Selection-rule blockade and rectification in quantum heat transport
SSelection-rule blockade and rectification in quantum heat transport
Teemu Ojanen ∗ Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany (Dated: November 21, 2018)We introduce a new thermal transport phenomenon, a unidirectional selection-rule blockade,and show how it produces unprecedented rectification of bosonic heat flow through molecular ormesoscopic quantum systems. Rectification arises from the quantization of energy levels of the con-duction element and selection rules of reservoir coupling operators. The simplest system exhibitingthe selection-rule blockade is an appropriately coupled three-level system, providing a candidate fora high-performance heat diode. We present an analytical treatment of the transport problem anddiscuss how the phenomenon generalizes to multilevel systems.
PACS numbers: 44.10.+i, 05.60.Gg, 63.22.-m, 44.40.+a
Heat conduction in nanoscale structures has become anactive field of research enjoying constantly increasing at-tention. Electronic properties have been studied in greatdetail in the last three decades and are better under-stood than thermal properties. The main reason for thishas been experimental challenges to measure and con-trol thermal properties accurately. Recently the field hasseen breakthroughs such as measurements of quantizedheat transport [1, 2] and realization of hybrid structuresprobing atomic-level heat transport properties [3].Experimental developments have increased theoreticalinterest to explore fundamental limits of thermal phe-nomena and devices based on their applications. Ther-mal rectification, i.e. asymmetry of heat current whenthe temperature bias is inverted, has been actively stud-ied in this context. Rectification has been observed so farin two experiments [4, 5] and has a significant applica-tion potential. There exists various theoretical proposalsof how to realize rectification in phononic [6, 7], pho-tonic [8] and other hybrid structures [9]. However, inmost proposals rectification is modest and limits of per-formance are unknown. The purpose of this paper is tointroduce a novel heat transport phenomenon, an asym-metric selection-rule blockade, enabling an unparalleledrectification performance. The phenomenon provides anexample of an intricate interplay of quantum-mechanicaland thermal properties in nanoscale structures.The simplest system exhibiting the selection-ruleblockade is a three-level system with a strongly unevenenergy-level separations. A crucial ingredient is that thetwo baths are coupled to the system so that one can onlyinduce transitions between the close-lying states and theother can effectively induce only the two larger transi-tions. The baths can exchange energy only when thebath coupling far-apart states has sufficiently high tem-perature to create excitations. In suitable temperatureregime the forward biasing leads to sequential heat flowwhile the reverse bias current, due to higher-order pro-cesses, is very strongly suppressed. This one-directional suppression of heat flow is the defining property of theselection-rule blockade. The paper is organized as follows. First we introducethe three-level model and show analytically how rectifi-cation arises from the selection rules. Then we estimatethe magnitude of the leakage current that sets the limitsof performance of rectification and discuss how the se-lection rules can be realized without special symmetries.We conclude by outlining how the selection-rule block-ade generalizes to multilevel systems and summarize ourresults. T L T R FIG. 1: Studied model consists of two heat baths at differenttemperatures coupled through a three-level system. The bluearrow represents transitions induced by the left reservoir andthe red arrows correspond to Golden-Rule transitions inducedby the right reservoir. The dashed arrow represents higher-order transitions via the virtual state by the right reservoir.
In the following we consider thermal conduction in thesetup depicted in Fig. 1. The system consists of threeparts, two reservoirs and a three-level system that me-diates heat. We call a configuration forward biased if T R > T L and reverse biased in the opposite case. Intu-itively one expects that if temperatures T L / R are muchsmaller than the allowed transition energies, heat flow iseffectively blocked. This is confirmed below as we showhow the intrinsic asymmetry in the strength of the block-ade gives rise to strong rectification. The Hamiltonian ofthe system is H = H L + H R + H C + H LC + H RC (1)where H L / R characterize the reservoirs and the centralHamiltonian is H C = (cid:15) | (cid:105)(cid:104) | + (cid:15) | (cid:105)(cid:104) | + (cid:15) | (cid:105)(cid:104) | . (2) a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l The level separation of the central part E = (cid:15) − (cid:15) , E = (cid:15) − (cid:15) is assumed to be unequal E /E (cid:28) H LC = X L ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) H RC = X R ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) , (3)where X L / R are Hermitian operators in the reservoirHilbert spaces. The specific form of the couplings im-ply that the left reservoir couples only the groundstateand the first excited state in contrast to the right reser-voir which couple all the energy levels. Applying theFermi Golden Rule one can calculate the transition ratesbetween the energy levelsΓ → = S X L ( ω ) / (cid:126) , Γ → = S X R ( ω ) / (cid:126) , Γ → = S X R ( ω + ω ) / (cid:126) , (4)where S X L / R ( ω ) = (cid:82) ∞−∞ dte iω t (cid:104) X L / R ( t ) X L / R (0) (cid:105) and ω / = E / / (cid:126) . The correlation functions are evaluatedin the absence of the couplings at the respective reser-voir temperatures. Information about H L and H R aswell as information about the structure of the reservoirsis encoded in the noise S X L / R ( ω ) so our discussion is in-dependent of the precise nature of the reservoirs so far. Itshould be noted that even though the right reservoir caninduce direct transitions between the ground state andthe first excited state it happens only in higher-orders.In the lowest order the allowed transitions are separate inthe different reservoirs. The corresponding inverse tran-sitions are obtained by inverting the sign of the frequencyargument, for example Γ → = S X L ( − ω ) / (cid:126) . Noise isrelated to the retarded X -correlation function and theBose-Einstein distribution n ( ω ) through the Fluctuation-Dissipation Theorem S X L / R ( ω ) = A L / R ( ω )(1 + n L / R ( ω )) , (5)where A L / R ( ω ) = − (cid:104) X L / R X L / R (cid:105) r ( ω ) are spectraldensities of the reservoirs defined through the Fouriertransform of the retarded function (cid:104) X L / R ( t ) X L / R (0) (cid:105) r = − iθ ( t ) (cid:104) [ X L / R ( t ) , X L / R (0)] (cid:105) [10]. Once the transitionrates are known, the steady-state occupation probabil-ities follow from the detailed-balance equations − P ( i ) (cid:88) j (cid:54) = i Γ i → j + (cid:88) j (cid:54) = i P ( j )Γ j → i = 0 . (6)Employing the rate-equation formulation [10], heat cur-rent can be calculated, say, between the left reservoir andthe central system yielding J = E [ P (1)Γ → − P (0)Γ → ] . (7)The probabilities required to evaluate (7) can be solvedin terms of the transition rates as P (0) = (Γ → Γ → + Γ → Γ → + Γ → Γ → ) /CP (1) = (Γ → Γ → + Γ → Γ → + Γ → Γ → ) /C, (8) with C = Γ Γ + Γ Γ + Γ Γ + Γ (Γ + Γ + Γ )+ Γ (Γ + Γ + Γ ) , (9)where we have introduced notation Γ i → j = Γ ij . Insertingthe probabilities and the transition rates in Eq. (7) weobtain an explicit expression for the current J = E Γ Γ Γ C ( e β L E − e β R E ) . (10)Result (10) indicate that the thermal window ( e β L E − e β R E ) determined by the reservoir temperatures and thesmaller energy separation E plays an important role inthe transport process. In the limit k B T R (cid:28) E current J/J T h / E T h / E J T R = T h J T R = T c FIG. 2: Rectification efficiency as a function of the higheroperation temperature. The different curves correspond tovalues T c = E , and E /E = 5 (red), E /E = 7 (blue), E /E = 9, (green) and E /E = 11 (black). Inset illus-trates the corresponding forward bias (solid lines) and re-verse bias currents (dashed lines, almost zero) in the unitsof J = E A / (cid:126) . (10) takes a simple form J T R (cid:28) E = E A R ( ω )1 + A R ( ω ) A R ( ω + ω ) e − β R ( E + E ) ( e β L E − e β R E )(1 + e β L E ) . (11)Result (11) shows that heat current is exponentially sup-pressed in the considered limit. This is an indication thatthe system acts as a high-performance thermal rectifierwhen the lower operation temperature is much smallerthan E while the higher temperature is comparable to it.To make rectification more explicit, consider a situationwhere the hot reservoir is at temperature k B T h ∼ E andthe cold one is at k B T c ∼ E . Furthermore, to simplifyexpressions, let us assume that the effective couplings areof the same order A L ( ω ) ≈ A R ( ω ) ≈ A R ( ω + ω ) = A .Starting from the expression (10) one can estimate thatthe fraction of the bias-inverted heat currents is (cid:12)(cid:12)(cid:12)(cid:12) J T R = T c J T R = T h (cid:12)(cid:12)(cid:12)(cid:12) = c e − E /E ∼ c e − T h /T c , (12)where c is a numerical factor of the order of unity. ThusEq. (12) shows that in the studied temperature regimethe fraction of forward and reverse bias currents is expo-nentially small in T h /T c . Results (10)-(12) illustrate theessence of the selection rule blockade and confirm quanti-tatively the intuitive physical picture. More comprehen-sive picture of transport properties can be obtained byplotting current (10) and the rectification ratio at differ-ent forward and reverse bias values, see Fig. 2. Reversebias current is rapidly suppressed as the ratio E /E is increased and effectively vanishes in the temperaturewindow k B T c ≤ E , k B T h > E for E /E >
3. Impor-tantly, forward bias current is finite and increasing in theregion of optimal rectification.In the derivation of Eq. (10) we only took into ac-count the Golden-Rule transitions. Deep in the block-aded regime k B T R (cid:28) E the lowest-order current van-ishes exponentially as shown in Eq. (11) and one is nat-urally lead to consider higher-order transitions. The sit-uation is similar to electronic transport in the Coulombblockade model where charging effects suppress conduc-tance (with the difference that the selection-rule blockadeis one directional). At low temperatures and biases elec-tric current through a small island vanishes exponentiallyin the lowest order in the perturbation theory. However,in the next order one obtains cotunneling processes whereelectrons are transferred via virtual intermediate statespartly lifting the blockade [11]. In our model the leadinghigher-order processes contributing to the reverse leak-age current in the blockaded regime correspond to theright reservoir-induced transitions via the highest-energystate as depicted in Fig. 1. These rates can be calculated,for example, by the standard T -matrix expansion to thesecond order [10]. Let us assume that the right reservoirconsists of noninteracting bosonic modes and that thecoupling operator is of the form X R = (cid:80) j ∈ R c j ( b j + b † j )where b j , b † j are canonical bosonic operators of the mode ω j . In the low-temperature limit T R → (2)1 → = 2 π (cid:126) (cid:90) ω dωA R ( ω ) A R ( ω − ω ) × (cid:12)(cid:12)(cid:12)(cid:12) E − (cid:126) ω + 1 E − E + (cid:126) ω (cid:12)(cid:12)(cid:12)(cid:12) ≈ π (cid:126) E (cid:90) ω dωA R ( ω ) A R ( ω − ω ) . (13)The rate is proportional to a typical second-order energydenominator ∼ /E suppressing the transition. As-suming that the spectral density has a power-law form A R ( ω ) ∝ ω d ( d = 1 for an ohmic bath), the leading con- tribution becomesΓ (2)1 → = 8 π d ( (cid:126) Γ → | T R =0 ) E (cid:18) E E (cid:19) d ω . (14)The second and the third factor are much smaller thanunity so (14) is small compared to the level separation ω . However, the rate is only algebraically suppressed by( E /E ) d . At finite temperature T R ∼ E the relevantpower law exhibits a crossover to ( k B T R /E ) d . Now wecan estimate the leakage current under the blockade by J r = − E P (1)Γ (2)1 → = − E Γ (2)1 → e β L E + Γ (2)1 → Γ → . (15)Result (15) shows that the true rectification efficiency isalgebraically, not exponentially, suppressed. It is clearthat the leakage current remain finite as long as there isthermal coupling between the two reservoirs and that agood rectifier is characterized by its ability to suppressthe parasitic reverse bias processes. Nevertheless, cur-rent (15) can be made arbitrarily small by decreasingmax { E /E , k B T R /E } .Usually selection rules are associated with symmetriesof the Hamiltonian leading to vanishing matrix elementsof the perturbation operator between unperturbed states.In applications finding a candidate system with suitablesymmetries and couplings is a nontrivial task. However,since the transitions of the different reservoirs are wellseparated in energy, there exists an alternative route toobtain the selection rules without invoking symmetries ofthe Hamiltonian. Golden-Rule rates consists of matrix el-ements of perturbation operators and a summation overa relevant phase space. Both factors can effectively im-pose selection rules in the system. Typically phase-spaceselection rules arise from the fact that the reservoir spec-tral density is nonvanishing only in a restricted frequencywindow outside which the reservoir cannot induce tran-sitions. In solid-state applications suitable reservoirs canbe realized by dissipative vibrational modes of phononicor photonic nature which filter out frequencies far fromthe resonance. Assuming that the center element cou-ples linearly to the modes, the reservoir parts of the cou-plings (3) take the form X L / R = c L / R ( b † L / R + b L / R ), where b † L / R , b L / R are the creation and annihilation operators ofthe reservoir modes. The retarded Green’s function of avibrational mode is G r ( ω ) = (( g r ) − + Σ r ( ω )) − , where( g r ) − = ( ω − ω ) / ω is the inverse free Green’s func-tion, ω is the frequency of the mode and Σ r ( ω ) is the re-tarded self-energy due to dissipation. Dissipation can berealized by coupling the mode to a bosonic bath, in whichcase the spectral densities A L / R ( ω ) = − g / R Im G r ( ω )of the reservoirs can be solved exactly [12] yielding A L / R ( ω ) = c / R (2 (cid:126) ω L / R ) A BL / R ( ω ) (cid:126) ( ω − ω / R ) + ( A BL / R ( ω ) ω L / R ) , (16)where ω L / R are the resonant frequencies of the reservoirsand A BL / R ( ω ) are the spectral densities responsible for dis-sipation of the reservoir modes. As illustrated in Fig. 3(left), if the left reservoir has a resonance frequency at ω L = ω and the right reservoir at ω R = ω + ω / ω ,the spectral densities have no overlap. Thus the reser-voirs can effectively induce transitions only as indicatedin Fig. 1. If the reservoirs are electromagnetic in naturethe desired spectral densities (16) are achieved by cou-pling the center element to dissipative LC-circuits whoseelectromagnetic fluctuations are restricted to a narrowband around the resonance [13]. In phononic systemsthe same effect could be realized by coupling large bathsto the central element through small vibrating bridges. H RC E E E E w w w w A L A R H LC } FIG. 3: Left: Spectral density of the left (right) reservoiris completely suppressed at frequencies ω (cid:29) ω ( ω (cid:28) ω ),effectively imposing the desired selections rules. Right: Low-lying spectrum of the Jaynes-Cummings model and the al-lowed transitions induced by the left (blue arrows) and theright (red arrows) reservoir. The selection-rule blockade is a not restricted to thestudied three-level system. Basic requirements are thatlevel separations of a system can be divided to smalland large intervals coupled to separate baths, and thatcombined effect of two large transitions corresponds toa small one as in the tree-level model. A prominent ex-ample fulfilling the requirements is a two-level systemcoupled to a harmonic oscillator described by a resonantJaynes-Cummings- type Hamiltonian [14] H JC = (cid:126) Ω (cid:18) a † a + 12 (cid:19) + (cid:126) Ω2 σ z + (cid:126) g a † + a ) σ x , (17)where the interaction is assumed to be small g (cid:28) Ω. Ex-cited states of the Hamiltonian (17) consists of doublets | E n (cid:105) = | n, ↓(cid:105) ± | n − , ↑(cid:105) where the energy separation ofcenters of adjacent doublets is approximately E = (cid:126) Ωand the separation within a doublet is E n = √ n (cid:126) g , seeFig. 3 (right). Since for the low-lying doublets energysplittings are small E n (cid:28) E , the selection-rule blockadecan be established if the left reservoir couples levels onlywithin doublets and the right reservoir couples states indifferent doublet. This requirement is satisfied in thelowest order by H RC = X R ( a † + a ) and H LC = X L σ z ,yielding immediately the desired selection rules. Anal-ogous to the three-level model, if the temperatures of the reservoirs are k B T L ∼ (cid:126) g , k B T R ∼ (cid:126) Ω heat flow isefficient, while in the reverse biased case k B T L ∼ (cid:126) Ω, k B T R ∼ (cid:126) g excitations induced by the right reservoir areforbidden and there exists only a weak leakage currentdue to higher-order processes.In conclusion, we introduced a fundamental heat trans-port phenomenon, the selection-rule blockade, based onthe quantum nature of the central system and selectionrules of the bath coupling operators. The phenomenonenables a high rectification efficiency in molecular andmesoscopic structures making it promising for future de-vice applications. The simplest system exhibiting anasymmetric blockade is a three-level model which we ana-lyzed in detail. We proposed a scheme to impose requiredselection rules without special symmetry properties of theHamiltonian and discussed how the selection-rule block-ade can be generalized to multilevel systems. ∗ Correspondence to [email protected][1] K. Schwab, E. A. Henriksen, J. M. Worlock, and M. L.Roukes, Nature (London) , 974 (2000).[2] M. Meschke, W. Guichard, and J. P. Pekola, Nature ,187 (2006).[3] V. P. Carey, G. Chen, C. Grigoropoulos, M. Kaviany, andA. Majundar, Nanoscale and Microscale ThermophysicalEngineering , 1 (2008).[4] C. W. Chang, D. Okawa, A. Majumdar and A. Zettl,Science , 1121 (2006).[5] R. Scheibner, M. K¨onig, D. Reuter, A. D. Wieck,C. Gould, H. Buhmann, and L. W. Molenkamp, NewJ. Phys. , 083016 (2008).[6] M. Terraneo, M. Peyrard, and G. Casati , Phys. Rev.Lett. , 094302 (2002); B. Li, L. Wang and G. Casati,Phys. Rev. Lett. , 184301 (2004).[7] Bambi Hu, Lei Yang, and Yong Zhang, Phys. Rev. Lett. , 124302 (2006).[8] T. Ruokola, T. Ojanen, and A.-P. Jauho, Phys. Rev. B , 144306 (2009); T. Ojanen and T. T. Heikkil¨a, Phys.Rev. B , 073414 (2007).[9] D. Segal and A. Nitzan, Phys. Rev. Lett. , 034301(2005); D. Segal, Phys. Rev. Lett. , 105901(2008); L. -A. Wu and D. Segal, Phys. Rev. Lett. ,095503 (2009); L. -A. Wu, C. X. Yu, and D. Segal,arXiv:0905.4015.[10] H. Bruus and K. Flensberg, Many-Body Quantum Theoryin Condensed Matter Physics (Oxford University Press,Oxford, 2004).[11] D. V. Averin and Yu. V. Nazarov, Phys. Rev. Lett. ,2446 (1990).[12] T. Ojanen and A. -P Jauho, Phys. Rev. Lett. , 155902(2008).[13] A. O. Niskanen, Y. Nakamura, and J. P. Pekola, Phys.Rev. B , 174523 (2007).[14] H. P. Breuer and F. Petruccione, The Theory of OpenQuantum Systems (Oxford Univeristy Press, Oxford,2002).[15] Widths of the resonances of A L / R ( ω ) are determined bythe strength of dissipation through A BL / R ( ωω