Notes on oldest jump unravelling of spatial decoherence master equation

TL;DR

This paper investigates the oldest known jump unraveling of the spatial decoherence quantum master equation, demonstrating its localization properties and comparing it to more recent diffusive approaches.

Contribution

It provides a rigorous proof that the orthojump unraveling localizes the quantum state, extending understanding of different stochastic unravellings of the master equation.

Findings

Orthojump unraveling localizes the quantum state.

It is analytically tractable despite requiring numerical methods.

Comparison with diffusive unravellings highlights similarities in localization.

Abstract

Solution of free particle quantum master equation with spatial decoherence can be unravelled into stochastic quantum trajectories in many ways. The first example, published in 1985, proposed a piecewise deterministic jump process for the wave function. While alternative unravellings, diffusive ones in particular, proved to be tractable analytically, the jump process 1985, also called orthojump, allows for few analytic results, it needs numeric methods as well. Here we prove that, similarly to diffusive unravellings, it is localizing the quantum state.

Equations

$$\frac{d\hat{\rho }}{dt}=-\frac{i}{\hslash }[\hat{H},\hat{\rho }]-\frac{D}{{\hslash }^2}[\hat{x},[\hat{x},\hat{\rho }]],$$

$$\frac{d\rho (x,p)}{dt}=-\frac{p}{m}{\partial }_x\rho (x,p)-D{\partial }_p^2\rho (x,p).$$

$$\mathbf{M}{\Psi }_t{\Psi }_t^{†}={\hat{\rho }}_t$$

$$\tilde{\Psi }=\exp \left(i⟨\hat{x}⟩\hat{p}-i⟨\hat{p}⟩\hat{x}\right){\Psi }_t,$$

$$\mathbf{M}{\tilde{\Psi }}_t{\tilde{\Psi }}_t^{†}={\hat{\tilde{\rho }}}_t.$$

$$d\Psi =-\frac{i}{\hslash }\hat{H}\Psi dt-\frac{D}{{\hslash }^2}(\hat{x}-⟨\hat{x}⟩{)}^2\Psi +\frac{\sqrt{2D}}{\hslash }(\hat{x}-⟨\hat{x}⟩)\Psi dW,$$

$${\tilde{\Psi }}_{\infty }(x)=\frac{1}{(2\pi {\sigma }_{\infty }^2{)}^{1/4}}\exp \left(-(1-i)\frac{x^2}{4{\sigma }_{\infty }^2}\right)$$

$${\sigma }_{\infty }^2=\sqrt{\frac{{\hslash }^3}{2Dm}}.$$

$${\hat{\tilde{\rho }}}_{\infty }={\tilde{\Psi }}_{\infty }{\tilde{\Psi }}_{\infty }^{†}.$$

$$(\Delta \tilde{x}{)}^2={\sigma }_{\infty }^2,(\Delta \tilde{p}{)}^2=\frac{{\hslash }^2}{2{\sigma }_{\infty }^2}.$$

$$d⟨\hat{x}⟩=\frac{1}{m}⟨\hat{p}⟩dt+\sqrt{\frac{2\hslash }{m}}dW,d⟨\hat{p}⟩=\sqrt{2D}dW.$$

$$d\Psi =-\frac{i}{\hslash }\hat{H}\Psi dt-\frac{D}{{\hslash }^2}[(\hat{x}-⟨\hat{x}⟩{)}^2-{\sigma }^2]\Psi dt+\left(\frac{x-⟨\hat{x}⟩}{\sigma }-1\right)\Psi dN,$$

$$\frac{d\Phi }{dt}=-\frac{i}{\hslash }\hat{H}\Phi -\frac{D}{{\hslash }^2}[(\hat{x}-⟨\hat{x}⟩{)}^2]\Phi .$$

$$\frac{d\Vert \Phi {\Vert }^2}{dt}=-\frac{2D}{{\hslash }^2}{\sigma }^2.$$

$$\Phi ⟶(\hat{x}-⟨\hat{x}⟩)\Phi ,$$

$$⟨x|{\hat{\rho }}_t|y⟩=\frac{1}{\sqrt{2\pi }\Sigma (t)}\exp \left\{-\frac{1}{8{\Sigma }^2(t)}(x+y{)}^2-\frac{1+2\sqrt{2}t+2t^2+\frac{2\sqrt{2}}{3}{t}^3+\frac{1}{3}{t}^4}{8{\Sigma }^2(t)}(x-y{)}^2-i\frac{1+\sqrt{2}t+t^2}{4{\Sigma }^2(t)}(x^2-y^2)\right\}$$

$${\Sigma }^2(t)=\frac{1}{\sqrt{2}}+t+\frac{t^2}{\sqrt{2}}+\frac{t^3}{3}.$$

$$⟨x|{\hat{\rho }}_{MC,t}|y⟩=\frac{1}{{\sum }_{n}1}\sum _n{\Psi }_t^{(n)}(x){\Psi }_t^{(n)\ast }(y).$$

$$\frac{{\left(\mathrm{Tr}(\hat{\rho }-{\hat{\rho }}_{MC}{)}^2\right)}^{1/2}}{(\mathrm{Tr}{\hat{\rho }}^2{)}^{1/2}}$$

$$⟨x|{\hat{\tilde{\rho }}}_{CM,t}|y⟩=\frac{1}{{\sum }_{n}1}\sum _n{\tilde{\Psi }}_t^{(n)}(x){\tilde{\Psi }}_t^{(n)\ast }(y)$$

$$\Delta {\tilde{x}}_{\infty }=1.62\pm 0.01,\Delta {\tilde{p}}_{\infty }=1.63\pm \mathrm{0.01.}$$

Full text

Introduction. A single Schrödinger particle becomes a simple open quantum system if the particle is interacting with a thermal reservoir. Its dynamics is given by a master equation which can take the following simple form valid typically at high temperatures:

[TABLE]

where $\hat{H}=(\hat{p}^{2}/2m)$ is the particle’s Hamiltonian, $\hat{x},\hat{p}$ are its coordinate and momentum resp., and $D$ is the diffusion constant. Joos and Zeh suggested this equation as the simplest model of spatial decoherence JooZeh85 while at the time similar single particle master equations were known from various fields, cf., e.g., Dek81 ; CalLeg83 . The Wigner function of $\hat{\rho}$ satisfies the classical Fokker-Planck equation in the high-temperature limit:

[TABLE]

This elucidates the importance of the master equation (1) as the quantized version of diffusion. Accordingly, $D$ is the coefficient of spatial decoherence as well as of momentum diffusion: the two effects are alternative interpretations of the non-Hamiltonian mechanism in the master equation. It is well-known that the classical diffusion (2) can be equivalently described by random trajectories $(x_{t},p_{t})$ in phase space. The same concept applies to the master equation (1) as well. The stochastic quantum trajectories are featured by state vectors $\Psi_{t}$ evolving by a stochastic process such that the stochastic mean

[TABLE]

satisfies the master equation (1). Then the quantum trajectories $\Psi_{t}$ are said to unravel the master equation.

The unravelling is never unique, one can choose diffusive unravellings, jump unravellings, or even their combinations. The earliest unravelling was the orthojump process Dio85 . It turned out subsequently that any master equation possesses a standard jump and a standard diffusive unravelling Dio86a . All possible diffusive unravellings can be parametrized uniquely RigMotOma97 ; WisDio01 , each of them corresponds to a given structure of time-continuous monitoring the system in question WisDio01 . Similar classification is still missing for jump unravellings.

While quantum trajectories became instrumental soon for quantum optics DalCasMol89 ; Car93 ; MilBresWis95 , their invention happened earlier in studies of foundations. In the nineteen-eighties, diffusive quantum trajectories were invented by Gisin to model quantum state collapse in a discrete system Gis84 . One of the present authors constructed jump Dio85 and diffusive Dio88a unravellings of the master equation (1) for his gravity-related spontaneous state collapse theories Dio86b and Dio89 , respectively. (On three decades of various spontaneous collapse theories, all based on unravellings, see the recent review by Bassi et al Basetal13 ).

Analytic proof was found for the wave function localization in diffusive quantum trajectories Dio88b . The wave function is approaching a steady localized shape for long times, as we recapitulate it below. Localization in the specific jump unravellings Dio85 has, however, never been studied. The problem is more complicated than the diffusive case because jumps will never allow for a steady shape. An analytic proof of localization has not yet been found, we shall rely on numeric (Monte-Carlo) simulations. Jump quantum trajectories of spatial decoherence were carefully studied by Gisin and Rigo RigGis96 , and in a sequence of works by Hornberger and co-workers BusHor0910 ; LucHor13 ; SorHor15 for modifications of the master equation (1) which included friction. Due to friction, quantum trajectories did reach a localized steady shape, calculable analytically. The effect and proof was bound to the presence of friction. Localization in the frictionless case (1) has remained to be studied in the present work.

We are going to study localization of quantum trajectories in both position and momentum. Consider the unitary transformation of a state $\Psi$ to its centre-of-mass frame:

[TABLE]

where the centre-of-mass state satisfies $\langle\widetilde{\Psi}|\hat{x}|\widetilde{\Psi}\rangle=0$ and $\langle\widetilde{\Psi}|\hat{p}|\widetilde{\Psi}\rangle=0$ by construction. Now we can define the centre-of-mass density matrix as follows:

[TABLE]

This matrix is non-negative and of unit trace, like common density matrices. Its evolution, however, is non-linear, completely different from the master equation (1) of the common density matrix $\hat{\rho}_{t}$. We use $\hat{\widetilde{\rho}}_{t}$ to characterize average localization of quantum trajectories $\Psi_{t}$ around their individual centre-of-mass $\left<\hat{x}\right>_{t},\left<\hat{p}\right>_{t}$. We can define centre-of-mass (squared) spreads by $(\Delta\widetilde{x})^{2}=\mathrm{Tr}(\hat{x}^{2}\hat{\widetilde{\rho}})$ and by $(\Delta\widetilde{p})^{2}=\mathrm{Tr}(\hat{p}^{2}\hat{\widetilde{\rho}})$.

Diffusive unravelling. Following Dio88a ; Dio88b , consider the stochastic Schrödinger equation Dio88b :

[TABLE]

where $dW$ is the Ito-differential of the Wiener stochastic process, satisfying $\mathbf{M}dW=0$, $(dW)^{2}=dt$. The solutions satisfy the condition (3) of unravelling. For long, the centre-of-mass solutions converge to the following complex Gaussian wave packet:

[TABLE]

li of squared width

[TABLE]

According to (7), the centre-of-mass density matrix (9) turns out to converge to a pure state:

[TABLE]

The coordinate and momentum spreads are given by

[TABLE]

The centre-of-mass of $\hat{\widetilde{\rho}}_{\infty}$ keeps to perform the following diffusive motion:

[TABLE]

Observe that the diffusion of the momentum is the classical one. On the contrary, the diffusion of the coordinate cannot happen classically, it is purely quantum. Asymptotic localization is thus the analytically calculable feature of the diffusive quantum trajectories of the simple spatial decoherence master equation (LABEL:MeE).

Orthojump unravelling. For the sake of comparison with the diffusive unravelling, let us cast the jump unravelling of Dio85 into the form of a stochastic Schrödinger equation:

[TABLE]

where $\sigma^{2}=\langle(\hat{x}-\left<\hat{x}\right>)^{2}\rangle$. $dN$ stands for the Ito-differential of a Poisson process, satisfying $\mathbf{M}dN=2D\sigma^{2}dt$, $(dN)^{2}=dN$. This equation corresponds to a piece-wise deterministic evolution of $\Psi_{t}$, interrupted by jumps at random times. In elementary terms, the mechanism is the following. Consider the deterministic non-linear Schrödinger equation

[TABLE]

[Note that this equation coincides with the deterministic part of the diffusive stochastic Schrödinger equation (6) and they share $\widetilde{\Psi}_{\infty}$ (7) as (normalized) steady-shape centre-of-mass solution.] Solve this non-linear Schrödinger equation for the initial condition $\Phi_{0}=\Psi_{0}$ and define the physical quantum state by $\Psi_{t}=\Phi_{t}/\|\Psi_{t}\|$. Note that the norm of $\Phi$ is strictly decreasing:

[TABLE]

The probability of jump-free deterministic evolution is decreasing exactly with the norm $\|\Phi\|^{2}$, i.e., the probability rate of jump is $(2D/\hbar^{2})\sigma^{2}$. When a jump occurs, the smooth deterministic evolution of $\Psi/\|\Psi\|$ is interrupted by the sudden change

[TABLE]

rendering the new state orthogonal to what it was before the jump (cf. also MilBresWis95 ). After the jump, the deterministic evolution (13) re-starts and continues until the next jump, etc.

Numeric tests of orthojumps. We have performed MC simulations of the orthojump quantum trajectories. While analytic solutions for individual trajectories are not (yet) known, the analytic solution of the master equation (1) is easy UnrZur89 ; DioKie02 , especially for Gaussian initial states Giuetal96 . To check the robustness of our MC simulation, we shall compare the MC-simulated density matrix $\hat{\rho}_{MC}$ to the analytic solution $\hat{\rho}$ of the master equation (1).

With suitable choice of physical units, we can always take trivial parameters $\hbar=m=D=1$ and that is what we do. For the rest of our work, we choose the pure state (7) deliberately (just for concreteness) as the initial pure state for (1). The analytic solution in coordinate representation reads:

[TABLE]

where the squared spatial spread is

[TABLE]

With the same initial pure state, we MC-generated $3\times 5000$ quantum trajectories $\{\Psi_{t}^{(n)};n=1,2,\dots,15000\}$ and determined $\hat{\rho}_{MC,t}$ numerically:

[TABLE]

Displaying its normalised distance

[TABLE]

from the analytic $\hat{\rho}_{t}$ (16) in the range $t\in(0,5)$, on different statistics, one confirms the stability and precision of simulation on 15 000 trajectories (Fig. 1). For qualitative comparison, Fig. 2 shows the MC-simulated Wigner function and the exact one at $t=5$.

These checks confirm that $15000$ trajectories will suffice to test the basic feature of interest: localization by orthojump unravelling.

We determined the centre-of-mass density matrix

[TABLE]

on three increasing statistics. Our main results are shown in Fig. 3, where the time-evolution of spatial and momentum spreads $\Delta\widetilde{x},\Delta\widetilde{p}$ are displayed for $t\in(0,5)$. Initial values are known analytically: $\Delta\widetilde{x}_{0}=\Delta\widetilde{p}_{0}=1/2^{1/4}\approx 0.84$. For times longer than the characteristic time scale $1$ (when $\hbar=m=D=1$) of the master equation (1), localization takes place asymptotically, both in coordinate and momentum. Both $\Delta\widetilde{x}$ and $\Delta\widetilde{p}$ converge to constants, their conservative estimates are

[TABLE]

This is the first numeric evidence, in lack of analytic ones, for localization of orthojump trajectories in frictionless spatial decoherence.

Summary. We have studied the localization of wave function in orthojump unravelling of the simplest and paradigmatic spatial decoherence master equation of a free particle. Localization in diffusive unravellings became proved analytically long ago. This time we were able to prove and calculate localization of the orthojump unravelling — using MC simulations. We used 15 000 MC-simulated quantum trajectories to confirm localization both in coordinate ($\Delta\widetilde{x}$) and momentum ($\Delta\widetilde{p}$), which we demonstrated on the centre-of-mass density matrix $\hat{\widetilde{\rho}}$. The obtained numeric values (21) are about twice as large as those (10) in diffusive unravelling. Such slightly looser localization may be explained heuristically. The asymptotic centre-of-mass density matrix $\hat{\widetilde{\rho}}_{\infty}$ contains randomness because the centre-of-mass wave function $\widetilde{\Psi}_{t}$ never ceases to undergo jumps, i.e., it is “breathing” at random times, whereas in diffusive unravelling $\Psi_{t}$ acquires a constant shape for large $t$ hence $\hat{\widetilde{\rho}}_{\infty}$ does not contain randomness, diffusive features contribute to the centre-of-mass motion (11) only.

Our work was restricted for the demonstration of stability and localization of the orthojump trajectories for the frictionless decoherent dynamics of a Schrödinger particle. Further studies should explore more details of orthojump trajectories’ rich structure. Numeric methods seem instrumeekntal. However, similar to the diffusive case (6), a possible power of the Ito formalism (12) remains to be explored for analytic calculations.

LD was supported by the Hungarian Scientific Research Fund under grant no. 103917, and by the EU COST Actions MP1209 and CA15220.