A question about atomic clocks

Question

I have a rather simple question about atomic clocks. I have read that:

Microwave radiation with a frequency of exactly 9.192.631.770 cycles per second causes the outermost electron of cesium-133 atom to reverse its spin direction. An atomic clock uses this phenomenon to tune moicrowaves to this exact frequency. It then counts 1 second for each 9.192.631.770 cycles.

So does that mean that in an interval of 1 second the outermost electron of the cesium-133 atom changes its spin direction 9.192.631.770 times? And let 1 and 2, be the two possible directions of spin. Does one change mean a change from 1 to 2 , or a change 1 to 2 and back to 1?

Answer 1

If you measure the absorption of microwaves by a gas of caesium atoms you'll get a spectrum looking something like this:

_{NB not a real spectrum - I drew this as an illustration}

When the microwave frequency is 9,192,631,770 Hz the microwave photons have exactly the right energy to flip the outer electron spin, so at this frequency they are more strongly absorbed than at nearby frequencies. Energy is absorbed from the microwaves to flip the electron spin, then the electron either reradiates a photon and relaxes back or it loses the energy and relaxes due to collisions with other caesium atoms.

The caesium clock works by tuning the frequency for maximum absorption i.e. exactly at the peak of the absorption line. Then it's just a matter of counting each cycle of your microwave generator, and after 9,192,631,770 of them one second has passed.

Response to comment:

The electron flips every time a caesium atom absorbs a microwave photon. So if we can work out how many photons per second are absorbed this will tell us how many electrons per second are flipped.

Suppose the microwave power is $P$, and the absorptance (the depth of the trough in the diagram above) is $f$, where $f = 0$ means no microwaves are absorbed and $f = 1$ means all the microwaves are absorbed. The the power absorbed is $Pf$. The energy of a photon is $h\nu$, where $\nu$ is the frequency of the hyperfine absorption 9,192,631,770 Hz. Then the number of photons absorbed per second, which is the same as the number of electrons flipped per second, is:

$$ N = \frac{Pf}{h\nu} $$

Alternatively, if $N_C$ is the number of caesium atoms in the clock then the number of times the electron is flipped per second per caesium atom is:

$$ N_a = \frac{Pf}{N_Ch\nu} $$

However I have no idea what power microwaves are used in an atomic clock or what fraction of the microwaves are absorbed, so I'm afraid I can't calculate the numbers $N$.

Answer 2

It does not mean that it switches the hyperfine state that often. I means that, if you hit the atom with a photon that (by $E = \hbar\omega$) corresponds to exactly that frequency, then the energy of that photon will precisely be the energy to lift the electron from the lower hyperfine state to the upper hyperfine state. If hit, it will then, as with all other orbital excitations, stay a short while in the upper state before dropping into the lower again, sending out a photon of exactly (modulo uncertainty) the same energy again.

All an atomic clock does is now to tune some kind of circuit so that it oscillates exactly with the same frequency as the photons/microwaves associated to this transition. (Not actually that easy)

Answer 3

edit: Actually I don't think the picture below for the different hyperfine states is quite correct. I'll try to update it with a more correct picture at a later time.

Consider the Cs atom to have two states. A ground state and an excited state. The ground state corresponds to the nuclear spin pointing down (along the $-z$ axis) and the excited state corresponds to the nuclear spin point up*. These starts are separated in energy by $E = h \times 9,192,631,770 \text{ Hz}$ where $h$ is Plank's constant. In both of these states we can also imagine electron to have a total spin and orbital angular momentum in the $+z$ direction as well. To imagine this angular momentum, imagine the electron is orbiting the nucleus in the $x-y$ plane. This means it is creating a current loop around the nucleus that creates a magnetic field at the location of the nucleus. Since the nucleus has a magnetic moment and it is in a magnetic field this means there will be an energy difference between the nuclear spin pointing up along the magnetic field of the electron and the nuclear spin pointing down against the magnetic field created by the electron*.

It is known that light composed of photons with the same energy (that is microwave radiation at $9.192... \text{ GHz}$) can cause a transition from the ground to the excited state. However, additionally it is possible to use a special pulse of light of that same frequency to put the atom into a superposition of ground and excited states. See Rabi Cycle.

One quick point on how the radiation is able to affect the internal quantum state of the atom. Recall that the nuclear spin has a magnetic moment. Good. Also recall the light is composed in a part of a magnetic field. Good. The reason the light is able to change the state of the nucleus is because the magnetic field of the light is able to put a force on the magnetic moment of the nucleus. That is, the light is literally tugging on the nucleus a little bit.

If a spin is in a superposition of point down and pointing up then that means it can be thought of as pointing sideways. This is a little trick of quantum spin physics.

Which direction sideways is the spin pointing? Well that depends the direction (and phase) of the magnetic field vector of light when it interacted with the atom. This is something which can be controlled experimentally. Let's say the experimentalists choose to have the atom point in the $+x$ direction after the first pulse of light.

Now the spin of the nucleus is pointing along the $+x$ direction. However, recall that the nuclear spin is sitting in the magnetic field of the electron (which is still whizzing about in the $x-y$ plane.) If a magnetic moment is transverse to a magnetic field then it will under go Larmor precession. Like a gyroscope, the magnetic moment will rotate around the magnetic field. At what speed will the magnetic moment rotate? It will rotate at the timescale set by the energy difference between the two states. The period of oscillation will be

$$ T = \frac{h}{E} = \frac{1}{9,192,631,770 \text{ Hz}} $$

It has occasionally been asked 'what is ticking in an atomic clock?', trying to make a comparison to a grandfather clock, for example, in which case it is obvious that the pendulum's motion under gravity is the source of the ticking. The answer is that in microwave atomic clocks it is the nuclear spin precessing about the magnetic field of the electron which 'ticks'.

To answer your question briefly: $9,192,631,770 \text{ Hz}$ is the timescale corresponding to the precession of the nuclear spin about the magnetic field of the electron. In addition, this also corresponds to the frequency of light which will cause a transition of the atom from the ground to the excited state. It is easy to see how these two concepts might get confused. Because these two frequencies are the same, if we are able to measure the transition frequency we have also measured the precession or ticking frequency.

*Note that I am giving an intuitive description of the hyperfine splitting which gives rise to the energy difference between the two states we are discussing here. Having a microscopic picture for the two different states greatly helps in understanding the functioning of an atomic clock.