N-Photon wave packets interacting with an arbitrary quantum system

-Photon wave packets interacting with an arbitrary quantum system

Abstract

We present a theoretical framework that describes a wave packet of light prepared in a state of definite photon number interacting with an arbitrary quantum system (e.g. a quantum harmonic oscillator or a multi-level atom). Within this framework we derive master equations for the system as well as for output field quantities such as quadratures and photon flux. These results are then generalized to wave packets with arbitrary spectral distribution functions. Finally, we obtain master equations and output field quantities for systems interacting with wave packets in multiple spatial and/or polarization modes.

pacs:
03.67.-a,42.50.Ct, 42.50.Lc, 03.65.Yz

Nonclassical states of light are important resources for quantum metrology GioLor11 (); Leroux11 (), secure communication BevGrangier02 (), quantum networks Kimble08 (); MoeMauOlm07 (); Agh11 (), and quantum information processing Nielsen05 (); KLM01 (). Of particular interest for these applications are traveling wave packets prepared with a definite number of photons in a continuous temporal mode, known as continuous-mode Fock states BlowLouden90 (); Lou_book00 (); GarChi_book08 (); Roh07 (). As the generation of such states becomes technologically feasible BullColl10 (); Varcoe04 (); Belthangady09 (); Yamamoto06 (); Zeilinger06 (); Walmsley08 (); Silberberg10 (); Kuhn11 (); LeePatPar12 (); McKBocBoo04 (); SpecBocMuc09 (); KolBelDu08 () a theoretical description of the light-matter interaction GarPar94 () becomes essential, see Fig. 1.

Previously, aspects of continuous-mode single-photon states interacting with a two-level atom have been examined. Others have investigated master equations GheEllPelZol99 (); two-time correlation functions GheEllPelZol99 (); DomHorRit02 (); properties of scattered light DroHavBuz00 (); DomHorRit02 (); SheFan05 (); Kos08 (); CheWubMor11 (); ZheGauBar10 (); SheFan07b (); ZhoGonSun08 (); LonSchBus09 (); Roy11 (); Roy10 (); Ely12 (); and optimal pulse shaping for excitation StoAlbLeu07 (); WanSheSca10 (); StoAlbLeu10 (); RepSheFan10 (); Ely12 (). The results in these studies were produced with a variety of methods which have not been applied to many systems other than two-level atoms or Fock states where , however see YudRei08 ().

One way to approach such problems is through the input-output formalism of Gardiner and Collett ColGar84 (); GarCol85 (); GarZol00 (); YurDen84 (); Caves82 (). A central result of input-output theory is the Heisenberg-Langevin equation of motion driven by quantum noise that originates from the continuum of harmonic oscillator field modes GarZol00 (); qnoise_rmp10 (). The application of input-output theory to open quantum systems has historically been restricted to Gaussian fields GarCol85 (); DumParZol92 (); GarZol00 () —vacuum, coherent, thermal, and squeezed— with several notable exceptions Gar93 (); Car93 (); GheEllPelZol99 (); GJNphoton (); GJNCgen ().

Figure 1: (Color online) Schematic depiction of a traveling wave packet interacting with an arbitrary quantum system. The temporal wave packet is described by a slowly-varying envelope which modulates fast oscillations at the carrier frequency. We consider the case where the wave packet is prepared in a nonclassical state of definite photon number.

In this article we present a unifying method, based on input-output theory, for describing the interaction between a quantum system and a continuous-mode Fock state. Consequently our formalism encapsulates and extends previous results. Specifically our method allows one to derive the master equations and output field quantities for an arbitrary quantum system interacting with any combination of continuous-mode -photon Fock states.

This article is organized as follows: In Sec. I we introduce the white noise Langevin equations of motion, the mathematical description of quantum white noise, and the formal definition of continuous-mode Fock states. In Sec. II we present the first main results: the method for deriving master equations for systems interacting with continuous-mode Fock states and related output field equations. This result is then extended in Sec. III to continuous-mode “-photon states,” where the spectral density function is not factorizable. Then, in Sec. IV we apply our formalism to the study of a two-level atom interacting with wave packets prepared in -photon Fock states. This application is intended to serve as an instructive example that reproduces and extends results in previous studies StoAlbLeu07 (); WanSheSca10 (); StoAlbLeu10 (). In Sec. V, we present the second main result: master equations and output field quantities for a system interacting with Fock state wave packets in two modes (e.g. spatial or polarization). This sets the stage for the study of many canonical problems in quantum optics. As a two-mode example, we examine the scattering of Fock states from a two-level atom in Sec. VI. Finally, we conclude in Sec. VII with discussion and possible applications.

I Model and Methods

A description of a system interacting with a traveling wave packet naturally calls for a formulation in the time domain. The input-output theory developed in the quantum optics community provides such a description GarCol85 (); YurDen84 (); Caves82 (); DumParZol92 (); GarZol00 (); Gar93 (); Car93 (). Often input-ouput theory is formulated for a one-dimensional electromagnetic field, although this is not a necessary restriction DumParZol92 (). (Such effective one-dimensional models are typically thought about in the context of optical cavities Aoki09 () or photonic waveguides CheWubMor11 (); Spillane08 (); Vetsch10 (); Chang07 ().) In this formalism the rotating wave approximation, the weak-coupling limit (the Born approximation), and the Markov approximation are made VanHove55 (); vanHStoMab05a (). Strict enforcement of these approximations is known as the quantum white noise limit Accardi ().

In Appendix (A.1) we review the quantum white noise limit; other introductory material can be found in Refs. GarCol85 (); GarZol00 (); YanKim03a (); vanHStoMab05a (). The main result is a quantum stochastic differential equation (QSDE) for the unitary time evolution operator that governs the system-field dynamics. From this equation one can derive QSDEs for system and field operators driven by white noise, also known as white-noise Langevin equations. It is these equations of motion that lie at the heart of the derivation of Fock-state master equations.

The Langevin equations derived in the white noise limit are in Stratonovich form GarZol00 (); Louisell1973book (); GarChi_book08 (). Stratonovich QSDEs obey the standard rules of calculus, but expectations can be hard to calculate because the quantum noises do not commute with the operators to which they couple. Stratonovich QSDEs can be converted to an equivalent form known as the Itō QSDEs. In Itō form the quantum noises commute with the operators to which they couple, which facilitates taking expectations. However, differentials must be calculated to second order GarZol00 (). To derive master equations we will be taking expectations over field states and consequently will work solely with Itō QSDEs.

i.1 Derivation of the vacuum master equation from the Itō Langevin equations

Consider an arbitrary system operator in the interaction picture, , with the initial condition . The time evolution of is given by the Itō Langevin equation [see Appendix (A.3)]

(1)

where the action of the superoperator is

(2)

The operators act on the system Hilbert space. The quantum noise increments , , and are field operators, discussed in more detail shortly.

The first two terms in Eq. (I.1) describe smooth evolution from an external Hamiltonian on the system and from a Lindblad-type dissipator. The second two terms describe the influence of quantum noise through coupling of a system operator linearly to the field operators, e.g. dipole-type coupling. The final term arises from coupling of a system operator to a quantity quadratic in the field operators, such as photon number. Such effective couplings appear in optomechanical systems Van11 () and arise after adiabatic elimination of the excited states in multi-level atoms DeuJes09 (), for example.

Let us return to the discussion about the quantum noise increments , , and . These field operators are defined in terms of the fundamental field operators and whose time arguments are mode labels rather than indicators of time evolution. They are often referred to as white noise operators because they satisfy the singular commutation relations . This is akin to classical white noise which is -correlated in time. Due to the singular nature of and , it is preferable to work with the quantum noise increments:

(3)
(4)

which drive the Heisenberg dynamics in Eq. (I.1).

Under vacuum expectation, the calculus rules for manipulating QSDEs are summarized by the relations

(5)

These composition rules are often referred to as the vacuum Itō table.

As a prelude to the derivation of the Fock-state master equations, we derive the vacuum master equation. First, we take vacuum expectations of Eq. (I.1) using the following notation (to be explained in Sec. II): . Consequently, we need the action of the quantum noise increments on vacuum,

(6)
(7)

All of the quantum noise terms in Eq. (I.1) vanish under vacuum. Then, using the cyclic property of the trace we obtain the vacuum master equation:

(8)

where the Lindblad superoperator is defined as

(9)

and the subscripts on denote that Eq. (8) is a vacuum master equation.

i.2 Continuous-mode Fock states

A continuous-mode single-photon state Lou_book00 (); GarChi_book08 (); BlowLouden90 () can be interpreted as a single photon coherently superposed over many spectral modes Mil07 (); Mil08 () with weighting given by the spectral density function (SDF) ,

(10)

We focus on quasi-monochromatic wave packets, where the spectral spread is much smaller than the carrier frequency, 1. This holds for optical carriers, whose bandwidths are small relative to the carrier frequency. Then, we can define a slowly-varying envelope rotating at the carrier frequency,

(11)

where is near any relevant system frequencies. The Fourier transform of the slowly-varying envelope, , characterizes a square-normalized temporal wave packet, . In the time domain, and within the quasi-monochromatic approximation, the single-photon state in Eq. (10) becomes Lou_book00 (),

(12)

where we have absorbed the possible detuning from the system frequency into . The operator creates a single photon in the wave packet . Equation (I.2) can be interpreted as a superposition of instantaneous photon creation times weighted by the temporal wave packet. Since the white noise operators are defined in the interaction picture, it is clear that is a slowly-varying temporal envelope rotating at the carrier frequency. By focusing on quasi-monochromatic wave packets we ensure the approximations made in the quantum white noise limit are not violated.

A straightforward extension leads to the definition of normalized, continuous-mode Fock states (referred to hereafter as Fock states) in the wave packet with photons BlowLouden90 (),

(13a)
(13b)

The Fock states in Eq. (13) are a subset of more general -photon states for which the SDF is not factorizable  Roh07 (). In Sec. III, we define these states and use them to derive master equations.

Ii Fock State Master Equations

In this section we derive master equations for a quantum system interacting with a field prepared in a Fock state. The derivation is performed in the interaction picture where the time-dependent operators evolve according to Eq. (I.1). To facilitate the derivation we first introduce notation convenient for representing expectations with respect to a particular field state. It should be noted that our method is a generalization to -photon states of a method introduced in Refs. GJNphoton (); GJNCgen () for a single photon.

Assuming no correlations before the interaction, the total system is described by the product state

(14)

with the system in the state and the field in the Fock state . Using the Hilbert-Schmidt inner product for operators and ,

(15)

one can take expectations with respect to system and/or field states. For the following derivation it is necessary to define the asymmetric expectation value,

(16)

where is a joint operator on the system and field and is not necessarily separable. We use a convention where capital letters, denote the number of photons in the input field. Lowercase letters, that is, where , label “reference” Fock states to which the system couples. Using the Hilbert-Schmidt inner product, we define a set of generalized density operators , first introduced in Ref. GheEllPelZol99 (), by tracing over only the field in Eq. (16):

(17)

Such generalized density operators were also used in Refs. GJNphoton (); GJNCgen () for a single photon. We delay the interpretation of these generalized density operators until Sec. II.1.

As the trace in Eq. (16) is over both system and field, it gives a -number expectation value. Using the partial trace we also define an asymmetric partial expectation over the field alone which results in an operator. We define this operation with the notation 2,

(18)

We base our derivation on the Itō Langevin equations of motion for system operators. In this picture, the state remains separable and the expectations will always have the form of Eq. (16) and Eq. (18).

At this point we must mention an important technical issue. The composition rules for the quantum noise increments, expressed in Eq. (5), are generally modified for non-vacuum fields GarZol00 (); WisMilBook (). However, it is shown in Appendix B.2 that the Itō table for Fock states is identical to that for vacuum. This allows the techniques from input-output theory to be extended to Fock states.

ii.1 Fock-state master equations for the system

Recall the first step towards deriving the vacuum master equation, Eq. (8), was taking the expectation of Eq. (I.1) with respect to vacuum, i.e. . Analogously, to derive the Fock-state master equations we must take the asymmetric expectations, i.e. Eq. (16) or Eq. (18). The only explicit field operators in Eq. (I.1) are the quantum noise increments and . Consequently the action of the quantum noise increments on Fock states is needed:

(19a)
(19b)

In Appendix B.1 we show how to derive these relations. Equations (19) show how “reference” Fock states of different photon number couple through the quantum noise increments.

We are now equipped to derive the Fock-state master equations. From Eq. (18), we take the partial trace over Fock states for an arbitrary system operator , whose equation of motion is given by Eq. (I.1). Doing so yields the Heisenberg master equations:

(20)

To extract the Schrödinger-picture master equations, we make use of Eq. (17): . Then, using the cyclic property of the trace, we can write down the master equations for the system state:

(21)

This set of coupled differential equations is the main result of this section. The initial conditions for these equations are: the diagonal equations should be initialized with the initial system state , while the off-diagonal equations should be initialized to zero. In order to calculate expectation values of system operators for an -photon Fock state one needs only the top-level density operator . However, extracting requires propagating all equations between 0 and to which it is coupled. We note some special cases of Eq. (21) have been derived previously in Refs. GheEllPelZol99 (); GJNphoton (); GJNCgen () however little intuition or physical interpretation was given to these equations.

The master equations in Eq. (21) require further explanation. The diagonal terms, , are valid state matrices describing the evolution of the system interacting with an -photon Fock state for . For example, when we recover the vacuum master equation: which is the only closed-form equation in Eq. (21). For , the diagonal equations couple “downward” towards the vacuum master equation via the off-diagonal equations where . These off-diagonal operators are non-Hermitian of trace-class zero GheEllPelZol99 (); consequently they are not valid state matrices but do satisfy .

The fact that the equations couple downward means that we need only consider a finite set of equations, which can be integrated numerically and in some cases, analytically. For a field in an -photon Fock state there are equations. From the symmetry , the number of independent coupled equations reduces to .

Finally, we comment on the physical interpretation of these equations. Absorption of a photon by the system significantly changes a field prepared in a Fock state, so its dynamics are non-Markovian GheEllPelZol99 (); GJNphoton (). This necessitates propagating a set of coupled master equations. (In contrast, for coherent states photons can be removed while leaving the field state unchanged and a single master equation suffices.) Before the wave packet has interacted with the system is zero and only the top level equation contributes to the evolution of the system. In other words, the system evolves solely under the terms on the first line of Eq. (21), which describe evolution from an external Hamiltonian and decay due to coupling to the vacuum. When the wave packet begins to interact with the system, becomes nonzero and the other coupled equations contribute to the evolution of the system. Then, the information flow propagates upwards from to because the equations couple downwards.

So far we have discussed the dynamics of the system before and during the interaction. The last physically important observation is related to the correlation between the system and the outgoing field during and after the interaction. Consider the case where is bimodal. When the temporal spacing between the peaks is much greater than the characteristic decay time of the system and since is zero at these intermediate times, the coherence between the first peak of the wave packet and the system is lost before the second peak begins to interact. Thus only the top-level equation must be propagated at these times, and the only nonzero terms describe external Hamiltonian drive and decay into the vacuum. When the temporal spacing between the two peaks is on the order of the system decay time or shorter, then the initial temporal coherence between the peaks can affect the system.

ii.2 Output field quantities

In addition to system observables, we may also be interested in features of the output field 3. Consider a field observable with initial condition . We insert the Itō Langevin equation of motion for into the asymmetric expectations. Using Eq. (18) for the partial trace , the result is operator-valued Heisenberg master equations. We focus here on expectation values, , which are found by tracing over the system as well, as in Eq. (16). For two field quantities of interest – photon flux and field quadratures – we produce a set of coupled differential equations similar in form to Eq. (21). The initial conditions are and similarly .

Photon flux

The photon flux is given by , which counts the number of photons in the field in the infinitessimal time increment to (GarZol00, , Sec. 11.3.1). The rules of Itō calculus are used in Appendix A.2 to give the equation of motion for the output photon flux ,

(22)

Taking expectations over Fock states using Eq. (16) yields an equation for the mean photon flux,

(23)

The solution to this equation gives the integrated mean photon number up to time .

Field quadratures

A Hermitian field quadrature measurable via homodyne detection is described by

(24)

Following the same prescription, the equation of motion for the quadrature after the interaction is

(25)

Taking expectations over Fock states using Eq. (16) gives the mean homodyne current,

(26)

ii.3 General input field states in the same wave packet

So far we have considered the case where the input field is a “pure” Fock state. These results can be generalized to field states described by an arbitrary combination (superposition and/or mixture) of Fock states in the same wave packet. As the Fock states span the full Hilbert space, they form a basis for arbitrary states in the wave packet ,

(27)

The coefficients are constrained by the requirements of valid quantum states: , and .

When the input field is described by Eq. (27) the system state is

(28)

where are the solutions to the master equations. Generating the full, physical density operator for an arbitrary field requires combining the appropriate solutions from the hierarchy of coupled equations in Eq. (21) with associated weights . The Heisenberg master equation is found in the same manner,

(29)

Finally, the expectation value of a system operator is given by

(30)
(31)

This technique also applies to the output field quantities in Sec. II.2. Note that the definition of the Hilbert-Schmidt inner product, Eq. (15), gives rise to the conjugate coefficients in Eq. (28) but not in Eqs. (29, 31).

Iii General N-Photon Master Equations

In many experimental settings multiple photons are not created in Fock states. Fock states are a subset of more general -photon states, which have a definite number of photons but an arbitrary SDF . Indeed, a quantum tomography protocol for characterizing the SDF was recently proposed RohdeSep06 () and implemented WasKolRob07 (). This motivates the derivation of master equations for such fields.

In a single spatial and polarization mode, a general -photon state is

(32)

Again we assume quasi-monochromatic wave packets such that is a slowly-varying envelope with respect to the carrier frequency. Then, in the time domain a general -photon state can be written as

(33)

These states are not amenable to our analysis directly. Thankfully, a formalism for dealing with such -photon states has been developed Roh07 (); Ou06 ().

To describe -photon states we make use of the occupation number representation developed by Rohde et al. Roh07 (), which we review in Appendix C. Using Eq. (98), Eq. (33) can be written in a basis of orthogonal Fock states,

(34)

where is a normalized Fock state described by Eq. (13) with photons in basis function . Counting the number of subscripts on in Eq. (34) gives the total number of photons , and the value of any subscript reveals the basis function that photon is in.

In order to derive the master equation, we must first write down the action of the quantum noise increments on Eq. (34):

(35)
(36)

where is defined as

(37)

and is interpreted to mean that a single photon in one of the basis Fock states has been annihilated.

To derive the master equation for a system interacting with the field , an asymmetric expectation value needs to be defined for such states: . As before this defines the generalized density operators . Using these definitions the master equations for the generalized density operators are

(38)

Each master equations couples to a set of equations enumerated by the indices . The total number of equations required to describe such a state depends on the overlap of the initial wave packet with the particular choice of basis. Equations for the output field can also be derived for -photon states, but we omit them for brevity. Equations similar to Eq. (38) were derived in Ref. GheEllPelZol99 () for two photons but did not include or .

Finally, we can consider input fields in combinations (superpositions and/or mixtures) of different -photon states. In particular we allow the total state to be a combination of different states with the same photon number and a combination of states with different photon numbers. To describe such a state first we need to consider a general combination of -photon states. That is,

(39)

where the summation is over different states with the same photon number . Then we can sum over photon numbers to obtain the most general input field:

(40)

The coefficients and are constrained by the requirement that the input state be a valid quantum state. Using Eqs. (40) and (28), the equations for the system and output field can be found.

Iv Example: Fock-state master equations for a two-level atom interacting with a Gaussian wave packet

Efficient photon absorption is important for information transfer from a flying to a stationary qubit. In this section we analyze this problem with a study of the excitation probability and output field quantities for Fock states interacting with a two-level atom. This problem has been studied before in much detail for a single photon in Refs. StoAlbLeu07 (); WanSheSca10 (); StoAlbLeu10 (). Our intention is to make a direct connection to established results and then to extend those results to higher photon numbers. Consequently, we do not focus on optimizing wave packet shapes as other studies have StoAlbLeu07 (); WanSheSca10 (); StoAlbLeu10 (); RepSheFan10 ().

The single-mode approximation in Sec. I is rooted in the presumption that the wave packet can be efficiently coupled to the two-level atom. This has been considered in the case of a mode-matched wave packet covering the entirety of the solid angle in free space StoAlbLeu07 (); WanSheSca10 (). A more widely applicable context is that of strongly confined 1D photonic waveguides RepSheFan10 (). In such systems the coupling rate into the guided modes can be much larger than into all other modes , where the total spontaneous emission rate is  CheWubMor11 (); Chang07 (). In the following analysis, we take the idealized limit that coupling to all other modes can be fully suppressed and we set . To properly account for losses, a second mode can be introduced using the tools of Sec. V and finally traced over.

In Sec. IV.1, we examine the form of the master equation for the simple case of a two-photon Fock state. Next in Sec. IV.2 we numerically examine a two-level atom interacting via a dipole Hamiltonian with a wave packet prepared with at most two photons. First we reproduce the single-photon excitation results from prior studies, then we broaden these results to include two photons and output field quantities. Finally in Sec. IV.3 we present a numerical study for large-photon-number Fock states. This allows us to explore the relationship between excitation probability, bandwidth, interaction time, and photon number. For photon numbers , we identify a region of strong coupling.

iv.1 Two-photon Fock state master equations

Figure 2: (Color online) Comparison of a Gaussian wave packet of bandwidth in three initial field states: a single-photon Fock state (solid), a two-photon Fock state (dashed), and an equal superposition (dash-dot). The wave packet is shown in black filled grey. (a) Excitation probability of a two-level atom. (b) Photon flux. It is distinctly modified by interaction with the atom. (c) Integrated photon flux. For comparison the integrated single-photon flux is plotted when there is no atom.

It is instructive to examine the form of the master equation for the simple case of interaction with a two-photon Fock state where both photons are created in the same temporal wave packet , . From Eq. (21), the two-photon Fock state master equations are,

(41a)
(41b)
(41c)
(41d)
(41e)
(41f)

with the initial conditions:

(42)
(43)

Similar equations to Eqs. (41) were originally derived in Ref. (GheEllPelZol99, , Equations 71 (a)-(f)) for a two-level atom but without the operator and the term proportional to . For an arbitrary quantum system and single photon equations which include and the term proportional to were later derived in Ref. GJNphoton (). Then Ref. GJNCgen () showed how to propagate these equations for any superposition or mixture of one photon and vacuum.

Now suppose the input field is in a superposition of one and two photons, with . From Eq. (28) we combine the solutions to the master equations, Eq. (41), to get the physical state,

(44)

Notice that the last two terms of Eq. (44) originate in the coherences of the input field. It is interesting that the “off-diagonal,” traceless, generalized density operators (e.g. ) contribute to the calculation of physical quantities, albeit in Hermitian combinations. Had the field been a “pure” Fock state or a statistical mixture of one and two photons, these terms would not appear.

Output field quantities are calculated in the same fashion as Eq. (44). For example, the mean photon flux is,

(45)

where Eq. (31) was used to calculate .

iv.2 A two-level atom interacting with one- and two-photon Gaussian wave packets

Now we specialize to a wave packet prepared with up to two photons interacting on a dipole transition with a two-level atom initially in the ground state . In the absence of an external system Hamiltonian the master equation parameters are: , , , and the coupling rate is chosen for simplicity to be . We focus on a square-normalized Gaussian wave packet, as defined in Ref. WanSheSca10 (), whose peak arrives at time ,

(46)

with no detuning and frequency bandwidth . For Gaussian wave packets the simple relationship between bandwidth and temporal width enables us to explore the tradeoff between interaction time and spectral support around resonance 4.

To study the excitation probability we numerically integrate the master equations (41a)–(41f). Then, for a given input field state we calculate the excitation probability,

(47)

where is given by Eq. (28).

Figure 2(a) presents the excitation probability for a two-level atom interacting with a Gaussian wave packet Eq. (46) prepared in a “pure” Fock state of one and two photons as well as an equal superposition; in Eq. (44). In the simulations we use a bandwidth known to be optimal for single-photon Gaussian wave packets: StoAlbLeu07 (). This gives a maximum excitation probability of for as found in other works StoAlbLeu07 (); WanSheSca10 (); StoAlbLeu10 (). Putting a second photon in the wave packet slightly increases this to ; however, we see in Sec. IV.3 that this is not universal behavior for all bandwidths and photon numbers.

In Fig. 2(b) we plot the mean photon flux of the output field, , after interaction with the atom. For the single-photon wave packet, we see a drastic change in the output photon flux when the photon is being absorbed by the atom. For two photons, however, much of the wave packet travels through the atom undisturbed, since a two-level atom can absorb at most one photon. The related integrated mean photon flux, , is plotted in Fig. 2(c). For these “pure” one- and two-photon Fock states there exist a definite number of excitations. Any excitation induced in the atom through absorption of a photon eventually decays back into the field. This is shown in Fig. 2(c) where the integrated mean photon flux for long times approaches the number of initial excitations . During the absorption of the single-photon wave packet, the integrated intensity flattens out since the photon has been transferred to an atomic excitation and arrives only later after decay.

For a single-photon wave packet, the Schrödinger equation can be solved analytically for the excitation probability StoAlbLeu10 (); CheWubMor11 ():

(48)

The simulations in Fig. 2 agree with the analytic expression in Eq. (48). However, it is not clear that the method used to derive Eq. (48) can be extended to higher photon numbers.

iv.3 Excitation for large photon numbers

Figure 3: (Color online) (a) Maximum excitation probability of a two-level atom interacting with Gaussian wave packets of bandwidth for photon numbers . Small (large) bandwidths correspond to long (short) temporal wave packets. (b) Scaling of with photon number (red circles). The fit shown is (blue line). (c ) Scaling of with optimal bandwidth for each photon number (red circles). The fit is . Details of the fits can be found in the main text.

In this section we expand the numerical study of excitation probability to Gaussian wave packets of the form of Eq. (46) prepared Fock states with photon number .

Scaling

For small bandwidths (), see the left side of Fig. 3(a), one would expect a high probability of excitation from the substantial spectral support near the transition frequency of the atom. However, the long temporal extent of the wave packet means the photon density over the relevant interaction time scale is too small to significantly excite the atom WanSheSca10 (). A complementary way of understanding this is that the dissipative terms in the master equations [terms on the first line of Eq. (21)] prevail over the coherent coupling (terms on the other lines). By extending the analysis in Ref. GheEllPelZol99 (), we find a recursive scaling of the excitation probability for very wide wave packets: , where with .

In the other asymptotic regime where bandwidths are large (), see the right side of Fig. 3(a), the maximum excitation probability is small even for large photon numbers. This is due to the wave packet being so short that its bandwidth is spread over frequencies far from the atomic resonance. We numerically find the asymptotic scaling for with for photon numbers .

At intermediate bandwidths, we note several interesting features. First, the maximum excitation probabilities are not universally ordered by photon number and adding photons to the field can decrease . In fact, there exists a bandwidth region in Fig. 3 where a single photon in the wave packet is optimal for excitation, .

Second, for each photon number there exists an optimal bandwidth for excitation. In Fig. 3 (b) we have plotted the absolute maximum of (maximized over and ) as a function of the number of photons. We find excellent agreement () by fitting to the model over the range with coefficients (95% confidence): . Therefore the absolute maximum of does monotonically increase with , but with diminishing returns.

In Fig. 3 (c) we investigate the optimal bandwidth for excitation for each photon number . Fitting to the model gives and with 95% confidence and . Thus, to achieve this scaling for photon number , the optimal bandwidth of the wave packet is . Thus, the optimal width seems to be proportional to the single-photon optimal bandwidth, .

Dynamics

Finally we illustrate the excitation probability dynamics. Figure (4) shows for bandwidths , chosen to illustrate three types of behavior. In each subplot (a)-(c), excitation curves are plotted for photon numbers .

In Fig. 4(a) a short pulse quickly excites the atom, which then decays into vacuum with rate after the wave packet leaves the interaction region. Larger photon number corresponds directly to larger maximum excitation. In the intermediate bandwidth regime, , excitations can be coherently exchanged between the atom and field, leading to oscillations in the excitation probabilities. This continues until the wave packet leaves the interaction region as shown in Fig. 4(b). Similar damped Rabi oscillations were observed for large-photon-number coherent state wave packets in Ref. (WanSheSca10, , Fig. 5). For a single photon in the field, these oscillations are never seen due to the tradeoff between spectral bandwidth and photon density DomHorRit02 (); SilDeu03 (). At the chosen bandwidth , a single photon achieves the highest maximum excitation with maximum excitation falling off roughly with photon number in agreement with Fig. 3. Finally, in Fig. 4(c) we see that an atom interacting with a long wave packet is excited and then decays well within the wave packet envelope and the curves are nearly symmetric around the peak of the wave packet for all photon numbers .

Figure 4: (Color online) Excitation probability of a two-level atom interacting with Gaussian wave packets of bandwidth prepared with photons. Highlighted are (solid), (dashed), and (dash-dot). The wave packet is plotted in black filled grey (normalized in (a) for clarity). (a) Behavior of short temporal wave packets (large bandwidths) shows is ordered by photon number. (b) For intermediate bandwidths, we see damped Rabi oscillations, discussed in Sec. IV.3.3. Note that is not necessarily ordered. (c) Behavior of long temporal wave packets (small bandwidths) where is again ordered. Note the different time scales in (a), (b), and (c).

Strong coupling

Figure 5: (Color online) Comparison of the numerically-calculated (dark blue) and analytically-predicted (dashed orange) Rabi oscillations for rectangular wave packets (normalized for clarity) with photons. (a) Wave packet length large compared to . (b) Wave packet length approaching the limit . We see increasing agreement between prediction and our numerics.

The damped Rabi oscillations seen in Fig. 4(b) suggest that there is a regime where coherent processes dominate over dissipation, known in cavity QED as the strong coupling regime. The authors of Ref. SilDeu03 () defined a strong coupling parameter (for very short rectangular wave packets): where . Specifically the wave packet was taken to be for times and zero otherwise. In this limit they showed that full Rabi oscillations for photons occur at frequency . In Fig. 5 we compare their analytically-predicted excitation oscillations with our numerical calculations for photons. In (a), the wave packet is long compared to and, while the oscillation frequencies match, the amplitudes do not due to dissipation. For short wave packets, as seen in (b), coherent coupling prevails over dissipation, we see excellent agreement with the predicted frequency (in our parameters: ) and good agreement with the predicted amplitude.

For non-rectangular pulses the frequency of the Rabi oscillations is time-dependent as seen in Fig. 4(b). We must account for the time variation of the wave packet in order to define a more general strong coupling parameter. To achieve strong coupling, the coherent coupling rate into the guided modes must dominate the total relaxation rate . We can immediately define the condition for instantaneous strong coupling: . However, in order to see interesting dynamics such as a complete Rabi oscillation, the coupling must remain strong over a characteristic timescale . From this argument we define an average strong coupling parameter,

(49)

If, for any wave packet , there is a value of such that Eq. (49) is much greater than one, then average strong coupling has been achieved over the time window .

A natural choice for is the characteristic decay time of the atom, . In Fig. 6(a) we present a contour plot of the average strong coupling parameter for Gaussian wave packets prepared in a single-photon Fock state (). Ideal coupling to the guided mode is assumed, . We see that, for any bandwidth, maximum coupling occurs when the time window is centered at the Gaussian peak (indicated by the vertical, dashed white line) and that the strongest coupling is achieved for Note that although the average strong coupling parameter for a single photon never exceeds one, for larger photon numbers the factor can lead to significant coupling. In Fig. 6(b) the excitation probability dynamics are shown for an optimal bandwidth wave packet. We see the appearance of damped Rabi oscillations when the wave packet has photons that are completely absent when only a single photon is in the field. For comparison, a wave packet of bandwidth is shown in Fig. 6(c). Even at this bandwidth, damped Rabi oscillations appear for photons, albeit with reduced contrast and frequency.

Figure 6: (Color online) (a) Contour plot of the average strong coupling parameter for a Gaussian wave packet prepared with a single-photon as a function of center of the time window () and bandwidth (where ). (b) and (c): Excitation probability of a two-level atom interacting with a wave packet of bandwidths for (b) and for (c). Only and photons are shown. The normalized wave packets are shown in black filled grey.

V Two-mode Fock state master equations

In this section we derive the master equations for a system interacting with an arbitrary combination of continuous-mode Fock states in two modes (spatial or polarization). This generalization allows one to consider wave packets scattering off of atoms or addressing multiple dipole transitions, for instance. The analysis for two modes is conceptually identical to but algebraically more complicated than the single-mode case.

v.1 Multi-mode Itō Langevin equations

The evolution of a system operator driven by multiple quantum noises is given by the multi-mode Itō Langevin equation,

(50)

where the modes are labeled by the subscripts and repeated indices are summed. is an external system Hamiltonian, the operator couples the system to the th field mode, and the scattering operator is constrained by: and (see (GouGohYan08, , Appendix A), (GouJam09, , Sec. IV) and GouJam09b () and the references therein for more details on multi-mode QSDEs). Note that the subscript on the multi-mode quantum noise increments has been dropped for notational compactness in favor of the mode labels . The multi-mode quantum noise increments are defined,

(51)

The composition rules for these quantum noises increments under Fock state expectation are