Work extraction via quantum nondemolition measurements of qubits in cavities: Non-Markovian effects

Work extraction via quantum nondemolition measurements of qubits in cavities: Non-Markovian effects. D. Gelbwaser-Klimovsky1, N. Erez1 , R. Alicki2 and G. Kurizki1 arXiv:1211.1772v2 [quant-ph] 16 Aug 2013 Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100, Israel 2 Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Wita Stwosza 57, PL 80-952 Gdańsk, Poland; Weston Visiting Professor, Weizmann Institute of Science, Rehovot 76100, Israel We show that frequent nondemolition measurements of a quantum system immersed in a thermal bath allow the extraction of work in a closed cycle from the system-bath interaction (correlation) energy, a hitherto unexploited work resource. It allows for work even if no information is gathered or the bath is at zero temperature, provided the cycle is within the bath memory time. The predicted work resource may be the basis of quantum engines embedded in a bath with long memory time, such as the electromagnetic bath of a high-Q cavity coupled to two-level systems. I. Introduction Information acquired by an observer on the system can be commuted into work in a single-bath engine [1–9]. To this end, the observer (“Maxwell’s demon”) must manipulate the system according to the results of measurements performed on it. The measurements have therefore to be selective, i.e., their outcomes must be read and discriminated to determine the observer’s course of action [7, 10]. The balance of work and information is embodied by the Szilard-Landauer (SL) principle [1, 2] whereby work obtainable from a measurement must not exceed the energy cost of erasing its record from the observer’s memory. Notwithstanding ongoing efforts to expand information-based thermodynamics (IT) so as to include measurement-cost effects [10–12] beyond the original SL balance (see Appendix), an important aspect has been little addressed thus far [13]: How essential is the thermodynamic paradigm of system-bath separability[14, 15] to the analysis of work extraction by measurements? This question is here investigated in the context of non-Markovian quantum thermodynamics “under observation” [16–21]. Namely, we show that frequent measurements can induce changes in system-bath correlations, unaccounted for by the separability paradigm, and thereby change the tradeoff of information and work. Consequently, selective (read) measurements can extract more work in a closed cycle than what the SL principle allows. This effect requires the cycle to be completed within a non-Markovian (bath-memory) time-scale. We further show that non-selective quantum nondemolition (QND) measurements of the system energy (which we shall also denote as unread measurements, i.e., measurements whose outcomes do not matter) enable the system to do work in a cycle, although they provide no information on the system, nor do they change its state (by their definition as QND measurements) and thus do not act as a “demon.” Work is shown to be obtainable within the bath memory time even at zero temperature (T = 0), i.e., for the vacuum state of the bath. This finding is in stark contrast to the expectation that the extractable work at T = 0 should vanish by the SL principle[1–8] or its current generalizations [10, 11] that are valid in the Markovian limit. The predicted effects cannot be ascribed to quantum coherence in the system, which is the source of work in recently explored quantum heat engines (QHEs) [22], since coherence is absent from this scenario. Rather, these effects follow up on the anomalous temperature effects of frequent QND measurements on non-Markovian time scales [16–21]. Here, we show that these effects may well determine the performance of quantum engines that operate on such time scales. Clearly, one could instead consider energy pumping into the sytem by non-QND measurements, e.g., projections onto the x -y plane of the qubit Bloch sphere. Yet we are interested in minimal intrusion into the system that would have no effect on an isolated qubit (in the absence of a bath). To this end, we analyze the following simple yet unconventional protocol: A brief unread QND measurement of the energy of a thermalized qubit neither changes its state nor yields information, yet inevitably decorrelates it from its bath and therefore requires energy investment that is absent in SL considerations (Sec. II). A postmeasurement cycle produced by sinusoidal modulation of the qubit frequency is shown (both analytically and numerically) to yield work (despite the fact that no information is obtained) from the unread measurement, provided the cycle is completed within the bath memory time (Sec. III). Post-measurement system-bath correlation change is shown to be the work resource. The analysis culminates in a revised work-information relation (Sec. IV). A demonstration of consistency with the second law (Sec. V) is followed by a discussion of the cost of multiple cycles (Sec. VI) and an analysis of feasible cavity-based experimental scenarios (Sec. VII). The findings are summarized in Sec. VIII. 2 II. System-bath decorrelation via QND measurements We consider a QND measurement of the energy of a thermalized qubit by a quantum probe (P) consisting of two degenerate yet distinguishable states (HP = 0) (e.g., photon-polarization states). This situation is modeled by the Hamiltonian Htot = HS + HB + HSB + HSP (t) (1) where S labels a two-level system (TLS) with energy states |ei and |gi, B denotes a (bosonic or fermionic) bath, HSB is the coupling of S to a bath operator B̂ and HSP (t) is the S-P interaction that effects the measurement. We choose the coupling Hamiltonians to have the form: HSB = σx B̂, where σx does not commute with HS and neither does B̂ with HB , in order to describe S-B equilibration, i.e., transitions between the levels of S and B. Since the measurement is to have a QND effect on S [23], in order for the density matrix of S, ρS , to retain the same σz -diagonal form it had in equilibrium, one should choose a projective measurement in the σz (qubit-energy) basis, i.e., HSP ∝ σz . The impulsive QND measurement of the energy of S by P is well described R τm HSP (t) = UC , where UC is the by [16, 18–20] e−i 0 dt ~ Controlled-NOT (CNOT) operation (with the probe acting as the target qubit). Its duration τm is assumed much shorter than all other time scales of evolution generated by Htot . The time-dependent system-probe coupling aimed at a CNOT operation has the form (|0i and |1i being the probe states) HSP (t) = h(t) |ei he| (|0i h0| + |1i h1| − |0i h1| − |1i h0|) (2) where we may choose π h(t) = 4τm     t − tm 2 tanh −1 τm (3) as a smooth temporal profile of the system coupling to the probe qubits during the measurement at time tm which lasts over time τm . The measurement outcomes are averaged over (for nonselective measurements-NSM), by tracing out the probe degrees of freedom. As long as the probe state is of the form ρP = I + dσz (4) where d is real, the measurement will not affect ρS , which is diagonal in the energy basis:   ρS 7−→ T rP UC ρS ⊗ ρP UC† = |ei he| ρS |ei he| + |gi hg| ρS |gi hg| (5) i.e., the diagonal elements of ρS are unchanged, and the off-diagonal elements are erased. Since the system is entangled with the bath, the effect of the measurement on ρS+B is:   ρS+B 7−→ T rP UC ρS+B ⊗ ρP UC† = |ei he| ρS+B |ei he| + |gi hg| ρS+B |gi hg| ≡ Bee |ei he| + Bgg |gi hg| (6) where Bee(gg) are bath states correlated to |ei and |gi respectively. Thus, the post-measurement ρS+B is blockdiagonal in the energy states of the system. It can be shown[16, 19] to be close to a product state of ρS and ρB . We assume at this stage that the state of P is unread (averaged out) after this measurement. The entire measuring process can be summarized as ρtot = ρP ⊗ ρS+B → ρS+P ⊗ ρB ; T rP ρtot → ρS ⊗ ρB . (7) If B effects were treated classically, or S-B correlations were ignored, this measurement would have no effect at all, since it does not change the energy-diagonal state of S. Yet, because of the non-commutativity of HSP and HSB , an impulsive NSM does change the S+B “supersystem”. Such change is absent from Markovian treatments wherein the measurement is assumed slow enough to warrant energy conservation of the supersystem[24], in contrast to the present fast one that breaks this conservation. This crucial point is beyond the system-bath separability paradigm[15, 24], and stems from the fact that at equilibrium S is correlated with B: they are in a Gibbs −β(HS +HB +HSB ) , β being the inverse temstate ρEq = e Z perature, and their mean correlation energy is negative to ensure stable equilibrium: hHSB iEq < 0 [16, 18, 19]. The impulsive NSM changes the Gibbs state and its mean observables into their post-measured counterparts[18] hHSB iEq < 0 7→ hHSB i = 1 1 hHSB iEq + Tr[B̂σz σx σz ρEq ] = 0, 2 2 (8) where we have used the identity σz σx σz = −σx . Concurrently, hHSP iEq = 0 7→ hHSP i = hHSB iEq , and hHP i = 0 both before and after the measurement. Hence, the decrease in the S+P correlation energy hHSP i reflects an equal increase in the correlation energy hHSB i. The foregoing expressions have the following meaning: Any NSM must increase the mixedness and thus the 3 entropy of ρS+B [25]. Since prior to the measurement S+B was in a maximal-entropy (Gibbs) state among all states with the same mean energy hHSB i, the entropy increase implies an increase of hHSB i. Since the QND NSM changes ρS+B into ρS ⊗ ρB , yet leaves ρS and ρB unchanged , it changes neither hHS i nor hHB i. The work invested in performing the impulsive NSM is therefore ∆Emeas = −∆hHSP i = −hHSB iEq > 0. (9) Equation (9) reveals the discrepancy between the energy cost (work investment) required for a brief QND measurement and that expected by the SL principle: the meas SL work investment in the measurement is WSL = 0, since it does not account for system-bath correlations. As shown in what follows, Eq. (9) can be a useful work resource only on non-Markovian time-scales at any temperature of the bath, including T = 0. By contrast, the measurement cost in current Markovian treatments[10, 11] vanishes at T = 0 (see Appendix). III Work extraction in a non-Markovian cycle As shown below, it is crucial to ensure that the cycle duration tcycle satisfies tc ≥ tcycle , i.e., work extraction must occur on the non-Markovian (memory) time-scale tc of S+B. Furthermore, it should be much longer than the duration τm of the measurement that triggers the cycle (Sec. II). If these conditions hold, the maximal work extractable in a post-measurement optimal cycle is (see App.) (WNext SM )Max = ∆Emeas − T ∆Smeas , (10) where ∆Emeas is given by Eq. (9), and ∆Smeas is the NSM-induced entropy increase of the supersystem S+B. This entropy increase reflects the destruction of S-B correlations (off-diagonal elements of ρSB ) and the lack of information gain by the NSM. The bound on the maximal extractable work discussed above does not suffice to demonstrate that work is indeed obtainable from S in a post-measurement cycle: we must ext show that the extracted work Wcycle > 0 in a feasible cycle. This work may be extracted by a classical, coherent (zero-entropy) off-resonant piston (harmonic oscillator) that is dispersively coupled to S via σZ , and modulates its energy levels [24], allowing S to undergo a closed cycle, after which ρS (tcycle ) = ρS (0). The piston coherent excitation-change then expresses the extracted work (see Discussion). The standard expression for the extractable work (i.e., the negative of the invested work) over a cycle is [14, 24]: ext Wcycle =− I T r{ρS ḢS }dt = − I s(t)ω̇(t)dt. (11) Here ω(t) is the level-separation (frequency) of the piston-driven TLS, s(t) is the polarization (population difference) of its energy-states |ei, |gi and the cyclic integral is over a closed trajectory in the frequencypolarization plane. According to the standard (Markovian) expression of the second law in open quantum systems [14, 15, 24], no work is extractable from the system: conversely, only the piston can do work on the system. This rule can be proved to be strictly obeyed if the bath-induced evolution is Markovian (App.). Yet, our analytical results and numerical simulations (Fig.1) show that while the system interacts with a bath on non-Markovian time scales, net work can be performed in a cycle by the system, i.e., the piston can be coherently amplified. Namely, under ext non-Markovian dynamics, we can ensure Wcycle > 0 in Eq. (11) by choosing ω̇(t) to oscillate out of phase with s(t). For weak S-B coupling and T = 0 we will explicitly ext show that Wcycle > 0 is indeed possible only in the nonMarkovian limit on the modulation rate, Ω ≫ t1c . More elaborate analysis, whereby the piston field “dresses” the qubit states with which the bath interacts[26], yields qualitatively similar results. We wish to evaluate the work performed by a qubit in contact with a bath at temperature T after a nonselective measurement of its energy, over a period (cycle) of its Stark-shift modulation of the form ω(t) = ωa + δSinΩt. (12) To this end we shall use the results of the weak-coupling, non-Markovian master equation[18, 27, 28], whereby Z ′ s(t) = e−J(t) ( ∆R(t′ )eJ(t ) dt′ + s(0)). (13) Here the relaxation integrals J(t) = Jg (t) + Je (t) , Rt Jg(e) (t) = 0 Rg(e) (t′ )dt′ and ∆R(t) = 21 (Rg (t) − Re (t)), depend on the effect of the non-Markovian (B) bath: these are the bath-induced transition rates Re (t) (|ei 7→ |gi) and Rg (t) (|gi 7→ |ei). Both J(t) and R(t) are partly oscillatory on non-Markovian time scales, reflecting the partial reversibility of S-B dynamics. They are proportional to the square of the system-bath coupling strength, η. Since the coupling is assumed weak, s(t) can be expanded as s(t) ≈ s(0)(1 − J(t)) + ∆J(t) + O(η 3 ), where Rt ∆J(t) = 0 ∆R(t′ )dt′ . The first term in the expansion, R 2π s(0), does not contribute to the work ( 0 Ω s(0)cosΩtdt = 0). The other terms are O(η 2 ), so we set s(0) ≈ −1/2, at T ≈ 0. The universal formula [18, 27] yields the relaxation integrals J(t) and ∆J(t) through Jg(e) (t) as the spectral overlap of the bath response GT (ω) and the modulation spectrum Ft (ω) : 4 WextCycle/(WextNSM)Max (a) Je(g) (t) = Z −∞ 1 2π Z 0.4 0.2 ∞ ∞ dωGT (ω)| dωGT (ω)Ft (ωa ∓ ω) = Z t ′ dt′ ei(ω+ωa )t ǫ(t′ )|2 , -0.2 -0.4 -0.6 -0.8 (14) 0 −∞ Rt ′ ′ where the modulated phase factor ǫ(t) = ei 0 dt ω(t ) . Usδ ing expansion of ǫ(t) for our ω(t) : e−i Ω CosΩt = P∞the Bessel δ inΩt n , and assuming a weak modulan=−∞ i Jn (− Ω )e δ tion, Ω ≪ 1, the expression for work in a cycle reduces to tc Cycle time (b) 3 <ΗB> δ ≈ 2π where G0 (ω) is the zero-temperature bath response, and ω + = ω + ωa . This expression may be approximated as ext Wcycle ≈ −δ Z 2π Ω Jg (t)ΩcosΩtdt. (16) 0 It shows that in the strongly non-Markovian limit Ω ≫ ext δ ∼ ωa , the sign of Wcycle oscillates with Ω, for a fixed ωa , and thus allows for either positive or negative work extraction, as opposed to the Markovian limit (App.). Hence the work invested by the NSM (Eq. (9)) can be partly extracted in a non-Markovian cycle. We next clarify how S can regain the energy deposited by the measurement, using our analytical results and simulations (Fig. 1–main panel): The source of work is seen to be only the change of hHSB i, the system-bath correlation energy. The rapid variation of the extractable work with the cycle duration tcycle (Fig. 1a) proves that work retrieval from hHSB i is limited to non-Markovian time scales : tcycle = 2π Ω should be shorter than tc , the bath memory time, to ensure work performance by the system enabled by an unread QND measurement. The reason for this anomalous effect is the (partly reversible) S+B dynamics expressed by the oscillatory relaxation integrals Jg (t) and Je (t) on non-Markovian time scales triggered by the measurement. IV The revised work-information relation We are now in a position to address the fundamental questions that motivate this paper: What is the difference between the maximal work extracted in a cycle via ext a selective (read) measurement, (Wsel )Max , and its nonselective (unread) counterpart, (WNext SM )Max ? How do they differ from their SL counterparts? Let us define the measurement basis as |ji, j = e, g. Then pj is the probability of finding the state ρS+B in the state j, and ρjS+B is the state of the supersystem af- 0.3∆Emeas 1 ∞ 2π G0 (ω) + 2 × (ω ) −∞   2π + 2π + sinc( (ω + Ω)) + sinc( (ω − Ω)) dω (15) Ω Ω ext Wcycle End 2 <ΗS> <ΗPiston > Z tend=tcycle+tm tm Start ∆Emeas -1 1 2 3 0.2∆Emeas ωat <ΗSB> FIG. 1. Work extraction by measurements. Main panel: Simulations of the first-cycle evolution of the energy of the system (solid thick blue line), the bath (solid thin purple line), the classical piston (dashed thin green line), and the system-bath correlations (dashed thick brown line). The piston periodically modulates the TLS frequency, ω(t) = ωa + δsinΩt. The period of the modulation starts with an unread measurement of the TLS energy. The parameters for the curves are ωa = 1, δ = 1/4, Ω = 5/2. The S-B coupling spectrum is a Lorentzian centered at ω0 of width t−1 (inverse correc lation or memory time of the bath), with ω0 − ωa = 3/7 , tc = 10 and the inverse bath temperature is β = 3.74. The cycle starts with a non-selective measurement at time ωa tm = 1 and ends at ωa tend = 3.51. The measurement invests ∆Emeas = −hHSB (t = tm )i in the system. Here the piston hHP iston i is seen to have gained energy during the cycle, but not at the expense of hHS i which returns to its initial value. Since hHB i has also gained energy, the source of work is the change in hHSB i, the system-bath correlation. The ext ext simulations imply that Wtot = Wcycle − ∆Emeas < 0, alext though Wcycle = hHP iston (tend )i − hHP iston (tm )i > 0. Inset: ext extractable work in a cycle Wcycle normalized to the maxiext mum (WNSM )M ax as a function of the cycle duration tcycle . ext It is seen that Wcycle > 0 (work done by the system) requires tcycle . tc , tc being the bath memory time. Same parameters as in main panel 1. ter the measurement. The maximum extractable work by a nonselective measurement is given by Eq. (10). Its ext counterpart forPselective measurement is (Wsel )Max = P pj ∆Ej − T pj ∆Sj . where ∆Ej and ∆Sj are the respective changes when the state j is measured, and P ∆Emeas = j pj ∆Ej . The difference between work extraction based on selective and nonselective measurement is 5 ext (Wsel )Max − (WNext SM )Max = X X j pj S (ρjS+B )) = T H ({pj }) pj ρS+B ) − T (S ( j j = T S (ρS ) 1.0 (17) 0.8 The last step follows from the equality of the Shannon entropy H of the system and its von Neumann entropy S (ρS ) when ρS is diagonal, as in our case, if we set kB = 1 and H = ln2HShannon, HShannon being the standard definition of the Shannon entropy. Explicitly, in our case H = −plnp − (1 − p)ln(1 − p). The foregoing expression may be recast in the form 0.6 0.4 0.2 Wext NSM WSL 0.05 ext )Max = (WNext (Wsel SM )Max + WSL . ext Wsel = ρee (0)We + ρgg (0)Wg = [Jg (t)ρgg (0)ω̇g (t) − Je (t)ρee (0)ω̇e (t)]dt 0.15 0.20 0.25 0.30 KBT (18) Here the first term on the r.h.s. is Eq. (10) and WSL = T H (p) denotes the standard SL work extraction: the energy required to reset the probe to its initial state. Equation (18) is our main result: it implies that the maximal work extractable via a selective measurement is higher than what is expected from the SL relation between work and information: the S-B correlations increase this extractable work by (WNext In the stanSM )Max . dard (Markovian) case, where the system-bath interaction energy is assumed negligible, (WNext SM )Max = 0 and ext )Max = T H (p). Hence, in the Markovian case, the (Wsel selective measurement (in this scenario) does not yield any work beyond the Landauer cost[2] WSL = T H (p) of resetting (“cleaning”) the memory of the probe after each measurement. In the non-Markovian case, the measurement process requires higher investment of work, since it changes the system-bath correlation energy, but also allows more work to be extracted than in the Markovian case. The NSM effect discussed above allows work extraction at T = 0 and without any information gain: both features take us beyond the SL [1, 2] information-work balance or its IT extensions[10, 11]. Namely, even at T = 0, hHSB iEq < 0 and its change by ∆Emeas powext )Max = (WNext ers the cycle, yielding (Wsel SM )Max > 0 (Fig. 2). The entire work then originates from the nonMarkovian change of system-bath correlations. Whereas WNext SM has been shown above to exceed the ext SL bound, it can be argued that Wsel can be even higher. To this end, consider that in the case of selective measurements, the work extraction will be the weighted sum of that obtained by each measurement result I 0.10 (19) FIG. 2. The Szilard-Landauer work WSL (thin purple) and ext ext (WNSM )M ax (solid blue), both normalized to (Wsel )M ax (Eq. (18)), as a function of the single-bath temperature. Even at T=0, where Szilard-Landauer work vanishes, work can be extracted via a measurement in the non-Markovian timedomain. where the two modulations ω̇e(g) (t) may be different. Suppose we choose ω̇e(g) so that they maximize the total ext ext work Wsel = (Wsel )Max . If the corresponding modulations happen to coincide, (ω̇e (t)Max = ω̇g (t)Max ), then the resulting expression is the same as (WNext SM )Max (Eq. (11)). If, on the contrary, (ω̇e (t)Max 6= ω̇g (t))Max , then the two bounds differ. Clearly, we may then have ext (Wsel )Max > (WNext SM )Max , (20) since by choosing ω̇e (t) and ω̇g (t) to be out of phase throughout the cycle, the two terms in Eq. (19) acquire the same sign, i.e., add up. By contrast, in WNext SM = I [Jg (t)ρgg (0) − Je (t)ρee (0))]ω̇(t)dt, (21) ext the two terms have opposite signs, so that (Wsel )Max can exceed (WNext ) , q.e.d. SM Max V Consistency with the second law The second law is upheld (in the sense that perpetual motion[29] becomes forbidden) only when we account for the energy and entropy cost of changing the “supersystem” state ρS+B from its correlated Gibbs form at equilibrium ρEq to its post-measurement product-state form ρS ⊗ ρB . This cost becomes evident only in a description of the evolution in terms of the total Hamiltonian and the corresponding state ρtot that encompass the degrees of freedom of the probe and the supersystem, P+S+B. 6 Explicitly, Htot = H0 + HBP + HSP , (22) where HSP is the system-probe coupling term, HBP allows a fast decorrelation between the system and probe, and H0 describes the “supersystem”, system+bath. Using the fact that the total Hamiltonian is cyclic, Htot (τ ) = Htot (0), the total extractable work in a cycle (by P+S+B) can be written as ext Wtot (τ ) = − Z τ   T r U (t)ρtot (0)U † (t) Ḣtot (t)dt = 0 −T r [ρtot (τ )Htot (τ ) − ρtot (0)Htot (0)] Z τ + T r [ρ̇tot (t)Htot (t)] dt (23) 0 Upon inserting the expression for ρ̇tot (t) and calculating the trace we find that the second term on the RHS is zero. The first term thus represents the energy change of the supersystem. Since initially the supersystem and the probe were at thermal equilibrium, any energy change should be positive. We then find: ext Wtot (τ ) =  −T r U (τ )ρtot (0)U † (τ )Htot (0) − ρtot (0)Htot (0) . (24)  Here the first term is the final mean energy and the second is the initial one. Because the total dynamics is unitary, the entropy of ρtot is fixed. This implies that the final mean energy (first term) must be greater or equal to the initial one (second term), as the thermal-equilibrium initial state minimizes the mean energy at fixed entropy. per the SL principle [2]. Clearly, the probe cannot circumvent the SL resetting cost WSL when selective measurements are required (Eq. (18)). Yet, this is not the case for a NSM, which requires no resetting because the bath can rapidly decorrelate P and the supersystem following the measurement (Eq. (7)), but prior to the next cycle: ρtot 7→ ρS+P ⊗ ρB 7→ ρ′P ⊗ ρ′S+B . The absence of resetting cost after a NSM follows from a remarkable observation: a single probe qubit has the same NSM effect on any number of system cycles. This holds since it does not matter how each cycle changes the state of the probe, ρP , because the resulting state commutes with the probe’s σzP (Eq. (4)). In particular, work is extractable even if P is in the fully mixed (infinitetemperature) state: for ρP = 21 I P , the CNOT leaves the probe qubit unchanged, i.e., it cannot be read out and yet the same probe qubit can still perform the required NSM on the system qubit as often as we like, i.e. our probe is never used up. This is because Eq. (26) still holds in this case: S+P become correlated by the measurement (after the previously described fast modulation period), but then the correlations between the system and the probe decay via thermal relaxation and revert to a product state (Fig. 3). After this relaxation the probe can be reused in the next cycle and have the same effect as in the first cycle. Hence, there is no need of resetting the probe for further use in consecutive cycles, provided it performs repeated NSM. ( i|ρtot|j ) System-Probe correlations 0.010 0.005 0 ext Wtot − I = ext < 0. T r{ρtot Ḣtot }dt = −∆Emeas + Wcycle (26) 1 2 3 4 t/tmod -0.005 (25) The negativity of Eq. (25) under a cyclic unitary evolution of the total Hamiltonian, starting from equilibrium of the supersystem and the probe, can be proved completely generally. It shows that the second law that forbids drawing work from a single bath only applies to the entangled evolution of S+B+P and that their standard separability assumption[1–8] fails for sufficiently fast cycles. VI Multiple cycles: Resetting cost for nonselective measurement? It might be suspected that resetting (purifying) the probe is necessary if we wish to reuse it in successive cycles and that would add to the thermodynamic cost as -0.010 Measurement time FIG. 3. The role of probe-system correlations and their destruction by the bath. Decay of the off-diagonal system-probe (S-P) elements (correlations) hi|ρtot |ji under Htot (Eq. 1) where |ii = |e, 1, ni and |ji = |g, 0, n ± 1i, the entries denoting the system, probe and bath quantum numbers, respectively. The parameters are the same as in Fig. 1. The probe frequency is 10/7 ωa . The decay time of the correlations is that of the oscillations envelope, here ∼ 4 modulation periods (4tmod ). After this time the probe can be reused for the next cycle. VII Experimental Scenario A feasible experimental test of these predictions may 7 P Piston ( ) S B FIG. 4. Possible experimental setup: The measured atomic TLS (S) in a cavity bath (B) does work on a frequencymodulating piston mode of the cavity . The time-modulated probe effects brief, non selective, QND measurements. involve (Fig. 4) the following ingredients: A) An an ensemble of N two-level atoms (S) in a cavity (resonator) of length L, whose high-Q field modes that are near-resonant with the atomic resonance frequency ωa constitute the non-Markovian intracavity bath (B) −4 with memory time tc ∼ LQ sec are feasible at c : tc ∼ 10 present [30, 31]. The interaction energy hHSB i and thus the NSM induced work (Eqs. (9), (10)) may attain the GHz range, as they scale with√the collectively-enhanced N-atom coupling to the bath, N . B) An off-resonant, coherent (classical) signal constitutes the piston with amplitude E0 (1 + pCosΩt), (p < 1) that modulates the atom level-distance ω( t) at a rate Ω ≫ 1/tc by periodic Stark shift. C) Injected pulses, much shorter than tc , can probe the atomic-state population in a QND fashion, on a subnsec/psec timescale. Specifically, the CNOT protocol of QND measurements in Eqs (1)-(6) demands entangling the polarization of a probe photon with the magnetic sublevels of one of the TLS states. If this photon is to be reused, the photon-atom states are to be disentangled by a depolarizing environment as per Sec. V. Yet even a classical probe may effect a QND measurement by its polarization action on the atomic sublevel population[23]. As discussed in Sec VI, NSMs may be performed by a probe that is arbitrarily noisy in its polarization. Both the weak-modulation (Eqs. (12) and (13) ) and the more elaborate exact solution of this S+B model [26] ext predict that WNext SM (Eqs.(16)) and Wsel (Eqs. (19)) can extract work via NSM at T = 0 (an empty cavity, or field-bath vacuum and for ground-state atoms) only if the cycle duration is much shorter than tc . The extracted work will be manifest by the amplification (lasing) of the off-resonant coherent piston mode despite the absence of atomic population-inversion or bath heat energy, at the expense of the S-B interaction (correlation) energy (Eq.(9)). The distinctive signature of this am- plification is that it is restricted to tcycle = 2π Ω ≤ tc : as tcycle starts exceeding tc , amplification will revert to loss. The described process is akin to intracavity parametric conversion of external driving (probe pulses), resulting in signal (“piston” mode) amplification [22, 32], but it is unique in its reliance on system-bath correlations, and in its insensitivity to the probe noise. VII Discussion We have shown the possibility of extracting useful work from an open quantum system following either a non-selective (unread) QND measurement (NSM) (Eq. (16)) or a selective (read) measurement (SM) (Eq. (19)). In both cases, a modulator (piston) can take work and gain energy from the system (be coherently-amplified) in a closed cycle. This work originates neither from the probe free-energy[10, 11] nor from the heat energy of the bath (as in Szilard’s engine[1–8]) but from a hitherto unexploited (and little-discussed) source: the inevitable change of the system-bath (S-B) correlation (interaction) energy (see [29]) by a brief QND measurement [16–21]. Only non-Markovian supersystem (S+B) dynamics can yield extractable work following such a measurement, as opposed to its Markovian limit that ignores system-bath correlations (Fig. 1 a). When discussing these effects, certain misunderstandings must be dispelled: (i) The proposed work resource cannot be explained by viewing either the unread probe or the piston as a fictitious additional “bath”. Neither constitutes a proper heat bath: the piston is a zerotemperature and zero-entropy classical drive that only gains work and energy from the system, while the probe must act impulsively, unlike usual sources of noise of heat. (ii) Nor can one deny the cycle is triggered by a measurement: even if the measurement is unread, it is still a measurement, as evidenced by the S-P correlations (see Eq. (26)). (iii) Recently considered measurement-cost (Markovian) effects [10, 11] are beyond the scope of our scenario (see SI2). The colloquial maxim “there are no free lunches” applies to the predicted effect, i.e., the surplus work is allowed only by extra investment of energy consistently with the first law (otherwise it would enable a “perpetuum mobile” machine)[29, 33] and the second law is also upheld (Eq.(25)). Yet this effect may allow us to study the possibilities of transforming energy input (e.g., electromagnetic probe pulses which may be very noisy as argued in Sec. VI, similarly to [22].) into useful work [coherent signal (piston) amplification] via rapid modulations of thermalized quantum systems The present engine model, in which the system is always coupled to a single bath and yet may perform useful work, is potentially important for systems totally embedded in a single bath, such as a cavity, so that conventional heat-engine (two-bath) thermodynamic cycles may be impossible to implement. Further investigation may include brief disturbances other than measurements, 8 e.g., phase flips of a TLS in a bath [34]. This research was supported by DIP, ISF, BSF and CONACYT. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] L. Szilard, Zeitschrift Physik 53, 840 (1929). R. Landauer, IBM J. Res. Dev. 5, 183 (1961). C. M. Caves, Phys. Rev. Lett. 64, 2111 (1990). M. O. Scully, Phys. Rev. Lett. 87, 220601 (2001). S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. 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Rev. A 87, 013841 (2013). T. M. Nieuwenhuizen and A. E. Allahverdyan, Phys. Rev. E 66, 036102 (2002). D. Petrosyan, G. Bensky, G. Kurizki, I. Mazets, J. Majer, and J. Schmiedmayer, Phys. Rev. A 79, 040304 (2009). [31] J. Verdú, H. Zoubi, C. Koller, J. Majer, H. Ritsch, and J. Schmiedmayer, Phys. Rev. Lett. 103, 043603 (2009). [32] H. E. D. Scovil and E. O. Schulz-DuBois, Phys. Rev. Lett. 2, 262 (1959). [33] G. W. Ford and R. F. O’Connell, Phys. Rev. Lett. 96, 020402 (2006). [34] D. D. Bhaktavatsala Rao and G. Kurizki, Phys. Rev. A 83, 032105 (2011). [35] N. Erez, Physica Scripta 2012, 014028 (2012). [36] K. Takara, H.-H. Hasegawa, and D. Driebe, Physics Letters A 375, 88 (2010). APPENDIX 1 Measurement cost in our scenario compared to Maxwell’s demon models A) Measurement cost for asymmetric detector The SL principle has been extended to a detector/memory modeled by a quantum particle in an asymmetric double-well potential (the Sagawa-Ueda (SU) model[10]). Such asymmetry may reduce the expense of resetting the detector to its initial state at the cost of increasing the work required to perform the measurement, but the sum of the two costs remains the same as the cost set by the SL principle. In that case, the mean energy of the device (D) may be altered by the measurement, and in particular the device may exchange energy with the measured system. For such a device the resetting cost may differ from Landauer’s. SU assume a total Hamiltonian of the form: H(t) = HD (t) + HB + HDB (t); HD (0) = HD (τ ) = HD ; HDB (0) = HDB (τ ) = 0 (A1) The SU Hamiltonian does not describe the measurement itself, only the detector-bath interaction HDB (t). By contrast, in our scenario the HSD (t) term in Eq. (1) causes a change of hHSB i a S+B correlation term, which is missing from the SU analysis: they adopt the S+B separability paradigm, whereas we do not. Thus, in the SU model, the measurement conserves hHSB i + hHD i, as opposed to our dynamics, hence the difference in measurement cost and post-measurement work extraction. In our case the final state of the D has the same energy as its initial state. Yet our measurement cost is given by Eq. (2) and is nonzero due to S+B correlation change. Similarly, the work invested in the measurement in our scenario is performed not on D, as in the SU model, but on S+B, and is equal to the change in hHSB i in Eq. (8). In the SU model, the initial state of D is assumed to be a thermal state corresponding to a particular subspace, HD 0 of its total Hilbert space , whereas the final state of D may have changed. This leads to the following expression for the measurement cost in the SU model, which is the 9 change in the free energy of D due to the measurement process (A2) ∆FD ≡ Σk pk FkD − F0D where pk is the post-measurement probability for D to be found in subspace HD k . By contrast, in our case the final state of the D has the same energy as its initial state and p0 = 1, giving: ∆FD = 0. Yet our measurement cost is given by Eq. (9) and is nonzero due to S+B correlation change. Similarly, the work performed on D by the measurement is assumed by SU [10] to be the negative of the change in energy of D, which in our scenario τ D Wmeas ≡ T r {(HD + HB ) ρDB |0 } = 0 (A3) Namely, in our scenario this change in the mean energy of D vanishes. By contrast, we have post-measured work extraction due to the change in S+B correlations by a measurement after time τ : D W ext = hHS+B (τ )iD − hHS+B (0)iD 6= Wmeas = 0. (A4) b) Measurement cost for degenerate detector In Ref [11] a detector D consisting of degenerate states, initially in thermal equilibrium, performs a selective measurement (SM) on S in order to extract work in a cycle as Maxwell’s demon. Since a SM reduces the entropy of S, the entropy of D must correspondingly rise. Hence, D must not be in a maximal entropy state prior to the measurement. Neither can its temperature be the same as that of S, i.e., T. The required lowering of the temperature and entropy of D are achieved by isothermally (quasistatically) lifting the degeneracy of the levels of D at a cost ∆ED . The maximal work extraction by D is [11] the free energy lost by D WD = ∆ED + T ∆S (A5) Although (A5) superficially looks similar to our Eq. (9), it is essentially different in that ∆ED is determined by the detector temperature, which is irrelevant for the impulsive NSM used in our scenario to change hHSB i. In particular, ∆ED = WD = 0 at T = 0 in Ref. [11], as opposed to our Eqs. (3) and (9), where the measurement cost does not change hHD i = 0, but hHSB i changes even at T = 0. The temperature and entropy restrictions on D in Ref. [11] do not exist in our model (see Sec. VIII). The reason for these differences is that by venturing beyond the S-B separability paradigm we enable the entropy of S to be reduced at the expense of ∆SS+B reflecting S-B correlation change by the measurement, whereas in Ref. [11] the bath B does not affect the entropy balance during the measurement under the S-B separability paradigm. 2 Maximal work in a post-measurement cycle The post-measured S+B supersystem is thus in a nonequilibrium state that can be harnessed to perform work on its way back to equilibrium. The maximal work possible is extractable in a cycle that is thermodynamically reversible apart from the measurement “stroke”[35]. Were ρ′S+B a thermal (Gibbs) state (for some temperature), we could use standard processes[24] to “close the cycle” by a reversible process, and the maximal extractable work would then be given by the difference in the Helmholtz free energy between ρ′S+B and the original equilibrium state[23]. However, since ρ′S+B is not a Gibbs state, it is not clear that this upper bound on work is appropriate. To find a thermodynamically reversible process that would bring the post-measured state back to equilibrium, we resort to a nonstandard procedure that allows maximal work extraction. Namely, we envision that the supersystem S + B is embedded in a Markovian bath BM , at the same temperature as B, T = β1 . The supersystem S + B equilibrates with BM at time tEq , say via coupling between B and BM . Since BM is Markovian we can neglect its correlation with S + B. Yet the correlations between S and B persist much longer, because B is nonMarkovian, with correlation (memory) time tc ≫ tEq . The stages of this nonstandard, optimal cycle are as follows (Fig. 5): (1) The initial equilibrium state −βHSB ρBM ⊗ ρSB , where ρSB = ρEq = e Z , undergoes at time t = 0 a measurement of S (Eqs. (8) and (9)) that leaves S + B + BM in (approximately) the product state ρBM ⊗ ρS ⊗ ρB . (2) We next stabilize ρS ⊗ ρB by making a sudden change of the S+B Hamiltonian: HS+B → ′ −βHS+B ′ HS+B , so that the overall state becomes ρBM ⊗ e Z ′ . ′ The change of work is Wstab = hHS+B i−hHS+B i. We are guaranteed that such stabilization is possible [35, 36], but it may not be feasible if we only act on S (by modulating the qubit level-distance). (3) Subsequently, we change ′ HS+B → HS+B by modulation over time τS+B ≫ tEq , i.e. quasistatically and isothermally as concerns BM , un−βHS+B til we attain the original equilibrium state ρBM ⊗ e Z and thereby close the cycle. The work change during the isothermal stage is Wisot = ∆Eisot − T ∆Sisot . The overall optimal cycle is described as follows: (i) In the first stroke, the energy cost of the measurement is (see Eq (9)) ∆Emeas = hHiρ′ −hHiρ . The NSM increases the VN entropy: ∆Smeas = S (ρ′ ) − S (ρ) . (ii)In the next (return) stroke, the stabilization (sudden) Hamiltonian change implies that work is performed by the system: Wsudden = hHiρ′ − hH ′ iρ′ and the entropy is unchanged. (iii) In the last stroke, the energy change of the supersystem is ∆Eisotherm = hHiρ − hH ′ iρ′ , ∆Sisotherm = −∆Smeas and the extracted work during this stroke is [24]: Wisotherm = −∆Eisotherm + T ∆Sisotherm . (iv) Fi- 10 tcycle<tc Η'S+B ρS+B ΗS+B ρS⊗ρB tionary state (detailed thermal balance) at temperature kB T = β1 eq Re (t)ρeq ee (t) = Rg (t)ρgg (t) , 3 2 ρ'S+B −1 ρeq (t) exp{−βEj (t)} jj (t) = Z tEq ΗSD τS+B 1 (ρS+B)Eq j ∈ (g, e) (A7) Z(t) being the normalization constant. To prove this result (which is consistent with known results) consider the following auxiliary expression X ρ̇jj (ln ρjj − ln ρeq jj ) = j ρee ρgg − (Rg ρgg − Re ρee ) ln eq = ρeq ρgg ee eq Rg ρgg (x ln y − x ln x + x − y)+ (Rg ρgg − Re ρee ) ln FIG. 5. Work extraction by measurements from S-B correlations: Optimal cycle that consists of 3 stages (see text): 1measurement, 2-stabilization, 3-modulation. nally, combining these results for all strokes one gets the expression in Eq.(10) for Wextracted = Wsudden + Wisotherm 3 No work can be extracted from a single Markovian-bath engine in a closed cycle We consider the evolution of the TLS state, ρS (t), that is diagonal in the energy basis, with parametrically timedependent energy levels Ee (t) − Eg (t) = ω(t): ρ̇ee (t) = Rg (t)ρgg − Re (t)ρee ρ̇ee (t) = −ρ̇gg (t) (A6) Let us now assume Markovian properties: A) Rg(e) (t) ≥ 0; B) Gibbs probability distribution in a sta- Re ρeq ee (y ln x − y ln y + y − x) ρgg and y = ρρee eq . Notice, that ρeq gg ee ≤0 (A8) Rg ρeq gg (x − y) + where x = Re ρeq ee (y −x) = 0 due to assumption B). The inequality in (A8) is obtained from the relation a ln a−a ln b+b−a ≥ 0 (for a, b ≥ 0) and assumption A). It implies P the following inequality for the entropy S(t) = −kB j ρjj (t) ln ρjj (t): Ṡ = −kB X j ρ̇jj ln ρjj ≥ −kB Q̇ = X X ρ̇jj ln ρeq jj = j 1 Q̇ , T (A9) ρ̇jj Ej j where we used the fact that d P ln Z(t) dt j ρjj = 0. P j ρ̇jj ln Z(t) = Since for a closed cycle the entropies and internal energies in the initial and final states of the system are equal, W = Q ≤ 0 (which is the second law of thermodynamics). This means that we cannot extract work from a single Markovian bath engine.