A Qubit as a Sensor

A Qubit as a Sensor / high school

Quantum for high school · page 3 of 3

The same arrow, used the other way round

A qubit as a sensor

A qubit's hidden direction turns at a rate set by the field around it. On page 2 that was a nuisance. Here it is the whole point. Start the arrow, let the world turn it, read how far it went, and you have measured the field. This is how the clocks behind GPS work, and how a diamond can sense the magnetic field of a beating heart.

What you need to know already

The arrow and its hidden direction. From page 1. Looking reads the height. Tipping sideways turns the hidden direction into height.
A magnet has a field. A compass needle turns to line up with it. Stronger magnet, stronger field. We will use magnetic fields, but everything works the same for electric ones.
A pile of coin flips tells you the odds. One flip does not. From page 1, section 02.

seen from above, the arrow is a clock hand

A field turns the arrow, like a clock hand

A compass needle swings to point along a field and stops. A qubit's arrow does something different. It keeps turning around the field, and the stronger the field, the faster it turns.

Seen from above, a sideways arrow is a clock hand. The field sets how fast it goes round.

The dial below is the sphere seen from the top, so you are watching the hidden direction directly. Turn up the field and the hand speeds up. Double the field and it goes round twice as fast. Every qubit does this, and the rate for a given field is a fixed number for each kind of qubit. That number is what turns a qubit into a ruler for fields.

Try it The clock hand

the sphere from above

Field strength

The number physicists use. For an electron, the hand goes round about 28 billion times per second in a field of one tesla, which is a strong laboratory magnet. In the Earth's field, about fifty millionths of a tesla, it still goes round more than a million times a second. Qubits are very fast clocks.

For teachers

Larmor precession. The Bloch vector rotates about the field axis at ω = γB, with γ/2π ≈ 28 GHz/T for a free electron and about 42.6 MHz/T for a proton. The dial is the equatorial plane of the Bloch sphere, so its angle is the phase φ from page 1. Precession, not alignment, is the point students most often miss, since a compass needle is the only field-following object they have handled.

Start the clock, wait, read it

You cannot see the hand turn. You can only look, and looking reads height. So tip the hand back up before you look, and the height tells you how far it turned.

Tip sideways, wait, tip back, look. The chance of reading 0 goes up and down as the wait gets longer, and the field sets how fast.

angle turned = rate × field × wait
chance of 0 = (1 + cos(angle turned)) / 2

This four-step recipe is the same fork-in-the-road from page 2, with the delay set by the world instead of by you. Slide the wait time. The chance swings between 100 percent and 0 percent, one full swing for every full turn of the hand. Now change the field. The swings get faster or slower. If you know the wait and count the swings, you know the field.

Try it The stopwatch experiment

the hand at the end of the wait

Field strength

Wait time

Why this is a sensor. The chance depends on the field through the angle turned. If you can measure the chance, you can work backwards to the field. And measuring a chance is what the next section is about.

For teachers

The Ramsey sequence: π/2 pulse, free evolution for τ, π/2 pulse, measure. P(0) = cos²(γBτ/2) = (1 + cos γBτ)/2. It is the H · R_z(φ) · H circuit of page 2 with φ = γBτ set by the environment. Atomic clocks run exactly this sequence and adjust their oscillator until the fringe sits at its steepest point. Field units on the bench are relative; the footer of section 01 gives the real scale.

Beating randomness by counting

One look gives a 0 or a 1, which says almost nothing about the chance. A thousand looks pin the chance down to a few percent, and that pins down the field.

A hundred times more looks makes the answer ten times better. The error shrinks with the square root of the count.

The page hides a field strength. Choose how many times to run the stopwatch experiment and it counts the zeros, works out the chance, and turns that into a field estimate with an error bar. Watch the error bar shrink as you add looks, and notice that it shrinks slowly. Ten times the looks buys only about three times the precision.

Try it Estimate a hidden field

Number of looks

The square-root rule. With N looks the chance is known to about 1/(2√N). For N = 100 that is 5 percent. For N = 10,000 it is half a percent. Real quantum sensors run the experiment millions of times a second, which is how they reach such precision.

For teachers

Binomial statistics: the standard error of an estimated probability is √(p(1−p)/N), at most 1/(2√N). Propagated through the fringe at its steepest point, the phase uncertainty is 1/√N radians, which is the standard quantum limit. The bench biases the hidden field so the operating point sits on a steep part of the fringe; a real sensor does this deliberately. Entangled sensors can beat 1/√N, but that is beyond this page.

The hand forgets

A longer wait means a bigger angle and a more sensitive measurement. But stray fields from everything nearby jiggle the hand, and after a while its direction is scrambled.

Wait longer for more sensitivity, but not too long, because the swings fade away. There is a best waiting time.

Every atom near the qubit makes its own tiny field, and they all wobble. The hand picks up a little random extra turn each moment. Average over many runs and the swings in the chance get shallower and shallower. Physicists call the time it takes for them to fade the coherence time. Turn up the noise and watch the swings die.

Try it Noise kills the swings

Noise from the surroundings

The materials problem. Making the hand remember longer is a question of what surrounds the qubit. Fewer nearby atoms with their own little magnets, purer crystals, and colder temperatures all help. A large part of quantum research is exactly this: chemistry and materials science in the service of a longer memory.

For teachers

Dephasing with coherence time T₂. The fringe contrast decays as e^−τ/T₂, and the best interrogation time for sensitivity is τ = T₂/2. The undergraduate page "Why a Qubit Forgets" develops this fully, including why fast-moving surroundings are less harmful than slow ones and how a spin echo undoes slow noise.

Where this is already in use

Quantum sensors are not a future technology. Some of them have been running for decades, and one of them is in orbit above you right now.

Atomic clocksGPSEvery GPS satellite carries a clock that runs the stopwatch experiment on atoms billions of times a second. Your phone's location comes from comparing those clocks. An error of a millionth of a second would put you 300 metres off.

MRIMedical imagingThe hydrogen nuclei in your body are qubits. An MRI scanner tips them sideways, lets them turn in a strong field, and listens to the signal as they lose their direction. Different tissues forget at different rates, and that is the contrast in the picture.

DiamondMagnetometersA single electron trapped in a diamond can sense the magnetic field of a nerve firing or a heart beating, from outside the body, at room temperature.

NavigationWhere GPS cannot reachSubmarines, and aircraft when GPS is jammed, are starting to navigate by measuring tiny changes in gravity and in the Earth's magnetic field with atom-based sensors.

Why this matters

The same fragility that makes quantum computers hard to build makes quantum sensors easy to make sensitive. A qubit forgets its direction because it feels every field around it. A sensor is a qubit whose forgetting you have learned to read. Which is why the people who build better qubits for computing and the people who build better sensors are often the same people, working on the same materials.

Try this at home a field turns a needle, and a phone can measure it

You need

A compass, or a compass app
A fridge magnet
A phone with a magnetometer app (most have one; search for "magnetometer" or "tesla meter")
A ruler

Steps

Open the magnetometer app. Away from anything magnetic it reads the Earth's field, roughly 25 to 65 microtesla depending on where you live.
Bring the fridge magnet toward the phone from 30 cm away, 2 cm at a time, and write down the reading at each distance.
Plot reading against distance. The field of a small magnet falls off very fast, roughly as one over the distance cubed. Halving the distance gives about eight times the field.
Now hold the compass near the magnet. The needle swings and stops. A qubit's arrow would not stop. It would keep turning, faster the closer you got.

Why this one. The phone's magnetometer is a real field sensor, and the one-over-distance-cubed rule is why a qubit only feels its nearest neighbours, which is the whole story of section 04.

That is the end of the three pages. A bit is a switch. A qubit is an arrow with a hidden direction. Turn the direction on purpose and you can make wrong answers cancel, which is computing. Let the world turn it and read how far it went, which is sensing. Everything else is detail, and the detail is where the careers are.