Double-Fock Superposition Interferometry for Differential Diagnosis of Decoherence
Malte C. Tichy, Young-Sik Ra, Hyang-Tag Lim, Clemens Gneiting, Yoon-Ho Kim, Klaus Mølmer
DDouble-Fock Superposition Interferometry for Differential Diagnosis of Decoherence
Malte C. Tichy, Young-Sik Ra, Hyang-Tag Lim, Clemens Gneiting, Yoon-Ho Kim, and Klaus Mølmer Department of Physics and Astronomy, University of Aarhus, DK–8000 Aarhus, Denmark Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang, 790-784, Korea Physikalisches Institut der Albert-Ludwigs-Universit¨at, D–79104 Freiburg, Germany (Dated: November 21, 2018)Interferometric signals are degraded by decoherence, which encompasses dephasing, mixing andany distinguishing which-path information. These three paradigmatic processes are fundamentallydifferent, but, for coherent, single-photon and N N -states, they degrade interferometric visibilityin the very same way, which impedes the diagnosis of the cause for reduced visibility in a singleexperiment. We introduce a versatile formalism for many-boson interferometry based on double-sided Feynman diagrams, which we apply to a protocol for differential decoherence diagnosis: Twin-Fock states | N, N (cid:105) with N ≥ | N : M (cid:105) = ( | N, M (cid:105) + | M, N (cid:105) ) / √ N > M >
PACS numbers: 42.50.Ar, 03.65.Yz, 42.50.St, 42.50.Dv
I. INTRODUCTION
The coherent superposition of physically exclusivesingle- or many-particle path amplitudes is exemplifiedbest with the minimalistic paradigm of a single particleprepared in the state | (cid:105) ≡ ( | , (cid:105) + | , (cid:105) ) / √
2, a coher-ent superposition of the upper and lower arms of a Mach-Zehnder interferometer. Fringes appear in the output sig-nal, the probability to find the particle in the upper de-tector, as a function of the relative phase η that the par-ticle acquires between the two arms (Fig. 1). We definethe fringe visibility of the output signal ( s , s ) = (1 , V | (cid:105) (1 , = P | (cid:105) (1 , , max − P | (cid:105) (1 , , min P | (cid:105) (1 , , max + P | (cid:105) (1 , , min , (1)where P | N : M (cid:105) ( s ,s ) is the probability of the signal with ( s , s )detected particles for the bosonic initial state | N : M (cid:105) ,defined below. Interference is jeopardized by several de-teriorating effects summarized as decoherence [1–5], and,in practice, the visibility V | (cid:105) (1 , never reaches unity. Theinterferometer in Fig. 1 illustrates three prominent mech-anisms for decoherence: Dephasing, mixing, and pathdistinguishability. By tracing out the internal degrees offreedom of the particle and performing the classical aver-age over phase fluctuations, the state of a single particlein the two arms is described by an effective two-statedensity matrixˆ (cid:37) = 12 (cid:32) e iη V | (cid:105) (1 , e − iη V | (cid:105) (1 , (cid:33) , (2)where the visibility V | (cid:105) (1 , coincides with the classicalensemble-average of the scalar product (cid:104) φ | ˜ φ (cid:105) of the statesof the particle in the upper and the lower arms, and thereby quantifies the coherence between the arms. Thethree decoherence effects are, thus, not differentiated inpractice by the single-particle interferometric signal. For– possibly strong – coherent states, for which the signalintensity replaces the event probability in Eq. (1), thereis no possibility for a qualitative differentiation of deco-herence mechanisms either, nor for N N -states, as weshow below. ⌘ | i !| ˜ i| i| i DistinguishabilityDephasing Mixing ˆ a † , ˆ a † , ˜ ˆ b † , ˆ b † , | N : M i S t a t e p r e p a r a ti on s s | i phase ! | i phase ! e i⌘ rand | i| i phase ! | i phase ! e i⌘ rand | i | ˜ i mix ! | ˜ ih ˜ | mix ! ˆ d Phase acquisition | i mix ! | ih | mix ! ˆ d | ˜ i mix ! | ˜ i mix ! | ˜ rand i| ˜ i mix ! | i mix ! | rand i FIG. 1: Interferometer subject to decoherence. Particles ineither arm start in the same internal single-particle state | φ (cid:105) .In the lower arm, each particle acquires the phase η , whichis measured by combining the arms at a beam splitter andrecording the number of particles s and s in the two de-tectors. The interferometric measurement is jeopardized bydephasing (a random phase η rand , acquired here in the up-per arm with probability 1 − γ phase ), path distinguishabil-ity (a change in the internal states of the particles travellingthrough the lower arm, resulting in a finite scalar product |(cid:104) φ | ˜ φ (cid:105)| = γ dist and leading to decoherence [6]) and mixing(with probability 1 − γ mix for each arm, the particles are leftin an unknown randomly chosen state). The parsimonious description inherent to (2) is cer-tainly sufficient to predict the combined impact of de-phasing, distinguishability and mixing on an interferom-eter [7, 8], and an impressive level of understanding ofdecoherence in nature has been achieved through stud-ies that essentially monitor only the visibility, as demon-strated for large molecules [8]. However, the precise cause a r X i v : . [ qu a n t - ph ] J a n of decoherence remains unknown in circumstances thatare not well understood, and three interferometers thatare affected by three qualitatively different decoherenceprocesses may exhibit the very same signal. For the char-acterization and eventual alleviation of decoherence, dif-ferential diagnosis , i.e. detailed information about the na-ture of decoherence, is crucial: Only a well-characterizedcause of the signal deterioration can be thoroughly ad-dressed and eventually removed.For example, in nuclear magnetic resonance, it is cru-cial to distinguish truly irrevocable dephasing from inho-mogeneous spin precessions, which can be diagnosed andreversed by spin-echo measurements [9]. In the presentcontext of interferometry, the key to a more differenti-ated picture of decoherence processes lies in the com-plexity inherent to entangled many-boson states. Here,we introduce a versatile treatment of bosonic double-Focksuperpositions, which encompasses single-particle states, N N -states and double-Fock states of the form | N, M (cid:105) as special cases. The formalism naturally allows us totreat mixed states and to thereby incorporate decoher-ence processes such as mixing, distinguishability and de-phasing. As a result, a four-particle double-Fock state | , (cid:105) allows a clear diagnosis of mixing against distin-guishability, while the entangled double-Fock superposi-tion | (cid:105) ≡ ( | , (cid:105) + | , (cid:105) ) / √ II. DOUBLE-FOCK INTERFEROMETRYA. Pure states
We consider double-Fock superpositions of the form | N : M (cid:105) = 1 √ (cid:16) | N (cid:105) ,φ | M (cid:105) , ˜ φ + | M (cid:105) ,φ | N (cid:105) , ˜ φ (cid:17) (3) ≡ √ | N, M (cid:105) + | M, N (cid:105) ) , where 0 ≤ M < N , and | K (cid:105) l,θ denotes the Fock stateof K bosons in the interferometric arm l and in the in-ternal state | θ (cid:105) . The latter pools all remaining relevantdegrees of freedom by which particles can possibly bedistinguished (besides the mode number): For a photon, | θ (cid:105) typically describes the polarization and the spatio-temporal mode function. Double-Fock superpositionscomprise single-photon ( N = 1 , M = 0) and N N -states ( N > , M = 0) as special cases.After propagation through the interferometer, thecomponent | M, N (cid:105) in (3) acquires the relative phase( N − M ) η with respect to | N, M (cid:105) . This phase canbe inferred by combining the two arms at a beam-splitter and measuring the probability P | N : M (cid:105) ( s ,s ) to find ( s , s = N + M − s ) particles in the two output modes[10]. In practice, the paths might not be fully indistin-guishable, which is reflected by a non-unity scalar prod-uct (cid:104) φ | ˜ φ (cid:105) , i.e. by partial distinguishability, which compli-cates the computation of event rates. One approach topartially distinguishable bosons consists in replacing theideal bosonic permanent by a sum of more general im-manants [11, 12]. Alternatively, the initial state can bedecomposed into orthogonal components of different de-grees of distinguishability [13–16]. Neither method, how-ever, offers a straightforward extension to mixed states –a prerequisite in our context – because the resulting ex-pressions for the event probabilities feature a complicateddependence on the scalar product (cid:104) φ | ˜ φ (cid:105) .Here, we overcome this shortcoming by treating the co-herent many-particle propagation via double-sided Feyn-man diagrams [17]. Our starting point is the expectationvalue of the projector ˆ Q ( s ,s ) , whose eigen-space is de-fined by the desired particle numbers in the output modesof the beam-splitter [Fig. 1], P | N : M (cid:105) ( s ,s ) = (cid:12)(cid:12)(cid:12) ˆ Q ( s ,s ) ˆ U | N : M (cid:105) (cid:12)(cid:12)(cid:12) (4)= (cid:104) N : M | ˆ U † ˆ Q ( s ,s ) ˆ Q ( s ,s ) ˆ U | N : M (cid:105) (5)= (cid:104) N, M | ˆ U † ˆ Q ( s ,s ) ˆ U | N, M (cid:105) (i)+ (cid:60) (cid:104) e iη ( N − M ) (cid:104) N, M | ˆ U † ˆ Q ( s ,s ) ˆ U | M, N (cid:105) (cid:105) , (ii)where ˆ U describes the many-particle beam-splitter trans-formation in Fock-space, induced byˆ a † k,θ → √ (cid:16) i ˆ b † k,θ + ˆ b † − k,θ (cid:17) , (6)where k refers to the beam splitter mode (see Fig. 1) anda phase-shift of π/ | N, M (cid:105) and | M, N (cid:105) lead to the same eventprobability. Eq. (5) contains two contributions to theevent probability: A main contribution (i) for which thebra- and ket-vectors are the same, and a swapped con-tribution (ii) with different bra- and ket-vectors. Thesetwo terms can be interpreted as double-sided Feyman-diagrams that combine propagation forwards and back-wards in time, implicit in Eq. (5): The state | N : M (cid:105) ispropagated in time via ˆ U , projected onto the measure-ment outcome described by the projector ˆ Q , and propa-gated back via ˆ U † [see Fig. 2].Inserting the transformation (6), we identify the per-mutations of the particles in the modes that yield thesame summands. All the possibilities for distributing theparticles among the modes need to be taken into account;using φ (cid:104) K | K (cid:105) ˜ φ = (cid:104) φ | ˜ φ (cid:105) K , we write the signal probabilityas a polynomial in the scalar product (cid:104) φ | ˜ φ (cid:105) , P | N : M (cid:105) ( s ,s ) = M ! N ! s ! s !2 M + N M (cid:88) J =0 C J (i) (cid:122) (cid:125)(cid:124) (cid:123) |(cid:104) φ | ˜ φ (cid:105)| J + (ii) (cid:122) (cid:125)(cid:124) (cid:123) ( − s |(cid:104) φ | ˜ φ (cid:105)| M − J ) (cid:60) (cid:104) ( i (cid:104) φ | ˜ φ (cid:105) e iη ) N − M (cid:105) , (7)where the explicit form of the combinatorial factor C J together with an illustration are given in Appendix A. h , | ˆ U † ˆ P (1 , ˆ U | , i h , | ˆ U † ˆ P (1 , ˆ U | , i h , | ˆ U † ˆ P (1 , ˆ U | , i (a) h , | ˆ U † ˆ P (1 , ˆ U | , i (b) h , | ˆ U † ˆ P (1 , ˆ U | , i (c) (i)(ii) (i)(ii) (i) h , | ˆ U † ˆ P (1 , ˆ U | , i ˆ Q (1 , h , | ˆ U † ˆ P (1 , ˆ U | , i ˆ Q (1 , ˆ Q (1 , h , | ˆ U † ˆ P (1 , ˆ U | , i ˆ Q (1 , h , | ˆ U † ˆ P (1 , ˆ U | , i FIG. 2: Full double-sided Feynman diagrams, illustrated for(a) | (cid:105) , ( s , s ) = (1 , | (cid:105) , ( s , s ) = (1 , | , (cid:105) , ( s , s ) = (1 , U , projected onto ˆ Q (1 , (orange frame, the projector doesnot differentiate the internal state, hence the gray coloring)and propagated back via ˆ U † . The upper rows correspond tothe term (i) in Eq. (7), the lower rows to (ii); the latter isabsent for the twin-Fock state | , (cid:105) . For states with possi-ble bosonic exchange processes, there are several competingpaths; in general, all paths need to be summed up. Eq. (7) reveals the dependence of the probability P | M : N (cid:105) ( s ,s ) for the event ( s , s ) on powers of the indistin-guishability parametrized by (cid:104) φ | ˜ φ (cid:105) up to order N + M .Absolute-square powers of the form |(cid:104) φ | ˜ φ (cid:105)| J [term (i)]contribute without any dependence on η . A particle“starting” and “ending” in the same arm [horizontalsingle-colored arrows in Fig. 3] contributes a certain am-plitude, a particle ending in a different arm [diagonaltwo-colored arrows] yields an amended amplitude that isattenuated by a factor (cid:104) φ | ˜ φ (cid:105) or (cid:104) ˜ φ | φ (cid:105) . The many-particlepaths for which all particles end in the arm they startedfrom (marked by dotted edges) constitute the “classical”contribution, which can be understood via interference-free classical combinatorics. The other terms containnon-vanishing powers of (cid:104) ˜ φ | φ (cid:105) and describe different ex-change processes. Bosonic exchange processes are thosewith J (cid:54) = 0, i.e. J counts how many particles were ac-tually exchanged between the arms within one compo-nent | N, M (cid:105) , such that, naturally, J ≤ M . Exchangesbetween the | N, M (cid:105) and | M, N (cid:105) -components lead to aphase-dependence in η [term (ii)].For single-photon and N N -states, M = 0, and thesum (7) reduces to one term, P | N :0 (cid:105) ( s ,s ) = P dist( s ,s ) (cid:20) − s (cid:60) (cid:20)(cid:16) i (cid:104) φ | ˜ φ (cid:105) e iη (cid:17) N (cid:21)(cid:21) , (8)where P dist( s ,s ) = (cid:0) Ns (cid:1) / N is the “classical” combinato-rially obtained probability to find ( s , s ) distinguish- able particles in the output modes. Consistently with J ≤ M = 0, no bosonic exchange processes take place.For twin-Fock states | N, N (cid:105) , the event probability canalso be obtained using Eq. (7), but neglecting the sec-ond summand (ii), i.e. the phase-dependent contribu-tion: Twin-Fock states do not carry any phase-relationbetween the two modes, therefore, a phase acquired inone arm manifests itself only as a global, non-observablephase. As a consequence, twin-Fock states are immuneto dephasing. The input state | , (cid:105) leads to Hong-Ou-Mandel interference [18]; for higher occupation, we ob-tain terms proportional to |(cid:104) φ | ˜ φ (cid:105)| J with J = 0 , , . . . , N [15].Double-Fock superpositions with 0 < M < N combinethe best of both worlds: phase-sensitivity and bosoniceffects. Since Eq. (3) is a superposition of two two-modeFock states, bosonic bunching governs the general statis-tics of the particles in the output modes [10, 16, 19].Simultaneously, interference between the two compo-nents | N, M (cid:105) and | M, N (cid:105) permits to measure the phase η . The phase-sensitivity of the | N : M (cid:105) -state is en-hanced by a factor N − M with respect to the single-photon case, just like for N N -states [20], which fea-ture an enhancement of a factor N . The richness of in-terference effects is reflected by the four different con-tributions depicted in Fig. 3(d). In general, the state | N : M (cid:105) leads to M + N + 1 distinguishable events:( N + M, , ( N + M − , , . . . (0 , N + M ), each of whichexhibits a certain dependence on higher powers of thescalar product (cid:104) φ | ˜ φ (cid:105) . B. Mixed states
In the previous section, we took into account the possi-ble deterioration of interference due to path distinguisha-bility ( |(cid:104) φ | ˜ φ (cid:105)| (cid:54) = 1), but we assumed that the state of theparticles is always the same when they reach the beamsplitter. Due to non-unitary random processes, however,we need to assume that the particles in the upper (lower)arm are in the internal state | ψ j (cid:105) ( | ˜ ψ k (cid:105) ) with probabil-ity p j (˜ p k ), i.e. in a mixed state. One then experiencesevent probabilities corresponding to the classical average(weighted by the p j and ˜ p k ) of the quantum-mechanical probability evaluated for | ψ j (cid:105) and | ˜ ψ k (cid:105) , P | N : M (cid:105) ( s ,s ) , mix = (cid:88) j,k p j ˜ p k P | N : M (cid:105) ψj, ˜ ψk ( s ,s ) , (9) h | ˜ i| , i |h | ˜ i| J =0(a) + | , i e i ⌘ | , i |h | ˜ i| h | ˜ i|h | ˜ i| h | ˜ ih | ˜ i | , i + e i ⌘ | , i + | , i + e i ⌘ | , i | , i + e i ⌘ | , i + | , i + e i ⌘ | , i | , i J =2 |h | ˜ i| J =1 |h | ˜ i| | i| i| i (b)(c)(d)(e) (i)(ii)(i)(ii) (i)(ii)(i) FIG. 3: Reduced double-sided Feynman diagrams illustrat-ing Eq. (5), divided up with respect to their contribution toEq. (7): For double-Fock superpositions, the first row corre-sponds to the phase-independent term in Eq. (7)(i), the sec-ond row denotes the phase-dependent term (ii), naturally ab-sent for twin-Fock-states without phase-dependence. In con-trast to Fig. (2), we omit the intermediate projector and onlyshow the initial state. The columns correspond to processeswith different numbers J of bosonic exchange processes. Theclassical contributions are marked by dotted edges. Particlesstarting in different modes are possibly distinguishable, re-flected by their different colors. Arrows connecting particlesof different colors contribute the scalar product (cid:104) φ | ˜ φ (cid:105) or (cid:104) ˜ φ | φ (cid:105) to the total amplitude. where we made the dependence on | ψ j (cid:105) , | ˜ ψ k (cid:105) explicit.In other words, we can still use Eq. (7), but each powerof a scalar product |(cid:104) φ | ˜ φ (cid:105)| m (cid:104) φ | ˜ φ (cid:105) k in (7) needs to bereplaced by the ensemble-averaged scalar product power (ASPP), denoted by curly brackets {} , (cid:110) |(cid:104) φ | ˜ φ (cid:105)| m (cid:104) φ | ˜ φ (cid:105) k (cid:111) = (cid:88) j,l p j ˜ p l |(cid:104) ψ j | ˜ ψ l (cid:105)| m (cid:104) ψ j | ˜ ψ l (cid:105) k , (10)which mathematically corresponds to a higher mo-ment of the scalar product. ASPPs of higher order,( m, k ) (cid:54) = (0 , ρ and ˜ ρ that describe the particles in the upper and lowerarm, because the coherences between the arms single outparticular bases {| ψ k (cid:105)} , {| ˜ ψ k (cid:105)} . Consider, for example,a qubit-like particle prepared in | φ (cid:105) = | (cid:105) and a ran-dom process that acts on the upper arm, which leavesthe qubit in | (cid:105) or | (cid:105) with probability 1/2. The averagescalar product with an unaffected qubit in the lower armis then ( (cid:104) | (cid:105) + (cid:104) | (cid:105) ) / /
2. A process that leavesthe qubit in the upper arm in | + (cid:105) = ( | (cid:105) + | (cid:105) ) / √ |−(cid:105) = ( | (cid:105) − | (cid:105) ) / √
2, however, leads to an average scalarproduct of ( (cid:104) + | (cid:105) + (cid:104)−| (cid:105) ) / single -particle density matrix is the fullymixed state ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) / | + (cid:105)(cid:104) + | + |−(cid:105)(cid:104)−| ) / / |(cid:104) φ | ˜ φ (cid:105)| m , the coherences between the arms are irrel-evant [Fig. 3], which makes the corresponding ASPPsindependent of the single-particle bases, e.g. for theensemble-averaged absolute-square of the scalar product, (cid:110) |(cid:104) φ | ˜ φ (cid:105)| (cid:111) = Tr( ρ ˜ ρ ) . The ASPPs of different orders arewidely independent of each other; averaged absolute val-ues merely fulfil {|(cid:104) φ | ˜ φ (cid:105)| m } k/m ≤ {|(cid:104) φ | ˜ φ (cid:105)| k } ≤ {|(cid:104) φ | ˜ φ (cid:105)| m } , (11)for every k ≥ m ≥
1, where the lower bound is due toJensen’s inequality and the upper bound follows from |(cid:104) ψ k | ˜ ψ j (cid:105)| ≤
1. For two mixed states with the same eigen-vectors, | ψ k (cid:105) = | ˜ ψ k (cid:105) ([ ρ, ˜ ρ ] = 0), the upper bound of (11)becomes exact; two pure, possibly distinguishable states ρ = | φ (cid:105)(cid:104) φ | , ˜ ρ = | ˜ φ (cid:105)(cid:104) ˜ φ | saturate the lower bound. III. DECOHERENCE MODEL
In general, non-unitary maps [3] that induce deco-herence processes in high dimensions can be arbitrar-ily complicated, reflecting the possibly complex dynam-ics in the two interferometric arms. Here, we focus onthe decoherence model illustrated in Fig. 1, which al-lows us to model the immediate impact of distinguisha-bility, mixing and dephasing via three survival probabil-ities γ dist , γ mix , γ phase , respectively. A. Path distinguishability
Distinguishability has various causes: On the onehand, we consider an observer with a meter ini-tially prepared in the state | (cid:105) meter , coupled to thelower arm. If the particle takes the upper path, | φ (cid:105)| (cid:105) meter → | φ (cid:105)| (cid:105) meter ; if it takes the lower path, | φ (cid:105)| (cid:105) meter → | φ (cid:105)| β (cid:105) meter . Formally, the leakage of which-path information can be accommodated in an amendedinternal state | ˜ φ (cid:105) of the particle in the lower arm thatincorporates the meter [21], such that (cid:104) φ | ˜ φ (cid:105) := (cid:104) | β (cid:105) .On the other hand, mis-alignment of the setup or anyother influence on the interferometric arms that permitsto distinguish a particle in the upper arm from a parti-cle in the lower arm via its internal state ( | φ (cid:105) and | ˜ φ (cid:105) ,respectively) leads to the same effect. We neglect herethe systematic acquisition of a relative phase between thetwo arms, which induces a shift of the overall signal in η , and assume (cid:104) φ | ˜ φ (cid:105) = γ dist ≥
0. The overall impact ofpath distinguishability then leads to {|(cid:104) φ | ˜ φ (cid:105)| m (cid:104) φ | ˜ φ (cid:105) k } = γ m + k dist . (12) B. Mixing
Mixing can be due to classical noise that disturbs theinternal state of the particle in an incoherent manner.Here, we model mixing as follows: With probability γ mix ,all particles in an arm remain unaffected; with probabil-ity (1 − γ mix ), all particles are left in an unknown statethat is chosen randomly for each run. That is, our mixingprocess corresponds to the addition of white noise withstrength 1 − γ mix , which leads to the following attenua-tion of the ASPPs: {|(cid:104) φ | ˜ φ (cid:105)| m (cid:104) φ | ˜ φ (cid:105) k } → {|(cid:104) φ | ˜ φ (cid:105)| m (cid:104) φ | ˜ φ (cid:105) k } γ , (13) {|(cid:104) φ | ˜ φ (cid:105)| m } → {|(cid:104) φ | ˜ φ (cid:105)| m } γ + 1 − γ d , where k ≥
1. The last term proportional to 1 /d reflectsthat the average absolute-squared scalar product of tworandom states is finite in a finite-dimensional Hilbert-space, whereas the average complex scalar product van-ishes due to isotropy; in the following, we assume d → ∞ ,which allows us to neglect the corresponding terms pro-portional to 1 /d . In contrast to distinguishability, whichaffects different powers in a different manner [Eq. (12)],mixing amends each ASPP of any power by the samefactor γ . C. Dephasing
Dephasing is ubiquitous: To name two examples, un-stable optical setups lead to phase fluctuations in pho-tonic experiments, while atomic interferometers are af-fected by background gas collisions. We incorporate theloss of phase coherence between the arms of the inter-ferometer by assuming that, with probability γ phase , allphases remain unaffected; with probability 1 − γ phase , allparticles in the lower mode acquire a uniformly randomphase 0 ≤ η rand ≤ π . The survival rate γ phase is, thus,independent of the number of particles [45]. Althougheach value of η rand induces an interference pattern in η with high visibility, the origin of that pattern is shiftedby ( N − M ) η rand . Since the shift is unknown and variesfrom run to run, the experimentally observed interferencepattern is washed out.We account for dephasing by amending the phase-dependent ASPPs, {|(cid:104) φ | ˜ φ (cid:105)| m (cid:104) φ | ˜ φ (cid:105) k } → {|(cid:104) φ | ˜ φ (cid:105)| m (cid:104) φ | ˜ φ (cid:105) k } γ phase , {|(cid:104) φ | ˜ φ (cid:105)| m } → {|(cid:104) φ | ˜ φ (cid:105)| m } , (14)where k ≥
1, i.e. phase-independent terms remain natu-rally unaffected by dephasing.
D. Overall impact of decoherence
The three decoherence mechanisms commute, and theresulting ASPPs after all processes become the productof the survival rates, {|(cid:104) φ | ˜ φ (cid:105)| m (cid:104) φ | ˜ φ (cid:105) k } = γ phase γ γ m + k dist , (15) {|(cid:104) φ | ˜ φ (cid:105)| m } = γ γ m dist , (16)for k ≥ , m ≥
0. Summarizing, on the one hand, observ-able signals P | N : M (cid:105) ( s ,s ) depend on various ASPPs [Eq. (7)].On the other hand, different ASPPs reflect the decoher-ence parameters in a different way [Eqs. (15),(16)]. Thisnourishes the hope that we can observe differences be-tween decoherence mechanisms in many-particle inter-ference signals. IV. DECOHERENCE DIAGNOSISA. Fringe visibility for single-photon, N N andtwin-Fock states The three decoherence mechanisms described in theprevious section reduce the fringe visibility of every in-terferometric signal. For the single-particle state | (cid:105) ,the visibility (1) is reduced to V | (cid:105) (1 , = (cid:110) (cid:104) φ | ˜ φ (cid:105) (cid:111) = γ phase γ γ dist , (17)i.e. one can only infer the product of definite powers ofthe three model parameters. Geometrically speaking, agiven value of the visibility inferred from the observedsignal [Fig. 4(a)] leaves room for a surface in the three-dimensional space ( γ phase , γ mix , γ dist ) [Fig. 4(c)].For N N -states | N : 0 (cid:105) , only the process in whichall particles are exchanged between the modes is relevant[Fig. 3(b)], and we find V | N :0 (cid:105) ( s ,s ) = (cid:110) (cid:104) φ | ˜ φ (cid:105) N (cid:111) = γ phase γ γ N dist . (18)Even though the phase-sensitivity of | N : 0 (cid:105) is enhancedwith respect to | (cid:105) , there are no contributions frombosonic exchange processes ( J >
0) in Eq. (7), and onlythe product of definite powers of the three model param-eters can be inferred from the experimental data.For the double-Fock state | , (cid:105) , the depth of the re-sulting phase-independent Hong-Ou-Mandel dip [18] isproportional to {|(cid:104) φ | ˜ φ (cid:105)| } = Tr( ρ ˜ ρ ) = γ γ , (19)which does not allow any differentiation between mixingand distinguishability, which makes the actual purity of asingle photon only accessible through advanced analyses[22]. In general, two identical mixed states ρ = ˜ ρ (cid:54) = | φ (cid:105)(cid:104) φ | cannot be distinguished from two pure distinguishablestates | φ (cid:105) , | ˜ φ (cid:105) that lead to the same ASPP.In principle, the visibilities for the three initial states | (cid:105) , | (cid:105) and | , (cid:105) depend on the three deco-herence model parameters in a different way [compareEqs. (17),(18) and (19)] and, by combining the data fromthree experiments with different initial states, the param-eters γ dist , γ mix , γ phase can be extracted. However, whenthe initial state is changed, it is difficult to assess whichdeteriorating effects are due to the possibly imperfectstate preparation and which are caused by the actualdecoherence in the interferometer. In the following, weshow how to circumvent this problem by extracting thedecoherence parameters in one single experiment. ⌘/⇡ dist phase mix S i gn a l (a)(b) (c) P | i (1 , P | i (0 , P | i (1 , P | i FIG. 4: (a) Single-particle interference signal with visibility V | (cid:105) (0 , ≈ .
21. (b) In double-Fock superposition interferom-etry with | (cid:105) , three independent parameters are accessi-ble experimentally. (c) The observed single-particle visibility V | (cid:105) (0 , constrains the three parameters ( γ dist , γ phase , γ mix ) to asurface defined by Eq. (17). For the double-Fock superposi-tion | (cid:105) , the three parameters unambiguously determine γ phase = 0 . , γ dist = 0 . , γ mix = 0 . B. Twin-Fock state | , (cid:105) Twin-Fock states [23] with M = N > M = N = 2. For twin-Fock states, the phase-dependent summand (ii) in (7) is absent, and we find P | , (cid:105) (0 , = 116 (cid:16) (cid:110) |(cid:104) φ | ˜ φ (cid:105)| (cid:111) + (cid:110) |(cid:104) φ | ˜ φ (cid:105)| (cid:111)(cid:17) , (20) P | , (cid:105) (1 , = 14 (cid:16) − (cid:110) |(cid:104) φ | ˜ φ (cid:105)| (cid:111)(cid:17) , (21) P | , (cid:105) (2 , = 18 (cid:16) − (cid:110) |(cid:104) φ | ˜ φ (cid:105)| (cid:111) + 3 (cid:110) |(cid:104) φ | ˜ φ (cid:105)| (cid:111)(cid:17) , (22)which match the results obtained via the orthonormal-ization of single-particle wave-functions [14, 15]. Thedependence of event probabilities on different powers of |(cid:104) φ | ˜ φ (cid:105)| stems from different bosonic exchange processes[Fig. 3(e)]. The absence of a second-order term in P | , (cid:105) (1 , is responsible for the narrowing of the width of the (1,3)-signal [15] with respect to the single-photon coherencelength, the alternating signs in P | , (cid:105) (2 , induce the non-monotonicity of the (2,2)-signal [14].Under the decoherence model above, we use (15), inaddition to Eq. (19), we have (cid:110) |(cid:104) φ | ˜ φ (cid:105)| (cid:111) = γ γ , (23)which allows us to read off γ dist and γ mix from the com-bined signal ( P | , (cid:105) (1 , , P | , (cid:105) (2 , ) (note that P | , (cid:105) (0 , is fixed bythe latter two), as illustrated in Fig. 5. m i x = . m i x = . m i x = . m i x = d i s t = d i s t = . d i s t = . T r ( ⇢ ˜ ⇢ ) = . T r ( ⇢ ˜ ⇢ ) = . T r ( ⇢ ˜ ⇢ ) = . p u r e m i x i n g p u r e d i s ti ngu i s h a b ilit y P | , i (2 , P | , i (1 , FIG. 5: Physical range of ( P | , (cid:105) (1 , , P | , (cid:105) (2 , ) in twin-Fock stateinterferometry with | , (cid:105) . Red solid lines show constantmixing ( γ mix = 1 , . , . , . γ dist = 1 , . , . , . P | , (cid:105) (1 , , P | , (cid:105) (2 , ) = (0 , / P | , (cid:105) (1 , , P | , (cid:105) (2 , ) = (1 / , / ρ ˜ ρ ) = 0 . , . , . γ dist , γ mix ,fulfilling P | , (cid:105) (1 , = (1 − γ γ ) /
2. The experiments re-ported in Refs. [14, 15] explore the distinguishability-inducedquantum-to-classical transition, corresponding here to the redsolid line with γ mix = 1. As a result, the quantum-to-classical transitions in-duced by mixing and by distinguishability differ strongly:Pure mixing (in general: p j = ˜ p j and | ψ j (cid:105) = | ˜ ψ j (cid:105) ; here,in our model: γ dist = 1) implies that ASPPs of higherpowers {|(cid:104) φ | ˜ φ (cid:105)| m } all take the same value, saturatingthe upper bound of Eq. (11). Mixing therefore alwaysinduces a linear interpolation between quantum and clas-sical probabilities, as evident from Eq. (7) [straight bluedashed line denoted by γ dist = 1 in Fig. 5]. In con-trast, pure distinguishability (i.e. the particles are de-scribed by pure states | φ (cid:105) and | ˜ φ (cid:105) , the lower bound ofEq. (11) is saturated; here, γ mix = 1), leads, in gen-eral, to more intricate, non-monotonic transitions [14][curved red solid line for γ mix = 1 in Fig. 5]. This qual-itative difference between these two decoherence mecha-nisms reinforces the role of non-monotonicity as a wit-ness of a distinguishability-induced quantum-to-classicaltransition [6, 27]. C. Double-Fock superposition | (cid:105) A clear and unambiguous differentiation of distin-guishability, mixing and dephasing is possible using thedouble-Fock superposition | (cid:105) . Such a state can begenerated experimentally by annihilating a single pho-ton in a twin-Fock state | , (cid:105) , where the photon is ex-tracted from either mode with the same probability [28],a technique that was experimentally demonstrated forthe | (cid:105) state [29]. Using (7), the two pertinent prob-abilities become P | (cid:105) (0 , = 18 (cid:16) ∓ {|(cid:104) φ | ˜ φ (cid:105)| (cid:104) φ | ˜ φ (cid:105)} sin( η ) (cid:17) (24)+ 14 (cid:16) {|(cid:104) φ | ˜ φ (cid:105)| } ∓ {(cid:104) φ | ˜ φ (cid:105)} sin( η ) (cid:17) , P | (cid:105) (1 , = 38 (cid:16) ± {|(cid:104) φ | ˜ φ (cid:105)| (cid:104) φ | ˜ φ (cid:105)} sin( η ) (cid:17) (25) − (cid:16) {|(cid:104) φ | ˜ φ (cid:105)| } ± {(cid:104) φ | ˜ φ (cid:105)} sin( η ) (cid:17) , where the upper signs refer to the event ( s , s ) and thelower ones to ( s , s ). The dependence on three differentASPPs can be understood from Fig. 3(d): The phase-independent classical contribution is modified by a phase-independent bosonic exchange contribution, weighted by {|(cid:104) φ | ˜ φ (cid:105)| } , and two differently weighted phase-dependentterms, corresponding to exchange of one or three parti-cles.The three independent observables that characterizeinterference are the visibility of (3 , V | (cid:105) (3 , , thevisibility of (2 , V | (cid:105) (2 , and the total probabilityto find all particles in one mode, P | (cid:105) = P | (cid:105) (3 , + P | (cid:105) (0 , (the total probability to find the particles in the (2,1)or (1,2)-channel is the complement 1 − P | (cid:105) ). An ad-ditional binary degree of freedom is the phase relationbetween the (1 ,
2) and the (3 , V | (cid:105) (2 , is set to itsnegative value when the (1 ,
2) and (0 ,
3) signals are out ofphase (we excluded complex scalar products (cid:104) φ | ˜ φ (cid:105) , suchthat we never encounter phase-shifts other than 0 and π ). We combine Eqs. (17) and (19) with Eq. (16), (cid:110) |(cid:104) φ | ˜ φ (cid:105)| (cid:104) φ | ˜ φ (cid:105) (cid:111) = γ γ phase γ , (26)to find the observables as a function of the decoherencemodel parameters. Inserting Eqs. (17,19,26) into theprobabilities (24, 25), we can express the decoherenceparameters ( γ dist , γ phase , γ mix ) as a function of the ob-servables (cid:16) V | (cid:105) (2 , , V | (cid:105) (3 , , P | (cid:105) (cid:17) . p u r e d e p h a s i n g p u r e f u ll y d e ph a s e d pu re m i x i n g d i s t i n g u i s h a b ili t y V | : i ( , ) = . V | : i ( , ) = . V | : i ( , ) = . m i x = d i s t = m i x = p h a s e = ph a s e = d i s t = V | i (3 , P | : i + V | i (2 , ph a s e = FIG. 6: Physical range of ( V | (cid:105) (2 , , V | (cid:105) (3 , , P ) for thedouble-Fock-superposition | (cid:105) . The wedge-like volume isconfined by three surfaces: The blue surface in the back-ground (with the solid black mesh) describes γ dist = 1,i.e. only dephasing and mixing arise; the red surface (solidlight-red mesh) corresponds to γ phase = 1, i.e. only mix-ing and distinguishability. The light green dotted mesh inthe foreground insinuates the third surface, characterized by γ mix = 1, i.e. dephasing and distinguishability. The fouredges (shown in light blue) correspond to pure dephasing,pure mixing, pure distinguishability, or full dephasing. Theyellow surfaces are areas of constant single-particle visibility V | (cid:105) (1 , = 0 . , . , . , . , . , .
9. Perfect interference takesplace for V | (cid:105) (3 , = V | (cid:105) (2 , = 1 , P | (cid:105) = 3 /
4, fully classical be-havior for V | (cid:105) (3 , = V | (cid:105) (2 , = 0 , P | (cid:105) = 1 / The volume of physically allowed combinations of( V | (cid:105) (2 , , V | (cid:105) (3 , , P | (cid:105) ) is shown in Fig. 6. Dephasing, dis-tinguishability and mixing lead to very different trajecto-ries, which define the edges of the volume. When one de-coherence mechanism fully destroys coherence ( γ dist = 0or γ phase = 0 or γ mix = 0), it is not possible to differen-tiate the other two: Full mixing or full distinguishability( γ dist = 0 or γ mix = 0, respectively) lead to classicalbehavior, independently of the value of the other deco-herence parameters. When phase coherence is fully lost( γ phase = 0), distinguishability cannot be differentiatedfrom mixing, as also evident from Eqs. (19): When theexpectation values of the scalar product (17) and thethird power (26) both vanish, only the product of γ dist and γ mix can be inferred. If the relationship between thedecoherence parameters and the observables were linear,we would observe a cube-like volume in Fig. 6; the patho-logical cases explain why we instead deal with a four-sided wedge. Outside the realm of full decoherence, eachchoice of ( γ dist > , γ phase > , γ mix >
0) leads to exactlyone point ( V | (cid:105) (2 , , V | (cid:105) (3 , , P | (cid:105) ), i.e. decoherence ratescan be inferred unambiguously from the observables, anddifferential diagnosis is possible [Fig. 4(b,c)]. In partic-ular, each value of the single-particle visibility V | (cid:105) (1 , re-sulting from a single-particle interference experiment iscompatible with a surface in the three-dimensional space(yellow surfaces in Fig. 6), the exact position on thatsurface then clearly reveals all three decoherence param-eters. Remarkably, the visibility of the (2 , V | (cid:105) (2 , can fully vanish for non-vanishing values of the decoher-ence parameters, due to the competition of the differentphase-dependent terms of opposite sign in Eq. (25) [28].In general, full dephasing does not lead to the classicalbehavior of distinguishable particles: Even though bothvisibilities vanish for γ phase →
0, bosonic statistics sur-vive, favouring the (3 , , D. General diagnosis
For the state | (cid:105) , the three measured observablesmatch the three physical parameters of the decoherencemodel presented in Section III, such that the latter canbe extracted with confidence outside pathological cases.This bijective relationship, however, is not a trivial arti-fact of scaling to larger particle numbers: In N N -stateinterferometry, one also measures several independentsignals, but due to the unique dependence on {(cid:104) φ | ˜ φ (cid:105) N } [Eq. (8)], different decoherence processes cannot be dis-tinguished.Decoherence processes that act on many particles mayimpact on the many-body density matrix in a complexfashion, beyond the three-parameter model of Section III:The mixing process may affect the upper and lower armdifferently and act in a more intricate way than by theaddition of white noise and dephasing can occur in a non-linear fashion that impinges on different particle numbersin a different way. Moreover, non-ideal beam splitters,particle loss and imperfect detectors with finite detec-tion efficiency and dark counts will additionally degradethe measured signals. As a general framework, a deco-herence model predicts the ASPPs {|(cid:104) φ | ˜ φ (cid:105)| m (cid:104) φ | ˜ φ (cid:105) k } asa function of its model parameters.By increasing the number of particles N and M , we cancontrol a larger set of observables, which allows us to keepup with the complexity of more sophisticated decoher-ence models and eventually infer the model parameters:The double-Fock superposition | N : M (cid:105) yields signals that permit to infer 2 M + 1 different ASPPs [Eq. (7)].We checked for N = M + 1 that all 2 M + 1 indepen-dent ASPPs can be inferred unambiguously from exper-imental observables up to M = 11. For twin-Fock states | N, N (cid:105) , all powers {|(cid:104) φ | ˜ φ (cid:105)| m } for m = 1 . . . N can be in-ferred, which we checked up to N = 10. It remains open,however, whether the relationship between experimentalobservables and ASPPs is always invertible. V. CONCLUSIONS
Many-boson states of the form | N : M (cid:105) provide re-markable features: Due to the dependence of event prob-abilities on several powers of scalar products inherent to(7), the experimental observables are sensitive to the ac-tual decoherence mechanism. Such double-Fock superpo-sitions therefore provide an inexpensive way to diagnosethe processes that deteriorate interferometric power. Inprinciple, any interferometer – be it optical, atomic ormolecular – can be diagnosed by feeding it with double-Fock-superpositions and analyzing the resulting visibil-ities. The alternative to differential decoherence diag-nose is quantum process tomography [30]. Since the in-ternal state of the particle | φ (cid:105) typically lives in a high-dimensional Hilbert-space, such reconstruction of the fulldensity matrix is infeasible in the current scenario.For large molecules [8, 31], the current paradigm fordecoherence, double-Fock superpositions are admittedlyextremely challenging to generate, let alone to interfereand detect. We may alternatively gain better insight intodecoherence processes with the help of other physical sys-tems: Cold atoms in few-well-lattices provide a feasiblemeans to test the discussed effects, since granular two-particle Hong-Ou-Mandel interference has recently beendemonstrated [32] and cold atoms can be subject to vari-ous decoherence mechanisms in a controllable way. Withphotons, the three decoherence processes discussed abovecan be simulated by using the polarization as the dis-tinguishing degree of freedom, and artificially inducingmixing, e.g. in the path delay. On the other hand, thediscussed methods may also help to characterize single-photon sources in a more precise way than by the usualHong-Ou-Mandel dip [33], which, as we have shown inSection IV A, does not reveal the cause of imperfect inter-ference. As a further extension, the diagnostic power ofdouble-Fock superpositions may also be used as a probefor other processes, for example, to quantify the non-Markovianity of an environment [34].In practice, only a finite number of events can be ob-served, leaving the visibilities uncertain, while the map-ping between model parameters and experimental ob-servables – with the ASPPs as intermediate step – mightbe quite intricate. Such more complex scenarios can betreated via Bayesian methods, which may also allow todesign optimized measurement strategies to quickly andreliably reveal the actual values of decoherence param-eters [35, 36]. Using double-sided Feynman diagrams,our analysis can be taken further to general states ofthe form | Φ( (cid:126)α ) (cid:105) = (cid:80) N tot n =0 α n | n, N tot − n (cid:105) . On the onehand, the (cid:126)α = { α , . . . α N tot } can be adjusted to achievethe best sensitivity to the type of decoherence process,i.e. the best differential diagnosis. On the other hand,given a fully diagnosed interferometer, the optimal (cid:126)α that achieves the best phase-sensitivity [37] may itselfdepend on the actually occurring decoherence processes.It remains to be studied to which extent the methodsof [38–40], which rely on post-selecting a desired outputstate in order to synthesize phase-super-resolving inter-ference signals, can be extended to the present purposeof decoherence diagnosis. From a more fundamental per-spective, the complicated dependence of visibilities ondecoherence parameters challenges any attempt to for-mulate a complementarity relation [21, 41–43] betweenparticle-like and wave-like behavior as well as to quan-tify macroscopic interference [31, 44], which remain greatdesiderata. Acknowledgements
M.C.T. would like to thank Pinja Haikka, StevenKolthammer, Benjamin Metcalf and Ian Walmsley for in-spiring discussions and useful comments, and ChristianKraglund Andersen, Alexander Holm Kiilerich, RobertKeil, Andrew Wade and Qing Xu for very valuable feed-back on the manuscript. M.C.T and K.M. acknowledgefunding by the Villum foundation and by the DanishCouncil for Independent Research. H.-T.L. acknowledgesthe financial support from the National Junior ResearchFellowship (Grant No. 2012- 000642). C.G. acknowl-edges financial support by DAAD. Y.-H.K. acknowledgesfinancial support by the National Research Foundation ofKorea (Grant No. 2013R1A2A1A01006029).
Appendix A: Computation of event probabilities
The coefficient C J in Eq. (7) is given by C J = max(0 ,s − M )+min( M,s ) (cid:88) r,r ∗ =max(0 ,s − M ) ( − r + r ∗ min( s , J ) (cid:88) j =max(0 , J − s ) ,j ! = r − r ∗ mod 2 M ( r,s − r )( r ∗ ,s − r ∗ ) ( j ) M ( N − r,s − N + r )( N − r ∗ ,s − N + r ∗ ) (2 J − j ) , (A1)where M p,n − pq,n − q ( j ) = (cid:18) np (cid:19)(cid:18) p ( j + p − q ) / (cid:19)(cid:18) n − p ( j − p + q ) / (cid:19) . (A2)The sum Eq. (A1) is illustrated in Fig. 7 and can be in-terpreted as follows: The particles are redistributed fromthe input to the output modes, with r ( N − r ) particlesfrom the first input mode found in the first (second) out-put mode. In order to eventually measure s and s par-ticles in the first and second output modes, respectively, s − r ( s − ( N − r )) particles from the second inputmode must be found in the first (second) output mode.Since only non-negative particle numbers are allowed forthe four processes, r is restricted to a certain range ofvalues. To yield the probability Eq. (4), we remain withtwo sums, over r and r ∗ . The relative phase acquired bysuch a process is ( − r + r ∗ , an additional relative phasearises for the exchange term (ii) in Eq. (7). The sum over j accounts for the bosonic exchange processes in the firstoutput mode, i.e. j | φ (cid:105) -particles are exchanged with | ˜ φ (cid:105) -particles; consequently, 2 J − j exchange processes occurin the second output mode. The overnormalization dueto the multiple creation of bosons in the same mode isaccounted for by M p,n − pq,n − q ( j ).Eq. (A1) can alternatively be derived using a decompo-sition of single-particle wave-functions in an orthonormal basis [14–16], for which, however, the clear separations incombinatorial factors and scalar products in Eq. (7) onlyemerges after lengthy algebraic manipulations. h | ˜ ih ˜ | ih ˜ | ˜ i h | ih | i r = 3 r ⇤ = 2 h N, M | ˆ U † ˆ Q ( s ,s ) ˆ Q ( s ,s ) ˆ U | N, M ih N, M | ˆ U † ˆ Q ( s ,s ) ˆ Q ( s ,s ) ˆ U | N, M ih N, M | ˆ U † ˆ Q ( s ,s ) ˆ Q ( s ,s ) ˆ U | N, M ih N, M | ˆ U † ˆ Q ( s ,s ) ˆ Q ( s ,s ) ˆ U | N, M ih N, M | ˆ U † ˆ Q ( s ,s ) ˆ Q ( s ,s ) ˆ U | N, M ih N, M | ˆ U † ˆ Q ( s ,s ) ˆ Q ( s ,s ) ˆ U | N, M i FIG. 7: One summand of Eq. (A1): We set N = 3 , M = 2, s = 4 , s = 1, and consider the process with r ∗ = 2 and r = 3reflected particles from the first input mode in time-forwardand time-backward direction, respectively. There are J = 1pairs of exchanged particles, leading to a weight |(cid:104) φ | ˜ φ (cid:105)| , andone ( j = 1) exchange occurs in the first output mode. [1] W. H. Zurek, Rev. Mod. Phys. , 715 (2003).[2] M. Schlosshauer, Rev. Mod. Phys. , 1267 (2005).[3] H. Breuer and F. Petruccione, The Theory of Open Quan-tum Systems (Oxford University Press, Oxford, 2006).[4] A. Buchleitner, M. Tiersch, and C. Viviescas,
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