Blackbody Radiation Noise Broadening of Quantum Systems
Eric B Norrgard, Stephen P Eckel, Christopher L Holloway, Eric L Shirley
BBlackbody Radiation Noise Broadening of Quantum Systems
Eric B. Norrgard,
1, 2
Stephen P. Eckel, Christopher L. Holloway, and Eric L. Shirley Sensor Science Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA RF Technology Division, National Institute of Standards and Technology (NIST), Boulder, CO 80305 (Dated: 2021-01-11)Precision measurements of quantum systems often seek to probe or must account for the interac-tion with blackbody radiation. Over the past several decades, much attention has been given to ACStark shifts and stimulated state transfer. For a blackbody in thermodynamic equilibrium, thesetwo effects are determined by the expectation value of photon number in each mode of the Planckspectrum. Here, we explore how the photon number variance of an equilibrium blackbody generallyleads to a parametric broadening of the energy levels of quantum systems that is inversely propor-tional to the square-root of the blackbody volume. We consider the the effect in two cases whichare potentially highly sensitive to this broadening: Rydberg atoms and atomic clocks. We find thateven in blackbody volumes as small as 1 cm , this effect is unlikely to contribute meaningfully totransition linewidths. Precision spectroscopy of atomic and molecular sys-tems is the basis of numerous metrological applications[1, 2] and tests of fundamental physics [3, 4] and symme-tries [5, 6]. A correct determination of transition energiesrequires careful accounting of Stark shifts due to am-bient blackbody radiation (BBR). High accuracy BBRshift calculations account for higher order electric andmagnetic multipolar moments [7] in the scalar and tensorlight shifts [8]. For optical lattice clocks, shifts induced bythe trapping laser must also be considered to attain highaccuracy (including shifts which are quadratic and higherorder in electromagnetic field energy density, known ashyperpolarizability) [9].In addition to shifting energy levels, blackbody radi-ation can drive transitions to other levels. This effectcan be considered a non-parametric line broadening, asthe BBR-stimulated transition rate Γ
BBR adds to thespontaneous decay rate Γ sp [10]. BBR-stimulated de-cay is prominent in Rydberg atomic systems (typicallyΓ BBR (cid:38) s − at room temperature), and has also beenobserved in trapped molecules [11, 12], and molecularions [13]. Recently, we surveyed the BBR-induced statetransfer and levels shifts in molecular and atomic Ryd-berg systems, showing these are highly promising plat-forms for blackbody thermometry applications [2].Here, we consider an additional BBR-induced linebroadening mechanism that appears to have been over-looked until now. For a blackbody at temperature T encompassing a volume V , fluctuations in the BBR en-ergy produce a root-mean-square (RMS) deviation σ ∆ E i in the BBR shift ∆ E i . The broadening is proportionalto (cid:112) T /V when the energy differences (cid:126) ω ij associatedwith strongest transitions are all large compared to thethermal energy scale (i.e. (cid:126) ω ij (cid:29) k B T ). Unlike BBR-stimulated transitions, the BBR noise induces a para-metric broadening (i.e. the quantum state remains un-changed). This effect is general to all quantum systemswhich are radiatively coupled to a thermal bath. Here,we derive the effect of BBR noise on a quantum system,and then consider the size of the effect on two experimen- tal systems, atomic clocks and circular Rydberg atoms.For both systems, we find the BBR noise is too smallto significantly contribute to line broadening under anycurrently feasible scenario.Before deriving the BBR noise broadening, we sum-marize the relevant photon statistics and AC Stark in-teraction (see Appendix A for more details on photonstatistics). For photons with angular frequency ω , wewill use x = (cid:126) ω/k B T to write the partition function Z = (1 − e − x ) − . The m th moment of the photon num-ber n is (cid:104) ˆ n m (cid:105) = ∞ (cid:88) n =0 n m e − nx = 1 Z ∂ m ∂ ( − x ) m Z ( x ) . (1)The variance in photon number for each BBR mode is σ n = (cid:104) ˆ n (cid:105) − (cid:104) ˆ n (cid:105) = e x / ( e x − . In this work, a hat overa symbol indicates an operator.We consider an ideal blackbody of effective volume V and temperature T which surrounds a quantum systemin state i . In this work, we define the effective length (cid:96) and volume V of the cavity through the mode spac-ing. Physically, a blackbody cavity is created througha combination of a resistive wall material, light trap-ping geometry [14], and surface micro-structure [15, 16].A plane wave incident on a surface with non-zero resis-tance incurs a phase shift according to the Fresnel equa-tions. Effectively, these phase shifts increase the cavitysize and decrease the mode spacing compared to thatof an ideal conductor. Similarly, surface micro-structurecan increase the physical length of each incident mode.For this work, we will not concern ourselves with thesecomplications and instead compare cavities with equal ef-fective size – that is, cavities with identical mode spacing– rather than cavities with equal physical dimensions.The AC Stark interaction is characterized by a realand an imaginary component [17–19]: E BBR i = ∆ E i − i (cid:126) BBR i = − (cid:90) ∞ dω (cid:104) ˆ E (cid:105) α si ( ω ) , (2) a r X i v : . [ phy s i c s . a t o m - ph ] J a n where ˆ E is the electric field operator, α si ( ω ) is the scalarpolarizability of level iα si ( ω ) = (cid:88) j | µ zij | (cid:126) (cid:16) ω ij − ω − i Γ ij + 1 ω ij + ω + i Γ ij (cid:17) . (3)Here, µ zij and Γ ij are the z component of the dipole ma-trix element and the partial decay rate, respectively, be-tween state i to state j . The imaginary part Γ BBR i ofEq. (2) is associated with stimulated state transfer fromlevel i to other levels j of the system:Γ BBR i = (cid:88) j µ zij ω ij (cid:15) (cid:126) πc (cid:104) ˆ n ( ω ij ) (cid:105) . (4)The real part of Eq. (2) is a shift in the energy of level i ,given by [10, 20]:∆ E i = − P (cid:90) ∞ dω (cid:126) ω (cid:104) ˆ n (cid:105) ε π c α si ( ω ) (5)where P denotes the Cauchy principal value.In the absence of other broadening mechanisms, theinteraction of a quantum system with the mean BBRfield leads to a Lorentzian lineshape with full width athalf maximum (FWHM)Γ i = Γ sp i + Γ BBR i , (6)where Γ sp i and Γ BBR i are the rates of spontaneous decayand BBR stimulated depopulations for level i , respec-tively. Here, we seek to quantify the small, additionalbroadening of the quantum level induced by fluctuationsin the BBR electric field. These fluctuations lead to RMSdeviations σ ∆ E i in the AC Stark shift ∆ E i . These fluc-tuations occur with typical timescale of the blackbodycoherence time t c = h/ k B T ( t c ≈
40 fs at room temper-ature) [21]. Therefore, the expected lineshape for a sin-gle measurement with duration t m (cid:29) t c is a Voigt profilewith Lorentzian width γ = Γ i and Gaussian half width athalf maximum (HWHM) σ = σ ∆ E i (cid:112) t c /t m . The FWHMof a Voigt profile is approximatelyFWHM V ≈ γ (cid:114) γ σ ln 2 . (7)In the limit γ (cid:29) σ , the Voigt FWHM becomesFWHM V ≈ γ + 8 ln 2 σ γ . (8)In order to calculate σ ∆ E i , we begin by consider-ing each photon mode as contributing ∆ (cid:15) i to the to-tal shift, i.e. ∆ E i = (cid:80) modes ∆ (cid:15) i . Assuming uncorre-lated modes, σ E i = (cid:80) modes σ (cid:15) i , it suffices to findthe contribution of each mode independently. The sum-mation (cid:80) modes corresponds to P (cid:82) V D mode in the con-tinuous limit, where D mode is the density of modesper unit volume per unit angular frequency dω , and D mode = D FSmode = ω dω/π c for free space. Combiningthis with Eq. (5) yields∆ (cid:15) i = − (cid:126) ω (cid:104) ˆ n (cid:105) ε V α si ( ω ) , (9)and the variance of the shift is given by σ (cid:15) i = (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∆ (cid:15) i ∂ ˆ n (cid:12)(cid:12)(cid:12)(cid:12) σ n = (cid:12)(cid:12)(cid:12)(cid:12) (cid:126) ω ε V α si ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) σ n . (10)Finally we add the contributions of each mode inquadrature to arrive at the variance of the AC Stark shiftdue to BBR noise: σ E i = (cid:88) modes σ (cid:15) i = P (cid:90) ∞ V D mode (cid:126) ω σ n ε V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α si ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = P (cid:90) ∞ dω (cid:126) ω ε π c V e (cid:126) ω/k B T ( e (cid:126) ω/k B T − × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j (cid:126) (cid:16) | µ zij | ω ij − ω − i Γ ij + | µ zij | ω ij + ω + i Γ ij (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (11)where in the last line we take D mode = D FSmode . Notethat the summation within the modulus in Eq. (11) isidentical to that found in the Kramers-Heisenberg for-mula for differential scattering cross-section of light (e.g.Ref. [22] Section 8.7); that is, we can consider the BBRnoise broadening of each level i to be due to variance inBBR Rayleigh scattering rate due to fluctuating photonnumber in each mode. Also of note is the RMS fluctu-ation in the BBR shift σ ∆ E i is proportional to 1 / √ V .While the RMS electric field of a blackbody is indepen-dent of volume, smaller volumes contain fewer modeswhich contribute to the field, and thus are subject tolarger field variations. This relationship between BBRfluctuations and volume was first noted by Einstein [23],and Eq. (11) can alternately be derived using standardthermodynamic relations (see Appendix B).Often, the quantum observable of interest is not theAC Stark shift, but the differential AC Stark shift∆ E (cid:48) ij = ∆ E j − ∆ E i , such as when measuring the tran-sition frequency ω ij between two states i, j . Conven-tionally, negative shifts imply a decrease in the observedtransition frequency. Likewise, for the BBR noise linebroadening, we define the differential shift per mode∆ (cid:15) (cid:48) ij = ∆ (cid:15) j − ∆ (cid:15) i :∆ (cid:15) (cid:48) ij = (cid:126) ω (cid:104) ˆ n (cid:105) ε V α sij (cid:48) ( ω ) . (12)with α sij (cid:48) ( ω ) = α sj ( ω ) − α si ( ω ) the differential dynamicscalar polarizability. TABLE I. BBR Stark shifts and noise broadening for optical clock systems. All temperature-dependent quantities are evaluatedat T = 300 K. Static differential polarizabilities α sij (cid:48) (0) from Ref. [17] are the recommended calculated value, or mean of valueswhen multiple calculations using the highest accuracy method were presented. For measurement time t m , the BBR noisebroadening is σ = σ ∆ E (cid:48) ij (cid:112) t c /t m . ( V = 10 − m ) ( V = 10 − m ) ν ij (PHz) y ij Γ sp (Hz) α sij (cid:48) (0) ( a ) ∆ E (cid:48) ij (Hz) σ ∆ E (cid:48) ij (Hz) σ ∆ E (cid:48) ij / Γ sp σ ∆ E (cid:48) ij (Hz) σ ∆ E (cid:48) ij / Γ sp Ca + [17] 0.411 65.7 1.4 × − -44.1 3.8 × − × − × − × − × − Sr + [17] 0.445 71.2 4.0 × − -29.3 2.5 × − × − × − × − × − Yb + a [17] 0.642 102.7 1.0 × − × − × − × × − × Yb + b [17] 0.688 110.1 3.1 42 -3.6 × − × − × − × − × − Al + [17] 1.121 179.3 8.0 × − × − × − × − × − × − In + [17] 1.27 203.2 8.0 × − < > -2.6 × − < × − < × − < × − < × − Lu + [24, 25] 0.354 56.6 5.1 × − × − × − × − × − × − × − × − × − × − × − × − × − × Sr [17] 0.429 68.6 1.4 × − × − × − × − × Yb [17] 0.518 82.9 8.0 × −
155 -1.33 3.40 × − × − × − × − × − × − × − × − × − Cd [17] 0.903 144.5 1.0 × − × − × − × − × − × − Hg [17] 1.129 180.6 1.1 × −
21 -1.81 × − × − × − × − × − Tm [26] 0.263 42.1 1.2 -0.063 5.4 × − × − × − × − × − W [27] 0.539 86.2 1.5 × -0.024 2.1 × − × − × − × − × − Ir [27] 0.750 120.0 4.0 × -0.01 8.6 × − × − × − × − × − Pt [27] 0.745 119.2 3.9 × − × − × − × × − × Th [28, 29] 2.002 319.9 6.3 × − < × − > -3.4 × − < × − < × − < × − < × − a − f s − f s F / . b − f s − f d D / . In many cases, is it appropriate to make a static ap-proximation y ij = (cid:126) ω ij /k B T (cid:29)
1. In this limit,∆ E (cid:48) ij = (cid:126) ε π c π (cid:16) k B T (cid:126) (cid:17) α sij (cid:48) (0) , (13) σ E (cid:48) ij = (cid:126) ε π c V π (cid:16) k B T (cid:126) (cid:17) (cid:12)(cid:12)(cid:12) α sij (cid:48) (0) (cid:12)(cid:12)(cid:12) , (14)where we have used the identities (cid:90) ∞ dx x e x − π , (cid:90) ∞ dx x e x ( e x − = 4 π . (15)To the best of our knowledge, the BBR noise broad-ening effect described by Eq. (11) has not been observedexperimentally. That is unsurprising, as in most cases,this effect is quite small. If we consider a “typical”atomic transition to have frequency ω ij = 10 s − anddifferential polarizability α sij (cid:48) = 4 πε × a (where a ≈ . σ = σ ∆ E (cid:48) ij (cid:112) t c /t m ≈ . × − (cid:112) t c T /t m V Hz m / K − / . For T = 300 Kand V = 1 L = 10 − m , σ ∆ E (cid:48) ij (cid:112) t c /t m ≈ h ×
14 nHz.In nearly all practical measurements, the fluctuation σ = σ ∆ E (cid:48) ij (cid:112) t c /t m averages down the numerically smallvalue of σ ∆ E (cid:48) ij by a considerable factor (e.g. for t m = 1 s, (cid:112) t c /t m ≈ × − at room temperature). We thensee using Eq. (8) that the BBR noise broadening be-comes a vanishingly small correction to the the linewidth( σ ≈ T = 300 K. We calculatethe RMS differential BBR shift deviation σ ∆ E (cid:48) ij usingEq. (14) as well as the differential BBR Stark shift ∆ E (cid:48) ij using Eq. (13) as a check of consistency with the atomictransition data references [17, 27–29]. For the BBR shiftof optical clock transitions, the static approximation isgenerally accurate to within a few percent [17]. Differen-tial polarizablities in Table I are given in their respectivereferences in atomic polarizability units. These may beconverted to SI units (Hz/(V/m) ) by multiplying by afactor of 4 πε a /h , with a the Bohr radius.The results of Table I show that detection of BBR noisebroadening in atomic frequency standards is not possiblewithout several orders of magnitude improvement overcurrent frequency sensitivity. Consider as examples theSr and Yb systems, which now routinely achieve frac-tional uncertainty of δω/ω ∼ − [30–32]. Ignoringnumerous technical noise sources (e.g. lattice phononscattering in optical lattice clocks [33]), even in an excep-tionally small BBR volume V = 10 − m at T = 300 K, σ ∆ E (cid:48) ij is only approximately 30 mHz in Sr and 10 mHzin Yb. For a measurement time t m = 1 s, the BBRnoise broadening is then σ ≈ σ ≈ FIG. 1. Blackbody radiation noise σ ∆ E i as a function of the length (cid:96) of a cubic cavity. From left to right, panels show thecircular state of Rb with principal quantum number n = 52 for cavity temperature 4 K, 77 K, and 300 K, respectively. Thedashed, solid black, and solid colored lines calculate σ ∆ E i with the density of modes D mode equal to free space, perfectlyabsorbing walls, and copper walls, respectively. exceed Γ sp by approximately (1 × − )Γ sp for Sr, or(4 × − )Γ sp for Yb. The small differential polarizabil-ity characteristic of ions [17, 24, 27], inner-shell transi-tions [26, 27], and the nuclear transition of Th [28, 29]makes the BBR noise broadening in these systems evensmaller than the Sr and Yb cases.We next consider the possibility of observing BBRnoise-limited linewidths in circular Rydberg states | n C (cid:105) ≡ | n, L = n − , J = n − / (cid:105) . Here we use the Al-kali Rydberg Calculator python package [34] to calculatetransition matrix elements and energies for circular statesof Rb. The largest transition dipole matrix elements forRydberg atoms with principal quantum number n areto states with n (cid:48) = n ±
1. For our circular Rydbergstate calculations, we include electric dipole transitionsto n (cid:48) = n − , n + 1 , n + 2 , n + 3.Rydberg transition wavelengths may be as large orlarger than the length scale of its surroundings, and wemust consider cavity effects on the mode density D mode .Figure 1 shows σ ∆ E i for the | (cid:105) state of Rb in an ef-fective cubic volume V = (cid:96) for three key conceptualcases. The dashed colored lines depict σ ∆ E i calculatedusing D mode = D FSmode ; in this case σ ∆ E i is strictly pro-portional to (cid:96) − / . The solid black line depicts an idealblackbody (i.e. perfectly absorbing walls); the mode den-sity for a blackbody D BBmode is found by quantizing thecavity modes in the usual manner and assigning eachmode a finesse F = 1 / D Cumode is complicated by the fact thatthe resistivity ρ of cryogenic copper may vary by twoorders of magnitude depending on purity [35]; we as-sume residual-resistance ratios typical of oxygen-free highconductivity Cu: ρ ( T = 300 K) /ρ (T = 77 K) = 10 and ρ ( T = 300 K) /ρ (T = 4 K) = 100, with ρ ( T = 300 K) =1 . × − Ω · m.Figure 2 considers Rb circular states for principalquantum number n ≤
80 in a cubic ideal blackbody.The black dashed lines depict the Lorentzian partiallinewidth due to spontaneous decay Γ sp i . Colored dashed lines depict the partial linewidth due to both sponta-neous and BBR-stimulated decay Γ i = Γ sp i + Γ BBR i .Solid lines depict BBR noise Gaussian width σ ∆ E i √ t c Γ i ,where we have assumed a measurement time t m = 1 / Γ i .The magnitude of the BBR noise relative to the decayrate is increased in cryogenic environments, as σ ∆ E i de-creases more slowly than Γ sp i + Γ BBR i with decreasingtemperature. For effective volumes V = 1 m , σ ∆ E i ] T t c ( ) FIG. 2. Decay rate Γ sp + Γ BBR (dashed lines) and blackbodyradiation noise σ ∆ E i (cid:112) t c (Γ sp + Γ BBR ) assuming decay rate-limited measurement time (solid lines) for circular states ofRb as a function of principal quantum number n . From leftto right, panels show a cubic ideal blackbody with effectivevolume V of 1 m , 10 − m , and 10 − m , respectively. is smaller than Γ i by roughly six orders of magnitude,even at T = 4 K and n = 80. For blackbodies with V = 10 − m , σ ∆ E i is smaller than Γ i by roughly two or-ders of magnitude at T = 4 K and n = 80. We cautionthat for such large n , typical relevant transition wave-lengths are similar to or exceed (cid:96) = 1 cm for this case.The assumption of an ideal blackbody for high n andsmall V is likely invalid, with the BBR noise reducedfrom these estimates by one or more orders of magnitudeas in Fig. 1. The observed FWHM for circular Rydbergstates is therefore unlikely to exceed Γ i by more thanroughly 10 − × Γ i in the most favorable cases.In this work, we have assumed isotropic polarizationof the BBR, and thus only considered the scalar polariz-ability. As each mode has two independent polarizations,polarization fluctuations in the BBR can lead to broaden-ing which involves the vector and tensor polarizabilitiesas well, and could be considered in future work.We have derived a parametric broadening, general toall quantum systems, which is due to interactions withfluctuations in a blackbody radiation field. This BBRnoise broadening is most significant in applications whichinvolve small effective BBR volumes, large transitiondipole moments, and/or high frequency precision. How-ever, our presented calculations for several atomic clocktransitions and for circular Rydberg states of Rb showthat BBR noise broadening is typically too small to bedetected in these systems with modern sensitivity by atleast several orders of magnitude. These calculations cat-alogue a novel quantum noise source as being well belowcurrent sensitivity for many ongoing experiments, such asatomic frequency standards and quantum sensing exper-iments using circular Rydberg atoms. For future exper-iments with substantially improved frequency precision,this work also sets a benchmark for testing fundamentalthermodynamics of photons. ACKNOWLEDGEMENT
The authors thank Dazhen Gu, Andrew Ludlow, KyleBeloy, Michael Moldover, Marianna Safronova, Cl´ementSayrin, Wes Tew, and Howard Yoon for insightful con-versations, and thank Nikunjkumar Prajapati, Joe Rice,and Wes Tew for careful reading of the manuscript.
Appendix A: Photon Statistics
The partition function Z for a blackbody radiationmode with frequency ω is Z = ∞ (cid:88) n =0 e − n (cid:126) ω/k B T = 11 − e − (cid:126) ω/k B T . (A1) A standard trick to calculate the m th moment of thephoton number n is (cid:104) n m (cid:105) = ∞ (cid:88) n =0 n m e − nx = 1 Z ∂ m ∂ ( − x ) m Z ( x ) . (A2)The first few moments of the photon number are listedin Table II. TABLE II. First four moments of of the photon number n forblackbody radiation, with x = (cid:126) ω/k B T . (cid:104) n m (cid:105)(cid:104) n (cid:105) e x − (cid:104) n (cid:105) e x +1( e x − (cid:104) n (cid:105) e x +4 e x +1( e x − (cid:104) n (cid:105) e x +11 e x +11 e x +1( e x − Appendix B: Thermodynamic Derivation
Here we rederive the main result of the main text, Eq.(11), from thermodynamic principles. The total energy E BBR ( ω ) contained within the volume V of a blackbodycavity per unit angular frequency is E BBR ( ω ) = V (cid:126) ω π c e (cid:126) ω/k B T − . (B1)The variance of E BBR ( ω ) is given by σ E BBR ( ω ) = k B T (cid:16) ∂E BBR ( ω ) ∂T (cid:17) = V (cid:126) ω π c e (cid:126) ω/k B T ( e (cid:126) ω/k B T − = V (cid:126) ω π c σ n ( ω ) . (B2)Because the spectral energy density U ( ω ) = ε E = E BBR ( ω ) /V , the variance of the spectral energy densityis given by σ U ( ω ) = (cid:126) ω V π c σ n ( ω ) . (B3)Note that the fluctuations in U ( ω ) are inversely propor-tional to the volume of the cavity.Since ∆ E i = − (cid:90) ∞ dω (cid:104) U ( ω ) (cid:105) ε α si ( ω ) , (B4)then σ E i = P (cid:90) ∞ dω (cid:12)(cid:12)(cid:12) α s ( ω )2 ε (cid:12)(cid:12)(cid:12) σ U ( ω ) = P (cid:90) ∞ dω (cid:126) ω ε V π c σ n | α s ( ω ) | , (B5)which matches Eq. (11) of the main text. Appendix C: Treating a Blackbody as an OpticalCavity
Here we derive the characteristic cavity parameters fora blackbody cavity. For a cavity mode of length (cid:96) , thefree spectral range is ∆ f FSR = c (cid:96) . (C1)If a photon is emitted into a mode of length (cid:96) , the char-acteristic length traveled is precisely (cid:96) , since blackbod-ies are by definition perfect absorbers. Therefore, thespectral width of a blackbody cavity mode (LorentzianHWHM) is ∆ f BBcav = c(cid:96) . (C2)The finesse of a cavity is defined as F =∆ f FSR / f BBcav . For an ideal blackbody, apparently allmodes have a finesse of F = 1 / (cid:96) are f q = q ∆ f FSR . Here, q = q x + q y + q z , and q x , q y , q z are the number of nodes in the x, y, z dimen-sion plus 1, excluding the boundaries. Allowed modeshave q x , q y , q z ≥
0, with at least one of q x , q y , q z ≥
1. Thequality factor of mode q is then Q ≡ f q / f BBcav = q/ f BBcav , the spectral density of mode q is d q ( f ) = 2 1 π f BBcav ( f BBcav ) + ( f q − f ) df, (C3) FIG. 3. Density of modes D mode for a perfect absorber as afunction of frequency. An excess mode density exists at lowfrequencies for perfect absorbers compared to free space. where the prefactor of 2 accounts for two possible po-larizations per mode. A related quantity is the “modedensity” D mode ( f ) = (cid:88) q d q ( f ) /V, (C4)which is the density of modes per unit frequency per unitvolume. In the limit of q (cid:29)