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Cavity quantum electrodynamics - Laboratoire Kastler Brossel

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Accueil du site > Atomes, cavités et photons > Principes de l’electrodynamique quantique en cavité > Relaxation du champ

Relaxation du champ

The main relaxation channel is the cavity damping. We show here how it can be incorporated in the theoretical description of the experiment, in the master equation framework or using quantum Monte Carlo trajectories.

Master equation

Decoherence is the result of a dissemination of information about a system via its entanglement with an environment $E$ . For all practical purposes, this information is buried in $E$ on which no measurement can be performed in practice. The system density operator $\rho$ is obtained by tracing over $E$ the system/environment entangled state.

The system relaxation is then described by a master equation, differential equation for $\rho$ alone. This equation can be derived from a simple environment model, a bath of harmonic oscillators, for instance, spanning a wide range of frequencies around the system’s eigenfrequencies, in thermal equilibrium at a finite temperature $T$ . Quite remarkably, the final form of the master equation is model-independent.

For the sake of simplicity, we consider only the zero temperature case. The master equation then writes in the standard Lindblad form :

$\frac{d{\rho}}{dt}=-\frac{i}{\hbar}[{H'_c},{\rho}]+\frac{\kappa}{2}\left[ 2a\rho a^\dagger-a^\dagger a\rho-\rho a^\dagger a\right]\ ,$

where $\kappa=1/T_c=\omega_c/Q$ the damping rate of the cavity mode energy. The first term in the r.h.s. describes the hamiltonian evolution. The second describes the effect of photon escape into $E$ .

Remarkably, this equation depends only upon the classical energy damping time $T_c=1/\kappa$ , which can be measured with macroscopic fields. At finite temperature, additional terms, proportional to $n_t$ , the mean number of thermal photons per mode, describe the creation of thermalexcitations in the mode.

It is easy to derive from the master equation the evolution of the photon number distribution $p(n)=\rho_{nn}=\langle{n}|{\rho}|{n}\rangle$ :

$\frac{d{p(n)}}{dt}=\kappa(n+1)p(n+1)-\kappa np(n)\ .$

This equation describes a cascade in the Fock states ladder. The lifetime of state $|{n}\rangle$ is thus of the order of $T_c/n$ , decreasing when the number of photon increases. We show below that the lifetime of a coherent state is independent upon its amplitude. That non-classical Fock states are more fragile than semi-classical coherent ones is a first insight into decoherence.

Monte Carlo trajectories

The master equation does not provide a detailed insight into the mechanisms of relaxation. The Monte Carlo wavefunction approach is more convenient for computational purposes and also more insightful. We give here the recipes for the Monte Carlo method in the cavity case and discuss briefly its physical contents.

The master equation is obtained by tracing over all possible results of a virtual measurement performed over the environment. Let us imagine instead that this measurement is explicitely made and its results recorded. What happens then ? For the sake of definiteness, the cavity will be owned by a first operator, called Alice, and Bob has a complete control over the environment.

Bob is to monitor relaxation events, corresponding to the loss of a photon by the mode. He could, for instance (see figure above) couple the cavity with a detector, registering a click whenever a photon escapes.

Let us consider a short time interval $\tau$ . At the beginning, the cavity is in a pure state $|{\phi^{(A)}}\rangle$ and Bob’s detector in a neutral state $|{0^{(B)}}\rangle$ . During the time interval $\tau$ , the system evolves according to the infinitesimal unitary transformation :

$|{\phi^{(A)}}\rangle\otimes|{0^{(B)}}\rangle\rightarrow\left[1-\frac{1}{2}\kappa \tau (a^{\dagger}a)\right]|{\phi^{(A)}}\rangle\otimes|{0^{(B)}}\rangle+\sqrt{\kappa \tau}a|{\phi^{(A)}}\rangle\otimes |{1^{(B)}}\rangle\ ,$

in which $|{1^{(B)}}\rangle$ , orthogonal to $|{0^{(B)}}\rangle$ , is the detector’s state after photon detection. The time interval $\tau$ being very short, the probability for loosing two photons is negligible. The cavity/environment state is entangled, an essential feature of decoherence.

At the end of this time interval, Bob performs a measurement on its detector, in the $\{|{0^{(B)}}\rangle$ , $|{1^{(B)}}\rangle\}$ basis. The probabilities $p_1$ and $p_0$ for finding the environment (equivalent to a single qubit in this simple situation) in these states are :

$p_1=1-p_0=\kappa \tau \langle \phi^{(A)}|a^{\dagger}a |{\phi^{(A)}}\rangle\ .$

Depending on the outcome of the measurement, the cavity state is projected in one of the states defined as :

$|{\phi_1^{(A)}}\rangle=\frac{a|{\phi^{(A)}}\rangle}{\sqrt{p_1}}\ ,$

for a photon click and :

$|{\phi_0^{(A)}}\rangle=\frac{\left[1-\frac{1}{2}\kappa \tau a^{\dagger}a\right]|{\phi^{(A)}}\rangle}{\sqrt{p_0}}\ .$

when no click is recorded. The last expression justifies the `jump operator’ name coined for $\sqrt{\kappa}a$ , since it describes the discontinuous change of the mode state when $B$ is found in $|{1^{(B)}}\rangle$ . In the event that $B$ does not change, the first equation shows that the oscillator state is also modified. It evolves under the effect of an infinitesimal non-unitary transformation produced by the anti-hermitian pseudo-hamiltonian $i\hbar\kappa a^{\dagger} a$ . This non-unitary transformation can be described as a renormalization of the oscillator frequency by the addition of an imaginary term in its frequency ( $\omega_c\rightarrow\omega_c-i\kappa/2$ ). It does not conserve the norm of the state, hence the necessity to normalize it by the $(p_0)^{-1/2}$ factor.

That the cavity state evolves when no photon is recorded by Bob might seem counterintuitive. Recall however that a null measurement provides information on the system and, hence, modifies its state. Recording no photon during $\tau$ is an indication that the number of photons is more likely to be small. The pseudo-hamiltonian describes the reduction of the field energy associated to this information. We give below a more detailed insight into this evolution.

At the end of the time interval, the cavity is yet in a pure state, depending upon the measurement result recorded by Bob. A Monte Carlo trajectory is then defined by a series of random measurement results associated to successive time bins of duration $\tau$ . At each step, the field state undergoes the action of either $[1-\kappa\tau a^{\dagger}a/2]$ or $a$ , depending upon the measurement result. It is computed by iterating the process, making a random decision according to the probability law to determine whether $B$ is found in $|{1^{(B)}}\rangle$ or $|{0^{(B)}}\rangle$ at each stage. The state of $A$ at the beginning of the $(n+1)^{{th}}$ step is determined by the outcome of the $n^{{th}}$ measurement and $B$ is initialized to $|{0^{(B)}}\rangle$ at the beginning of each step.

The master equation result is recovered by assuming that Bob’s measurements are left unread. The cavity mode ends up, at the end of the first time interval $\tau$ , in the density operator averaged over the two possible outcomes :

$\rho(\tau)=p_0|{\phi_0^{(A)}}\rangle\langle{\phi_0^{(A)}}|+p_1|{\phi_1^{(A)}}\rangle\langle\bra{\phi_1^{(A)}}|\ .$

Plugging in this equation the expressions of the final state and keeping the first order terms in $\tau$ , we get :

$\rho(\tau)-|{\phi^{(A)}}\rangle\langle{\phi^{(A)}}|=-\frac{\kappa\tau}{2}\left[a^{\dagger}a,|{\phi^{(A)}}\rangle\langle{\phi^{(A)}}|\right]_+ +\kappa\tau a|{\phi^{(A)}}\rangle\langle{\phi^{(A)}}| a^{\dagger}\ ,$

where $[,]_+$ is an anti-commutator. Identifying the left hand side with $d\rho/dt$ , we recover the master equation

The Monte-Carlo approach has many advantages. Assuming that Bob is observing the environment and communicating the results of his measurements to Alice means that she can, at all times, describe her system by a wavevector which evolves randomly, according to the unpredictable outcomes of the measurements. Calculating a set of Monte Carlo trajectories can be much more economical, for large photon numbers, or more generally for high-dimensionality systems, than solving the master equation.

Moreover, this approach is adapted to the description of a single realization of an experiment manipulating a unique quantum system. In many modern experiments, an information is continuously extracted from the system’s environment and its wave function follows a random trajectory exhibiting explicit quantum jumps. The statistics of these jumps is reproduced by a Monte Carlo simulation involving an environment with as many states as the number of possible exclusive measurements outcomes. In most cases, this number is small and a few quantum states of the environment detectors owned by Bob are enough to simulate the experimental results.

The Monte Carlo procedure is easily applied to the relaxation of a Fock state $|n\rangle$ . Being an eigenstate of the pseudo-hamiltonian, a Fock state does not evolve between quantum jumps. Each quantum jump is a transition between two adjacent Fock states. The cavity is always in a Fock state and the photon number undergoes a staircase evolution, with random jumps. The usual exponential decay is recovered by averaging many trajectories. We now turn to the interesting case of an initial coherent state.

The coherent state paradox

Let us describe the Monte Carlo trajectories starting from a coherent state $|{\beta}\rangle$ . The probability for counting a photon in the first time interval is $p_1=\kappa\tau|\beta|^2=\grk\tau\overline n$ . If a photon is counted, the state is unchanged since $|{\beta}\rangle$ is an eigenstate of the jump operator $\sqrt{\kappa}a$ . An evolution occurs if no photon is recorded. The imaginary frequency contribution to the pseudo hamiltonian produces a decrease of the state amplitude :

$|{\beta}\rangle\rightarrow |{\beta e^{-\kappa\tau/2}}\rangle\ .$

This situation is paradoxical, since the field looses energy only if no photons are counted ! How comes that the state does not change when one photon has been lost ? We have the combination of two effects, which tend to change the photon number in opposite directions. One the one hand, the loss of a quantum reduces the average energy in the cavity. On the other hand, the photon click provides an information about the state in which the field was just before the photon was emitted, which tends to increase the photon number.

To understand this point, let us ask a simple question : knowing that it has emitted a photon at a given time, what is the probability $p_{c}(n)$ that the cavity contained $n$ photons just before the click ? If the photon emission probability were independent on $n$ , $p_{c}(n)$ would be equal to $p(n)=e^{-\overline n}\overline n^n/n!$ , the photon number distribution in the coherent state. In fact, the probability for losing a photon is proportional to $n$ . It follows that $p_{c}(n)=knp(n)$ , where $k=1/\overline n$ is determined by normalization. Thus :

$p_{c}(n)=\frac{np(n)}{\overline n}=e^{-\overline n}\frac{\overline n^{n-1}}{(n-1)!}=p(n-1)\ .$

For a coherent state, $p_{c}(n)$ is equal to $p(n-1)$ , the unconditional probability that the initial field contains $n-1$ photons. The maximum of $p(n-1)$ , hence of $p_c(n)$ , occurs for $n=\overline n +1$ . The knowledge that a click has occured biases the photon number distribution before this click towards larger $n$ values, with a photon number expectation exceeding the a priori average value by one unit. The loss of the photon signaled by the click brings this number back down to $\overline n$ . The two effects exactly cancel and the state of the field has not changed !

This is a peculiar property of coherent states. For non-poissonian fields with larger fluctuations, the effect could be even more counter-intuitive and result in an overall increase of the photon expectation number, just after the loss of a photon ! This situation requires fluctuations in the initial field energy. It does not occur for a Fock state, which looses its energy in an intuitive way.

Why, now, are coherent states loosing their energy when no photon clicks are registered ? If no photon is detected, it is more likely that the photon distribution has less photons that was a priori assumed. To consider an extreme situation, a coherent field has a small probability, $e^{-\overline n}$ , for containing no photon at all. This is the probability that the detector will never click, however long one waits. The longer the period without click, the more likely it becomes that the field is effectively in vacuum. Its wave function evolves continuously, without jump, under the effect of the non-unitary evolution and ends up in $|0\rangle$ !

The insensitivity of coherent states to jumps make their Monte Carlo trajectories certain. To determine the state at time $t$ , we have concatenate evolutions during the successive intervals between jumps $t_1$ , $t_2,\ldots t_i,\ldots t_N$ adding up to $t$ . Whatever the distribution of these intervals, the final state, $|{\beta e^{-\grk (t_1+t_2+....+t_N)/2}}\rangle=|{\beta e^{-\grk t/2}}\rangle$ remains pure : coherent states are impervious to entanglement with the environment. They are the `pointer states’ of cavity decoherence.

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