Quantum carpets: a tool to observe decoherence

Quantum carpets—the spatio-temporal de Broglie density profiles—woven by an atom or an electron in the near-field region of a diffraction grating bring to light, in real time, the decoherence of each individual component of the interference term of the Wigner function characteristic of superposition states. The proposed experiments are feasible with present-day technology.

Although in experiments involving a cavity field [10][11][12], superconducting quantum circuits [13], an ion stored in a Paul trap [14], C 70 fullerenes [15] and a neutron in an interferometer [16] one has observed certain features of the decoherence of Fock states or of the superposition of two coherent states [17], the quest [18] to record the decay of the total Wigner function [19] of such non-classical states remains an important issue in studies in quantum mechanics. In the present work, we show that the space-time structures in the position probability distribution of a de Broglie wave after a diffraction grating, i.e. quantum carpets [20], allow us to observe the decay of each individual component of the Wigner function in real time.
Under appropriate conditions the quantum mechanical probability density of a nonrelativistic particle displays [20], when represented in space-time, characteristic valleys and ridges as shown in figure 1. These striking patterns have led to the name quantum carpet. Its design is most pronounced when the energy spectrum, determining the propagation of the particle, produces a strong degeneracy of intermode traces [21], as is the case for the particle caught in a box [22] or diffracted from an infinite periodic grating. The latter situation is closely related to the Talbot effect [20,22]. Moreover, some features, such as the full and fractional revivals, also persist in the presence of small relativistic corrections [23] or even in a fully relativistic case [24], described by the Dirac equation. The ridges and valleys do not follow classical trajectories but the world lines created by interference. Consequently, the design of the carpet is purely an interference effect and is, therefore, sensitive to decoherence.
We show that a quantum carpet in the presence of decoherence allows us to observe the decay of each momentum component of the interference Wigner contribution. Our proposal relies on three guiding principles: (i) a quantum carpet, which is the position probability density P(x, t) to find the particle at the transverse position x at a distance z = v z t after a diffraction Quantum carpets in the absence of decoherence represented by the position probability density P(x, t) of a beam of atoms or electrons after it has been diffracted from a grating aligned along the x-axis. To enhance the pattern visibility, the plot is in the logarithmic scale. Bright or dark colours represent high or low probability densities as indicated by the thermometer located on the right-hand side of the figure. Time translates into distance z = v z t from the grating by the macroscopic velocity v z of the beam orthogonal to the x-axis. Here we have chosen an initial Gaussian wave packet centred inx = L/2, having an average momentump = 10h/L and width x = 0.03L. grating, is given by an incoherent sum of the Wigner functions at the grating [22], (ii) a diffraction grating can create a superposition state and (iii) de Broglie waves based on atoms and electrons provide a variety of mechanisms of decoherence [25].
We consider two types of carpets corresponding to two distinctly different mechanisms of decoherence: a carpet woven by (i) a two-level atom that can spontaneously go to its ground state and thereby receives a single random momentum kick, and (ii) an electron that passes close to a metal plate and thereby gets damped by its image charge and undergoes Brownian motion. In both cases the design of the carpet gets washed out as the distance to the grating increases.
Present-day technology even allows us to implement our proposal. Indeed, quantum carpets for several types of de Broglie waves [26][27][28][29][30][31] and of light [32] have been observed. Moreover, the influence of a single spontaneously emitted photon on the far-field interference pattern of an atomic wave has been measured [33]. Likewise, the field of electron optics [34,35] has made remarkable progress and a double-slit experiment in the presence of a metal plate has already been carried out [36] at the University of Tübingen. Hence, the proposed observation of the space-time patterns in the probability density can serve as a flexible tool to monitor decoherence in a variety of experimental settings and conditions. A deeper understanding of decoherence not only is of interest to investigate the quantum to classical transition, but also has practical implications for the development of reliable quantum technologies.
This paper is organized as follows. In section 2 we connect the design of a quantum carpet to the individual components of the initial Wigner function. We then, in section 3, consider two different mechanisms of decoherence and study their effect on the space-time structures of the probability density. Finally, we summarize and discuss our results in section 4. For technical details of the derivation of the Wigner function produced by a periodic array of wave functions, see the appendix.

Quantum carpets woven by Wigner functions
We start our discussion by connecting the design of a carpet to the individual components of the initial Wigner function. For this purpose, we study the diffraction of a non-relativistic quantum mechanical particle from a grating and emphasize that many such experiments have been performed with atoms [26][27][28], molecules [8], electrons [29][30][31]36], neutrons [16] or light [32].

Carpet represented as a superposition of Wigner function components
Here we express the probability density in space-time forming the quantum carpet as an infinite superposition of Wigner functions corresponding to an array of Schrödinger cat states. This state corresponds to the initial state. The time dependence enters as a displacement of the original Wigner function along straight lines in space-time.
A mechanical or optical grating of period 2L produces a periodic array of wave functions. Here x denotes the coordinate along the grating and φ = φ(x) is the wave packet created by a single slit. When we substitute this wave function into the definition [19] W (x, p; t) ≡ 1 2πh of the Wigner function, we find in view of the results obtained in the appendix that the Wigner function of such an array with period 2L consists of a periodic array of Wigner functions separated by L.
Due to the spatial periodicity, the momentum is discrete. We also note, in (3), the presence of the alternating weight factor (−1) n·l for the individual contributions of the Wigner function.
After the grating the particle evolves freely as expressed by the Liouville equation [19] L W ψ (x, p; t) = 0, whereL denotes the Liouville operator of free motion for a particle of mass M. It is easy to verify that the expression in terms of the initial Wigner function is the solution of the Liouville equation (6).
With the help of expression (3), we find that with the straight space-time trajectories The position distribution P(x, t) at a distance z ≡ v z t from the grating forms the carpet. Here v z is the velocity component orthogonal to the grating. Since definition (2) of the Wigner function implies the identity we find P(x, t) from (9) by integration over p, which yields Hence, the cut of the Wigner distribution along the x-axis for a fixed momentum p n moves along the straight space-time trajectory χ n,l with a fixed speed determined by p n . Thus as time increases, the individual cuts of the Wigner distribution at neighbouring momenta separate. We can distinguish them when the size of the Wigner structure is smaller than the separation between two of them. The fastest separation occurs for almost vertical space-time trajectories, that is, for small momenta contained in the Wigner function of the original state. Small momenta are usually a characteristic feature of superposition states. A consequence of momentum quantization is the periodic revival of the probability density P(x, t). Indeed, there exists a revival time T 1 such that P(x, t + T 1 ) = P(x, t). This rephasing takes place when all components of the Wigner functions W φ [χ n,l (x, t), p n ] have travelled a transverse distance, which is an integer multiple of the period 2L of the grating. Therefore, the revival time is set by the smallest momentum component p 1 ≡ πh/(2L), that is, by

Schrödinger cats
Next we recall [17] the Wigner function corresponding to the anti-symmetric superposition built from the wave packet ϕ = ϕ(x). Indeed, when we substitute this superposition into definition (2) of the Wigner function we obtain, owing to the bilinearity of W in φ, the expression where represents the Wigner function of the basis packet, and the interference term W int is given by Hence, the Wigner function of the superposition state (14) is the sum not only of the Wigner functions of the basis packet and its phase space inversion, that is, of W ϕ (x, p) and W ϕ (−x, − p), but also contains the term W int . It is this interference term which is extremely sensitive to decoherence. The elementary example of a Gaussian wave function yields the Gaussian Wigner distribution centred aroundx and momentump. Here the widths x and p ≡h/ x are measures of the extension of the quantum state in phase space indicating when the Gaussian has decayed to 1/e. In contrast, the interference term is always located at the origin of phase space independent of the initial positionx and momentump. Moreover, it assumes negative values whereas W ϕ is everywhere positive.

Quantum carpet for a Schrödinger cat
Indeed, for the example of the superposition state φ with a Gaussian wave packet ϕ of a macroscopic average momentump p 1 , we find a clear separation into momenta aroundp giving rise to rather flat trajectories in space-time and small momenta creating steep world lines. The latter result from the interference term W int . Hence, W int translates itself into the design of the carpet.
But how to create a superposition state such as φ from a grating? Many possibilities offer themselves. In the case of atoms [19], we can use an optical grating which automatically creates the desired superposition. In the case of a mechanical grating [21], we can send two de Broglie waves under an appropriate angle onto the grating. In figure 1 we show a quantum carpet woven by a particle prepared in the superposition state defined by (14) and (18). Due to the parity of the initial wave function, it suffices to study the probability density in the space region from 0 to L and up to the revival time T rev = T 1 /2.

Quantum carpets in the presence of decoherence
We now derive expressions for quantum carpets in the presence of decoherence. Here we consider two quantum systems with different mechanisms of decoherence: (i) a two-level atom undergoing spontaneous emission and (ii) an electron moving over a metal surface creating image charges.

Decoherence due to spontaneous emission
Our discussion starts with a two-level atom moving under the influence of spontaneous emission. The Wigner functions W e and W g corresponding, respectively, to atoms in the excited and ground states satisfy the generalized optical Bloch equations, which describe the coupled dynamics of internal and external atomic degrees of freedom [37] L W e (x, p; t) = −γ W e (x, p; t) (21) andL where γ is the rate of spontaneous emission andL denotes the Liouville operator of free motion defined by (7). The term in (21) proportional to γ describes the relaxation due to spontaneous emission. Moreover, the term at the right-hand side of (22) describes the transfer of population from the excited to the ground state by spontaneous emission.
reflects the momentum component along the x-axis. When the atoms exit the grating at time t = 0 in the excited state, the solutions of these equations read and with the displacement in position during the decay time 1/γ due to the recoil of the atom giving it a velocityhk/M. A comparison between expressions (24) and (8) for the Wigner functions W e (x, p; t) and W (x, p; t) in the excited state and the standard one shows that the influence of decoherence is just a multiplication of W by the decay factor exp(−γ t). As a result, we find the quantum carpet formed by the atoms in the excited state to be the original quantum carpet, (12), in the absence of decoherence. However, its intensity decays as a function of time, that is, separation from the grating.
Next we turn to the carpet woven by the ground state atoms. For this purpose, we substitute the initial Wigner function (3) into (25) and obtain the formula which after integration over momentum with the help of the delta function yields the expression Hence, while the intensity of the carpet formed by the excited atoms fades away, the carpet formed by the atoms in the ground state builds up in intensity. However, we now have to average the original Wigner function W φ over the displacement δx s due to the spontaneously emitted photon, that is, The averaged Wigner function W (s) φ depends explicitly on time. In particular, after a specific time the characteristic structures of the Wigner function are averaged out.
In figure 2 we illustrate these features by showing the quantum carpets for a beam of two-level atoms, initially prepared in the excited state. The left column displays the same ideal quantum carpet in the absence of decoherence (γ = 0), which we use as a benchmark and a guide to the eye. In the other columns, we gradually increase the rate of spontaneous emission from γ = 1 (centre) to γ = 5 (right). In all our numerical calculations, we seth = 1, L = 1 and M = 1. Moreover, we choose the photon momentumhk = 1 and use a constant distribution f s = 1/2.
In the top and bottom rows, we show the carpets corresponding to the excited and ground states, respectively. We clearly observe the exponential damping of the probability density P (s) e to find the atom in the excited state (top sequence) indicated by (27). Indeed, the design of the quantum carpet remains identical to the case without decoherence (left), but the contrast in the space-time structures becomes fainter as the atom moves away from the grating. Decoherence of an atomic quantum carpet. Position probability density P(x, t) of a beam of atoms after it has been diffracted from a grating aligned along the x-axis. To enhance the pattern visibility, the plots are in the logarithmic scale. Bright or dark colours represent high or low probability densities as indicated by the thermometer located on the right-hand side of the left figure. Time translates into distance z = v z t from the grating by the macroscopic velocity v z of the beam orthogonal to the x-axis. The left column displays the ideal quantum carpet in the absence of decoherence (γ = 0). This plot provides us with a benchmark against which we compare the quantum carpets for increasing values of the damping rate: γ = 1 (centre) and γ = 5 (right). Top: probability density P (s) e (x, t) to find the atom in the excited state given by (27). Bottom: probability density P (s) g (x, t) to find the atom in the ground state determined by (29) and (30). Here we have chosen an initial Gaussian wave packet centred inx = L/2, having an average momentump = 10h/L and a width x = 0.03L. At the same time, the probability P (s) g of finding the atom in the ground state given by (29) and (30) (bottom centre and right) builds up in intensity. However, the random recoil due to spontaneous emission produces a less distinct pattern in the corresponding quantum carpets. In this case, one can only appreciate the revival of the Gaussian wave packet at T rev /4, T rev /2, 3T rev /4 and T rev . The pattern of the quantum carpet appears blurred already after T rev /4.
In order to obtain the carpet for the excited state, we have directly used the expression for P (s) e , (27), and computed the series using a convergence tolerance of 10 −4 . We found that this tolerance was sufficient to bring out the structures in the quantum carpet.
To plot P (s) g , one could, in principle, use the analytical expression given by (29). However, in practice, we found the convergence of the series to be too slow. Thus, we numerically evaluate the expression that is, we interchange summations and integrations. We now first perform the summation over n and l and then we average over ε and τ .
Here we use an adaptive Simpson quadrature rule to calculate the integral and an error tolerance of 10 −4 for the convergence of the series in the integrand. We found that this procedure is sufficient to illustrate the effect of decoherence in the carpet structures.

Decoherence due to image charges
We now turn to the second model of decoherence: an electron propagates above a metal surface as it creates the quantum carpet. The image charge in the metal [38] leads to a damped Brownian motion of the electron described by the Langevin equation, i.e. a Heisenberg equation of motion for the position operatorx(t) of the particle in equilibrium with a linear passive heat bath. For the case of a free particle interacting with an Ohmic bath, this equation takes on the well-known form [39] Here andF denote the damping constant and the Gaussian quantum force, respectively. The explicit solution of the Langevin equation, (32), with the force operator allows us to analyse the corresponding quantum carpet in the presence of Brownian motion. In order to evaluate the expectation value exp[ikx(t)] with the time-dependent position operatorx(t), we recall that the noise operatorf (t) at time t commutes with the operatorsx(0) andp(0) of the electron at the initial time t = 0, when it starts interacting with its image charge. Hence, the average exp[ikx(t)] factors into one over the electron and the other over the reservoir, that is, We can perform the average of the symmetrically ordered operator pertaining to the electron using the Wigner function, (3), and arrive at which after integration over p(0) using the delta function and the new integration variable In the final step, we substitute this expression into the space-time distribution (35) for the carpet and find that where we have introduced y ≡ y − x and the space-time trajectory which is curved due to the damping. It is convenient to arrange the terms in the form with the averaged Wigner function and the time-dependent kernel of Brownian motion. Hence, the Brownian motion averages the initial Wigner function over a time-dependent kernel K and the characteristic interference features of the Wigner function disappear.
The similarities and differences between the carpets of spontaneous emission and Brownian motion stand out most clearly in the classical limit where K reduces to a Gaussian [5,39] with a time-dependent width and D ≡ k b T /(M ). Here, k b and T denote Boltzmann's constant and the temperature of the metal surface, respectively. At first glance, it might appear that this result is different from the corresponding results in [4]. However, the work by Ford et al [4] is concerned with 'entanglement at all times', in which case no divergences appear, even at zero temperature.
On the other hand, the work by Ford and O'Connell [5] is concerned with the solution of the exact master equation (which was shown to be equivalent to the solution of the Langevin equation for the initial value problem) and, in this case, it was shown that serious divergences arise for low bath temperatures. However, as stressed in the abstract of this paper, worthwhile results may be obtained for high temperatures but one must distinguish between two cases: (a) a particle at zero (or low) temperature which is suddenly coupled to a bath at high temperature and (b) a particle whose initial temperature is the same as the bath temperature (the case where the initial temperature and the bath temperature are both high but different was not analysed in detail but it follows closely the analysis for the previous case).
It is scenario (a) that is relevant in the present context, where we envisage an electron at a relatively low temperature being brought into the confines of a metal surface at a much higher temperature.
When we substitute the Gaussian kernel (44) into (42), we find the averaged Wigner function due to the Brownian motion, with the Gaussian weight function In figure 3 we show the probability density P (B) (x, t) to find the electron at a transverse position x and a distance v z t from the grating given by (41) and (46). The left column displays again the ideal quantum carpet in the absence of decoherence. We then increase the value of the damping rate from = 0.1 (middle) to = 0.5 (right). As a result the Brownian motion quickly destroys the space-time structures in P (B) (x, t) and after T rev /2 the design of the quantum carpet gets completely washed out.

Discussion
We conclude by comparing the quantum carpets of an atom undergoing spontaneous emission and an electron moving over a metal surface. According to (46) the averaged Wigner function W

Summary and outlook
In conclusion, we have analysed the influence of decoherence on quantum carpets woven by de Broglie waves. Spontaneous emission or Brownian motion, the sources of decoherence, manifest themselves directly in the destruction of the design of the carpet. In this way, we can observe in real time the decay of the interference terms of the Wigner function. We emphasize that the proposed technique to bring out the influence of decoherence is fundamentally different from the one used in the landmark experiments using photons in a cavity [10][11][12], an ion stored in a trap [14] or the circuit QED analogues [13]. Whereas they require the reconstruction [40,41] of the Wigner function from measurements, such as the occupation probabilities of atomic, phononic or photonic states, the present method of using a quantum carpet measures directly the parts of the Wigner function most susceptible to decoherence. Indeed, here we take advantage of three facts: (i) we can prepare a de Broglie wave in an array of superposition states corresponding to a Schrödinger cat by diffracting it from a grating, (ii) the free propagation of the matter wave after the grating forming the carpet is classical when described by the Wigner function of the array of superposition states and (iii) the design of the carpet is intimately connected to the shape of the interference terms in the Wigner function.
Indeed, the structures of the individual world lines with their valleys and crests crisscrossing the carpet are the Wigner function components of the interference term caused by the superposition state at a given low momentum. Since we consider an infinitely long periodic grating, these momenta are discrete and structure develops in the carpet only at discrete inclinations with respect to the axes defining space-time, that is, in the near-field diffraction pattern.
Decoherence attacks predominantly the interference terms of the Wigner function and thereby fills the valleys. As a result, the design of the carpet gets washed out. However, it is only due to the discreteness of the momenta and the time evolution of a free particle after the grating that these components separate in the form of a fan and we can observe the filling-up of each component.
Our proposal to use the design of a quantum carpet as a tool to observe decoherence is particularly suited for diffraction experiments such as the pioneering ones based on atoms [42], large molecules [6-8, 15, 25], light [32] or electrons [36]. Although most of these articles have analysed the apparent decoherence based on specific models, they have not made use of the direct connection between the near-field diffraction pattern defining the carpet and the Wigner function. However, it is in this way that we gain deeper insight into the nature of decoherence and, in particular, in crossing the border between the microscopic and the macroscopic world. where we have defined the Wigner function