Abstract

The study of decoherence properties improves our understanding of the fundamental principles of quantum mechanics and advances the study of quantum information processing. Herein, we report a wave-optical experiment that can simulate the decoherence process of a Schrödinger’s cat state (SCS) by photon loss. The method is based on an analogy between image rotation in wave optics and a beam splitter in quantum optics. Experimental results show that the SCS rapidly decays into a statistical mixture of two Gaussian states with approximately 10% photon loss. This behavior can be well described within the framework of quantum optics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

To provide intuitive understanding in several physical systems, a continuous variable (CV) and its canonically conjugate variable are frequently compared to “position” and “momentum,” respectively, in a one-particle system. The probability distribution of a macroscopic state in a position–momentum phase space is typically Gaussian, and its widths are limited by the uncertainty relation between two CVs. A Schrödinger’s cat state (SCS) is a superposition of different macroscopic states [1]. The probability distribution of an SCS is not just a sum of Gaussians, and it can contain negative values. This indicates that an SCS cannot be decomposed to an ensemble of classical objects. SCSs have been experimentally generated in various systems, such as those consisting of a trapped ion [24], a standing microwave [5,6], and traveling light [7,8], in which the corresponding CVs are displacements from a trap center, the amplitude of a standing wave, and the in-phase amplitude with a local oscillator, respectively. These systems are good candidates for verifying fundamental principles in quantum mechanics, and they act as building blocks in quantum information processing (QIP). In particular, SCSs have been useful for investigating the quantum-classical boundary [9], quantum computing with CVs [10,11], and measurement sensitivity improvements [12].

Despite their fascinating behaviors, SCSs are fragile states [13]. The decay of SCSs, referred to as decoherence, is caused by interaction with other subsystems. For example, consider the superposition of two Gaussian wave packets in position space, Ψc. The position distribution, |Ψc|2, is split into two separate Gaussians, while the momentum distribution, |Ψ˜c|2, oscillates around the origin, where the tilde (˜) represents the Fourier transform. Upon interaction with external subsystems, the oscillation of the momentum distribution rapidly vanishes (decays), while the splitting of the position distribution remains. After decoherence, the SCS becomes a statistical mixture of two Gaussian wave packets. In fact, these decoherence processes and properties have been precisely observed in microwave systems [14,15]. In the case of traveling light, decoherence is caused by interaction with materials with imperfect transparency, which results in a decrease of the mean photon number. This process can be modeled as the mixing of an SCS with a vacuum state using a beam splitter (Fig. 1) [16]. Quantum feedback controls (QFCs) are helpful in protecting an SCS from decoherence [17,18]; however, the fragile nature of the SCS itself makes the development of QFC systems difficult.

 

Fig. 1. Model of decoherence caused by photon loss. An SCS is mixed with a vacuum state using a beam splitter (BS). The vacuum state and an odd SCS are expressed using the coherent state basis |α as |B=|0 and |A=|α|α. Photon loss at the finite transmission rate of optical components degrades the quality of the SCS.

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Multiple analogies exist between the spatial and temporal wave functions of light, also known as wave optics and quantum optics [19,20]. The spatial wave function, Ψ, obeys the paraxial Helmholtz equation, which resembles the Schrödinger equation (Appendix A) [21]. When the spatial coherence function, Γi, is regarded as a matrix element of the density operator in position space, the spatial Wigner function (WF), Wi, introduced in wave optics [22] resembles the two-mode WF in a two-dimensional (2D) quantum system (Appendix B). On the focal plane of a Gaussian laser beam propagating in free space, Ψ corresponds to the wave function of a coherent state (Appendix C) [23,24]. On the focal plane of two parallel Gaussian laser beams, Ψ corresponds to the wave function of an SCS [25,26]. With respect to optical components, two waveguides coupled by their evanescent field [27] or Kerr medium [28] have been proposed as a physical object corresponding to a beam splitter. However, if the SCS in wave optics did not decohere without these objects, the SCS would be considered extremely robust, unlike the SCS in quantum optics. To fully reproduce the fragile nature of an SCS, another object in wave optics should correspond to the beam splitter in quantum optics.

In this study, we proved a theoretical analogy between image rotation in wave optics and a beam splitter in quantum optics. Then, we demonstrated a wave-optical experiment that corresponds to the progressive measurement of a decohering SCS by photon loss in quantum optics. As the decoherence proceeded, we observed that the negative part of the spatial WF rapidly vanished, and purity rapidly decreased toward that for a statistical mixture of two Gaussian states. Because the theoretical frameworks are identical, the behavior can be accurately described by the beam-splitter model of photon loss (i.e., a vacuum state, |0, and an odd SCS, |α|α, are mixed by a beam splitter, where |±α expresses a coherent state of complex amplitude ±α). Thus, wave-optical experiments based on this analogy could be useful for developing QIP protocols for protecting an SCS against decoherence.

2. SPATIAL WF OF LIGHT

In this study, the spatial WF of light on the z=zi plane was represented by Wi(x,p¯u;y,p¯v), where the propagation direction of light, say, the z axis, was orthogonal to the x and y axes. Wi was regarded as a statistical weight when light was decomposed to an ensemble of virtual particles with positions (x,y) and momenta (p¯u,p¯v) on the z=zi plane as

Wi(x,p¯u;y,p¯v)=δ(xqA)δ(p¯upA)δ(yqB)δ(p¯vpA)i,
where qA,pA,qB, and pB are the CVs of the system, and the double angle bracket (i) represents the expectation value calculated from Wi(qA,pA;qB,pB). It is written as
fidqAdpAdqBdpBfWi(qA,pA;qB,pB).

Note that qA,pA,qB, and pB are treated as operators, while x,p¯u,y, and p¯v are treated as eigenvalues in Appendices C and F. The probability distribution for a few variables was represented by reducing the dependencies of other variables on Wi. For example, the 2D intensity distribution, Ii(x,y), the one-mode WF, Fi(x,p¯u), and the probability distribution in one-dimensional (1D) position space, Pi(x), were obtained as follows (Appendix B):

Ii(x,y)=δ(xqA)δ(yqB)i,
Fi(x,p¯u)=δ(xqA)δ(p¯upA)i,
Pi(x)=δ(xqA)i.

Note that probabilities Wi, Ii, Fi, and Pi were normalized as 1i=Iidxdy=p¯Fidxdu=Pidx=1. The temporal evolution of Wi was represented by the Hamiltonian of system H and the Liouville equation (Appendix D):

Wt=k=A,B[WqkHpkWpkHqk],
where t is the temporal duration of H acting on the system.

The Hamiltonian of light propagating in free space under the paraxial approximation is presented as follows [21]:

Hfree=cp¯[1+pA2+pB22p¯2],
where c is the velocity of light and p¯ is the average value of the z-component momentum of light. As a consequence of Eqs. (4)–(7), the 1D intensity distribution Pi(x) evolves in free space as follows (Appendix E) [29]:
Pi(x)=p¯F0(x(ziz0)u,p¯u)du.

Equation (8) allowed for the reconstruction of the one-mode WF, F0(x,p¯u), from a data set of the intensity distributions of light propagating in free space, P0(x),P1(x),P2(x), [29,30]. All one-mode WFs used in this study were experimentally obtained based on Eq. (8). The reconstruction procedure can be mathematically written as an inverse Radon transform [29], resembling homodyne tomography in quantum optics [16]. Note that two other methods have been demonstrated for determining the spatial WF: (i) direct production of the WF as an optical image [31] and (ii) the measurement of each value of the WF at any point in a phase space using a Sagnac interferometer [32,33].

3. ANALOGY BETWEEN IMAGE ROTATION AND THE BEAM SPLITTER

We now propose the main idea of this study, which plays an important role in simulating the progressive measurements of a decohering SCS by photon loss in quantum optics. The rotation of a series of images on the x-y plane along the z axis can be associated with a beam splitter in quantum optics. Suppose the Hamiltonian is represented as

Hrot=Ω(qBpAqApB),
where Ω is a real constant. As the Liouville equation (6) becomes W/t=Ω[qB/qApB/pA+qA/qB+pA/pB]W, the WFs before and after the interaction are connected as
Wi(ϕ)(qA,pA;qB,pB)=Wi(qAτqBϱ,pAτpBϱ;qAϱ+qBτ,pAϱ+pBτ),
where ϕΩt, τcosϕ, and ϱsinϕ. Because the 2D intensity distribution defined as Eq. (3) becomes Ii(ϕ)(x,y)=Ii(xτyϱ,xϱ+yτ), the interaction, Hrot (9), was realized by rotating a series of images with angle ϕ. A Dove prism is a physical object that realizes Hrot. The relation (10) has the same form as the input–output of the WF for a beam splitter in quantum optics (Appendix F) [16]. The correspondence between the rotation angle of the images, ϕ, and the transmittance of the beam splitter, T, is
T=(cosϕ)2.

The one-mode WF after the beam splitter-like interaction, F0(ϕ)(x,u), can be reconstructed from the experimental data set of the 1D intensity distributions after image rotation, P0,1,2,(ϕ)(x)=I0,1,2,(ϕ)dy. Note that the one-mode WFs in front of the beam splitter correspond to F0(0) and F0(π/2), which can be reconstructed from experimental data sets P0(0),P1(0),P2(0), and P0(π/2),P1(π/2),P2(π/2),, respectively.

The analogy can be further clarified using fictitious complex amplitudes a and b instead of (qA,pA) and (qB,pB), respectively, defined as [23]

aq¯p¯2[qAq¯+ipAp¯],bq¯p¯2[qBq¯+ipBp¯],
where is the Planck constant and q¯ is a constant with the length dimension. The Hamiltonian (9) can be rewritten as Hrot=iΩ(a*bb*a) [34]. The input–output relation of the WF (10) can be rewritten as Wi(a;b)=Wi(ϕ)(a(H);b(H)), where a(H)=aT+b1T and b(H)=a1T+bT. The expressions of a(H) and b(H) using a and b resembled the input–output relations of the complex amplitude of the beam splitter in quantum optics [34]. The density operator corresponding to F0(ϕ) can be written as ρ^dec=TrB[U^rotρ^AB(0)U^rot], where ρ^AB(0) is the density operator corresponding to W0, TrB[] denotes the partial trace to reduce the B-mode, and U^rotexp[ϕ(a^b^a^b^)] (Appendices B and F). Note that similarities were discussed in [35] between the decoherence described by the beam-splitter model, that adopted in this work, and that described by the Fokker–Planck equation [36,37].

4. EXPERIMENT

Figure 2 shows a schematic diagram of the experimental setup. A light beam at λ=780nm, which was emitted from an external cavity diode laser, passed through a single-mode fiber (780 HP). The Gaussian-like beam shape was converged to an intensity-1/e2 radius of w0=86μm using a triplet lens collimator (TC18APC-780, Thorlabs) and two doublet lenses. The Mach–Zehnder interferometer (MZI) was composed of mirrors (i.e., M1 and M2) and beam splitters (i.e., BS1 and BS2), which produced interference between two Gaussian beams. The beam profile of an output port was acquired to monitor the phase difference between the two Gaussian beams, θ (Phase Monitor). The voltage of a piezoelectric transducer (PZT) was manually adjusted so that the phase difference between the two beams was nearly equal to θ=π before each intensity measurement. Note that, unlike the typical setup of the MZI, the central axes of the output beams were horizontally displaced by 2d=0.38mm. The Dove prism is a symbolic expression of image rotation. Similarly, the slit along the y axis and the photo detector are symbolic expressions of intensity measurements. We used a charge-coupled device (CCD) camera in the actual experiment, instead of these items (★). The pixel size of the CCD camera corresponding to the slit aperture was Δ=6.8μm. The z=z0 plane was set around the focal plane of the two beams, and we set z0=0mm. The one-mode WFs corresponding to the vacuum state, SCS, and decohered SCS were experimentally reconstructed from the data set of the intensity distributions, {Pi(ϕ)(x)}.

 

Fig. 2. Experimental apparatus. A beam with a spatial WF for the vertical mode, which acts like a vacuum state, and a spatial WF for the horizontal mode, which acts like an SCS, is created using a MZI. A Dove prism induces a beam splitter-like interaction between these modes. The 1D intensity distributions, {Pi(ϕ)(x)}, were measured to reconstruct the spatial WF.

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5. RESULTS

Figure 3(a) shows the experimentally measured intensity distribution after the π/2 rotation, Pi(π/2)(x). Figure 3(b) presents the one-mode WFs, Fi(π/2), for the z=(50,0,50)mm planes, which were reconstructed from the data set of {Pi(π/2)(x)} using the inverse Radon transform based on Eq. (8) [29]. For comparison, the intensity distribution, Pi(π/2)(x), and those calculated from the reconstructed WFs, p¯F0(π/2)(x(ziz0)u,p¯u)du, are depicted in Fig. 3(c), which shows good agreement with the experimental result. Analogies between a Gaussian laser beam propagating in free space and a coherent state in quantum optics were discussed (Appendix C) [23,24]. In actuality, the one-mode WF on z=z0 (i.e., F0(π/2)) exhibited a Gaussian shape centered at the origin in a phase space, which resembled a vacuum state in quantum optics.

 

Fig. 3. Experimental results of the intensity measurements. (a) Intensity distribution after the π/2 rotation, Pi(π/2). (b-1, b-2, and b-3) Reconstructed WFs, Fi(π/2). (c-1, c-2, and c-3) Observed intensity distributions, Pi(π/2) (red bar), and the distributions calculated from the reconstructed WFs, Fi(π/2)du (blue line).

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Figures 4(a-1, a-2, and a-3) and 4(b-1, b-2, and b-3) show the observed intensity distributions after rotating the images at ϕ=0,13°, and 18° (i.e., {Pi(ϕ)}) and the corresponding one-mode WFs on z=z0 (i.e., F0(ϕ)), respectively. Note that ϕ=(0,13,18)° corresponds to T=(1.00,0.95,0.90) using Eq. (11). The analogies between two parallel Gaussian laser beams and an SCS in quantum optics were discussed (Appendix C) [25,26]. In actuality, F0(0) shown in Fig. 4(b-1) oscillated along the momentum axis between the two Gaussian peaks with a strong negative peak close to the origin, resembling an SCS in quantum optics.

 

Fig. 4. Observed intensity distributions, {Pi(ϕ)}, with an image rotation angle of (a-1) ϕ=0 (T=1.00), (a-2) ϕ=13° (T=0.95), and (a-3) ϕ=18° (T=0.90). (b-1), (b-2), and (b-3) denote the corresponding WFs to (a-1), (a-2), and (a-3), respectively, on the z=0mm planes, F0(ϕ).

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The amplitude of the oscillation along the momentum axis decreased as decoherence proceeded [Figs. 4(b-2 and b-3)]. The negative part of the WF almost vanished when decoherence approached T=0.90, while the splitting of the position axis remained. The fact that the WF contains a negative value indicates that the object cannot be decomposed to an ensemble of classical particles, which is not always a necessary condition. However, it is a sufficient condition for verifying that the state is nonclassical in quantum optics. Two strong positive Gaussians implies that the state can be regarded as a statistical mixture of two macroscopic states. Our results indicated that the progressive measurements of the macroscopic superposition decaying to the statistical mixture were successfully simulated in a wave-optical experiment. The nonclassical features decreased rapidly with the fictitious transmittance defined in Eq. (11), which agrees with the fragile nature of the SCS in quantum optics.

6. DISCUSSIONS

We quantitatively evaluated the degree of decoherence by calculating the purity defined by the one-mode WF as follows:

Π(ϕ)2πp¯2k[F0(ϕ)(x,p¯u)]2dxdu,
where k2π/λ. The purity, Π(ϕ), could take values 0Π(ϕ)1. Moreover, Π(ϕ) varied from 1 to 1/2 when the pure state decohered into a statistical mixture of two pure states. Note that the purity is a poor measure of decoherence when T<1/2 because the dominant component of the observing mode in such a case becomes the vacuum state rather than the SCS. In other words, the purity increases from approximately 1/2 to 1. Figure 5 shows the purity for the transmittance calculated from the experimental data. The observed purity decays exponentially as the transmittance of the virtual beam splitter, T, decreases.

 

Fig. 5. Purity with respect to transmittance obtained through experiments (closed circle) and theory (solid line).

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Figure 5 also depicts the theoretical values of purity with respect to transmittance for comparison. These values were obtained by assuming that the two beams were diffraction-limited Gaussian beams converging on the same z=z0 plane after the MZI, and displacement d was constant for any z=zi planes. The theoretical two-mode WF under these assumptions is written as follows (Appendix C):

W0(x,p¯u;y,p¯v)=S(x,p¯u)G(y,p¯v),
where S(x,p¯u) is the one-mode WF of an SCS with displacement d and relative phase θ expressed as [25]
S(x,p¯u)=G(xd,p¯u)+G(x+d,p¯u)+2G(x,p¯u)cos(2kdu+θ).

Here, G(x,p¯u) is the one-mode WF of a Gaussian beam on the focal plane with a waist size of w0 expressed as [29]

G(x,p¯u)=exp[2w02x2k2w022u2].

The theoretical one-mode WF of the decohered SCS, Sdec, was numerically obtained by applying Eq. (10) and the partial tracing of Eq. (4) with θ=π in Eq. (15). The purity was finally calculated from Sdec using Eq. (13). Note that this is a common method of calculating the SCS of light after it passes through a lossy medium in quantum optics [35]. Figure 5 shows that the experimental values have a good qualitative agreement with the theory. The slight difference between experiments and theory could arise from the uncertainty of pixel size, deterioration of the signal-to-noise ratio when measuring the intensity distributions close to the focal plane, mismatch between the focal plane of the beams and the z=z0 plane, phase fluctuations of the MZI, or imperfect beam quality, which would degrade the purity of the Gaussian beam.

The variance of the fictitious quadrature amplitude for the vacuum state can be calculated as

(Re[b])20=p¯w028q¯,
(Im[b])20=q¯p¯2k2w02.

Thus, the constant, q¯, in Eq. (12) should be set as

q¯=kw022
to make the one-mode WF of the vacuum state (i.e., δ(βb)0) isotropic in complex plane β. Under this condition, the fictitious in-phase amplitude of the A-mode becomes Re[a]=qA/w0. The displacement was d/w0=2.2 in the experiment. Therefore, the simulated SCS corresponded to an SCS with a complex amplitude of |α|=2.2.

7. CONCLUSION

In this study, we demonstrated a wave-optical experiment that simulated progressive measurements of a decohering SCS in quantum optics. The experiment was performed with standard optical experimental elements, such as coherent (classical) light, an MZI, and a CCD camera. Spatial WFs were reconstructed from the data set of the intensity distributions. The reconstructed WFs indicated that two states, such as a vacuum state and an SCS in quantum optics, were generated in two different modes. Image rotation was used to mix the two modes, similar to a beam splitter in quantum optics. The behavior of the decohering SCS was well described by the same framework as that in quantum optics. Notably, this experiment required a relatively easier technique than those required in a standard quantum optical system, such as a superconducting microwave cavity, ion traps, and squeezed light. If the Dove prism were replaced by an inhomogeneous index medium, such as a multimode optical fiber, a complicated Hamiltonian would be obtained instead of Hrot [20]. A Hamiltonian written as more than quadratic polynomial could be obtained instead of Hfree [38], which plays important roles in quantum computing with CVs [10]. Therefore, the use of the analogies between the spatial and temporal wave functions of light would allow for the simulation of several QIP protocols.

APPENDIX A: SPATIAL COHERENCE FUNCTION OF LIGHT

When light is monochromatic and linearly polarized, its electric field is expressed as

E(x,t)=E0{Ψ(x)exp[i(kzωt)]+c.c.}/2,
where E(x,t) is the electric field at position x=(x,y,z) at time t, ω is the angular frequency of light, and E0 is a constant with the dimensions of the electric field. The envelope of the oscillating electric field, Ψ(x), obeys the paraxial Helmholtz equation as
iΨz=LΨ,
where L is differential operator defined as
L12k[2x2+2y2]+kcp¯V(x,y).

Here, V(x,y) is a position-dependent function defined by the refractive index, n(x,y,z), as

V(x,y)cp¯2[1ω2c2k2{n(x,y,z)}2].

Briefly, the electric field obeys the wave equation, [2(n2/c2)2/t2]E=0. By substituting Eq. (A1), the wave equation becomes [(/x)2+(/y)2k2+2ik/z+(/z)2+n2ω2/c2]Ψ=0. When the term (/z)2Ψ is negligible, the wave equation becomes the paraxial Helmholtz equation (A2). Note that Eq. (A2) resembles the Schrödinger equation, which enables us to represent the temporal evolution of the wave function as Ψ(x,y,zi)=exp[i(ziz0)L]Ψ(x,y,z0).

In general, the phase coherence of light is imperfect. To express the electric field as a statistical mixture of phase coherent light, we introduce a spatial coherence function as [22]

Γi(x+,y+;x,y)jλjψj(i)(x+,y+){ψj(i)(x,y)}*,
where λj represents real coefficients and ψj(i)(x,y) is
ψj(i)(x,y)=exp[i(ziz0)L]ψj(x,y).
Here, ψ1(x,y),ψ2(x,y), are a set of normalized orthogonal functions (i.e., ψl*ψmdxdy=δlm). By calculating Γ/z=limziz0[(ΓiΓ0)/(ziz0)], we find that the coherence function (A5) obeys the transport equation written as
Γz=i(L+L)Γ,
where L± denotes differential operators defined as
L±12k[2x±2+2y±2]+kcp¯V(x±,y±).
Note that the spatial WF, Wi, is originally introduced as the Fourier transform of the coherence function [22]:
Wi(x,p¯u;y,p¯v)k2(2πp¯)2drdseik(ru+sv)×Γi(x+r/2,y+s/2;xr/2,ys/2).

APPENDIX B: EXPRESSIONS USING DENSITY OPERATORS

The intensity distributions, Ii,Pi, and one-mode WF, Fi, expressed by the probability distribution function, Wi, as Eqs. (3)–(5) can be also expressed using density operators similar to those in quantum mechanics. For this purpose, we associate the momentum with the wavenumber as

p¯=k.
The value of the coherence function defined as Eq. (A5) can be associated with a matrix element of a density operator, ρ^AB(i), in 2D position space as
Γi(x+,y+;x,y)x+,y+|ρ^AB(i)|x,y.
At the same time, we introduce the smallest unit of length, Δ, so that the condition
Δ2lmlΔ,mΔ|ρ^AB(i)|lΔ,mΔ=1
is established, where l and m are integers. Under these assumptions, the spatial WF defined as Eq. (A9) becomes identical to the two-mode WF in a 2D quantum system.

The 2D intensity distribution, Ii, in Eq. (3), the one-mode WF, Fi, in Eq. (4), and the 1D intensity distribution, Pi, in Eq. (5) can be expressed as

Ii=x,y|ρ^AB(i)|x,y,
Fi=12πdreikrux+r/2|ρ^A(i)|xr/2,
Pi=x|ρ^A(i)|x,
where x+|ρ^A(i)|x can be written as
x+|ρ^A(i)|x=dqBx+,qB|ρ^AB(i)|x,qB
=Δmx+,mΔ|ρ^AB(i)|x,mΔ.
Equation (B8) indicates that ρ^A(i) corresponds to the density operator of the subsystem represented as ρ^A(i)=TrB[ρ^AB(i)], where TrB[] denotes the partial trace introduced after Eq. (12).

Πi(2πp¯2/k)Fi2dxdu, which we refer to as “purity” [Eq. (13)], exactly corresponds to the purity of the subsystem in quantum mechanics represented as TrA[{ρ^A(i)}2]. This correspondence can be confirmed as follows:

Πi=2πp¯2kdxdu{Fi}2
=k2πdxdudrdseik(sr)u×x+r/2|ρ^A(i)|xr/2xs/2|ρ^A(i)|x+s/2
=dxdrx+r/2|ρ^A(i)|xr/2×xr/2|ρ^A(i)|x+r/2
=Δ2lmlΔ|ρ^A(i)|mΔmΔ|ρ^A(i)|lΔ=ΔllΔ|{ρ^A(i)}2|lΔ.

APPENDIX C: COHERENT STATE IN WAVE OPTICS

The wave function on the focal plane of a Gaussian laser beam propagating in free space corresponds to the wave function of a coherent state [23]. We briefly review this correspondence by introducing the operator version of the positions and momenta (i.e., q^A,q^B,p^A, and p^B) as follows:

x,y|q^A|jxψj(x,y),
x,y|q^B|jyψj(x,y),
x,y|p^A|jixψj(x,y),
x,y|p^B|jiyψj(x,y),
where ψj is one of the normalized orthogonal functions defined after Eq. (A6).

Using Eqs. (C1)–(C4), the operator version of the fictitious complex amplitudes defined in Eq. (12) can be written as x,y|a^|j=Ax,y|j and x,y|b^|j=Bx,y|j, where differential operators A and B are

A=xw0+w02x,
B=yw0+w02y.
Here, we select the values of q¯ and p¯ to satisfy Eqs. (19) and (B1). Thus, the wave function of a Gaussian beam propagating on the z axis,
ΨG(x,y,z)exp[ikx2+y22(zz0)ikw02],
satisfies AΨG(x,y,z0)=BΨG(x,y,z0)=0, which are the conditions for the vacuum state in a two-mode system.

The wave function on the focal plane of two parallel Gaussian beams corresponds to the wave function of an SCS, and its WF becomes Eq. (14) [25]. Briefly, the wave functions of two parallel Gaussian beams are written as

Ψ1(x,y,z)=eiθ1ΨG(xd,y,z),Ψ2(x,y,z)=eiθ2ΨG(x+d,y,z),
where 2d is the spacing of the two beams, and θ1,2 is the phase of each beam. Note that AΨ1(x,y,z0)=(d/w0)Ψ1(x,y,z0), AΨ2(x,y,z0)=(d/w0)Ψ2(x,y,z0), and BΨ1,2(x,y,z0)=0 are established, which are the conditions for the coherent states of complex amplitudes ±d/w0 in the A-mode and 0 in the B-mode. When the relative phase is adjusted to θ1θ2=θ, its superposition is written as
Ψc(x,y,z)=Ψ1(x,y,z)+eiθΨ2(x,y,z).
Using Eq. (A9), we find that the spatial WF for Ψc(x,y,z0) is written as Eq. (14).

APPENDIX D: LIOUVILLE EQUATION FOR SPATIAL WF

The transport equation (A7) is identical to the Liouville equation for the WF (6) with the Hamiltonian of

H=Hfree+V(qA,qB)
and interaction time t=z/c [22]. The derivation can be given as follows: As V(x,y) can be Taylor expanded as
V(x,y)=[1+(xqA)qA+(yqB)qB+(xqA)222qA2+]V(qA,qB),
the transport equation (A7) becomes
Γz=i2k(2x+22x2+2y+22y2)Γikcp¯[VqA(x+x)+VqB(y+y)+2VqA2(x++x2qA)(x+x)+]Γ.
In accordance with Eq. (A9), the coherence function can be expressed as the inverse Fourier transform of the WF as
Γi(x+,y+;x,y)=p¯2dudveik(ru+sv)Wi(qA,p¯u;qB,p¯v),
where (x++x)/2=qA, x+x=r, (y++y)/2=qB, and y+y=s. By substituting Eq. (D3) into the following equations, we confirm
drdseik(rpA+spB)/p¯(2x±2)Γi(x+,y+;x,y)=(12qA±ikp¯pA)2Wi(qA,pA;qB,pB),
drdseik(rpA+spB)/p¯(x±)Γi(x+,y+;x,y)=(qAp¯2ikpA)Wi(qA,pA;qB,pB).
Using Eq. (A9), (D5), and (D6), the transport equation (D3) can be rewritten as
Wz=k=A,B[pkp¯Wqk+1cVqkWpk],
where the terms including the higher-order derivatives for V (i.e., 2V/(qAqB), …) are neglected. Equation (D7) is identical to the Liouville equation (6) with the Hamiltonian of Eq. (D1) and interaction time t=z/c.

APPENDIX E: DERIVATION OF EQ. (8)

We briefly review the derivation of Eq. (8) here. The Liouville equation (6) for the Hamiltonian (7) becomes

Wt=cp¯[pAqA+pBqB]W.
The solution is
Wi(qA,pA;qB,pB)=W0(qActpA/p¯,pA;qBctpB/p¯,pB).
By substituting ct=ziz0 and applying Eq. (4), we can obtain
Fi(x,p¯u)=F0(x(ziz0)u,p¯u).
In accordance with Eqs. (4) and (5), Fi and Pi are related as
Pi(x)=p¯Fi(x,p¯u)du.
By substituting Eq. (E3) into Eq. (E4), we can obtain Eq. (8).

APPENDIX F: ANOTHER DEVIATION OF EQ. (10)

Quantum mechanics predicts that the beam splitter described by the Hamiltonian (9) rotates the coordinates of the wave function, which was proved in [35]. As a result, the input–output relation of the WF at the beam splitter is given by Eq. (10). In this section, we briefly derive the same result by mediating the Schrödinger equation instead of the Liouville equation.

The temporal evolution of the wave function (A6) can be rewritten as

ψj(i)(x,y)=x,y|exp[i(t/)H^]|j,
where H^ is the operator version of H defined in Eq. (D1), ct=ziz0, and Eqs. (B1) and (C1)–(C4) are assumed. The Schrödinger equation, i(/t)ψj=x,y|H^|j, can be obtained by differentiating Eq. (F1) with respect to t. By substituting the Hamiltonian of the beam splitter, H=Hrot, defined in Eq. (9) and interaction time t=ϕ/Ω into Eq. (F1), we obtain
x,y|exp[iϕΩH^rot]|j=x,y|exp[iϕ(q^Bp^Ap^Aq^B)]|j
=exp[ϕ(yx+xy)]ψj(x,y)
=ψj(xcosϕysinϕ,xsinϕ+ycosϕ)
=xτyϱ,xϱ+yτ|j,
which indicates that the coordinates are rotated after the beam splitter for an arbitrary wave function. Therefore, the coherence function of the output state after the beam splitter can be written as
Γi(ϕ)(x+,y+;x,y)x+,y+|U^rotρ^AB(i)U^rot|x,y
=x+τy+ϱ,x+ϱ+y+τ|ρ^AB(i)|xτyϱ,xϱ+yτ
=Γi(x+τy+ϱ,x+ϱ+y+τ;xτyϱ,xϱ+yτ),
where U^rotexp[iϕ/(Ω)H^rot]. By substituting Eq. (F8) into Eq. (A9), we find that the input–output relation of the WF written as Eq. (10) is valid. Note that U^rot is rewritten by the operator version of the fictitious complex amplitudes, a^,b^, defined in Eq. (12) as U^rotexp[ϕ(a^b^a^b^)].

Funding

Matsuo Academic Research Foundation; Japan Society for the Promotion of Science (JSPS) KAKENHI (JP25800224).

REFERENCES

1. E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–812 (1935). [CrossRef]  

2. C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A “Schrödinger cat” superposition state of an atom,” Science 272, 1131–1136 (1996). [CrossRef]  

3. D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005). [CrossRef]  

4. T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011). [CrossRef]  

5. M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. Raimond, and S. Haroche, “Observing the progressive decoherence of the “meter” in a quantum measurement,” Phys. Rev. Lett. 77, 4887–4890 (1996). [CrossRef]  

6. B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013). [CrossRef]  

7. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science (New York) 312, 83–86 (2006). [CrossRef]  

8. T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010). [CrossRef]  

9. W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715–775 (2003). [CrossRef]  

10. S. Lloyd and S. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999). [CrossRef]  

11. T. Ralph, A. Gilchrist, G. Milburn, W. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003). [CrossRef]  

12. A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, “Schrödinger cats and their power for quantum information processing,” J. Opt. B Quantum Semiclass. Opt. 6, S828–S833 (2004). [CrossRef]  

13. D. F. Walls and G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985). [CrossRef]  

14. S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature 455, 510–514 (2008). [CrossRef]  

15. G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013). [CrossRef]  

16. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).

17. P. Goetsch, P. Tombesi, and D. Vitali, “Effect of feedback on the decoherence of a Schrödinger-cat state: a quantum trajectory description,” Phys. Rev. A 54, 4519–4527 (1996). [CrossRef]  

18. D. Vitali, P. Tombesi, and G. Milburn, “Protecting Schrödinger cat states using feedback,” J. Mod. Opt. 44, 2033–2041 (1997). [CrossRef]  

19. D. Dragoman, “Phase space correspondence between classical optics and quantum mechanics,” Prog. Opt. 43, 433–496 (2002). [CrossRef]  

20. M. Man’ko, V. Man’ko, and R. V. Mendes, “Quantum computation by quantumlike systems,” Phys. Lett. A 288, 132–138 (2001). [CrossRef]  

21. D. Gloge and D. Marcuse, “Formal quantum theory of light rays,” J. Opt. Soc. Am. 59, 1629–1631 (1969). [CrossRef]  

22. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986). [CrossRef]  

23. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993). [CrossRef]  

24. O. Steuernagel, “Equivalence between focused paraxial beams and the quantum harmonic oscillator,” Am. J. Phys. 73, 625–629 (2005). [CrossRef]  

25. K. Wódkiewicz and G. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815–821 (1998). [CrossRef]  

26. D. Dragoman and M. Dragoman, “On the similarities between the Wigner distribution function in classical and quantum optics,” Optik 112, 497–501 (2001). [CrossRef]  

27. J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004). [CrossRef]  

28. R. Mar-Sarao and H. Moya-Cessa, “Optical realization of a quantum beam splitter,” Opt. Lett. 33, 1966–1968 (2008). [CrossRef]  

29. J. Jaguś, J. K. Jabczyński, W. Żendzian, and J. Kwiatkowski, “Application of Wigner transform for characterization of aberrated laser beams,” Opt. Appl. 35, 33–41 (2005).

30. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995). [CrossRef]  

31. K.-H. Brenner and A. Lohmann, “The Wigner distribution function display of complex 1D signals,” Opt. Comm. 42, 310–314 (1982). [CrossRef]  

32. E. Mukamel, K. Banaszek, I. A. Walmsley, and C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317–1319 (2003). [CrossRef]  

33. B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005). [CrossRef]  

34. G. S. Agarwal, Quantum Optics (Cambridge University, 2013).

35. U. Leonhardt, “Quantum statistics of a lossless beam splitter: SU(2) symmetry in phase space,” Phys. Rev. A 48, 3265–3277 (1993). [CrossRef]  

36. P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, “Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques,” Adv. Phys. 57, 363–455 (2008). [CrossRef]  

37. A. Polkovnikov, “Phase space representation of quantum dynamics,” Ann. Phys. 325, 1790–1852 (2010). [CrossRef]  

38. A. L. Rivera, S. M. Chumakov, and K. B. Wolf, “Hamiltonian foundation of geometrical anisotropic optics,” J. Opt. Soc. Am. A 12, 1380–1389 (1995). [CrossRef]  

References

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  • |

  1. E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–812 (1935).
    [Crossref]
  2. C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A “Schrödinger cat” superposition state of an atom,” Science 272, 1131–1136 (1996).
    [Crossref]
  3. D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
    [Crossref]
  4. T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
    [Crossref]
  5. M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. Raimond, and S. Haroche, “Observing the progressive decoherence of the “meter” in a quantum measurement,” Phys. Rev. Lett. 77, 4887–4890 (1996).
    [Crossref]
  6. B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013).
    [Crossref]
  7. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science (New York) 312, 83–86 (2006).
    [Crossref]
  8. T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
    [Crossref]
  9. W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715–775 (2003).
    [Crossref]
  10. S. Lloyd and S. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
    [Crossref]
  11. T. Ralph, A. Gilchrist, G. Milburn, W. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
    [Crossref]
  12. A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, “Schrödinger cats and their power for quantum information processing,” J. Opt. B Quantum Semiclass. Opt. 6, S828–S833 (2004).
    [Crossref]
  13. D. F. Walls and G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985).
    [Crossref]
  14. S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature 455, 510–514 (2008).
    [Crossref]
  15. G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
    [Crossref]
  16. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).
  17. P. Goetsch, P. Tombesi, and D. Vitali, “Effect of feedback on the decoherence of a Schrödinger-cat state: a quantum trajectory description,” Phys. Rev. A 54, 4519–4527 (1996).
    [Crossref]
  18. D. Vitali, P. Tombesi, and G. Milburn, “Protecting Schrödinger cat states using feedback,” J. Mod. Opt. 44, 2033–2041 (1997).
    [Crossref]
  19. D. Dragoman, “Phase space correspondence between classical optics and quantum mechanics,” Prog. Opt. 43, 433–496 (2002).
    [Crossref]
  20. M. Man’ko, V. Man’ko, and R. V. Mendes, “Quantum computation by quantumlike systems,” Phys. Lett. A 288, 132–138 (2001).
    [Crossref]
  21. D. Gloge and D. Marcuse, “Formal quantum theory of light rays,” J. Opt. Soc. Am. 59, 1629–1631 (1969).
    [Crossref]
  22. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [Crossref]
  23. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
    [Crossref]
  24. O. Steuernagel, “Equivalence between focused paraxial beams and the quantum harmonic oscillator,” Am. J. Phys. 73, 625–629 (2005).
    [Crossref]
  25. K. Wódkiewicz and G. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815–821 (1998).
    [Crossref]
  26. D. Dragoman and M. Dragoman, “On the similarities between the Wigner distribution function in classical and quantum optics,” Optik 112, 497–501 (2001).
    [Crossref]
  27. J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
    [Crossref]
  28. R. Mar-Sarao and H. Moya-Cessa, “Optical realization of a quantum beam splitter,” Opt. Lett. 33, 1966–1968 (2008).
    [Crossref]
  29. J. Jaguś, J. K. Jabczyński, W. Żendzian, and J. Kwiatkowski, “Application of Wigner transform for characterization of aberrated laser beams,” Opt. Appl. 35, 33–41 (2005).
  30. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [Crossref]
  31. K.-H. Brenner and A. Lohmann, “The Wigner distribution function display of complex 1D signals,” Opt. Comm. 42, 310–314 (1982).
    [Crossref]
  32. E. Mukamel, K. Banaszek, I. A. Walmsley, and C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317–1319 (2003).
    [Crossref]
  33. B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
    [Crossref]
  34. G. S. Agarwal, Quantum Optics (Cambridge University, 2013).
  35. U. Leonhardt, “Quantum statistics of a lossless beam splitter: SU(2) symmetry in phase space,” Phys. Rev. A 48, 3265–3277 (1993).
    [Crossref]
  36. P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, “Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques,” Adv. Phys. 57, 363–455 (2008).
    [Crossref]
  37. A. Polkovnikov, “Phase space representation of quantum dynamics,” Ann. Phys. 325, 1790–1852 (2010).
    [Crossref]
  38. A. L. Rivera, S. M. Chumakov, and K. B. Wolf, “Hamiltonian foundation of geometrical anisotropic optics,” J. Opt. Soc. Am. A 12, 1380–1389 (1995).
    [Crossref]

2013 (2)

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013).
[Crossref]

G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
[Crossref]

2011 (1)

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref]

2010 (2)

T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[Crossref]

A. Polkovnikov, “Phase space representation of quantum dynamics,” Ann. Phys. 325, 1790–1852 (2010).
[Crossref]

2008 (3)

S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature 455, 510–514 (2008).
[Crossref]

R. Mar-Sarao and H. Moya-Cessa, “Optical realization of a quantum beam splitter,” Opt. Lett. 33, 1966–1968 (2008).
[Crossref]

P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, “Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques,” Adv. Phys. 57, 363–455 (2008).
[Crossref]

2006 (1)

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science (New York) 312, 83–86 (2006).
[Crossref]

2005 (4)

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
[Crossref]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
[Crossref]

J. Jaguś, J. K. Jabczyński, W. Żendzian, and J. Kwiatkowski, “Application of Wigner transform for characterization of aberrated laser beams,” Opt. Appl. 35, 33–41 (2005).

O. Steuernagel, “Equivalence between focused paraxial beams and the quantum harmonic oscillator,” Am. J. Phys. 73, 625–629 (2005).
[Crossref]

2004 (2)

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[Crossref]

A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, “Schrödinger cats and their power for quantum information processing,” J. Opt. B Quantum Semiclass. Opt. 6, S828–S833 (2004).
[Crossref]

2003 (3)

T. Ralph, A. Gilchrist, G. Milburn, W. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[Crossref]

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715–775 (2003).
[Crossref]

E. Mukamel, K. Banaszek, I. A. Walmsley, and C. Dorrer, “Direct measurement of the spatial Wigner function with area-integrated detection,” Opt. Lett. 28, 1317–1319 (2003).
[Crossref]

2002 (1)

D. Dragoman, “Phase space correspondence between classical optics and quantum mechanics,” Prog. Opt. 43, 433–496 (2002).
[Crossref]

2001 (2)

M. Man’ko, V. Man’ko, and R. V. Mendes, “Quantum computation by quantumlike systems,” Phys. Lett. A 288, 132–138 (2001).
[Crossref]

D. Dragoman and M. Dragoman, “On the similarities between the Wigner distribution function in classical and quantum optics,” Optik 112, 497–501 (2001).
[Crossref]

1999 (1)

S. Lloyd and S. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[Crossref]

1998 (1)

K. Wódkiewicz and G. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815–821 (1998).
[Crossref]

1997 (1)

D. Vitali, P. Tombesi, and G. Milburn, “Protecting Schrödinger cat states using feedback,” J. Mod. Opt. 44, 2033–2041 (1997).
[Crossref]

1996 (3)

P. Goetsch, P. Tombesi, and D. Vitali, “Effect of feedback on the decoherence of a Schrödinger-cat state: a quantum trajectory description,” Phys. Rev. A 54, 4519–4527 (1996).
[Crossref]

M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. Raimond, and S. Haroche, “Observing the progressive decoherence of the “meter” in a quantum measurement,” Phys. Rev. Lett. 77, 4887–4890 (1996).
[Crossref]

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A “Schrödinger cat” superposition state of an atom,” Science 272, 1131–1136 (1996).
[Crossref]

1995 (2)

1993 (2)

U. Leonhardt, “Quantum statistics of a lossless beam splitter: SU(2) symmetry in phase space,” Phys. Rev. A 48, 3265–3277 (1993).
[Crossref]

G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[Crossref]

1986 (1)

1985 (1)

D. F. Walls and G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985).
[Crossref]

1982 (1)

K.-H. Brenner and A. Lohmann, “The Wigner distribution function display of complex 1D signals,” Opt. Comm. 42, 310–314 (1982).
[Crossref]

1969 (1)

1935 (1)

E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–812 (1935).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, Quantum Optics (Cambridge University, 2013).

Allen, L.

G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[Crossref]

Ballagh, R. J.

P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, “Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques,” Adv. Phys. 57, 363–455 (2008).
[Crossref]

Banaszek, K.

Barreiro, J. T.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
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S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature 455, 510–514 (2008).
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Chwalla, M.

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Clement, T. S.

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P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, “Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques,” Adv. Phys. 57, 363–455 (2008).
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Dotsenko, I.

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J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
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P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, “Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques,” Adv. Phys. 57, 363–455 (2008).
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G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
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G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
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T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
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A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, “Schrödinger cats and their power for quantum information processing,” J. Opt. B Quantum Semiclass. Opt. 6, S828–S833 (2004).
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T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
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T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
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M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. Raimond, and S. Haroche, “Observing the progressive decoherence of the “meter” in a quantum measurement,” Phys. Rev. Lett. 77, 4887–4890 (1996).
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T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
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D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
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D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
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King, B. E.

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B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013).
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T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
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J. Jaguś, J. K. Jabczyński, W. Żendzian, and J. Kwiatkowski, “Application of Wigner transform for characterization of aberrated laser beams,” Opt. Appl. 35, 33–41 (2005).

Langer, C.

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science (New York) 312, 83–86 (2006).
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B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013).
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G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
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D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
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K.-H. Brenner and A. Lohmann, “The Wigner distribution function display of complex 1D signals,” Opt. Comm. 42, 310–314 (1982).
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M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. Raimond, and S. Haroche, “Observing the progressive decoherence of the “meter” in a quantum measurement,” Phys. Rev. Lett. 77, 4887–4890 (1996).
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T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
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T. Ralph, A. Gilchrist, G. Milburn, W. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
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A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, “Schrödinger cats and their power for quantum information processing,” J. Opt. B Quantum Semiclass. Opt. 6, S828–S833 (2004).
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T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
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B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013).
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A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, “Schrödinger cats and their power for quantum information processing,” J. Opt. B Quantum Semiclass. Opt. 6, S828–S833 (2004).
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T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref]

Nigg, S. E.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013).
[Crossref]

G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
[Crossref]

Ourjoumtsev, A.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science (New York) 312, 83–86 (2006).
[Crossref]

Ozeri, R.

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
[Crossref]

Paik, H.

G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
[Crossref]

Polkovnikov, A.

A. Polkovnikov, “Phase space representation of quantum dynamics,” Ann. Phys. 325, 1790–1852 (2010).
[Crossref]

Raimond, J.

M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. Raimond, and S. Haroche, “Observing the progressive decoherence of the “meter” in a quantum measurement,” Phys. Rev. Lett. 77, 4887–4890 (1996).
[Crossref]

Raimond, J.-M.

S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature 455, 510–514 (2008).
[Crossref]

Ralph, T.

T. Ralph, A. Gilchrist, G. Milburn, W. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[Crossref]

Ralph, T. C.

A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, “Schrödinger cats and their power for quantum information processing,” J. Opt. B Quantum Semiclass. Opt. 6, S828–S833 (2004).
[Crossref]

Raymer, M. G.

Reichle, R.

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
[Crossref]

Rivera, A. L.

Sayrin, C.

S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature 455, 510–514 (2008).
[Crossref]

Schindler, P.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref]

Schoelkopf, R. J.

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013).
[Crossref]

G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
[Crossref]

Schrödinger, E.

E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–812 (1935).
[Crossref]

Seidelin, S.

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
[Crossref]

Si, Z.

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[Crossref]

Smith, B. J.

Steuernagel, O.

O. Steuernagel, “Equivalence between focused paraxial beams and the quantum harmonic oscillator,” Am. J. Phys. 73, 625–629 (2005).
[Crossref]

Tang, S.

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[Crossref]

Tombesi, P.

D. Vitali, P. Tombesi, and G. Milburn, “Protecting Schrödinger cat states using feedback,” J. Mod. Opt. 44, 2033–2041 (1997).
[Crossref]

P. Goetsch, P. Tombesi, and D. Vitali, “Effect of feedback on the decoherence of a Schrödinger-cat state: a quantum trajectory description,” Phys. Rev. A 54, 4519–4527 (1996).
[Crossref]

Tualle-Brouri, R.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science (New York) 312, 83–86 (2006).
[Crossref]

Vitali, D.

D. Vitali, P. Tombesi, and G. Milburn, “Protecting Schrödinger cat states using feedback,” J. Mod. Opt. 44, 2033–2041 (1997).
[Crossref]

P. Goetsch, P. Tombesi, and D. Vitali, “Effect of feedback on the decoherence of a Schrödinger-cat state: a quantum trajectory description,” Phys. Rev. A 54, 4519–4527 (1996).
[Crossref]

Vlastakis, B.

G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
[Crossref]

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013).
[Crossref]

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D. F. Walls and G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985).
[Crossref]

Walmsley, I. A.

Wineland, D. J.

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
[Crossref]

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A “Schrödinger cat” superposition state of an atom,” Science 272, 1131–1136 (1996).
[Crossref]

Wódkiewicz, K.

K. Wódkiewicz and G. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815–821 (1998).
[Crossref]

Wolf, K. B.

Wunderlich, C.

M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. Raimond, and S. Haroche, “Observing the progressive decoherence of the “meter” in a quantum measurement,” Phys. Rev. Lett. 77, 4887–4890 (1996).
[Crossref]

Zendzian, W.

J. Jaguś, J. K. Jabczyński, W. Żendzian, and J. Kwiatkowski, “Application of Wigner transform for characterization of aberrated laser beams,” Opt. Appl. 35, 33–41 (2005).

Zurek, W. H.

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715–775 (2003).
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Adv. Phys. (1)

P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, “Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques,” Adv. Phys. 57, 363–455 (2008).
[Crossref]

Am. J. Phys. (1)

O. Steuernagel, “Equivalence between focused paraxial beams and the quantum harmonic oscillator,” Am. J. Phys. 73, 625–629 (2005).
[Crossref]

Ann. Phys. (1)

A. Polkovnikov, “Phase space representation of quantum dynamics,” Ann. Phys. 325, 1790–1852 (2010).
[Crossref]

J. Mod. Opt. (1)

D. Vitali, P. Tombesi, and G. Milburn, “Protecting Schrödinger cat states using feedback,” J. Mod. Opt. 44, 2033–2041 (1997).
[Crossref]

J. Opt. B Quantum Semiclass. Opt. (1)

A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn, “Schrödinger cats and their power for quantum information processing,” J. Opt. B Quantum Semiclass. Opt. 6, S828–S833 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nature (3)

S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature 455, 510–514 (2008).
[Crossref]

G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495, 205–209 (2013).
[Crossref]

D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of a six-atom ‘Schrödinger cat’ state,” Nature 438, 639–642 (2005).
[Crossref]

Naturwissenschaften (1)

E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–812 (1935).
[Crossref]

Opt. Appl. (1)

J. Jaguś, J. K. Jabczyński, W. Żendzian, and J. Kwiatkowski, “Application of Wigner transform for characterization of aberrated laser beams,” Opt. Appl. 35, 33–41 (2005).

Opt. Comm. (1)

K.-H. Brenner and A. Lohmann, “The Wigner distribution function display of complex 1D signals,” Opt. Comm. 42, 310–314 (1982).
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Opt. Lett. (4)

Optik (1)

D. Dragoman and M. Dragoman, “On the similarities between the Wigner distribution function in classical and quantum optics,” Optik 112, 497–501 (2001).
[Crossref]

Phys. Lett. A (1)

M. Man’ko, V. Man’ko, and R. V. Mendes, “Quantum computation by quantumlike systems,” Phys. Lett. A 288, 132–138 (2001).
[Crossref]

Phys. Rev. A (8)

P. Goetsch, P. Tombesi, and D. Vitali, “Effect of feedback on the decoherence of a Schrödinger-cat state: a quantum trajectory description,” Phys. Rev. A 54, 4519–4527 (1996).
[Crossref]

G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993).
[Crossref]

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[Crossref]

K. Wódkiewicz and G. Herling, “Classical and nonclassical interference,” Phys. Rev. A 57, 815–821 (1998).
[Crossref]

U. Leonhardt, “Quantum statistics of a lossless beam splitter: SU(2) symmetry in phase space,” Phys. Rev. A 48, 3265–3277 (1993).
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T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J. Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill, “Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum,” Phys. Rev. A 82, 031802 (2010).
[Crossref]

D. F. Walls and G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985).
[Crossref]

T. Ralph, A. Gilchrist, G. Milburn, W. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[Crossref]

Phys. Rev. Lett. (3)

S. Lloyd and S. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999).
[Crossref]

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref]

M. Brune, E. Hagley, J. Dreyer, X. Maître, A. Maali, C. Wunderlich, J. Raimond, and S. Haroche, “Observing the progressive decoherence of the “meter” in a quantum measurement,” Phys. Rev. Lett. 77, 4887–4890 (1996).
[Crossref]

Prog. Opt. (1)

D. Dragoman, “Phase space correspondence between classical optics and quantum mechanics,” Prog. Opt. 43, 433–496 (2002).
[Crossref]

Rev. Mod. Phys. (1)

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715–775 (2003).
[Crossref]

Science (1)

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A “Schrödinger cat” superposition state of an atom,” Science 272, 1131–1136 (1996).
[Crossref]

Science (New York) (2)

B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-photon Schrödinger cat states,” Science (New York) 342, 607–610 (2013).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science (New York) 312, 83–86 (2006).
[Crossref]

Other (2)

U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).

G. S. Agarwal, Quantum Optics (Cambridge University, 2013).

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Figures (5)

Fig. 1.
Fig. 1. Model of decoherence caused by photon loss. An SCS is mixed with a vacuum state using a beam splitter (BS). The vacuum state and an odd SCS are expressed using the coherent state basis | α as | B = | 0 and | A = | α | α . Photon loss at the finite transmission rate of optical components degrades the quality of the SCS.
Fig. 2.
Fig. 2. Experimental apparatus. A beam with a spatial WF for the vertical mode, which acts like a vacuum state, and a spatial WF for the horizontal mode, which acts like an SCS, is created using a MZI. A Dove prism induces a beam splitter-like interaction between these modes. The 1D intensity distributions, { P i ( ϕ ) ( x ) } , were measured to reconstruct the spatial WF.
Fig. 3.
Fig. 3. Experimental results of the intensity measurements. (a) Intensity distribution after the π / 2 rotation, P i ( π / 2 ) . (b-1, b-2, and b-3) Reconstructed WFs, F i ( π / 2 ) . (c-1, c-2, and c-3) Observed intensity distributions, P i ( π / 2 ) (red bar), and the distributions calculated from the reconstructed WFs, F i ( π / 2 ) d u (blue line).
Fig. 4.
Fig. 4. Observed intensity distributions, { P i ( ϕ ) } , with an image rotation angle of (a-1)  ϕ = 0 ( T = 1.00 ), (a-2)  ϕ = 13 ° ( T = 0.95 ), and (a-3)  ϕ = 18 ° ( T = 0.90 ). (b-1), (b-2), and (b-3) denote the corresponding WFs to (a-1), (a-2), and (a-3), respectively, on the z = 0 mm planes, F 0 ( ϕ ) .
Fig. 5.
Fig. 5. Purity with respect to transmittance obtained through experiments (closed circle) and theory (solid line).

Equations (68)

Equations on this page are rendered with MathJax. Learn more.

W i ( x , p ¯ u ; y , p ¯ v ) = δ ( x q A ) δ ( p ¯ u p A ) δ ( y q B ) δ ( p ¯ v p A ) i ,
f i d q A d p A d q B d p B f W i ( q A , p A ; q B , p B ) .
I i ( x , y ) = δ ( x q A ) δ ( y q B ) i ,
F i ( x , p ¯ u ) = δ ( x q A ) δ ( p ¯ u p A ) i ,
P i ( x ) = δ ( x q A ) i .
W t = k = A , B [ W q k H p k W p k H q k ] ,
H free = c p ¯ [ 1 + p A 2 + p B 2 2 p ¯ 2 ] ,
P i ( x ) = p ¯ F 0 ( x ( z i z 0 ) u , p ¯ u ) d u .
H rot = Ω ( q B p A q A p B ) ,
W i ( ϕ ) ( q A , p A ; q B , p B ) = W i ( q A τ q B ϱ , p A τ p B ϱ ; q A ϱ + q B τ , p A ϱ + p B τ ) ,
T = ( cos ϕ ) 2 .
a q ¯ p ¯ 2 [ q A q ¯ + i p A p ¯ ] , b q ¯ p ¯ 2 [ q B q ¯ + i p B p ¯ ] ,
Π ( ϕ ) 2 π p ¯ 2 k [ F 0 ( ϕ ) ( x , p ¯ u ) ] 2 d x d u ,
W 0 ( x , p ¯ u ; y , p ¯ v ) = S ( x , p ¯ u ) G ( y , p ¯ v ) ,
S ( x , p ¯ u ) = G ( x d , p ¯ u ) + G ( x + d , p ¯ u ) + 2 G ( x , p ¯ u ) cos ( 2 k d u + θ ) .
G ( x , p ¯ u ) = exp [ 2 w 0 2 x 2 k 2 w 0 2 2 u 2 ] .
( Re [ b ] ) 2 0 = p ¯ w 0 2 8 q ¯ ,
( Im [ b ] ) 2 0 = q ¯ p ¯ 2 k 2 w 0 2 .
q ¯ = k w 0 2 2
E ( x , t ) = E 0 { Ψ ( x ) exp [ i ( k z ω t ) ] + c.c. } / 2 ,
i Ψ z = L Ψ ,
L 1 2 k [ 2 x 2 + 2 y 2 ] + k c p ¯ V ( x , y ) .
V ( x , y ) c p ¯ 2 [ 1 ω 2 c 2 k 2 { n ( x , y , z ) } 2 ] .
Γ i ( x + , y + ; x , y ) j λ j ψ j ( i ) ( x + , y + ) { ψ j ( i ) ( x , y ) } * ,
ψ j ( i ) ( x , y ) = exp [ i ( z i z 0 ) L ] ψ j ( x , y ) .
Γ z = i ( L + L ) Γ ,
L ± 1 2 k [ 2 x ± 2 + 2 y ± 2 ] + k c p ¯ V ( x ± , y ± ) .
W i ( x , p ¯ u ; y , p ¯ v ) k 2 ( 2 π p ¯ ) 2 d r d s e i k ( r u + s v ) × Γ i ( x + r / 2 , y + s / 2 ; x r / 2 , y s / 2 ) .
p ¯ = k .
Γ i ( x + , y + ; x , y ) x + , y + | ρ ^ A B ( i ) | x , y .
Δ 2 l m l Δ , m Δ | ρ ^ A B ( i ) | l Δ , m Δ = 1
I i = x , y | ρ ^ A B ( i ) | x , y ,
F i = 1 2 π d r e i k r u x + r / 2 | ρ ^ A ( i ) | x r / 2 ,
P i = x | ρ ^ A ( i ) | x ,
x + | ρ ^ A ( i ) | x = d q B x + , q B | ρ ^ A B ( i ) | x , q B
= Δ m x + , m Δ | ρ ^ A B ( i ) | x , m Δ .
Π i = 2 π p ¯ 2 k d x d u { F i } 2
= k 2 π d x d u d r d s e i k ( s r ) u × x + r / 2 | ρ ^ A ( i ) | x r / 2 x s / 2 | ρ ^ A ( i ) | x + s / 2
= d x d r x + r / 2 | ρ ^ A ( i ) | x r / 2 × x r / 2 | ρ ^ A ( i ) | x + r / 2
= Δ 2 l m l Δ | ρ ^ A ( i ) | m Δ m Δ | ρ ^ A ( i ) | l Δ = Δ l l Δ | { ρ ^ A ( i ) } 2 | l Δ .
x , y | q ^ A | j x ψ j ( x , y ) ,
x , y | q ^ B | j y ψ j ( x , y ) ,
x , y | p ^ A | j i x ψ j ( x , y ) ,
x , y | p ^ B | j i y ψ j ( x , y ) ,
A = x w 0 + w 0 2 x ,
B = y w 0 + w 0 2 y .
Ψ G ( x , y , z ) exp [ i k x 2 + y 2 2 ( z z 0 ) i k w 0 2 ] ,
Ψ 1 ( x , y , z ) = e i θ 1 Ψ G ( x d , y , z ) , Ψ 2 ( x , y , z ) = e i θ 2 Ψ G ( x + d , y , z ) ,
Ψ c ( x , y , z ) = Ψ 1 ( x , y , z ) + e i θ Ψ 2 ( x , y , z ) .
H = H free + V ( q A , q B )
V ( x , y ) = [ 1 + ( x q A ) q A + ( y q B ) q B + ( x q A ) 2 2 2 q A 2 + ] V ( q A , q B ) ,
Γ z = i 2 k ( 2 x + 2 2 x 2 + 2 y + 2 2 y 2 ) Γ i k c p ¯ [ V q A ( x + x ) + V q B ( y + y ) + 2 V q A 2 ( x + + x 2 q A ) ( x + x ) + ] Γ .
Γ i ( x + , y + ; x , y ) = p ¯ 2 d u d v e i k ( r u + s v ) W i ( q A , p ¯ u ; q B , p ¯ v ) ,
d r d s e i k ( r p A + s p B ) / p ¯ ( 2 x ± 2 ) Γ i ( x + , y + ; x , y ) = ( 1 2 q A ± i k p ¯ p A ) 2 W i ( q A , p A ; q B , p B ) ,
d r d s e i k ( r p A + s p B ) / p ¯ ( x ± ) Γ i ( x + , y + ; x , y ) = ( q A p ¯ 2 i k p A ) W i ( q A , p A ; q B , p B ) .
W z = k = A , B [ p k p ¯ W q k + 1 c V q k W p k ] ,
W t = c p ¯ [ p A q A + p B q B ] W .
W i ( q A , p A ; q B , p B ) = W 0 ( q A c t p A / p ¯ , p A ; q B c t p B / p ¯ , p B ) .
F i ( x , p ¯ u ) = F 0 ( x ( z i z 0 ) u , p ¯ u ) .
P i ( x ) = p ¯ F i ( x , p ¯ u ) d u .
ψ j ( i ) ( x , y ) = x , y | exp [ i ( t / ) H ^ ] | j ,
x , y | exp [ i ϕ Ω H ^ rot ] | j = x , y | exp [ i ϕ ( q ^ B p ^ A p ^ A q ^ B ) ] | j
= exp [ ϕ ( y x + x y ) ] ψ j ( x , y )
= ψ j ( x cos ϕ y sin ϕ , x sin ϕ + y cos ϕ )
= x τ y ϱ , x ϱ + y τ | j ,
Γ i ( ϕ ) ( x + , y + ; x , y ) x + , y + | U ^ rot ρ ^ A B ( i ) U ^ rot | x , y
= x + τ y + ϱ , x + ϱ + y + τ | ρ ^ A B ( i ) | x τ y ϱ , x ϱ + y τ
= Γ i ( x + τ y + ϱ , x + ϱ + y + τ ; x τ y ϱ , x ϱ + y τ ) ,

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