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Strongly Coupled Quantum Heat Machines David Gelbwaser-Klimovsky, and Alán Aspuru-Guzik J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b01404 • Publication Date (Web): 18 Aug 2015 Downloaded from http://pubs.acs.org on August 23, 2015

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The Journal of Physical Chemistry Letters

Strongly Coupled Quantum Heat Machines David Gelbwaser-Klimovsky∗ and Alán Aspuru-Guzik Department of Chemistry and Chemical Biology Cambridge, MA 02138 E-mail: [email protected]

∗

To whom correspondence should be addressed

1 ACS Paragon Plus Environment


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Abstract Energy conversion of heat into work at the quantum level is modeled by quantum heat machines (QHMs) generally assumed to operate at weak coupling to the baths. This supposition is grounded in the separability principle between systems and allows the derivation of the evolution equation. In the weak coupling regime, the machine’s output is limited by the coupling strength, restricting their application. Seeking to overcome this limitation, we here analyze QHMs in the virtually unexplored strong coupling regime, where separability, as well as other standard thermodynamic assumptions, may no longer hold. We show that strongly coupled QHMs may be as efficient as their weakly coupled counterparts. In addition, we find a novel turnover behavior where their output saturates and disappears in the limit of ultra-strong coupling.

Table of Contents (TOC) Graphic

Driving Hot bath

Cold bath

Strong coupling

Strong coupling Working

Keyworkds: Energy conversion, Quantum heat machines, Strong coupling, Open quantum systems, Polaron transformation 2 ACS Paragon Plus Environment

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One of the basic tenets of standard thermodynamics is the principle of separability, which allows to clearly define and distinguish systems that interact with each other, assuring extensivity of thermodynamic variables. When the surface to volume ratio is small, surface effects are negligible, and thermodynamic variables only depend on the volume and not on the shape. This argument implicitly assumes a weak coupling, restricting the interaction space to a small interface between the systems. 5,12,21 The assumption of weak coupling was essential for the development of open quantum system theory, 43 in particular for the development of the Lindblad-Gorini- Kossakowaski-Sudarshan master equation (LGKS), 8,22,43 that describes the evolution of a system interacting with a thermal bath. The conversion of heat into work is modeled by quantum heat machines (QHMs) 3,7,10,15,16,33,34,44,57 which use LGKS to describe the evolution of the “working fluid” under the influence of the hot and cold baths. Progress in this field has been recently reviewed. 19,28 QHMs may operate either as engines, by converting heat into work (power extraction), or as refrigerators, by investing work power and cooling the cold bath. In both cases, quantum resources have been proposed 17,18,45,47 in order to boost their output and efficiency. Nevertheless, these models assume a weak coupling to the baths, resulting in limited QHMs outputs and consequently restricting their applications. The potential technological implications of high-output QHMs, such as faster and more powerful laser cooling, 20,55 call for a prompt way to overcome the limitation set by the weak coupling assumption. Heat exchange in the strong coupling limit has been explored in the context of quantum transport. 4,11,38,48–50 In this study, we explore at the strong coupling, how efficiently the input heat is transformed into work, termed energy conversion efficiency, which although a related topic, has been virtually unexplored. One of the few exceptions 14 considers the case of Hamiltonian quench, which involves the switching “on” and “off” of the system-bath interaction Hamiltonian, introducing an energy and efficiency cost that reduces the machine efficiency below the Carnot bound. In this letter we take a different approach by putting forward a strongly coupled continuous QHM model, that does not require the coupling to and uncoupling from the baths, which may not



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be possible at nanoscale, where the system is totally embedded in thermal baths. We investigate its output and efficiency in order to determine its performance limits and which thermodynamic principles, e.g., Carnot bound, still hold at the strong coupling regime, even though the extensivity of thermodynamic variables is not longer valid. Addressing these issues becomes more relevant in the light of the large progress achieved in the field of strongly coupled superconductors, 2,24,25,41 which makes the realizations of strongly coupled QHMs potentially tractable in the near future. We employ a model for a continuous QHM similar to the one studied previously in the weak coupling limit. 53 This system can operate either as an engine or a refrigerator depending on the spectrum of the reservoirs and the engine’s driving frequency. This model is comprised by a driven two-level quantum system, that represents the working fluid, permanently coupled to the heat baths (hot and cold). The evolution of this model is governed by the Hamiltonian ∑ ∗ ak ) + + Ω2 (σ+ e−iωl t + σ− eiωl t ) + σz ⊗ k ξC (gC,k a†k + gC,k ∑ ∑ ∑ ∗ bk ) + k ωC,k a†k ak + k ωH,k b†k bk , σx ⊗ k ξH (gH,k b†k + gH,k

H=

ω0 σ 2 z

(1)

where ξC(H) is the strength parameter of the cold (hot) bath, gi,k is a dimensionless parameter that defines the relative coupling strength of the TLS to the mode k of the i-bath, σj are the standard Pauli matrices, and a†k , ak (b†k ,bk ) are the creation and annihilation operator of the cold (hot) bath mode k. While the cold bath is a dephasing bath, the hot bath generates decay. This is in contrast to other models 48,49 where only two dephasing baths are considered, preventing work extraction. The election of the hot and cold bath is somehow arbitrary and a similar analysis could be performed if they are interchanged. The system+driving Hamiltonian is not diagonal in the σz basis, thereby allowing energy exchange with the cold bath. The weakly coupled counterpart of this asymmetric QHM has been experimentally implemented. 20,55 The coupling is consider weak if γτcor ≪ 1, where γ is the decay rate and is equivalent to the resonant coupling spectrum (γ = G(ω0 )), and τcor is the bath correlation time. 51 Is it possible to extract work or to cool down in the strong coupling regime? To elucidate this question, we consider that both couplings are strong. While the reduced dynamics is analytically solvable in the weak regime, in this case the perturbation expansion on the coupling strength contains infinite 4 ACS Paragon Plus Environment

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no-neglectable terms. 30 Nevertheless for H, this obstacle may be overcome by solving the problem in a more appropriate basis, where the system is effectively weakly coupled to the two baths. This is achieved ∑ by using the polaron transformation, 6,9,26,32,37,40,52 eS , where S = σZ ⊗ k (αk a†k − αk∗ ak ) and g

αk = ξC ωC,k . Physically, this transformation shifts the equilibrium position of the cold bath osC,k cillators by a factor proportional to the TLS free energy. The transformed bath still exchange energy with the TLS through the driving. It has been previously applied for studying heat 50 and e = eS He−S , is electronic 29 transport in non-driven setups. The transformed Hamiltonian, H

e= H

+ σ− eiωl t ) + Ω2 (e−iωl t σ+ ⊗ (A+ − A) + eiωl t σ− ⊗ (A− − A)) + ∑ ∑ ∑ ∗ (σ+ ⊗ A+ + σ− ⊗ A− ) ⊗ k ξH (gH,k b†k + gH,k bk ) + k ωC,k a†k ak + k ωH,k b†k bk , ω0 σ 2 z

+

Ωr (σ+ e−iωl t 2

(2) †

∗

where A± = Πk D(±2αk ), D(αk ) = eαk ak −αk ak is the displacement operator, A = ⟨A± ⟩ = e

2 −2ξC

∑ β ω

gC,k 2

coth( C2 k ) k ω k

and Ωr = ΩA. The terms on the Hamiltonian proportional to the identity

have been neglected. In the transformed Hamiltonian, the coupling operators are different. A+ − A and A− − A, instead of a†k , ak and extra terms are added to the hot bath coupling (see Eq. (2) and Suppl.). As we show below, the new couplings may be effectively weak even for high values of the original coupling strengths, ξC(H) . Therefore, in the transformed basis, the assumptions derived from the weak coupling are correct (e.g. the transformed baths are at thermal equilibrium) and the master equation may be derived using standard techniques 1,43 also for values of ξC(H) that break the weak coupling assumption in the original basis. 6,9,26,32,37,40 The transformed cold bath, now interacts with the TLS through two different operators, Fe1 (t) = ( ) ∑ † Ω ∗ e (A− (t) − A) and F2 = A− (t) ⊗ k ξH gH,k bk (t) + gH,k bk (t) . The correlation function of the 2 first is

f1 (0)⟩ ⟨Fe1 (t)† F

( =

ΩA 2

)2

2 4ξC

(e

∑ k

Λk (t) ω2 k


− 1),

(3)


where

ξC2

∑

Λk (t) =

⟨F1† (t)F1 (0)⟩

k

=

∑ k

ξC2

) ( βC ωk ∥gC,k ∥ cos(ωk t) coth( ) − i sin(ωk t) 2 2

is the time correlation of the original coupling operator, F1 (t) =

∑

(4)

( ) † ∗ ξ g a (t) + g a (t) C C,k k C,k k k

and βC is the equilibrium temperature of the transformed cold bath. The coupling spectra that ei (ω) = govern the evolution are derived from the correlations of the transformed operators, G ´ ∞ itω e ⟨Fei (t)† Fei (0)⟩dt, i ∈ 1, 2. −∞ a) G1(ω0) ~ G1(ω0)

Power [a.u.]

b)

Coupling spectrum [tc-1]

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Coupling strength[ξc/ω0]

weak coupling validity limit

Coupling strength[ ξc/ω0]

Figure 1: (Color online) Effects of the coupling strength. a) Coupling spectrum in the original e1 (ω0 ) (continuous line) as a function of basis G1 (ω0 ) (dotted line) and in the transformed basis G the coupling strength ξC ∼ ξH . The weak coupling assumption holds only for coupling spectrum below the dashed line. b) Power for a QHM as a function of the coupling strength, ξC ∼ ξH . A turnover is observed and the power decays for ultra-strong coupling. At this limit, the lack of system-bath separability prevents work extraction.

(

In Fig. 1, the dependence on the coupling strength, ξC , of the coupling spectrum in the original ) ( ) ´∞ e1 (ω), continuous line G1 (ω) = −∞ eitω ⟨F1 (t)† F1 (0)⟩dt, dotted line and transformed basis G

are compared. While both coupling spectra are proportional to the square of the coupling for small coupling strengths, in other regimes their behavior diverge. The validity of the “weak” coupling e1 (ω) has been broadly shown. 6,9,26,32,37,40,56 In a similar manner, the assumption for the spectrum G e2 (ω) to the weak coupling regime as long as ξH ∼ ξC (See Suppl.) operator A± (t), will constraint G 6 ACS Paragon Plus Environment

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. e1 (ω) keeps the standard KMS condition, G e1 (−ω) = e−βC ω G e1 (ω), 31,36 which implies that G the rate between absorption and emission processes is equal to a Boltzmann factor, with the bath temperature, which is independent of the frequency. e1 (ω > ωcutof f ) ̸= 0 as long as ω is a linear combination of bath modes harmonics. This G propriety lets the use of highly detuned baths in strongly coupled QHMs, unlike for weakly coupled QHMs that require resonant baths (or at least resonant with linear combinations of ω0 and ωl , the TLS and driving frequency, respectively 16 ). The polaron transformation allows the derivation of the QHM evolution for a wide range of values of the coupling strengths. Nevertheless, this simplification entails other complications, as the loss of separability. In the transformed basis, the second spectrum is far from standard. It (†)

involves exchange of excitation with both baths (the operators bk and A± for the hot and cold bath respectively). Through the second coupling, the energy exchange involves both baths, preventing to treat them independently. The lack of separability breaks the standard KMS condition, casting doubt on the validity of other thermodynamic principles, as the Carnot bound. The rate between absorption and emission processes is more complicate than the standard, requiring the introduction of frequency dependent “local” temperatures β(ω) (see Eq. (5) and Suppl.). They are analogous e2 (ω) the KMS condition is to the non-equilibrium position-dependent local temperatures. 27 For G generalized (see Suppl.):

e2 (ω), e2 (−ω) = e−β(ω)ω G G

β(ω) = βC λ(ω) + βH (1 − λ (ω)) , (5)

e2 (ω) and may take where λ(ω) measures the relative contribution of the transformed cold bath to G any positive or negative value (see Suppl.). Therefore β(ω) is not restricted to the range [βH , βC ]. It depends on both baths coupling strength distribution and modes, making β(ω) frequency dependent, blurring its physical interpretation. As we show later, it allows to establish clear ther-



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modynamic bounds to the efficiency of the QHMs and to relate them to the Carnot bound. The precise value taken by β(ω), depends on how the exchange energy ω is divided between the hot and the cold baths. For weak coupling, A± → 1, the bath separability is recovered, λ(ω) → 0 and β(ω) → βH . e1 (ω), G e2 (ω) not only includes harmonics of the cold bath, but comIn a similar manner as G binations of them with modes of the hot bath. Therefore, also the hot bath may be highly detuned from the TLS frequency (and from any linear combination with the driving frequency). In the transformed basis, we use the standard weak coupling master equation based on the general Floquet theory of open systems. 1 Here, we just stress the main steps, but the detailed derivation may be found in the supplementary information and in Ref 35. The reduced evolution of the TLS density matrix, ρ, is given by a linear combination of Lindblad generators obtained from the Fourier components in the interaction picture of the working fluid coupling operators ∑ ∑ ∑ ∑ e∓iωl t σ± (t) = q∈Z ω S±1,q (ω)e−i(ω+qωl )t and σ± (t) = q∈Z ω S±2,q (ω)e−i(ω+qωl )t . In the interaction picture,

] [ ]) ∑ ∑ dρ Gi (ω + qωl ) ∑ ([ † † Sj,q (ω)ρ, Sj,q (ω) + Sj,q (ω), ρSj,q (ω) . = Lρ, L = L1qω + L2qω , Liqω = × dt 2 q,ω q,ω j=±i (6) The TLS density matrix evolves until it reaches a steady state (limit cycle), L¯ ρ = 0. As it has been shown, 19,37 memory effects may be neglected for long time dynamics. At this point any transient effect averages out and one may calculate the steady state work power and heat flows between the system and the baths produced by the i-coupling operator,

Ji =

∑

[ ] sgn(ω) (ω + qωl ) T r Liqω ρ¯ ,

P = −J1 − J2 ,

(7)

qω

where sgn(ω) = 1 for ω > 0 and sgn(ω) = −1 for ω < 0. These are net heat flows between


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the system and the baths. J1 is the partial heat flow from the cold bath. It only involves the heat flow produced by the first coupling. On the other hand, the second coupling concerns both baths, therefore J2 is a mix of heat flows from the hot, as well as the cold bath. Because the baths only interact between them through the system, any energy exchange between them, flows through the system and is already accounted by the heat flows defined in Eq. (7). Therefore, the extracted power may be directly calculated using energy conservation (see Eq. (7)). In particular we are interested in the ultra-strong coupling regime (ξC ∼ ξH ≫ ω0 ) to find out if QHMs may have an ultra-high output. Nevertheless at this limit, and assuming a weak driving with positive detuning (δ = ω0 − ωl ≫ Ωr > 0), the work power dependence on the coupling strength goes as (see Suppl.)

P ∝ ωl

2 −4ξC

2

A −βC δ e (e − e−β(ω0 )ω0 ) ∝ 2 ξC

∑ β ω

gC,k 2

coth( C2 k ) k ω k

ξC2

.

(8)

The conditions for work extraction (P < 0), ωl β(ω0 ) 0. ξC2

(10)

For ultra-strong coupled baths, the work power will decay with the coupling strength as shown in Eq. (8). The exact counterpart of Eq. (8) for any value of ξC ∼ ξH is plotted on figure 1-Inset. Opposite to what may be expected from previous results in the weak coupling regime, work power does not increase indefinitely with the coupling strength. Not only it saturates, but at some point, decays and vanishes. At the ultra-strong limit, the system and the baths are no longer independent, preventing work extraction which requires some degree of separability.



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The determination of the energy conversion efficiency, as well as the cooling power (in the refrigerator operation mode), is a more subtle issue. A naive guess would be to consider J2 , which is positive, as the incoming heat flow from the hot bath and to define the efficiency of energy conversion as

η=

−P ωl β(ω0 ) = ≤1− , J2 ω0 βC

(11)

which can take any value, even above Carnot limit. Nevertheless, the lack of separability complicates the determination of how much energy is exchanged with each bath through the coupling e2 (ω). Only a fraction, 1 − λ(ω), of J2 is originated in the hot bath. Therefore the spectrum G correct efficiency expression is

η=

−P ωl βH = ≤1− = ηCar . J2 (1 − λ (ω0 )) ω0 (1 − λ(ω0 )) βC

(12)

From Eq. (12) we conclude that the Carnot bound may be reached, but not surpassed, by the appropriate choice of driving frequency. Therefore, strongly coupled machines are as efficient as their weakly coupled counterpart. 23 In the weak coupling, bath separability is recovered, J2 → JH and λ → 0, retrieving the standard efficiency expression. In a similar way, one can calculate the cooling power for the refrigeration operation. This sets the opposite condition on the frequencies:

ωl ω0

≥1−

β(ω0 ) , βC

making J1 > 0, which can erroneously

be confused with the cooling power. The lack of separability between both baths (the dependence of Fe2 on both baths operators) mixes the heat flows between both baths and part of J2 is heat flowing to the cold bath. The correct expression for the cooling power is JC = J1 − λ(ω0 )J2 and is limited by the Carnot bound for refrigerators. The cooling power has a similar dependence on the coupling strength as the work power and also decays and vanishes for ultra-strong coupling. An ideal platform to test our results are superconducting quantum circuits, where the almost unexplored strong coupling regime has been recently achieved, 13,39 showing astounding ξi /ω0 ratios of around 0.12. Moreover, recent theoretical studies have shown that a σx −coupling between


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quantum microwaves and artificial Josephson-based atoms can be pushed up to ξi /ω0 ∼ 2, 42 which is well beyond the critical point (ξi /ω0 = 1) at which the power efficiency is maximum. Our proposal consists of a periodically driven superconducting flux qubit with tunable gap, 46 where the main loop is coupled to the hot bath (σx coupling), and the α−loop is coupled to the cold bath (σz coupling). In order to bring the σz coupling to the strong regime, we galvanically couple the α−loop to the open transmission line that plays the role of the cold bath. The possibility of converting heat into work (work extraction) and cooling in the strong coupling regime was shown. Even though some thermodynamic principles, as the standard KMS condition, do not longer hold and some thermodynamic variables are not longer extensive, we found that the energy conversion is limited by the standard Carnot bound. The introduction of frequency-local temperatures, that account for the different baths contributions to the heat flows, are useful to determine how the heat flows are divided between baths and to correctly calculate the energy conversion efficiency. As we have shown, continuous strongly coupled QHMs, as their weakly coupled counterparts, avoid the efficiency reduction due to the coupling turning on and off and keep the Carnot bound which can be reached under the appropriate driving frequency. The appearance of the “non-equilibrium” temperatures is related to the loss of separability. Even though both baths, in the transformed basis, are in equilibrium, the heat flows mixes the contribution of both of them, causing an effective deviation from equilibrium. There are similarities between weakly and strongly coupled QHMs, but the differences should not be overlooked. The unique signature of the strong coupling may be found in two effects: (i) The effective deviation from equilibrium, reflected in the KMS condition local temperatures, caused by the perturbation of one bath to the other, through the system that interconnects them. (ii) The possibility of operating strongly coupled QHMs with highly detuned baths, in contrast to their weakly counterparts, that require resonant baths. 16 This represents a new paradigm with potential applications in any area related to open quantum systems, besides energy conversion and QHMs. An important feature of strongly coupled QHMs is that, differently to their weakly coupled



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counterpart, where the outputs are proportional to the square of the coupling strength, work and cooling power saturate at some point and for ultra-strongly coupled machines they fall down as the coupling strength increases. This is a consequence of the lost of separability as the coupling strength increases, and shows that QHMs require some degree of separability to operate. This effective decoupling between the system and the bath has been observed in other models, 54 and may be an universal feature related to the dynamics of open quantum systems at this regime. In order to optimize QHMs output the “right” coupling strength is needed, resembling the quantum Goldilocks effect 35 found in photosynthetic systems. The latter should be further investigated to determine if evolution fine-tuned the coupling strength to the baths in order to maximize their chemical power output. Alternatively, the turnover behavior may be corroborated experimentally using superconducting qubits. 2,24,25,41

Acknowledgement We acknowledge Borja Peropadre and Joonssuk Huh for useful discussions. We acknowledge the support from the Center for Excitonics, an Energy Frontier Research Center funded by the U.S. Department of Energy under award DE-SC0001088 (Polarons and energy conversion process). D. G-K. also acknowledges the support of the CONACYT (non-equilibrium thermodynamics) and the COST Action MP1209 (Quantum thermodynamics). The authors declare no competing financial interest. Supporting information The supporting information add details on: A) the system bath couplings in the original as well as the transformed basis; B)the generalized KMS condition; C) the power at the ultra-strong regime and the effective weak coupling; D) master equation, heat flows and power. This material is available free of charge via the Internet at http://pubs.acs.org.

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Strongly Coupled Quantum Heat Machines - ACS Publications

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