Saturday, 8 March 2014

Spins, magnets, and quantum mechanics

Quantum mechanics is often described as an area of physics that deal with energy and matter at the atomic scales, where different weird, unusual stuff happen. To some extent, it is true that quantum objects behave in ways that seem counter to our everyday, common-sense intuition. Unfortunately, focusing on these particular aspects of quantum theory might give the impression that it is mysterious, mystical, and difficult to understand. And that is simply not true. Things like superposition require a little getting used but, for the most part, quantum mechanics works in ways you expect and that naturally make sense. 

I hope to demonstrate this by discussing an experiment with atoms and magnets that is explained in fairly simple terms using quantum mechanics.  This is largely how I was  introduced to the subject many years ago. But to start, we will need to go over some basic ideas regarding magnets and magnetic fields.

Most of us are familiar with magnets from their ability to attract iron and similar metals but in physics, a magnet is just any material that produces a magnetic field. A magnet influences its surroundings through the magnetic field it creates and reacts to the magnetic fields it experiences from other magnetic objects.

A nice thing about magnets is we can understand how they work without having to be very precise. I'm sure many of you have played around with a bar magnet before, which is often found bent into a horseshoe shape, to create a region of particularly strong magnetic field in between the ends labeled north pole and south pole. You probably also know about and maybe experienced first-hand how opposite poles attract and similar ones repel, and how they attract or repel more when two magnets are brought closer together.  
Something less familiar is what determines the force at which a magnet attracts and repels objects. The strength of a magnet is measured by a property called magnetic moment, which is responsible for a magnet's tendency to align with magnetic fields.

According to Maxwell's equations describing electricity and magnetism, a moving electric charge creates a magnetic field, so in terms of physics, they behave like magnets in many situations. If we consider a loop carrying an electric current as shown in the figure below, we find that the magnetic moment depends on the current flow and the area enclosed by the loop, specifically their product.

(a) For a loop forming a circle with area A and carrying an electric current I that flows counter-clockwise, the magnetic moment is given by m = I x A and directed upwards. (b) The magnetic moment of a bar of magnet is the combined effect of the magnetic properties of all its electrons, and points from the south pole towards the north pole of the magnet.

The magnetic moment is a vector quantity like velocity so it comes with a direction. Its direction is given by what's called the right-hand rule, which means you twist your fingers on your right hand in the same direction as the current flow and your thumb will point to the direction of the magnetic moment.  

An electron possesses a magnetic moment like current-carrying loops. It will react to magnetic fields in the same way except that an electron's magnetic moment is a built-in property: because an electron is considered a pointlike object, there is no loop or current that gives rise to the magnetic moment. 

Sometimes it is useful to imagine electrons as small round objects rotating like tops. It is exactly this picture of a spinning electron that gave rise to the term "electron spin," and an electron's magnetic moment is due to its spin. It should be emphasized that while this is a nice way to visualize an electron, it's not meant to be taken literally: if the electron did actually spin, the measured value of the magnetic moment suggests that it spins at a rate much faster than the speed of light, which is inconsistent with Einstein's relativity.

The magnetism of an ordinary bar magnet is produced by the accumulated effect of the magnetic moment of its electrons. In this case, the magnetic moments are mostly aligned along the direction from the south pole to the north pole, which also sets the direction of the  bar's magnetic moment.

Bar magnet in a uniform magnetic field. If its magnetic moment is at an angle compared to the direction of the field as shown, we see that the bar does not move up or down because it experiences equal forces on both poles. However, the bar will start to rotate around an axis that is parallel to the magnetic field due to the torque acting on the bar.

Consider a bar of magnet placed in a uniform magnetic field as shown above. If the source of uniform magnetic field (that is, the north and south poles of the magnet producing it) is located far away from the bar of magnet, then the same amount of force pushes or pulls at it along the direction of the field. The bar magnet remains in place. 

But if the bar is placed at a certain angle to the direction of the field, it will spin around an axis oriented in the same direction as the field. The rotating motion is due to the torque acting on the bar magnet by virtue of its magnetic moment. That pretty much explains all that can happen in a uniform field.

[For completeness, the torque is given by T = m B sin x, where m is the magnetic moment of the bar magnet, B is the strength of the magnetic field, and x is the angle between the directions of the magnetic moment and uniform field.  The direction of the torque is perpendicular to both the magnetic moment and uniform field.]

Bar magnet in a non-uniform magnetic field. The left side shows the bar magnet placed at an angle to the magnetic field, which is generally pointing upwards. Because the south pole of the bar magnet meets the south pole side of the field, it is repelled. The same goes for the north pole in the bottom. However,  the repulsion is stronger at the top because the field is stronger there. So the net effect is to slowly drag the bar magnet down, which is indicated by the green arrow. The right side shows the relative positions of different bar magnets after passing the non-uniform field. The amount and direction in which the bar is displaced depends on the orientation of their magnetic moment.

Now suppose we place the bar magnet in a non-uniform magnetic field, which can be produced by bending a huge bar magnet such that one of the pointy ends of the south pole is aligned to a flat edge of the north pole, like in the figure above. In this case, the magnetic field is stronger near the pointy end side than it is at the flat side. 

If the bar's magnetic moment points in the same direction as the field, its north pole experiences a stronger attraction towards the upper part of the field (since the field pointing that way means it points towards a south pole) then it is pulled upwards as it moves across the field.

If the bar's magnetic moment points opposite to the direction as the field, its south pole experiences a stronger repulsion away from the upper part of the field then it is pulled downwards as it move across the field. This arrows on the screen in the figure indicate the relative position that the bar magnet lands on according to how it is oriented. If the electron were a classical object, the magnetism it acquires from its spin would cause it to behave in a similar fashion.

We make our connection with quantum mechanics by using what we've learned about magnets to describe what's known as the Stern-Gerlach experiment.

Schematic for the Stern-Gerlach experiment. A beam of silver atoms produced from a furnace is sent through a non-uniform magnetic field and collected on a detecting screen.If the magnetic properties of atoms behave like bar magnets, we expect a continuous line of spots on the screen since the magnetic moment of the silver atoms should not have any preferred orientation.

In the original experiment done by Otto Stern and Walther Gerlach, pieces of silver were heated in a furnace until it boils. The resulting metallic vapor is sent through a collimating device, which produces a narrow beam of atoms. The beam of silver atoms then pass through a strong, non-uniform magnetic field. Finally, the atoms are collected on a photographic plate or some other type of detecting screen.

If the magnetic properties of silver atoms are like ordinary bar magnets, we expect the atoms on the screen to spread out on a continuous line because if we pick a few atoms at random, there is no preferred direction for its magnetic moment.

A silver atom with its valence electron with its magnetic moment pointing up, the atom get displaced to the top spot on the screen. If it were pointing down, it would be deflected to the bottom spot indicated.

When they did the experiment, Stern and Gerlach observed that the atoms mostly went into one of two spots on the screen. If we did think of the silver atoms as magnets, it seems that their magnetic moment, for whatever reason, points up or down only.

The fact that there are two beam of atoms is split into two parts is already unexpected but what's really surprising is that you can change the orientation of the non-uniform magnetic field and in general you will get the same kind of splitting into two spots. Somehow the atoms have a magnetic property that has only two possible values whatever the alignment of the field is.

Of course, nowadays we understand why this happens: it's because almost all of the contribution to the magnetic moment of a silver atom comes from the single electron in its valence or outermost shell. In effect, a silver atom (or any atom with a single valence electron) responds to the magnetic field basically the same way an electron would. The Stern-Gerlach experiment actually proved that a quantum property called spin exists for electrons although this was not immediately realized at the time.

For our purposes, we will identify the magnetic moment of the electron in the Stern-Gerlach experiment with the spin of the electron. (Indeed, they are directly related although strictly speaking, because the charge of an electron is negative, the spin actually points opposite the direction of the magnetic moment.)

Through the Stern-Gerlach apparatus, we can prepare, control, and measure the spin of electrons in various ways and see a description in terms of qubits aids our understanding of what's happening.

As we have doing in some of the last few entries, we consider an electron spin can point up or down, which we write as states |u) and |d) for the electron. In the Stern Gerlach experiment, the two spots we get on the screen correspond to measuring the spin of the valence electron in the silver atoms in the states |u) and |d).

It is possible to change the alignment of the non-uniform magnetic field. Each orientation essentially corresponds to a measurement of the the electron spin pointing up or down in that direction. The left side shows one way to label the orientations using quantum states and the right side shows  how one might implement a "|+) or |-)" measurement using a Stern-Gerlach apparatus.

If atoms that are deflected up are labelled |u) and atoms that are deflected down are labeled |d), we can also have superpositions of |u) and |d) if we rotate the orientation of the non-uniform magnetic field. For instance, in the figure above, we can put the field sideways so that it splits the beam of atoms into "|+) or |-)". 

Any other orientation in the circle splits the beam of atoms into two groups whose electrons are in one of two possible spin states, 

a |u) + b|d) or a|u) - b|d)

for some numbers a and b. It might be worth recalling here that a = b = 1/sqrt(2) for the|+) and |-) states. Also, the numbers a and b correspond to probabilities for measuring the electron with spin pointing up and down, where the specific probabilities are given by |a|^2 and |b|^2, respectively.

Schematic for |u) and |d) selectors. The top part shows how the selectors are implemented with the Stern-Gerlach apparatus and the bottom part shows a box notation for each selector.

There are two main types of devices we can utilize in the experiment. The first one is called a selector. A selector choose one of the two possible spin states for any particular orientation. The selectors for |u) and |d) are shown in the figure above. 

As you can see, since the atoms coming out of a selector are guaranteed to be in the state |u) or |d) (or whatever other state is chosen), it allows us to prepare a beam of atoms in a known, definite state.

In the figure we've also introduced a box notation that describes what state a selector selects. The box notation will be useful when we describe situations where a number of devices are used in sequence.

A flipper reverses the direction of the spin using a uniform magnetic field. As shown on the left box, if you have a spin pointing up and uniform magnetic field pointing out towards you, then spin rotates in a clockwise manner. Applying the field for a sufficient amount of time and the spin's direction is reversed. The box notation on the bottom right shows that our flipper here flips |u) and |d).

Another device is called a flipper and what it does is to flip the spin value, so |u) is turned into |d) and vice-versa. With atoms, this can be achieved by applying a uniform magnetic field of the right strength and duration, as depicted in the figure above. 

Using combinations of flippers and selectors, we can explore different ways to prepare, manipulate, and measure beams of atoms. One of the simplest situations to look at is using a flipper and a selector in sequence, which can be ordered in two ways. 

If you have a selector followed by a flipper, then the selector picks out the atoms in the |u) state and the flipper flips those spins down into the |d) state. So the atoms that come out of this particular arrangement will be all have spins pointing down.

If you have a flipper followed by a selector, then the flipper switches the spins of the atoms first, so those pointing up will flip down and those pointing down will flip up, and then the selector picks out all the atoms that ended up in state |u). So the atoms resulting from this arrangement will all have spins pointing up.

This leads us to our first lesson regarding operations we perform on quantum objects: the order in which they are applied is important.  

Consider a slightly more advanced situation involving two flippers and one selector, shown in the figure above. What happens to a beam of atoms when we let them pass through an up-selector, sandwiched by two up-down flippers? Here is helpful to consider an incoming beam of atoms in the state a|u) + b|d). We choose this state only because we want to be able to say something about any arbitrary state. 

Now, the beam first encounters the up-down flipper, which turns the state of the atoms into b |u) + a|d), since it swaps the states |u) and |d). 

Next, we have the up-selector, which picks up atoms pointing up. Since the state at this point is  b |u) + a|d), the probability the spin is measured as pointing up is |b|^2. If we only care about the state of the atoms that go through and not how many, then all that matters is that the atoms that get transmitted will be in the |u) state.

Finally, the spins are flipped and since all the atoms at this point are in the state |u), they get flipped to state |d). If we interpret the probability as the ratio of atoms that come out of the devices from those that went in, then we can also say that |b|^2 of the original atoms in the incoming beam are left.

Compare this with what happens when we have a down-selector. If we again start with an incoming beam of atoms in the state  a|u) + b|d), we expect a fraction |b|^2 of them to be transmitted by the down-selector, and of course, the selected atoms will be in the |d) state. 

Since we made our observation for an arbitrary spin state, we learn the following: an up-selector sandwiched by two up-down flippers works in the exact same way as a down-selector. For now, the most general thing we can say about it is this: flippers and selectors can be combined in different ways to achieve the same result.  

The last example we wish to consider are sequences of selectors, like the two shown above. The top sequence is an up-selector followed by a down-selector. It's pretty easy to determine what happens here since the first selector produces a beam of |u) atoms and the second selector transmits only |d) atoms. Clearly, none of the atoms come out of this sequence.

In the bottom sequence, we insert a plus-selector between the two selectors. Recall that a plus-selector transmits only atoms with spin in the |+) state. To follow what happens in this case, it is helpful to suppose an incoming beam of atoms in the state a|u) + b|d)

First, the up-selector transmits the fraction |a|^2 of atoms that are measured in the |u) state. Next, the plus-selector transmits atoms in state |+). Because  

|u) = 1/sqrt(2) |+) + 1/sqrt(2) |-), 

this means that |1/sqrt(2)|^2 = 1/2 of the atoms that pass through the up-selector get through the plus-selector as well. Finally, the down-selector transmits half of the beam transmitted by the first two selectors, since  

|+) = 1/sqrt(2) |u) + 1/sqrt(2) |d)

All in all, if we start with an beam of atoms in state a|u) + b|d), we expect the fraction (|a|^2)/4 of them to come out of the bottom sequence in the |d) state.

This is just an illustration of how a selector "disturbs" the quantum state of our beam of atoms. It's already apparent in how a single selector works, since you could say that any incoming beam forgets its previous state and is converted into the state being selected. The above example just shows that the amount of disturbance can be rather significant. 

After playing around with beams of atoms, flippers, and selectors, what now? Perhaps it has not been obvious since the only quantum stuff we've really used are quantum states for describing the beam of atoms as they pass through devices but we've actually pretty much covered all the essential features of the quantum mechanics of qubits. All that I've really left out of the discussion is the mathematical ingredients that you would need if you wanted to be more rigorous and systematic. 

Some of the mathematics used for describing qubit states and operations.

A few of these ingredients are shown above. Those familiar with the technical details know that states are represented by column vectors, the flippers are represented by matrices known as Pauli operators, and the selectors are represented by projection operators. 

It's been a long journey but the main point we're trying to make in this post is this: sure, quantum mechanics involves plenty of stuff that are quite unexpected, bizarre even but despite these elements, it is sound, logical, and easy to understand once you become acquainted with its rules. It's quite cumbersome without the math, but if you just wanted a feel for what the essence of quantum physics is, you can go a long way with spins, magnets, flippers and selectors.

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