Factorization of numbers with Gauss sums: II. Suggestions for implementations with chirped laser pulses
W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, G. G. Paulus
aa r X i v : . [ qu a n t - ph ] O c t Factorization of numbers with Gauss sums:II. Suggestions for implementations with chirpedlaser pulses
W Merkel , S W¨olk , W P Schleich , I Sh Averbukh , BGirard and G G Paulus Institut f¨ur Quantenphysik, Universit¨at Ulm,Albert-Einstein-Allee 11, D-89081 Ulm, Germany Department of Chemical Physics,Weizmann Institute of Science, Rehovot 76100, Israel Laboratoire de Collisions, Agr´egats, R´eactivit´e, IRSAMC (Universit´e deToulouse/UPS; CNRS) Toulouse, France Institut f¨ur Optik und Quantenelektronik, Friedrich-Schiller-Universit¨at Jena,Max-Wien-Platz 1, D-07743 Jena, GermanyE-mail: [email protected]
Abstract.
We propose three implementations of the Gauss sum factorizationschemes discussed in part I of this series [S W¨olk et al., preceding article]: ( i ) atwo-photon transition in a multi-level ladder system induced by a chirped laser pulse,( ii ) a chirped one-photon transition in a two-level atom with a periodically modulatedexcited state, and ( iii ) a linearly chirped one-photon transition driven by a sequenceof ultrashort pulses. For each of these quantum systems we show that the excitationprobability amplitude is given by an appropriate Gauss sum. We provide rules how toencode the number N to be factored in our system and how to identify the factors of N in the fluorescence signal of the excited state. Submitted to:
New J. of Physics
1. Introduction
In [1] we have shown that Gauss sums are excellent tools to factor numbers. Throughoutthis article we have concentrated on the general principles and the underlyingmathematical foundations. However, we have not addressed physical implementations ofGauss sums. The present article complements [1] and proposes three distinct realizationsof Gauss sum factorization.Several experiments [2–11] have already successfully demonstrated factorizationwith the help of Gauss sums. However, all of them have implemented the truncatedGauss sum A ( M ) N ( ℓ ) ≡ M + 1 M X m =0 exp (cid:18) − π i m Nℓ (cid:19) , (1) actorization with Gauss sums: Implementations N/ℓ had to be precalculated.In contrast the three implementations of Gauss sum factorization proposed in the presentpaper, encode the number N to be factored and the trial factor ℓ in two independentvariables. As a consequence, the ratio N/ℓ does not to be precalculated.The first suggestion utilizes a two-photon transition in an equidistant ladder systemdriven by a chirped laser pulse. Since in this system the excitation probability amplitudeis given by the continuous Gauss sum S N ( ξ ) ≡ M X m = − M w m exp (cid:20) π i (cid:18) ± m + m N (cid:19) ξ (cid:21) , (2)discussed in [12], it can factor numbers.In addition, we suggest two alternative approaches based on one-photon transitionsin a laser-driven two-level system. Both rely on quantum interference of multiple opticalexcitation paths, but lead to different quadratic phase factors.We consider a two-level system with a permanent dipole moment in the excitedstate. A cw-microwave field interacting with this dipole modulates the energy of theexcited state and induces in this way an equidistant set of sidebands. The probabilityamplitude for a one-photon transition caused by a chirped laser pulse is the sum overall possible excitation channels involving one optical photon as well as multiple quantaof the microwave field. The Gaussian nature of this sum arises from the quadratic chirpof the laser pulse.The second approach uses a multi-pulse excitation of a two-level system with alinearly chirped resonance frequency. The probability amplitude of excitation is a sumof contributions arising from each pulse. The quadratic phase dependence characteristicof a Gauss sum originates from the linear variation of the resonance frequency of thetwo-level system.Throughout the article we neglect spontaneous emission since the interaction timewith the pulses is much shorter than the decay time of the atomic level. This fact allowsus to describe the system by the Schr¨odinger equation rather than a density matrix.Our article is organized as follows: Since we rely heavily on the physics of chirpedlaser pulses we first summarize in Sec. 2 the basic elements of this branch of optics. InSec. 3 we then recall the main results of Ref. [12] and, in particular, the expression forthe excitation probability amplitude corresponding to a chirped two-photon transition inan harmonic ladder system. Starting from this formula we demonstrate that this systemallows us to factor numbers, and that in principle a single realization of a factorizationexperiment can be employed to reveal the factors of another number.Section 4 provides the basic elements of Secs. 5 and 6 by engineering a one-photontransition. As a second physical system implementing Gauss sums we investigate inSec.5 a two-level system with a permanent dipole moment driven by a microwave field.A chirped laser pulse interacts with the so-engineered Floquet ladder. The resultingprobability amplitude is again given by a Gauss sum. Section 6 is devoted to the actorization with Gauss sums: Implementations
2. Chirped pulses: Essentials
Throughout the article we take advantage of the technology of chirped laser pulses. Forthis reason we briefly summarize in the present section the key ideas and formulas ofthis field.A chirped laser pulse E ( t ) ≡ E (cid:2) e − i ω L t f ( t ) + c.c. (cid:3) (3)consists of an amplitude E , a carrier frequency ω L and a pulse shape function f ( t ) ≡ f exp (cid:20) −
12 (∆ ωf ) t (cid:21) . (4)Here we have introduced the complex-valued amplitude f ≡ r a a , (5)and the dimensionless parameter a ≡ ∆ ω φ ′′ (6)represents the second order dispersion. Moreover, ∆ ω denotes the bandwidth of thepulse and φ ′′ ≡ d φ ( ω ) / d ω is a measure of the quadratic frequency dependence of thephase of the laser pulse.When we substitute (5) into the exponent of the Gaussian in (4) we find that thepulse shape f ( t ) = f exp (cid:18) − ∆ ω a ) t (cid:19) exp (cid:18) − i a ∆ ω a ) t (cid:19) . (7)of a chirped pulse consists of the product of a real-valued Gaussian and a phase factorwhose phase is quadratic in time.Since the instantaneous frequency ν ( t ) ≡ dd t (cid:18) a ∆ ω a ) t (cid:19) = a ∆ ω a t (8)of the pulse is the derivative of this phase with respect to time, the frequency changeslinearly in time as the pulse switches on and off. For a positive value of φ ′′ we find anincreasing frequency, whereas a negative φ ′′ corresponds to a decreasing frequency. actorization with Gauss sums: Implementations ›› δω ω | g i| e i | m i Figure 1.
Model of ladder system. The ground state | g i is connected by a two-photontransition to the excited state | e i . We include an harmonic manifold of D ≡ M ′ + M +1intermediate states | m i with M ′ ≤ m ≤ M which are shifted by the offset δ m ≡ δ + m ∆with respect to the central frequency ω . The offset of the central state with m = 0 is δ , whereas neighboring states in the harmonic manifold are separated by ∆.
3. Chirping a two-photon transition
In Ref. [12] we have considered a two-photon transition in the ladder system of Figure 1driven by a chirped laser pulse. In the weak field limit, the probability amplitude tobe in the excited state results from the interference of multiple quantum paths eachcontributing a quadratic phase factor. Since the population in the excited state hasthe form of a continuous Gauss sum we can use this observable to factor numbers assuggested [1] in part I of this series.We first briefly summarize the essential results of Ref. [12]. Then we turn to ademonstration of the factorization capability and bring out the physical origin of thescaling property derived in Ref. [1] from mathematical arguments.
We consider a quantum system with a ground state | g i and an excited state | e i separatedby an energy 2 ~ ω . In the neighborhood of the midpoint of the energy difference weassume to have a manifold of equidistant energy levels as shown in Figure 1. Theiroff-set δ m ≡ δ + m ∆ (9)with respect to the central frequency ω is the sum of the off-set δ to the central stateand integer multiples m of the separation ∆ of two neighboring states of the manifold.At this point it is important to note that this decomposition of δ m is not unique.Indeed, we could have chosen a ”central” level which is different from the one indicatedin Figure 1. This choice would have changed the integers m . This ambiguity in thelabeling of the states is the deeper physical origin of the scaling property of the Gauss actorization with Gauss sums: Implementations c ( T P T ) e = e i γ S N (10)to be in the excited state after such a two-photon transition is given [12], apart fromthe phase factor exp( iγ ) by the Gauss sum S N ( ξ ) ≡ M X m = − M ′ w m exp (cid:20) π i (cid:18) m + m N (cid:19) ξ (cid:21) (11)discussed in part I of this series. Here the variable ξ ≡ δ ∆ π φ ′′ (12)is expressed in terms of the parameters δ and ∆ of the harmonic manifold of intermediatestates and the dimensionless chirp φ ′′ . Hence, by varying the chirp we can tune ξ . Forthis reason we call the variable ξ the dimensionless chirp.The number N ≡ δ ∆ (13)to be factored is represented by the ratio of the two characteristic frequencies of theladder.The weight factors w m ≡ ˜ ω m erfc (cid:18) i δ m ∆ ω √ − i a (cid:19) exp " − (cid:18) δ m ∆ ω (cid:19) (14)contain the complementary error function [13]erfc( z ) ≡ √ π ∞ Z z d u e − u (15)of complex argument z , and the abbreviation˜ ω m ≡ − π em Ω mg ∆ ω (16)involves the Rabi frequencies Ω mg and Ω em connecting the ground and the excited statewith the intermediate states, respectively. We are now in the position to discuss our factorization scheme. For this purpose wefirst address the experimental requirements and limitations and then present an exampledemonstrating the capability of this system to factor numbers. actorization with Gauss sums: Implementations The probability amplitude c ( T P T ) e , (10), of populating the excitedstate is of the form of a continuous Gauss sum discussed in part I of this series. Thus thedependence of the population | c ( T P T ) e | in the excited state on the dimensionless chirp ξ can be employed to reveal the factors of an integer N . For this purpose we encode N according to (13) in the parameters δ and ∆ of the harmonic manifold of intermediatestates. In order to apply the factorization scheme to a broad range of numbers N , werequire control over δ and ∆.We emphasize that the equidistant spacing within the harmonic manifold is essentialfor obtaining the Gauss sum and for our factorization scheme. Moreover, the dimension D ≡ M ′ + M + 1 of the intermediate levels has to be adapted to the number N tobe factored. The larger N the more intermediate states are required for a meaningfulsignal. For the factorization of N we require N ≤ D ≤ N where the lowest quantumnumber is bound by M ′ > N/ N : If the signal shows a distinctmaximum around the integer ξ = ℓ then ℓ is a factor of N . In order to resolve the signalin the vicinity of candidate prime numbers, we require sufficient stability of and accuracyin the chirp φ ′′ .According to Ref. [1] we also need to impose a restriction on the weight factor w m ,(14) of the contribution arising from the quantum path through the m th intermediatestate: Indeed, w m must be slowly varying as a function of m in order to ensurethat no specific excitation path is favored or discriminated. As a result the dipolematrix elements associated with all possible transitions should be of the same order ofmagnitude. Next we present numerical results for an artificial ladder systemconsisting of D intermediate states. Here, we have made the idealized assumption thatthe Rabi frequencies associated with the sequential path | g i → | m i → | e i are identical,that is Ω em Ω mg = const..In Figure 2 we display the population | c ( T P T ) e | of the excited state for the number N = 15 = 3 · ξ . For a single intermediatestate, there exist [12] several interfering quantum paths only if ξ <
0. However,for an equidistant manifold, the population is symmetric with respect to ξ , that is | c ( T P T ) e ( ξ ) | = | c ( T P T ) e ( − ξ ) | .The insets magnify the signal in the vicinity of trial factors ξ = 2 , , ξ = 3 and 5 the factors of N = 15. Incontrast, the signal at non-factors such as ξ = 2 or 7 does not show any characteristicfeatures. actorization with Gauss sums: Implementations ξ | c ( T P T ) e | . . . . . . . .
01 1 . . . . . . .
99 7 7 . . . . . . . . .
01 1 . . . . . . . . N = 15 Figure 2.
Factorization of N = 15 = 3 · | c ( T P T ) e | ofthe excited state given by (10) and (11) after a chirped two-photon transition throughthe intermediate levels of the ladder system of Figure 1. In the center we provide anoverview over the complete signal as a function of the dimensionless chirp ξ . Theinsets magnify the signal in the vicinity of candidate prime factors. Pronouncedmaxima at the prime factors ξ = 3 and 5 are clearly visible. In contrast, at non-factors ξ = 2 and 7 the signal does not exhibit any peculiarities. Parameters are δ = 0 . − , △ = 0 . − , △ w = 0 . − , a = − M + 1 = 23. In part I of this series [1] we have already shown that the realization of the Gauss sum S N for N is sufficient to reveal the factors of another number N ′ . Whereas the proofpresented in Ref. [1] had relied on mathematical arguments we now use the freedom inlabeling the levels of the equidistant ladder system to verify this surprising feature.For this purpose we recall from (13) that N is encoded in the ratio 2 δ/ ∆ describedby the offset δ of the reference level | m = 0 i and the level spacing ∆. However, whenwe choose a different reference level | k i with the associated offset δ ′ ≡ δ + k ∆ , (17)the number to be factored is N ′ ≡ δ ′ ∆ = N + 2 k. (18) actorization with Gauss sums: Implementations N to be factored in terms of theoffset δ and the detuning ∆ we find from (12) that the dimensionless chirp ξ = N ∆ π φ ′′ (19)is proportional to N .As a result, the dimensionless chirp ξ ′ ≡ N ′ ∆ π φ ′′ (20)corresponding to N ′ is related to ξ and N by the scaling transformation ξ ′ = N ′ N ξ. (21)Therefore, we can factor the number N ′ by analyzing the signal c ( T P T ) e , which wasrecorded in its dependence on ξ for N , using the new scale ξ ′ ≡ ( N ′ /N ) · ξ .
4. Engineering a one-photon transition
In the preceding section we have used a chirped two-photon transition going through anequidistant ladder system to factor numbers. Unfortunately, the requirements on thesystem are rather stringent and it is hard to identify quantum systems with such anarrangement of levels. For these reasons we now study more elementary models basedon a two-level system with a ground state | g i and an excited state | e i separated by anenergy ~ ω .We assume the excited state to have a permanent dipole moment ℘ ee which interactswith a time-dependent modulating field E m = E m( t ). In the following sections weconsider two cases: ( i ) a sinusoidal time dependence manifesting itself in a periodicmodulation of the excited state, and ( ii ) a quadratic chirp reflecting itself in a linearshift.In addition to E m we have a time-dependent weak driving field E d = E d( t ) causingtransitions between the ground and the excited state. Depending on the two cases E dis either a single chirped laser pulse, or a sequence of pulses.This arrangement corresponds to the interaction Hamiltonian V ≡ − ℘ ee E m( t ) | e ih e | − ( ℘ ge | e ih g | + c.c.) E d( t ) (22)where ℘ ge denotes the dipole moment of the two-level transition. Here we have assumedthat the frequencies of E m and E d are clearly separated. Hence, E m only acts on theexcited state and E d only on the transition.In the interaction picture the equations of motion for the probability amplitudes c e = c e ( t ) and c g = c g ( t ) to be in the excited and ground state read [14]i ddt c e ( t ) = − Ω ee ( t ) c e ( t ) − Ω ge ( t )e i ω t c g ( t ) (23)or i ddt c g ( t ) = − Ω eg ( t )e − i ω t c e ( t ) . (24) actorization with Gauss sums: Implementations time-dependent Rabi-frequenciesΩ ee ( t ) ≡ ℘ ee E m( t ) / ~ and Ω eg ( t ) ≡ ℘ eg E d( t ) / ~ (25)associated with the two electric fields E m and E d, respectively.In order to solve (23) and (24) we first recall that the strong field E m which causesthe modulation of the excited state appears through Ω ee in (22) and multiplies c e . Onlythe weak driving field E d which enters (22) via Ω ge and multiplies c g induces transitions.For this reason it suffices to describe this process by perturbation theory of first order.At time t the two-level system occupies the ground level, that is c g ( t ) =1 and c e ( t ) = 0. In the weak field limit the probability amplitude for the excitedstate does not change significantly under the action of a weak chirped pulse which yields c g ( t ) ≈
1. Hence, (23) reduces to the inhomogeneous differential equationi ddt c e ( t ) ∼ = − Ω ee ( t ) c e ( t ) − Ω ge ( t )e i ω t (26)where the interaction with the chirped laser pulse acts as an inhomogeneity.It is easy to verify that the solution of (26) reads c e ( t ) = i e i β ( t ) t Z t dt ′ exp[ − i β ( t ′ )] exp(i ω t ′ ) Ω ge ( t ′ ) (27)where we have introduced the phase β ( t ) ≡ t Z t dt ′ Ω ee ( t ′ ) = ℘ ee ~ t Z t dt ′ E m ( t ′ ) . (28)When we substitute the electric field E d( t ) ≡ E d (cid:2) e − i ω L t h ( t ) + c.c. (cid:3) (29)of amplitude E d , carrier frequency ω L and envelope h = h ( t ) into the solution (27) forthe probability amplitude c e we find in rotating wave approximation c e ( t ) = i Ω ge e i β ( t ) t Z t dt ′ exp[ − i β ( t ′ )] exp( iδt ′ ) h ( t ′ ) . (30)Here we have defined the time-independent Rabi frequencyΩ ge ≡ ℘ ge E d / ~ (31)associated with the electric field E d of the transfer pulse and the detuning δ ≡ ω − ω L between the atomic and the carrier frequency. actorization with Gauss sums: Implementations PSfrag replacements | g i| e i E d ( t ) E m ( t ) ∆ δ ω ( t ) ω ω L t Figure 3.
Engineering the Floquet ladder. We consider a two-level system withthe ground state | g i and the excited state | e i separated by the energy ~ ω . Theexcited state is modulated by a strong sinusoidal field E m = E m( t ), (32) giving riseto equidistant sidebands separated by ~ ∆. The one-photon transition is driven bya chirped laser pulse E d = E d( t ), (29), characterized by a linear variation of theinstantaneous frequency ω = ω ( t ).
5. Floquet ladder
So far we have neither specified the modulating nor the driving field. In the presentsection we consider a sinusoidal modulation of the excited state by a strong cw fieldcreating a set of equidistant sidebands, very much in the spirit of the harmonic manifoldof Figure 1. Moreover, we include a weak chirped laser pulse driving the transition,that is the envelope h = h ( t ) of E d is given by f ( t ) of (4). Figure 3 summarizes thisengineering of the Floquet ladder. We now evaluate the probability amplitude c ( F L ) e given by (28) and (30) for the excitationof the Floquet ladder. Here we proceed in two steps: we first include the modulationand then calculate the remaining integral for the case of a chirped pulse. In the case of the modulation field E m( t ) ≡ F cos(∆ t + ϕ ) , (32)with period 2 π/ ∆, amplitude F and phase ϕ , the time-dependent phase β = β ( t )defined by (28) takes the form β ( t ) ≡ κ sin(∆ t + ϕ ) . (33)Here we have chosen the lower integration limit t ≡ ( n π − ϕ ) / ∆ and have introducedthe dimensionless ratio κ ≡ Ω ee ∆ (34) actorization with Gauss sums: Implementations time-independent Rabi frequency Ω ee ≡ ℘ ee F / ~ and ∆.When we apply the generating function [13]exp (i κ sin θ ) = ∞ X n = −∞ J n ( κ ) e i nθ (35)of the Bessel function J n to evaluate the phase factor in the integrand of (30) we findthe probability amplitude c ( F L ) e ( t ) = i Ω ge e i β ( t ) X n J n ( κ ) e − i nϕ h n ( t ) (36)to be in the excited state. Here we have interchanged the order of integration andsummation and have introduced the integral h n ( t ) ≡ t Z t d t ′ exp (i δ n t ′ ) h ( t ′ ) (37)with the offset δ n ≡ δ − n ∆ (38)of the n -th satellite of the excited state in the splitted manifold.Hence, the modulation of the excited state causes equidistant sidebands and J n ( κ )determines the weight of the n -th sideband. In the language of Floquet theory we havereplaced the time dependent Hamiltonian by an infinite dimensional Floquet matrix. So far our calculation is valid for an arbitrary pulse shape h = h ( t ). We now perform the integration for the case of a chirped pulse where theenvelope h is given by the complex-valued Gaussian f defined by (4).Since we are interested in times after the pulse has interacted, that is for √ a / ∆ ω ≪ t we extend the upper and lower limits of the integration in (37) to+ ∞ and −∞ , respectively, which yields h n ( t ) ∼ = h n ≡ f ∞ Z −∞ d t e − (∆ ω f t ) +i δ n t , (39)or h n = √ π ∆ ω exp " − f (cid:18) δ n ∆ ω (cid:19) . (40)When we recall the definition (5) of f we can decompose this Gaussian into a real-valuedone and into a quadratic phase factor, that is h n = √ π ∆ ω exp " − (cid:18) δ n ∆ ω (cid:19) exp (cid:20) i δ n φ ′′ (cid:21) . (41)Here we have also made use of (6). actorization with Gauss sums: Implementations In the expression for h n given by (41) the off-set δ n which depends linearly on n entersquadratically. Hence, h n contains quadratic phase factors. In the present section wecast the probability amplitude c ( F L ) e given by (36) into the form of a Gauss sum whichallows us to factor numbers.With the help of (41) we find from (36) the formula c ( F L ) e = iΩ ge e i β ( t ) √ π ∆ w X n exp " − (cid:18) δ n ∆ w (cid:19) J n ( κ ) e − i nϕ exp (cid:20) i δ n φ ′′ (cid:21) . (42)Next we recall the definition (38) of δ n and express it by δ n ∆ w = − n − δ/ ∆∆ w/ ∆ ≡ − n − N ∆ n , (43)where in the last step we have introduced the abbreviations N ≡ δ ∆ and ∆ n ≡ ∆ w ∆ . (44)Likewise, we obtain from (38) the identity δ n = δ − ( n − n ∆2 δ )2 δ ∆ . (45)When we substitute (43) and (45) into (42), define the dimensionless chirp ξ ≡ δ ∆ π φ ′′ (46)and recall the definition (44) of N we arrive at the probability amplitude c ( F L ) e ( t ) = N ( t ) X n ˜ w n exp (cid:20) − i π (cid:18) n − n N (cid:19) ξ (cid:21) , (47)to be in the excited state in the Floquet-ladder scheme. Here, we have used theabbreviation N ( t ) ≡ π i Ω ge ∆ e i β ( t ) e i δ φ ′′ / (48)together with the weight factor˜ w n ≡ √ π ∆ n exp " − (cid:18) n − N ∆ n (cid:19) J n ( κ )e − i nϕ . (49)In order to reduce the influence of the Bessel function J n in ˜ w n we adjust themodulation index κ ≡ Ω ee ∆ = F ℘ ee ~ ∆ (50)determined by the microwave field, (32), such that ˜ w n is slowly varying as a function of n . For this purpose we recall [13] the asymptotic expansion J n ( z ) ∼ = r πz cos (cid:16) z − n π − π (cid:17) (51) actorization with Gauss sums: Implementations n ≪ z . Indeed, we find for thechoice κ ≡ πs + π , (52)where s is a large integer the approximation J n ( κ ) ∼ = r πκ ( ( − m for n = 2 m n = 2 m + 1 . (53)Thus all weight factors ˜ w n with odd index n vanish and the probability amplitude c ( F L ) e given by (47) reduces to c ( F L ) e ∼ = N X m w m exp (cid:20) − π i( m − m N ) ξ (cid:21) , (54)where w m ≡ π ∆ n √ κ exp " − (cid:18) m − N/ n (cid:19) e i m ( π − ϕ ) . (55)We note that for the choice of | ϕ | = π/ w m is unity and theonly m -dependence left results from the Gaussian. For an appropriate choice of ∆ n ,which according to (44) is determined by the bandwidth ∆ ω of the chirped pulse, thisGaussian is slowly varying.When we compare the probability amplitude c ( F L ) e to be in the excited state givenby (54) to the generic representation S ( ξ ; A, B ) ≡ X m w m exp (cid:20) π i (cid:18) mA + m B (cid:19) ξ (cid:21) (56)of a Gauss sum discussed in part I of this series [1] we find c ( F L ) e = N S ( ξ ; − , N ) . (57)Since the fluorescence signal of the excited state is proportional to the population | c ( F L ) e | in this state, it is proportional to the Gauss sum |S ( ξ ; − , N ) | . In order to gain information on the factors of an appropriately encoded number N we present now two schemes: The first one requires a continuous measurement of thefluorescence signal as a function of the dimensionless chirp ξ . For the second one itsuffices to acquire the fluorescence signal at integer values ξ = ℓ . actorization with Gauss sums: Implementations According to Ref. [1] ℓ is a factor, or a multiple ofa factor, of N if S ( ξ ; − , N ) given by (56) shows a pronounced maxima at ξ = ℓ . InFigure 4 we present numerical results for the factorization of N = 21 = 3 · M + 1 = 78 harmonics for two choices of the relativephase ϕ . Here, we display | c ( F L ) e | based on (47). Again we indicate candidate primefactors by vertical lines.For ϕ = π/ ℓ = 3and 7. In contrast, the signal does not show any peculiarities at non-factors such as ℓ = 2 and 5 as shown by the insets at the bottom.For ϕ = 0 the phase factor e i mπ = ( − m leads to oscillatory weight factors andour criterion of finding pronounced maxima at factors of N is not applicable here.Nevertheless, the signal | c ( F L ) e | still contains information on the factors of N . Indeed,the corresponding signal shown at the bottom of Figure 4 vanishes at the factors ℓ = 3 and 7 as indicated by the insets on the top but displays no peculiarities at non-factors such as ℓ = 2 and ℓ = 5 depicted at the bottom. Next we present another approach towards factorizationwith the help of the Gauss sum, (56). For this technique we assume that we havesufficient control over the rescaled chirp ξ to tune it precisely to an integer ξ = ℓ . Asa consequence, the term linear in the summation index in the phase factor drops outand the probability amplitude c ( F L ) e , approximately given by (57), is proportional to theGauss sum S N ( ℓ ) ≡ X m w m exp (cid:20) π i m ℓN (cid:21) . (58)Hence, we deal with a Gauss sum over purely quadratic phases [15].In Ref. [1] we have analyzed the properties of the function S N = S N ( ℓ ). Inparticular, we have shown that S N ( ℓ ) allows us to factor numbers in a ratherstraightforward way. Since S N ( ℓ ) approximates the excitation probability amplitude c ( F L ) e of a Floquet ladder given by (47), the occupation probability | c ( F L ) e | at integervalues ℓ of the dimensionless chirp ξ should yield information about the factors of anappropriately encoded number N .In Figure 5 we verify this statement by presenting numerical results for thefactorization of N = 105 = 3 · · | c ( F L ) e | is depicted only for integer values of the rescaled chirp ξ = ℓ . Moreover,data points with ℓ being a factor of N and their products arrange themselves on astraight line through the origin. Data points corresponding to integer multiples of afactor are characterized by identical values. On the other hand the signal is suppressedat non-factors of N in complete accordance with the predictions of Ref. [1]. actorization with Gauss sums: Implementations ξ | c ( F L ) e | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ξ | c ( F L ) e | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N = 21 ϕ = π/ N = 21 ϕ = 0 Figure 4.
Factorization of N = 21 = 3 · | c ( F L ) e | , (47), as a function ofthe continuous rescaled chirp ξ for the phase | ϕ | = π/ ϕ = 0 (bottom) of thecw-field. The electric field parameters are chosen to yield the width of the weight factordistribution ∆ n = 12 .
71 and the modulation index κ = 100 · π + π/
4. The positionsof candidate prime factors are indicated by vertical lines. The insets demonstrate thatthe signal exhibits pronounced maxima at the prime factors ℓ = 3 and 7 (top) but notat ℓ = 2 and 5 (bottom). For the phase ϕ = 0 the factorization criterion of observingdistinct maxima at factors of N does not apply. Here the factors are identified by zeros(top) rather than maxima. The non-factors have a non-vanishing signal (bottom). actorization with Gauss sums: Implementations ℓ | c ( F L ) e | N = 105 ϕ = π/ Figure 5.
Factorization of N = 105 = 3 · · | c ( F L ) e | , (47), for integer values ξ = ℓ of the rescaled chirp. At the primefactors ℓ = 3 , ℓ = 15 , ,
35 the signal displays maxima. Atnon-factors the signal is suppressed. Data points corresponding to factors of N aresituated on a line through the origin. Integer multiples of a factor are characterized bythe same value of the signal as illustrated by the two horizontal lines.. Since c ( F L ) e isonly an approximation of S N ( ℓ ), there are small deviations of this behavior. In orderto satisfy the criterion of slowly varying weight factors we have chosen the parameters∆ n = 90 and κ = 10 · π + π/
6. Pulse train
In this section we turn to yet another realization of a Gauss sum in a physical system. Incontrast to the method of the preceding section now the quadratic phase factors are notdue to a chirped laser pulse, but arise from the combination of a linear time-variationof the resonance condition and a pulse train as shown by Figure 6. The probabilityamplitude c ( P T ) e of excitation after a sequence of laser pulses follows from the sum overthe contributions from the individual pulses and is of the form of a Gauss sum. Againthe system is capable of factoring numbers. However, the roles of the trial factor andthe number to be factored are interchanged. We modulate the energy of the excited state by an electric field E m( t ) ≡ F tT , (59)which increases linearly in time. Here F denotes the amplitude of the field and T is atime scale.When we substitute this field into the definition (28) of the phase β we find theexpression β ( t ) = 12 Ω ee T t − β ≡ α ( t ) − β (60) actorization with Gauss sums: Implementations PSfrag replacements | g i| e i E d ( t ) ∆ ∼ E m ( t ) δ ω ( t ) ω ω L t − T + TT Figure 6.
Excitation by a pulse train. We consider a two-level system with groundstate | g i and excited state | e i . The modulation field E m = E m( t ), (59), causes a linearvariation of the excited state energy. Simultaneously a sequence of 2 M +1 delta-shapedweak laser pulses E d with envelope h = h ( t ), (61) and carrier frequency ω L = ω − δ induces a transfer to | e i . Consecutive pulses here depicted for M = 1 are separated by T . which contains the time-independent Rabi frequency Ω ee ≡ ℘ ee F / ~ and β ≡ Ω ee t / (2 T ).Moreover, we drive the one-photon transition with the electric field E d, given by(29) and consisting of a train h ( t ) ≡ M + 1 M X n = − M δ ( t − n T ) (61)of 2 M + 1 delta-shaped pulses separated by T . Here, we have chosen a normalization ∞ Z −∞ d t h ( t ) = 1 (62)The approximation of the pulse by a delta function reflects the fact that the temporalwidth of the individual pulses has to be small compared to T .When we substitute the pulse train h = h ( t ), (61), into (30) and perform theintegration we arrive at c ( P T ) e ( t ) = i Ω ge e i α ( t ) M + 1 M X n = − M exp (cid:20) i (cid:18) δ T n − Ω ee T n (cid:19)(cid:21) . (63)Here we have assumed that the range of integration in (30) is large enough to coverthe whole pulse train.Again the probability amplitude c ( P T ) e of excitation involves the sum over quadraticphase factors and is therefore a Gauss sum. Though in principle we could apply the same factorization scheme as in Sec. 5 wepropose here a more powerful technique for factorization. Indeed, by a proper choice of actorization with Gauss sums: Implementations n . For this purpose we relate thedetuning δ and the pulse separation T to the number N to be factored by N ≡ δT π . (64)With this choice we find for the quadratic phase12 Ω ee T n = 2 πn Nξ (65)where we have introduced the dimensionless variable ξ ≡ δ Ω ee . (66)As a consequence, the probability amplitude c ( P T ) e for the pulse train given by (63)reduces to c ( P T ) e ( t ) = i Ω ge e i α ( t ) A N ( ξ ) (67)and is governed by the Gauss sum A N ( ξ ) ≡ M + 1 M X n = − M exp (cid:20) − π i n Nξ (cid:21) . (68)In contrast to the Gauss sums of the preceding sections the roles of N and ξ areinterchanged. Indeed, now the variable ξ appears in the denominator and the number N to be factored in the numerator. In Sec.5.3 we have shown that the Gauss sum arising in the excitation of the Floquetladder reveals the factors of N for a continuous tuning of the chirp ξ as well as forinteger values ξ = ℓ . Likewise, the Gauss sum A N = A N ( ξ ) defined by (68) provides uswith the factors for continuous as well as integer values of ξ . However, the analysis forcontinuous ξ is more complicated and has been presented in Ref. [16]. For this reasonwe focus in the present section only on the discrete case.Since the Rabi frequency Ω ee is a free parameter we can adjust ξ to be an integer ℓ . As a result we arrive at the sum ‡A N ( ℓ ) ≡ M + 1 M X n = − M exp (cid:20) − π i n Nℓ (cid:21) . (69)We now demonstrate that A N is even more suited to factor numbers than thetwo Gauss sums S or S N given by (56) and (58), respectively. Whenever the integerargument ℓ is a factor q of N the phase of each phase factor of A N is an integer multipleof 2 π . As a consequence, each term in the sum is unity. Since the sum contains 2 M + 1terms the signal at a factor q of N takes on the maximum value of |A N ( q ) | = 1 . (70) ‡ In Ref. [1] we have shown that the Gauss reciprocity relation establishes the connection between thetwo types of Gauss sums S N and A N of (58) and (69), respectively. actorization with Gauss sums: Implementations ℓ | c ( P T ) e | N = 1911 Figure 7.
Factorization of N = 1911 = 3 · ·
13 using a train of 21 pulses. Themodulus | c ( P T ) e | of the signal, (67), exhibits clear maxima indicated by dashed lines atinteger arguments ℓ corresponding to factors of N . In Figure 7 we illustrate the power of this read-out mechanism of factors using theexample N = 1911 = 3 · ·
13. We find that already 21 pulses allow us to decidewhether ℓ is a factor of N or not.
7. Comparison of factorization schemes
We devote this section to a brief comparison of the factorization schemes based on theFloquet ladder and the pulse train discussed in Secs.5 and 6. Here we first concentrateon the methods of readout and then briefly address experimental requirements and thenecessary resources.In the Floquet-ladder approach two techniques to analyze the fluorescence signaldetermined by the population | c ( F L ) e | in the excited state and given by (47), offerthemselves: (i) We measure | c ( F L ) e | as a function of the continuous chirp ξ . In thiscase pronounced maxima at trial factors indicate factors of N . (ii) An alternative read-out relies on the measurement of the signal at integer values ξ = ℓ . Here we find thatthe signals at factors of N form a straight line through the origin.For the pulse-train approach the proposed read-out scheme is based on ameasurement of the signal | c ( P T ) e | , (67), at integer values of the argument ℓ . Factorsof N are characterized by the same maximal value, whereas the signal at non-factors issuppressed.Next we address the experimental requirements for these schemes to work. Toreveal the factors of a given number N it is sufficient to analyze the fluorescence signalfor values of the dimensionless chirp ξ in the interval [0 , √ N ]. For the continuous versionthe resolution in ξ has to be sufficiently high to resolve the shape of the signal in thevicinity of candidate primes. For the discrete scheme the signal has to be acquired onlyfor integer arguments ℓ . Nevertheless, we require precise control of ξ . When we compare actorization with Gauss sums: Implementations S N and A N necessary to find factors. In the approach based on the Floquet ladder this numberis determined by the width ∆ n of the weight factor distribution w m , (55). Indeed, thisdistribution has to be sufficiently broad in order to achieve a signal with an appropriatecontrast. For the pulse-train approach the number of terms contributing to A N isdetermined by the number of pulses in the train. Already with a few terms the signalhas enough contrast to bring out the factors.One may wonder whether the elimination of the phase linear in n in the pulse-trainapproach is also possible in the Floquet-ladder system. The basic idea was to chose thenumber N to be factored such that the term linear in the summation index drops outof the phase factor. Here, we had three parameters at our disposal: two are required toencode the number N and one parameter is free to vary the argument ℓ .In the Floquet-ladder approach we also have control over the three parameters δ, ∆ and φ ′′ for encoding both the number to be factored N and the dimensionlessargument ℓ . However, if all three would have been used to encode N , we would nothave a parameter left for controlling ℓ .
8. Conclusions
In the present article we have proposed three physical systems to implement three typesof Gauss sums. Our ultimate goal was to construct an analogue computer which wouldcalculate these Gauss sums. We have then analyzed the signal to deduce from it thefactors of an appropriately encoded integer N .Our first system is based on a two-photon transition in a ladder system driven by achirped laser pulse. Though this factorization scheme performs well for small numbersits performance for larger numbers is questionable since the required dimension D ofthe harmonic ladder needs to be of the order of N . Moreover, it is rather difficult tofind such an equidistant ladder system in nature.For this reason we have investigated two other systems. In the approach of theFloquet ladder a cw-field modulates the excited state of a two-level atom giving rise toa manifold of equidistant sidebands. When driven by a chirped laser pulse the resultingexcitation probability amplitude is a Gauss sum. The second technique is based on alinear chirp of the excited state energy. A pulse train of delta-shaped pulses ensuresthat the excitation probability amplitude is of the form of a Gauss sum. The origin ofthe quadratic phase factors is different in these two realizations of Gauss sums. In thefirst one they are due to the chirped laser pulse, whereas in the ladder approach theyoriginate from a linear chirp of the resonance condition.In all three examples the excited state probability is experimentally accessible via adetection of the fluorescence signal. Moreover, for each system we have developed rules actorization with Gauss sums: Implementations N . This feature is in contrastto Shor’s algorithm which achieves an exponential speed-up due to entanglement. Thenext challenge is to combine these ideas with entanglement and create a Shor-algorithmwith Gauss sums. However, this task goes beyond the scope of the present article andhas to await a future publication. Acknowledgments
We have profited from numerous and fruitful discussions with M Arndt, W B Case, BChatel, C Feiler, M Gilowski, D Haase , M Yu Ivanov, E Lutz, H Maier, M Mehring,A A Rangelov, E M Rasel, M Sadgrove, Y Shih, M ˇStefa´nˇak, D Suter, V Tamma, SWeber, and M S Zubairy. W M. would like to thank A Wolf for stimulating discussions.W M. and W P S acknowledge financial support by the Baden-W¨urttemberg Stiftung.Moreover, W P S also would like to thank the Alexander von Humboldt Stiftung and theMax-Planck-Gesellschaft for receiving the Max-Planck-Forschungspreis. I Sh A thanksthe Israel Science Foundation for supporting this work. Our research has also benefitedimmensely from the stimulating atmosphere of the
Ulm Graduate School MathematicalAnalysis of Evolution, Information and Complexity under the leadership of W Arendt.
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Optical Resonance and Two-Level Atoms (Dover Publications, NewYork)[15] Another quantity where purely quadratic phase factors occur is the autocorrelation function ofthe two-dimensional quantum rotor. See for example Mack H, Bienert M, Haug F, Straub f S,Freyberger M and Schleich W P 2002 in:
Proc. of the Enrico Fermi Summer School, Course actorization with Gauss sums: Implementations CXLVIII: Experimental Quantum Computation ed P Mataloni and F De Martini (Elsevier,Amsterdam); Merkel W, Crasser O, Haug F, Lutz E, Mack H, Freyberger M, Schleich W P,Averbukh I Sh, Bienert M, Girard B, Maier H and Paulus G G 2006
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