Quantum Filtering (Quantum Trajectories) for Systems Driven by Fields in Single Photon States and Superposition of Coherent States

Quan tum Filtering (Quan tum T ra jectories) for Systems Driv en b y Fields in Single Photon and Sup erp osition of Coheren t States John E. Gough, 1 , ∗ Matthew R. James, 2, 3 , † Hendra I. Nurdin, 3 , ‡ and Josh ua Combes 3, 4 , § 1 Institute for Mathematics and Physics, A b erystwyth University, SY23 3BZ, Wales, Unite d Kingdom 2 ARC Centr e for Quantum Computation and Communic ation T e chnolo gy 3 R ese ar ch School of Engineering, Austr alian National University, Canberr a, ACT 0200, Austr alia 4 Dep artment of Physics and Astr onomy, University of New Mexic o, A lbuquer que NM 87131-0001, USA (Dated: Nov ember 12, 2018) W e deriv e the sto chastic master equations, that is to sa y , quantum filters, and master equations for an arbitrary quantum system prob ed by a contin uous-mo de b osonic input field in tw o types of non-classical states. Sp ecifically , w e consider the cases where the state of the input field is a sup erposition or combination of: (1) a con tinuous-mode single photon wa ve pack et and v acuum, and (2) an y con tinuous-mode coheren t states. I. BA CKGR OUND AND MOTIV A TION The pro duction and v erification of non-classical states of light, such as single-photon states [1] and sup erp osi- tions of coherent states (also known as Schr¨ odinger cat states) [2 – 4], has become routine. In particular, the pro- duction of single photon states has b een ac hieved in a v a- riet y of experimental arc hitectures such as: cavit y quan- tum electrodynamics (QED) [5, 6], quan tum dots in semi- conductors [7], and recently in circuit QED [8]. Suc h non-classical states hav e b een considered in connection with quantum computing [9, 10] and secure communica- tion [11] o ver quan tum netw orks [12]. A basic problem in quantum optics concerns the ex- traction of information ab out a system of interest (t wo- lev el atom, cavit y mode, etc) from light scattered b y the system, Figure 1. Based on measurements of the scat- tered, or output, light, one can determine a conditional state from whic h one can mak e estimates of observ ables of the system. A general approach to estimation prob- lems of this kind, called filtering problems, w as dev elop ed b y Belavkin [13]-[16] within a framework of contin uous non-demolition quan tum measuremen t in the case where the input prob e field, B ( t ) in Figure 1, is a quan tum white noise with v acuum state (or more generally Gaus- sian state, see [17]- [20]). Belavkin’s form ulation, which generalizes the classical nonlinear filtering theory [21], is quite general. F or example, in the schematic repre- sen tation of a contin uous measurement process shown in Figure 1, the measurement signal Y ( t ) pro duced by a de- tector (e.g. photon counter or homodyne detector) may b e the num b er of quanta in the output field, or alterna- tiv ely it may be a quadrature of the output field. One obtains a filtering equation whic h is a sto c has- tic differential equation of the state % ( t ) conditioned on ∗ jug@aber.ac.uk † Matthew.James@anu.edu.au ‡ School of Electrical Engineering and T elecomm unications, The Universit y of New South W ales, Sydney , NSW 2052, Australia; h.nurdin@unsw.edu.au § combes@unm.edu measurement signal estimates system detector input output filter FIG. 1. A sc hematic representation of a contin uous measure- men t pro cess, where the measurement signal produced by a detector is filtered to pro duce estimates ˆ X ( t ) = π t ( X ) = tr[ ρ ( t ) X ] of system op erators X at time t . Y ( t ): in the later terminology employ ed in quan tum op- tics, the output is referred to as a quan tum tra jectory and the filtering equation as a sto c hastic master equation [22 – 24]. Av eraging o ver the measured output is equiv a- len t to a non-selectiv e measurement, and the corresp ond- ing state will satisfy the corresp onding master equation. The choice of detection sc heme on the output field deter- mines the particular selectiv e ev olution, usually referred to as an unra velling of the master equation in quantum optics. T o date, quantum tra jectories and quantum fil- tering hav e only b een dev elop ed for input fields that are in a Gaussian state, with sp ecific cases b eing coherent state fields (this includes v acuum fields as special case), thermal fields, and squeezed fields [25 – 27]. While the re- sulting equations allow us to estimate non-commutating observ ables of the monitored system, the Gaussian nature of the inputs ensure that they app ear formally similar to the classical equations. The aim of the present paper is to extend the theory to classes of non-classical inputs. In this article we extend Bela vkin’s quantum filtering theory and the input-output theory of quantum optics [22, 27] to non-Gaussian contin uous-mo de states ρ field whic h are sup erpositions or combinations of a) a con tinuous-mode single photon and v acuum, and b) contin uous-mo de coherent states, i.e. contin uous- mo de cat-states. The problem to b e tackled here is to deriv e the master and stochastic master equations for a “system” ⊗ “field” 2 with initial state ρ 0 ⊗ ρ field and unitary evolution process U ( t ) when the field state is one of the states ab ov e. T o mak e the problem tractable, we seek a larger representa- tion of the form “extended system” ⊗ “field” where “extended system”=“ancilla” ⊗ “system” suc h that, for all system observ ables X , tr system ⊗ field n ρ 0 ⊗ ρ field U ( t ) † ( X ⊗ I ) U ( t ) o = tr ancilla ⊗ system ⊗ field  ρ a ⊗ ρ 0 ⊗ P v ac ˜ U ( t ) † ( R ( t ) ⊗ X ⊗ I ) ˜ U ( t )  (1) where ˜ U ( t ) is a unitary evolution pro cess coupling the ancilla, system, and field, ρ a is a fixed state of the an- cilla, P v ac = | 0 ih 0 | is the v acuum state (pro jection) for the contin uous-mo de field, and R ( t ) is some pro cess tak- ing v alues in the observ ables of the ancilla. The filter- ing problem may then b e solved for the extended system with reference to the v acuum state for the field using traditional techniques. The extension to single photon states is interesting for foundational reasons [28] as well as the aforementioned tec hnological reasons [9]. Lik ewise, quantum filtering for cat states is of foundational imp ortance, while practical uses would b e tow ards quan tum enhanced metrology [29]. One p ossible application would b e to quantum enhanced metrology of a time v arying parameter [30, 31]. This article is structured as follo ws. In Section II w e review standard input-output theory . Sp ecifically w e consider the idealized quantum white-noise mo del of a quan tum sto c hastic differential equation (QSDE) and use it to derive the master equations and quan tum tra- jectories for Gaussian fields. Then we review a general parametrization to sp ecify the system en vironment cou- pling for input-output systems. Using this parametriza- tion we review the metho ds, recently in tro duced [32 – 34], to simplify and formalize the netw ork theory of cascaded op en quantum sys tems and quantum feedback netw orks. Section II I is fo cused on deriving the master equation and sto c hastic master equation (quan tum filter) driven b y contin uous-mo de single photon w av e pack ets. W e generalize the single photon filter to an y sup erposition or combination of single photon and v acuum input field. The system that is prob ed is left arbitrary so in general our filter can apply to qubits, qudits and mec hanical os- cillators. As an example we calculate the single photon filter for a tw o level atom (or qubit) dispersively cou- pled to the field. W e derive the tra jectories for b oth a homo dyne type measuremen t and a photon counting measuremen t. In Section IV we present the extension to sup erposi- tions of coheren t states. W e deriv e the cat-state-filter for an arbitrary quantum system and an arbitrary cat state. Again we illustrate the filtering equations with a qubit system and homo dyne and photon coun ting mea- suremen ts. In Section V we conclude and discuss our future re- searc h and some op en questions. Notation The comm utator and an ti-commutator will be denoted as [ A, B ] = AB − B A and [ A, B ] + = AB + B A , resp ectiv ely . W e set D A B ≡ A † B A − 1 2 ( A † AB + B A † A ) and D ? A B ≡ AB A † − 1 2 ( A † AB + B A † A ). The scattering, coupling and Hamiltonian op erators describing a giv en Mark ovian open system coupling will b e written as a triple G = ( S, L, H ), to b e explained in more detail in Section I I A, and this provides an op erator- v alued parameterization of the system. The asso ciated sup eroperators are Lindbladian : L G X ≡ − i [ X , H ] + D L X, Liouvil lian : L ? G ρ ≡ − i [ H , ρ ] + D ? L ρ, and note that, for traceclass ρ and b ounded X , tr { ρ L G X } = tr { X L ? G ρ } . I I. MODELS OF OPEN QUANTUM SYSTEMS In this section w e briefly review quan tum stochas- tic calculus (input-output theory) and quantum filtering (tra jectories) for a system coupled to a heat bath mo d- elled as a b oson field in the v acuum state. A. Input-Output Mo del Using QSDEs Hudson and Parthasarath y [35, 36] sho wed how to di- late a dissipated completely p ositive semigroup evolution, with Lindblad generator, to a unitary mo del on the sys- tem space with a (Bose) F ock space ancilla. Here they dev elop ed an analogue to the It¯ o theory of sto chastic in- tegration with resp ect to creation, annihilation and scat- tering pro cess B † ( t ) , B ( t ) and Λ ( t ). They show ed the existence and uniqueness of solutions to unitary quan tum sto c hastic differential equations of the form dU ( t ) =  ( S − 1) d Λ ( t ) + LdB † ( t ) − L † S dB ( t ) − ( 1 2 L † L + iH ) dt  U ( t ) . (2) where G = ( S, L, H ) consists of a unitary S describing photon scattering phase, a b ounded op erator L describing coupling to the creation mode of the field, and a b ounded Hermitean op erator H describing the system Hamiltonian. (The result has b een extended to non-b ounded co efficien ts.) The increments are future p ointing op erator-v alued It¯ o incremen t, that is dB ( t ) ≡ B ( t + dt ) − B ( t ) is a for- w ard of the quantum noise. In particular, we hav e. 3 [ U ( t ) , dB ( t )] = [ U ( t ) , dB † ( t )] = [ U ( t ) , d Λ( t )] = 0. The full quantum It¯ o table is × dt dB d Λ dB † dt 0 0 0 0 dB 0 0 dB dt d Λ 0 0 d Λ dB † dB † 0 0 0 0 . (3) More generally , for quan tum sto c hastic in tegral pro cesses X ( t ) , Y ( t ), one has the It¯ o product rule d ( X ( t ) Y ( t )) = ( dX ( t )) Y ( t ) + X ( t ) d Y ( t ) + dX ( t ) d Y ( t ) . Indep enden tly , Gardiner and Collett dev elop ed an equiv alent quan tum input-output theory [26, 27] based on Lehmann-Symanzik-Zimmermann scattering theory of Bose white noise pro cesses. F ormally one b egins with singular fields satisfying  b ( t ) , b † ( s )  = δ ( t − s ) , with the connection to the regular processes being for- mally B † ( t ) = Z t 0 b † ( s ) ds, B ( t ) = Z t 0 b ( s ) ds, Λ ( t ) = Z t 0 b † ( s ) b ( s ) ds. The quantum sto c hastic calculus may then b e under- sto od as effectively arising through Wick ordering of the singular fields. The multiple input version is relatively straightfor- w ard. W e hav e n indep enden t inputs b j and with B j ( t ) = R t 0 b j ( s ) ds , Λ j k ( t ) = R t 0 b † j ( s ) b k ( s ) ds , etc., w e ha v e dU ( t ) =    X j k ( S j k − δ j k ) d Λ j k ( t ) + X j L j dB † j ( t ) − X j k L † j S j k dB k ( t ) −  1 2 X j L † j L j + iH  dt    U ( t ) , where we now hav e parameterizing op erators S =    S 11 . . . S 1 n . . . . . . . . . S n 1 . . . S nn    , L =    L 1 . . . L n    , H with S unitary and H self-adjoin t. F or simplicit y we treat the case of a single input and output. B. Heisen b erg-Langevin Equations The Heisen b erg dynamics of arbitrary system operator X is defined by transforming to the Heisenberg picture j t ( X ) = U † ( t )( X ⊗ I field ) U ( t ) . (W e will usually drop the subscripts “system” and “field’ when there is no confusion.) F rom the quan tum It¯ o prod- uct rule and table one deduces the QSDE for a system op erator j t ( X ) = X ( t ): with all system op erators trans- formed to the Heisenberg picture. d j t ( X ) = j t ( L X ) dt + dB † ( t ) j t ( S † [ X, L ]) + j t ([ L † , X ] S ) dB ( t ) + j t ( S † X S − X ) d Λ( t ) . (4) C. Deriv ation of the Master Equation Supp ose that the system is in an initial state ρ (0) = ρ 0 and that the join t state of the system and bath is ρ 0 ⊗ P v ac where P v ac = | 0 ih 0 | is pro jection on to the v acuum state of the field. The state of the system, % ( t ), obtained by a veraging ov er the en vironment at a giv en time t is then % ( t ) = tr field  U ( t )( ρ 0 ⊗ P v ac ) U † ( t )  . (5) W e wish to obtain a differen tial equation for the av erage of an observ able X of the system at time t : $ t ( X ) = tr system ⊗ field { j t ( X ) % 0 ⊗ P v ac } ≡ tr system { ρ ( t ) X } , and from the Heisenberg-Langevin equation (4) w e ha ve d$ t ( X ) = tr system ⊗ field { d j t ( X ) ρ 0 ⊗ P v ac } = tr system ⊗ field { j t ( L G X ) ρ 0 ⊗ P v ac } dt, as the incremen ts dB , dB † , d Λ v anish in the v acuum state. W e therefore obtain the equation d$ t ( X ) dt = $ t ( L G X ) , $ 0 ( X ) = tr system { ρ 0 X } whic h ma y then b e expressed as the master equation d% ( t ) dt = L ? G % ( t ) ≡ − i [ H, % ( t )] + D ? L % ( t ) , (6) with initial data ρ 0 . Note that the master equation (6) is a consequence of the QSDE model. D. The Input-Output Relations The output field B out is obtained from the input by mo ving in to the Heisenberg picture: B out ( t ) = U ( t ) † ( I system ⊗ B ( t )) U ( t ) ≡ U ( τ ) † ( I system ⊗ B ( t )) U ( τ ) for any τ ≥ t . Again from the quan tum It¯ o calculus we find dB out ( t ) = j t ( S ) dB ( t ) + j t ( L ) dt. (7) Note that the output field again satisfies the canonical comm utation relations. 4 system output input FIG. 2. An op en quantum system. The input field (before in teraction) is represented b y the operator B ( t ) and output field (after interaction) is denoted by B out ( t ). E. Deriv ation of the Quan tum Filter (Sto c hastic Master Equation) - Quadrature Case W e supp ose that we contin uously monitor the quadra- ture phase using p erfect (100% efficiency) homo dyne de- tection. This entails measurement, for each t ≥ 0, of the field Y ( t ) = B out ( t ) + B † out ( t ) ≡ U ( t ) † ( I system ⊗ Q ( t )) U ( t ) where Q ( t ) = B ( t ) + B † ( t ). W e note that the set of ob- serv ables { Y ( t ) : t ≥ 0 } is self-comm uting and we may si- m ultaneously diagonalize (and measure!) all observ ables. A t any time t , we ma y additionally estimate an observ- able that commutes with the observ ables up to time t . This includes observ ables X ( τ ) for τ ≥ t , since [ X ( τ ) , Y ( t )] = U ( τ ) ∗ [ X ⊗ I field , I system ⊗ Q ( t )] U ( τ ) ≡ 0. This is the non-demolition prop erty . Quantum filtering is the estimation of j t ( X ) based on observ ations of the output pro cesses { Y ( s ) : 0 ≤ s ≤ t } . Fig. 1 depicts the scenario w e are considering. F rom the It¯ o calculus we see that d Y ( t ) = ( L ( t ) + L † ( t )) dt + dQ ( t ) . Defining the expectation E [ · ] = tr { ρ 0 ⊗ ρ field ( · ) } for a giv en state ρ 0 ⊗ ρ field , we seek to minimize E [( ˆ X ( t ) − j t ( X )) 2 ] o ver all observ ables ˆ X ( t ) in the algebra Y t generated by { Y ( s ) : 0 ≤ s ≤ t } . The minimizer is called the least- squares estimator for X ( t ) given { Y ( s ) : 0 ≤ s ≤ t } and will b e denoted as ˆ X ( t ) = π t ( X ) = E [ j t ( X ) | Y t ] . (8) The later notation suggest that in π t ( X ) is the condi- tional exp ectation of j t ( X ) given the past history , which w ould b e the classical interpretation. While conditional exp ectations generally do not exist in the quantum prob- abilistic setting, the nondemolition prop erty ab o ve suf- fices to allow one to realize precisely this interpretation, see for instance [19, 37]. The conditional exp ectation can indeed b e interpreted as an orthogonal pro jection onto a subspace of commuting op erators Y t . This means that j t ( X ) − π t ( X ) is orthogonal to this measurement sub- space Y t , that is, E [( j t ( X ) − π t ( X )) C ] = 0 (9) for all op erators C b elonging to the measuremen t sub- space Y t , [19]. Setting C = I shows that E [ π t ( X ))] = E [ j t ( X )] . Qua ntu m S toc hast ic Dif fer ential Eq uat ion Mas ter Eq uati on Sto chastic M ast er Equ ati on (Q uan tum Fi lter ) Qua ntu m c ond iti onal exp ect ati on Qua ntu m e xpe cta tion cla ssical exp ect ati on FIG. 3. Relationship b etw een QSDEs for the unitary , system and environmen t; master equation; and sto chastic master equation (filter) for op en quan tum systems ( H L is given by (14). The difference b etw een the master equation and stochastic master equation is due to the difference in the type of exp ectation take, i.e. unconditioned or conditioned, resp ectiv ely . 5 No w let us return to the v acuum state for the field: ρ field = P v ac . W e shall recall a simple deriv ation of the filter using an analogue of a the characteristic function tec hnique of classical filtering [38]. W e in tro ducing a pro- cess C ( t ) satisfying the QSDE dC ( t ) = g ( t ) C ( t ) d Y ( t ) , (10) with initial condition C (0) = I . Here w e assume that g is integrable, but otherwise arbitrary . The technique is to make an ansatz of the form dπ t ( X ) = α t dt + β t ( X ) d Y ( t ) (11) where we assume that the pro cesses α t and β t are adapted and lie in Y t . These co efficien ts may be deduced from the iden tity E [( π t ( X ) − j t ( X )) C ( t )] = 0 whic h is v alid since C ( t ) is in Y t . W e note that the It¯ o pro duct rule implies I + I I + I I I = 0 where I = E [( dπ t ( X ) − d j t ( X )) C ( t )] , = E  α t C ( t ) + β t j t  L + L †  C ( t )  dt − E [ j t ( L G X ) C ( t )] dt, I I = E [( π t ( X ) − j t ( X )) dC ( t )] , = E  ( π t ( X ) − j t ( X )) g ( t ) C ( t ) j t  L + L †  dt, I I I = E [( dπ t ( X ) − d j t ( X )) dC ( t )] = E [ β t g ( t ) C ( t )] dt + E  g ( t ) j t ([ L † , X ]) C ( t )  dt. No w from the iden tity I + I I + I I I = 0 we ma y extract separately the co efficients of g ( t ) C ( t ) and C ( t ) as g ( t ) w as arbitrary to deduce π t  ( π t ( X ) − j t ( X )) j t  L + L †  + π t ( β t ) = 0 , π t  α t + β t j t  L + L †  − j t ( L G X )  = 0 . Using the pro jectiv e prop ert y of the conditional exp ecta- tion π t (( π t X )) = π t ( X ) and the assumption that α t and β t lie in Y t , we find after a little algebra that β t = π t  X L + L † X  − π t ( X ) π t  L + L †  , α t = π t ( L G X ) − β t π t  L + L †  , so that the equation (11) reads as dπ t ( X ) = π t ( L G X ) dt (12) +( π t ( X L + L † X ) − π t ( L + L † ) π t ( X )) dW ( t ) , where the innovations pr o c ess W ( t ) is a Wiener pro cess. It is related to the measurement pro cess Y ( t ) by the equa- tion d Y ( t ) = π t ( L + L † ) dt + dW ( t ) (13) and has the interpretation as giv en the difference betw een the observed c hange d Y ( t ) and the exp ected change π t ( L + L † ) dt in the measured field immediately after time t . Note that the increment dW ( t ) of the innov ations pro- cess is independent of π s ( X ) for all 0 ≤ s ≤ t . It imp ortan t to note that Q ( t ) (equiv alent to a Wiener pro cess) and W ( t ) (also a Wiener process) are distinct, and that Q ( t ) is not in the comm utative observ ation sub- space. Some care is needed in in terpreting equation (12) for the quan tum filter. All of the terms in this equa- tion b elong to the comm utative subspace Y t , and so (by the sp ectral theorem [19]) are statistically equiv alent to classical sto chastic pro cesses. The sto chastic master equation may b e expressed in terms of the densit y op erator-v alued sto chastic pro cess ρ ( t ): dρ ( t ) = L ? G ρ ( t ) dt + H L ρ ( t ) dW ( t ) . where we introduce H L ρ = Lρ + L † ρ − tr  L + L †  ρ  ρ. (14) The incremen ts dW ( t ) can b e generated independently of ρ ( s ) for all 0 ≤ s ≤ t and the stochastic master equation ab o v e driv en b y the generated incremen ts can th us act as a simulated quantum tr aje ctory of the state conditioned up on the measuremen t outcomes { y ( s ); 0 ≤ s ≤ t } . F. Photon Coun ting Case If instead we measure the num b er observ able Y ( t ) = U † ( t )Λ( t ) U ( t ) = Λ out ( t ) = R t 0 b † out ( s ) b out ( s ) ds then the quan tum filter is (see the surv ey pap er [19] for the deriv a- tion), dρ ( t ) = L ? G ρ ( t ) dt + J L ρ ( t ) dN ( t ) where J L ρ = LρL † tr { ρL † L } − ρ, and the inno v ations pro cess in this case is giv en by dN ( t ) = d Y − tr { ρ ( t ) L † L } dt and is a comp ensated P ois- son pro cess of intensit y tr { ρ ( t ) L † L } . G. Cascade Connections A simple quan tum net w ork may b e formed b y con- necting the output of one system to the input of another system, [32, 34, 39, 40]. Fig. 4 illustrates the op en quan- tum system G = ( S, L, H ) equiv alent to the cascade of systems G 1 = ( S 1 , L 1 , H 1 ) and G 2 = ( S 2 , L 2 , H 2 ). This 6 equiv alent system can b e describ ed in terms of the series pr o duct G T = G 2 / G 1 [34], defined b y G 2 / G 1 = ( S 2 S 1 , L 2 + S 2 L 1 , H 1 + H 2 + Im { L † 2 S 2 L 1 ] } ) . (15) Note in equation (15) the order of the op erators is impor- tan t. The series pro duct provides the three parameters for the combined or total open system G in terms of the parameters for eac h of the systems G 1 and G 2 . FIG. 4. Tw o quan tum systems cascaded, so that the output of system G 1 b ecomes the input of system G 2 . Imp ortan tly the flow of information is directional. In (b) the circuit has b een simplified using quantum netw ork theory to an equiv a- len t system G = G 2 / G 1 . This t yp e of net work topology is called a cascade or series connection. I II. SINGLE PHOTON FIELDS The master equation for a Mark ovian coupling of a system to a b oson field in a con tinuous-mode one or t wo photon state was first treated in [41]. In this section, w e review the problem of determining the associated fil- ter (stochastic master equation) for an arbitrary system G = ( S, L, H ) driv en by a single photon field [42]. In section I I I G we generalize the master equation and filter to include any com bination of single photon and v acuum as a prob e field. The final section, section I II H, is an explicit example of the homo dyne single photon filtering equations for a tw o level atom. A. Con tinuous-Mode Single Photon States There are man y wa ys to generate single photon states [47]. One common technique for creating heralded single photon states is by sp ontaneous parametric downcon v er- sion (SPDC). The photons from suc h a pro cess are inher- en tly multimodal [43], and sp ectral filtering is typically p erformed to get a single mo de photon. The creation op erator for a photon with one-particle state ξ is B † ( ξ ) = Z ∞ 0 ξ ( t ) dB † ( t ) (16) normalized so that k ξ k 2 = R ∞ 0 | ξ ( t ) | 2 dt = 1. The single photon state is then defined to be | 1 ξ i = B † ( ξ ) | 0 i . (17) One may in terpret this is the frequency domain as | 1 ξ i = R ∞ −∞ ˆ ξ ( ω ) ˆ b † ( ω ) | 0 i where ˆ ξ is the F ourier transform of ξ and ˆ b ( ω ) the formal transform of the input process. This represen tation is often referred to as the multimode, or con tinuous-mode, single photon state, see for instance [44, Sec. 6.3], [45, Sec. 14.2], [46, Eq. (9)]. Muc h of the calculations that follo w will inv olv e the iden tities dB ( t ) | 1 ξ i = ξ ( t ) | 0 i dt, d Λ( t ) | 1 ξ i = ξ ( t ) dB † ( t ) | 0 i , (18) and this will b e the origin of the departure of the master and filter equations from the v acuum case. B. Single Photon Master Equation Without loss of generalit y we fix the initial state of the system to b e a pure state ρ 0 = | η ih η | and our aim is to obtain a differen tial equation for the expectation $ 11 t ( X ) = h η 1 ξ | j t ( X ) | η 1 ξ i , for arbitrary system op erator X . Starting from the Heisen b erg-Langevin equation as b efore, but now using the identities (18) w e find d dt $ 11 t ( X ) = E 11 [ j t ( L G X ))] + E 01 [ j t ( S † [ X, L ])] ξ ∗ ( t ) + E 10 [ j t ([ L † , X ] S )] ξ ( t ) + E 00 [ j t ( S † X S − X )] | ξ ( t ) | 2 = $ 11 t ( L G X ) + $ 01 t ( S † [ X, L ]) ξ ∗ ( t ) + $ 10 t ([ L † , X ] S ) ξ ( t ) + $ 00 t ( S † X S − X ) | ξ ( t ) | 2 where E j k [ A ] = h η φ j | A | η φ k i $ j k t ( X ) = E j k [ j t ( X )] with φ j =      | 0 i , j = 0; | 1 ξ i , j = 1 . ) Rather than finding a single master equation as in the v acuum case, we end up with a system of equations ˙ $ 11 t ( X ) = $ 11 t ( L X ) + $ 01 t ( S † [ X, L ]) ξ ∗ ( t ) + $ 10 t ([ L † , X ] S ) ξ ( t ) + $ 00 t ( S † X S − X ) | ξ ( t ) | 2 , ˙ $ 10 t ( X ) = $ 10 t ( L X ) + $ 00 t ( S † [ X, L ]) ξ ∗ ( t ) , ˙ $ 01 t ( X ) = $ 01 t ( L X ) + $ 00 t ([ L † , X ] S ) ξ ( t ) , ˙ $ 00 t ( X ) = $ 00 t ( L X ) , (19) with initial conditions $ 11 0 ( X ) = $ 00 0 ( X ) = h η , X η i , $ 10 0 ( X ) = $ 01 0 ( X ) = 0 . (20) 7 The main feature here is that the differen tial equation for exp ectations $ j k dep ends on low er order $ j k , allowing us to solv e for $ 11 inductiv ely . Lik ewise, defining the traceclass op erators % j k via tr  % j k ( t ) † X  = $ j k t ( X ) , (21) w e obtain a system of equations ˙ % 11 ( t ) = L ? % 11 ( t ) + [ S ρ 01 ( t ) , L † ] ξ ( t ) + [ L, % 10 ( t ) S † ] ξ ∗ ( t ) + ( S ρ 00 ( t ) S † − % 00 ( t )) | ξ ( t ) | 2 , ˙ % 10 ( t ) = L ? % 10 ( t ) + [ S ρ 00 ( t ) , L † ] ξ ( t ) , ˙ % 01 ( t ) = L ? % 01 ( t ) + [ L, % 00 ( t ) S † ] ξ ∗ ( t ) , ˙ % 00 ( t ) = L ? % 00 ( t ) , (22) with % 11 (0) = % 00 (0) = | η ih η | , % 10 (0) = % 01 (0) = 0 . Note % j k ( t ) † = % kj ( t ). C. An Input-Output Model of Single Photon Signal Generation In section I I I E we will set up a general technique for deriving the filtering equations for situations including the single photon input field. It is p ossible to give an alternate deriv ation in this case motiv ated by the idea of using a pre-interaction preparation where a v acuum input is first passed through a fixed system in order to generate the one photon field. Our motiv ation for consid- ering suc h a scenario stems from statistical and engineer- ing mo delling where it is common practice to use ‘signal generating filters’ [38] driven by white noise to represent colored noise. Analogously , in this section, we construct a quantum signal generating filter M = ( S M , L M , H M ). Cascading the single photon generating filter M with the quan tum system G we wish to prob e, Figure 5, we cre- ate an extended system. Because this extended system G T = G / M is driv en b y v acuum, the master equation and quan tum filter follo w from the known v acuum case up on substitution of the parameters for the cascade sys- tem (Section II I E). W e stress that the signal generation mo del here (and in Section IV C for the case of a system driv en b y a sup erposition of contin uous-mo de coherent states) serv es only as a conv enien t theoretical mathemat- ical device to derive the quantum filtering (or sto c hastic master) equations. It is not suggested that single pho- tons with a given w a vepac ket shape are to be generated in practice with physical devices that implement this par- ticular generator. The idea b ehind the signal generating filter M is sim- ple. W e take the filter to b e a t wo level atom initially prepared in its excited state | ↑i . The interaction with the v acuum input is taken to be ( S M , L M , H M ) = ( I , λ ( t ) σ − , 0) , (23) signal model HD measurement signal white noise vacuum detector FIG. 5. An ancilla system M is used to mo del the effect of the single photon state for B ( t ) on the system G . whic h means that at some stage the atom decays in to its ground state | ↓i creating a single photon in the out- put. The mechanism for pro ducing the single photon is therefore sp ontaneous emission due to the coupling to the v acuum fluctuations. Here σ − is the low ering op er- ator from the upp er state | ↑i to the ground state | ↓i . The Schr¨ odinger equation for | ψ t i = V ( t ) | ↑i ⊗ | 0 i then b ecomes d | ψ t i = h λ ( t ) σ − dB ∗ t − 1 2 | λ ( t ) | 2 σ + σ − dt i | ψ t i , and it is an elementary calculation to see that this has the exact solution | ψ t i = p w ( t ) | ↑i ⊗ | 0 i + | ↓i ⊗ B ∗ t ( ξ ) | 0 i (24) where B ∗ t ( ξ ) = R t 0 ξ s dB ∗ s , and (to preserv e normaliza- tion) w ( t ) = R ∞ t | ξ ( s ) | 2 ds with the complex-v alued func- tion ξ ( · ) related to λ ( · ) by λ ( t ) = 1 p w ( t ) ξ ( t ) . (25) Since w (0) = k ξ k 2 = 1, we therefore generate the limit state | ψ ∞ i = | ↓i ⊗ B † ( ξ ) | 0 i ≡ | ↓i ⊗ | 1 ξ i . Th us the generator mo del will output the desired sin- gle photon state | 1 ξ i provided that we c ho ose the (time- dep enden t) coupling strength λ ( t ) according to (25). D. The Extended System W e no w define our extended system as the cascade system G T = G / M , as in Figure 5, where using the cascade connection formalism from Section I I G we hav e G T = G / M = S, L + ξ ( t ) p w ( t ) S σ − , H + ξ ( t ) p w ( t ) Im( L † S σ − ) ! . (26) Let us denote by ˜ U ( t ) the unitary for the extended system driv en b y v acuum for the parameters G T on the ancilla+system Hilb ert space. Sp ecifying an initial state | ↑i ⊗ | η i ⊗ | 0 i , w e consider the exp ectation ˜ $ t ( A ⊗ X ) = E ↑ η 0 [ ˜ U † ( t )( A ⊗ X ) ˜ U ( t )] , (27) (here A is an ancilla op erator, and X is a system op era- tor). 8 In order to b e useful, the extended system G T (driv en b y v acuum) m ust be capable of capturing exp ectations of X ( t ), for arbitrary op erator X of the system G , at time t as if it were driven by the single photon field. That is, w e m ust ha ve E η ξ [ X ( t )] = E ↑ η 0 [ ˜ U † ( t )( I ⊗ X ⊗ I ) ˜ U ( t )] , (28) that is w e ha ve the situation outlined in equation (1) with ρ a = | ↑ih↑ | , R ( t ) = I . W e are required to show that E η ξ [ X ( t )] = E ↑ η 0 [ ˜ U † ( t )( I ⊗ X ) ˜ U ( t )] (29) holds for an y op erator X of the system G . Our v erification of (29) is to compare the differen tials of b oth sides. No w the left hand side of (29) is just the single photon expectation $ 11 t ( X ) = E 11 [ X ( t )] = E η ξ [ X ( t )], whose differen tial equation is determined from the system (19). The differential of the righ t hand side of (29) may b e found using the Lindblad sup eroperator L G T [ A ⊗ X ] for the extended system, which ma y b e ex- pressed in the form L G T [ A ⊗ X ] = A ⊗ L G X + ( D L M A ) ⊗ X + L † M A ⊗ S † [ X, L ] + AL M ⊗ [ L † , X ] S + L † M AL M ⊗ ( S † X S − X ) , for an y ancilla operator A and system op erator X . W e first observe that D L M ( I ) = 0 , D L M ( σ − ) = − | ξ ( t ) | 2 2 w ( t ) σ − , D L M ( σ + ) = − | ξ ( t ) | 2 2 w ( t ) σ + , D L M ( σ + σ − ) = − | ξ ( t ) | 2 w ( t ) σ + σ − , where w ( t ) = R ∞ t | ξ ( s ) | 2 ds . Then we hav e d dt E ↑ η 0 [ ˜ U † ( t )( I ⊗ X ) ˜ U ( t )] = ˜ $ 11 t ( L G X ) + ˜ $ 01 t ( S † [ X, L ]) ξ ∗ ( t ) + ˜ $ 10 t ([ L † , X ] S ) ξ ( t ) + ˜ $ 00 t ( S † X S − X ) | ξ ( t ) | 2 , (30) where ˜ $ j k t ( X ) = ˜ $ t ( Q j k ⊗ X ) w j k ( t ) , (31) with ( Q j k ) = Q 00 Q 01 Q 10 Q 11 ! = σ + σ − σ + σ − I ! , ( w j k ) = w 00 w 01 w 10 w 11 ! = w ( t ) p w ( t ) p w ( t ) 1 ! . Notice that equation (30) for ˜ $ 11 t ( X ) has the same form as the $ 11 t ( X ) equation in (19). In general, the equa- tions for ˜ $ j k t ( X ) hav e the same form as equations (19) for $ j k t ( X ). Since at time t = 0 w e hav e ˜ $ j k 0 ( X ) = $ j k 0 ( X ), it follo ws that ˜ $ j k t ( X ) = $ j k t ( X ) for all t . This estab- lishes the iden tity (29). E. Single Photon Stochastic Master Equation (Filter) for Quadrature Phase Measuremen ts In this section w e explain ho w the quantum filter for the conditional expectation π 11 t ( X ) = E η ξ [ X ( t ) | Y ( s ) , 0 ≤ s ≤ t ] for the system G driven b y a single photon field ma y now b e obtained from the quan tum filter for the conditional exp ectation ˜ π t ( A ⊗ X ) = E ↑ η 0 [ ˜ U † ( t )( A ⊗ X ) ˜ U ( t ) | I ⊗ Y ( s ) , 0 ≤ s ≤ t ] (32) for the extended system G T = G / M driv en by v acuum. Indeed, we hav e d ˜ π t ( A ⊗ X ) = ˜ π t ( L G T ( A ⊗ X )) dt +( ˜ π t ( A ⊗ X L T + L † T A ⊗ X ) − ˜ π t ( L T + L † T ) ˜ π t ( A ⊗ X )) dW ( t ) , (33) where dW ( t ) = d Y ( t ) − ˜ π t ( L T + L † T ) dt . If we define π j k t ( X ) = ˜ π t ( Q j k ⊗ X ) w j k ( t ) , (34) where Q j k and w j k ( t ) w ere defined in the previous sec- tion, we obtain the coupled system of nonlinear sto chastic differen tial equations dπ 11 t ( X ) =  π 11 t ( L X ) + π 01 t ( S † [ X, L ]) ξ ∗ ( t ) + π 10 t ([ L † , X ] S ) ξ ( t ) + π 00 t ( S † X S − X ) | ξ ( t ) | 2  dt +  π 11 t ( X L + L † X ) + π 01 t ( S † X ) ξ ∗ ( t ) + π 10 t ( X S ) ξ ( t ) − π 11 t ( X ) K t  dW ( t ) , dπ 10 t ( X ) =  π 10 t ( L X ) + π 00 t ( S † [ X, L ]) ξ ∗ ( t )  dt +  ( π 10 t ( X L + L † X ) + π 00 t ( S † X ) ξ ∗ ( t ) − π 10 t ( X ) K t  dW ( t ) , dπ 01 t ( X ) =  π 01 t ( L X ) + π 00 t ( S † [ X, L ]) ξ ∗ ( t )  dt +  ( π 01 t ( X L + L † X ) + π 00 t ( S † X ) ξ ∗ ( t ) − π 01 t ( X ) K t  dW ( t ) , dπ 00 t ( X ) = π 00 t ( L X ) dt +  π 00 t ( X L + L † X ) − π 00 t ( X ) K t  dW ( t ) . (35) 9 Here, K t = π 11 t ( L + L † ) + π 01 t ( S ) ξ ( t ) + π 10 t ( S † ) ξ ∗ ( t ) (36) and the inno v ations pro cess W ( t ) (given abov e) may be expressed as dW ( t ) = d Y ( t ) − K t dt. (37) W e hav e π 01 t ( X ) = π 10 t ( X † ) † , and the initial conditions are π 11 0 ( X ) = π 00 0 ( X ) = h η , X η i , π 10 0 ( X ) = π 01 0 ( X ) = 0 . In order to see that the single photon quan tum fil- ter is giv en b y the system of coupled equations (35), w e m ust show that the conditional exp ectation for the sys- tem driven by the single photon field is given b y π t ( X ) = E η ξ [ X ( t ) | Y ( s ) , 0 ≤ s ≤ t ] = E ↑ η 0 [ ˜ U † ( t )( A ⊗ X ) ˜ U ( t ) | I ⊗ Y ( s ) , 0 ≤ s ≤ t ] = π 11 t ( X ) . (38) T o obtain the filter, we again apply the c haracteristic function technique, setting C = C g ( t ) as b efore with dc g ( t ) = g ( t ) c g ( t ) d Y ( t ). W e need to verify that E η 0 [ j t ( X ) c g ( t )] = E η 0 [ π t ( X ) c g ( t )] (39) F or the extended system we hav e E ↑ η 0 [ ˜ U † ( t )( A ⊗ X ) ˜ U ( t ) c g ( t )] = E η 0 [ ˜ π t ( A ⊗ X ) c g ( t )] , (40) for all functions g , arbitrary ancilla, and system op erators A and X respectively . Hence (39) will follo w provided w e can show that E j k [ X ( t ) c g ( t )] = E eη 0 [ ˜ U † ( t )( Q j k ⊗ X ) ˜ U ( t ) c g ( t )] w j k ( t ) . (41) Ho wev er, equation (41) ma y b e verified in exactly the same w ay we prov ed that ˜ $ j k t ( X ) = $ j k t ( X ) in the pre- vious section, that is, by comparing the differentials of b oth sides of (41). The details of this calculation are omitted. No w, write π j k t ( X ) = tr(( % j k ( t )) † X ). Then from the differen tial equations for π j k t ( X ) and the definition % j k ( t ) w e immediate get the differential equations for the evo- lution of % j k ( t ), as follo ws: dρ 11 ( t ) =  L ? ρ 11 ( t ) + [ S ρ 01 ( t ) , L † ] ξ ( t ) + [ L, ρ 10 ( t ) S † ] ξ ∗ ( t ) + ( S ρ 00 ( t ) S † − ρ 00 ( t )) | ξ ( t ) | 2  dt +  Lρ 11 ( t ) + ρ 11 ( t ) L † + ρ 10 ( t ) S † ξ ∗ ( t ) + S ρ 01 ( t ) ξ ( t ) − K t ρ 11 ( t )  dW ( t ) , dρ 10 ( t ) =  L ? ρ 10 ( t ) + [ S ρ 00 ( t ) , L † ] ξ ( t )  dt +  Lρ 10 ( t ) + ρ 10 ( t ) L † + S ρ 00 ( t ) ξ ( t ) − K t ρ 10 ( t )  dW ( t ) , dρ 01 ( t ) =  L ? ρ 01 ( t ) + [ L, ρ 00 ( t ) S † ] ξ ∗ ( t )  dt +  Lρ 01 ( t ) + ρ 01 ( t ) L † + ρ 00 ( t ) S † ξ ∗ ( t ) − K t ρ 01 ( t )  dW ( t ) , dρ 00 ( t ) = L ? ρ 00 ( t ) dt +  Lρ 00 ( t ) + ρ 00 ( t ) L † − K t ρ 00 ( t )  dW ( t ) , (42) where K t ≡ tr { ( L + L † ) ρ 11 ( t ) } + tr { S ρ 01 ( t ) } ξ ( t ) + tr { S † ρ 10 ( t ) } ξ ∗ ( t ) , with the initial condition ρ 11 (0) = ρ 00 (0) = | η ih η | , ρ 10 (0) = ρ 01 (0) = 0 . F. Single Photon Stochastic Master Equation (Filter) for Photon Coun ting Measuremen ts In this section w e briefly deriv e the filtering equations for photon counting measurements. The quan tum filter for the photon counting case is given by the system of equations dπ 11 t ( X ) = { π 11 t ( L X ) + π 01 t ( S † [ X, L ]) ξ ∗ ( t ) + π 10 t ([ L † , X ] S ) ξ ( t ) + π 00 t ( S † X S − X ) | ξ ( t ) | 2 } dt +  ν − 1 t  π 11 t ( L † X L ) + π 01 t ( S † X L ) ξ ∗ ( t ) + π 10 t ( L † X S ) ξ ( t ) + π 00 t ( S † X S ) | ξ ( t ) | 2  − π 11 t ( X )  dN ( t ) , dπ 10 t ( X ) = { π 10 t ( L X ) + π 00 t ( S † [ X, L ]) ξ ∗ ( t ) } dt +  ν − 1 t  π 10 t ( L † X L ) + π 00 t ( S † X L ) ξ ∗ ( t )  − π 10 t ( X )  dN ( t ) , dπ 01 t ( X ) = { π 01 t ( L X ) + π 00 t ([ L † , X ] S ) ξ ( t ) } dt +  ν − 1 t  π 01 t ( L † X L ) + π 00 t ( L † X S ) ξ ( t )  − π 01 t ( X )  dN ( t ) , dπ 00 t ( X ) = π 00 t ( L X ) dt +  ν − 1 t  π 00 t ( L † X L )  − π 01 t ( X )  dN ( t ) , 10 or in the Schr¨ odinger-picture dρ 11 ( t ) =  L ? ρ 11 ( t ) + [ S ρ 01 ( t ) , L † ] ξ ( t ) + [ L, ρ 10 ( t ) S † ] ξ ∗ ( t ) + ( S ρ 00 ( t ) S † − ρ 00 ( t )) | ξ ( t ) | 2  dt +  ν − 1 t  Lρ 11 ( t ) L † + Lρ 10 ( t ) S † ξ ∗ ( t ) + S ρ 10 ( t ) L † ξ ( t ) + S ρ 00 ( t ) S † | ξ ( t ) | 2  − ρ 11 ( t )  dN ( t ) , dρ 10 ( t ) =  L ? ρ 10 ( t ) + [ S ρ 00 ( t ) , L † ] ξ ( t )  dt +  ν − 1 t  Lρ 10 ( t ) L † + S ρ 00 ( t ) L † ξ ( t )  − ρ 10 ( t )  dN ( t ) , dρ 01 ( t ) =  L ? ρ 01 ( t ) + [ L, ρ 00 ( t ) S † ] ξ ∗ ( t )  dt +  ν − 1 t  Lρ 01 ( t ) L † + Lρ 00 ( t ) S † ξ ∗ ( t )  − ρ 01 ( t )  dN ( t ) , dρ 00 ( t ) = L ? ρ 00 ( t ) dt +  ν − 1 t  Lρ 00 ( t ) L †  − ρ 00 ( t )  dN ( t ) , (43) where ν t = π 11 t ( L † L ) + π 01 t ( S † L ) ξ ∗ ( t ) + π 10 t ( L † S ) ξ ( t ) + π 00 t ( I ) | ξ ( t ) | 2 , = T r  ρ 11 ( t ) L † L  + T r  ρ 10 ( t ) S † L  ξ ∗ ( t ) + T r  ρ 01 ( t ) L † S  ξ ( t ) + T r  ρ 00 ( t ) I  | ξ ( t ) | 2 , and the inno v ations process N ( t ) is giv en by dN ( t ) = d Y ( t ) − ν t dt. G. Com bination of One Photon and V acuum States In this section we take the state of the field to be in a state defined b y the densit y operator ρ field = X j k γ kj | φ j ih φ k | (44) where we use the notation introduced ab ov e for the pho- ton | φ 1 i = | 1 ξ i and v acuum | φ 0 i = | 0 i states. The co ef- ficien ts γ j k m ust of course satisfy the condition that the 2 × 2 complex matrix ρ a = X j k γ kj | j ih k | = γ 11 γ 10 γ 01 γ 00 ! (45) is a density matrix, i.e. ρ a ≥ 0, T r[ ρ a ] = 1. By c ho os- ing the co efficients γ j k appropriately w e can mo del an input field that is an y combination of single photon and v acuum. F or example: the single photon field is given b y γ 11 = 1 and all other coefficients are zero, a sup erp o- sition lik e | ψ i f = α 1 | 1 ξ i + α 0 | 0 i is obtained by setting γ 11 = | α 1 | 2 , γ 10 = α 1 α ∗ 0 , γ 01 = α 0 α ∗ 1 , γ 00 = | α 0 | 2 ; and a simple combination is ρ field = η | 1 ih 1 | + (1 − η ) | 0 ih 0 | where γ 11 = η , γ 00 = 1 − η and γ 10 = γ 01 = 0. 1. The Master Equation The expectation $ t ( X ) = h X ( t ) i of the system op era- tor X ( t ) when the system and field are initialized in the state | η ih η | ⊗ ρ field is given by $ t ( X ) = E η ρ field [ X ( t )] = X j k γ j k E j k [ X ( t )] = X j k γ j k $ j k t ( X ) , (46) where $ j k t ( X ) are defined in section I II B. While there is no differential equation for $ t ( X ), it can b e computed from the weigh ted sum (46), Figure 6. F rom equation (46) we see that the density op erator for the exp ectation $ t ( X ) = tr { % ( t ) X } is given b y % ( t ) = X j k γ kj % j k ( t ) , (47) where the % j k ( t ) are the density op erators in troduced in section I I I B. 10 2 × 2 comp l ex mat r i x ρ a = ! jk γ kj | j "# k | = " γ 11 γ 10 γ 01 γ 00 # ( 43) i s a d en si t y mat r i x , i . e. ρ a ≥ 0, T r [ ρ a ] = 1. By c ho os- i n g t h e co e ffi ci en t s γ jk ap p r op r i at el y w e can mo d el an i n p u t fi el d t h at i s an y com b i n at i on of s i ngl e p h ot on an d v acu u m. F or ex amp l e: t h e si n gl e p h ot on fi el d i s gi v en by γ 11 = 1 an d al l ot h er co e ffi ci en t s ar e ze ro, a su p er p o- si t i on l i k e | ψ " f = α 1 | 1 ξ " + α 0 | 0 " i s ob t ai n ed b y set t i n g γ 11 = | α 1 | 2 , γ 10 = α 1 α ∗ 0 , γ 01 = α 0 α ∗ 1 , γ 00 = | α 0 | 2 ; an d a si mp l e mi x t u r e i s ρ fi e ld = η | 1 "# 1 | + ( 1 − η ) | 0 "# 0 | wh er e γ 11 = η , γ 00 =1 − η an d γ 10 = γ 01 = 0. 1 . T h e m a s ter e q u a ti o n Th e ex p ect at i on & t ( X )= # X ( t ) " of t h e sy st em op er a- t or X ( t ) wh en t h e sy st em an d fi el d ar e i n i t i al i zed i n t h e st at e | η "# η | ⊗ ρ fi e ld i s gi v en b y & t ( X )= E ηρ f i el d [ X ( t ) ] = ! jk γ jk E jk [ X ( t )] = ! jk γ jk & jk t ( X ) , ( 44) wh er e & jk t ( X ) ar e d efi n ed i n sect i on I I B. W h i l e t h er e i s n o d i ff er en t i al e qu at i on f or & t ( X ) , i t can b e comp u t ed f r om t h e w ei gh t ed su m ( 44) , F i gu r e 6. F r om eq u at i on ( 44) w e see t h at t h e d en si t y op er at or f or t h e ex p ect at i on & t ( X ) = t r { ρ ( t ) X } i s gi v en b y ρ ( t )= ! jk γ kj ρ jk ( t ) , ( 45) wh er e t h e ρ jk ( t ) ar e t h e d en si t y op er at or s i n t r o d u ced i n sect i on I I B. 2 . T h e s to ch a s ti c m a s ter e qu a ti o n T u r n i n g n o w t o t h e p r ob l em of d et er mi n i n g t h e fi l t er , w e agai n mak e u se of t h e cascad e ex t en d ed sy st em f r om S ect i on I I D . No w w e h a v e & jk t ( X )= ˜ & t ( Q jk ⊗ X ) w jk ( t ) , ( 46) an d so i f w e de fi ne t h e mat r i x R ( t )= ! jk γ jk w jk ( t ) Q jk , ( 47) wh er e w jk ( t ) an d Q jk ar e as d efi n ed i n S ect i on I I D , w e h a v e ( u si n g ( 46) , ( 44) an d ( 26) ) & t ( X ) = ˜ & t ( R ( t ) ⊗ X ) . ( 48) master equations weighting F I G . 6 . Th e exp ec t a t io n ! t ( X )= ! X ( t ) " o f t h e syst em o p - era t o r X ( t ) wh en t h e syst em a n d fi eld are in i t ia liz ed in t h e st a t e | η "! η | ⊗ ρ fi el d m a y b e c a l c u l a t ed b y w eig h t in g t h e so lu - tions ! jk t ( X ) fro m t h e si n g l e p h o t o n ma st er equ a t io n s ( 1 8 ) . Not e t h at t h e d efi n i t i on ( 26) of ˜ & t ( R ( t ) ⊗ X ) i n v ol v es t h e an ci l l a sy st em i n i t i al i zed i n t h e ex ci t ed st at e | e " = | ↑" . Th e con d i t i on al ex p ect at i on π t ( X )= E ηρ f i el d [ X ( t ) | Y ( s ) , 0 ≤ s ≤ t ] ( 49) cor r esp on d i n g t o t h e fi el d i n t h e s t at e ρ fi e ld i s r el at ed t o t h e con d i t i on al ex p ect at i on ˜ π t ( A ⊗ X ) f or t h e ex t en d ed sy st em ( se e ( 31) ) b y t h e Ba y es r el at i on π t ( X )= ˜ π t ( R ( t ) ⊗ X ) ˜ π t ( R ( t ) ⊗ I ) . ( 50) D i v i si on b y t h e d en omi n at or i n ( 50) is n eed ed t o en su r e n or mal i zat i on π t ( I ) = 1. T o p r o v e ( 50) , w e n ee d t o sh o w t h at ˜ π t ( R ( t ) ⊗ X ) = ˜ π t ( R ( t ) ⊗ I ) π t ( X ) , or eq u i v al en t l y E ↑ η 0 [˜ π t ( R ( t ) ⊗ X ) c g ( t ) ] = E ↑ η 0 [˜ π t ( R ( t ) ⊗ I ) π t ( X ) c g ( t )] f or al l c h oi ce of c h ar act er i st i c f u n ct i on s c g ( t ) . Ho w- ev er , E ↑ η 0 [˜ π t ( R ( t ) ⊗ X ) c g ( t ) ] eq u al s E ↑ η 0 [ ˜ U † ( t )( R ( t ) ⊗ X ⊗ I ) ˜ U ( t ) c g ( t ) ] , b u t b y t h e ex t en d ed sy st em r ep r esen - t at i on t h i s i s j u st E ηρ f i el d [ X ( t ) c g ( t ) ] wh i c h i n t u r n eq u al s E ηρ f [ π t ( X ) c g ( t )] ≡ E ↑ η 0 [ ˜ U † ( t )( R ( t ) ⊗ I ) ˜ U ( t ) π t ( X ) c g ( t )] wh i c h e st ab l is h es t h e Ba y es r el at i on ( 50) . S i n ce ˜ π t ( R ( t ) ⊗ X )= ! jk γ jk π jk t ( X ) , ( 51) wh er e π jk t ( X ) i s d efi n ed b y ( 33) , t h e d esi r ed con d i t i on al ex p ect at i on ma y b e ex p r essed as π t ( X )= $ j ,k γ jk π jk t ( X ) $ j ,k γ jk π jk t ( I ) . ( 52) Agai n , t h er e i s n o d i ff er en t i al eq u at i on f or π t ( X ) ; i n - st ead i t i s com pu t ed f r om a n or mal i zed w ei gh t ed su m, FIG. 6. The expectation $ t ( X ) = h X ( t ) i of the system op- erator X ( t ) = j t ( X ) when the system and field are initialized in the state | η ih η | ⊗ ρ field ma y b e calculated by weigh ting the solutions $ j k t ( X ) from the single photon master equations (19). 11 2. The Sto chastic Master Equation T urning now to the problem of determining the filter, w e again make use of the cascade extended system from Section I I I D. No w w e ha ve $ j k t ( X ) = ˜ $ t ( Q j k ⊗ X ) w j k ( t ) , (48) and so if we define the matrix R ( t ) = X j k γ j k w j k ( t ) Q j k , (49) where w j k ( t ) and Q j k are as defined in Section II I D, we ha ve (using (48), (46) and (27)) $ t ( X ) = ˜ $ t ( R ( t ) ⊗ X ) . (50) Note that the definition (27) of ˜ $ t ( R ( t ) ⊗ X ) in volv es the ancilla system initialized in the excited state | e i = | ↑i . The conditional expectation π t ( X ) = E η ρ field [ X ( t ) | Y ( s ) , 0 ≤ s ≤ t ] (51) corresp onding to the field in the state ρ field is related to the conditional exp ectation ˜ π t ( A ⊗ X ) for the extended system (see (32)) by the Bay es relation π t ( X ) = ˜ π t ( R ( t ) ⊗ X ) ˜ π t ( R ( t ) ⊗ I ) . (52) Division b y the denominator in (52) is needed to ensure the normalization π t ( I ) = 1. T o prov e (52), we need to sho w that ˜ π t ( R ( t ) ⊗ X ) = ˜ π t ( R ( t ) ⊗ I ) π t ( X ), or equiv a- len tly E ↑ η 0 [ ˜ π t ( R ( t ) ⊗ X ) c g ( t )] = E ↑ η 0 [ ˜ π t ( R ( t ) ⊗ I ) π t ( X ) c g ( t )] for all choice of characteristic functions c g ( t ). How- ev er, E ↑ η 0 [ ˜ π t ( R ( t ) ⊗ X ) c g ( t )] equals E ↑ η 0 [ ˜ U † ( t )( R ( t ) ⊗ X ⊗ I ) ˜ U ( t ) c g ( t )], but by the extended system represen- tation this is just E η ρ field [ X ( t ) c g ( t )] which in turn equals E η ρ f [ π t ( X ) c g ( t )] ≡ E ↑ η 0 [ ˜ U † ( t )( R ( t ) ⊗ I ) ˜ U ( t ) π t ( X ) c g ( t )] whic h establishes the Bay es relation (52). Since ˜ π t ( R ( t ) ⊗ X ) = X j k γ j k π j k t ( X ) , (53) where π j k t ( X ) is defined b y (34), the desired conditional exp ectation may b e expressed as π t ( X ) = P j,k γ j k π j k t ( X ) P j,k γ j k π j k t ( I ) . (54) Again, there is no differential equation for π t ( X ); in- stead it is computed from a normalized weigh ted sum, 11 ( 52) , an d t he fi l t er i n g eq u at i on s ( 34) , F i gu r e 7. Th e cor - r esp on d i n g con d i t i on al d en si t y op e rat or i s gi v en b y ! ( t )= ! jk γ kj ! jk ( t ) ! jk γ kj tr { ! jk ( t ) } , ( 53) wh er e t h e con d i t i onal q u an t i t i es ! jk ( t ) ma y b e comp u t ed f r om t h e si n gl e p h ot on fi l t er i n g eq u at i on s ( 41) . filtering equations weighting and normalization measurement signal F I G . 7 . R ela t i o n sh ip b et w een t h e m ea su red si g n a l a n d t h e fi l t ered est i m a t e. Th e d i ff eren t ia l equ a t io n s ( 3 4 ) m u st b e in - t eg ra t ed t o c o mp u t e π jk t ( X ) . Th en d ep en d i n g o n t h e st a t e o f t h e in p u t fi el d , t h e p ro b e, t h ese est i ma t es m u st b e w ei g h t ed b y t h e a p p ro p ri a t e c o e ffi c ien t s a n d n o rm a liz ed a s sp ec i fi ed in ( 5 2 ) t o p ro d u c e t h e d esired c o n d it io n a l exp ec t a t io n π t ( X ). Th ese ex p r essi on s al l o w fi l t er i ng on an y com b i n at i on of si n gl e p h ot on an d v acu u m. O n e n ot ab l e case i t t h at of si mp l e mi x t u r e of on e p h ot on an d v acu u m ( ρ p ro b e = p | 1 !" 1 | + ( 1 − p ) | 0 !" 0 | ) wh i c h i s an ex p er i men t al l y accu - r at e mo d el f or t h e ou t p u t of t h e S P D C p r o cess [ 1] . H. E x ampl e Her e w e gi v e an ex p l i ci t ex amp l e of t h e fi l t er i n g met h o d d er i v ed ab o v e t o a sy st em d r i v en b y a p u r e s i ngl e p h ot on wi t h h omo d y n e measu r emen t of t h e amp l i t u d e q u ad r at u r e of t h e fi el d . W e t ak e t h e sy st em of i n t er est t o b e a t w o- l ev el sy st em ( a q u b i t ) . In p ar t i cu l ar , w e con - si d er a d amp ed q u b i t L = √ κσ − wi t h i n t er n al d y n ami cs gi v en b y H s = ωσ z an d n o scat t e ri n g i . e. S = I . Her e κ > 0 i s a scal ar p ar amet er oft e n r ef er r ed t o as t h e mea- su r emen t st r en gt h an d ω i s a f r eq u en cy . 1 . T h e m a s ter e q u a ti o n As t h er e ar e on l y t w o c omp on en t s t o t he i np u t fi el d | Ψ ! w e on l y n eed t o con si d er a gen er al i zed Bl o c h r ep r e- sen t at i on of t h e f or m ρ jk =( c jk I + x jk σ x + y jk σ y + z jk σ z ) ( 54) wi t h j , k ∈ { 0 , 1 } . Not e t h at x 00 , y 00 , z 00 an d x 11 , y 11 , z 11 ar e r eal , wh i l e x 01 , y 01 , z 01 ma y b e comp l ex . F or t h e mast er eq u at i on t h e co e ffi ci en t s c jk = 1, w e wi l l h o w ev er r eq u i r e t he m l at er . Al so n ot e, f or ex amp l e, t r " ! 00 σ x # = x 00 an d t r " ! 01 σ x # = x 01 ∗ . Usi n g eq u at i on s ( 21) d er i v ed i n sect i on I I B, w e n o w comp u t e t h e gen er al i zed Bl o c h comp on en t eq u at i on s f or t h e u n con d i t i on ed ev ol u t i on of ou r sy st em. W e ob t ai n n i n e cou p l ed eq u at i ons f or t h e n i n e comp on en t s: ˙ x 00 = − 2 ω y 00 − κ 2 x 00 , ˙ y 00 =2 ω x 00 − κ 2 y 00 , ˙ z 00 = − κ ( 1 + z 00 ) , ˙ x 01 = − κ 2 x 01 − 2 ω y 01 − √ κξ ( t ) ∗ z 00 , ˙ y 01 =2 ω x 01 − κ 2 y 01 − i √ κξ ( t ) ∗ z 00 , ˙ z 01 = − κ z 01 − √ κ x 00 ξ ( t ) ∗ + i √ κ y 00 ξ ( t ) ∗ , ˙ x 11 = − κ 2 x 11 − 2 ω y 11 + √ κ z 10 ξ ( t )+ √ κ z 01 ∗ ξ ( t ) ∗ , ˙ y 11 =2 ω x 11 − κ 2 y 11 + i √ κ z 01 ξ ( t ) − i √ κ z 01 ∗ ξ ( t ) ∗ , ˙ z 11 = − κ − κ z 11 − √ κ x 10 ξ ( t ) − i √ κ y 10 ξ ( t ) − √ κ x 01 ∗ ξ ( t ) ∗ + i √ κ y 01 ∗ ξ ( t ) ∗ . Th ese eq u at i ons m u st com b i n ed u si n g ( 54) t o cal cu l at e r el ev an t q u an t i t i es. 2 . T h e s to ch a s ti c m a s ter e qu a ti o n No w w e t u r n ou r at t en t i on t o cal cu l at i n g t h e fi l t er - i n g eq u at i on s f or t h i s ex amp l e. Recal l t h at c jk = 1 i n t h e gen er al i zed Bl o c h r ep r esen t at i on f or t h e mast er eq u a- t i on . F or t h e fi l t er , t h e co e ffi ci en t s c jk ar e n ot n ecessar y u n i t y , t h ou gh w e al w a y s h a v e c 11 = 1 at al l t i mes. Ho w- ev er , u n l i k e t h e mast er eq u at i on , t hi s wi l l n ot b e so f or c 01 ,c 10 ,c 00 , as t h ese co e ffi ci en t s wi l l n o w ev ol v e i n t i me. No w, su b st i t u t i n g ou r p ar t i cu l ar c hoi c e of ( S , L , H ) i n t o eq u at i on s ( 34) , t h e q u an t u m fi l t er f or t h e t w o-l ev el sy s- t em i s gi v en b y t h e fi n i t e set of cou p l ed eq u at i on s I I I. SU PE R PO SITIO N O F C O HE R E NT F IEL D ST A TE S In t h i s sect i on w e t u r n t o t h e p r ob l em of d et er mi n - i n g t he mast er eq u at i on an d t h e q uan t u m fi l t er f or sy s- t ems dr i v e b y a b oson fi el d wh ose st at e i s a su p er p osi t i on of c onti nu ou s m o de coh er en t st at es. In se ct i on I I I A w e d escr i b e con t i n uou s- mo d e coh er en t st at es an d su p er p o- si t i on s of t h em, as w el l as t h e act i on of t h e q uan t u m n oi ses on su c h s t at es. S ect i on I I I B i s d ev ot ed t o t h e d er i v at i on of t h e mas te r eq u at i on f or su p er p osi t i on s of coh er en t st at es. In sect i on I I I C w e d ev el op a cas cad ed sy st em si gn al mo d el . Th i s mo d el al l o ws u s t o u se t h e met h o d ol ogy f r om S ect i on I I, wi t h ap p r op r i at e c h an ges d u e t o t h e n at u r e of t h e su p er p osi t i on of coh er en t st at es, FIG. 7. Relationship b etw een the measured signal and the filtered estimate. The differential equations (35) must b e in- tegrated to compute π j k t ( X ). Then dep ending on the state of the input field, the prob e, these estimates m ust be w eighted b y the appropriate co efficien ts and normalized as sp ecified in (54) to pro duce the desired conditional exp ectation π t ( X ). (54), and the filtering equations (35), Figure 7. The cor- resp onding conditional densit y op erator is giv en b y ρ ( t ) = P j k γ kj ρ j k ( t ) P j k γ kj tr { ρ j k ( t ) } , (55) where the conditional quan tities ρ j k ( t ) ma y be computed from the single photon filtering equations (42). These expressions allow filtering on an y combination of a single photon and a v acuum state. One notable case it that of simple combination of one photon and v acuum ( ρ probe = p | 1 ih 1 | + (1 − p ) | 0 ih 0 | ) whic h is an exp erimen- tally accurate mo del for the output of the SPDC pro cess [1]. H. Illustrativ e Example of Single Photon Master and Filtering Equations Here apply the filtering method derived abov e to the problem of exciting a t w o lev el atom, in free space, with a contin uous mo de single photon. This problem has re- ceiv ed m uch attention recen tly [50 – 53]. Until now it has only b een possible to calculate ensemble av eraged quan ti- ties. Here w e show the individual tra jectories asso ciated with a particular exp erimental run. This problem can b e parametrized in our mo del as fol- lo ws. W e take the coupling op erator to b e L = √ κσ − , the internal dynamics of the atom are sp ecified by the Hamiltonian H = 0 and there is no scattering i.e. S = I . Here κ > 0 is the coupling rate (often referred to as the measuremen t strength) and is chosen to be κ = 1. The atom is take to be in the ground state initially | g ih g | , then a single photon in the wa vepac k et ξ ( t ) interacts with the atom. W e tak e the w a vepac ket to be a Gaussian 12 parametrized as ξ gau ( t ) =  Ω 2 2 π  1 / 4 exp  − Ω 2 4 ( t − t c ) 2  , (56) where t c sp ecifies the peak arriv al time and Ω is the fre- quency bandwidth of the pluse. No w we wish to calculate the excited state population of the tw o level atom as a function of time. Other stud- ies hav e only b een able to calculate the master equation ev olution of the atomic state [50 – 53]. In our formalism this corresp onds to propagating the master equations and taking the expectation P e ( t ) = T r  % 11 ( t ) | e ih e |  , (57) where % 11 ( t ) is the solution to Eq. (22). In Fig. 8 Eq. (57) is plotted, the dotted line (red), as a function of time for a tw o level atom interacting with a gaussian pulse. W e c ho ose Ω = 1 . 46 κ whic h is kno wn to be optimal for excitation via a single photon in a Gaussian pulse [50 – 52]. Our n umerics agree with the prior results that max t P e ( t ) ≈ 0 . 8 [50–52]. FIG. 8. The excited state p opulation, P e , of a tw o-level atom in teracting with one photon in a Gaussian wa v epack et. The dashed line is the Gaussian wa vepac ket | ξ ( t ) | 2 with bandwidth Ω = 1 . 46 κ . The dotted (red) line is P e as calculated b y the master equation. The grey lines are the individual tra jecto- ries P c e . The solid line is the ensemble a verage of sixt y four tra jectories plotted with error bars (the shaded ligh t green region). Ho wev er, in our formalism w e can also calculate the conditional state of the system using the quan tum fil- tering equations deriv ed ab o ve. The conditional excited state p opulation is denoted by P c e ( t ) = T r  ρ 11 ( t ) | e ih e |  , (58) where ρ 11 ( t ) is the solution to the filtering equations Eq. (42) or Eq. (43) for homo dyne or photon counting measuremen ts resp ectively . In what follows we will fo- cus on the homodyne measurement filtering equations i.e. Eq. (42). In Fig. 8, 64 differen t tra jectories giv en b y Eq. (58) are plotted as grey lines. F or this particular bandwidth there is very little spread in the tra jectories for t < 3. After the bulk of the wa v epack et has passed, at t = 4, many of the tra jectories start to decay , as evidenced by the man y grey lines b elo w P c e = 0 . 5 for t > 4. Nevertheless there are a n um b er of tra jectories whic h con tinue to rise tow ards P c e = 1 for t > 4. This means in a particular run of an exp eriment the atom may b ecome fully excited. Such b eha vior can not be seen through the master equation approac h of Refs. [50 – 53]. It is possible to confirm the consistency of the tra jec- tories with the master equation solution b y calculating a n umerical a verage of the tra jectories. W e plot the ensem- ble av erage of the tra jectories as the solid line in Fig. 8 with error bars smeared around this line. The numeri- cally calculated ensemble av erage agrees with the master equation behavior given that a small ensemble was used to calculate this mean v alue. IV. SUPERPOSITION OF COHERENT FIELD ST A TES In this section w e turn to the problem of determin- ing the master equation and the quantum filter for sys- tems driv e b y a boson field whose state is a superp osition of contin uous-mo de coherent states. In section IV A w e describ e contin uous-mo de coherent states and sup erp o- sitions of them, as w ell as the action of the quan tum noises on such states. Section IV B is dev oted to the deriv ation of the master equation for superp ositions of coheren t states. In section IV C we dev elop a cascaded system signal mo del. This mo del allows us to use the metho dology from Section I II, with appropriate c hanges due to the nature of the sup erp osition of coherent states, to derive the filtering equations in section IV D. Then w e giv e the filter for the case of photon coun ting in sec- tion IV E and generalize to mixed input states. A. Sup erpositions and Combinations of Coherent States T ypically single mo de coherent states of a field are de- noted b y | α i . In this paper w e shall often refer to a sup erposition of contin uous-mo de coheren t states as a (con tinuous-mode) cat-state [44, 49]. F ormally , the su- p erpositions of contin uous-mo de coherent states is giv en b y | ψ i = n X j =1 s j | α j i , (59) where | α j i are coherent states, determined by func- tions α j ( t ) with α j 6 = α k if j 6 = k . The superp osi- tion w eights s j are complex n umbers such that h ψ | ψ i = P j,k s ∗ j s k h α j | α k i = 1 (i.e., ψ is normalized and is a pure 13 state v ector of the field). Given a function α , the co- heren t state | α i of a con tinuous-mode field is giv en by the displacement or W eyl op erator D ( α ) applied to the v acuum state of the con tinuous-mode field: | α i = D ( α ) | 0 i . (60) The inner product of tw o coheren t states | α i , | β i in the F ock space is giv en b y h α | β i = exp  − 1 2 k α k 2 − 1 2 k β k 2 + h α, β i  , (61) where h g , f i = R ∞ −∞ g ( s ) ∗ f ( s ) ds and k · k = h· , ·i are the L 2 inner product and norm, respectively . The normalization condition for the sup erp osition state (59) means that the co efficien ts must satisfy P j,k s ∗ j s k g j k = 1, where g j k = h α j | α k i . More generally , we may consider a field density op era- tor ρ field = X j k γ kj | α j ih α k | , (62) that generalizes the sup erposition state | ψ i to allow for statistical com binations of coherent states. The normal- ization for the state ρ field is P j,k γ j k g j k = 1. In what follows the action of the quantum noises dB and d Λ on coherent states will b e imp ortan t: dB ( t ) | α i = α ( t ) | α i dt, d Λ( t ) | α ( t ) i = dB ∗ ( t ) α ( t ) | α i . (63) B. Master Equation for Systems Driv en by a Field in a Com bination or Superp osition of Coherent States Again, b efore w e deriv e the master equation w e in tro- duce some notation that helps to formulate the master equation. Recall that w e defined the asymmetric exp ec- tation E j k [ X ⊗ F ] ≡ h η | X | η ih φ j | F | φ k i . In section II I w e to ok the field states | φ j i , | φ k i to b e either v acuum or one photon. In this section w e use this same notation but the field states are understo o d to b e contin uous-mo de coher- en t states i.e. | α j i , | α k i . The indices j, k now tak e the v alues 1 , . . . , n . The exp ectation of an arbitrary system observ able, with resp ect to the state | η ih η | ⊗ ρ field , at time t is $ t ( X ) = E η ρ field [ X ( t )] . (64) Using the notation (similar to the single photon case) $ j k t ( X ) = E j k [ X ( t )] = h η α j | X ( t ) | η α k i , (65) with ρ field as given in 62, we may write (64) as $ t ( X ) = X j k γ j k $ j k t ( X ) . (66) As in section I I I B, we can derive the Heisenberg mas- ter equation by taking the exp ectation of the equation of motion for an arbitrary system operator dX ( t ), i.e. (4). Doing so yields the equations ˙ $ j k t ( X ) = $ j k t ( G j k t X ) , (67) where we define a new sup erop erator G j k t X ≡ L X + S † [ X, L ] α ∗ j ( t ) + [ L † , X ] S α k ( t ) +( S † X S − X ) α ∗ j ( t ) α k ( t ) , (68) with initial conditions $ j k 0 ( X ) = h η | X | η i g j k . Note that equations (67) are uncoupled. The corresp onding densit y op erator is % ( t ) = X j k γ j k % j k ( t ) (69) where ˙ % j k = G j k? t [ % j k ] ≡ L ? % + [ S % j k , L † ] α j ( t ) + [ L, %S † ] α ∗ k ( t ) +( S % j k S † − % j k ) α j ( t ) α ∗ k ( t ) , (70) and % j k (0) = | η ih η | g j k . The master equations (67) and (70) consist of a w eighted sum of cross-exp ectations. Clearly these equa- tions reduce to the v acuum master equation if the only term in the sup erposition or combination is the v acuum. C. Extended System In this section w e describ e a cascade extended system G T = G / M that will be used in section IV D to deter- mine the quantum filtering equations for the mixed or sup erposition of coherent state field. The ancilla system M will b e an n -level system, with orthonormal basis | j i , j = 1 , . . . , n . The parameters for this system are M = ( I , L M , 0) , where L M = X j α j ( t ) | j ih j | , (71) and w e take the initial state of the ancilla to b e the den- sit y matrix ρ a = 1 N a X j k γ kj | j ih k | , (72) where N a = P l γ ll is a normalization factor. The ex- tended system is G T = G / M = ( S, L + S L M , H + Im  L † S L M  ) . Define Q j k = | j ih k | . Then a straightforw ard calcula- tion shows that L L M ( Q j k ) = m j k ( t ) Q j k , (73) 14 where m j k ( t ) = α ∗ j ( t ) α k ( t ) − 1 2 | α j ( t ) | 2 − 1 2 | α k ( t ) | 2 . (74) No w consider the extended system G T initialized in the state ρ a ⊗ | η ih η | ⊗ | 0 ih 0 | (driven by v acuum | 0 i ). Then the metho ds used in Sections II I D and II I G may b e adapted to the presen t case to show that $ j k t ( X ) = ˜ $ t ( Q j k ⊗ X ) w j k ( t ) , (75) where w j k ( t ) is defined to be the solution of ˙ w j k ( t ) = m j k ( t ) w j k ( t ) , w j k (0) = 1 N a g j k , (76) and E η ρ field [ X ( t )] = E ρ a η 0 [ ˜ U † ( t )( R ( t ) ⊗ X ) ˜ U ( t )] , (77) where R ( t ) = X j,k γ j k w j k ( t ) Q j k , (78) These expressions are very similar to the photon case, but with some imp ortan t differences. F or instance, the ancilla w as initialized in the excited state for the photon case, while here for the mixed coherent case the initial ancilla state is the densit y ρ a . D. The Sto c hastic Master Equation (Filter) for Amplitude Quadrature Measuremen ts The quan tum filter for the general com bination of co- heren t state case ma y no w be derived in exactly the same w ay as w as done for the combination of single photon and v acuum in Section I I I G. The conditional exp ectation we are interested in is π t ( X ) = E η ρ field [ X ( t ) | Y ( s ) , 0 ≤ s ≤ t ] , (79) where no w ρ field is giv en b y (62). Equations (52), (54), and (55) again hold, but with mo difications to the terms as describ ed abov e. The filtering equations are as follows. The conditional quan tities π j k t ( X ) satisfy the coupled system of equations dπ j k t ( X ) = π j k t ( G j k X ) dt + H j k t ( X ) dW ( t ) where the inno v ations pro cess W ( t ) is a Wiener pro cess and is giv en by dW ( t ) = d Y ( t ) − X l γ ll N a π ll t ( L + S α l ( t ) + L † + S † α ∗ l ( t )) dt. and the new sup erop erator H j k l ( · ) is defined by H j k l ( X ) ≡ π j k t  X ( L + S α k ( t )) + ( L † + S † α ∗ j ( t )) X  − π j k t ( X ) X l γ ll N a π ll ( L + L † + S α l ( t ) + S † α ∗ l ( t )) . As b efore, w e may write π j k t ( X ) = tr  % j k ( t ) † X  , where % j k ( t ) satisfies the coupled differen tial equations (for j, k = 1 , 2 , . . . , n ): dρ j k ( t ) = G j k? t [ ρ j k ( t )] dt + H j k? t  ρ j k ( t )  dW ( t ) (80) where H j k? t  ρ j k  ≡ ( L + S α k ( t )) ρ j k + ρ j k ( L † + S † α ∗ j ( t )) − ρ j k X l γ ll N a tr  ( L + L † + S α l ( t ) + S † α ∗ l ( t )) ρ ll  , with initial conditions ρ j k 0 ( t ) = | η ih η | g j k (recall that g j k = h α j | α k i ). The conditional density op erator is given b y (55), with the ρ j k ( t ) given instead b y (80). W e remark that the innov ations for the cat case now dep ends on the w eights, in contrast to the mixed pho- ton/v acuum case. E. The Sto c hastic Master Equation (Filter) for Photon Coun ting Measuremen ts Analogously , we ma y also compute the quantum filter- ing equations for a system driven by a coheren t sup er- p osition in the case where the measurement p erformed on the output field, Y ( t ), is photon counting. The filter- ing equations in the Heisenberg form are given by (for j, k = 1 , 2 , . . . , n ): dπ j k t ( X ) = π j k t ( G j k ( X )) dt +  π j k t ( L † X L + α k ( t ) L † X S + α ∗ j ( t ) S † X L + α ∗ j ( t ) α k ( t ) S † X S ) P n j =1 γ j j N a π j j t ( L † L + α j ( t ) L † S + α ∗ j ( t ) S † L + | α j | 2 I ) − π j k t ( X )  dN ( t ) , 15 where dN ( t ) = d Y ( t ) − n X j =1 γ j j N a π j j t ( L † L + α j ( t ) L † S + α ∗ j ( t ) S † L + | α j ( t ) | 2 I ) dt, (81) and with initial conditions π j k 0 ( X ) = h η | X | η i g j k . The corresp onding Schr¨ odinger-picture filter is dρ j k ( t ) = G j k? t [ ρ j k ] dt +  N − 1 [ Lρ j k L † + α k ( t ) S ρ j k L † + α ∗ j ( t ) Lρ j k S † + α ∗ j ( t ) α k ( t ) S ρ j k S † )] − ρ j k  dN ( t ) , where N = n X j =1 γ j j N a T r  ρ j j ( L † L + α j ( t ) L † S + α ∗ j ( t ) S † L + | α j | 2 I )  (82) and dN ( t ) = d Y ( t ) − N dt , with initial conditions ρ j k (0) = | η ih η | g j k . V. CONCLUSION W e hav e sho wn that quantum filtering may b e ex- tended b ey ond the Gaussian input situation to consider a range of non-classical states that are of current in ter- est. Photon w a ve pack et shaping is already b eing applied exp erimen tally and our filtering equations for the single photon input completes the problem addressed by Gheri et al. in [41] by giving the quantum tra jectories asso- ciated to the master equation they deriv e. W e extend this general combinations of the v acuum an a one pho- ton state through a straigh tforward w eighting procedure. The filter equations themselv es hav e p otential applica- tions to areas such as shaping wa v e pack et for maximal / minimal absorption by , for instance, a tw o lev el atom, or to controlling the system so as to shap e the outgoing field. W e hav e also derived the quantum filter for cat states. While the concept of an environmen t b eing in a sup erpo- sition of states ma y seem unphysical from the p ersp ectiv e of macroscopic sup erselection rules, as w e hav e seen this ma y effectively be what happ ens in ternally once a stan- dard input is first fed through an appropriate filter sys- tem M . This leads naturally to questions of decoherence [48], and whether preparing input in a cat state is ad- v antageous in preven ting decoherence of cat states for a giv en system. It is now exp erimen tally possible to isolate quan tum systems sufficiently well to create cat states in a lab oratory [2 – 4]. The cat-state filtering equation will be of imp ortance for in vestigating questions as to whether suc h superp ositions ma y protected via appropriate envi- ronmen t engineering. A cknow le dgements. The authors wish to thank J. Hop e for helpful discussions and for p ointing out reference [47] to us. W e also wish to thank A. Doherty , H. Wiseman, E. Huntington and an anon ymous referee (of an earlier v ersion of this man uscript) for helpful discussions and suggestions; G. Zhang for carefully reading and earlier v ersion of this manuscript; and B. Baragiola for discus- sions about sec. II I H. MJ and HIN gratefully ac kno wl- edge the supp ort of the Australian Researc h Council. JC ackno wledges supp ort from National Science F oun- dation Grant No. PHY-0903953 and Office of Na v al Re- searc h Grant No. N00014-11-1-008. JG gratefully ac- kno wledges the supp ort of the UK Engineering and Ph ys- ical Sciences Researc h Council through Research Pro ject EP/H016708/1 [1] A. I. Lv ovsky , H. Hansen, T. Aic hele, O. Benson, J. Mlynek, and S. Schiller, Phys. Rev. Lett. 87 , 050402 (2001). [2] J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. P olzik, Phys. Rev. Lett. 97 , 083604 (2006). [3] A. Ourjoumtsev, R. T ualle-Brouri, J. Laurat, and P . Grangier, Philipp e, Science 312 , 83 (2006). [4] A. Ourjoumtsev, H. Jeong, R. T ualle-Brouri, and P . Grangier, Philipp e, Nature 448 , 784 (2007). [5] A. Kuhn, M. Hennrich, and G. Remp e, Phys. Rev. Lett. 87 , 067901 (2002). [6] J. McKeever, A. Bo ca, A. D. Bo ozer, R. Miller, J. R. Buc k, A. Kuzmic h, and H. J. Kim ble, Science 303 1992 (2004). [7] Z. Y uan, B. E. Kardynal, R. M. Stevenson, A. J. Shields, C. J. Lob o, K. Co op er, N. S. Beattie, D. A. Ritchie, and M. Pepper, Science 295 , 102 (2002). [8] C. Eic hler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. W allraff, Phys. Rev. Lett. 106 , 220503 (2011). [9] E. Knill, R. LaFlamme, and G. J. Milburn, Nature (Lon- don) 409 , 46 (2001). [10] T. C. Ralph, A. Gilchrist, and G. J. Milburn, Ph ys. Rev. A 68 , 042319 (2003). [11] N. Gisin, G. Rib ordy , W. Tittel, and H. Zbinden, Re- views of mo dern ph ysics, 74 , 145 (2002). 16 [12] J. I. Cirac, P . Zoller, H. J. Kimble, and H. Mabuchi, Ph ys. Rev. Lett. 78 , 3221 (1997). [13] V. P . Bela vkin, In Le ctur e notes in Contr ol and Inform Scienc es 121 , 245–265, Springer–V erlag, Berlin 1989. [14] V. P . Belavkin, In Sto chastic Metho ds in Mathematics and Physics 310–324, W orld Scientific, Singap ore 1989. [15] P . Staszewski and G. Staszewsk a, Op en Systems & In- formation Dynamics, 3 , 275 (1995). [16] A. Barc hielli and V. P . Belavkin, Phys. A Math. Gen. 24 , 1495 (1991). [17] J.E. Gough, A. Sob olev, Op en Sy s. & Inf. Dynamics, 11, 1-21, (2004) [18] L. Bouten, M. Guta, and H. Maassen, J. Phys. A: Math. and Gen. 37, 3189 (2004). [19] L. Bouten, R. v an Handel and M. R. James, SIAM Jour- nal on Control and Optimization 46 , 2199 (2007). [20] J. Gough, C. K¨ ostler, Comm un. Sto c h. Anal., 4, No. 4, 505-521 (2010) [21] R.L Stratonovic h, Radio Engineering and Electronic Ph ysics, 5:11, pp.1-19, (1960). [22] H. J. Carmic hael. An op en systems appr o ach to quantum optics (Springer: lecture notes in ph ysics vol. 18, 1993). [23] J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68 , 580 (1992). [24] N. Gisin and I. C. Perciv al, J. Ph ys. A: Math. Gen. 25 , 567 (1992). [25] R. Drum, A. S. P arkins, P . Zoller, and C. W. Gardiner, Ph ys. Rev. A 46 , 4382 (1992). [26] C. W. Gardiner and P . Zoller. Quantum Noise (Springer Berlin, 2000). [27] C. W. Gardiner and M. J. Collett, Ph ys. Rev. A 31 , 3761 (1985). [28] S. M. T an, D. F. W alls, and M. J. Collett, Phys. Rev. Lett. 66 , 252-255 (1991) [29] H. M. Wiseman and G. J. Milburn, Quan tum Measure- men t and Control , (Cambridge Univ. Press, Cambridge, 2010) [30] G. J. Milburn W. J. Munro, K. Nemoto and S. L. Braun- stein, Phys. Rev. A 66 , 023819 (2002). [31] W. J. Munro, T. C. Ralph, S. Glancy , S. L. Braunstein, A. Gilchrist, K. Nemoto, and G. J. Milburn, Journal of Optics B: Quantum and Semiclassical Optics, 6 , 828 (2004). [32] M. Y anagisaw a and H. Kimura, IEEE T rans. Automat. Con trol 48 , 2107 (2003), and M. Y anagisaw a and H. Kim ura, IEEE T rans. Automat. Control 48 , 2121 (2003). [33] J. Gough, M. R. James, Commun. Math. Phys. 287 , 1109 (2009). [34] J. Gough, M. R. James, IEEE T rans. on Automatic Con- trol 54 , 2530 (2009). [35] R. L. Hudson and K. R. Parthasarath y , Commun. Math. Ph ys. 93 , 301 (1984). [36] K.R. Parthasarath y . A n intr o duction to quantum sto chas- tic c alculus (Birkhauser, 1992). [37] V. Belavkin, Theory Probab. Appl. 3 8, 573 (1994) [38] B.D.O Anderson, J. Moore, Optimal Filtering , (Prentice- Hall, 1979) [39] C. W. Gardiner, Phys. Rev. Lett. 70 , 2269 (1993). [40] H. J. Carmic hael, Phys. Rev. Lett. 70 , 2273 (1993). [41] K. M. Gheri, K. Ellinger, T. P ellizzari, and P . Zoller, F ortsc hr. Phys. 46 , 401 (1998). [42] J. Gough, M. James, and H. Nurdin, Quantum master equation and filter for systems driven b y fields in a single photon state, IEEE Conference on Decision and Control, (2011). [43] M. G. Raymer, J. Noh, K. Banaszek and I. A. W almsley , Ph ys. Rev. A 72 , 023825 (2005); A. M. Bra´ nczykT. C. Ralph, W. Helwig and C. Silb erhorn, New Journal of Ph ysics 12 , 063001 (2010). [44] R. Loudon, The Quantum The ory of Light , 3rd ed. (Ox- ford Universit y Press, Oxford, 2000). [45] G. J. Milburn, in Springer Handb o ok of Lasers and Op- tics, edited by F. T r¨ ager (Springer, 2007) Chap. 14, pp. 1053 -1078. [46] G. J. Milburn, Eur. Ph ys. J. Sp ecial T opics 159, 113 (2008). [47] B. Lounis and M. Orrit, Rep. Prog. Ph ys. 68 , 1129 (2005); S. Scheel, Journal of Mo dern Optics 56 , 141 (2009). [48] J. Kupsc h, J. Kupsch, in De c oher enc e and the App e ar anc e of a Classic al World in Quantum The ory , edited by D. Guilini et al. (Springer, Berlin, 1997). [49] J.C. Garrison, R.Y. Chiao, Quantum optics , (Oxford Univ ersity Press, 2008). [50] M. Stobi´ nsk a, G. Alber, and G. Leuc hs, EPL 86 , 14007 (2009). [51] Y. W ang, J. Min´ a ˇ r, L. Sheridan, and V. Scarani, Ph ys. Rev. A 83 , 063842 (2011). [52] M. Stobi´ nsk a, G. Alb er, and G. Leuchs, Chapter 8 - Quantum Ele ctr o dynamics of One-Photon Wave Pack- ets Pages 457-483, in Unstable States in the Continuous Sp e ctr a, Part I: A nalysis, Concepts, Methods, and R e- sults , Edited by Cleanthes A. Nicolaides and Erkki Brn- das, V olume 60, Pages 1-549 (2010). Also av ailable as [53] E. Rephaeli, Jung-Tsung Shen, and S. F an, Phys. Rev. A 82 , 033804 (2010).