Maxwell's lesser demon: a quantum engine driven by pointer measurements
MMaxwell’s lesser demon: a quantum engine driven by pointer measurements
Stella Seah, Stefan Nimmrichter, and Valerio Scarani
1, 3 Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore (Dated: March 9, 2020)We discuss a self-contained spin-boson model for a measurement-driven engine, in which a de-mon generates work from thermal excitations of a quantum spin via measurement and feedbackcontrol. Instead of granting it full direct access to the spin state and to Landauer’s erasure strokesfor optimal performance, we restrict this demon’s action to pointer measurements, i.e. random orcontinuous interrogations of a damped mechanical oscillator that assumes macroscopically distinctpositions depending on the spin state. The engine can reach simultaneously the power and efficiencybenchmarks and operate in temperature regimes where quantum Otto engines would fail.
Conventionally, thermal machines operate through theinteraction of a working medium with hot and cold reser-voirs. In the context of quantum thermodynamics, inter-est has been raised in finding non-thermal resources suchas coherence [1–5], squeezed baths [6–8] or measurementchannels [9–12] that could induce advantages to standardthermal machines.Specifically, the role of measurement in relation tothermodynamics and information flow has been studiedrigorously. For example, models of thermal machines fa-cilitated by Maxwell’s demon – an external agent thatacquires information of the system and performs appro-priate feedback – have been proposed in order to provideaccurate thermodynamic description of information flow[13–19]. More recently, a measurement channel has beendeemed a source of “quantum heat” [9] due to the in-creased entropy following a measurement, which couldbe exploited for both cooling [11] and work extraction[10, 12, 20, 21]. However, proper treatment of actual era-sure cost of pointers [22–24] as well as the interpretationof incoherent measurement schemes as a form of heat and work exchange [25–27] still remain a contentious topic forsuch measurement-based thermal machines.In this paper, we reveal the mechanisms underlyingMaxwell’s demon by considering a self-contained enginebuilt from the standard ingredients (hot and cold reser-voirs and a working medium) as well as an embed-ded macroscopic pointer. Specifically, we revisit defini-tions of work, heat, and information flow in a practicalmeasurement-feedback scheme. In contrast to regularMaxwell-demon type engines where the demon has ac-cess to the state of the working medium and stores it inits memory, we restrict our demon’s access to the pointeronly, modelled by a damped mechanical degree of free-dom. Work can then be extracted from the medium byreading off the pointer position and applying appropri-ate feedback. We show that such a setup generates anew type of engine with features different from standardquantum engines. In particular, we see that it is possibleto attain simultaneous high powers and efficiencies basedon the model’s benchmarks. The regime of operation isalso wider than that of a quantum Otto engine.
FIG. 1. (Color online) Sketch of the demon system consist-ing of a qubit (working medium) and a harmonic oscillator(pointer). The qubit can be thermally excited by a hot bathat the rate κ h and temperature T h , and it displaces the equi-librium position of the pointer to ± x depending on its state.A cold bath of temperature T c thermalizes the pointer aroundits equilibrium point at the rate κ c . Work can be extractedcoherently or incoherently from the excited spin by the de-mon’s interrogation of the pointer position. Spin-boson model.—
We consider a qubit with baretransition frequency Ω representing the working mediumfor heat-to-work conversion. A harmonic oscillatorpointer of frequency ω couples to the qubit and is dis-placed to the left or right depending on the internal stateof the qubit, see Fig. 1. The model Hamiltonian reads asˆ H = (cid:126) Ω2 ˆ σ z + (cid:126) ω (cid:18) ˆ a † ˆ a + 12 (cid:19) + (cid:126) ωx ˆ σ z ˆ a + ˆ a † √ (cid:126) Ω2 ˆ σ z + (cid:126) ω ˆ b † ˆ b + const , (1)with ˆ σ z = | e (cid:105)(cid:104) e | − | g (cid:105)(cid:104) g | , ˆ a the oscillator’s mode operatorand ˆ b = ˆ a + ˆ σ z x / √ | g, n g (cid:105) := | g (cid:105) ⊗ ˆ D | n (cid:105) , | e, n e (cid:105) := | e (cid:105) ⊗ ˆ D † | n (cid:105) , (2)where the energy eigenvalues are E e,gn = ± (cid:126) Ω / (cid:126) ωn a r X i v : . [ qu a n t - ph ] M a r modulo a constant, and ˆ D = exp (cid:0) x ˆ a † / √ − x ˆ a/ √ (cid:1) isthe displacement operator.A hot thermal reservoir with mean occupation number¯ n h = 1 / [exp( (cid:126) Ω /k B T h ) −
1] injects heat and randomlyexcites the qubit, as mediated by the dissipators L h ρ = (cid:88) k κ h (Ω + kω ) (cid:40) [¯ n h (Ω + kω ) + 1] (3) ×D (cid:34)(cid:88) n d ∗ n, − k | g, ( n − k ) g (cid:105)(cid:104) e, n e | (cid:35) ρ + ¯ n h (Ω + kω ) D (cid:34)(cid:88) n d n,k | e, ( n + k ) e (cid:105)(cid:104) g, n g | (cid:35) ρ (cid:41) , with D [ ˆ A ] ρ = ˆ Aρ ˆ A † − { ˆ A † ˆ A, ρ } / d n,k = (cid:104) n | ˆ D | n + k (cid:105) . We derive (3) from a secular approxi-mation of the weak coupling master equation (see Ap-pendix).A cold reservoir with ¯ n c = 1 / [exp( (cid:126) Ω /k B T c ) −
1] con-tinuously couples to the pointer to erase/reset the infor-mation encoded in it. We employ thermal dissipatorsacting on the displaced mode operator ˆ b [28], L c ρ = κ c (¯ n c + 1) D [ˆ b ] ρ + κ c ¯ n c D [ˆ b † ] ρ. (4)Before introducing a demon for measurement-feedback,let us discuss the operation regime for this engine. Ide-ally, we want to work in the limit Ω (cid:29) ω (cid:29) κ c (cid:29) κ h ,which describes a separation of energy scales betweenthe working medium and the pointer in the regime ofresolved sidebands and weak thermal couplings. Thepointer does not contribute appreciably to the energybalance (Ω (cid:29) ω ), but it reacts quickly to any change inthe qubit state ( κ c (cid:29) κ h ). Moreover, we require suffi-ciently large x compared to the thermal width x th =coth / (cid:126) ω/ k B T c ≥ x > ρ ∞ ≈ (1 − p ∞ ) | g (cid:105)(cid:104) g | ⊗ ˆ Dρ g ˆ D † + p ∞ | e (cid:105)(cid:104) e | ⊗ ˆ D † ρ e ˆ D. (5)In particular, we have p ∞ ≈ ¯ n h / (2¯ n h + 1) and ρ e,g ≈ exp( − (cid:126) ω ˆ a † ˆ a/k B T c ) /Z c to lowest order in κ h /κ c , i.e. a T h -thermal mixture of displaced T c -thermal pointer statesencoding the qubit state. The demon will access thepointer position and perform conditioned feedback op-erations to extract energy from the qubit, and it willbe functional so long as it possesses the ability to re-solve the separated pointer states. This is unlike thecase of a finite-dimensional pointer (e.g. a qubit), whosestates would not remain distinguishable in the presenceof noise. Furthermore, if the demon were able to mea-sure a qubit, it could measure the system directly andthe pointer would be redundant [29]. We remark that our setup incorporates the practicalcost of resetting the measurement apparatus: when thepointer reacts to a change in the qubit state and movestowards its new equilibrium point, the energy expelledto the cold bath amounts to 2 (cid:126) ωx ≥ (cid:126) ωx > k B T c .This is always greater than the energy loss k B T c ln 2 ofan ideal Landauer erasure protocol.Having set the model, we now introduce two demonconfigurations for work extraction: (1) an active agentperforming random measurement-feedback and (2) a pas-sive agent in the form of a coherent control field contin-uously monitoring the pointer. Active demon.—
We first consider an active demonthat interrogates the pointer position and performs nec-essary feedback at a rate γ based on the following pro-tocol: (i) a dichotomic projective measurement ( ˆ P and1 − ˆ P ) of the pointer to detect whether it is on the left( (cid:104) ˆ x (cid:105) < σ x induced by a strong control pulse if the pointeris on the left, i.e. the qubit is most probably excited.Notice that the measurement step (i) induces transitionsbetween the energy eigenstates of the pointer due to mea-surement backaction, since [ ˆ H, ˆ P ] (cid:54) = 0. While this canbe interpreted as a form of “quantum heat” [9], the netenergy change will be small compared to the extractionstep (ii). For infinitesimally short and sufficiently sparsePoisson-distributed events, the process can be effectivelydescribed by the coarse-grained generator [31–34] L m ρ = γ D [ˆ σ x ˆ P ] ρ + γ D [ ˆ P ] ρ, (6)which leads to a minor perturbation of the steady state ρ ∞ as long as γ (cid:28) κ c .Assuming vanishing overlap between the two displacedpointer states such that (5) is reached, the demon wouldideally generate a maximum energy output of W max = (cid:126) (Ω − ωx ) p ∞ by application of a spin flip on (5). In fact,such an intuitive scheme is sufficient for extracting energyclose to the ergotropy [35] (maximum extractable energyfrom a quantum system by means of a cyclic unitarytransformation) contained in (5), W erg ≈ (cid:126) (Ω − ω ) p ∞ − (cid:126) ω ¯ n c for ¯ n h > ¯ n c .The present scheme does not rely on externally im-posed engine strokes with synchronized switching of con-trol pulses or couplings to thermal reservoirs. The ran-dom measurement process not only facilitates a conve-nient assessment of stationary energy flows, ˙ Q c,h,m =tr { ˆ H L c,h,m ρ ∞ } , but it also does not depend on the pre-cise timing of “measurement strokes”. Specifically, thesteady-state power due to L m consists of two terms,˙ Q m = ˙ Q ba − ˙ W , with˙ Q ba = 2 γ tr (cid:110) ˆ H D [ ˆ P ] ρ ∞ (cid:111) = 2 γ (cid:126) ω tr (cid:110) ˆ b † ˆ b D [ ˆ P ] ρ ∞ (cid:111) , ˙ W = γ tr (cid:110) ˆ P ρ ∞ ˆ P (cid:104) ˆ H − ˆ σ x ˆ H ˆ σ x (cid:105)(cid:111) = γ tr (cid:110) ˆ P ρ ∞ ˆ P [ (cid:126) Ω + 2 (cid:126) ωx ˆ x ] ˆ σ z (cid:111) . (7)Here, ˙ Q ba describes the pure backaction effect of pointermeasurement without feedback coming from a unital FIG. 2. (Color online) Steady-state output power (a) andefficiency (b) as a function of x /x th = x (cid:112) tanh( (cid:126) ω/ k B T c )( T c = 0 on the right). The blue (thick), red, purple, and green(thin) curves correspond to measurement rates γ/ω = 10 − ,10 − , 10 − , and 10 − , respectively. The horizontal dottedlines represent the approximations γW max and (8), the shadedregions mark ¯ n c ≥ ¯ n h , while the black solid line in (b) showsthe Carnot efficiency η Carnot = 1 − T c /T h . The Otto efficiency η Otto = 1 − ω/ Ω = 0 .
99 is constant. We fix Ω = 100 ω , x =2 . κ h = 10 − ω , κ c = 0 . ω , ¯ n h = 1 i.e. T h = (cid:126) Ω /k B ln 2. channel that increases the system’s entropy, and this en-ergy would have to come from the source implement-ing the projectors. Meanwhile, ˙ W stems from the ˆ σ x -feedback and can be understood as the average rate of useful energy extracted by performing a spin flip on thepost-measurement state ˆ P ρ ∞ ˆ P .When the measurement rate γ (cid:28) κ c and the pointerseparation x (cid:29)
1, the projector would reduce the stateto the excited branch in (5) resulting in the benchmarkpower γW max . The repeated measurements however di-minish the branch weight, p ∞ ≈ ¯ n h / (2¯ n h + 1 + γ/κ h )(see Appendix). For the efficiency, η = ˙ W / ˙ Q h , we findthe approximate upper bound[36] η max ≈ − ωx / Ω1 + 2[1 + (2¯ n h + 2) κ h /γ ] ωx / Ω . (8)Both the output power and efficiency grow with γ untilan optimum is reached around γ (cid:46) κ c . At higher γ , weeventually reach a Zeno limit where frequent measure-ments hinder the pointer from moving between the leftand the right equilibrium, essentially freezing the engineoperation.Figure 2 shows (a) the output powers and (b) efficien-cies as a function of T c for various rates γ . Here, T c isexpressed in terms of the ratio between pointer displace-ment x and characteristic thermal width x th . This is anexemplary case where x = 2 .
5, which should lead to aclear separation of the ground- and excited-state distribu-tions so long as the cold bath temperature is sufficientlylow ( x > x th ). As our demon scheme captures the mea-surement and erasure costs through a mechanical pointercontinuously reset by the cold bath, the engine operationis consistent with the second law of thermodynamics andthe efficiencies do not exceed the Carnot bound.In the low- T c limit, the efficiencies and output pow-ers approach the analytical benchmarks given by (8) and γW max respectively, especially for small γ where the mea-surement effect is negligible and the steady state can byapproximated by (5). At high T c , the efficiencies andpowers fall below the benchmark and the output powereventually becomes negative due to the larger overlapbetween the two displaced thermal states, which leads toinaccurate readout of the qubit state.Should the macroscopic pointer be replaced with aqubit, the operation would be restricted to the stan-dard Otto window (¯ n h > ¯ n c ). This is because feed-back errors leading to work consumption instead ofextraction would proliferate with growing ¯ n c and thenet work output per interrogation would be limited by (cid:126) (Ω − ω )(¯ n h − ¯ n c ) / (2¯ n h + 1)(2¯ n c + 1) and lead to anOtto efficiency η Otto = 1 − ω/ Ω [37]. In our model witha macroscopic pointer, we see that the engine operateswell beyond the Otto window (shaded region in Fig. 2)so long as the pointer states are spatially distinguish-able, i.e. when x (cid:38) x th . At vanishing κ h , the systemreaches a maximum efficiency that is lower than Otto, η ≈ − x ω/ Ω < η Otto . It can be attained simultane-ously with the maximum power γW max . Passive demon.—
Instead of an incoherent schemebased on random monitoring by an external agent, itwould be insightful to formulate an integrated setup inwhich the measurement-feedback takes place internallyand all energy exchanges become transparent: we do nothave to deal with work cost associated to ˙ Q ba . To thisend, we consider a position-dependent driving field ofstrength ζ with detuning ∆, which now plays the role ofthe demon that probes the qubit in a non-invasive, coher-ent manner. We can describe the effect of such a demonby a time-dependent Rabi termˆ V ( t ) = (cid:126) ζf (ˆ x ) e − i (Ω − ∆) t | e (cid:105)(cid:104) g | + h.c.. (9)The field serves as an interface for continuous work ex-traction depending on the position-dependent function f (ˆ x ), bearing similarities to work extraction via coherentpulses from a cyclic demon engine previously consideredin [18]. Possible choices of f ( x ) include a Heaviside func-tion Θ( − x ) or a Gaussian centred around x = − x .To assess the scheme’s steady-state performance, weconsider the weak driving limit, ζ (cid:28) ω, Ω, where cor-rections to the thermal dissipators L h,c can be omitted[38, 39]. In the frame rotating at the driving frequency,the time dependence due to (9) conveniently disappearsand the time evolution follows from ˜ L h,c = L h,c andˆ˜ H/ (cid:126) = ∆ˆ σ z / ω ˆ b † ˆ b + ζf (ˆ x )ˆ σ x . The correspondingsteady state ˜ ρ ∞ describes the engine’s limit cycle andyields the average output power [40]˙ W = − tr (cid:110) ρ ∞ ( t ) ∂ t ˆ V ( t ) (cid:111) = − (cid:126) ζ (Ω − ∆)tr { f (ˆ x )ˆ σ y ˜ ρ ∞ } . (10)The heat fluxes from the hot and cold reservoirs read as˙ Q h,c = tr (cid:26)(cid:20) ˆ˜ H + (cid:126) Ω − ∆2 ˆ σ z (cid:21) L h,c ˜ ρ ∞ (cid:27) . (11) FIG. 3. (Color online) Output power (a) and efficiency (b)against the detuning ∆ of the driving field for f ( x ) = Θ( − x )(solid) and f ( x ) = 1 (dotted). We fix Ω = 100 ω , x = 2 . κ h = 10 − ω , κ c = 0 . ω , ζ = 0 . ω and the same hot andcold bath occupancy ¯ n c = ¯ n h = 1. The Carnot and Ottoefficiencies are both 0.99. Figure 3 shows the engine’s output powers and effi-ciencies at its limit cycle as a function of the detun-ing for an exemplary set of engine parameters and var-ious cold bath temperatures. Here, the optimal outputpower is much smaller than the driving rate times theextractable excitation energy, ζW max ≈ (cid:126) Ω κ h . Thiswas not the case for the previously discussed incoher-ent measurement-feedback scheme, which exhibits a workpower of up to γW max , because that scheme implicitly as-sumes a large driving strength and short feedback timesuch that the feedback is essentially described by a condi-tional spin flip depending on the position of the pointer.In the current scheme, the driving field would not causea full spin flip. Nevertheless, the output power can becomparable to what the measurement-feedback schemepredicts for similar settings, see also Fig. 4.Here, we achieve a maximum work power (and effi-ciency) when ∆ ≈ ωx . This is because the frequencyof the qubit is modulated by the pointer position, andat this driving frequency, the field addresses predomi-nantly the qubit only when the pointer is located at − x ,i.e. the qubit is excited and the field is able to extract apositive net energy from it. Hence one can modify thescheme by removing the position dependence f ( x ) andconsider a non-invasive interrogation of the qubit statesolely through the application of a red-detuned field of∆ ≈ ωx . This does not cause a backaction-induceddirect flow of energy to the pointer, a minor contributionto the energy balance when Ω (cid:29) ω , which is inherentto the position-dependent case and appears explicitly as˙ Q ba in the previous measurement-feedback scheme.The dotted line in Fig. 3 shows the output power andefficiency achievable by non-invasive interrogation as afunction of the detuning. Close to the optimal workingpoint, the performance is almost the same as the position-dependent case, but the position-independent driving willcease to produce work as the detuning approaches zero;indeed, we would obtain a heat pump consuming workat negative detunings.Finally, Figure 4 compares the active and passive de- FIG. 4. (Color online) Output power (a) and efficiency (b)against the rate γ or ζ , comparing the active measurement-feedback scheme (black solid) with the passive scheme (bluedotted) at optimal detuning ∆ = 2 ωx and f ( x ) = Θ( − x ).The other parameters are taken from Fig. 3. Note that un-derlying master equation model may no longer be reliable for ζ ∼ ω . mon at optimal detuning and position-dependent drivingin terms of their powers and efficiencies. We plot themas a function of the respective interrogation rates γ and ζ . The active scheme performs well over a broad rangeof small measurement rates γ , but it stops working whenthe Zeno effect kicks in at γ > κ c . The passive schemeeventually catches up at strong driving rates ζ . Experimental platforms.—
Regarding implementa-tions, the proposed Hamiltonian (1) would describemolecular batteries [41]: molecules with an optical elec-tronic transition strongly coupled to an infrared vibra-tion mode. It also resembles the Holstein Hamiltonianfor a molecule undergoing fast vibrational relaxation [42],where displacements can reach magnitudes x ∼
1, whilethe vibrational relaxation time is short compared to theoptical lifetime, i.e. κ h (cid:28) κ c . A broadband opticallight source (e.g. filtered sunlight) could serve as thehot bath exciting the electron, and a resonant IR cav-ity mode could be employed to monitor the vibrationmode displacement [43, 44]. Alternatively, hybrid op-tomechanical systems would be a natural platform to in-corporate a macroscopic pointer in the ultrastrong regime[45–47]. Our scheme could also be realized in a tailoredtrapped-ion setup similar to the recently demonstratedspin-flywheel engine [48]. Conclusions.—
We presented a self-contained enginemodel in which useful energy is extracted from thermalexcitations of a quantum spin by a restricted demon thatcan only interrogate the spin state through the positionof a macroscopic pointer attached to the spin. Our workreveals the fundamental energy fluxes for an autonomousMaxwell’s demon engine including work extraction, mea-surement backaction and information transfer. Specifi-cally, we evaluated the engine performance both for anactive demon performing measurement-feedback eventsat random times and for a passive demon in the formof a stationary control field. While the use of a macro-scopic pointer shows that the energy loss associated witherasure/reset would exceed Landauer erasure in reality,it also allows the engine to operate beyond typical op-eration windows in quantum engines, putting forth theparadigm of continuous measurement-driven engines.
Acknowledgments.—
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Here we compare the local and global secular form of the hot bath dissipator that arises in the usual manner froma linear exchange interaction of the qubit with a thermal oscillator bath. For the local model, one simply employs thestandard dissipator for an isolated qubit, L loc h ρ = κ h (Ω)[¯ n h (Ω) + 1] D [ˆ σ − ] ρ + κ h (Ω)¯ n h (Ω) D [ˆ σ + ] ρ, (12)assuming that the qubit-pointer coupling and thus the influence of the pointer on the qubit energy are negligible.For an isolated qubit, the jump operators ˆ σ ± can mediate only a single transition of frequency Ω, given the thermalcoupling rate κ h and the mean thermal bath occupation ¯ n h at this frequency. In the combined qubit-pointer system,the same operators now induce a family of transitions Ω + kω with k ∈ Z . Specifically, we can expand in terms of thecombined energy basis (2),ˆ σ + = | e (cid:105)(cid:104) g | = ∞ (cid:88) m,n =0 (cid:104) m | ˆ D | n (cid:105)| e, m e (cid:105)(cid:104) g, n g | = ∞ (cid:88) n =0 ∞ (cid:88) k = − n (cid:104) n + k | ˆ D | n (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) ≡ d n,k | e, ( n + k ) e (cid:105)(cid:104) g, n g | , (13)with the weight coefficients d n,k . The above local dissipator contains cross-terms between different transitions k (cid:54) = k (cid:48) ,which means that it preserves a certain amount of coherences between different energy levels of the system. Moreover,using it implies that one can neglect the frequency dependence of the bath parameters, κ h (Ω + kω ) ≈ κ h and¯ n h (Ω + kω ) ≈ ¯ n h , which is only valid when Ω (cid:29) ω .The global secular model does not preserve any coherences between different Fock numbers, because it containsonly resonant jump terms, L glo h ρ = (cid:88) k κ h (Ω + kω ) [¯ n h (Ω + kω ) + 1] D (cid:34)(cid:88) n d ∗ n, − k | g, ( n − k ) g (cid:105)(cid:104) e, n e | (cid:35) ρ + (cid:88) k κ h (Ω + kω )¯ n h (Ω + kω ) D (cid:34)(cid:88) n d n,k | e, ( n + k ) e (cid:105)(cid:104) g, n g | (cid:35) ρ. (14)For the demon models studied in the main text, we find that both dissipators yield approximately the same results.The reason is, on the one hand, that we indeed consider Ω (cid:29) ω and can thus assume constant κ h and ¯ n h . On theother hand, our model also includes a cold bath with stronger damping rate κ c > κ h , which suppresses any coherencesbetween Fock states of the pointer that L loc h alone would have preserved.The steady-state heat input for κ h (cid:28) κ c can then be approximated using the local dissipator, too,˙ Q h ≈ tr (cid:26)(cid:18) (cid:126) Ω2 + (cid:126) ωx ˆ x (cid:19) ˆ σ z L loc h ρ ∞ (cid:27) ≈ (cid:126) κ h (cid:2) ¯ n h (1 − p ∞ )(Ω + 2 ωx ) − (¯ n h + 1) p ∞ (Ω − ωx ) (cid:3) . (15)For the qubit excitation probability, the same approximation yields ∂ t p e ( t ) ≈ tr (cid:8) | e (cid:105)(cid:104) e | ( L loc h + L m ) ρ (cid:9) = − κ h (¯ n h + 1) p e ( t ) + κ h ¯ n h [1 − p e ( t )] − γ tr (cid:110) ˆ σ z ˆ P ρ ˆ P (cid:111) ≈ − κ h (¯ n h + 1) p e ( t ) + κ h ¯ n h [1 − p e ( t )] − γp e ( t ) . (16)Here the second line holds in the ideal operation regime of γ (cid:28) κ c and x (cid:29)
1, when ˆ P reduces the state to itsexcited branch. At steady state, we obtain p ∞ = p e ( ∞ ) = ¯ n h / (2¯ n h + 1 + γ/κ hh