Strongly Coupled Quantum Heat Machines - The Journal of Physical

Letter pubs.acs.org/JPCL

Strongly Coupled Quantum Heat Machines David Gelbwaser-Klimovsky* and Alán Aspuru-Guzik Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, United States S Supporting Information *

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 analyze QHMs in the virtually unexplored strong coupling regime here, 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 ultrastrong coupling.

O

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 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, for example, 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

ne 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 proposed17,18,45,47 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 exceptions14 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. © 2015 American Chemical Society

/=

ω0 Ω σz + (σ+e−iωlt + σ −eiωlt ) 2 2 + σz ⊗ ∑ ξC(g ak† + g * ak ) C ,k

C ,k

k

+ σx ⊗

∑ ξH(gH ,kbk† + gH*,kbk) + ∑ ωC ,kak†ak k

+

∑

ωH , kbk†bk

k

k

(1)

Received: July 2, 2015 Accepted: August 18, 2015 Published: August 18, 2015 3477

DOI: 10.1021/acs.jpclett.5b01404 J. Phys. Chem. Lett. 2015, 6, 3477−3482

Letter

The Journal of Physical Chemistry Letters

F̃2 = A−(t) ⊗∑kξH(gH,kb†k (t) + g*H,kbk(t)). The correlation function of the first is

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. Although the cold bath is a dephasing bath, the hot bath generates decay. This is in contrast to other models48,49 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. Although the reduced dynamics is analytically solvable in the weak regime, in this case the perturbation expansion on the coupling strength contains infinite no-neglectable terms.30 Nevertheless for / , 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 g S = σZ ⊗ ∑k(αka†k −α*k ak) and αk = ξC ωC ,k . Physically, this

⎛ ΩA ⎞2 4ξC2 ∑ Λk(t )/ ωk2 ∼ k ⎟ (e ⟨F1̃(t ) †F1(0)⟩ = ⎜ − 1) ⎝ 2 ⎠

(3)

where ξC2 ∑ Λk(t ) = ⟨F1(t )† F1(0)⟩ k

=

∑ ξC2 k

⎛ ⎞ ⎛ β ωk ⎞ gC , k 2 ⎜cos(ωkt )coth⎜ C ⎟ − i sin(ωkt )⎟ ⎝ 2 ⎠ ⎝ ⎠ (4)

is the time correlation of the original coupling operator, F1(t) = ∑kξC(gC,ka†k (t) + g*C,kak(t)) and βC is the equilibrium temperature of the transformed cold bath. The coupling spectra that govern the evolution are derived from the correlations of the itω ̃ † ̃ transformed operators, G̃ i(ω) = ∫ ∞ −∞e ⟨Fi(t) Fi(0)⟩dt, i ∈ 1, 2.

C ,k

transformation shifts the equilibrium position of the cold bath oscillators by a factor proportional to the TLS free energy. It has been previously applied for studying heat50 and electronic29 transport in nondriven setups. The transformed bath still exchange energy with the TLS through the driving. The transformed Hamiltonian, H̃ = e S /e−S , is H̃ =

ω0 Ω σz + r (σ+e−iωlt + σ −eiωlt ) 2 2 Ω −iωlt σ+ ⊗ (A+ − A) + eiωlt σ − ⊗ (A − − A)) + (e 2 +(σ+ ⊗ A+ + σ − ⊗ A −) ⊗ ∑ ξH(g bk† + g * bk ) H ,k

Figure 1. (Color online) Effects of the coupling strength. (a) Coupling spectrum in the original basis G1(ω0) (dotted line) and in the transformed basis G̃ 1(ω0) (continuous line) as a function of 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 ultrastrong coupling. At this limit, the lack of system−bath separability prevents work extraction.

H ,k

k

+

∑ ωC ,kak†ak + ∑ ωH ,kbk†bk k

(2)

k

In Figure 1, the dependence on the coupling strength, ξC, of the coupling spectrum in the original (G1(ω) = itω † ∫∞ −∞e ⟨F1(t) F1(0)⟩dt, dotted line) and transformed basis ̃ (G1(ω), continuous line) are compared. Although both coupling spectra are proportional to the square of the coupling for small coupling strengths, in other regimes, their behavior diverges. The validity of the “weak” coupling assumption for the spectrum G̃ 1(ω) has been broadly shown.6,9,26,32,37,40,56 In a similar manner, the operator A±(t), will constraint G̃ 2(ω) to the weak coupling regime as long as ξH ∼ ξC (See Supporting Information). G̃ 1(ω) keeps the standard KMS condition, G̃ 1(−ω) = −βCω ̃ e G1(ω),31,36 which implies that the rate between absorption and emission processes is equal to a Boltzmann factor, with the bath temperature, which is independent of the frequency. G̃ 1(ω > ωcutoff) ≠ 0 as long as ω is a linear combination of bath modes harmonics. This 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, respectively16).

†

where A± = ΠkD(±2αk), D(αk) = eαkak−αk*ak is the displacement 2

2

βC ωk

operator, A = ⟨A±⟩ = e−2ξC ∑k ∥ gC ,k / ωk ∥ coth( 2 ) and Ωr = Ω A. 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 Supporting Information). 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 techniques1,43 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 Ω through two different operators, F1̃(t ) = 2 (A −(t ) − A) and 3478


Letter


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

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 b(†) k 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 Supporting Information). They are analogous to the nonequilibrium position-dependent local temperatures.27 For G̃ 2(ω) the KMS condition is generalized (see Supporting Information)

Ji =

(5)

2

ωl β(ω0) 0 ξC2

(10)

For ultrastrong 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. It not only saturates but also, at some point, decays and vanishes. At the ultrastrong limit, the system and the baths are no longer independent, preventing work extraction, which requires some degree of separability. 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

q,ω

∑ ([Sj ,q(ω)ρ , Sj†,q(ω)] j =±i

+ [Sj , q(ω), ρS†j , q(ω)])

βC ωk 2 )

The conditions for work extraction (P < 0)

∑ 31qω + ∑ 3q2ω, G (ω + qωl) × 2

coth(

(8)

i

3iqω =

gC , k 2

A2 e−4ξC ∑k ωk P ∝ ωl 2 (e−βCδ − e−β(ω0)ω0) ∝ ξC ξC2

dρ = 3ρ , dt q,ω

(7)

where sgn(ω) = 1 for ω > 0 and sgn(ω) = −1 for ω < 0. These are net heat flows between 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 ultrastrong coupling regime (ξC ∼ ξH ≫ ω0) to find out if QHMs may have an ultrahigh 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 Supporting Information)

where λ(ω) measures the relative contribution of the transformed cold bath to G̃ 2(ω) and may take any positive or negative value (see Supporting Information). 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 establishing clear thermodynamic bounds to the efficiency of the QHMs and relating 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. In a similar manner as G̃ 1(ω), G̃ 2(ω) not only includes harmonics of the cold bath, but combinations 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 Supporting 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∓iωltσ±(t) = ∑q∈Z∑ωS±1,q(ω) e−i(ω+qωl) t and σ±(t) = ∑q∈Z∑ωS±2,q(ω)e−i(ω+qωl)t. In the interaction picture

3=

P = −J1 − J2

qω

G̃2( −ω) = e−β(ω)ωG̃2(ω), β(ω) = βC λ(ω) + βH (1 − λ(ω))

∑ sgn(ω)(ω + qωl)Tr[3iqωρ ̅ ],

η= (6)

ω β(ω0) −P = l ≤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 spectrum G̃ 2(ω). Only a fraction, 1 − λ(ω), of J2 is

The TLS density matrix evolves until it reaches a steady state (limit cycle), 3ρ ̅ = 0 . As it has been shown,19,37 memory effects may be neglected for long time dynamics. 3479


Letter


interconnects them. (ii) The possibility of operating strongly coupled QHMs with highly detuned baths, in contrast to their weakly counterparts, that require resonant baths.16This 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 counterpart, where the outputs are proportional to the square of the coupling strength, work, and cooling power saturate at some point and for ultrastrongly 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 effect35 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

originated in the hot bath. Therefore, the correct efficiency expression is η=

β ωl −P = ≤ 1 − H = 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 F̃2 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 ultrastrong 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 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, no longer hold and some thermodynamic variables are no 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 mix 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

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b01404. 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 ultrastrong regime and the effective weak coupling; (D) master equation, heat flows, and power. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS 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 (nonequilibrium thermodynamics) and the COST Action MP1209 (Quantum thermodynamics). The authors declare no competing financial interest.

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REFERENCES

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