Abstract. A generalized analytical tripartite loss model is posited for Mach-Zehnder interferometer (MZI)

Transcription

1 Tripartite loss model for Mach-Zehnder interferometers with application to phase sensitivity : Complete expressions for measurement operator mean values variances cross correlations A. D. Parks S. E. Spence J. E. Troupe arxiv:physics/ v1 22 Nov 2006 Quantum Processing Group Electromagnetic Sensor Systems Department Naval Surface Warfare Center Dahlgren Virginia Center for Quantum Studies George Mason University Fairfax Virginia Astract A generalized analytical tripartite loss model is posited for Mach-Zehnder interferometer MZI phase sensitivity which is valid for oth aritrary photon input states aritrary system environmental states. This model is shown to susume the phase sensitivity models for the lossless MZI the ground state MZI. It can e employed to develop specialized models useful for estimating phase sensitivities as well as for performing associated design trade-off analyses for MZIs which operate in environmental regimes that are not contained within the ground state MZI s envelope of validity. As a simple illustration of its utility the model is used to develop phase sensitivity expressions for an MZI with excited internal arms an MZI with excited output channels. These expressions yield a conditional relationship etween the expected numer of photons entering an MZI its efficiency parameters which - when satisfied - predicts an enhanced phase sensitivity for the MZI with excited output channels relative to that for the MZI with excited internal arms. 1

2 I. INTRODUCTION A loss model for Mach-Zehnder interferometers MZIs has recently een reported y us in the literature1] hereafter referred to as PI. This model is used to develop a phase sensitivity expression for aritrary photon input states that is generally applicale for lossy MZIs employing suunit efficiency homodyne detection schemes. In particular it is shown there that this phase sensitivity 2 ϕ is given y κ 2 α 2 C α κ 2 γ 2 C γ κ 2 ε 2 C ε κ 2 αγ 2 C αγ κ 2 αε 2 C αε κ 2 γε 2 C γε κ α κ αγ Dα;αγ κ α κ αε Dα;αε κ γ κ αγ Dγ;αγ κ γ κ γε Dγ;γε κ ε κ αε Dε;αε κ ε κ γε Dε;γε κ αγ κ αε Dαγ;αε κ αγ κ γε Dαγ;γε κ αε κ γε Dαε;γε 2 ϕ = κ C α C α κ αγ C ϕ αγ κ αε 2 1 ϕ αε ϕ ϕ is the associated phase angle to e measured; the suscripts x {α γ ε} refer to the three regions used in the model 0 < x 1 is the value of the efficiency parameter for region x; the suscripted κ s are constants which depend upon the regional efficiency 2 parameters. The quantities 2 C x = Ĉ2 x Ĉx Dx;y = Ĉx ĈyĈx Ĉ y 2 Ĉx Ĉy are measurement operator variances cross correlations the operator Ĉ measures the difference in the numer of photons exiting the output ports respectively Ŷ = Ψ Ŷ Ψ is the mean value of the operator Ŷ for the system state Ψ = ψ ain in ψ c ψ d ψ e ψ f ψ g ψ h. Here ψ ain in is the preloss input state ψ x x {c d e f g h} is the normalized system environmental state associated with the system environmental annihilation operator x. Precise definitions for the suscripted κ s Ĉx operators appearing aove are provided y eqs.1-5 in PI. The reader is referred to section II in PI for a description of the physical assumptions the regional architecture upon which the model is ased. The primary focus of PI was the development of the phase sensitivity expression for the ground state MZI. Such an MZI is defined in terms of our model as one which exhiits loss via non-unit regional efficiency parameter values for which all system environmental states are vacuum states i.e. ψ x = 0 x {c d e f g h} so that Ψ = ψ ain in In this case the ground state phase sensitivity 2 ϕ gs is given y eq.1 with all ut the first fourth fifth terms in the numerator the 2

3 first term in the denominator set equal to zero see eqs in PI. This expression for 2 ϕ gs is a useful result ecause - as discussed in PI - it represents to a good approximation MZI phase sensitivity for a wide range of MZI environmental temperatures for frequencies in the near-ir to the near-uv region of the electromagnetic spectrum. However for certain MZI applications it may e important to model the phase sensitivity for conditions the system environmental states are not vacuum states i.e. the MZI is not in its ground state so that eq.12 in PI does not apply. When using eq.1 to model 2 ϕ for such cases it is necessary to have the complete analytical expressions for each term appearing in the right h side of this equation. The purpose of this article is to extend the results of PI y providing the expressions for these terms so that eq.1 is a complete model that may e more generally useful for MZI phase sensitivity analyses involving system environmental regimes which are not contained within the envelope of validity for the ground state model. Complete expressions for the partial phase derivatives of the measurement operator mean values the measurement operator variances the measurement operator cross correlations appearing in the right h side of eq.1 are provided respectively in the following three sections of this paper. These expressions are validated in the final section of this paper y demonstrating that - when used in conjunction with eq.1 - they yield the phase sensitivities for the lossless MZI the ground state MZI previously developed in PI. As an additional illustration of its utility as an analytical tool the model is employed to provide the phase sensitivity expressions for an MZI with excited internal arms i.e. an excited γ-region configuration for an MZI with excited output channels i.e. an excited ε-region configuration. These expressions are used to estalish an associated conditional relationship etween the expected numer of photons entering an MZI its regional efficiency parameter values. When this condition is satisfied then the phase sensitivity of the excited γ-region configuration is more degraded than that of the excited ε-region configuration. 3

4 II. PARTIAL PHASE DERIVATIVES OF MEASUREMENT OPERATOR MEAN VALUES The expressions for the measurement operator mean values needed to evaluate the partial derivatives appearing in the denominator of eq.1 are Ĉαε Ĉα Ĉαγ { = 4 ˆρ ˆρ ˆρ aˆρ a cosϕ ˆρ aˆρ } ˆρ ˆρ a sin ϕ 2 = 2 i ˆρ ê ˆρ ˆf ˆf ˆρ e iϕ ê ˆρ e iϕ = {ˆρ aĝ 1 e iϕ ĝ ˆρ a 1 e iϕ } i ˆρ ˆf a ˆf ˆρ a ˆρ aê e iϕ ê ˆρ a e iϕ 3 { 1 ˆρ ĥ e iϕ 1 ĥˆρ } e iϕ 4 { 1 i ˆρ aĥ e iϕ 1 ĥˆρ } { a e iϕ 1 i ˆρ ĝ e iϕ ĝ ˆρ } 1 e iϕ ρ a = αâ in 1 αĉ ρ = α in 1 α d. Here 0 < α 1 is the efficiency parameter for the α region of the model â in in are the input port annihilation operators ĉ d are the α region environmental annihilation operators ê f are the γ region environmental annihilation operators ĝ ĥ are the ε region environmental annihilation operators. - 4 : The partial derivatives appearing in eq.1 are readily otained as follows from eqs. 2 Ĉα ϕ Ĉαγ ϕ Ĉαε ϕ = 4{ ρ ρ ρ a ρ a sin ϕ ρ a ρ ρ ρ a cosϕ} 5 { = 2 ρ ê i ρ aê e iϕ ê ρ } i ê ρ a e iϕ ρ aĥ ρ ĝ i ρ aĝ = ĥ ρ a ĝ ρ i ] ρ ĥ e iϕ ĝ ρ a ĥ ρ ] e iϕ. 4

5 III. MEASUREMENT OPERATOR VARIANCES Expressions for the measurement operator variances needed for the evaluation of the numerator of eq.1 are 2 2 ˆρ ˆρ ˆρ aˆρ a ˆρ ˆρ ˆρ aˆρ a cos 2 ϕ C α = 16 ˆρ aˆρ ˆρ ˆρ a ˆρ aˆρ ˆρ ˆρ a sin 2 ϕ { } 6 ˆρ ˆρ ˆρ aˆρ a ˆρ aˆρ ˆρ ˆρ a 2 ˆρ ˆρ ˆρ aˆρ a ˆρ aˆρ ˆρ ˆρ sin ϕ cosϕ a the term enclosed in curley races is the anti-commutator defined y { X Ŷ } XŶ Ŷ X; ê ê 2 C γ = 4 ˆf ˆf 2 êê ˆf ˆf ê ˆf ê ˆf 2 ê ê ˆf ˆf êê ˆf ˆf 2 ê ˆf ê ˆf ; 7 2 ĝĝĝ ĝ ĝ ĝ ĥ 2 ĥ ĥ 2 C ε = ĥĥ ĥ 2 ĝ ĝĥ ĥ ĝ ĝ ; ĥ ĥ ˆF1 êê ˆF 1ê ê ˆF2 ê ê ˆF3 êê ˆF4 ˆf ˆf ˆF ˆf 4 ˆf ˆF5 ˆf ˆf ˆF6 ˆf ˆf ˆF7 ê ˆf ˆF 7ê ˆf ˆF8 ê ˆf ˆF 8ê ˆf ˆρ ˆf 2 2 ˆρ ˆf ˆρ ˆf 2 2 a ˆρ a ˆf 2 ˆρ ˆf ˆρ ˆf ˆρ ˆf a ˆρ a ˆf ˆρ ê ˆρ ê ˆρ aê ˆρ a ê ˆρ aê 2 2 ˆρ ê e 2iϕ ˆρa ê 2 ˆρ ê 2 e 2iϕ 2 C αγ = 4 2 ˆρ ˆf a ˆρ ê ˆρ a ˆf ˆf ˆρ ê ˆρ ˆρ aê ˆρ ˆf ˆρ aê e iϕ 2 ˆρ a ˆf ˆρ ê ˆf ˆρ a ˆρ ê ˆρ ˆf ˆρa ê ˆf ˆρ ˆρa ê e iϕ 2i ˆρ ˆf a ˆρ ˆf ˆρ ˆf a ˆρ ˆf ˆρ a ê ˆρ ê ˆρ a ˆf ˆρ ˆf ˆρ a ˆf ˆρ ˆf ˆρ aê ˆρ ê ˆρ aê ˆρ ê e 2iϕ ˆρ a ê ˆρ ê e 2iϕ ˆρ a ˆf ˆρ aê ˆf ˆρ a ˆρ aê ˆρ ˆf ˆρ ê ˆρ ˆf ˆρ ê e iϕ ˆρ ˆf ˆρ ê ˆf ˆρ ˆρ ê ˆρ a ˆf ˆρa ê ˆf ˆρ a ˆρa ê e iϕ ] ˆF 1 = ˆρ aˆρ a ˆρ ˆρ 2iˆρ aˆρ e 2iϕ = 8 5

6 2 C αε = ˆF 2 = ˆρ aˆρ a ˆρ ˆρ i ˆρ aˆρ ˆρ aˆρ ˆF 3 = ˆρ aˆρ a ˆρ ˆρ i ˆρ aˆρ ˆρ aˆρ ˆF 4 = ˆρ aˆρ a ˆρ ˆρ 2iˆρ aˆρ ˆF 5 = ˆρ aˆρ a ˆρ ˆρ i ˆρ aˆρ ˆρ ˆρ a ˆF 6 = ˆρ aˆρ a ˆρ ˆρ i ˆρ aˆρ ˆρ ˆρ a ˆF 7 = 2i ˆρ aˆρ a ˆρ ˆρ e iϕ ] ˆF 8 = 2 ˆρ aˆρ ˆρ aˆρ i ˆρ aˆρ a ˆρ aˆρ a ˆρ ˆρ ˆρ ˆρ ˆR1 ĝĝ ˆR 1ĝ ĝ ˆR2 ĝ ĝ ˆR3 ĝĝ ˆR4 ĥĥ ˆR5 ĥ ĥ ˆR6 ĥĥ ˆR7 ĝĥ ˆR 7ĝ ĥ ˆR8 ĝ ĥ e iϕ ; ˆR 4ĥ ĥ ˆR 8ĝĥ ˆρ aĝ2 2 ˆρ ĥ 2 ˆρ aĝ ˆρ ĥ 1 e iϕ 2 2 ˆρa ĝ 2 ˆρ ĥ 2 ˆρa ĝ ˆρ ĥ 1 e iϕ ˆρ aĥ ˆρ ĝ 2 ˆρ aĥ ˆρ ĝ 1 e iϕ 2 2 ˆρ a ĥ ˆρ ĝ 2 2 ˆρ a ĥ ˆρ ĝ 1 e iϕ 2 ˆρ 2 a ĝ ˆρ a ĝ ˆρ ĥ ˆρ ĥ ˆρ aĝ ˆρ ĥ ˆρ a ĝ ˆρ 1 e iϕ 1 e iϕ ĥ ˆρ 2 aĥ ˆρ a ĥ ˆρ ĝ ˆρ ĝ ˆρ aĥ ˆρ ĝ 1 e ˆρ iϕ 1 e iϕ a ĥ ˆρ ĝ ˆρ 2i a ĝ ˆρ aĥ ˆρ ĥ ˆρ aĥ ˆρ aĝ ˆρ 1 e iϕ 1 e iϕ ĝ ˆρ ĥ ˆρ ĝ ˆρa ĝ ˆρ ĝ ˆρ ĥ ˆρ ĝ ˆρ a ĝ ˆρ 1 e a ĥ ˆρ ĥ ˆρ iϕ 1 e iϕ a ĥ ˆρ a ĝ ˆρ ĝ ˆρ ĥ ˆρ ĝ ˆρ aĝ ˆρ 1 e a ĥ ˆρ ĥ ˆρ iϕ 1 e iϕ a ĥ ˆρa ĝ ˆρ aĥ ˆρ ĥ ˆρ aĥ ˆρ a ĝ ˆρ 1 e ĝ ˆρ iϕ 1 e iϕ ] ĥ ˆρ ĝ ˆR 1 = ˆρ aˆρ a 1 e iϕ 2 ˆρ ˆρ 1 e iϕ 2 2iˆρ aˆρ 1 e iϕ 1 e iϕ 9 6

7 ˆR 2 = ˆρ aˆρ a 1 eiϕ 1 e iϕ ˆρ ˆρ 1 e iϕ 1 e iϕ iˆρ aˆρ 1 eiϕ 1 e iϕ iˆρ aˆρ 1 e iϕ 1 e iϕ ˆR 3 = ˆρ aˆρ a 1 e iϕ 1 e iϕ ˆρ ˆρ 1 e iϕ 1 e iϕ iˆρ aˆρ 1 e iϕ 1 e iϕ iˆρ aˆρ 1 e iϕ 1 e iϕ ˆR 4 = ˆρ ˆρ 1 e iϕ 2 ˆρ aˆρ a 1 e iϕ 2 2iˆρ aˆρ 1 e iϕ 1 e iϕ ˆR 5 = ˆρ ˆρ 1 eiϕ 1 e iϕ ˆρ aˆρ a 1 eiϕ 1 e iϕ iˆρ aˆρ 1 e iϕ 1 e iϕ iˆρ aˆρ 1 eiϕ 1 e iϕ ˆR 6 = ˆρ ˆρ 1 e iϕ 1 e iϕ ˆρ aˆρ a 1 e iϕ 1 e iϕ iˆρ aˆρ 1 e iϕ 1 e iϕ iˆρ aˆρ 1 e iϕ 1 e iϕ ˆρ ˆR 7 = 2 1 e iϕ 2 ˆρ aˆρ 1 e iϕ 2 iˆρ aˆρ a 1 e iϕ 1 e iϕ iˆρ ˆρ 1 e iϕ 1 e iϕ ] 2 ˆρ aˆρ ˆR 8 = 1 e iϕ ˆρ aˆρ 1 e iϕ 1 e iϕ ] i ˆρaˆρ a ˆρ aˆρ a 1 e iϕ 1 e iϕ ˆρ ˆρ ˆρ ˆρ ; 1 e iϕ 1 e iϕ ˆQ1 êê ˆQ 1ê ê ˆQ2 ê ê ˆQ3 êê ˆQ4 ˆf ˆf ˆQ ˆf 4 ˆf ˆQ5 ˆf ˆf ˆQ6 ˆf ˆf ˆQ7 ê ˆf ˆQ 7ê ˆf ˆQ8 ê ˆf ˆQ 8ê ˆf ê ĝ êĝ 2 ˆf ĥ ˆfĥ ê ĥ 2 êĥ ˆf ĝ 2 2 ˆfĝ 2 êĝ êĝ êĥ ˆf 2 C γε = ĥ ˆfĥ ê ĥ ˆf ĝ ˆfĝ ê ĥ ˆf ĝ ˆfĝ êĥ ê ĝ ˆf ĥ êĝ ˆfĥ ê ĝ ˆfĥ êĝ ˆf ĥ ê ĥ ˆfĝ ˆf ĝ êĥ 2i ê ĝ ê ĥ êĝ ê ĥ êĝ ˆfĝ ê ĝ ˆfĝ êĥ ˆfĥ ˆf êĥ ĥ ˆf ĥ ˆf ĝ ˆfĥ ˆf ĝ êĝ êĥ ê ĝ êĥ ê ĝ ˆf ĝ êĝ ˆf ĝ ˆf ĥ ê ĥ ˆfĥ ê ĥ ˆfĥ ˆfĝ ˆf ĥ ˆfĝ ˆQ 1 = ĝ ĝ ĥ ĥ 2iĝ ĥ 7

8 ˆQ 2 = ĝĝ ĥĥ i ĝ ĥ ĝĥ ˆQ 3 = ĝ ĝ ĥ ĥ i ĝ ĥ ĝĥ ˆQ 4 = ĥ ĥ ĝ ĝ 2iĝ ĥ ˆQ 5 = ĝĝ ĥĥ i ĝĥ ĝ ĥ ˆQ 6 = ĝ ĝ ĥ ĥ i ĝĥ ĝ ĥ ˆQ 7 = 2i ĝ ĝ ĥ ĥ ˆQ 8 = 2 ĝĥ ĝ ĥ i ĝĝ ĝ ĝ i ĥĥ ĥ ĥ. IV. MEASUREMENT OPERATOR CROSS CORRELATIONS The expressions for the measurement operator cross correlations that are required to evaluate the numerator in eq.1 are { Ŝ1 Ŝ 1 Ŝ2 Ŝ 2 } cosϕ { Ŝ3 Ŝ 3 Ŝ4 Ŝ 4 } sinϕ ˆDα;αγ = 8 2{ ˆρ ˆρ ˆρ aˆρ a cos ϕ ˆρ aˆρ ˆρ aˆρ sin ϕ} { ˆρ ˆf i ˆρ ˆf a ˆρ ˆf i ˆρ a ˆf ˆρ aê i ˆρ e e iϕ ˆρ a ê i ˆρ ê e iϕ } Ŝ 1 = ˆρ ˆρ ˆρ ˆρ ˆρ ˆρ 2ˆρ aˆρ aˆρ i 2ˆρ aˆρ ˆρ ˆρ aˆρ aˆρ a a] ˆρ aˆρ aˆρ ˆf ] Ŝ 2 = 2ˆρ aˆρ ˆρ ˆρ aˆρ aˆρ a ˆρ aˆρ aˆρ a i 2ˆρ aˆρ aˆρ ˆρ ˆρ ˆρ ˆρ ˆρ ˆρ êe iϕ Ŝ 3 = 2ˆρ aˆρ ˆρ ˆρ aˆρ ˆρ ˆρ aˆρ ˆρ i 2ˆρ aˆρ aˆρ ˆρ aˆρ aˆρ ˆρ aˆρ aˆρ ] ˆf ] Ŝ 4 = 2ˆρ aˆρ aˆρ ˆρ aˆρ aˆρ ˆρ aˆρ aˆρ i 2ˆρ aˆρ ˆρ ˆρ aˆρ ˆρ ˆρ aˆρ ˆρ êe iϕ ; { Û1 Û 1 Û2 Û 2 } cosϕ { Û3 Û 3 Û4 Û 4 } sin ϕ ˆDα;αε = 4 2{ ˆρ ˆρ ˆρ aˆρ a cosϕ ˆρ aˆρ ˆρ aˆρ sin ϕ} { ˆρ aĝ ˆρ ĥ 1 e iϕ ˆρa ĝ ˆρ ĥ 1 e iϕ i ˆρ aĥ ˆρ ĝ 1 e iϕ i ˆρ ĝ ˆρ a ĥ 1 e iϕ } 8

9 Û 1 = 2ˆρ aˆρ ˆρ ˆρ aˆρ aˆρ a ˆρ aˆρ aˆρ a ĝ 1 e iϕ iĥ 1 e iϕ] Û 2 = 2ˆρ aˆρ aˆρ ˆρ ˆρ ˆρ ˆρ ˆρ ˆρ ĥ 1 e iϕ iĝ 1 e iϕ] Û 3 = 2ˆρ aˆρ aˆρ ˆρ aˆρ aˆρ ˆρ aˆρ aˆρ ĝ 1 e iϕ iĥ 1 e iϕ] Û 4 = 2ˆρ aˆρ ˆρ ˆρ aˆρ ˆρ ˆρ aˆρ ˆρ ĥ 1 e iϕ iĝ 1 e iϕ] ; ˆV1 ˆV 1 ˆV2 ˆV 2 ˆV3 ˆV 3 ˆV4 ˆV 4 ˆDγ;αγ = 4 2 ê ˆf ê ˆf ˆρ a ˆf i ˆρ ˆf ˆρ ˆf a i ˆρ ˆf ˆρ ê i ˆρ aê e iϕ ˆρ ê i ˆρ a ê e iϕ ] ˆV 1 = ˆρ a iˆρ ê ˆf ˆf ˆV 2 = ˆρ a iˆρ ê ˆf ˆf ˆV 3 = 2 ˆρ a iˆρ ê ˆf ˆf ˆV 4 = ˆρ iˆρ a 2ê ê ˆf êê ˆf êê ˆf e iϕ ; Ŵ1 Ŵ 1 Ŵ2 Ŵ 2 2 ê ˆf ê ˆf ˆDγ;γε = 2 ˆf ĝ i ˆf ĥ ˆfĝ i ˆfĥ ê ĥ i ê ĝ êĥ i êĝ ] Ŵ 1 = ê ˆf ˆf ê ˆf ˆf 2ê ˆf ˆf ĝ iĥ Ŵ 2 = ê ê ˆf êê ˆf 2ê ê ˆf ĥ iĝ ; Ĥ1 Ĥ 1 Ĥ2 Ĥ 2 Ĥ3 Ĥ 3 Ĥ4 Ĥ 4 2 ĝĝ ĥ ] ĥ ˆρ aĝ ˆρ ˆDε;αε = ĥ 1 e iϕ ˆρa ĝ ˆρ ĥ 1 e iϕ i ˆρ aĥ ˆρ ĝ 1 e iϕ i ˆρ a ĥ ˆρ ĝ 1 e iϕ ] Ĥ 1 = ˆρ 2ĝ ĝĥ ĥ ĥĥ ĥ ĥ ĥ 1 e iϕ Ĥ 2 = ˆρ a 2ĝ ĥ ĥ ĝ ĝĝ ĝ ĝ ĝ 1 e iϕ Ĥ 3 = iˆρ a 2ĝ ĝĥ ĥ ĥĥ ĥ ĥ ĥ 1 e iϕ 9

10 1 Ĥ 4 = iˆρ 2ĝ ĥ ĥ ĝ ĝĝ ĝ ĝ ĝ e iϕ ; ˆL1 ˆL 1 ˆL2 ˆL 2 ˆL3 ˆL 3 ˆL4 ˆL 4 ˆDε;γε = 2 ĝĝ ĥ ĥ ê ĝ êĝ ˆf ĥ ˆfĥ êĥ i ê ĥ i ˆf ĝ ˆfĝ ] ˆL 1 = ê 2ĝ ĥ ĥ ĝ ĝĝ ĝ ĝ ĝ ˆL 2 = ˆf 2ĝ ĝĥ ĥ ĥĥ ĥ ĥ ĥ ˆL 3 = iê 2ĝ ĝĥ ĥ ĥĥ ĥ ĥ ĥ ˆL 4 = i ˆf 2ĝ ĥ ĥ ĝ ĝĝ ĝ ĝ ĝ ; ˆK1 ˆK 1 ˆK2 ˆK 2 ˆK3 ˆK 3 ˆK4 ˆK 4 ˆK5 ˆK 5 ˆK6 ˆK 6 2 ˆρ ˆf ˆρ ˆf ˆρ aê e iϕ ˆρ a ê e iϕ ˆDαγ;αε = 2 i ˆρ ˆf a i ˆρ a ˆf i ˆρ ê e iϕ i ˆρ ê e iϕ ] ˆρ aĝ ˆρ ĥ 1 e iϕ ˆρa ĝ ˆρ ĥ 1 e iϕ i ˆρ aĥ ˆρ ĝ 1 e iϕ i ˆρ ĝ ˆρ a ĥ 1 e iϕ ] ˆK 1 = 2ˆρ ˆρ ˆfĥ 2ˆρ aˆρ ˆfĝ 2ˆρ aˆρ ˆf ĝ ˆρ ˆρ ˆf ĥ ˆρ ˆρ ˆf ĥ i2ˆρ aˆρ a ˆfĝ ˆρ aˆρ a ˆf ĝ ˆρ aˆρ a ˆf ĝ 2ˆρ aˆρ ˆfĥ 2ˆρ aˆρ ˆf ĥ 1 e iϕ ] 1 ˆK 2 = 2 ˆρ aˆρ êĥ ˆρ ˆρ êĝ iˆρ aˆρ aêĥ ˆρ aˆρ êĝ e iϕ e iϕ ˆK 3 = 2ˆρ aˆρ ˆfĥ a ˆρ aˆρ ˆf a ĥ ˆρ aˆρ ˆf a ĥ 2ˆρ aˆρ ˆfĝ 2ˆρ aˆρ ˆf ĝ i2ˆρ aˆρ ˆfĥ 2ˆρ aˆρ ˆf ĥ 2ˆρ ˆρ ˆfĝ ˆρ ˆρ ˆf ĝ ˆρ ˆρ ˆf 1 e iϕ ĝ 1 ˆK 4 = 2 ˆρ aˆρ a ê ĝ ˆρ aˆρ ê ĥ iˆρ aˆρ ê ĝ ˆρ ˆρ ê ĥ ] e iϕ e iϕ 1 ˆK 5 = 2ˆρ aˆρ êĥ ˆρ ˆρ êĝ ˆρ ˆρ êĝ iˆρ aˆρ aêĥ ˆρ aˆρ aêĥ 2ˆρ aˆρ êĝ ] e iϕ e iϕ 1 ˆK 6 = 2ˆρ aˆρ êĥ ˆρ aˆρ a êĝ ˆρ aˆρ aêĝ i2ˆρ aˆρ êĝ ˆρ ˆρ êĥ ˆρ ˆρ ] e iϕ êĥ e iϕ ; 10

11 Ô1 Ô 1 Ô2 Ô 2 Ô3 Ô 3 Ô4 Ô 4 Ô5 Ô 5 Ô6 Ô 6 ˆDαγ;γε = 2 2{ ˆρ ˆf ˆρ ˆf ˆρ aê e iϕ ˆρ a ê e iϕ i ˆρ ˆf a ˆρ a ˆf ˆρ ê e iϕ ˆρ ê ] e iϕ } { ê ĝ êĝ êĥ ˆf ĥ ˆfĥ i ê ĥ ˆf ĝ ˆfĝ ]} ˆρ ê ˆfĝ ê ˆfĝ ê ˆf ĝe iϕ ê ˆfĝ ] e iϕ Ô 1 = 2 iˆρ a ê ˆfĝ ê ˆfĝ ê ˆf ĝe iϕ ê ˆfĝ ] e iϕ Ô 2 = ˆρ iˆρ a 2 ˆf ˆfĥ ˆf ˆf ĥ ˆf ˆfĥ ˆρ a ê ˆfĥ ê ˆfĥ ê ˆf ] ĥe iϕ ê ˆfĥ e iϕ Ô 3 = 2 iˆρ ê ˆfĥ ê ˆfĥ ê ˆf ] ĥe iϕ ê ˆfĥ e iϕ Ô 4 = ˆρ a iˆρ 2 ˆf ˆfĝ ˆf ˆf ĝ ˆf ˆfĝ Ô 5 = ˆρ iˆρ a 2êêĥ êê ĥ ê êĥ e iϕ 2êêĝ Ô 6 = ˆρ a iˆρ êê ĝ ê êĝ e iϕ ; ˆM1 ˆM 1 ˆM2 ˆM 2 2{ ˆρ aĝ ˆρ ˆDαε;γε = ĥ 1 e iϕ ˆρ a ĝ ˆρ ĥ 1 e iϕ i ˆρ aĥ ˆρ ĝ 1 e iϕ i ˆρ a ĥ ˆρ ĝ 1 e iϕ } { ê ĝ êĝ êĥ ˆf ĥ ˆfĥ i ê ĥ ˆf ĝ ˆfĝ } ˆρ a2ê ĝĝ êĝĝ êĝ ĝ 2 ˆf ĝĥ 2 ˆfĝĥ ˆρ ˆM 1 = 2ê ĝĥ 2êĝ ĥ 2 ˆf ĥĥ ˆfĥĥ ˆfĥ ĥ] 1 e iϕ iˆρ a 2ê ĝĥ 2êĥ ĝ 2 ˆf ĝĝ ˆfĝĝ ˆfĝ ĝ ˆρ 2ê ĥĥ êĥĥ êĥ ĥ 2 ˆf ĝĥ 2 ˆfĝ ĥ] ˆρ a 2 ˆf ĝĥ 2 ˆfĝ ĥ 2ê ĥĥ êĥĥ êĥ ĥ ˆρ ˆM 2 = 2ê ĝĥ 2êĝĥ 2 ˆf ĝĝ ˆfĝĝ ˆfĝ ĝ] 1 e iϕ iˆρ a 2ê ĝĥ 2êĝ ĥ 2 ˆf ĥĥ. ˆfĥĥ ˆfĥ ĥ ˆρ 2 ˆf ĝĥ 2 ˆfĝĥ 2ê ĝĝ êĝĝ êĝ ĝ] 11

12 V. APPLICATIONS For the purpose of validation it is demonstrated in this section that when the expressions for the loss model given in the previous three sections are used in conjunction with eq.1 they yield the required phase sensitivity results that have een previously developed in PI for the lossless MZI the ground state MZI. As an additional illustrative application this model is also used to derive the phase sensitivity 2 ϕ γ for an MZI with its two internal arms the γ region in the excited state ψ e ψ f = 1 1 the phase sensitivity 2 ϕ ε for an MZI with its two channels etween the output ports ideal detectors the ε region in the excited state ψ g ψ h = 1 1. These results are used to determine a condition which relates the expected numer of photons entering an MZI to its efficiency parameters such that 2 ϕ γ > 2 ϕ ε when this condition is satisfied. A. Phase sensitivity for a lossless MZI Although this case is trivial only requires the evaluation of eqs.5 6 it is included here for completeness. When the MZI is lossless then each of the regional efficiencies has unit value so that - with the exception κ α = all of the powers products of suscripted κ s in eq.1 are zero valued ρ a = â in ρ = in. In this case eqs.2 6 Ĉ0 reduce to the quantities α 2 C α 0 defined in section IV of PI eq.1 reduces to the expression for the phase sensitivity 2 ϕ lossless for a lossless MZI given y eq.10 therein. B. Phase sensitivity for the ground state MZI The ground state MZI is defined when the regional efficiency parameters in the MZI model have their values in the open real interval 0 1 the system state is Ψ gs = ψ ain in In this case eventhough each of the suscripted κ s in eq.1 is non-vanishing - with the exception of the first fourth fifth terms in the numerator the first term in the denominator - each term in eq.1 is zero when its value is determined using Ψ = Ψ gs. Consequently eq.1 reduces to the expression for the ground state MZI phase sensitivity 2 ϕ gs given y eq.11 in PI. These terms are zero ecause they are sums of terms which vanish due to the fact that 12

13 each contains a factor X vac = 0 X is either a single environmental operator or a juxtaposition of several like environmental annihilation or creation operators the suscript vac refers to the fact that the mean value is evaluated using only the associated environmental vacuum state. For example 2 C γ = 0 ecause each term in eq.7 vanishes i.e. for the first term ê ê f f = Ψ gs ê ê f f Ψ gs = ψ ain in ψ ain in ψ c ψ c ψ d ψ d ψ e ê ê ψ e ψ f f f ψ f ψ g ψ g ψ h ψ h = ψ e ê ê ψ e ψ f f f ψ f = 0 ê ê 0 0 f f 0 = 0 0 f f 0 = 0 the reader will also recognize ê ê = N as the numer operator N with the property 0 N 0 = 0 similarly for terms two through seven. It is also easily verified that 2 C α 2 C αγ = 4 Ψ gs F3 F ] 6 Ψ gs 2 C αε = Ψ gs R3 R ] 6 Ψ gs C α are as specified in section V of PI. ϕ C. Phase sensitivities for simple excited state MZIs : A sensitivity trade-off condition Consider now the phase sensitivities 2 ϕ γ 2 ϕ ε for MZIs in the system states Ψ γ = ψ ain in Ψ ε = ψ ain in respectively. When Ψ γ is used as the model s system state it is found that all ut the first second fourth fifth sixth terms in the numerator the first term in the denominator of eq.1 vanish. Using eq.8 it is also found that 2 C αγ = 2 C gs αγ 2 C γ αγ 2 Cαγ gs 4 Ψ γ F3 F ] 6 Ψ γ = 4 Ψ gs F3 F ] 6 Ψ gs 2 Cαγ γ 4 Ψ γ F2 F 3 F 5 F ] ] 6 Ψ γ = 4 α â inâin in in 1. 13

14 Since each of the expressions for 2 C α 2 C αε C α remains invariant when the system ϕ states Ψ gs Ψ γ are used for their valuations then 2 ϕ γ = 2 ϕ gs 2 ϕ e γ 2 ϕ gs κ2 α 2 C α κ 2 αγ 2 Cαγ gs κ2 αε 2 C αε κ C α 2 α ϕ is the phase sensitivity for the ground state MZI 2 ϕ e γ κ2 γ 2 C γ κ 2 αγ 2 Cαγ γ κ2 γε 2 C γε κ C α α ϕ 2. Evaluation of the right h side of the last equation yields this corrects the expression for 2 ϕ es given in section VI of PI ] 2 1 γ 2 ϕ γ = 2 ϕ gs α 2 γ 2 ε ε in in αγ ] in in 1 γ 1 sin ϕ â in in cosϕ â inâin â inâin inâin Thus if the γ region is in the excited state 1 1 γ 0 1 then 2 ϕ γ > 2 ϕ gs. Oserve that when there are no losses in the γ region i.e. when γ = 1 then - as required - 2 ϕ e γ = 0 so that 2 ϕ γ = 2 ϕ gs. When Ψ ε is used as the model s system state then all terms in eq.1 vanish except the first term in the denominator the first fourth fifth sixth terms in the numerator. Also it is determined from eq.9 that 2 C αε = 2 C gs αε 2 C ε αε ] Cαε gs Ψ ε R3 R ] 6 Ψ ε = Ψ gs R3 R ] 6 Ψ gs 2 Cαε ε Ψ ε R2 R 3 R 5 R ] ] 6 Ψ ε = 8 α â inâin in in 1 it is easily verified that the quantities 2 C α 2 C αγ C α ϕ yield identical expressions when the system states Ψ gs Ψ ε are used to evaluate them. Thus 2 ϕ ε = 2 ϕ gs 2 ϕ e ε 14

15 2 ϕ gs κ2 α 2 C α κ αγ 2 C αγ κ 2 αε 2 C gs κ C 2 α α ϕ αε is the phase sensitivity for the ground state MZI so that ] 2 1 ε 2 ϕ ε = 2 ϕ gs α 2 γ 2 ε 2 ϕ e ε κ2 αε 2 Cαε ε κ 2 γε 2 C γε κ C α α ϕ in in â inâin 2 αγ â inâin in in 1 sin ϕ â in in inâin cosϕ It is clear from this expression that when the ε region is in the excited state 1 1 ε 0 1 then 2 ϕ ε > 2 ϕ gs when there are no losses in the ε region so that ε = 1 then - as required - 2 ϕ e ε = 0 2 ϕ ε = 2 ϕ gs. For the sake of further illustrating the utility of the model note that eqs yield the difference 2αγ ε 1 γ 1 ε] 2 ϕ γ 2 ϕ ε = α 2 γ 2 ε in in â inâin â inâin ] 2 11 in in 2 ε 1 γ 2 ε γ ] sin ϕ â in ] 2. in cosϕ inâin. Thus for γ 1 or ε 1 it can e concluded that when the condition γ ε ε 1 γ 2 ] α â inâin in in > γ ε 1 γ 1 ε] 12 is satisfied then 2 ϕ γ > 2 ϕ ε i.e. phase sensitivity is more degraded when an MZI is in state Ψ γ than when it is in state Ψ ε. This is an intuitively pleasing conclusion since - unlike the case for the excited ε region - the degradation in sensitivity due to the excited γ region would e expected to experience additional degradation induced y the unexcited ε region. As a special case of this result oserve that if γ = ε 1 then - since the right h side of 12 has unit value - the relationship 2 ϕ γ > 2 ϕ ε also prevails when â inâin in in > 1 α. 15

16 VI. CLOSING REMARKS The aove provides a generalized analytical tripartite loss model for MZI phase sensitivity which is valid for oth aritrary photon input states aritrary system environmental states. This model susumes the phase sensitivity models for the lossless MZI the ground state MZI is useful for developing specialized models for estimating the phase sensitivity as well as for performing associated design trade-off analyses for MZIs MZIlike instruments which operate in environmental regimes which are not contained within the envelope of validity for the ground state model. 1] A. D. Parks S. E. Spence J. E. Troupe N. J. Rodecap Rev. Sci. Instrum