Caldeira-Leggett model
| Amir Ordacgi Caldeira (2010), Scholarpedia, 5(2):9187. | doi:10.4249/scholarpedia.9187 | revision #137033 [link to/cite this article] |
The Caldeira-Leggett Model is a semi-empirical model for dealing with the dynamics of a system coupled to its environment. Depending on the constraint imposed by the classical equation of motion of the system variable, it is possible to establish a minimal Hamiltonian model of a closed system (system of interest plus its environment) which mimics the realistic physical phenomenon and is analytically treatable. One of its most important consequences is the possibility of dealing with the quantum mechanics of dissipative systems.
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Dissipative Systems
There are many systems in Nature whose dynamics does not conserve energy. Among several examples that could be given the one which is considered the paradigm of this kind of motion is the well-known Brownian motion.
Classical Brownian motion
In the nineteenth century, the English botanist R. Brown observed that small particles immersed in a viscous fluid exhibited an extremely irregular motion. If no external force is applied on the particle, its average velocity \(\langle\vec{v}\rangle = 0\) and its variance \(\langle v^{2}\rangle\) is finite. The averages are taken over an ensemble of identically prepared systems. This phenomenon has since been known as Brownian motion.
The theoretical approach to treat this problem is through the so-called Langevin equation which reads \[\tag{1} M\ddot{q}+\eta\dot{q}+V^{\,\prime}(q)=f(t) \]
where \(f(t)\) is a fluctuating force such that \(\langle f(t)\rangle=0\) and \(\langle f(t)f(t^{\prime})\rangle= 2\eta k_{B}T\delta(t-t^{\prime})\ ,\)where \(M\) is the mass of the particle, \(\eta\) is the damping constant, \(V(q)\) is an external potential and \(k_{B}\) is the Boltzmann constant. This equation is a good description of the phenomenon only if
- the mass \(M\) of the Brownian particle is much larger than the mass \(m\) of the molecules composing the viscous fluid, and
- one is interested in the behavior of the particle for time intervals much longer than the average time \(\tau\) between molecular collisions.
Actually, the Langevin equation can generally be used to describe the dynamics of physical variables in many different systems. For example, the dynamics of the charge stored in the capacitor of a RLC circuit is written as \[\tag{2} L\ddot{Q}+ R\dot{Q}+ \frac{Q}{C}=V_{f}(t), \]
where \( V_{f}(t)\) is a fluctuating voltage such that \(\langle V_{f}(t)\rangle=0\) and \(\langle V_{f}(t)V_{f}(t^{\prime})\rangle= 2k_{B} T R \delta(t-t^{\prime})\ .\)
The dynamical variables of conventional circuits usually exhibit a purely classical motion all the way down to extremely low temperatures. However, there are special circuits, in particular those containing superconducting devices, in which things can be very different when one reaches the appropriate domain of the circuit parameters and quantum mechanics comes into play. In this circumstance one has to deal with dissipation, fluctuations and quantum effects on the same footing.
Quantum Brownian motion
The quantum mechanics of dissipative systems posed a very hard problem for physicists for many years. The origin of this problem lay in the fact that the standard procedures of quantization are based on the existence of either a Hamiltonian or a Lagrangian function for the system in question. On the other hand, it is well- known that it is not possible to obtain a Langevin equation from the application of the classical Lagrange's or Hamilton's equations to any Lagrangian or Hamiltonian which has no explicit time dependence. The employment of time dependent functions would allow one to use the standard procedures of quantization directly but it would create problems with the uncertainty principle.
Over many decades, people have tried to solve this problem. In spite of the variety of methods used, all these attempts fall into two main categories: They either look for new schemes of quantization or use the system-plus-reservoir approach as mentioned in Caldeira and Leggett (1983a). The former approaches always rely on some questionable hypotheses and lead to results dependent on the method used. The way out of this dilemma is to explicitly consider the fact that the dissipative system is always coupled to a given thermal environment (the latter approach). There is no dissipative system in Nature which is not coupled to another system responsible for its losses. Then, before one tries to modify the canonical scheme of quantization it is wiser to apply the traditional methods to more realistic situations.
Conceptually the idea is very simple. However, in practice, its implementation requires a little labor. Once one explicitly considers the coupling of the system of interest to the environment it must be known what sort of system the latter is and how their mutual coupling takes place. This can be a very hard task.
The treatment of a realistic model for the environment to which the system of interest is coupled, is not necessarily the best path to be taken. It would only encumber the intermediate steps of the calculations and hide the essence of the important physics of the problem.
Nevertheless, fundamentally different composite systems - system of interest-plus-environment - might have the former obeying Brownian dynamics in the classical limit. Although this appears to be an additional complication to this approach it actually allows one to argue in favor of some simplifying hypotheses. For instance, a possible assumption is that different reservoirs may share some common characteristics such as the behavior of their spectrum of excitations or the way they are acted by their Brownian particles. This idea is further explored with the specific model that will be developed next.
It must be a simple model which under certain conditions reproduces Brownian motion in the classical regime. Thus, the justification for the choice of the model will be provided a posteriori. However, it is worth mentioning that the employment of detailed microscopic models for some environments may show different quantum mechanical behavior which turns out be very important in some cases. Although the motivation to develop the model below has been the possibility of dealing with the quantum limit of the Brownian motion, the reader must be warned that it can be generalized to describe more general dissipative systems in the quantum regime. This point will be emphasized again when appropriate.
The model
Development
The composite system is assumed to be described by the Lagrangian \[\tag{3} L = L_{S} + L_{I} + L_{R} + L_{CT} , \]
where \[\tag{4} L_{S} = \frac{1}{2} \, M \, \dot{q}^{2} - V(q) , \]
\[\tag{5} L_{I} = \, q \sum_{k} C_{k} \, q_{k} , \]
\[\tag{6} L_{R} = \sum_{k} \frac{1}{2} \, m_{k} \, \dot{q}_{k}^{2} - \sum_{k} \frac{1}{2} \, m_{k} \, \omega_{k}^{2} \, q_{k}^{2} , \]
\[\tag{7} L_{CT} = - q^{2}\sum_{k} \frac{1}{2} \, \frac{C_{k}^{2}}{m_{k} \, \omega_{k}^{2}} \, , \]
are respectively the Lagrangians of the system of interest, interaction, reservoir, and counter-term (see below). The reservoir consists of a set of non-interacting harmonic oscillators of coordinates \(q_{k}\ ,\) masses \(m_{k}\) and natural frequencies \(\omega_{k}\ .\) Each one of them is coupled to the system of interest by a coupling constant \(C_{k}\ .\)
Despite appearing naïve and very academic, this model is a very good approximation for a realistic composite system whenever ( see Caldeira and Leggett (1983b);
- the system of interest is only weakly disturbing the environment, which implies that the dynamics of the latter can be described in the lowest order of perturbation theory or,
- the interaction of the system of interest with its environment can be treated within the adiabatic approximation, which is the case when the variable of the system of interest is much slower than the degrees of freedom of the environment.
Initially one shall study the classical equations of motion
resulting from (3). Writing the Euler-Lagrange equations of
the composite system one has
\[\tag{8}
M \ddot{q} = - V^{\prime}(q) + \sum_{k} C_{k} \, q_{k} - \, q\sum_{k}
\frac{C_{k}^{2}}{m_{k} \, \omega_{k}^{2}} ,
\]
\[\tag{9} m_{k} \, \ddot{q}_{k} = - m_{k} \, \omega_{k}^{2} \, q_{k} + C_{k} \, q . \]
Taking the Laplace transform of (9) one gets \[\tag{10} \tilde{q}_{k}(s) = \frac{\dot{q}_{k}(0)}{s^{2} + \omega_{k}^{2}} + \frac{s \, q_{k}(0)}{s^{2} + \omega_{k}^{2}} + \frac{C_{k} \, \tilde{q}(s)}{ m_{k} \, \left( s^{2} + \omega_{k}^{2} \right)} , \]
which after the inverse transformation can be taken to (8) yielding \[\tag{11} M \ddot{q} + V^{\prime}(q) + \, q\sum_{k} \frac{C_{k}^{2}}{m_{k} \, \omega_{k}^{2}} = \frac{1}{2 \, \pi \, i} \int_{\varepsilon - i \, \infty}^{ \varepsilon + i \, \infty} \sum_{k} C_{k}\left\{ \frac{\dot{q}_{k}(0)}{s^{2} + \omega_{k}^{2}} + \frac{s \, q_{k}(0)}{s^{2} + \omega_{k}^{2}} \right\} \, e^{s \, t} \, ds + \sum_{k} \frac{C_{k}^{2}}{m_{k}} \frac{1}{2 \, \pi \, i} \int_{\varepsilon - i \, \infty}^{ \varepsilon + i \, \infty} \frac{\tilde{q}(s)}{s^{2} + \omega_{k}^{2}} \, e^{s \, t} \, ds . \]
Thus using the identity \[ \frac{1}{s^{2} + \omega_{k}^{2}} = \frac{1}{\omega_{k}^{2}} \left\{ 1 - \frac{s^{2}}{s^{2} + \omega_{k}^{2}} \right\} , \] one can show that the last term on the rhs of (11) generates two other terms, one of which exactly cancels the last one on its lhs, and the resulting equation is \[\tag{12} M \ddot{q} + V^{\prime}(q) + \sum_{k} \frac{C_{k}^{2}}{m_{k} \, \omega_{k}^{2}} \, \frac{1}{2 \, \pi \, i} \int_{\varepsilon - i \, \infty}^{ \varepsilon + i \, \infty} \frac{s^{2} \, \tilde{q}(s)}{s^{2} + \omega_{k}^{2}} \, e^{s \, t} \,ds = \frac{1}{2 \, \pi \, i} \int_{\varepsilon - i \, \infty}^{ \varepsilon + i \, \infty} \sum_{k} C_{k}\left\{ \frac{\dot{q}_{k}(0)}{s^{2} + \omega_{k}^{2}} + \frac{s \, q_{k}(0)}{s^{2} + \omega_{k}^{2}} \right\} \, e^{s \, t} \, ds. \]
Then one sees that the inclusion of \(L_{CT}\) in (3) was solely
to cancel one extra harmonic contribution that would come from the
coupling to the environmental oscillators.
The last term on the lhs of (12) can be rewritten as \[ \frac{d\,\,}{dt} \left\{ \sum_{k} \frac{C_{k}^{2}}{m_{k} \, \omega_{k}^{2}} \, \frac{1}{2 \, \pi \,i} \int_{\varepsilon - i \, \infty}^{ \varepsilon + i \, \infty} \frac{s \, \tilde{q}(s)}{s^{2} + \omega_{k}^{2}} \, e^{s \, t} \, ds \right\} \] which with the help of the convolution theorem reads \[\tag{13} \frac{d\,\,}{dt} \left\{ \sum_{k} \frac{C_{k}^{2}}{m_{k} \, \omega_{k}^{2}} \, \int_{0}^{t} \cos \left[ \omega_{k} \, \left( t - t^{\prime} \right) \right] \, q(t^{\prime}) \, dt^{\prime} \right\} . \]
In order to replace \(\sum_{k} \longrightarrow \int d\omega\) one
introduces the spectral function \(J(\omega)\) as
\[\tag{14}
J(\omega) =
\frac{\pi}{2} \sum_{k} \frac{C_{k}^{2}}{m_{k} \, \omega_{k}} \,
\delta \left( \omega - \omega_{k} \right) ,
\]
which allows one to write \[\tag{15} \sum_{k} \frac{C_{k}^{2}}{m_{k} \, \omega_{k}^{2}} \, \cos \left[ \omega_{k} \, \left( t - t^{\prime} \right) \right] = \frac{2}{\pi} \int_{0}^{\infty} d\omega \, \frac{J(\omega)}{\omega} \cos \left[ \omega \, \left( t - t^{\prime} \right) \right] . \]
The function \(J(\omega)\) is nothing but the imaginary part of the
Fourier transform of the retarded dynamical susceptibility of the
bath of oscillators, namely,
\[\tag{16}
J(\omega) = {\textrm{Im}}{\mathcal{F}} \left\{ - i \, \frac{\theta
\left( t - t^{\prime} \right)}{\hbar} \left\langle \left[ \sum_{k}
C_{k} \, q_{k}(t) \, , \, \sum_{k^{\prime}} C_{k^{\prime}} \,
q_{k^{\prime}}(t^{\prime}) \right] \right\rangle \right\} .
\]
Now, assuming that \[\tag{17} J(\omega) = \begin{cases} \eta \, \omega \,\,\textrm{if} \,\,\omega < \Omega \,\,\textrm{and} \\ 0 \,\,\text{if}\,\, \omega > \Omega , \end{cases} \]
where \(\Omega\) is a high frequency cutoff that fixes the frequency (or time) scale of the problem, one can rewrite (15) as \[\tag{18} \sum_{k} \frac{C_{k}^{2}}{m_{k} \, \omega_{k}^{2}} \, \cos \left[ \omega_{k} \, \left( t - t^{\prime} \right) \right] = \frac{2}{\pi} \int_{0}^{\Omega} \eta \, \cos \left[ \omega \, \left( t - t^{\prime} \right) \right] = 2 \, \eta \, \delta \left( t - t^{\prime} \right) , \]
where the limit \(\Omega \rightarrow \infty\) was taken. This result allows one to rewrite Eq.(13) as \[\tag{19} \frac{d\,\,}{dt} \int_{0}^{t} 2 \, \eta \, \delta \left( t - t^{\prime} \right) q(t^{\prime}) dt^{\prime} = \eta \, \dot{q} + 2 \, \eta \, \delta \left( t \right)q(0). \]
Notice that the hypothesis (17) was crucial for the obtainment of a force proportional to the velocity of the particle. This particular choice is usually referred to in the literature as the Ohmic spectral function. In general, the spectral function \(J(\omega) \propto \omega^{\alpha}\ ,\) and it is called super-Ohmic for \(\alpha > 1\) and sub-Ohmic for \(\alpha < 1\ .\)
Finally, the rhs of (12) can be interpreted as a force
\(f(t)\) depending on the initial conditions imposed on the
oscillators of the bath. Suppose that each environmental
oscillator is initially in thermodynamical equilibrium about the
position, \(\bar{q}_{k}(0)\equiv C_{k}q(0) /m_{k}\omega_{k}^{2}\ .\)
Thus, using the equipartition theorem one has, in the classical
limit,
\[
\left\langle q_{k}(0) \right\rangle
=\frac{C_{k}q(0)}{m_{k}\omega_{k}^{2}}\quad \textrm{and} \quad
\left\langle \dot{q}_{k}(0) \right\rangle = \left\langle
\dot{q}_{k}(0) \, \Delta q_{k}(0) \right\rangle = 0 ,
\]
\[
\left\langle \dot{q}_{k}(0) \, \dot{q}_{k^{\prime}}(0)
\right\rangle = \frac{k_{B} \, T}{m_{k}} \, \delta_{k k^{\prime}} ,
\]
\[\tag{20}
\left\langle \Delta q_{k}(0) \, \Delta q_{k^{\prime}}(0)
\right\rangle = \frac{k_{B} \, T}{m_{k} \, \omega_{k}^{2}} \,
\delta_{k k^{\prime}},
\]
where \(\Delta q_{k}(0)\equiv q_{k}(0)-\bar{q}_{k}(0) \ .\) Using these relations and the rhs of (12) it can be shown that \[ \left\langle f(t) \right\rangle = 0 \qquad\textrm{and} \] \[\tag{21} \left\langle f(t) \, f(t^{\prime}) \right\rangle = 2 \, \eta \, k_{B} \, T \, \delta \left( t - t^{\prime} \right) . \]
If one now inserts (19) in (12) one finally has
\[\tag{22}
M \, \ddot{q} + \eta\, \dot{q} + V^{\prime}(q) = f(t) ,
\]
where \(f(t)\) satisfies (21). This is the well-known Langevin equation for the classical Brownian motion.
So, one sees that the hypothesis that the environmental oscillators are in equilibrium about \(\bar{q}_{k}(0)\) was essential for one to get rid of the spurious term \(2 \, \eta \, \delta \left( t \right)q(0)\) in (19). Another possibility to reach (22) is to assume that the oscillators are in equilibrium independently of the initial position of the external particle, that is \(\langle q_{k}(0)\rangle=0\ ,\) and drop that term by considering the particle motion as starting at \(t=0^{+}\ .\) A more thorough analysis of this problem can be found in Rosenau et al.(2000)
As a last remark it must be mentioned that another completely equivalent way to write this model (3) is to replace the interaction Lagrangian (5) by \[\tag{23} \tilde{L}_{I} = q \,\sum_{k} \tilde{C}_{k} \, \dot{q}_{k} , \]
and omit \(L_{CT}\ .\) In this way one can define the canonical momenta \(p_{k}\) as \[\tag{24} p_{k} = \frac{\partial L\,\,}{\partial \dot{q}_{k}} = m_{k} \, \dot{q}_{k} + \tilde{C}_{k} \, q , \]
and write \[\tag{25} \tilde{H} = p \, \dot{q} + \sum_{k}\,p_{k} \, \dot{q}_{k} - L = \frac{p^{2}}{2 \, M} + V(q) + \sum_{k} \left\{ \frac{1}{2 \, m_{k}} {\left( p_{k} - \tilde{C}_{k} \, q \right)}^{2} + \frac{1}{2} m_{k} \, \omega_{k}^{2} \, q_{k}^{2} \right\} , \]
Now, performing a canonical transformation \(p \rightarrow p\ ,\) \(q \rightarrow q\ ,\) \(p_{k} \rightarrow m_{k} \, \omega_{k} \, q_{k}\ ,\) and \(q_{k} \rightarrow p_{k} / m_{k} \, \omega_{k}\) and defining \(C_{k} \equiv \tilde{C}_{k} \, \omega_{k}\) one has \[\tag{26} H = \frac{p^{2}}{2 \, M} + V(q) - q \sum_{k} C_{k} q_{k} + \sum_{k} \left\{ \frac{p_{k}^{2}}{2 \, m_{k}} + \frac{1}{2} m_{k} \, \omega_{k}^{2} \, q_{k}^{2} \right\} + q^{2}\sum_{k} \frac{C_{k}^{2}}{2m_{k}\omega_{k}^{2}} , \]
which has (3) as its corresponding Lagrangian. Therefore, the electromagnetic Lagragian with \(\tilde{L}_{I}\) replacing \(L_{I}\) in (3) and no counter-term is completely equivalent to (3) itself.
Now that one knows a treatable model that generates the classical Brownian motion for the variable of interest (\(q(t)\) in the present case) one can study the quantum mechanics of the composite system and extract from it only the relevant dynamics referring to the system of interest.
Final remarks
Although the model presented above, or more particular forms thereof, had been used by several authors to explain the dynamics of dissipative systems from a more microscopic point of view (see references in Caldeira and Leggett (1983a)), it was only in its present form that it was successfully applied to describing the quantum mechanical limit of Brownian motion in general circumstances as well as problems involving quantum tunnelling in general dissipative systems. Probably this is the main reason why the model is nowadays known in the literature as the " Caldeira-Leggett" model.
The procedure to treat the quantum dynamics of dissipative systems makes use of the fact that since the non-interacting oscillators are bilinearly coupled to the particle of interest, their effect on the latter can be easily obtained by properly tracing out the oscillators´ coordinates from either the time dependent full density operator of the whole system (as in Feynman and Vernon (1963)) or its equilibrium density operator (as in Feynman (1972)). Both procedures generate expressions which depend only on the variables of the system of interest either in real or "imaginary" time, respectively. The former approach is useful for treating problems such as the motion of wavepackets in the classically accessible region of a given potential (see Caldeira and Leggett (1983a)) or the coherent tunnelling of a particle between two degenerate minima of a bistable potential (see Leggett et al (1987)) whereas the latter describes the tunnelling of a particle out of a local minimum of a metastable potential or the general properties of a particle in thermal equilibrium in a general potential (see Caldeira and Leggett(1983b)). Moreover, it should be emphasized that since all the characteristics of specific dissipative system are encapsulated in the spectral function \(J(\omega)\ ,\) the effect of the environment on the quantum mechanical behaviour of the system can be solely expressed in terms of phenomenological parameters (Leggett (1984)) already present in its semi – classical dynamics.
Although they seem purely academic, the above-mentioned examples can actually describe very realistic situations. In particular, they may appear in the study of the dynamics of appropriate variables of meso or nanoscopic superconducting devices which are useful for testing quantum mechanics on the macroscopic level through phenomena known as macroscopic quantum tunnelling (MQT) or macroscopic quantum coherence (MQC). These devices might also become very important as far as technological applications are concerned since they are viewed as good candidates for the implementation of a qubit.
Finally, it is worth mentioning that despite being much more general than it appears the model introduced here does not describe the action of a totally arbitrary environment on the system of interest. For instance, there are reservoirs whose dynamics can be mimicked by a set of non-interacting two-level systems instead of oscillators (as in Caldeira, Castro-Neto and Oliveira de Carvalho (1993) or Prokof'ev and Stamp (2000)) and these present effects on the sub-system dynamics quite different from the former model.
References
- Caldeira , A O; Castro-Neto, A H and Oliveira de Carvalho, T (1993). Dissipative Quantum Systems modelled by a Two-Level Reservoir Coupling. Physical Review B48: 13974.
- Caldeira(1983a). Path Integral Approach to Quantum Brownian Motion. Physica 121A: 587. doi:10.1016/0378-4371(83)90013-4.
- Caldeira(1983b). Quantum Tunnelling in a Dissipative System. Annals of Physics 149: 374. doi:10.1016/0003-4916(83)90202-6.
- Feynman(1963). The theory of a General Quantum System Interacting with a Linear Dissipative System. Annals of Physics 24: 1208. doi:10.1016/0003-4916(63)90068-x.
- Feynman, R P (1972). Statistical Mechanics. W A Benjamin Inc., Philippines.
- Leggett, A J (1984). Quantum Tunnelling in the Presence of an Arbitrary Linear Dissipation Mechanism. Physical Review B30: 1208.
- Leggett, A J et al. (1987). Dynamics of the Dissipative Two State System. Reviews of Modern Physics 59(1): 1. doi:10.1103/revmodphys.59.1.
- Prokof'ev(2000). Theory of the Spin Bath. Reports on Progress in Physics 63: 669. doi:10.1088/0034-4885/63/4/204.
- Rosenau da Costa, M; Caldeira, A O; Dutra, S M and Westfahl Jr., H (2000). Exact Diagonalization of Two Quantum Models for the Damped Harmonic Oscillator. Physical Review A61: 2210. doi:10.1103/physreva.61.022107.
Internal references
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
- Jean Zinn-Justin (2009) Path integral. Scholarpedia, 4(2):8674.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
Further reading
- Feynman, R P and Hibbs, A R (1965). Quantum Mechanics and Path Integrals. McGraw-Hill, New York.
- Breuer, H P and Petruccione, F (2006). The Theory of Open Quantum Systems. Claredon Press, Oxford.
- Weiss, U (1999). Quantum Dissipative Systems. World Scientific, Singapore.
