Lattice-depth measurement using continuous grating atom diffraction
LLattice-depth measurement using continuous grating atom di ff raction Benjamin T. Beswick, ∗ Ifan G. Hughes, † and Simon A. Gardiner ‡ Joint Quantum Centre (JQC) Durham–Newcastle, Department of Physics,Durham University, Durham DH1 3LE, United Kingdom (Dated: March 12, 2019)We propose a new approach to characterizing the depths of optical lattices, in which an atomic gas is givena finite initial momentum, which leads to high amplitude oscillations in the zeroth di ff raction order which arerobust to finite-temperature e ff ects. We present a simplified model yielding an analytic formula describing suchoscillations for a gas assumed to be at zero temperature. This model is extended to include atoms with initialmomenta detuned from our chosen initial value, before analyzing the full finite-temperature response of thesystem. Finally we present a steady-state solution to the finite-temperature system, which in principle makespossible the measurement of both the lattice depth, and initial temperature of the atomic gas simultaneously. I. INTRODUCTION
There is much interest in the precise measurement of op-tical lattice [1] depths in the field of atomic physics, particu-larly for accurate determination of transition matrix elements[2–6], better knowledge of these matrix elements can be usedto improve the black body radiation correction for ultrapre-cise atomic clocks [7, 8], and allows quantitative modelingof atom-light interaction [9]. Other areas of interest includeatom interferometry [10, 11] and many body quantum physics[12, 13], where knowledge of the lattice depth is essential forinterpreting experimental results.Commonly used lattice depth measurement schemes in-clude Kapitza–Dirac scattering [13–17], parametric heating[18], Rabi oscillations [19], and, more recently, the suddenphase shift method [20]. For the case of a weak lattice( V (cid:46) . E R for any atom, where V is the lattice depth and E R is the atomic recoil energy), methods based on multipulseatom di ff raction have been explored [21, 22], with a view toreducing signal-to-noise considerations in the measurementof the resultant di ff raction patterns. In previous work wehave presented improved models for the expected multipulsedi ff raction patterns for a given lattice depth. We have alsonoted that when considering a gas with initial momentum (cid:126) K /
2, the functional form of these models is markedly sim-pler and therefore easier to fit to data to make an accuratemeasurement of the lattice depth [23].In this paper we explore such a measurement scheme fora lattice which is not pulsed but instead continuously presentthroughout the experimental sequence, which we show to bemore robust to finite-temperature e ff ects than a multipulse ap-proach. In Sec. II, we describe our model system and exper-imental considerations. In Sec. III, we introduce a simplifiedanalytic approach for determining the time evolution of theatomic population in the zeroth di ff raction order, and make acomparison to exact numerical calculations. Finally, in Sec.IV, we present an approximate analytic model for the finite-temperature response of the system, and discuss how these ∗ [email protected] † [email protected] ‡ [email protected] may be used to determine both the lattice depth and initialtemperature of the atomic gas. II. MODEL SYSTEM: ATOMIC GAS IN AN OPTICALGRATINGA. Experimental setup and Hamiltonian
We consider a two-level atom in an assumed noninteract-ing Bose-Einstein condensate exposed to a far o ff resonanceoptical grating, the Hamiltonian of which is given by Eq. (1):˜ H Latt = ˆ p M − V cos (cid:16) K (cid:104) ˆ x + v φ t (cid:105)(cid:17) , (1)where ˆ p is the momentum operator along the lattice axis, V is the lattice depth, K is twice the laser wavenumber k L , M isthe atomic mass and v φ is the phase velocity of the grating inthe x direction ( v φ = p = (cid:126) K / As shown in Fig. 1(a), theBEC is di ff racted by the static optical grating for a time t , be-fore a time of flight measurement interrogates the populationof the gas in each of the allowed momentum states. In princi-ple there is an infinite ladder of such states, each separated byinteger multiples of (cid:126) K [24, 25], though here we show onlythe zeroth and first di ff raction orders. We note that an ini-tial state p = (cid:126) K / ff raction, or equivalently we may prepare the BEC in a statewith p = v φ t to the standing wave as in Fig. 1(b). We show thisequivalency in Sec. II B below. B. Gauge transformations and momentum kicks
The Hamiltonian of Eq. (1) can be transformed to a framecomoving with the walking grating by use of the unitary trans- The initial momentum p = (cid:126) K / ff raction orders with astrong sinusoidal character, as suggested in [23]. a r X i v : . [ qu a n t - ph ] M a r Atomic gas − ~ K / + ~ K / + ~ K / − ~ K − . . . p / ~ K E n e r gy / ~ | e i| g i ∆ + ω rec ∆ p = ~ K k = β = / k = − β = / FIG. 1. (Color online) (a) A BEC initially prepared in the p = + (cid:126) K / K is twice the laser wavenumber k L , is exposed it to astatic optical grating, causing it to di ff ract into an, in principle, in-finite number of momentum states separated by integer multiples of (cid:126) K , here we show only the first di ff raction order. Equivalently, theBEC may be prepared in the p = − (cid:126) K /
2. (c), semiclassical energy-momentumdiagram for a single two-level atom scattering photons from a staticoptical grating. The atom begins on the ground state energy parabola,with classical momentum p = (cid:126) K / p = − (cid:126) K / (cid:126) K / M , to reach thedetuned virtual state above, before undergoing stimulated emissionback to the ground state, resulting in a total momentum transfer of ∆ p = − (cid:126) K . This scattering process and its exact reversal are theonly processes which semiclassically conserve both the energy andmomentum of the atom grating system, indicating that populationtransfer between the p = (cid:126) K / p = − (cid:126) K / formationˆ U = ˆ U x ˆ U p ˆ U α = exp (cid:16) imv φ ˆ x / (cid:126) (cid:17) exp (cid:16) − iv φ ˆ pt / (cid:126) (cid:17) exp ( i α t / (cid:126) ) , (2)where we have chosen α = Mv φ / U p ˆ x ˆ U † p = ˆ x + v φ t and ˆ U x ˆ p ˆ U † x = ˆ p + Mv φ . This transformationyields: ˆ H Latt = ˆ p M − V cos ( K ˆ x ) . (3)The Hamiltonian of Eq. (3) describes the system in a framemoving with velocity − v φ , therefore, a gas moving with veloc-ity v = − v φ in the lab frame. Conversely, a gas moving with veloc-ity v = (cid:126) K / M in the comoving frame, moves with velocity v = ( (cid:126) K / M ) − v φ in the lab frame. Choosing v φ = v φ = (cid:126) K / M , we have thesituation shown in Fig. 1(b).The spatial periodicity of Eq. (3) allows us to invoke Blochtheory [26], by rewriting the momentum operator in the fol-lowing basis: ( (cid:126) K ) − ˆ p = ˆ k + ˆ β, (4a)ˆ k | ( (cid:126) K ) − p = k + β (cid:105) = k | ( (cid:126) K ) − p = k + β (cid:105) , (4b)ˆ β | ( (cid:126) K ) − p = k + β (cid:105) = β | ( (cid:126) K ) − p = k + β (cid:105) . (4c)We may speak of k ∈ Z as the discrete part of the momentum,and β ∈ [ − / , /
2) as the continuous part or quasimomentum [27]. Here β is a conserved quantity, as such, only momen-tum states separated by integer multiples of (cid:126) K are coupled[24, 25]. This simplification allows us to construct the timeevolution operator for a lattice pulse of duration t from thelattice Hamiltonian (3) as follows:ˆ U ( β, τ ) Latt = exp (cid:32) − i (cid:34) ˆ k + k β − V e ff cos(ˆ θ ) (cid:35) τ (cid:33) , (5)in which β is simply a scalar value such that overall phaseswhich depend solely on β can be neglected. Here V e ff = V M / (cid:126) K is the dimensionless lattice depth, ˆ θ = K ˆ x and τ = t (cid:126) K / M is the rescaled time.By using Eq. (5) to calculate | ψ ( τ ) (cid:105) = (cid:80) j c j ( τ ) | k = j (cid:105) , thepopulation in each discrete momentum state | k = j (cid:105) followingan evolution for a rescaled time of τ is given by the absolutesquare of the coe ffi cients P j ( τ ) = | c j ( τ ) | . In this paper weemploy the well-known split-step Fourier approach [25, 28]to determine | ψ ( τ ) (cid:105) , as well as an analytic approach based ona simpler two-state model.The dynamics of a single atom in the BEC standing-wavesystem can be understood in terms of the scattering processgiven by the semiclassical energy diagram of Fig. 1(c) (seealso [29–33]). A two-level atom begins in a state with mo-mentum p = (cid:126) K /
2, before absorbing a photon with momen-tum p = − (cid:126) K /
2, and subsequently emits a second photon withthe momentum p = (cid:126) K /
2. This is the only scattering pro-cess which classically conserves energy, whilst also conserv-ing the quasimomentum. We therefore expect that scatteringinto states with momentum p > | (cid:126) K / | ought to be stronglysuppressed even under the fully quantum time evolution. Weexplore this simplified picture in Sec. III. III. REDUCTION TO AN EFFECTIVE 2-STATE SYSTEMA. Simplification
We may test the conjecture that population transfer intostates with k < − k > k = − k = B. Two-state model analytics
We may represent the Hamiltonian (3) in the β = / | k = (cid:105) = (cid:32) (cid:33) , (6a) | k = − (cid:105) = (cid:32) (cid:33) , (6b)yielding: H × = (cid:32) / − V e ff / − V e ff / / (cid:33) . (7)We recognize Eq. (7) as a Rabi matrix with zero detuning, theeigenvectors and eigenvalues of which are well known [34],and can be used to straightforwardly determine the time evo-lution of the population in the | k = (cid:105) and | k = − (cid:105) states,respectively: P = cos ( V e ff τ/ , (8a) P − = sin ( V e ff τ/ , (8b)as outlined in Appendix A. This analytic result is compared toour exact numerics in Figs. 2 and 3, both of which show ex-cellent agreement for a wide range of experimentally relevantvalues of the e ff ective lattice depth V e ff . We note in particularthat the form of Eqs. (8a) and (8b) is such that there is an ex-act universality between τ and V e ff , which is elucidated in Fig.3(b), where all population curves fall on top of each other. IV. FINITE-TEMPERATURE RESPONSEA. Other values of β In the following section we consider the e ff ect of evolvinginitial states with quasimomentum di ff erent to β = / | k + β (cid:105) under the time evolution operator (5).We make the assumption from the outset that the initial mo-mentum distribution of the gas (centred at β = /
2) spans lessthan half of each of the k = k = − k = β = k = ff raction order centered around | β | = /
2, andlow amplitude but rapidly oscillating solutions as β is detuned from this value. We may also use our simplified semiclassicalmodel of Sec. III to derive an approximate analytic result forthe same calculation, in which the quasimomentum β is en-coded as a detuning to be included in our initial Rabi modelof Eq. (7). These additions yield the following 2 × H × ( β ) = (cid:32) β / − V e ff / − V e ff / − β + β ) / (cid:33) , (9)in which β is now a free parameter. The time evolution ofthe zeroth di ff raction order population governed by this matrixcan be found using the approach given in Appendix B, thus: P ( β ) = − V ff ( β − / + V ff sin (cid:18) (cid:113) ( β − / + V ff τ (cid:19) , (10)which is similar to the result reported in [15] for a zero tem-perature gas, and agrees excellently with the exact numericsfor physically relevant parameters as shown in Fig. 4. Wetherefore expect that thermal averaging of this result shouldproduce an accurate description of the full finite-temperatureresponse. B. Finite temperature analysis
To find the finite temperature response of the system weweight the contribution of Eq. (10) for each individual quasi-momentum subspace according to the Maxwell-Boltzmanndistribution: D k = ( β, w ) = w √ π exp (cid:32) − ( β − / w (cid:33) , (11)where the dimensionful temperature is given by T w = (cid:126) K w / Mk B [35]. Mathematically this corresponds to theintegral: P ( w ) = (cid:90) D k = ( β, w ) P ( β ) d β. (12)Inserting Eqs. (11) and (10), we have: P ( ρ ) = √ πρ (cid:90) − exp (cid:32) − γ ρ (cid:33) − γ + (cid:112) γ + φ d γ, (13)where we have introduced γ = ( β − / / V e ff , φ = V e ff τ and ρ = w / V e ff for simplicity. The exponential and trigonometricterms can be power expanded, and the integral (13) solvedterm by term, giving: P ( ρ ) = − ∞ (cid:88) s = s (cid:88) q = u s ( φ ) M s , q v q ( ρ ) , (14)where u s ( φ ) = ( − φ ) s + s ! / [2( s + M s , q = − (2 q )! / [2( q !) ( s − q )!] and v q ( ρ ) = ( ρ / q (see Appendix p w a l k i ng ( un it s o f ~ K ) V e ff = . τ/ (2 π )0 . . . . . . P (b) 0 5 10 15 20 25 30 V e ff = . τ/ (2 π )(d) 0 5 10 15 20 25 30 V e ff = . τ/ (2 π )(f) -5.5-3.5-1.50.52.54.56.5 p s t a ti c ( un it s o f ~ K ) P − − − − FIG. 2. (Color online) Time evolved momentum distributions for an atomic gas initially prepared in the | k = , β = / (cid:105) momentum state(corresponding to the | k = , β = (cid:105) state in the lab frame for a walking grating), as calculated numerically on a basis of 2048 momentumstates. The top row of false color plots [(a),(c),(e)] shows the population in the first 13 momentum states, to be read on the logarithmic colorbarto the right, a cuto ff population of P cuto ff = − has been applied to accommodate the log scale. The labels p static and p walking denote themomentum as measured in the lab frame for the case of a static and a walking grating respectively. The bottom row of plots [(b),(d),(f)] showsthe time evolution of the population in the | k = (cid:105) (red circles) and | k = − (cid:105) (blue squares) states, where the solid line through each curve isgiven by the analytic solution of Eqs. (8a) and (8b). Also shown is the population in the | k = (cid:105) state (green points). Each column of plotscorresponds to a simulation for a fixed value of the e ff ective lattice depth V e ff , here, from left to right V e ff = . , . , .
13 respectively.
C). Equation (14) can in principle be solved numerically byrecursively populating the elements of a su ffi ciently large pairof u ( φ ), v ( ρ ) vectors and M matrix, though the elements ofthe vectors will grow with s and q respectively unless φ and ρ are su ffi ciently small, and this condition is only satisfiedfor certain experimentally relevant regimes. Nonetheless,Eq. (14) yields some insight when expressed as a sum overderivatives of sinc functions (see Appendix D): P ( ρ ) = − ∞ (cid:88) q = (cid:18) ρ (cid:19) q (2 q )! q ! (cid:40)(cid:18) φ (cid:19) q + (cid:34)(cid:32) φ (cid:33) dd( φ/ (cid:35) q (cid:34) sin ( φ/ φ/ (cid:35)(cid:41) . (15)With q =
0, Eq. (15) reduces to the zero temperature resultof Eq. (8a), as such we should expect the finite temperaturebehavior of the system to be captured in terms with q > q is always convergent, the presenceof the ( φ/ q + term guarantees that all individual terms with q ≥ V e ff = . V e ff = . φ = V e ff τ and ρ = w / V e ff as free parameters, would give anaccurate value of the e ff ective lattice depth, if the time τ isknown to high precision and the lattice depth is su ffi cientlysmall.Further, we note that using standard integral results, wemay also extract the steady state solution to Eq. (13) as φ → ∞ : P ,φ →∞ ( ρ ) = ρ (cid:114) π (cid:32) ρ (cid:33) Erfc √ ρ , (16) τ/ (2 π )0 . . . . . . P (a)0.0 0.55 1.1 1.65 2.2 2.75 3.3 τ × V e ff / (2 π )0 . . . . . . P (b)FIG. 3. (Color online) (a): Plot of P , the population in the | k = (cid:105) state, versus number of pulses, as calculated on a basis of 2048 mo-mentum states using a split-step Fourier method (solid markers). Thesolid lines correspond to the analytic solution for P in a two statebasis, as given by Eq. (8a). Each set of markers corresponds to afixed value of the e ff ective lattice depth ranging from the slowest-oscillating curve at V e ff = .
01 to the fastest oscillating one at V e ff = .
11 in steps of 0 .
02. (b): Reproduction of (a), with the num-ber of pulses axis scaled by V e ff to reveal a universal curve both inthe analytics and the numerical simulations. The data have been ex-tended to span the full range of the horizontal axis. which depends only on ρ = w / V e ff . Here, ‘Erfc’ is the com-plementary error function [36]. In essence, by measuring thesteady state population experimentally, and numerically fit-ting Eq. (16), ρ = w / V e ff can be straightforwardly determinedand substituted into Eq. (13), leaving a fit in only one parame-ter φ = V e ff τ . The steady state population can be found eitherby allowing the atomic gas to evolve in the lattice for a su ffi -ciently long time, or taking the average value of P in time foran appropriate number of oscillations. In fact, this improvedfitting approach not only allows φ = V e ff τ , and therefore the When evaluating Eq. (16) for physically relevant values of ρ = w / V e ff ,the exponential term becomes large as the error function takes a corre-spondingly such that P ,φ →∞ ( ρ ) remains bounded between 0 and 1. Thiscomplication can present a problem for numerical evaluation using stan-dard numerical routines. In practice, we numerically implement Eq. (16)exclusively in terms of rational numbers in Mathematica, before requestinga numerical evaluation to a specified precision. -0.50 -0.25 0 0.25 0.50 β τ / ( π ) (a)0 8 16 24 32 40 τ/ (2 π )0 . . . . . . P (b) P . . . . . P as computed in a basis of 2048 momentum states for values ofthe dimensionless quasimomentum β [see Eqs. (4b,4c)] ranging from β = − . β = . β = . V e ff = . ff erent dynamical behaviors are made clear forthe chosen evolution time τ/ (2 π ) =
40. (b): Slices taken throughthe quasimomentum distribution parallel to the time axis for β = , . , . β = .
125 up toa maximum of β = .
5, enclosing the full range of dynamics in the k = β subspace [Eq. (10)]. e ff ective lattice depth V e ff to be determined more accurately,but also allows the initial e ff ective temperature to be deter-mined from w = ρ V e ff . V. CONCLUSIONS
We have presented a simplified model system yielding ananalytic zero-temperature formula for the evolution of the ze-roth di ff raction order population, and demonstrated the valid-ity of this approach across a wide range of lattice depths. Wehave extended this model to incorporate finite-temperature ef-fects and discussed from where they arrive mathematically.We have shown that there is excellent agreement between thisanalytic model and exact numerical calculations if the latticedepth is su ffi ciently small, and shown that a steady state so- .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 .
35 0 . . . . . . . P (a)0 .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 .
35 0 . τ × V e ff / (2 π )0 . . . . . . P (b) 0 . . . . . . . . . . . . . . . . . . τ × V e ff / (2 π )(d)FIG. 5. (Color online) Plot of the finite temperature response of P vs (number of pulses) × V e ff , where V e ff is the dimensionless lattice depth [seeEq. (5)], as calculated for an ensemble of 4001 particles each evolved in a basis of 2048 momentum states (hollow markers). The left column[(a), (b)] corresponds to the weak-lattice regime, and the right column [(c), (d)] to the strong-lattice regime. The top row of plots [(a), (c)] showsthe finite-temperature response of P at a temperature of w = . ff erent lattice depths, V e ff = . , . , .
05 (all curvesfall on top of each other) in the weak regime (a) and V e ff = . , . , . P at a di ff erent temperature ( w = . , . , . ff ective lattice depth is kept constant at V e ff = . V e ff = .
01 the weak-lattice case. In all panels, the solid lines represent the result yielded by numerically integrating Eq. (13). The horizontaldashed lines correspond to the result of the steady state solution of Eq. (16) for each set of parameters. lution exists, which may be useful for determining the latticedepth and initial temperature of a gas from a single set of pop-ulation measurements. With regard to potential experimentalimplementations, we note that the phase velocity of a walkingoptical lattice can be calibrated extremely precisely, however,does require optical elements to be in place which will reducethe intensity of the laser beam and therefore the lattice. Thealternative is to impart a specified momentum to an initiallystationary BEC; it is unlikely that this can be achieved withthe same level of precision, however there is no need for anyadditional optical elements a ff ecting the lattice depth. ACKNOWLEDGMENTS
B.T.B., I.G.H., and S.A.G. thank the Leverhulme Trustresearch program grant RP2013-k-009, SPOCK: ScientificProperties of Complex Knots for support. We would also liketo acknowledge helpful discussions with Andrew R. MacKel-lar.
Appendix A: Derivation of the two-state model
To calculate the time-evolution of the population in the ze-roth di ff raction order, we construct the time evolution operatorin the momentum basis from the Hamiltonian of Eq. (7), re-produced here for convenience: H × = (cid:32) / − V e ff / − V e ff / / (cid:33) . (A1)The diagonal terms simply represent an energy shift that canbe transformed away, thus the eigenvalues of Eq. (7) can sim-ply be read from the o ff -diagonal: E ± = ± V e ff /
2. We maynow solve the eigenvalue equation: (cid:32) − V e ff / − V e ff / (cid:33) (cid:32) v ± v ± (cid:33) = ± V e ff / (cid:32) v ± v ± (cid:33) . (A2)Equation (A2) leads directly to − v ± = ± v ± , yielding eigenvec-tors: | E + (cid:105) = √ (cid:32) − (cid:33) , | E − (cid:105) = √ (cid:32) (cid:33) . (A3)We may now construct our initial condition in the energy ba-sis, in which the matrix representation of the time evolutionoperator ˆ U ( τ ) = exp (cid:16) − i ˆ H Latt τ (cid:17) (A4)is diagonal: | ψ ( τ = (cid:105) = | k = (cid:105) = √ | E + (cid:105) + | E − (cid:105) ) . (A5)The time evolution of the population in the zeroth di ff ractionorder is given by: P = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
12 ( (cid:104) E + | + (cid:104) E − | ) ˆ U ( τ ) ( | E + (cid:105) + | E − (cid:105) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , = (cid:12)(cid:12)(cid:12) e − iE + τ + e − iE − τ (cid:12)(cid:12)(cid:12) , = (cid:12)(cid:12)(cid:12) e − iV e ff τ/ + e iV e ff τ/ (cid:12)(cid:12)(cid:12) , = cos ( V e ff τ/ , (A6)which corresponds to Eq. (8a). Appendix B: Derivation of β dependent two-state model To calculate the time-evolved population for a given quasi-momentum subspace, we follow the same procedure as in Ap-pendix A. Equation (9), reproduced here for convenience H × ( β ) = (cid:32) β / − V e ff / − V e ff / − β + β ) / (cid:33) , is nothing other than a Rabi matrix, the eigenvalues of whichare E ± = (cid:20) (1 / − β + β ) ± (cid:113) ( β − / + V ff (cid:21) /
2, and thecorresponding eigenvectors: | E + (cid:105) = (cid:32) cos( α/ α/ (cid:33) = √ (cid:104) (cid:112) + cos( α ) | k = (cid:105) + (cid:112) − cos( α ) | k = − (cid:105) (cid:105) , (B1a) | E + (cid:105) = (cid:32) − sin( α/ α/ (cid:33) = − √ (cid:104) (cid:112) − cos( α ) | k = (cid:105)− (cid:112) + cos( α ) | k = − (cid:105) (cid:105) , (B1b)where cos( α ) = ( β − / / (cid:113) ( β − / + V ff . This leads di-rectly to: | ψ ( τ = (cid:105) = | k = (cid:105) = cos( α/ | E + (cid:105) − sin( α/ | E − (cid:105) = √ (cid:104) (cid:112) + cos( α ) | E + (cid:105) − (cid:112) − cos( α ) | E − (cid:105) (cid:105) . We may now simply calculate the time-evolved state from theaction of the time evolution operatorˆ U ( τ, β ) = exp (cid:16) − i ˆ H ( β ) Latt τ (cid:17) , on this initial state thus: | ψ ( τ, β ) (cid:105) = exp (cid:16) − i ˆ H ( β ) Latt τ (cid:17) | k = (cid:105) = √ (cid:104) √ + c e − iE + τ | E + (cid:105) + √ − c e − iE − τ | E − (cid:105) (cid:105) . Here we have introduced c ≡ cos( α ). The time-evolved popu-lation in the zeroth di ff raction order for a given β subspace isthen given by: p ( τ, β ) = |(cid:104) k = | ψ ( τ, β ) (cid:105)| = (cid:12)(cid:12)(cid:12) (1 + c ) e − iE + τ + (1 − c ) e − iE − τ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) e E + τ/ e E − τ/ (cid:104) (1 + c ) e − i [ E + − E − ] τ/ + (1 − c ) e i [ E + − E − ] τ/ (cid:105)(cid:12)(cid:12)(cid:12)(cid:12) = cos ([ E + − E − ] τ/ + c sin ([ E + − E − ] τ/ = + ( c −
1) sin ([ E + − E − ] τ/ = − V ff ( β − / + V ff sin (cid:18) (cid:113) ( β − + V ff τ/ (cid:19) , (B2)which corresponds to Eq. (10). Appendix C: Derivation of finite-temperature matrix equation
To derive the matrix equation for the finite-temperature re-sponse of the zeroth di ff raction order population, we beginfrom Eq. (12), into which we insert Eqs. (11) and (10), yield-ing: P ( w ) = − √ π w (cid:90) ∞−∞ d α V ff α + V ff exp (cid:32) − α w (cid:33) sin (cid:18) (cid:113) α + V ff τ (cid:19) , = − P − ( w ) (C1)where we have introduced α ≡ ( β − / P − ( w ), the population in the | k = − (cid:105) state. The sinusoidal term can be rewritten using sin ( θ ) = [1 − cos(2 θ )] /
2, thus: P − ( w ) = V ff √ π w (cid:90) ∞ d α α + V ff exp (cid:32) − α w (cid:33) × (cid:20) − cos (cid:18) (cid:113) α + V ff τ (cid:19)(cid:21) , (C2)where we have used the fact that the integrand is an even func-tion. The term in cos (cid:18) (cid:113) α + V ff τ (cid:19) can then be power ex-panded, leading to: P − ( w ) = V ff √ π w (cid:90) ∞ d α − α + V ff exp (cid:32) − α w (cid:33) ∞ (cid:88) s = ( − s ( α + V ff ) s τ s (2 s )! , = V ff √ π w ∞ (cid:88) s = ( − s τ s + (2[ s + (cid:90) ∞ d α exp (cid:32) − α w (cid:33) ( α + V ff ) s , (C3)such that the square root in the argument no longer appears,and the ( α + V ff ) s term can be binomially expanded thus: P − ( w ) = V ff √ π w ∞ (cid:88) s = ( − s τ s + s !(2[ s + s (cid:88) q = V s − q ) q !( s − q )! (cid:90) ∞ d α α q exp (cid:32) − α w (cid:33) . (C4)Further, introducing ξ ≡ α / (2 w ), the remaining integral canbe rewritten as: (cid:90) ∞ d α α q exp (cid:32) − α w (cid:33) = w q + q − / (cid:90) ∞ d ξ exp( − ξ ) ξ q − / , = w q + q − / Γ ( q + / , which, when substituted into Eq. (C4) leads to: P − ( w ) = √ π ∞ (cid:88) s = ( − s ( V e ff τ ) s + s !(2[ s + s (cid:88) q = q !( s − q )! w V ff q Γ ( q + / . (C5)Finally, noting that Γ ( s + / = (2 s )! √ π/ (2 s s !), Eq. (C5)can be rewritten, thus: P − ( w ) = ∞ (cid:88) s = s (cid:88) q = ( − V ff τ ) s + s !(2[ s + (cid:16) − (cid:17) (2 q )!( q !) ( s − q )! w V ff q , (C6) = u s ( V e ff τ ) M s , q v q ( w / V e ff ) , or, equivalently, with φ = V e ff τ and ρ = w / V e ff : P ( ρ ) = − P − ( ρ ) = − ∞ (cid:88) s = s (cid:88) q = u s ( φ ) M s , q v q ( ρ ) , which corresponds to Eq. (14). Appendix D: Expression of Eq. (14) in terms of Sinc functions