Foundations of Physics
Open Systems' Density Matrix Properties in a Time Coarsened Formalism
Robert Englman
Asher Yahalom
First Online: 14 April 2015
The concept of time-coarsened density matrix for open systems has frequently featured in equilibrium and non-equilibrium statistical mechanics, without being probed as to the detailed consequences of the time averaging procedure. In this work we introduce and prove the need for a selective and non-uniform time-sampling, whose form depends on the properties (whether thermalized or not) of the bath. It is also applicable when an open microscopic sub-system is coupled to another finite system. By use of a time-periodic minimal coupling model between these two systems, we present detailed quantitative consequences of time coarsening, which include initial state independence of equilibration, deviations from long term averages, their environment size dependence and the approach to classicality, as measured by a Leggett–Garg type inequality. An interacting multiple qubit model affords comparison between the time integrating procedure and the more conventional environment tracing method.
Foundations of quantum mechanics Density matrix Time coarsened formalism
Thanks are due to Rafael Ruppin for substantial help in the calculations and to Ronnie Kosloff for a discussion. The article has benefited from the constructive comments of two referees, who also called our attention to several important references.
Appendix 1: Original Outline of the Three Steps to Construct the Density Matrix
In the quoted previous work [14] it was shown how the decohered-truncated phase of the full subsystem-environment state (their DM), leading into the reduced subsystem state, can be obtained by a time averaging approach. The abstract steps needed to achieve this and the main assumptions behind it were given by [14] in an "Outline", which is briefly reproduced in this Appendix. The Hamiltonians for which the system's states were determined were (i) a Rabi-model (a single-spin in interaction with a single vibrator), (ii) two mutually coupled spins, each coupled to a single vibrator. For completeness these Hamiltonians, formulated in a semi-classical, time-dependent language, are also shown in this article in Appendix 2. While it is clear that a Hamiltonian in which a time dependent term appears represents an open system, what our results have shown is that even with the arguably simplest form of the time dependent term (namely, a single subsystem-environment harmonic interaction term), one obtains results equivalent to those in the standard environment-tracing formalism. The simplifying (and approximative) steps necessary to pass from the Leggett et al's (spin-oscillatory ensemble) model [41] to ours were described in Sect. 2.3 of the earlier article. The present text has extended the time-averaging formalism to issues not considered before, including second moments (fluctuations) of the DM (in Sect. 6) and its approach through time averaging to classicality (Sect. 7).
We adopt the von Neumann definition
$$\begin{aligned} \rho _{ij}(t) = \frac{1}{(\sum _a 1)} \sum _{a} \langle i|\psi _{a}(t)\rangle \langle \psi _{a}(t)|j\rangle \end{aligned}$$
In this definition the summation index \(a\) represents the values of all coordinates, variables etc. external to the system (e.g., those of the environment affecting the system) and appearing also in the Hamiltonian. Thus the set \({\psi _{a}(t)}\) for all \(a\)'s forms a time dependent ensemble of states. The variables of the system themselves are implicit (not written out) in \(\psi _a (t)\).
We solve only for a single external condition thus dispensing with the \(a\) index in the wave function, but obtain \(\rho (t)\) as the average over an adequate set of adjacent times:
$$\begin{aligned} \rho _{ij}(t)= \frac{1}{2 \Delta t}\int _{t-\Delta t}^{t+\Delta t}d\tau \langle i|\psi (\tau ) \rangle \langle \psi (\tau )|j \rangle \end{aligned}$$
This should be equivalent to Eq. (14) if the ergodic hypothesis holds for the duration \(2\Delta t\). While for several cases the off-diagonal matrix elements are small or vanishing, \(\rho _{ij}(t)\), as defined above, represents in general a mixed state whose diagonalized form \(\rho _{ij}(t)\rightarrow \rho _{ii}^d(t)\delta _{ij}\) satisfies \(\sum _i (\rho _{ii}^d(t))^2<1\) (Sect. 5 in this paper).
Appendix 2: Minimal Coupling Hamiltonians
Single spin:
$$\begin{aligned} H(t)= e\sigma _z+k\sigma _x \sin (\omega t) \end{aligned}$$
Interacting spins:
$$\begin{aligned} \mathbf H _{total}(t)= & {} h(t) + \mathbf H _{int} \end{aligned}$$
$$\begin{aligned} h(t)\!= & {} \! \sum _i[E_i\sigma _{zi} +k_i\sigma _{zi}\cos (\omega _i t+\alpha '_i)\!+\!k'_i\sigma _{xi}\sin (\omega _i t+\alpha '_i)] \end{aligned}$$
$$\begin{aligned} \mathbf H _{int}= & {} \sum _{ij}[\gamma _{ij} (\sigma _{zi} \cdot \sigma _{zj})+\gamma ' (\sigma _{xi} \cdot \sigma _{xj}+\sigma _{yi}\cdot \sigma _{yj})] \end{aligned}$$
having written in the first line the total Hamiltonian, comprising the parts in the next two lines. First, the spin energy terms in which \(E_i\) are energies of the spin systems, the \(\sigma \)'s are Pauli matrices operating in the respective spin spaces; \(k,\alpha \), together with their tagged partners, are parameters of the spin-boson couplings and \(\omega \) is the frequency of the external source. This external, boson source is classical and for it the Hamiltonian need not be written out. The Hamiltonian does not include back-reaction on the source, which can at least approximately be justified for periodic coupling and energies. The last line is the spin-spin interaction term. A two-spin version of this interacting-spin Hamiltonian was treated in [14].
Appendix 3: Extremization of \(G(t)\) in Eq. (6)
We now prove that the maximum of \(G(t)\) is unity when the wave function components \(\psi _u(t)\) and \(\psi _l(t)\) are constant during the \(2\Delta t\) integration range.3 By implication, \(G(t)\) is less than unity (mixed state case) when the components are genuinely time dependent. The method of proof is a calculation of variations.
It has been stated in text that a maximum of \(G(t)\) occurs when the wave function components are throughout real and positive. We further simplify by writing
$$\begin{aligned} \psi _u(t)\equiv f(t),~~\psi _l(t)\equiv \sqrt{1-f^2(t)} \end{aligned}$$
Written as
$$\begin{aligned} G[f(t)]\!=\!\left[ \frac{1}{2\Delta t}\int _{t-\Delta t}^{t+\Delta t}(2f^2(t')\!-\!1)dt'\right] ^2\! +\! \left[ \frac{1}{\Delta t}\int _{t-\Delta t}^{t + \Delta t}f(t')\sqrt{(}1-f^2(t'))dt'\right] ^2\nonumber \\ \end{aligned}$$
a variation gives
$$\begin{aligned} \delta G[f(t)]= & {} \left[ \frac{1}{\Delta t}\int _{t-\Delta t}^{t+\Delta t}(2f^2(t')-1)dt'\right] \left[ \frac{1}{2\Delta t}\int _{t-\Delta t}^{t+\Delta t}4f(t')\delta f(t')dt'\right] \nonumber \\&+ \Big [\frac{2}{\Delta t}\int _{t-\Delta t}^{t + \Delta t}f(t')\sqrt{1-f^2(t')}dt'\Big ]\Big [\frac{1}{\Delta t}\int _{t-\Delta t}^{t + \Delta t}(\sqrt{1-f^2(t')}\nonumber \\&- \frac{f^2(t')}{\sqrt{1-f^2(t')}})\delta f(t')dt'\Big ] \end{aligned}$$
Remarkably, the cofactor of \(\delta f(t')\) vanishes when \(f(t)=f\), a constant throughout the range of integration, so that \(G[f(t)=f]\) is then an extremum. Its value is unity, independent of \(f\). That it is also a maximum can be shown by evaluating \(G(t)\) for the case that the function \(f(t)\) takes two values \(f_1,f_2\ne f_1\) in fractions \(F, (1-F)\) (respectively) of the integration interval and showing that the leading term in \(G[f_1,f_2]\) is negative for \(|f_1|,|f_2|\le 1\), namely
$$\begin{aligned} -\frac{F(1-F)}{8}(f_1-f_2)^2[4+(f_1+f_2)^2]\left[ 1+\frac{4+(f_1+f_2)^2}{(4-(f_1+f_2)^2)^2}\right] \end{aligned}$$
Appendix 4: Proof of Sum Formula in Eq. (7) for a Finite Bath
It is assumed that the subsystem and the large, but finite sized bath form together a microcanonical ensemble with mean energy of \(E_{total}\) and small energy uncertainty \(\nu \). The range of the subsystem's energies \(E(t)\), that vary in time, is divided up into N segments, numbered \(n(=1,\ldots ,N )\) each of spread \(\nu \). The segments contain \(r_{n+1}-r_n\) discrete (supposed non-degenerate) energy levels \(E^b_r\) of the bath, such that these bath state energy levels satisfy
$$\begin{aligned} E_{total}= E(t)+ E^b_r \end{aligned}$$
for some value of \(E(t)[\approx E(t_n)]\) situated within the segment spread \(\nu \). The corresponding time spread over the segment is \(t_{n+1}-t_n\), having assumed that for the short passage time over the narrow segment spread the subsystem energy is a monotonic, single-valued function of time. This time spread is of the order of the bath's relaxation time.
The subsystem's density operator ("the reduced" density operator) \(\rho ^S(t)\) is obtained in the bath tracing formalism by weighting the system's proper density operator \(\rho _n(t)\) in each segment by the above number of bath states, giving
$$\begin{aligned} \rho _n(t)(r_{n+1}-r_n)= & {} \rho _n(t)\sum _{r_n}^{r_{n+1}} \int _{E(t_n)}^{E(t_{n+1})}dE'\delta (E_{total}-E'-E^b_r)\nonumber \\= & {} \sum _{r_n}^{r_{n+1}} \rho _n(t)\int _{t_n}^{t_{n+1}}dt' \frac{dE(t')}{dt'}\delta (E_{total}-E(t')-E^b_r)\nonumber \\= & {} \sum _{r=r_n}^{r_{n+1}}\rho _n(t_r) \end{aligned}$$
having in the last line interchanged the order of the summation and the time integration and taken heed of the property of the Dirac delta function. We thus obtain for the weighted density operator in the time segment a discrete sum with the time sum going over all such times that the system energy is complemented to \(E_{total}\) by a bath state's energy. In the time-integrated formalism the system's density matrix is obtained by integrating over time windows \(t_w\) that are much wider than the time spread of a segment. Then the resultant matrix element is
$$\begin{aligned} \rho _{ij}\propto \sum _{t_r\epsilon t_w} \rho _{ij}(t_r) \end{aligned}$$
where the time sum goes over all such times \(t_r\) within the time window that the instantaneous subsystem energy \(E(t)\) is complemented by a bath level to add up to \(E_{total}\). The proportionality constant is fixed, by the requirement that the trace of density matrix is unity. In the case of an infinite bath with a continuous energy spectrum, the above sum is replaced by a time integral including the energy density of the bath levels \(D^b(E^b)\):
$$\begin{aligned} \rho _{ij}(t)\propto \int _{t}^{t+t_w}dt'D^b(E^b) \rho _{ij}(t') \end{aligned}$$
This formula is also applicable for the case that the bath is in thermal equilibrium at a temperature \(T\), so that the bath energy density is proportional to
$$\begin{aligned} e^{\frac{E^b}{k_B T}}= e^{\frac{E_{total}-E(t)}{k_B T}} \end{aligned}$$
by virtue of the expression in Eq. (24) for the bath energy. This result, originally due to [42], has already been used in [14], with the time independent factor \(e^{\frac{E_{total}}{k_B T}}\) having been absorbed in the proportionality factor.
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1.Soreq NRCYavneIsrael
2.Ariel UniversityArielIsrael
Englman, R. & Yahalom, A. Found Phys (2015) 45: 673. https://doi.org/10.1007/s10701-015-9894-5
First Online 14 April 2015