We show that the heat transport between two bodies, mediated by
electromagnetic fluctuations, can be controlled with an intermediate quantum
circuit - leading to the device concept Mesoscopic Photon Heat Transistor
(MPHT). Our theoretical analysis is based on a novel Meir-Wingreen-Landauer
type of conductance formula, which gives the photonic heat current through an
arbitrary circuit element coupled to two dissipative reservoirs at finite
temperatures. As an illustration we present an exact solution for the case when
the intermediate circuit can be described as an electromagnetic resonator. We
discuss in detail how the MPHT can be implemented experimentally in terms of a
flux-controlled SQUID circuit.Comment: 4 pages, 3 figure

Jauho, Antti-Pekka

Ojanen, Teemu

English

arXiv

We show that the heat transport between two bodies, mediated by electromagnetic fluctuations, can be controlled with an intermediate quantum circuit-leading to the device concept of a mesoscopic photon heat transistor (MPHT). Our theoretical analysis is based on a novel Meir-Wingreen-Landauer-type of conductance formula, which gives the photonic heat current through an arbitrary circuit element coupled to two dissipative reservoirs at finite temperatures. As an illustration we present an exact solution for the case when the intermediate circuit can be described as an electromagnetic resonator. We discuss in detail how the MPHT can be implemented experimentally in terms of a flux-controlled SQUID circuit

Ojanen, T.

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Mesoscopic photon heat transistorOjanen, T.; Jauho, Antti-PekkaPublished in:Physical Review LettersLink to article, DOI:10.1103/PhysRevLett.100.155902Publication date:2008Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Ojanen, T., & Jauho, A-P. (2008). Mesoscopic photon heat transistor. Physical Review Letters, 100(15), 155902.DOI: 10.1103/PhysRevLett.100.155902Mesoscopic Photon Heat TransistorTeemu Ojanen1,2,* and Antti-Pekka Jauho3,41Low Temperature Laboratory, Helsinki University of Technology, P.O. Box 2200, FIN-02015 HUT, Finland2Institut fu¨r Theoretische Physik, Freie Universitet Berlin, Arnimallee 14, 14195 Berlin, Germany3Department of Micro- and Nanotechnology, Technical University of Denmark, DTU Nanotech, Building 345 East,DK-2800 Kongens Lyngby, Denmark4Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland(Received 17 December 2007; published 17 April 2008)We show that the heat transport between two bodies, mediated by electromagnetic fluctuations, can becontrolled with an intermediate quantum circuit—leading to the device concept of a mesoscopic photonheat transistor (MPHT). Our theoretical analysis is based on a novel Meir-Wingreen-Landauer-type ofconductance formula, which gives the photonic heat current through an arbitrary circuit element coupledto two dissipative reservoirs at finite temperatures. As an illustration we present an exact solution for thecase when the intermediate circuit can be described as an electromagnetic resonator. We discuss in detailhow the MPHT can be implemented experimentally in terms of a flux-controlled SQUID circuit.DOI: 10.1103/PhysRevLett.100.155902 PACS numbers: 66.70.f, 44.40.+a, 72.70.+m, 73.23.bProblems involving interactions between quantum par-ticles and electromagnetic fields have a long and richhistory. During the last decades developments in meso-scopic physics have provided unprecedented possibilitiesto engineer the electron-photon interactions. A major ad-vantage in mesoscopic systems is their versatility whichallows a high level of control in their design and operation.Recent advances in this field include impressive quantumstate manipulations involving single photons in circuitcavity QED experiments [1] and a demonstration ofsingle-channel photon heat transport [2].At low temperatures, when phonon modes become ef-fectively frozen, photonic heat conduction becomes thedominant channel for thermal transport [2,3]. In thisLetter we study photonic heat transport in a structureconsisting of two reservoirs coupled via an intermediateelectric circuit. The reservoirs are assumed to behave aslinear dissipative circuit elements and are thereby fullycharacterized by their response functions and tempera-tures. Furthermore, at low temperatures the wavelengthsof relevant field fluctuations are much longer than a typicalsystem size so the reservoirs can be effectively consideredas lumped elements. Within these assumptions we applythe Caldeira-Leggett procedure and model the reservoirs ascontinuous distributions of harmonic oscillators. By apply-ing nonequilibrium Green’s function methods we derive aformally exact Meir-Wingreen-Landauer- type formula [4]for the heat current through the structure. Our formulainvolves the noise power of the intermediate circuit, inthe presence of the coupling to the leads, and serves as ageneral starting point for solving the heat transport prob-lem. We find an exact solution for the heat current flowingthrough an electromagnetic resonator circuit and showexplicitly that, in analogy with the transistor effect incharge transport problems, the heat flow through the struc-ture can be modulated by applying an external control tothe middle circuit. We suggest that an experimental dem-onstration of the heat-transistor action can be achieved byusing a superconducting quantum interference device(SQUID) circuit as the tunable resonator. Thus, the mag-netic flux-controlled heat current is a photonic analogue tothe gate voltage controlled electronic heat current recentlydemonstrated in Ref. [5].Now we turn to the technical derivation of the generalformula for the heat current in the system consisting of aleft and a right reservoir, and an arbitrary quantum circuitbetween them (Fig. 1). We treat the problem by employinga nonequilibrium Green’s function method analogous tothose used earlier in electron and heat transport problems[4,6–9]. The reservoirs are described by quadratic bosonfields and, according to the Caldeira-Leggett prescription,can be thought of as arbitrary linear electric circuits bychoosing specific distributions of frequencies !j and cou-plings gj (introduced below) [10,11]. The total Hamil-tonian is assumed to be of the form H  HL HR HM HC, where=TL TRYL (ω) YR (ω)FIG. 1 (color online). Temperature gradient between the leftand right reservoirs induces photonic heat current through anarbitrary quantum circuit coupled to them. Heat flows from leftto right when TL > TR. The reservoirs are assumed to behave aslinear dissipative elements and they couple only through themiddle circuit. In principle a direct coupling between the reser-voirs always exists but in practice it can be made negligible byan appropriate sample design.PRL 100, 155902 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending18 APRIL 20080031-9007=08=100(15)=155902(4) 155902-1 © 2008 The American Physical Society HL=R Xj2L=R@!ja^yj a^j  1=2; (1)and the inductive coupling term is HC  MI^i^L  i^R; (2)which involves the current operators for the central regionI^ and for the reservoirs i^L=R  Pj2L=Rgja^j  a^yj , respec-tively. The specific form of the Hamiltonian and the currentoperator of the middle region do not need to be specified atthis point (below we shall treat a specific example). Themutual inductances between the middle circuit and theleads are assumed to be equal, though this is not necessaryin the following derivation. A capacitive coupling betweenthe reservoirs and the middle system can be treated in closeanalogy with the inductive coupling studied here [12].The reservoir Hamiltonians HL=R do not commute withthe total Hamiltonian H, thus giving a rise to an energyflow in the structure. This energy flow is characterized by aheat current JL=R defined as JL=Rt  h _HL=Ri  iMXj2L=Rgj!jha^jtI^ti  H:c: 2MRe Xj2L=Rgj!jG<j t; t; (3)where G<j t; t0  iha^jt0I^ti. The transport problem isreduced to finding the lesser Green’s function G<j t; t0which can be derived by the equation-of-motion technique[6,13]. Following the standard prescription [6], we firstconsider the equilibrium zero-temperature time-orderedcorrelation functions, which obey i@t0 !jhTa^jt0I^ti Mgj@hTI^t0I^ti; (4)or, after a formal integration, hTa^jt0I^ti Mgj@Zdt1hTI^tI^t1iDjt1  t0; (5)where Djt1  t0 is the free reservoir Green’s function. Innonequilibrium, this equation holds on the Keldysh con-tour, and using the analytical continuation rules known asLangreth’s theorem [6], we obtain G<j t; t0 Mgj@Zdt1hI^tI^t1irD<j t1  t0 hI^tI^t1i<Daj t1  t0; (6)where the superscripts r, a, and < stand for ‘‘retarded,’’‘‘advanced,’’ and ‘‘lesser,’’ respectively. Explicitly, thecurrent correlation functions are hI^tI^t0ir  itt0hI^t; I^t0i and hI^tI^t0i<  ihI^t0I^ti. In asteady state G<j t; t0  G<j t t0, and it is convenientto introduce the Fourier transform G<j ! Mgj@hI^ I^ir!D<j !  hI^ I^i<!Daj !; (7)where Daj !  1=!!j  i, D<j ! i2n!j!!j, and n! is the Bose function.We thus obtain JL  2Xj2LM2g2j!j2@Z 11d!ImhI^ I^ir!2n!j	 !!j  ImhI^ I^i<!!!j: (8)To make a connection to quantities with a clear physicalinterpretation, it is useful to express Eq. (8) in terms of thenoise power. The noise power for an observable A^ isdefined as SA! R11 dtei!tt0hA^tA^t0i, so the cur-rent noise in the left lead is SiL! Xj2Lg2j n!j2!!j n!j  12!!j: (9)In terms of the noise power Eq. (8) becomes JL  2M2Z 10d!!2@ImhI^ I^ir!SiL! ImhI^ I^i<!	 12SiL!SiL!: (10)Also the I^-correlation functions can be written in terms ofnoise power: ImhI^ I^ir!  12 SI!  SI! andImhI^ I^i<!  SI!, which yields JL  M2Z 10d!!2@SI!SiL!  SI!SiL!:(11)A similar expression holds for JR; in steady-state situationsJ  JL  JR. Formulas (10) and (11) contain the freereservoir noise functions which can be obtained straight-forwardly from the admittances YL=R! and temperaturesTL=R by applying the fluctuation-dissipation theorem [10]: SiL=R!  ReYL=R!@!cothL=R@!=2  1: (12)With the help of Eq. (12), the heat current (11) can bewritten as JL Z 10d!!2M22f2SI!  SI!ReYL!nL! SI!2ReYL!g: (13)If the lead admittances share the same frequencydependence, YL!  cYR! with some constant c, thestationary heat current can be expressed as J  JL=c1  cJR=c 1 and cast into the Landauer form: J  M2Z 10d!!22SI!  SI!	 2ReYL!ReYR!ReYL!  ReYR! nL!  nR!: (14)Equations (13) and (14) are the main formal results of thisPRL 100, 155902 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending18 APRIL 2008155902-2Letter. The formulas are valid for a generic middle circuitHamiltonian which may contain an arbitrary nonlinearinteraction. The main restriction of our approach is theassumption of linearity of the leads. However, metallicleads satisfy the assumption of linearity very accuratelyin relevant frequency scales so our treatment is valid for awide range of applications. We emphasize the strong ana-logue between our results and the Meir-Wingreen conduc-tance formula: here the ‘‘bosonic thermal window’’(nL  nR) replaces the ‘‘fermionic voltage window’’ (nFL nFR), the real part of the admittance plays the role of thelinewidth function L=R, and the noise power SI! SI! of the central region, which needs to be evaluatedseparately, replaces the central region spectral function.Since SI contains information of the internal dynamics ofthe middle region in the presence of coupling to the reser-voirs, its evaluation may be difficult indeed, and only inspecial cases analytic progress can be expected. In thefollowing we calculate SI! for an electromagnetic reso-nator and illustrate how the reservoir admittances comeinto play through the self-energy of the middle circuit.Suppose now that the mediating quantum circuit is anelectromagnetic resonator with inductance L and capaci-tance C [see Fig. 2(a)]. The central region Hamiltonianthen takes the form HM  @!0b^yb^ 12, and the currentoperator is I^  I0b^ b^y, where I0 @!0=2Lpand!0  1=LCp; b^, b^y are bosonic creation and annihilationoperators, b^; b^y  1. To evaluate (14) one needs to findthe retarded function hI^tI^t0ir  it t0hI^t; I^t0iwhich can be expressed as the sum of four different re-tarded functions hI^ I^ir!  I20hb^b^yir!  hb^yb^ir!  hb^ b^ir! hb^yb^yir!: (15)Note the presence of the anomalous functions hb^tb^t0irand hb^ytb^yt0ir; they are important and come into playbecause the interaction term also includes the non-rotating-wave terms a^jb^ and a^yj b^y. By using an equation-of-motiontechnique [6,13] we find a closed set of equations !!0  ihb^b^yir  1 r!hb^b^yir  hb^yb^yir!!0  ihb^yb^yir  r!hb^b^yir  hb^yb^yir!!0  ihb^yb^ir  1r!hb^ b^ir  hb^yb^ir!!0  ihb^ b^ir  r!hb^ b^ir  hb^yb^ir; (16)where the retarded self-energy is given by r!  MI02@2Xj2L;Rg2j1!!j  i1!!j  i:(17)The reservoir admittances are given by the Kubo formula YL!  ihi^Li^Lir!@! i@!Xj2Lg2j1!!j i1!!j i; (18)so the self-energy is related to the reservoir admittancesthrough relation r!   iM2I20!@YL!  YR!: (19)The algebraic system (16) is readily solved yielding hI^ I^ir!  i@!!F! i M2!2YL!  YR!;(20)where we have introduced shorthand F!  i@=I20	!2 !20=2!0. By extracting the imaginary part ofEq. (20) and recalling the relation 2ImhI^ I^ir! SI!  SI! we find J Z 10d!24@!5M4ReYL!ReYR!j!F! M2!2YL!  YR!j2	 nL!  nR!: (21)The result (21) is valid for arbitrary reservoirs admittances,as long as the intermediate circuit can be described as anoscillator. It can be shown that the upper limit of heatcurrent (21) is given by the universal single-channel heatconductance quantum GQ  2k2BT=3h [2,3,9,14–16], anexact parallel with the electrical single-channel conduc-tance G0  2e2=h, which follows from the Meir-Wingreenformula for a noninteracting resonant level model at reso-nance [6].In the following we assume that the reservoirs areidentical and consist of a resistor, a capacitor, and aninductor in series. Thus the admittances can be written as+ +Φ =TL TRTL TRYL (ω) YR (ω)a)b)FIG. 2 (color online). An electromagnetic resonator as theintermediate circuit (a). In (b) the mediate resonator circuitwith a tunable inductance is realized by a dc SQUID with anexternal magnetic flux bias . The external flux can be used tomodulate the heat current through the structure and plays ananalogous role to the gate voltage in electronic transistor.PRL 100, 155902 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending18 APRIL 2008155902-3YL!  YR!  R11iQ!=!R!R=! , where R, Q, and !Rare an effective resistance, a quality factor, and a resonancefrequency of the reservoir. The net heat flow is determinedby five dimensionless parameters M2!0=LR, !0=!R, Q,@!0=kBTL, and @!0=kBTR, where TL and TR are tempera-tures of the reservoirs. By externally controlling the pa-rameters of the middle circuit it is possible to tune the heatflow through the structure. A practical way to realize this isto employ a dc SQUID as a middle circuit and apply amagnetic flux  through it [see Fig. 2(b)]. At low tem-peratures the system can be modeled by an LC oscillatorwith a tunable inductance L  L=j cos=0jwhere 0  h=2e is the flux quantum and L is determinedby the Ambegaokar-Baratoff critical current [17]. TheLC-oscillator description of the SQUID circuit is expectedto be accurate within realistic parameter values at theexperimentally verified crossover temperature Tcr 100 mK below which the photonic thermal conductanceshould dominate [2]. A numerical evaluation of Eq. (21) ispresented in Fig. 3, which shows the heat flow as a functionof the external bias flux . The maximum flow is obtainedat integer values of =0, whereas at half-integer valuesthe reservoirs are thermally decoupled. At low qualityfactors the reservoirs are more efficiently matched andthe system has a better thermal coupling. By increasingthe coupling parameter M2!0=LR the maximum value ofthe heat flow could be enhanced closer to the single-channel maximum value.In summary, we have studied photon heat transport innanoelectronic circuits based on a novel Meir-Wingreen-Landauer formula in a two-terminal geometry. This for-mula expresses the heat current in terms of the admittancesof the heat reservoirs, and the noise power of the interme-diated mesoscopic circuit. The formula can serve as astarting point for an analysis of photon heat transport in awide range of applications. As an example, we present anexact solution to the transport problem in the case of anelectromagnetic resonator playing the role of the mediatingcircuit. We propose a new device concept, a mesoscopicphoton heat transistor, where the heat current through thestructure can be strongly modulated by the external mag-netic flux through a dc-SQUID loop.We would like thank Antti Niskanen, Jukka Pekola, andTero Heikkila¨ for valuable discussions. The first authoracknowledges support from Academy of Finland.*To whom all correspondence should be addressed.teemuo@boojum.hut.fi[1] A. A. Houck et al., Nature (London) 449, 328 (2007).[2] M. Meschke, W. Guichard, and J. P. Pekola, Nature(London) 444, 187 (2006).[3] D. R. Schmidt, R. J. Schoelkopf, and A. N. Cleland, Phys.Rev. Lett. 93, 045901 (2004).[4] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512(1992).[5] O.-P. Saira et al., Phys. Rev. Lett. 99, 027203 (2007).[6] H. Haug and A.-P. Jauho, Quantum Kinetics in Transportand Optics of Semiconductors (Springer-Verlag, Berlin,Heidelberg, 1996).[7] J. S. Wang, J. Wang, and N. Zeng, Phys. Rev. B 74, 033408(2006).[8] T. Yamamoto and K. Watanabe, Phys. Rev. Lett. 96,255503 (2006).[9] T. Ojanen and T. T. Heikkila¨, Phys. Rev. B 76, 073414(2007).[10] M. H. Devoret, in Quantum Fluctuations, Proceedings ofthe Les Houches Summer School, Session LXIII (ElsevierScience, Amsterdam, 1997).[11] A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987).[12] A direct coupling between the reservoirs could easily beincluded in the formalism, but since it is can be madesmall by the device geometry, we will not discuss it here.[13] H. Bruus and K. Flensberg, Many-Body Quantum Theoryin Condensed Matter Physics (Oxford University Press,Oxford, 2004).[14] J. B. Pendry, J. Phys. A 16, 2161 (1983).[15] L. G. C. Rego and G. Kirczenow, Phys. Rev. Lett. 81, 232(1998).[16] K. Schwab, E. A. Henriksen, J. M. Worlock, and M. L.Roukes, Nature (London) 404, 974 (2000).[17] M. Tinkham, Introduction to Superconductivity (DoverPublications, New York, 2004).    0 1 2 3 40  1/32/31Φ/Φ0J/J max 0 0.5 1 1.5 20.20.40.60.8J/J maxkBTR /hω0FIG. 3 (color online). Heat flow through the SQUID structureas a function of applied flux , corresponding to parametersM2!0=LR  1, !0=!R  1 and TR  TL=2  @!0. The differ-ent curves correspond to different reservoir Q values (from top tobottom), Q  0, Q  0:1, Q  0:5, Q  2, and Q  10. Theheat flux is normalized with respect to the universal single-channel maximum value Jmax  2k2B6h T2L  T2R. In the insetthe heat flux is plotted as a function of the temperature of theright reservoir corresponding to the parameters !0=!R  1,TL  2@!0, Q  0:1, and   0 mod 0. The differentcurves correspond to parameter values (from bottom totop) M2!0=LR  0:5, M2!0=LR  1, M2!0=LR  3,M2!0=LR  5, and M2!0=LR  10.PRL 100, 155902 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending18 APRIL 2008155902-4

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