ (0) n E(cid:12)(cid:12)(cid:12) . (13)Here, ˆ U ( t f ) is ˆ U n ( t f ) = T > exp (cid:20) − i ¯ h Z t f dt ′ ˆ H n ( t ′ ) (cid:21) , (14)where T > denotes time ordering operator. Finally, we obtain h exp( − β ∆ E ) i = X n P (0) n X m w mn exp( − β ∆ E ) (15)with ∆ E = E ( f ) m − E (0) n . After some manipulations [40] we get h exp( − β ∆ E ) i = exp (cid:20) N βl ω (cid:0) L − L (cid:1)(cid:21) . (16)Comparing (16) with (8) its clear that JE is verified, what was expected (seeRef. [16]).
3. Visibility of Interference Fringes and its connection with JE
In this section, we verify the possibility of an experimental realization pro-cedure. We assume that the harmonic oscillator is a microwave field storedin a high- Q superconducting cavity. The field state can be monitored by es-tablishing an interaction with a Rydberg atom [31, 35, 41]. Rydberg atomshave suitable properties for use as probes of even weak electromagnetic fields,such as high dipole moments, which ensure high coupling strengths, and highmean lifetimes. We consider a non-demolition measurement procedure [42] ofthe number of photons contained in the electromagnetic field by setting a dis-persive interaction between it and each Rydberg atom. The number of photonsin the cavity is probed by Ramsey interferometry [43], and this measure allowsobtaining some information about the field state inside the cavity.A schematic representation of the experimental setup is illustrated in FIG.1. We consider a three-level Rydberg atom, as illustrated in Fig. 2. The three-level atom is sent through an apparatus as schematized in FIG. 1. The atom,5 igure 1: Schematic representation of the apparatus used in a typical Ramsey interferom-etry with Rydberg atoms. A Rydberg atom A , in general, an atom of an alkali element,is prepared a highly excited electronic level | g i and it is sent through the apparatus. Thetwo Ramsey zones, R and R , are low- Q cavities devised to change the atomic states as | g i → ( | g i + | f i ) / √ | f i → ( − | g i + | f i ) / √
2. Despite the low mean number of photonsinside the two Ramsey zones, from the practical point of view, the atom sees a classical fieldthere, so much that the atom leaves the Ramsey zones in a non-entangled state. In the su-perconducting microwave cavity C the atom interacts dispersively with the cavity field. Thisinteraction glues different phase shifts in each atomic state that depends on the number ofphotons of the cavity field. So, right after the atom leaves the superconducting cavity C , theglobal state of atom plus the field inside it remains entangled . The detector D measures theatom at | f i . Repeating the process under slight different conditions (for example, changingthe frequency of the mode inside the Ramsey zones) a interferometric pattern is produced,and the information about the number of photons of the mode inside the cavity C can beextracted. when passing through C , will interact dispersively with the atom inside it andthe interest Hamiltonian isˆ H = ¯ hω (cid:18) ˆ a † ˆ a + 12 (cid:19) + E e | e ih e | + E g | g ih g | + E f | f ih f | +¯ hω h(cid:0) ˆ a † ˆ a +1 (cid:1) | e ih e |− ˆ a † ˆ a | g ih g | i , (17)where ˆ a † (ˆ a ) is the creation (annihilation) operator acting on the field stateinside the cavity C , | i i is the i -th atomic level, defined as i = e , g and f , E i is the corresponding energy of the i th level and ω is the field frequencyin C , ω = Ω / δ is the coupling constant in the dispersive regime, Ω is thevacuum Rabi frequency inside cavity C and δ is the atom-field detuning betweentransition frequency of the energy levels | e i and | g i , ω eg = ( E e − E g ) / ¯ h , andthe frequency of the stored mode in C , ω .Without lost of generality, equation (17) can be presented asˆ H = ˆ H + ˆ H I , (18)6 igure 2: Three-level atom. The states | e i and | f i have the same parity and are opposed tothe parity of | g i . The field in the superconducting cavity has frequency ω and is de-tunedof δ = ω − ω eg from transition frequency ω eg = ( E e − E g ) / ¯ h between levels | e i and | g i .The Ramsey zones [43] have frequencies ω r and are close to the transition transition sintonyfrequency ω gf = ( E g − E f ) / ¯ h between levels | g i and | f i . where ˆ H = ¯ hω (cid:18) ˆ a † ˆ a + 12 (cid:19) + ( E e + ¯ hω ) | e ih e | + E g | g ih g | + E f | f ih f | , (19)ˆ H I = ¯ hω ˆ a † ˆ a (cid:0) | e ih e | − | g ih g | (cid:1) . (20)We observe that [ ˆ H , ˆ H I ] = 0, then, in the interaction picture we have anarbitrary state of the field in cavity C , it is given by ρ F (0) = X i,j ρ i,j | i ih j | , (21)an atom is sent to interact with the field in cavity C , this atom is previouslyprepared in the state ρ A (0) = g | g ih g | + f | f ih f | + (cid:2) x | g ih f | + c.h. (cid:3) , (22)with g + f = 1 and | x | ≤ gf . After a time interval ∆ t , the atom-field state inthe interaction picture, is given by ρ (∆ t ) = e − iH I ∆ t/ ¯ h ρ F (0) ρ A (0) e iH I ∆ t/ ¯ h (23)taking the trace in field variables in time ∆ t we obtain the atomic state that is ρ A (∆ t ) = Tr F (cid:2) ρ (∆ t ) (cid:3) . (24)7s the atom goes through R the states | g i and | f i will be entangled with arelative phase φ , as we can observe in FIG. 1. After that, the atom is measuredin D and as we change φ , the Ramsey interference fringes appear. The visibility V of the interference fringes pattern is proportional to the absolute value ofcoherence’s term of the state (24) and can be obtained by V (∆ t ) = 2 (cid:12)(cid:12)(cid:12) Tr (cid:2) ρ A (∆ t ) | g ih f | (cid:3)(cid:12)(cid:12)(cid:12) , (25)as we can see in reference [44].The visibility V depends on field state eq.(21), then the interference fringescarries field state information, it becomes clear as we analyze equation (24) indeeper, we have: ρ A (∆ t ) = X m h m | e − iH I ∆ t/ ¯ h ρ F (0) ρ A (0) e iH I ∆ t/ ¯ h | m i = X m h m | e − iH I ∆ t/ ¯ h ρ F (0) e iH I ∆ t/ ¯ h e − iH I ∆ t/ ¯ h ρ A (0) e iH I ∆ t/ ¯ h | m i = X m,n h m | e − iH I ∆ t/ ¯ h ρ F (0) e iH I ∆ t/ ¯ h | n ih n | e − iH I ∆ t/ ¯ h ρ A (0) e iH I ∆ t/ ¯ h | m i . (26)Taking (21) into (26), we obtain ρ A (∆ t ) = X m,n ρ m,n e − iω ∆ t ( m − n ) (cid:0) | e ih e |−| g ih g | (cid:1) δ n,m e − iω ∆ t ( n − m ) | e ih e | e iω ∆ tn | g ih g | ρ A (0) e − iω ∆ tm | g ih g | = X n,n ρ n,n e iω ∆ tn | g ih g | ρ A (0) e − iω ∆ tn | g ih g | = g | g ih g | + f | f ih f | + (cid:2) x X n ρ n,n e inω ∆ t | g ih f | + c.h. (cid:3) . (27)We observe that the term P n ρ n,n e inω ∆ t in the equation (27) can be written as X n ρ n,n e inω ∆ t = X n h n | ρ F (0) e iω ∆ ta † a | n i = Tr h ρ F (0) e iω ∆ ta † a i . (28)Now, introducing (28) into 27) the field state is given by ρ A (0) = g | g ih g | + f | f ih f | + (cid:2) ¯ x | g ih f | + c.h. (cid:3) , (29)where ¯ x = x Tr h ρ F (0) e iω ∆ ta † a i . We can obtain the Ramsey interference fringesvisibility, equation (25) for the atom in the state(29), then V (∆ t ) = V (cid:12)(cid:12)(cid:12)(cid:12) Tr h ρ F (0) e iω ∆ ta † a i(cid:12)(cid:12)(cid:12)(cid:12) . (30)The visibility clearly depends on field initial state. In equantion (30) we defined V = 2 | x | , it is the visibility of the vacuum state. Form now on, we consider theoptimum case where V = 1. This occurs when | x | = 1 / | g i + e iθ | f i ) / √
2, and θ is a arbitrary relative phase.8 .1. The field state in a equilibrium Thermal State Following the previous mathematical procedure, easily we can obtain thevisibility when we have in C a field initially in a thermal given by ρ T h = e − β ¯ H Z , (31)where ¯ H = ¯ hω (cid:0) a † a +1 / (cid:1) is the field Hamiltonian that is in thermal equilibriumand Z = Tr (cid:0) e − β ¯ H (cid:1) , as usual, is the partition function. Then, after some algebrawe obtain V T h (∆ t ) = sinh( β ¯ hω / q sinh ( β ¯ hω /
2) + sin ( ω ∆ t/ . (32)For practical proposes, the perfect choice of the interaction time can be obtainedif we adopt ω ∆ t = π , then equation(32) can now be written as V T h ( π/ω ) = tanh( β ¯ hω / . (33)It is interesting to note that equation (32) can be simplified in two limiting cases V T h ( π/ω ) ≈ β ¯ hω / , se β ¯ hω ≪ , , se β ¯ hω ≫ . (34) The Hamiltonian for a displaced field in a cavity C is given by˜ H = ¯ hω (cid:16) a † a + 12 (cid:17) + ¯ hω (cid:0) αa † + α ∗ a (cid:1) . (35)where α is the displacement magnitude, observe that it is directly related withthe protocol term ˜ L in equation (11). The Hamiltonian (35) can be written as˜ H = D † ( α ) ¯ HD ( α ) − ¯ hω | α | , (36)where D is the displacement operator defined as D ( α ) = exp (cid:0) αa † − α ∗ a (cid:1) . Inthis case, the thermal equilibrium state is ρ ( α ) T h = e − β ˜ H Z ( α ) , (37)now the partition function is Z ( α ) = Tr (cid:0) e − β ˜ H (cid:1) . Easily we can show that ρ ( α ) T h = D † ( α ) ρ T h D ( α ) , (38) Z ( α ) = Z e β ¯ hω | α | . (39)Now we can obtain the new visibility in the same way we obtained for theThermal State (previous subsection), but now we consider the state (37). Just9eplacing ρ F (0) with ρ ( α ) T h in eq.(30), then, after some algebra we find the visi-bility as V ( α ) T h (∆ t ) = V T h (∆ t ) exp n − | α | sin ( ω ∆ t/
2) sinh( β ¯ hω /
2) cosh( β ¯ hω / ( β ¯ hω /
2) + sin ( ω ∆ t/ o . (40)Again, assuming a interaction time as ∆ t = π/ω , then the equation (40) can besimplified as V ( α ) T h ( π/ω ) = V T h ( π/ω ) e − | α | tanh( β ¯ hω / . (41)Now we will investigate the limiting cases, as in the previous subsection. Themain difference is that the displacement magnitude plays an important role. Wehave now four situations, they are1. For | α | ∼ V ( α ) T h ( π/ω ) V T h ( π/ω ) = − | α | β ¯ hω , if β ¯ hω ≪ ,e − | α | , if β ¯ hω ≫ .
2. For | α | ≫ V ( α ) T h ( π/ω ) V T h ( π/ω ) = e −| α | β ¯ hω , if β ¯ hω ≪ ,e − | α | , se β ¯ hω ≫ . This can be summarized by V ( α ) T h ( π/ω ) V T h ( π/ω ) = e −| α | β ¯ hω , if β ¯ hω / ≪ e − | α | , if β ¯ hω / ≫ The Helmholtz free energy can be defined as F = − β ln( Z ) (43)where Z is the state partition function. The variation in Helmholtz free energy∆ F from the initial state given (31 to the final displaced state given by (37),can be obtained by the respective partition functions, given by Z = Tr (cid:0) e − β ¯ H (cid:1) Z ( α ) = Z e β ¯ hω | α | , F = − ¯ hω | α | . (44)Now, the term e − β ∆ F can be represented as e − β ∆ F = e | α | β ¯ hω . (45)From the equation (32) we can writesin ( ω ∆ t/
2) = 1 − (cid:2) V T h (∆ t ) (cid:3) (cid:2) V T h (∆ t ) (cid:3) sinh ( β ¯ hω / . (46)Substituting (46) into (40) we get the relation " V T h V ( α ) T h / (1 −V Th ) = e | α | sinh( β ¯ hω ) (47)between the visibilities independent of the interaction time ∆ t . In case that β ¯ hω ≪ e − β ∆ F = " V T h V ( α ) T h / (1 −V Th ) . (48)This result shows that the variation in Helmholtz free energy, for this exper-imental set-up, can be obtained in terms of visibility, it means that the term e − β ∆ F from Jarzynski [1] equality can be experimentally obtained from visibilitymeasurements. There are many possibilities for choosing the quantum work operator [48,52, 20], and the best definition still an open question [51, 49, 48, 52, 20]. If weconsider the definition ∆ E then the JE is validly and we can use the visibilityto estimate the work done on the cavity, in this case we have e − β ∆ F = " V T h V ( α ) T h / (1 −V Th ) = h e − β ∆ W i . (49)The Cavity quality factor plays an important role on this case, since the calcu-lations where carried out without considering dissipation. That means that theprotocol time should be small comparing with the life time [46] of the field inthe cavity.Another quantum work definition is constructed in [20], and it gives andifferent result from ∆ E , they demonstrate that quantum correlation function G ( u ) = Tr h U † ( t ) e iuH ( t ) U ( t ) e − iuH (0) ρ (0) i , (50)11ith U ( t ) solution of the equation i ¯ h∂U ( t ) /∂t = H ( t ) U ( t ) and U (0) = , when u = iβ contains all available statistical information about the work such asthe averaged exponentiated work h exp( − βW ) i . When we calculate the eq.(50)for our experimental proposal, again we find precisely that e − β ∆ F = h e − βW i ,again the visibility is a useful tool to determine the free energy variation andconsequently the averaged exponential work.
4. Conclusions
We have shown that Jarzynski’s theorem can be tested experimentally inthe context of superconducting cavities. In particular, we find a direct relation-ship between the visibility and the free energy variation. Considering the workoperator ∆ E we have shown that it is possible to use de Jarzynzki equality todetermine the work done or extracted at a superconducting cavity by measuringthe fringes visibility, that is commonly used in cavity experiments. In terms ofJE, we can also obtain the state of field in the cavity with the visibility measure,since we know the initial state. Taking into account that visibility measurementis simpler than usual state measurements, this approach is very efficient. Acknowledgements
RCF and ACO gratefully acknowledge the support of Brazilian agency Funda¸c˜aode Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG) through grantNo. APQ-01366-16.
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