Abstract. Welcher Weg (which-path) detectors where complementarity is enforced without altering the interfering beams are analysed in terms of the quantum ...
J. Opt. B: Quantum Semiclass. Opt. 1 (1999) 668–677. Printed in the UK
PII: S1464-4266(99)05149-6
Randomization of quantum relative phase in welcher Weg measurements ´ A Luis and L L Sanchez-Soto ´ Departamento de Optica, Facultad de Ciencias F´ısicas, Universidad Complutense, 28040 Madrid, Spain Received 14 June 1999, in final form 1 October 999 Abstract. Welcher Weg (which-path) detectors where complementarity is enforced without
altering the interfering beams are analysed in terms of the quantum relative phase. In such a case, the measurement disturbs the interference via random classical phase shifts. This applies when the interfering particles are atoms or photons. In the case of photons, the quantum relative phase coincides with the field phase difference. Keywords: Complementarity, quantum phase
1. Introduction
Complementarity is a purely quantum concept without classical analogue. In a broad sense, two observables are complementary if precise knowledge of one of them implies that all possible outcomes of measuring the other one are equally probable. Perhaps the best illustration of complementarity is that the observation of interference and the knowledge of the path followed by the interfering particle are mutually exclusive. This is often expressed as that all quantum objects exhibit particle-like or wave-like behaviour under different experimental conditions. Nevertheless, there is no strict border between them, and to some extent partial fringe visibility and partial welcher Weg (which-path) knowledge are compatible [1–3]. It is natural to argue that complementarity is enforced by the perturbation caused by the apparatus determining the path taken by the particle. The actual mechanism will vary from one experimental situation to another. In most cases, including classic examples such as Einstein’s recoiling slit [4], Feynman’s light-scattering arrangement [5] or Heisenberg’s microscope [6], the mechanism is a somehow random momentum transfer to the interfering particle [7, 8]. In addition to the loss of coherence, the exchange of momentum has further consequences: in double-slit interferometers it causes the broadening of the single-slit diffraction pattern. However, there are welcher Weg measurements where the interference is lost, seemingly without momentum transfer or any other alteration of the interfering beams [9]. Two experimental demonstrations of this class of welcher Weg detection have been performed for photons [10] and electrons [11]. In double-slit interferometers, for instance, the lack of alteration would be supported by the absence of any broadening of the diffraction pattern. As further evidence of this, a quantum eraser can be devised which allows an interference of high visibility to be extracted from within a 1464-4266/99/060668+10$30.00
© 1999 IOP Publishing Ltd
featureless pattern. This occurs if a suitable measurement erasing the path information is made on the welcher Weg detector. By construction, these examples resist a simple explanation in terms of momentum exchange [12]. However, the prevailing role played by this variable has prompted the re-examination of what can be understood as random momentum kicks [8, 13]. It has been shown that it is possible to discriminate between classical and quantum momentum transfer. If the particle experiences random classical momentum kicks, its output momentum probability distribution Pob (p) results from the convolution of the initial P (p) with a probability distribution P (p) of momentum transfer Z (1.1) Pob (p) = dp0 P (p − p 0 )P (p0 ). This definition fits properly with the classical intuition and leads to the broadening of the single-slit diffraction pattern (unless the slits have zero width, in which case there cannot be further broadening). On the other hand, a quantum transfer involves the convolution of momentum amplitudes instead of probabilities. Quantum kicks can destroy the interference without modifying the diffraction pattern, but they lack the simple classical picture that equation (1.1) provides. Instead of following this approach, we can try to explain these subtle examples of complementarity by using a different dynamical variable. In this context phase, and especially relative phase, are pre-eminent tools in solving and understanding classical interference problems. It can be expected that they should also be useful in the quantum domain. Their usefulness was already demonstrated by Stern et al in [14], where the loss of interference was explained in terms of phase uncertainty. However, in that work the phase is considered as a state parameter rather than as a dynamical variable. Our approach to this problem can be regarded as a development and substantiation of the ideas of [14]. This is done by introducing a quantum description
Randomization of quantum relative phase in welcher Weg measurements
space Hs is |ψi = cos(θ/2)|ψ1 i + sin(θ/2)eiφ |ψ2 i.
(2.1)
The parameter θ gives the probability of each path, and φ is the relative phase. The interference is recorded on a distant screen such that the position pattern on it is equivalent to the momentum distribution of |ψi on the plane of the apertures: P (p) = |ψ˜ 1 (p)|2 [1 + sin θ cos(pd + φ)]. Figure 1. A double-slit interferometer for atoms with welcher
Weg detection. The microwave cavities are initially empty and act as welcher Weg detectors if the atom de-excites depositing a photon in one of them.
In this expression |ψ˜ 1 (p)|2 is the single-aperture diffraction pattern. If, as usual, |ψ˜ 1 (p)|2 is a slowly varying function of p, the visibility of the fringes is V = sin θ.
of the relative phase as a dynamical variable. This allows one to compute the associated probability distribution and the way it transforms under the interaction with the apparatus. This quantum relative phase provides a simple explanation of complementarity when there is no alteration of the interfering paths. It has been recently shown that in such a case this variable experiences a randomization of the form (1.1) [15]. In this work we further investigate the role played by the relative phase in this kind of interference problems, especially in atomic interferometers (section 4). It is worth emphasizing that in the case of material systems this wave-like variable does not have a classical analogue, so its quantum translation must be derived from first principles. In addition, we look for the relation of this relative phase with the phase difference between two modes of the electromagnetic field by applying this formalism to the case of photons as interfering particles (section 5). Finally, in section 6 we examine whether the erasure of the welcher Weg information can also be understood in terms of the relative phase, in spite of the fact that this variable experiences a classical randomization. We begin with the description of the most general welcher Weg detection which does not modify the wavefunctions of the interfering beams (section 2). In section 3 and in the appendix we recall the quantum description of the relative phase and its general application to this context. 2. Welcher Weg measurements without classical momentum transfer
The internal paths in a two-beam interferometer will be represented by the normalized vectors |ψ1 i and |ψ2 i. Although the formalism we develop is valid regardless of the particular realization of the interferometer, for definiteness and without loss of generality we shall refer to a double-slit arrangement such as the one illustrated in figure 1. Then, ψ1 (x) and ψ2 (x) denote the position wavefunctions on the corresponding aperture. The two apertures are identical and spaced a distance d apart, so that ψ2 (x) = ψ1 (x + d). In the transverse-momentum representation this implies that ¯ = 1, which will be assumed ψ˜ 2 (p) = eipd ψ˜ 1 (p), with h throughout this paper. Since the two apertures do not overlap, we have that hψ1 |ψ2 i = 0. In the absence of any welcher Weg detector, the most general pure state for the interfering particle in its Hilbert
(2.2)
(2.3)
The determination of the path taken by the particle requires an apparatus. Its effect can be represented by a unitary transformation U acting on Hs ⊗ Ha , where Ha is the Hilbert space of the apparatus. We are interested in welcher Weg schemes in which there is at least one initial state of the apparatus |Ai such that there is no alteration of the wavefunctions associated with the particle passing through a single slit: i.e., U |ψj i|Ai = |ψj i|Aj i,
(2.4)
where |A1 i and |A2 i are the states of the apparatus carrying full path information. The final state after the interaction is U |ψi|Ai = cos(θ/2)|ψ1 i|A1 i + sin(θ/2)eiφ |ψ2 i|A2 i, (2.5) which leads to Pob (p) = |ψ˜ 1 (p)|2 [1+sin θ |hA1 |A2 i| cos(pd+φ+δ)], (2.6) where δ = arghA1 |A2 i. The interaction does not modify the diffraction pattern, which is still given by |ψ˜ 1 (p)|2 . However, the fringe visibility has changed to Vob = |hA1 |A2 i|V ,
(2.7)
which is always less than or equal to the visibility V . The loss of visibility is related to the performance of the welcher Weg detection, which depends on the inner product hA1 |A2 i. The smaller hA1 |A2 i, the better the accuracy achievable. Optimum welcher Weg detection occurs when hA1 |A2 i = 0, since in this case the states of the apparatus are perfectly distinguishable and the path followed can be discriminated unambiguously. In such a case Vob = 0, and the interference is completely lost. For intermediate situations there is an inequality relating visibility and welcher Weg knowledge [3]. Finally, it is worth noting that this kind of observation does not alter the probability of the two trajectories. The probabilities of |ψ1 i and |ψ2 i in the state (2.5) after the coupling with the apparatus are P1 = cos2 θ/2 and P2 = sin2 θ/2, which are the same as in the unobserved state (2.1). Then, this class of welcher Weg detectors are quantum nondemolition measurements of the trajectory [16]. 669
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3. Loss of interference and classical phase kicks
If neither the single-slit diffraction pattern |ψ˜ 1 (p)|2 nor the path probabilities are modified by the observation, we have to conclude that the interaction with the apparatus changes the relative phase φ (we are also excluding the possibility of any modification of the separation d of the apertures). In order to translate this reasoning into quantitative relations, it is necessary to examine how the probability distribution associated with this variable transforms under the interaction with the apparatus. This means that a quantum description of this dynamical variable is needed. The effective Hilbert space Hs of the interfering particle is always two-dimensional because we only focus on meaningful runs, where the particle actually goes through the slits. By definition, the welcher Weg arrangements we are dealing with do not modify the dependence on x of the wavefunctions ψ1 (x) and ψ2 (x), so that no other vectors are necessary, even after the interaction. Therefore, the system is always describable by an effective two-dimensional Hilbert space, analogous to a 21 angular momentum [17]. In these conditions, it is possible to represent the quantum relative phase (we shall denote by ϕ) by using the positive-operator measure (POM) 1 (|ψ1 ihψ1 | + |ψ2 ihψ2 | + e−iϕ |ψ1 ihψ2 | 2π +eiϕ |ψ2 ihψ1 |),
3(ϕ) =
(3.1)
where ϕ can take any value in a 2π interval [18, 19]. This and other approaches to the quantum relative phase are briefly examined in the appendix. This POM defines a probability distribution for the relative phase as P (ϕ) = tr[ρ3(ϕ)], where ρ is the density matrix of the system. For the unobserved state (2.1), we have P (ϕ) =
1 [1 + sin θ cos(ϕ − φ)], 2π
(3.2)
which is centred on φ, while for the observed state (2.5) Pob (ϕ) =
1 [1 + sin θ |hA1 |A2 i| cos(ϕ − φ − δ)], (3.3) 2π
which is centred on φ + δ and broader than (3.2). This can be demonstrated by using the dispersion Z 2 D 2 = 1 − |heiϕ i|2 = 1 − dϕ eiϕ P (ϕ)
(3.4)
as a suitable measure of the phase uncertainty [20]. The quantity |heiϕ i| is directly related to the visibility of the interference fringes: for the unobserved case we have |heiϕ i| = V /2, while after the interaction |heiϕ iob | = Vob /2. In consequence, if Dob > D then Vob < V , which means that larger phase uncertainty implies lesser visibility, as could be expected, even from a classical perspective. The effect of observation is to add fluctuations to the relative phase. Next, we examine whether this is the result of random classical kicks in a form analogous to (1.1). The states |A1 i and |A2 i can always be written as V1 |Ai and V2 |Ai, respectively, where V1 and V2 are unitary 670
operators acting on Ha . Provided that the initial state of the apparatus is |Ai, we can consider the following form for U : U = |ψ1 ihψ1 |V1 + |ψ2 ihψ2 |V2 .
(3.5)
With this expression we can compute how 3(ϕ) transforms under the interaction with the apparatus 1 (|ψ1 ihψ1 | + |ψ2 ihψ2 | + e−iϕ V1† V2 |ψ1 ihψ2 | U † 3(ϕ)U = 2π (3.6) +eiϕ V2† V1 |ψ2 ihψ1 |). The operator V1† V2 is unitary and can be regarded as the complex exponential of a Hermitian operator. This means that the observation shifts ϕ by a quantity depending on variables of the apparatus. To be more specific, we can introduce the eigenstates of V1† V2 V1† V2 |ξ i = eiξ |ξ i,
(3.7)
Z
so that V1† V2 =
dξ eiξ |ξ ihξ |.
(3.8)
With the help of these equations we can recast equation (3.6) in the form Z (3.9) U † 3(ϕ)U = dξ 3(ϕ − ξ )|ξ ihξ |, which leads to
Z
Pob (ϕ) = hψ|hA|U † 3(ϕ)U |Ai|ψi =
dξ P (ϕ − ξ )P (ξ ), (3.10)
where P (ξ ) = |hξ |Ai|2 ,
(3.11)
and we have assumed, without loss of generality, a continuous range of variation for ξ . If the phase kicks ξ (i.e. the spectrum of V1† V2 ) were a discrete set ξk , the probability P (ξ ) will be made of delta functions centred on ξk X P (ξ ) = P (k)δ(ξ − ξk ), (3.12) k
so that equation (3.10) becomes X P (ϕ − ξk )P (k), Pob (ϕ) =
(3.13)
k
where the discrete index k runs the range of allowed values for ξ . The form of equation (3.10) is completely general, irrespective of the particular measuring arrangement. The details of the actual mechanisms allowing the path The experimental observation only enter via P (ξ ). conditions determine the allowed values for ξ (in particular, whether ξ is continuous or discrete) and their probabilities. These points will be confirmed in the following sections showing that equations (3.10) or (3.13) are valid under very different experimental conditions. We can see, in equations (3.10) and (3.13), that the relative phase is randomized via phase kicks distributed according to P (ξ ). Actually, this is the way in which classical probability distributions transform under the
Randomization of quantum relative phase in welcher Weg measurements
action of random displacements. Then, although we are dealing with purely quantum phenomena, we shall refer to equations (3.10) and (3.13) as a classical randomization in order to emphasize that relative phase allows for a full explanation of the loss of visibility in terms of the most simple picture possible. Increasing the accuracy of the observation implies the convolution of P (ϕ) with a broader P (ξ ), since the dispersion of ξ is directly related to the accuracy of the observation Z dξ eiξ P (ξ ) = hA|V1† V2 |Ai = hA1 |A2 i. (3.14) The smaller |hA1 |A2 i|, the larger the dispersion for ξ . Then, convolution (3.10) leads to a Pob (ϕ) with larger phase uncertainty than P (ϕ), which in turn implies lesser visibility of the interference fringes. In other words, complementarity is dynamically enforced by the relative-phase uncertainty introduced by the interaction with the quantum apparatus. Why relative phase, instead of momentum, is the variable that experiences a classical randomization can be understood from the peculiarities of this kind of observation. The detection schemes we are considering cannot be regarded as a measurement of the position of the interfering particle (leaving aside the case of apertures with zero width). Position measurements are usually described by couplings of the form U = exp(ixB), where B is an observable depending on variables of the apparatus. If this were the case, the momentum of the particle will experience a transformation of the form (1.1). Instead of position, what is actually measured is just the observable described by the projection measure |ψj ihψj | (j = 1, 2). Then, the observable we can expect to be directly and unavoidably disturbed by observation is not momentum but the observable complementary to |ψj ihψj |, that is the relative phase, as it is shown in the appendix. The POM 3(ϕ) is not the unique possibility for the quantum description of this variable. For instance, it can be also described by means of an operator [19, 21]. The exponential of the relative phase is given by the unitary operator Eϕ0 = e−iϕ0 |ψ1 ihψ2 | + eiϕ0 |ψ2 ihψ1 |,
(3.15)
where ϕ0 plays the role of origin of phases. Under the system– apparatus coupling, this operator transforms as U † Eϕ0 U = e−iϕ0 V1† V2 |ψ1 ihψ2 | + eiϕ0 V2† V1 |ψ2 ihψ1 | Z (3.16) = dξ Eϕ0 −ξ |ξ ihξ |. Thus, the interaction produces a fluctuating phase shift distributed according to P (ξ ). Here, once again, the interaction with the apparatus leads to a convolution of the form (3.10), so this approach leads to the same conclusions analysed above. Finally, let us point out that this random-phase picture applies to all welcher Weg detectors that do not modify the state vectors |ψ1 i and |ψ2 i describing each path. This includes arrangements involving momentum recoil. For example, this can be the case of double-slit schemes when the slit width is less than the resolution of the observation (this is the wavelength of the scattered photons, for instance). In
such a case, this random-phase description can be applied to the Einstein’s recoiling slit as well as to scattering arrangements, for example. It has been demonstrated theoretically and experimentally that in these cases the loss of interference can also be understood as being caused by phase randomization [1, 7, 22, 23]. 4. Classical phase kicks in atomic interference
To illustrate these points we analyse, from the perspective of the relative phase, some practical realizations already examined by using other variables. In this section we focus on interferometric arrangements where the interfering particles are atoms. 4.1. Detection by photon emission We begin with by considering the two-slit arrangement introduced by Scully et al [9], where the interfering particle is an excited Rydberg atom (figure 1). Two identical microwave cavities C1 and C2 , initially empty, are placed in front of the slits. The path followed can be detected if the atom de-excites depositing a photon in one of them. In this example the apparatus involves two internal states of the atom (excited |ei and ground |gi) and the field state in the cavities. The initial state of the apparatus is |Ai = |0, 0i|ei, where |n1 , n2 i are the corresponding photon-number states. We assume that the width of the slits is small enough in comparison with the wavelength of the field modes, so that the deflection of the atom is negligible (i.e. the wavefunctions ψj (x) are not modified). In such a case, the atom–field interaction within each cavity can be described by the Hamiltonians (j = 1, 2) Hj = κ(|gihe|aj† + |eihg|aj ),
(4.1)
where aj and aj† are the annihilation and creation operators for the corresponding field modes and κ is a coupling constant. The total Hamiltonian can be written as H = H1 |ψ1 ihψ1 | + H2 |ψ2 ihψ2 |.
(4.2)
If τ denotes the time of passage through the cavities, the unitary transformation U = exp(−iτ H ) relating the atom– field state after and before the cavities is of the form (3.5), with Vj = exp(−iτ Hj ). The final state after the interaction is of the form (2.5), with |A1 i = cos(κτ )|0, 0i|ei − i sin(κτ )|1, 0i|gi, |A2 i = cos(κτ )|0, 0i|ei − i sin(κτ )|0, 1i|gi.
(4.3)
In this example we have hA1 |A2 i = cos2 (κτ ), and both the fringe visibility and the efficiency of the welcher Weg detection depend on the interaction time. The efficiency varies because there is a probability cos2 (κτ ) that the atom crosses the cavities without depositing the photon. When this happens, the path followed cannot be inferred. Optimum welcher Weg detection occurs provided that cos(κτ ) = 0. In figure 2 we have represented Pob (ϕ) as a function of ϕ and κτ for θ = π/2 and φ = 0. In order to find the possible phase kicks and their probabilities we can use the fact that V1† V2 commutes with 671
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Figure 2. The probability distribution for the relative phase difference as a function of the rescaled time κt spent by the atom within the cavities for θ = π/2 and φ = 0. Figure 3. The phase kick ξ0 for detection by photon emission
N = |eihe| + a1† a1 + a2† a2 . The initial state |Ai = |0, 0i|ei belongs to the subspace N = 1 and due to this commutation relation the state of the apparatus is always in this subspace. This allows us to reduce the whole problem to this threedimensional subspace where ! isc c s2 † V1 V2 |N=1 = 0 (4.4) c −is , is −isc c2 with c = cos(κτ ) and s = sin(κτ ). Since this matrix is unitary and its determinant is unity the eigenvalues are necessarily of the form 1, eiξ0 , e−iξ0 . The value of ξ0 can be determined by computing the trace of V1† V2 |N =1 , leading to the condition 1 + 2 cos ξ0 = 2c + c2 , (4.5) which gives cos ξ0 = 21 [cos2 (κτ ) + 2 cos(κτ ) − 1].
(4.6)
It can be easily seen that the eigenstate with eigenvalue 1 is r 1−c 1+c |1, 0i|gi + |0, 1i|gi + i |0, 0i|ei (4.7) 3+c s and then P (0) =
1 − cos(κτ ) . 3 + cos(κτ )
(4.8)
Moreover, since hA|V1† V2 |Ai = cos2 (κτ ) we have P (ξ0 ) = P (−ξ0 ) and, since P (ξ ) are probabilities, P (±ξ0 ) = 21 [1 − P (0)] =
1 + cos(κτ ) . 3 + cos(κτ )
(4.9)
The behaviour of ξ0 , P (0) and P (±ξ0 ) has been represented in figure 3 as functions of κτ . Under conditions for optimum welcher Weg detection, cos(κτ ) = 0, the possible phase shifts are ξ = 0, ±2π/3 with equal probabilities. This leads to a completely random relative phase X 1 Pob (ϕ) = 13 , (4.10) P (ϕ − m2π/3) = 2π m=−1,0,1 and the interference is completely washed out. Although to determine the path followed only the two states (4.3) are necessary, three possible phase kicks (0, ±ξ0 ) appear in this example. Since the phase kicks are the eigenvalues of V1† V2 the number of phase kicks depends on the effective dimension of the Hilbert space of the apparatus. The minimum number of phase kicks is two, but there is no upper bound since the Hilbert space of the apparatus can have an arbitrarily large dimension. 672
(welcher Weg detector in figure 1) and the probability distribution of phase kicks P (ξ0 ) = P (−ξ0 ) and P (0) as functions of κt.
Figure 4. A double-slit interferometer for atoms with welcher Weg detection performed by a standing light wave in a cavity immediately after the slits. The passage of the atom by one slit or the other produces two different phase shifts of the cavity field, which can be discriminated by a phase-dependent measurement such as homodyne detection, for instance.
4.2. Detection by phase shifting of a probe field Nonresonant atom–field interaction provides another possibility to devise a welcher Weg detector for a double-slit interferometer without introducing classical momentum kicks. To this end, the two slits can be immediately followed by a standing-wave light in a single cavity [24], as shown in figure 4. The mechanism for the inference of the atomic path is the phase change experienced by the standing wave. The path followed can be inferred by a phase-dependent measurement on the cavity field. Assuming the simplest possible coupling, we will have an effective two-level atom interacting with a single mode of the radiation field with complex amplitude operator a. The atom is initially in its ground state |gi and its transverse motion during the passage through the standing wave is assumed to be negligible (Raman–Nath regime). In the limit of a large detuning, 1, between the atomic transition and the cavity frequencies, the atom–field interaction Hamiltonian becomes H =
κ2 cos2 (kx + ϑ0 )(aa † |eihe| − a † a|gihg|), 1
(4.11)
where k = 2π/λ is the wavenumber of the cavity field, κ is the coupling constant and ϑ0 is a constant phase. In principle, this interaction Hamiltonian will modify the wavefunctions ψj (x) through the dependence of H on x. Here, we are interested in the case where such modification is negligible.
Randomization of quantum relative phase in welcher Weg measurements
This occurs when the width of the slits, δx, is negligible in comparison with the wavelength λ of the standing field, δx λ. Then, H does not vary noticeably within each of the two apertures. At each slit, placed at x = ±d/2, the atom–field interaction is H± '
state |Ai = |αi|gi, where α is the complex amplitude of the coherent state. In such a case, we have |Aj i = |αeiϑj i|gi, and
2
κ cos2 (±kd/2+ϑ0 )(aa † |eihe|−a † a|gihg|), (4.12) 1
and the total Hamiltonian can be written as κ2 (c1 |ψ1 ihψ1 | + c2 |ψ2 ihψ2 |)(aa † |eihe| − a † a|gihg|), 1 (4.13) where c1 = cos2 (kd/2 + ϑ0 ) and c2 = cos2 (−kd/2 + ϑ0 ). If τ denotes the interaction time, the unitary operator U = exp(−iτ H ) for the atom–field interaction is of the form (3.5) with H '
Vj = exp[−iϑj (aa † |eihe| − a † a|gihg|)],
(4.14)
(4.18)
hA1 |A2 i = e−2|α|
2
sin2 (δϑ/2) i|α|2 sin(δϑ)
e
(4.19)
.
The minimum phase shift that can be detected by using coherent states is of the order of 1/|α|, which is of the order of its phase uncertainty [25]. Meaningful welcher Weg detection implies δϑ > 1/|α|, leading to a severe degradation of the interference. Optimum conditions are reached when δϑ = π , so that the two phase shifts ϑ1 and ϑ2 are as different as possible (mod2π ). In this case there is no phase kick (for n even), with probability Peven , or ϕ is shifted by π (for n odd), with probability Podd . Then, equation (4.16) leads to Pob (ϕ) = P (ϕ)Peven + P (ϕ + π )Podd ,
where ϑj = τ κ 2 cj /1. Then
(4.20)
where V1† V2 = exp[−iδϑ(aa † |eihe| − a † a|gihg|)],
(4.15)
and δϑ = ϑ2 − ϑ1 . In this example the cavity field and the internal state of the atom form the apparatus. Since the atom is initially in its ground state, it will always stay in this state because the lack of resonance prevents atomic transitions. Then, we can see that U represents a field-phase shift, ϑj , depending on the slit crossed by the atom. It is possible to infer which slit is crossed by means of a phase-dependent measurement on the cavity field that should allow one to discriminate between ϑ1 and ϑ2 . The measurement of the appropriate field quadrature by means of a homodyne detector would serve for this purpose. The transformation that the relative phase experiences due to the interaction is of the form (3.10), but with a numerable set of possible phase shifts ξ = nδϑ: Pob (ϕ) =
∞ X
P (ϕ − nδϑ)P (n),
(4.16)
n=0
where P (n) is the photon-number probability distribution of the initial field state. The performance of the detection, as well as the ensuing degradation of the interference, strongly depends on the initial field state. An extreme situation is provided by a field number state |Ai = |ni|gi, where n is the number of photons stored in the cavity. After the interaction, we have that the atom–field state is of the form (2.5) with |Aj i = einϑj |Ai.
(4.17)
This leads to hA1 |A2 i = einδϑ and Vob = V . Therefore, there is no randomization of the relative phase Pob (ϕ) = P (ϕ − nδϑ) and there is no loss of visibility (merely a shift of the whole pattern). Accordingly, there is no possible inference of the slit crossed, since |A1 i and |A2 i are, in fact, the same state. Number states have completely random phase, so no phase shift can be detected. On the other hand, meaningful welcher Weg detection can be achieved if the cavity field is initially in a coherent
Peven = e−|α| cosh |α|2 ,
Podd = e−|α| sinh |α|2 . (4.21) The dependence of Pob (ϕ) on |α| is very similar to that of figure 3. When |α| → 0 the field state tends to be a number state (vacuum) and no inference of the path followed is possible, Podd → 0 and Pob (ϕ) → P (ϕ). In contrast, when |α| 1 the two possible phase shifts are clearly distinguishable: Podd ' Peven → 21 and Pob (ϕ) → 1/(2π ) tends to be a uniform distribution. The relative phase is completely random and there is no interference. 2
2
4.3. Ramsey interferometry As a further example, we consider a completely different arrangement where the internal paths of the interferometer are not spatial trajectories but internal states of the interfering particle; a two-level atom for instance. This situation corresponds to the ideas introduced and carried out experimentally in [26]. Here we just show that this different arrangement also fits into the framework developed in sections 2 and section 3. The atom is prepared in a superposition of its ground and excited levels, 1 |ψi = √ (|gi + |ei), 2
(4.22)
by a resonant microwave π/2-pulse in a low-Q cavity. Then it crosses a high-Q microwave cavity storing a coherent state that can act as a welcher Weg detector |Ai = |αi. The atomic transition and the cavity frequency are detuned enough so that the atom–field interaction Hamiltonian is H =
κ2 (aa † |eihe| − a † a|gihg|), 1
(4.23)
where κ is the coupling constant and 1 is the detuning. This interaction leads to the joint atom–field state 1 |ψiob = √ (|gi|αeiφ i + e−iφ |ei|αe−iφ i), 2
(4.24) 673
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where φ = κ 2 τ/1 and τ is the interaction time. Finally, another microwave cavity, like the first one, recombines the two internal paths. If the high-Q cavity is empty (α = 0), the interference is observed as the dependence on φ of the final population of the atomic levels. This is the unobserved interference. On the other hand, if the cavity stores a coherent state different from vacuum, the path followed can be inferred by a phasedependent measurement on the cavity field. Further details can be found in [26]. We can show that this scheme is another example of the detection arrangements we are examining in this work. The two paths are |ψ1 i = |gi and |ψ2 i = |ei. The detection is described by the unitary operator U = exp[−iφa † a(|eihe| − |gihg|)] = e−iφa a |eihe| †
+e
iφa † a
|gihg|,
(4.25)
which is clearly of the form (3.5). If the atom is prepared such that |ψi = |ei or |gi, it will remain undisturbed by the coupling with the apparatus. This is the analogue of the absence of broadening of the diffraction pattern in the doubleslit examples. In this case, the interferometric relative phase that is disturbed by the welcher Weg observation is precisely the phase of the atomic dipole associated with the transition |gi ↔ |ei [19]. Leaving this aside, everything occurs as in the previous example. 5. Complementarity in optical interference
In the previous section the relative phase has been applied to arrangements in which the interfering particles were atoms. However, this class of wave-like variables is more usually encountered when dealing with the electromagnetic field, where, in contrast to what happens with matter, there is a clear classical counterpart. The purpose of this section is to show that the relative phase becomes precisely the phase difference when the interfering particles are photons. In order to simplify the analysis and benefit from previous works [27], we shall focus on a Mach–Zehnder interferometer illuminated by a single photon (figure 5). The complex amplitude operators for the field modes propagating along the two internal paths are denoted by a1 and a2 . We are interested in welcher Weg detectors arranged to avoid the change of momentum of the photon. This can be accomplished by using the optical Kerr effect. Such a nonlinear coupling is the basis of some schemes for quantumnondemolition measurements [28,29]. In particular, the same arrangement we shall consider was proposed as a quantum nondemolition measurement of photon number in [30]. The feasibility of this class of quantum measurements has been demonstrated experimentally [29, 31]. The nonlinear medium is placed in one arm of the interferometer, coupling the mode a1 with an auxiliary probe field described by the complex amplitude operator b. This coupling can be described by the nonlinear interaction Hamiltonian H = χ a1† a1 b† b, (5.1) where χ is a parameter depending on the medium. Here, once again, the path followed will be recorded in the phase 674
Figure 5. Mach–Zehnder interferometer illuminated by a single photon. The welcher Weg detection is implemented by a Kerr interaction with a probe field b in one arm of the interferometer.
of the field mode b. This detection does not affect the spatial structure of the field modes and no photon will leave modes a1 and a2 due to the interaction with the apparatus. If the interfering photon follows paths a1 or a2 with certainty, the presence of the apparatus does not affect the field state at all. This is the translation of the conditions considered in section 2 to this interferometer, so this arrangement is an optical analogue of the atomic schemes discussed above. The field state after the first beam splitter, but before the Kerr medium, is |ψi = (t|1, 0i + r|0, 1i)|Ai,
(5.2)
where t and r (assumed to be real without loss of generality) are the transmission and reflection coefficients of the input beam splitter BS1 , |n1 , n2 i are photon-number states, and |Ai is the initial field state in mode b. After the Kerr medium and immediately before the output beam splitter BS2 , the field state is |ψi = t|1, 0i|A1 i + reiφ |0, 1i|Ai, (5.3) where |A1 i = e−iχτ b b |Ai, †
(5.4)
τ is the interaction time within the nonlinear medium, and φ is the phase introduced by the different optical length of the two internal paths. We can see that everything works as in the preceding section after suitable renaming of the vectors and parameters. However, in this case it would not be necessary to introduce the relative phase. The loss of interference can be analysed directly in terms of the field phase of mode a1 , which in turn enters in the phase difference between the two modes a1 and a2 governing the interference. In other words, the relative phase would be a variable derived from existing quantities. Nevertheless, in order to provide a more direct link with the analysis carried throughout this work, we will focus directly on the phase difference between the modes a1 and a2 . In spite of the fact that there is no universally accepted description of the field phase, the conclusions that can be obtained are independent of the particular formalism used, at
Randomization of quantum relative phase in welcher Weg measurements
least in this particular case. We will follow the description based on the Susskind–Glogower phase states ∞ 1 X |ϑi = √ einϑ |ni, 2π n=0
(5.5)
which define the POM 3(ϑ) = |ϑihϑ|,
(5.6)
and the corresponding probability distribution P (ϑ) = tr[ρ3(ϑ)], where ρ is the density matrix for the field [25]. Then, a POM for the phase difference ϕ = ϑ2 − ϑ1 can be introduced by Z 3(ϕ) = dϑ1 32 (ϕ + ϑ1 )31 (ϑ1 ), (5.7) where 3j (ϑj ) (j = 1, 2) are defined from equation (5.5) for each mode [32]. Let us compute how 3(ϕ) evolves under the action of the Hamiltonian (5.1). In this case 32 (ϑ2 ) is constant while 31 (ϑ1 ) transforms as †
†
eiχτ a1 a1 b b 31 (ϑ1 )e−iχτ a1 a1 b b ∞ X † † = eiχτ na1 a1 31 (ϑ1 )e−iχτ na1 a1 |nibb hn|, †
†
(5.8)
n=0
where |nib are the number states in mode b. equations (5.5) and (5.6) we have †
†
eiχτ a1 a1 b b 31 (ϑ1 )e−iχτ a1 a1 b †
†
b
=
∞ X
From
31 (ϑ1 + χ τ n)|nibb hn|. (5.9)
This leads to ∞ X
3(ϕ − χ τ n)|nibb hn|,
(5.10)
n=0
and then Pob (ϕ) =
∞ X
P (ϕ − χ τ n)P (n),
(5.11)
n=0
where P (n) = |hn|Ai|2 is the photon-number distribution of the initial field state |Ai in mode b and P (ϕ) is the probability distribution for the phase difference before the Kerr medium for an arbitrary field state in modes a1 and a2 . In the particular case of the one-photon state (5.2), we have P (ϕ) =
1 (1 + 2rt cos ϕ). 2π
(5.12)
The result obtained by using the field phase difference is the same found for atomic interferometers. It is easy to see that 3(ϕ) in (5.7) commutes with the total photon-number operator N = a1† a1 + a2† a2 . Since we are dealing with a single photon, we can restrict ourselves to the subspace with N = 1 and, in such a case, 3(ϕ) becomes 1 (|1, 0ih1, 0| + |0, 1ih0, 1| + e−iϕ |1, 0ih0, 1| 2π (5.13) +eiϕ |0, 1ih1, 0|),
3N=1 (ϕ) =
6. Erasure
Although the relative phase experiences a classical randomization, it also provides a simple explanation of the erasure of the path information. A pattern of high visibility can be recovered if the interference records are classified according to the outcomes of a suitable measurement made on the apparatus. Such measurement must provide no information about the path followed by the particle [9], which forces the state of the apparatus |ηi to satisfy hη|A2 i = eiη hη|A1 i,
(6.1)
and then one reaches the maximum visibility allowed by the initial state. We can introduce a joint probability distribution for ϕ and η after the interaction with the apparatus Pob (ϕ, η) = hψ|hA|U † 3(ϕ)|ηihη|U |Ai|ψi = P (ϕ − η)P (η),
n=0
3ob (ϕ) =
which is precisely the same as the POM for the relative phase in equation (3.1) with |ψ1 i = |1, 0i and |ψ2 i = |0, 1i. Finally, we can mention that it is also possible to describe the phase difference by means of an operator, which is the usual quantum description of observables [33]. The conclusions that can be obtained by using such an operator are the same as those already discussed, since for a one-photon state it has the same form (3.15).
(6.2)
where P (η) = |hη|A1 i|2 . If the outcomes η are discarded, we have to integrate equation (6.2) in η, which leads to equation (3.10) and the consequences already examined. On the other hand, if the outcomes are not discarded, for each η we get a conditional probability distribution 1 [1 + sin(2ϑ) cos(ϕ − φ − η)], 2π (6.3) which corresponds to the original interference pattern shifted by η and having the maximum visibility allowed by the initial state (2.1). When the welcher Weg information is contained in the phase of a probe field, a simple procedure for the erasure is a photon-number measurement of the probe field. Referring to the example involving the photon emission in a cavity, a possible procedure for implementing the erasure is as follows: after the passage of the atom, the two cavities can be coupled in such a way that the field modes would evolve under the mode-coupling Hamiltonian Pob (ϕ|η) = P (ϕ − η) =
H = κ(a1† a2 + a1 a2† ),
(6.4)
where the constant κ depends on the strength of the coupling. A photon deposited in one of the cavities can pass to the other and after a time τ 0 = π/(4κ), we have 1 |1, 0i → √ (|1, 0i − i|0, 1i), 2 1 |0, 1i → √ (|0, 1i − i|1, 0i), 2
(6.5)
675
A Luis and L L S´anchez-Soto
and the photon can be in either cavity 1 or 2 with equal probabilities, irrespective of where it was initially. The measurement of the photon number in the cavities will provide no information about the path followed, so it removes the welcher Weg information. The total system state at this moment is 1 |ψi = √ [cos(ϑ/2)|ψ1 i − ieiφ sin(ϑ/2)|ψ2 i]|1, 0i|gi 2 i − √ [cos(ϑ/2)|ψ1 i + ieiφ sin(ϑ/2)|ψ2 i]|0, 1i|gi, (6.6) 2 where we have assumed that the time τ spent by the atom within the cavities provides the optimum conditions for welcher Weg detection, cos(κτ ) = 0. The probability distributions for ϕ conditioned to the presence of the photon in cavity 1 or 2 are 1 [1 + sin ϑ cos(ϕ − φ + π/2)], 2π 1 [1 + sin ϑ cos(ϕ − φ − π/2)], Pob (ϕ|0, 1) = 2π Pob (ϕ|1, 0) =
(6.7)
respectively. These are the unobserved probability distribution P (ϕ) in equation (3.2) for the unobserved situation but shifted by ±π/2, in agreement with equation (6.3). 7. Conclusions
We have shown that the quantum relative phase provides a simple and useful tool for the analysis of welcher Weg detectors in which there is no alteration of the trajectories. In such a case, complementarity is enforced by random phase kicks. This conclusion is valid irrespective of the particular approach used to describe this variable. While the relative phase has a clear classical counterpart when dealing with electromagnetic field modes, in situations involving matter it could be regarded as a variable of quantum origin, since it enters via the superposition principle underlying quantum phenomena. This means that the possibility of describing the loss of interference as a classical randomization of the relative phase does not contradict the fact that the detection arrangements studied in this work are, in fact, full of quantum features. Appendix. Quantum relative phase for a two-dimensional system
In this appendix we provide the main arguments for supporting the POM (3.1) as a suitable quantum description of the relative phase. In the most general superposition of two orthogonal states |ψ1 i and |ψ2 i |ψi = cos(θ/2)|ψ1 i + e sin(θ/2)|ψ2 i, iφ
(A1)
the parameter φ is the relative phase of the superposition, and can be obtained as φ = arg[tr(ρ|ψ1 ihψ2 |)], 676
(A2)
where ρ is the density matrix. Taking into account that the operators Jx = 21 (|ψ1 ihψ2 | + |ψ1 ihψ2 |), Jy =
i (|ψ1 ihψ2 | − |ψ2 ihψ1 |), 2
(A3)
Jz = 21 (|ψ2 ihψ2 | − |ψ1 ihψ1 |), satisfy the SU (2) commutation relations, we have that equation (A2) is φ = arg[tr(ρJ− )], where J− = Jx − iJy . This allows one to identify the relative phase with the azimuthal angle in the Jx –Jy plane of an angular momentum, which is the variable conjugate to Jz . We are interested in the relative phase as a quantum variable instead of a state parameter. It is possible to describe this variable in quantum terms by means of a POM 3(ϕ) defining a probability distribution P (ϕ) = tr[ρ3(ϕ)]. A suitable 3(ϕ) can be obtained by imposing appropriate conditions [34]. Since P (ϕ) must be a true probability distribution for a 2π -periodic variable, we have the following conditions on 3(ϕ): 3(ϕ) > 0,
Z
3† (ϕ) = 3(ϕ),
dϕ 3(ϕ) = I,
3(ϕ + 2π ) = 3(ϕ),
(A4)
2π
where I = |ψ1 ihψ1 | + |ψ2 ihψ2 | is the identity in the twodimensional space spanned by |ψ1 i and |ψ2 i. Moreover, the relative phase is a variable conjugate to Jz , so we can impose [35] 0
0
eiϕ Jz 3(ϕ)e−iϕ Jz = 3(ϕ + ϕ 0 ).
(A5)
The most general 3(ϕ) satisfying these conditions is 1 (|ψ1 ihψ1 | + |ψ2 ihψ2 | + γ ∗ e−iϕ |ψ1 ihψ2 | 2π (A6) +γ eiϕ |ψ2 ihψ1 |),
3γ (ϕ) =
where γ is a complex parameter with |γ | 6 1. Let us note that 1 , (A7) tr[3(ϕ)|ψj ihψj |] = 2π as expected from the conjugate relation between Jz and ϕ. The argument of γ merely expresses an origin of angles. Its modulus |γ | is related with the fuzzy or noisy character of 3γ (ϕ), since P (ϕ) becomes broader as |γ | → 0, as it can be seen by computing the dispersion. We also have that Z 3γ (ϕ) = dϕ 0 Gγ (ϕ 0 )3(ϕ + ϕ 0 ), (A8) 2π
where Gγ (ϕ) =
1 (1 + γ e−iϕ + γ ∗ eiϕ ), 2π
(A9)
and 3(ϕ) = 3γ =1 (ϕ). Therefore, 3γ =1 (ϕ) can be identified as the best quantum description of ϕ by means of a POM [18]. Besides this approach, it is also possible to describe ϕ in the quantum domain by the more standard procedure of ascribing an operator to any dynamical variable. This
Randomization of quantum relative phase in welcher Weg measurements
operator can be introduced from a polar decomposition that defines the operator E complex exponential of ϕ as p p (A10) J− e−iϕ0 = J− J+ E = E J+ J− , where ϕ0 is a constant playing the role of the origin [21]. This equation has unitary as well as non-unitary solutions for E. The unitary solutions are Eϕ0 = e−iϕ0 |ψ1 ihψ2 | + eiϕ0 |ψ2 ihψ1 |.
[17]
(A11)
This operator has two eigenvectors |ϕ± i 1 |ϕ± i = √ (|ψ1 i ± eiϕ0 |ψ2 i), 2
(A12)
with eigenvalues Eϕ0 |ϕ± i = ±|ϕ± i. This description also has a proper conjugate relation with Jz since tr(|ϕ± ihϕ± ||ψj ihψj |) = 21 , and
0
0
eiϕ Jz Eϕ0 e−iϕ Jz = Eϕ0 +ϕ 0 .
[13] [14] [15] [16]
(A13)
[18] [19] [20] [21]
[22] [23]
(A14)
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