User:Bryanrichards13/sandbox/quantum sensor

Quantum sensors are devices that employ qubits to measure physical quantities of interest.

Definitions and criteria
The meaning of "quantum sensing" can differ by context. Two candidate definitions (each of which describe aspects of the quantum sensors discussed in this article) are


 * 1) A system with quantized energy levels employed to measure a physical value.
 * 2) A system exhibiting quantum coherence which is used to measure a physical value.

A third sense in which "quantum sensing" is sometimes used is in describing the measurement of a physical value using quantum entanglement, sometimes also referred to as "quantum metrology". Such sensors are not covered here.

There is a set of agreed-upon criteria for a system to be considered a quantum sensor:


 * 1) The system must be well-modeled by a qubit, or an ensemble of qubits. It must have two discrete energy levels, rather than continuous bands.
 * 2) There must be methods of initializing and reading out the state.
 * 3) The system must be able to be transformed between states in a coherent fashion. (Not necessary for all forms of quantum sensing.)
 * 4) The system must couple with a physical quantity to be measured.

It should be noted that criteria (1)-(3) are also in the set of DiVincenzo criteria listed for quantum computing devices.

Generic features
In general, the Hamiltonian for a quantum sensor takes the form

$$\hat{H}(t) = \hat{H}_0 + \hat{H}_V(t) + \hat{H}_\textrm{control}(t)$$,

where $$\hat{H}_0$$ is the Hamiltonian of the sensor qubit without external fields, $$\hat{H}_V(t)$$ describes the interaction of the qubit with the quantity to be measured, and $$\hat{H}_\textrm{control}(t)$$ describes the interaction with fields used to manipulate the state of the qubit (e.g. for initialization and readout). In what follows, we designate $$\omega_0$$ the transition frequency for the energies of $$\hat{H}_0$$, and $$\Gamma$$ the transition rate between those levels. The qubit Hamiltonian is often denoted

$$\hat{H}_0 = E_0 |0\rangle \langle0| + E_1 |1\rangle \langle 1|.$$

In applications where the signal is small compared to the differences in energy levels of the qubit's internal Hamiltonian, the signal Hamiltonian $$\hat{H}_V(t)$$ can be considered a perturbation on the qubit Hamiltonian $$\hat{H}_0$$. In such a case, $$\hat{H}_V(t)$$ can be resolved into a commuting and a noncommuting component:

$$\hat{H}_V(t) = \hat{H}_{V_{\bot }} + \hat{H}_{V_{\parallel}}.$$

These can be written

$$\hat{H}_{V_\parallel} = \frac{1}{\sqrt{2}}\gamma V_\parallel (t)\left( |1\rangle \langle 1| - |0 \rangle \langle 0| \right)$$

$$\hat{H}_{V_\bot} = \frac{1}{\sqrt{2}}\gamma \left( V_\bot (t) |1 \rangle \langle 0| - V_\bot^\dagger|0 \rangle \langle 1| \right)$$.

Here, $$\gamma$$ quantifies the coupling between the qubit and the signal of interest $$V(t).$$

The parallel term ("parallel" here is in reference to the quantization axis of the qubit) $$\hat{H}_{V_\parallel}$$ results in a shift of the energies of $$\hat{H}_0$$ (and thus a shift in $$\omega_0$$), while the transverse term $$\hat{H}_{V_\bot}$$ increases the transition rate $$\Gamma$$. This difference plays an important role in the sensing of vector fields.

General sensing protocol
Sensing protocols generally follow a sequence of qubit initialization, interaction, and readout. The qubit is initialized into a known basis state, and possibly then manipulated into a known superposition state. It is then allowed to evolve for a time $$t$$ under the full Hamiltonian $$\hat{H}(t)$$ (in some protocols, the control Hamiltonian is varied during this time), after which it is in a "final sensing state". This final sensing state is read out by projecting it onto a basis of "readout states" (often the same basis of eigenstates of the internal qubit Hamiltonian); an eigenvalue is measured, depending probabilistically on the coefficients of the state. This process is repeated, averaging (either over time or over a noninteracting ensemble) to determine these coefficients of the final state.

This sensing protocol is repeated over an interval of time, potentially while varying the control Hamiltonian. The measurements are used to reconstruct the signal $$V$$.

Measuring energy level shifts
Certain quantum sensors measure time-independent fields without exploiting quantum coherent evolution under the field. Static fields can shift the energy levels of an internal qubit Hamiltonian (as discussed above with regard to $$\hat{H}_{V_\parallel}$$). If the state of the system can be continuously read out and re-initialized while a control field is swept in frequency through a range around $$\omega_0$$, the shifted transition frequency between the qubit states, then $$\omega_0$$ can be determined upon observing a transition. This basic protocol can be used, for example, to measure the local magnetic field using a spin qubit in a nitrogen-vacancy center. This is discussed in more detail below.

Ramsey interferometry (slope or linear detection)
One protocol used for sensing static fields that does exploit quantum coherence is known as Ramsey interferometry. This process can be used with various qubit implementations (including photons, as its name suggests). We assume that a static field $$B$$ exists parallel to the qubit's quantization axis, and seek to measure the magnitude of this field. The Hamiltonian for the qubit and field is

$$\hat{H} = \frac{\hbar}{2}(\omega_0-\Delta)\hat{\sigma_z}$$,

where $$\omega_0$$ is the zero-field splitting of the qubit and $$\Delta = \gamma B$$ is the shift from the external field ($$\gamma$$ is a coupling constant between the qubit and field, e.g. the gyromagnetic ratio for a spin qubit and magnetic field ). By making a transformation to the rotating frame, this Hamiltonian can be written

$$\hat{H} = -\frac{\hbar}{2}\Delta \hat{\sigma_z} \quad \textrm{(rotating} \; \textrm{frame)}$$.

In a Ramsey measurement, the state of the qubit is initialized into $$|0\rangle$$. Recall that a qubit initialized into an energy eigenstate, upon being subjected to a control field tuned to the transition frequency, will undergo "Rabi oscillations" between energy eigenstates at the Rabi frequency $$\Omega$$ depending on the power of the control field. When such a control field is turned on, the Hamiltonian in the rotating frame, using the rotating wave approximation, is

$$\hat{H} = -\frac{\hbar}{2}\Delta \hat{\sigma_z} + \frac{\hbar \Omega}{2}\hat{\sigma_y}$$.

We assume that the control field is much stronger than the field $$B$$ to be measured, so that $$\Delta << \Omega$$. That is,

$$\hat{H} = \frac{\hbar \Omega}{2} \hat{\sigma_y} \quad \textrm{(control} \; \textrm{field} \; \textrm{on)}$$

$$\hat{H} = -\frac{\hbar \Delta}{2} \hat{\sigma_z} \quad \textrm{(control} \; \textrm{field} \; \textrm{off)}$$.

Before detailing the steps of the Ramsey protocol, we form the propagators for the Hamiltonians above to examine the time-evolution under each:

$$\exp\left[-i\frac{\Omega t}{2}\hat{\sigma}_y\right] \quad \textrm{(control} \; \textrm{field} \; \textrm{on)}$$

$$\exp\left[i\frac{\Delta t}{2}\hat{\sigma}_z\right] \quad \textrm{(control} \; \textrm{field} \; \textrm{off)}$$

The Bloch sphere representation can help in visualizing the evolution of the qubit state under these propagators. Recalling that $$\exp[i\theta\hat{\sigma}_i/2]$$ is a counterclockwise rotation about the $$i$$th axis ($$i = \{x,y,z\}$$) by an angle $$\theta$$, we see that these propagators represent rotations of the Bloch vector about the $$y$$- and $$z$$-axes by an angle that depends on the time $$t$$. In particular, with careful choice of a time for which the control field is on, one can form a $$\pi$$-pulse

$$\hat{\pi} = \exp\left[-i\frac{\Omega \tau_\pi}{2}\hat{\sigma}_y\right]$$

or a $$\pi/2$$-pulse

$$\frac{\hat{\pi}}{2} = \exp\left[-i\frac{\Omega \tau_{\pi/2}}{2}\hat{\sigma}_y\right]$$,

with $$\Omega \tau_\pi = \pi$$ and $$\Omega \tau_{\pi/2} = \pi/2$$,

which take

$$|0\rangle \mapsto |1\rangle$$, and

$$|0\rangle \mapsto |+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$$,

respectively.

The Ramsey protocol, then, is as follows:


 * 1) The qubit is initialized into the state $$|0\rangle$$.
 * 2) A $$\pi/2$$-pulse is applied, sending the state to the coherent superposition $$|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$$.
 * 3) The state is allowed to evolve coherently for a time $$t$$ according to the Hamiltonian of the applied field $$B$$. In doing so, it becomes $$(|0\rangle + e^{it\Delta}|1\rangle)/\sqrt{2}$$.
 * 4) A second $$\pi/2$$-pulse maps the relative phase accrued in step (3) to a probability amplitude; the state after the pulse is $$i\sin\frac{t\Delta}{2}|0\rangle + \cos\frac{t\Delta}{2}|1\rangle$$.
 * 5) Repeated measurements on an ensemble of this state give a measured value of the probability amplitude, from which a value for $$\Delta$$ can be obtained.

Note that the transition probability $$p \equiv |\langle 1 | \psi \rangle|^2$$, as a function of $$\Delta$$ and $$t$$, is

$$|\langle 1 | \psi \rangle |^2 = \cos^2 \frac{t\Delta}{2} = (1 + \cos(t\Delta))/2.$$

For sensing applications, it is most common to fix $$t$$ so that variations in $$\Delta$$ can be measured.

Intuitively, the sensor will be most sensitive where it exhibits the greatest change in the measured transition probability $$|\langle 1 | \psi \rangle|^2$$ for a given change in $$\Delta$$, that is, where the slope of the function above is a maximum. For this reason, these sensors are often biased by a fixed (known) field $$\Delta_0$$ to operate around the point of greatest slope. This is discussed in greater detail in the Sensitivity section.

Rabi measurement
In contrast to the Ramsey measurement discussed above, which is used to measure a static signal whose Hamiltonian commutes with the internal qubit Hamiltonian (i.e. a signal field parallel to the quantization axis), the Rabi protocol is designed to measure a time-dependent signal whose Hamiltonian does not commute with the qubit Hamiltonian (resulting from a signal field orthogonal to the quantization axis). More specifically, this process measures a value for the matrix element $$|V_\bot|$$, as defined in Definitions and criteria.The protocol is as follows:


 * 1) The qubit is initialized into the state $$|0\rangle$$.
 * 2) The Hamiltonian can be written $$\hat{H} = \frac{1}{2}V_\bot \hat{\sigma}_x = \omega_1 \hat{\sigma}_x$$, with $$\omega_1$$ denoting the Rabi frequency. (The internal Hamiltonian is neglected here.) Upon evolving for a time $$t$$, the state becomes $$|\alpha\rangle \equiv \frac{1}{2}(1+e^{-i\omega_1t})|0\rangle + \frac{1}{2}(1-e^{-i\omega_1t})|1\rangle.$$
 * 3) Repeated measurements in the basis $$\{|0\rangle,|1\rangle\}$$ on an ensemble of this state yield a value for the transition probability $$p \equiv 1- |\langle 0|\alpha\rangle|^2 = \sin^2(\omega_1t/2)$$.

When the internal Hamiltonian is not neglected, the transition probability takes the form

$$p = \frac{\omega_1^2}{\omega_1^2 + \omega_0^2}\sin^2\left(\sqrt{\omega_1^2+\omega_0^2}t\right)$$.

This resonance condition allows the measurement of both the strength $$V_\bot$$ of the applied field as well as its frequency.

Ramsey interferometry (variance or quadratic detection)
The Ramsey protocol discussed above can be altered slightly to measure time-dependent fields. It should be noted that the "slope detection" protocol discussed above is not capable of measuring a time-dependent signal $$\delta V$$ which fluctuates about the point of maximum slope with $$\langle \delta V \rangle = 0$$. However, if the measurement is biased to the point of zero slope where the transition probability $$p$$ is zero, then for small signals $$V$$, it can be shown that

$$\delta p \approx \frac{1}{4}\gamma^2 V_{rms}^2t^2 \quad (\gamma V_{rms}t<<1)$$.

This relation allows $$V_{rms}$$ to be found from a measurement of $$\delta p$$.

For signal fluctuations $$\delta V$$ that follow a Gaussian distribution, then it can be shown that

$$p = \frac{1}{2}[1-\exp(-\gamma^2V_{rms}^2t^2/2)]$$

without the small-signal restriction.

Dynamical decoupling
The sensitivity of the Ramsey protocol is limited in part by decoherence during the free interaction time of the qubit with the field; random phases are acquired due to other fields in the qubits' environment and due to inhomogeneities in the static field to be measured. Modifying the pulse sequence by introducing a $$\pi$$-pulse halfway between the two $$\pi/2$$-pulses mitigates this decoherence. This flips the qubit's Bloch vector to the opposite hemisphere, after which the accumulation of phase prior to the $$\pi$$-pulse is reversed. After this "Hahn echo" sequence is complete, the phase accumulated due to a constant applied field $$\Delta$$ will be zero. The coherence time of the qubit is extended, and is now limited by fluctuations over time in $$\Delta$$ fast enough to occur during the pulse sequence.

While a static field will not be measured by this sequence, an AC field will have an effect on the final state of the qubit and thus can be measured. This pulse sequence can be extended by including more than one $$\pi$$-pulse between the $$\pi/2$$-pulses. By tuning the spacing between these $$\pi$$-pulses, the frequency of an AC signal can be measured. This is known as "dynamical decoupling" or "quantum lock-in detection". Qualitatively, when the spacing in time between $$\pi$$-pulses is equal to a half-period of the signal, the qubit state is flipped every time the direction of the field changes, leading to the greatest accumulation of phase. It is this resonance condition that allows the signal frequency to be determined.

Pulse sequences of this kind employing equally spaced $$\pi$$-pulses include Carr-Purcell and periodic dynamical decoupling. Other, more intricate sequences may also be used, enjoying such advantages as response to a narrower range of frequencies and resistance to errors in pulse shaping.

Quantum projection noise
In measuring the final state of a sensing qubit, a value of $$x_0$$ or $$x_1$$ (corresponding to basis states $$|0\rangle$$ and $$|1\rangle$$, respectively) is read out. For a given final qubit state $$|\psi\rangle$$, if we denote

$$1-p \equiv |\langle 0 | \psi \rangle |^2$$

$$p \equiv |\langle 1 | \psi \rangle|^2$$,

then a set of $$N$$ measurements on an ensemble of qubits in the state $$|\psi\rangle$$ gives a binomial distribution. Supposing that $$N_1$$ of the $$N$$ measurements yield a value of $$x_1$$, then $$p$$ is estimated to be $$N_1/N$$, with variance

$$\sigma^2 = \frac{1}{N}p(1-p)$$.

This variance represents the uncertainty introduced into the measurement by the projective measurement.

Quantum decoherence
Coherent evolution and manipulation of qubits can be interrupted by random changes of phase or transitions between states. The probability of such events increases with sensing time. The effect of this decoherence and relaxation on the sensing process is often treated phenomenologically, by introducing a decoherence function $$ \chi(t)$$, such that

$$\delta p_{\textrm{observed}}(t) = \delta p_{\textrm{ideal}}(t)e^{-\chi(t)}$$

where $$p_{\textrm{observed}}$$ and $$p_{\textrm{ideal}}$$ represent the measured transition probabilities with and without decoherence, respectively. In many cases, $$\chi(t)$$ is well described by a power law in time $$t$$. In such a model, the decoherence (relaxation) rate $$\Gamma$$, where

$$\chi(t) = (\Gamma t)^a$$,

gives an approximate decoherence (relaxation) time

$$T_\chi = \frac{1}{\Gamma}$$

which is used to characterize the effect of the particular decoherence or relaxation mechanism on the qubit's evolution. Note that $$T_\chi$$ is the time after which the transition probability $$\delta p_{\textrm{ideal}}$$ has decayed by about 63%.

Qubit manipulation errors
In practice, qubits can be manipulated or initialized into states different than the targeted states. This often leads to a decrease in the observed transition probability $$\delta p$$, and can adversely affect the sensing process in other ways as well. It is important to note that this source of noise does not scale with sensing time, but is constant.

Classical noise
Classical noise enters the measurement during the readout process. In order to translate the data read out into an approximation of the transition probability $$p$$, it is important to compare the classical noise to the quantum projection noise. Cases in which the quantum projection noise dominates the classical readout noise are referred to as "single shot readout," and those in which the reverse is true are referred to as "averaged readout".

Single shot readout
Recall that the measurement of the qubit's final state yields one of two values, denoted $$x_0$$ and $$x_1$$, corresponding to the eigenstates $$|0\rangle$$ and $$|1\rangle$$, respectively, of the internal qubit Hamiltonian. If the noise introduced during the readout of these values is small compared to the projection noise, then a histogram of the values read out will show them to be distributed around $$x_0$$ and $$x_1$$ with a separate peak at each value. (We assume here that these distributions are each well-approximated by Gaussian distributions.) The distributions' tails between zero and one will overlap each other. To estimate $$p$$, each measured value must be associated with either $$x_0$$ or $$x_1$$. Those values in the area of overlap stand a chance of being associated with the wrong eigenvalue. This observation provides a way to quantify classical readout noise in the single-shot regime. Denoting by $$\kappa_0$$ and $$\kappa_1$$ the fractions of the measured values assigned to $$x_0$$ and to $$x_1$$, respectively, then the variance from the readout noise is given by

$$\sigma_{\textrm{readout}}^2 = \frac{1}{N}[\kappa_0(1-\kappa_0)p + \kappa_1(1-\kappa_1)(1-p)]$$.

If

$$\kappa_0 \approx \kappa_1 \equiv \kappa << 1,$$

then this becomes

$$\sigma_{\textrm{readout}}^2 \approx \frac{\kappa}{N}$$.

Averaged readout
If, on the other hand, the classical readout noise is large compared to the projection noise, then we no longer expect to see separated peaks around the eigenvalues $$x_0$$ and $$x_1$$; rather, the readout value fluctuates to such an extent that two distributions merge into a single distribution. Because there are no longer separate distributions to fit, there is no clear way to associate each measured value with one eigenvalue. Rather, the mean value of the entire distribution is found and used as the value for $$p$$.

It can be shown that in this averaged readout case, the variance of the readout noise is given by

$$\sigma_\textrm{readout}^2 = \frac{\sigma_x^2}{(x_1 - x_0)^2} = \frac{R^2}{4N},$$

where $$\sigma_x^2$$ is the variance of the distribution of measured values, $$N$$ is the number of measurements, and

$$R = \frac{\sigma_\textrm{readout}}{\sigma_\textrm{projection}} = \frac{2\sigma_x\sqrt{N}}{|x_1 - x_0|}$$.

Ramsey interferometry
Taking into account only projection noise, the uncertainty in a measurement made with the Ramsey protocol is given by the uncertainty $$\sigma_{\textrm{proj}}$$ due to the projection noise and the slope of $$|\langle 1 | \psi \rangle|^2$$ as a function of $$\Delta$$. When the sensor is biased to the point of greatest slope, this is

$$\sigma_\Delta = \sigma_p \frac{2}{\tau}$$,

where $$\tau$$ is the time between the $$\pi/2$$-pulses. At this point of greatest slope, $$\sigma_p = 1/2$$. However, this projection noise is reduced when many measurements are made and averaged over. If the measurement is repeated $$M$$ times on $$N$$ qubits, the uncertainty due to projection noise becomes

$$\sigma_p = \frac{1}{2\sqrt{MN}}$$,

and the overall uncertainty is

$$\sigma_\Delta = \frac{1}{\tau\sqrt{MN}}$$.

However, this is only correct for single-shot readout, when it is assumed that the result of each measurement can be associated with a particular eigenvalue ; for averaged readout, when $$M_0$$ measurements must be made and averaged to obtain a single value of $$p$$, the effective number of measurements is reduced:

$$\sigma_\Delta = \frac{1}{\tau\sqrt{MN/M_0}}$$.

This metric does not take into account the acquisition time required to achieve the given precision. Replacing $$M\tau \equiv T$$ above (the acquisition time is equal to the number of measurements times the time needed per measurement), we have

$$\sigma_\Delta = \frac{1}{\sqrt{\tau N T}}$$.

The uncertainty in the measured field, then, can be written

$$\sigma_B = \frac{\sigma_\Delta}{\gamma} = \eta_B\frac{1}{\sqrt{T}}$$,

where $$\eta_B = 1/(\gamma\sqrt{\tau N})$$. The value $$\eta_B$$ is called the sensitivity, and quantifies the achievable precision per averaging time.

Quantum projection noise and averaging have been taken into account when writing down this sensitivity, but the limit imposed by the finite coherence time has not. Denoting the coherence time $$T_2^*$$, it can be shown that the optimal sensitivity for a Ramsey-style sensor is given by

$$\eta_B = \frac{1}{\gamma \sqrt{NT_2^*}}$$.

Dynamic decoupling
As discussed above, the introduction of $$\pi$$-pulses in the "time-reversal" scheme of dynamical decoupling extends the coherence time of the sensing qubit, from $$T_2^*$$ to a time denoted $$T_2$$. It can be shown that the sensitivity in this case is

$$\eta_B = \frac{1}{\gamma\sqrt{NT_2}}$$.

Application: magnetic field sensing with spin qubits
Spin qubits implemented in atomic vapors and nitrogen-vacancy defects in diamond have been shown to be effective sensors of magnetic fields. These are described briefly below.

Atomic magnetometers
Atomic vapor magnetometers generally use electron spins in gas-phase alkali atoms (e.g. rubidium, cesium) as sensing qubits. A laser is used to pump the atoms into a suitable initial state, which interact with the magnetic field to be measured. The electronic states are continuously read out by monitoring the intensity of another laser beam that is shined through the vapor. The term "magneto-optical effects" is often used to describe such changes in the transmissivity of a medium due to the interaction of the atoms with a magnetic field.

As in many cases, interactions between qubits and their environment are a primary source of decoherence. In an atomic vapor, such interactions include the collision of an atom with the cell walls or with another atom. Collisions with the cell walls can be mitigated by using a buffer gas to keep more alkali atoms away from the walls. Additionally, coating the cell walls with substances such as paraffin can reduce the amount of time an atom spends in contact with the wall. Meanwhile, decoherence due to spin-exchange among alkali atoms can be reduced when the number of collisions grows large enough so that the effects average out. For this to work, the rate of collisions must exceed the rate of precession of the Bloch vector in the magnetic field.

NV-center magnetometers
The electronic ground state of an NV-center defect is a spin-triplet, the degeneracy of which is lifted by the crystal field, as well as an external magnetic field if present. This allows the state to be manipulated with microwave radiation and electron spin resonance techniques to be used. Additionally, illumination by green light can result in both fluorescence with an intensity depending on the electronic spin state, and in pumping the state to the $$m_s = 0$$ sublevel of the ground state. This allows continuous initialization and readout.

Decoherence can be caused by carbon atoms in the lattice or by other crystal defects. In particular, paramagnetic defects like P1-centers and nearby NV-centers can be sources of random phase. Dynamic decoupling techniques may be used to mitigate.