18. Quantum Trajectories for Photodetection¶
PDF pages 833–864
Chapter 18 Quantum Trajectories for Photodetection 18.1 Quantum Jumps In deriving the unconditioned master equation for spontaneous emission in Section 11.5,
(unconditioned master equation for spontaneous emission) (18.1) where the Lindblad superoperator D[\sigma]\rho is again given by
, (18.2) (Lindblad superoperator) we have explicitly ignored the state of the field by computing the partial trace. Now we will consider what happens when we measure it. In particular, we will assume that we make projective measurements of the field photon number in every mode, not distinguishing between photons in different modes. It is this extra interaction that will yield the continuous measurement of the atomic state. From the relation
(18.3) that we derived in the Weisskopf–Wigner treatment of spontaneous emission [Eq. (11.42)], the transition probability in a time interval of length dt is \Gamma\rhoee dt = \Gamma
dt, where we recall that \sigma\dagger\sigma = |e\rangle \langle e| is the excited-state projection operator. Then assuming an ideal detector that detects photons at all frequencies, polarizations, and angles, there are two possibilities during this time interval: 1. No photon detected. The detector does not ‘‘click’’ in this case, and this possibility happens with probability 1 −\Gamma
dt. The same construction as above for the master equation carries through, so we keep the equations of motion for \rhoee, \rhoeg, and \rhoge, where the last two are given by
i\omega0 −\Gamma \rhoge. (18.4) However, we do not keep the same equation for \rhogg: no photodetection implies that the atom does not return to the ground state. To see this, recall that the atom–field interaction Hamiltonian HAF in the rotating-wave approximation contains a term of the form \sigmaa\dagger. The only way for the atom to end up in the ground state (the action of \sigma) is for a photon to be created (the action of a\dagger). So if no photon is detected, then \sigma could not have acted on the atom. Thus, \partial t\rhogg = 0. This case is thus generated by the master equation
¯h[H, \rho] −\Gamma
(18.5)
Chapter 18. Quantum Trajectories for Photodetection This equation is the same as the unconditioned equation (18.1), except for the \sigma\rho\sigma\dagger term, which is precisely what we argued should be omitted. This evolution is unnormalized since Tr[\rho] decays to zero at long times. We can remedy this by explicitly renormalizing the state \rho(t + dt), which gives
¯h[H, \rho] −\Gamma
\rho. (18.6) This follows from adding an extra term to the master equation to satisfy the normalization Tr[\rho] = 1, which implies dTr[\rho] = Tr[d\rho] = 0. 2. Photon detected. A click on the photodetector occurs with probability \Gamma
dt. Again, the interaction Hamiltonian HAF in the rotating-wave approximation contains a term of the form \sigmaa\dagger, which tells us that photon creation (and subsequent detection) is accompanied by lowering of the atomic state. Thus, the evolution for this time interval is given by the reduction
(18.7) We can write this in differential form as
(18.8) so that \rho is subtracted from itself, and is thus replaced by the first term, which is the lowered and renormalized density operator. The overall evolution is stochastic, with either case occurring during a time interval dt with the stated probabilities. We can explicitly combine these two probabilities by using the counting process N(t) (see Sections 17.5.1 and 17.5.2). In any given time interval dt, we define dN such that it is unity with probability \Gamma
dt and zero otherwise. Thus, we can write the average over all possible photodetection histories as
dt. (18.9) (instantaneous photodetection probability) Now we can add the two above possible cases together, with a weighting factor of dN for the second case, to obtain a stochastic master equation (SME):
¯h[H, \rho] dt −\Gamma
dN. (SME for direct photodetection) (18.10)
here is smooth, except punctuated by individual events where the density operator changes discontinuously to the ground state. We will refer to such events as quantum jumps.1 Note that while we are using the notation dN(t) that suggests a Poisson process, because of the state dependence of the mean, N(t) is not Poisson-distributed in general (photon antibunching is a good example), and it is more properly called a counting process (Section 17.5.2). This SME gives the evolution of the quantum state conditioned on the results of the photodetection measurement. To see this explicitly, note that we can write the current from the idealized photodetector as
dN(t) dt , (18.11) where Qph is the total charge conducted by the detector for each photon detection event (which we assume to be deterministic and perfectly repeatable). The current thus appears ideally as a sequence of random 1E. Schrödinger, ‘‘Are There Quantum Jumps? Part I,’’ The British Journal for the Philosophy of Science 3, 109 (1952); E. Schrödinger, ‘‘Are There Quantum Jumps? Part II,’’ The British Journal for the Philosophy of Science 3, 233 (1952).
18.1.1 Ensemble Average¶
18.1 Quantum Jumps delta functions representing the detected photons, and dN is the same quantity that appears in the SME. We can thus write the SME directly in terms of the photocurrent as d\rho dt = −i ¯h[H, \rho] −\Gamma
˜Idet(t). (SME with measurement record) (18.12)
record, since it contains all the information from the measurement of the resonance fluorescence. The SME (18.12) thus shows how to incorporate the measurement record into the evolution of the quantum state—the observer’s state of knowledge of the atomic system. 18.1.1 Ensemble Average To work out the ensemble average of the SME (18.10), we will make use of the ensemble average \langle \langle dN\rangle \rangle from Eq. (18.9). It is tempting to assume the statistical independence of \rho(t) and dN(t),
(wrong!), (18.13) similar to the statement \langle \langle \rho dW\rangle \rangle = 0 from It¯o calculus. Although this assumption does in fact lead to the correct ensemble-averaged master equation, it does so for the wrong reasons: as we can see from Eq. (18.9) that dN(t) depends on \rho(t) [though \rho(t) depends only on dN(t′) for t′ < t, not on dN(t)]. However, we will still need to work out the expectation values like \langle \langle \rho dN\rangle \rangle , that appear in the last term of the ensemble average of Eq. (18.10). Treating the explicit possible outcomes n of dN(t) in terms of
** \rho(t) X n\in {0,1}
++ = **
++ = \Gamma DD \rho(t)
EE dt. (18.14) Similarly, it follows that \rho(t) dN(t)
= \Gamma
\rho(t)
dt. (18.15) Using these two relations, the ensemble average of Eq. (18.10) becomes
DD
\rho EE dt +
dN ++ = −i
DD
\rho EE dt + \Gamma
DD \rho
EE dt = −i
(18.16) The resulting ensemble-averaged master equation,
dt = −i
(18.17) (ensemble-averaged SME) has precisely the same form as the unconditioned master equation (18.1) that we derived earlier. The unconditioned master equation thus follows from a measurement process, where the information from the measurement is discarded, and the observer averages (traces) over all possible measurement outcomes.
18.1.3 Information Gain¶
Chapter 18. Quantum Trajectories for Photodetection 18.1.2 Quantum Trajectories and the Stochastic Schrödinger Equation It is a fairly easy exercise to verify that this master equation is equivalent to the stochastic Schrödinger equation (SSE)
\sigma p
−1 !
(SSE for direct detection) (18.18) by keeping terms to second order,
=
+
, (18.19)
the dissipation in the master equation—which can’t be written as a deterministic Schrödinger equation—can be written as a stochastic Schrödinger equation. Note that Markovian master equations can generally be unravelled in an infinite number of ways; we have only indicated one way so far here. A solution to the SSE is called a quantum trajectory,3 and the solution of the unconditioned master equation represents the average over all trajectories. (Solutions of the SME are also called quantum trajectories.) 18.1.3 Information Gain Clearly, the observer gains information about the atom when the photodetector clicks, but what is perhaps less obvious is that the observer also gains information about the atom even when a photon is not detected. We can see this from either the SME (18.10) or the SSE (18.18), since even during times when photons
‘‘active.’’ Let’s look at the evolution in two cases to see how this works. In particular, suppose the atom starts in a pure state, so that we can work with the SSE. Suppose also that the atom is not driven by an external field, so there are no Rabi oscillations. In this case, we can ignore all Hamiltonian evolution by trasforming to a suitable rotating frame (where the two levels are degenerate). Then the state vector of the general form
(18.20) obeys the equation, assuming no photon is detected,
(18.21) which in terms of the coefficients becomes
1 −|ce|2 ce
2 |ce|2cg. (18.22) Note that unlike the equations of motion for the unconditioned case, these equations are nonlinear in the coefficients due to the measurement process. Now suppose the atom starts in the superposition state
\sqrt
, (18.23) 2This term was coined by Howard Carmichael, An Open Systems Approach to Quantum Optics (Springer, 1993). For earlier work on photon counting statistics, see H. J. Carmichael, Surendra Singh, Reeta Vyas, and P. R. Rice, ‘‘Photoelectron waiting times and atomic state reduction in resonance fluorescence,’’ Physical Review A 39, 1200 (1989). 3Howard Carmichael, op. cit.
18.1.4 Monte Carlo Trajectories¶
18.1 Quantum Jumps and suppose that no photon is detected until some long time t. Since |ce|2 < 1, the excited state decays (somewhat more quickly than exponentially) away asymptotically to zero, and the population is thus trans- ferred to the ground state. The interpretation is this: if the observer knows the atom to be in either state with equal probability, and does not see an emitted photon after a very long time (compared to 1/\Gamma), then the observer concludes that the atom was in fact in the ground state. By not observing a photon, the observer still gains information about the state of the atom and ‘‘collapses’’ it to the ground state. On the other hand, if the initial atomic state is the excited state,
(18.24) then the observer knows that at some point, the atom must decay: the observer knows the atom to be in the excited state with certainty until the photon is detected. This is reflected by the above equations, since if |ce|2 = 1, the excited state does not decay: \partial tce = 0. The measurement terms have interpretations that are less clear if the atom also undergoes Rabi os- cillations. However, the measurement terms must act in the same way in a small time dt, regardless of other processes that influence the atom. The combined evolution of a monitored atom, conditioned on not
Wt/2p
The solid line is what we expect of ordinary, resonant Rabi oscillation without any spontaneous emission
severely distorted by the measurement, which tends to cause decay of the excited-state amplitude; thus the rise in excited-state population is slowed, and the fall is accelerated, leading to oscillations closer in shape to a sawtooth wave. The dotted line shows the probability of the atom not having decayed by that time. This is significant since we only see the distorted Rabi oscillations by keeping experiments where the photodetector does not click; any experiments where the detector clicks must be discarded, so that we post-select on the dN = 0 cases. Note that this nonlinear evolution is thus difficult to see past the first oscillation, since the nondecay probability becomes very small. 18.1.4 Monte Carlo Trajectories Stochastic Schrödinger equations of this form are popular for simulating master equations,4 since if the state vector has O(n) components, the density matrix will have O(n2) components, and thus is much more computationally expensive to solve.5 If s solutions (quantum trajectories) of the stochastic Schrödinger 4R. Dum, P. Zoller, and H. Ritsch, ‘‘Monte Carlo simulation of the atomic master equation for spontaneous emission,’’ Physical Review A 45 4879 (1992) (doi: 10.1103/PhysRevA.45.4879); C. W. Gardiner, A. S. Parkins, and P. Zoller, ‘‘Wave- function quantum stochastic differential equations and quantum-jump simulation methods,’’ Physical Review A 46 4363 (1992) (doi: 10.1103/PhysRevA.46.4363); Yvan Castin, Jean Dalibard, and Klaus Mølmer, ‘‘A Wave Function Approach to Dissipative Processes,’’ AIP Conference Proceedings 275, 143 (1993) (Thirteenth International Conference on Atomic Physics, ICAP-13, H. Walther, T. W. Hänsch, and B. Neizert, Eds.) (doi: 10.1063/1.43795); P. Marte, R. Dum, R. Taïeb, P. D. Lett, and P. Zoller, ‘‘Quantum wave function simulation of the resonance fluorescence spectrum from one-dimensional optical molasses,’’ Physical Review Letters 71 1335 (1993) (doi: 10.1103/PhysRevLett.71.1335); Klaus Mølmer and Yvan Castin, ‘‘Monte Carlo wavefunctions in quantum optics,’’ Quantum and Semiclassical Optics 8, 49 (1996) (doi: 10.1088/1355-5111/8/1/007). 5If you want to be fastidious, a state that can be represented in terms of n basis states will have a state vector that can be represented by 2n −2 real numbers, while the density-matrix representation will require n2 −1 real numbers. The count for
Chapter 18. Quantum Trajectories for Photodetection equation can be averaged together to obtain a sufficiently accurate solution to the master equation and s ≪n, then this Monte-Carlo-type method is computationally efficient for solving the master equation. However, note that the average of the simulated ensemble converges slowly, typically as 1/\sqrts, so this Monte-Carlo method is best used where n is very large (such as when simulating the center-of-mass motion of a quantum system), and very high accuracy is not required. This idea is illustrated here, where the first plot shows the evolution of the excited-state probability for a single atom (quantum trajectory) with jumps to the ground state, corresponding to a detected photon. Nine other trajectories are included to illustrate the dephasing due to the random nature of the jumps. The usual Rabi oscillations are visible here, since the atom is driven by a classical field of Rabi frequency Ω, but
Wt/2p
The second plot shows the ensemble-averaged excited-state probability computed from the unconditioned master equation (solid line), an average of 20 trajectories (dashed line), and an average of 2000 trajectories (dotted line). As many trajectories are averaged together, the average converges to the master-equation solution for the ensemble average. (About 20,000 trajectories are necessary for the Monte-Carlo average to be visually indistinguishable from the master-equation solution on the time scale plotted here.) Note that the ‘‘Rabi oscillations’’ apparent here are distorted slightly by the nonlinear renormalization term in Eq. (18.18) from the usual sinusoidal oscillations in the absence of spontaneous emission. However, the damping rate in the above plots is small, so the distortion is not visually apparent. Unravellings of this form are much easier to solve computationally than ‘‘quantum-state diffusion’’ unravellings involving dW that we will study later. Of course, it is important for more than just a numerical method, since this gives us a powerful formalism for handling the evolution of a quantum state, accounting for photodetection. Although quantum trajectories represent a useful simulation method for the master equation, they are interesting in their own right, since they model the measurement process itself and the resulting conditioned dynamics. In fact, some of the original work6 that motivated quantum trajectories was to understand |\psi\rangle follows from having n complex numbers, one of which can be discarded due to fixed normalization and an arbitrary overall phase. The count for \rho follows from having n real numbers on the diagonal, and then counting the complex numbers above the diagonal (the elements below the diagonal are redundant). The number of complex diagonal numbers is the sum of all integers
real number for Tr[\rho] = 1, and you get n2 −1. 6C. Cohen–Tannoudji and J. Dalibard, ‘‘Single-Atom Laser Spectroscopy. Looking for Dark Periods in Fluorecence Light.’’
18.1.5 Detector Efficiency¶
18.1 Quantum Jumps experiments7 on quantum jumps in vee atoms (a slightly different usage of the term from our usage above), where the fluorescence on a fast transition depends on—blinks on and off to indicate—the state of the atom with respect to the other (slow) transition. The quantum-jump results are understood as ‘‘single-shot’’ (single-trajectory) phenomena, not as ensemble averages. 18.1.5 Detector Efficiency To handle the case of photodetectors with less than ideal efficiency \eta, we simply combine the conditioned master equation (18.10) and unconditioned master equation (18.1), with weights \eta and 1−\eta, respectively:
dN\eta. (SME for inefficient detection) (18.25) The weighting for the counting-process term is absorbed into the counting process itself,
dt, (detection probability for inefficient detection) (18.26) to account for the fact that fewer photons are detected. We can also write this master equation in the form
dN\eta, (SME for inefficient detection) (18.27) where it is clear from the Lindblad-superoperator term that the total disturbance is equivalent to the un- conditioned case, and the subsequent terms, whose effects are proportional to \eta, represent the influence of detecting the fraction \eta of the photons. That sounds reasonable, but let’s do that more carefully. First, divide dN up into two parts, dN1 and dN2, such that
dt
dt. (18.28) The sum of two counting processes is still a counting process, and each counting process is characterized by its intensity (mean transition rate) in the same way that a Poisson process is fully characterized by its mean. Thus, we can write
(18.29) noting that while this is always true for standard Poisson processes, we are justified in writing this here only because the state dependence enters each of the counting processes in the same way here. We can thus write the SME (18.10) as
¯h[H, \rho]dt −\Gamma
dN1 +
dN2. (18.30) Europhysics Letters 1, 441 (1986); P. Zoller, M. Marte, and D. F. Walls, ‘‘Quantum jumps in atomic system,’’ Physical Review A 35, 198 (1987) (doi: 10.1103/PhysRevA.35.198). 7Warren Nagourney, Jon Sandberg, and Hans Dehmelt, ‘‘Shelved Optical Electron Amplifier: Observation of Quantum Jumps,’’ Physical Review Letters 56, 2797 (1986) (doi: 10.1103/PhysRevLett.56.2797); Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, ‘‘Observation of Quantum Jumps,’’ Physical Review Letters 57, 1696 (1986) (doi: 10.1103/PhysRevLett.57.1696); J. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland, ‘‘Observation of Quantum Jumps in a Single Atom,’’ Physical Review Letters 57, 1699 (1986) (doi: 10.1103/PhysRevLett.57.1699); W. M. Itano, J. C. Bergquist, and D. J. Wineland, ‘‘Photon Antibunching and Sub-Poissonian Statistics from Quantum Jumps in One and Two Atoms,’’ Physical Review A 38, 559 (1988) (doi: 10.1103/PhysRevA.38.559); R. G. Hulet, D. J. Wineland, J. C. Bergquist, and W. M. Itano, ‘‘Precise Test of Quantum Jump Theory,’’ Phys. Rev. A 37, 4544 (1988) (doi: 10.1103/PhysRevA.37.4544) W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, ‘‘Quantum Zeno effect,’’ Physical Review A 41, 2295 (1990) (doi: 10.1103/PhysRevA.41.2295); D. J. Berkeland, D. A. Raymondson, and V. M. Tassin, ‘‘Tests for non-randomness in quantum jumps,’’ Physical Review A 69, 052103 (2004) (doi: 10.1103/PhysRevA.69.052103).
Chapter 18. Quantum Trajectories for Photodetection If we detect photons with efficiency \eta, then not detecting a fraction 1 −\eta of the photons is equivalent to taking an ensemble average over dN2, because we are discarding the information provided by dN2. Taking this average, we let double angle brackets denote this ensemble average with respect to dN2, and again note that \rho(t) and dN2(t) are statistically independent:
dN1 +
= −i
dN1
= −i
dN1. (18.31) This is equivalent to Eqs. (18.25) if we understand \rho to be the ensemble average \langle \langle \rho\rangle \rangle and we relabel dN1 −\rightarrow dN\eta. In the case \eta < 1, it is not possible to unravel the SME into an equivalent SSE. It is only in the case \eta = 1 that this is possible, because only in this case does an initially pure state remain pure under the master-equation evolution—otherwise, the observer must trace over all possibilities for the undetected photons and thus necessarily ends up with a mixed state. Of course, the resulting density operator can be computed by simulating many trajectories of the SSE, and then computing the appropriate (partial) ensemble average. 18.2 Homodyne Detection Now we will see how we can get a completely different unravelling of the master equation by modifying the atomic field measurement. In particular, we will obtain a white-noise limit of a jump process in the master equation. To set up this measurement, suppose that all of the light radiated from an atom is collimated ‘‘somehow’’ (e.g., by exotic ‘‘4\pi’’ optics) into a directed beam. An alternative to simply feeding it into a photon-counting detector is to mix it on a beam splitter with a local oscillator. What exactly we mean by a local oscillator is a monochromatic field with an intensity much larger than the atomic field, but otherwise is somewhat context-dependent: if the atom is driven by a monochromatic field, then we would use part of the driving field as the local oscillator, while if the atom is undriven, we would simply use a laser field at the atomic resonance frequency. The point is, the atom will naturally be oscillating with a spectrum centered at some frequency, and we would choose the local oscillator to have the same frequency. Once the radiation field and local oscillator are mixed together by the beam splitter, the light is then detected. This measurement is called homodyne detection,8 and here we will consider the simplest case of homodyne detection, where only one output of the beam splitter is monitored. local oscillator detector Even though this setup detects some ‘‘irrelevant’’ light due to the local oscillator, the amount of information that we acquire via this setup (within the idealizations below) is equivalent to that of direct detection. If the 8Homodyne detection was treated in terms of quantum trajectories first by Howard Carmichael, op. cit. Our treatment here more closely follows that of Howard Mark Wiseman, Quantum Trajectories and Feedback, Ph.D. thesis (University of Queensland, 1994), and H. M. Wiseman and G. J. Milburn, ‘‘Quantum theory of field-quadrature measurements,’’ Physical Review A 47, 642 (1993) (doi: 10.1103/PhysRevA.47.642).
18.2 Homodyne Detection radiated field from the atom is detected directly, the operator that we associate with the field is proportional to C = \sqrt \Gamma\sigma, (18.32) so that the average number of photons detected in a time interval dt is
C\daggerC
dt = \Gamma
dt. (18.33) We can similarly associate the operator Cloc = \sqrt \Gammaa, (18.34) with the local-oscillator field, where a is the annihilation operator for the local-oscillator field mode (which is comparable to the atomic lowering operator \sigma—recall from Section 5.7 that the dipole radiation field is written in terms of the atomic dipole operator \sigma). The normalization here assures that photons in either the dipole or local-oscillator fields are detected in the same way by the detector. The combined field operator, after the beam splitter (on the detector side), is then Cr = \sqrt \Gamma
p 1 −r2 a , (18.35) where r \in [0, 1] is the field reflection coefficient of the beam splitter (as seen by the atomic field). Here, we are assuming that the fields of the collimated atomic radiation and the local oscillator are perfectly ‘‘mode matched.’’ We will model the local oscillator as a coherent state |\alpha\rangle of light—the quantum model for a classical, monochromatic field—with photon flux \Gamma|\alpha|2 (whose detection is also described by a counting process). Recalling that |\alpha\rangle is an eigenstate of the field annihilation operator a, we can write a|\alpha\rangle = \alpha|\alpha\rangle , and thus the operator for the total field at the detector effectively becomes Cr = \sqrt \Gamma
p 1 −r2 \alpha (18.36) whenever applied to the field state. Note also that \alpha is in general a complex number, representing the phase of the local oscillator field. In this homodyne scheme, any atomic radiation that transmits through the beam splitter is not de- tected, and the information associated with that light is ‘‘wasted.’’ Thus we will want to consider the limit r −\rightarrow 1, but we will then also take the limit |\alpha| −\rightarrow \inftysuch that the transmitted field amplitude
p 1 −r2 (18.37) remains nonzero. (In fact we will eventually take the limit \beta −\rightarrow \infty.) Thus, the detected field operator is
\sqrt
(18.38) and so we see that the effect of adding the local-oscillator field is to add a scalar constant to the atomic lowering operator. The average photodetection rate due to the combined field is
D C\dagger
E dt = \Gamma
dt = \Gamma h
+
dt. (18.39) As we expect, the photodetection rate is the sum of the individual photodetection rates for the local oscillator and for the atomic radiation, plus an interference term. Herein lies the advantage of homodyne detection: the atomic signal due to the interference terms is effect ‘‘boosted’’ by the local oscillator by a factor of |\beta|. This is an enormous advantage if the detector suffers from a low level of background noise (‘‘dark currents’’), since the homodyne scheme can raise the signal to a level much larger than background noise. We will also see that homodyne detection measures different aspects of the atom, as compared with direct detection.
18.2.2 Quantum-State Diffusion¶
Chapter 18. Quantum Trajectories for Photodetection 18.2.1 State Collapse When a photon is detected directly from the atom, recall that the state vector is reduced according to
p
=
p
, (18.40) or equivalently, the density operator is reduced according to \rho −\rightarrow
(18.41) The same reduction occurs when a homodyne photon is detected, but now using the operator C\beta, for the state vector,
rD C\dagger
E =
p
, (18.42) and of course the density operator, \rho −\rightarrow
\beta D C\dagger
(18.43) The reduction here is partial: when a photon is detected, it is not possible to distinguish whether the photon came from the atom or the local oscillator, and thus the atom is projected into a coherent superposition of being reduced as in the direct-detection case (if the photon came from the atom) and of being unaffected (if the photon came from the local oscillator). The two cases are weighted by the amplitudes of the atomic and local oscillator fields, respectively. 18.2.2 Quantum-State Diffusion Evidently, we obtain the master equation for homodyne detection from the master equation for direct detection by the replacement
(18.44) However, note that adding the local oscillator field cannot change the form of the unconditioned master equation
(18.45) since in the unconditioned case the local oscillator influences quantum information that we are discarding anyway. Noting that
+
¯h i¯h
, \rho . (18.46) Thus, the unconditioned master equation is invariant under the simultaneous replacement
H −\rightarrow H −i¯h\Gamma
. (transformation from jump to homodyne detection) (18.47) Thus, this is the total transformation we should make to go from direct to homodyne detection.
18.2 Homodyne Detection Recalling from Eq. (18.10) that the master equation for direct detection is
¯h[H, \rho]dt −\Gamma
(18.48) where we have defined the jump superoperator
=
. (18.49) (jump superoperator) Thus, the master equation for homodyne detection is
¯h[H, \rho]dt−\Gamma
, \rho dt−\Gamma
(18.50) Expanding out the middle three terms, we see that the |\beta|2 terms cancel, so that
= −i
(18.51) where we have defined the measurement superoperator
\rho. (18.52) (homodyne superoperator) The transition rate of the counting process is correspondingly modified to read
dt, (18.53) which matches the photodetection rate we computed above for the homodyne case. Now we will consider the limit |\beta| −\rightarrow \inftyof a strong local oscillator, so that most of the detected photons come from the local oscillator. The rate of information gain—and thus the rate at which we disturb the system—remains constant, but the rate at which photons are detected becomes arbitrarily large. Thus, the white-noise approximation for the counting process N(t) is appropriate (here the approximation has same form as for the Poisson process), and we can make the replacement dN −\rightarrow dN dt
dt + s dN dt
dW, (18.54) which in the homodyne case here becomes dN −\rightarrow \Gamma
dt + q
(18.55) to obtain an It¯o SDE for the state evolution. First, let’s work out the part proportional to dt:
\Gamma
dt
\rho dt
dt −\Gamma
\rho dt = \Gamma
(18.56)
Chapter 18. Quantum Trajectories for Photodetection Notice that we have dropped the ensemble-average symbols on the right-hand side, since the ensemble average is only for convenience of notation. Thus, the H[\beta∗\sigma]\rho terms cancel in the master equation (18.51), which thus becomes
¯h[H, \rho]dt −\Gamma
q
= −i
q
(18.57) In the part proportional to dt, we see that all the \beta-dependent parts cancel, so taking the limit of large |\beta| is trivial. We can now expand out the part proportional to dW, keeping only the lowest-order terms in |\beta|−1: p
q
=
\rho |\beta|2 −\rho ! p \Gamma|\beta|2 = \sqrt \Gamma h
\rho i |\beta| = \sqrt \GammaH
|\beta| \rho = \sqrt \GammaH
\rho. (18.58) Here, \phi is the phase of the local oscillator, defined by
(18.59) (local-oscillator phase) Thus, the master equation becomes
\sqrt \GammaH
\rho dW, (SME for homodyne detection) (18.60) where we note that due to the quadratic nature of the Lindblad superoperator, we have used
\rho. (18.61) The form (18.60) for the stochastic master equation is our main result here: adding the local oscillator field prior to detection completely changes the form of the SME. The form here, in terms of the Wiener process dW(t), is called the quantum-state diffusion form9 for the SME (as opposed to the quantum-jump form). In this form it is easy to see that the ensemble average recovers the unconditioned master equation (18.1), since the ensemble average in It¯o calculus amounts to setting dW = 0. 9N. Gisin and I. C. Percival, ‘‘The quantum-state diffusion model applied to open systems,’’ Journal of Physics A: Mathemat- ical and General 25, 5677 (1992) (doi: 10.1088/0305-4470/25/21/023). For other related early work, see A. Barchielli, L. Lanz, and G. M. Prosperi, ‘‘A Model for the Macroscopic Description and Continual Observations in Quantum Mechanics,’’ Nuovo Ci- mento B 72, 79 (1982); A. Barchielli and G. Lupieri, ‘‘Quantum stochastic calculus, operation valued stochastic processes, and continual measurements in quantum mechanics,’’ Journal of Mathematical Physics 26, 2222 (1985) (doi: 10.1063/1.526851); V. P. Belavkin, in Information Complexity and Control in Quantum Physics, A. Blaquiere, S. Diner, and G. Lochak, Eds. (Springer, 1987); V. P. Belavkin, ‘‘A new wave equation for a continuous nondemolition measurement,’’ Physics Letters A 140, 355 (1989) (doi: 10.1016/0375-9601(89)90066-2); V. P. Belavkin, ‘‘A posterior Schrödinger equation for continuous nondemo- lition measurement,’’ Journal of Mathematical Physics 31, 2930 (1990) (doi: 10.1063/1.528946); L. Diósi, ‘‘Stochastic Pure State Representation for Open Quantum Systems,’’ Physics Letters A 114, 451 (1986) (doi: 10.1016/0375-9601(86)90692-4); and L. Diósi, ‘‘Continuous Quantum Measurement and Itô Formalism,’’ Physics Letters A 129, 419 (1988) (doi: 10.1016/0375- 9601(88)90309-X).
18.2.3 Measurement Record¶
18.2 Homodyne Detection These trajectories are illustrated here, for the same paramters as for the quantum-jump unravelling
(quantum trajectory) with quantum-state diffusion, with each infinitesimal jump corresponding to a detected photon in the homodyne setup. Nine other trajectories are included to illustrate the dephasing due to the stochastic nature of the evolution, and the qualitative difference with respect to the quantum-jump evolution. The usual Rabi oscillations are still visible here, since the atom is driven by a classical field of Rabi frequency Ω, the the oscillations are distorted by influence of the quantum noise. The influence is visually greatest when the excited-state population is greatest, which makes sense intuitively: if the atom is in the ground state, then the observer knows that any detected photon is due only to the local oscillator. Wt/2p
The second plot shows the ensemble-averaged excited-state probability computed from the unconditioned master equation (solid line), an average of 20 trajectories (dashed line), and an average of 2000 trajectories (dotted line). As many trajectories are averaged together, the average converges to the master-equation solution for the ensemble average. (About 20,000 trajectories are necessary for the Monte-Carlo average to be visually indistinguishable from the master-equation solution on the time scale plotted here.) In all cases, the trajectories are plotted in discrete time increments of ∆t = 0.005, but the trajectories were calculated
18.2.3 Measurement Record Now we can ask, what exactly are we measuring here? Recall from Eq. (18.11) that the photodetector current is
dN(t) dt , (18.62) where we are in the limit where dN is given by the replacement (18.55). Thus,
- Qph q
(18.63)
with the detector photocurrent. In general, we will want to subtract off the large constant photocurrent due
18.2.4 Information Gain from the Measurement Record¶
Chapter 18. Quantum Trajectories for Photodetection to the local oscillator field, and then retain only the lowest-order terms in |\beta|−1 in the dt and dW parts:
- Qph p
(18.64) We can then define the normalized photocurrent by
Qph|\beta| = \Gamma
- \sqrt \Gamma \xi(t). (normalized photocurrent) (18.65) In the standard form of an It¯o SDE, we can write
dt + \sqrt \Gamma dW. (homodyne measurement record) (18.66) We can regard dr(t) as the (scaled) measurement record for homodyne detection, with r(t) proportional to the total accumulated charge conducted by the photodetector (total photon count). We thus see that on average, we gain information about either the real or imaginary part of the radiated field, or equivalently \sigma, depending on the local-oscillator phase:
dt. (18.67)
(18.68)
(18.69) In either case, clearly the homodyne measurement provides information about the mean atomic dipole moment, or the phase of the atomic dipole, as opposed to the direct measurement, which was more closely related to the atomic excitation\langle \sigmaz\rangle . Note that with the proper choice of local-oscillator frequency, the dipole field and local oscillator time dependences cancel, these expectation values are measured in the rotating frame of the local oscillator. Thus, the expectation values are (adiabatic) constants. Of course, due to the dW term in the measurement record (18.66), the information in the mean is masked by quantum noise, and so the information must be extracted, e.g., by signal averaging. Of course, the best (i.e., correct in the Bayesian sense) method for obtaining information about the system is to evolve the master equation, conditioned on the measurement record. Since the dW in (18.66) is the same as the dW in the SME (18.60), we can eliminate it and write the SME directly in terms of the measurement record:
\sqrt \GammaH
\rho
dr(t) −\Gamma
dt \sqrt \Gamma ! . (homodyne SME with measurement signal) (18.70) This equation is still an It¯o SDE, even though dW does not explicitly appear here. 18.2.4 Information Gain from the Measurement Record Now let’s make a statement that is somewhat subtle, but nevertheless helps to gain insight here. Suppose
That is, you two have different density operators at t = 0. We are thus regarding the density operator as being subjective information about the quantum state. We will return to this topic in depth later, but let’s just go with it for now. For technicalities of consistency, we will assume that both you and the other observer either both assign a particular measurement outcome (based on the density operator) to consistenly have
18.2.6 Balanced Homodyne Detection¶
18.2 Homodyne Detection either zero or nonzero probability (i.e., in any basis, both observers agree on whether or not any particular diagonal density-matrix element is zero). In this view, either state is as good as any other, so let’s view the measurement record as being related to the expectation value of the other observer B:
B dt + \sqrt \Gamma dW ′. (18.71) Here, the B subscript denotes that the expectation value is taken with respect to the density operator \rhoB of the second observer. Clearly, dW ′ is not the same as the original dW if the expectation values differ with respect to the two quantum states. Then, we may rewrite the SME (18.70) as
\sqrt \GammaH
\rho h\sqrt \Gamma
B −
dt + dW ′i . (18.72) Notice that the other expectation value here is taken with respect to your (observer A’s) state \rho. Thus, we see that the measurement term is sensitive to the difference between the estimates of the quantity
(18.73) according to you and to observer B. In fact, this difference will be suppressed during the evolution, so that as you and observer B gain more and more consistent information, the difference in your density operators will tend to vanish at long times: in light of new information, the density operator ‘‘forgets’’ its initial condition. It is also possible to insist on an objective view of this same situation, where your initial density operator is ‘‘wrong,’’ and observer B is an omniscient observer that is ‘‘right.’’ In this case, with more measurement information, your density operator will tend to converge to the ‘‘true’’ density operator. This is sometimes a useful way to think about things, but you must be careful since it will also get you into trouble (e.g., it leads to problems associated with wave-function collapse being a physical process). Note that the convergence of states here must be viewed with caution, since complete convergence can only occur if the measurement information in fact resolves the differences between the different states. For example, if you and I disagree about the uncertainty in the X1 quadrature, but the measurement only provides information about the X2 quadrature, then clearly our states need not converge to the same state as a result of the continuous measurement: the measurement won’t affect the X1 uncertainty, so we won’t agree on that aspect of the state. The convergence of states is in some sense more intuitive for the jump process without the local oscil- lator. Recall that a detected photon lowers the atom to the ground state |g\rangle . Thus, with this measurement, any initial state is mapped to the same final state via a single jump, and so any observers must subsequently agree on the final state. 18.2.5 Diffusion Form of the Stochastic Schrödinger Equation The SME (18.60) is equivalent to the SSE
\sigma + 1
\sqrt \Gamma \sigma −1
(SSE for homodyne detection) (18.74) as we can see again by expanding d\rho to second order in d|\psi\rangle and using the It¯o rule dW 2 = dt. Again, this ‘‘diffusion unravelling’’ of the SME is only valid for unit detection efficiency. Otherwise, we must use a modified SME, which we will derive below. 18.2.6 Balanced Homodyne Detection The homodyne technique above, while relatively easy to analyze, has some practical disadvantages. First is the large dc offset due to the local oscillator that must be subtracted to obtain the desired signal. Second is the beam splitter, which must have close to unit reflection, and the corresponding requirement that the local oscillator field be very strong. Both of these problems are solved in practice by balanced homodyne detection, which involves a 50/50 beam splitter, detecting both output ports of the beam splitter, and then subtracting the two photocurrents.
Chapter 18. Quantum Trajectories for Photodetection local oscillator detector 1 detector 2 Suppose that the beam splitter is lossless with reflection coefficient r \in [0, 1]. Then proceeding in the same way as in the last section, the operator associated with the field impinging on detector 1 is C1 = \sqrt \Gamma
p 1 −r2 \alpha , (18.75) where t = \sqrt 1 −r2 is the transmission coefficient of the (lossless) beam splitter, and again \alpha is the amplitude of the coherent state |\alpha\rangle of the local-oscillator field. The operator associated with the field impinging on detector 2 is C2 = \sqrt \Gamma p
, (18.76) which follows from the Stokes relation r′ = −r relating the reflection coefficient of the beam splitter from the two sides, or alternately that we may regard the beam splitter as inducing a unitary transformation on the input fields of the form t r −r t , (18.77) with r2 + t2 = 1, which in this case is an orthogonal transformation since we have assumed the coefficients to be real. We can now associate two counting processes dN1 and dN2 with detectors 1 and 2, respectively, to account for ‘‘clicks’’ on each detector. The average photodetection rate of detector 1 is
D C\dagger 1C1 E dt = \Gamma h r2
- r p 1 −r2
- 1 −r2 |\alpha|2i dt, (18.78) while the mean photodetection rate of detector 2 is
D C\dagger 2C2 E dt = \Gamma h1 −r2
−r p 1 −r2
dt. (18.79) Again, the contributions of the two fields, as well as the interference effects, are apparent in these expressions. Now the two detectors generate photocurrents as before according to Idet,1 = Qph,1 dN1(t) dt Idet,2 = Qph,2 dN2(t) dt , (18.80) where for the moment, we will assume the detectors have generally different ‘‘gains’’ Qdet,1 and Qdet,2 per photon. The subtracted photocurrent is then I−= Idet,1 −Idet,2 = Qph,1 dN1(t) dt −Qph,2 dN2(t) dt , (18.81)
18.2 Homodyne Detection where the mean is given by¶
Qph,1r2 −Qph,2 1 −r2
- (Qph,1 + Qph,2) r p 1 −r2
- Qph,1 1 −r2 −Qph,2r2 |\alpha|2 . (18.82) Note that with the condition Qph,1 1 −r2 = Qph,2r2, (18.83) the |\alpha|2 term vanishes; that is, the gains of the photodetectors can be adjusted to null out the large dc term. However, the small
term vanishes only if Qph,1r2 = Qph,2 1 −r2 . (18.84) The only way to satisfy both conditions is to take a completely symmetric, or balanced, setup with Qph,1 = Qph,2 = Qph and r2 = 1/2. In this case, the mean subtracted photocurrent takes on the simpler form
, (18.85) and thus in the balanced setup, only the interference terms contribute without further subtraction or ap- proximation. 18.2.6.1 Master Equation Now we will derive the master equation for balanced homodyne detection, and see that in the limit of a strong local oscillator, the result is the same as for simple homodyne detection. Recalling again from Eq. (18.10) that the master equation for direct detection is
¯h[H, \rho]dt −\Gamma
(18.86)
dt, and the relevant superoperators are once again
\rho. (18.87) We can decompose the counting process dN into two parts dN1 and dN2, as we did in Section 18.1.5, and correspondingly split the other damping term in the master equation, so that the split counting processes are determined by
dt
1 −r2 \Gamma
dt, (18.88) and the master equation is
¯h[H, \rho] dt −r2 \Gamma
− 1 −r2 \Gamma
(18.89) This equation is equivalent to the original direct-detection master equation (18.86), but now we can interpret this form as direct detection of the atomic fluorescence after a beam splitter, where terms on the second line represent detector 1, and the terms on the third line represent detector 2.
Chapter 18. Quantum Trajectories for Photodetection Now note that we currently have collapse operators C1 = \sqrt \Gamma r\sigma C2 = \sqrt \Gamma p 1 −r2 \sigma (18.90) associated with the two detectors, which give the expectation values (18.88). To incorporate the local- oscillator field, and obtain the proper homodyne collapse operators (18.75) and (18.76), we must make the replacement
\sqrt 1 −r2 r \alpha (18.91) in the parts of the master equation (18.89) associated with detector 1, and we must also make the replacement
r \sqrt 1 −r2 \alpha (18.92) in the parts of the master equation (18.89) associated with detector 2. In analogy with the calculation of Eq. (18.47), upon making these replacements, to keep the unconditioned master equation unchanged, we must also transform the Hamiltonian according to H −\rightarrow H −i¯h\Gamma 2 r p
- i¯h\Gamma 2 r p
= H, (18.93) and thus the Hamiltonian needs no modification under the above replacements. That is, the effects of the two replacements on the unconditioned master equation exactly cancel, unlike the case of simple homodyne detection. Thus, implementing the replacements in Eq. (18.89), we have
¯h[H, \rho]dt −r2 \Gamma 2 H "
\sqrt 1 −r2 r \alpha∗ !
\sqrt 1 −r2 r \alpha !#
"
\sqrt 1 −r2 r \alpha # \rho dN1 − 1 −r2 \Gamma 2 H
r \sqrt 1 −r2 \alpha∗ \sigma − r \sqrt 1 −r2 \alpha
\sigma − r \sqrt 1 −r2 \alpha \rho dN2. (18.94) Expanding out and simplifying the H[c]\rho terms, we see that all terms involving \alpha cancel, leaving the simpler expression
¯h[H, \rho]dt −\Gamma 2 H
\rho dt + J "
\sqrt 1 −r2 r \alpha #
\sigma − r \sqrt 1 −r2 \alpha \rho dN2. (18.95) Now, to take the white-noise limit, where the amplitude of the local oscillator is large (|\alpha| −\rightarrow \infty). In this case, we again make replacements of the form
r
(18.96) In particular, the contribution to the master equation proportional to dt from the dN1 term is J "
\sqrt 1 −r2 r \alpha #
p
(18.97) where the details are exactly as in the previous calculation of Eqs. (18.56). Similarly, the contribution to the master equation proportional to dt from the dN2 term is J \sigma − r \sqrt 1 −r2 \alpha
1 −r2 \Gamma
p
(18.98)
18.2 Homodyne Detection When these parts are included in the master equation, the first terms combine to form the usual dissipation term from the unconditioned equation, while the second terms cancel each other, and the result is
- v u u tr2\Gamma *
\sqrt 1 −r2 r \alpha∗ !
\sqrt 1 −r2 r \alpha !+ J "
\sqrt 1 −r2 r \alpha # \rho dW1 + v u u t (1 −r2) \Gamma
r \sqrt 1 −r2 \alpha∗ \sigma − r \sqrt 1 −r2 \alpha J \sigma − r \sqrt 1 −r2 \alpha \rho dW2, (18.99) where now dW1 and dW2 are the Wiener processes corresponding to dN1 and dN2, respectively. Unfortu- nately, the stochastic terms do not simplify nearly as well, and so we will now consider the limit of a strong local oscillator, |\alpha| −\rightarrow \infty, and keep only the lowest order terms in |\alpha|−1. The calculation is the same as in Eqs. (18.58),
\sqrt \GammaH
\rho dW, (balanced homodyne master equation) (18.100) where as before \phi is the phase of the local-oscillator field,
(18.101) (local-oscillator phase) and where the Wiener process dW is given in terms of its independent constituents by
p 1 −r2 dW2, (18.102) (composite noise process) and is thus itself a standard Wiener process. The form of the master equation is thus independent of r, but the two Wiener processes contribute to dW equally only in the balanced case r = 1/ \sqrt 2. 18.2.6.2 Measurement Record When we take the white-noise limit of the subtracted photocurrent (i.e., the measurement record), and again keep only lowest-order terms in |\alpha|−1, we find
p 1 −r2
dt + Qph,1 p \Gamma(1 −r2)|\alpha|2 dW1 −Qph,2 p \Gammar2|\alpha|2 dW2, (18.103) if we assume condition (18.83) is fulfilled so that the large |\alpha|2 dc offset term in the mean photocurrent vanishes. However, note that even with Eq. (18.83), there is no simple way to combine the two quantum- noise terms. If the two detectors have equal response, Qph,1 = Qph,2 = Qph, then I−= Qph\Gamma
dt + Qph \sqrt \Gamma|\alpha| dW ′, (18.104) where the composite Wiener process is dW ′ = p 1 −r2 dW1 + r dW2. (18.105) However, note that dW ′ is not equivalent to the composite Wiener process dW that appeared in the master equation (18.102). These two noise processes are only simply identified in the case of perfect balancing,
\sqrt 2, in which case dW ′ = dW, I−= Qph\Gamma
dt + Qph \sqrt \Gamma|\alpha| dW. (balanced homodyne photocurrent) (18.106) This is of the same form as for simple homodyne detection, but no extra subtraction is required to eliminate the dc offset term. The measurement photocurrent can then be rescaled as for simple homodyne detection.
18.2.7 Heterodyne Detection¶
Chapter 18. Quantum Trajectories for Photodetection 18.2.7 Heterodyne Detection In general, the local-oscillator field need not have the same nominal frequency as the atomic radition (i.e., it need not be the same field that drives the atom), and in such case the measurement setup corresponds to heterodyne detection. In the above analysis of homodyne detection, we have made no particular assumptions about the phase \phi of the local oscillator, and we can thus treat heterodyne detection by simply setting \phi = ∆t, where ∆= \omegalo −\omega is the detuning between the local oscillator frequency \omegalo and the driving field \omega of the atom. Thus, from the homodyne SME (18.89), the heterodyne-detection master equation becomes
\sqrt \GammaH \sigmaei∆t \rho dW, (18.107) (heterodyne SME) and from Eq. (18.66), the measurement record (scaled photocurrent) is
dt + \sqrt \Gamma dW. (heterodyne measurement record) (18.108) Heterodyne detection is often used in the case where the detuning ∆is large (compared to some relevant frequency scale, such as the inverse of a measurement averaging time), in which case the information in the measurement record is ‘‘encoded’’ at the high frequency ∆. This is often pragmatically useful in the laboratory, since technical noise tends to drop off as 1/\omega, and thus the encoding at high frequency is generally less susceptible to technical noise. After detection it is most useful to demodulate the photocurrent signal by shifting the useful information to near-dc. In post-processing this is done by multiplying by the harmonic function e−i∆t to obtain
dt + \sqrt \Gamma dV, (18.109) where dV := e−i∆tdW (18.110) (rotating Wiener process) is the frequency-shifted noise process. For large detunings, the rapidly rotating term at frequency 2∆is ignorable over any reasonable averaging time and is thus negligible, so we may write10
\sqrt \Gamma dV, (18.111) (heterodyne measurement record) Thus, the measurement record now contains information about \langle \sigma\rangle , and thus about both quadratures \langle \sigmax\rangle and\langle \sigmay\rangle . The reason that this works is that the detuning separates the components\langle \sigma\rangle and
\sigma\dagger to different frequencies \pm∆, and thus the information about only one of these expectation values may be detected. By contrast, in homodyne detection the information about the two operators is encoded at the same frequency, and thus we only obtain information in the quadrature combinations
. Note that demodulating the signal, as in analog electronics, by multiplying dr by the real harmonic function cos ∆t does not work in the same way, since it shifts the information for both \langle \sigma\rangle and
\sigma\dagger to zero frequency, and thus amounts simply to homodyne detection. In terms of the noise process dV , we may write the master equation as
\sqrt \GammaH \sigmaei∆t \rho ei∆tdV. (18.112) (heterodyne SME) In the noise term there are thus contributions that go as \sigmae2i∆tdV and \sigma\daggerdV . However, we may not make a rotating wave approximation here and neglect the former compared to the latter, because dW contains all 10For explicit solutions of the dynamics for homodyne and heterodyne detection, see Howard Wiseman, ‘‘Complementarity in Spontaneous Emission: Quantum Jumps, Staggers, and Slides,’’ in Directions in Quantum Optics, H. J. Carmichael, R. J. Glauber, and M. O. Scully, Eds. (Springer, 2001), p. 347.
18.2.8 Detector Efficiency and Multiple Observers¶
18.2 Homodyne Detection frequencies, and thus both terms are ‘‘fast’’ in the same sense. In treating this equation it is useful to note the relations (valid in the limit of large ∆)
dV 2 = e−2i∆tdt \approx 0
(18.113) Otherwise, there is no fundamental simplification to the master equation in this case, and even though the measurement information is different, it is simplest to think of the heterodyne and homodyne master equations as equivalent except for the time dependence of the phase \phi. 18.2.8 Detector Efficiency and Multiple Observers The SME (18.60) for homodyne detection,
\sqrt
(18.114)
any setup above), applies when the detector catches all of the photons emitted by the atom. To model a finite detection efficiency, suppose the radiated field is split by a beam splitter into two components weighted by \sqrt\eta1 and \sqrt\eta2, respectively (i.e., the intensities are weighted by \eta1 and \eta2), which are then monitored by homodyne detection on homodyne detectors 1 and 2, respectively. local oscillator local oscillator homodyne 1 detector homodyne 2 detector Æh1 Æh2 We are assuming the amplitudes \sqrt\eta1 and \sqrt\eta2 to be real, with \eta1 +\eta2 = 1. Then the SME above is modified to include the two measurement processes, where \sigma −\rightarrow \sqrt\eta1\sigma for the process on detector 1, and \sigma −\rightarrow \sqrt\eta2\sigma for the process on detector 2:
\sqrt
\sqrt
(18.115) We can combine the two dissipation terms, since they are linear in \eta1,2, and factor the \eta1,2 out of the measurement terms to obtain
p
p
(SME for two observers) (18.116) The dissipation term is precisely the same as if the light were not split: the disturbance does not depend on th details of the measurement. Of course, if we define
(18.117)
Chapter 18. Quantum Trajectories for Photodetection we recover the original master equation where the light was not split. The measurement record (18.66) is correspondingly modified into two measurement records for the two detectors:
dt + \sqrt \Gamma dW1
dt + \sqrt \Gamma dW2. (measurement records for two observers) (18.118) The original measurement record is recovered by taking the combination
dt + \sqrt
(18.119) We can rescale these records so that the expectation values have the same amplitudes as the original mea- surement record:
\sqrt\eta1 = \Gamma
dt + s \Gamma \eta1 dW1
\sqrt\eta2 = \Gamma
dt + s \Gamma \eta2 dW2. (measurement records for two observers) (18.120) In this case, the quantum noise is effectively amplified for each of the two detectors compared to the case of a single detector. As we will discuss below, the increased quantum noise is due to the presence of another information channel, which necessarily disturbs, or back-acts, on the quantum system. Now what we essentially have is the theory for two observers monitoring the same atomic fluorescence. Suppose that there are two observers 1 and 2, each of which has access to only their respective detector 1 or 2. Observer 1 does not have access to detector 2, and thus must trace over all possible results on detector 2. We do this by taking an ensemble average over all possible realizations of dW2, which we again do by effectively setting dW2 = 0, and thus obtain the SME for the state of knowledge \rho1 of observer 1:
p
- (SME for observer 1) (18.121) Here, we have used the notation
1 \rho, (18.122) so that the superscript on the H superoperator denotes that the expectation value is taken with respect to \rho1. The measurement record for observer 1 has the same form as in Eq. (18.120),
1 dt + s \Gamma \eta1 dW ′ 1, (measurement record for observer 1) (18.123) but now the expectation value is taken with respect to \rho1. Correspondingly, we cannot assume that the noise process dW ′ 1 according to observer 1 is the same as the original dW1. Similarly, the SME for observer 2 is
p
2, (SME for observer 2) (18.124) and the measurement record according to observer 2 is
2 dt + s \Gamma \eta2 dW ′ 2. (measurement record for observer 2) (18.125)
18.3 Conditioned Dynamics and Squeezing We can interpret the first SME (18.116) and corresponding measurement records (18.120) that include both detectors as those of another omniscient observer that has access to both detectors. We can now equate the two expressions for each measurement record to obtain dW ′ 1 = p \eta1\Gamma h
−
i + dW1 dW ′ 2 = p \eta2\Gamma h
−
i + dW2, (18.126) relating the noise processes dW ′ 1 and dW ′ 2 for the individual observers to the noise processes dW1 and dW2 for the omniscient observer. Note that we have derived the above equations assuming the two observers are making the same measurement on the light, but this is easily generalized to the case of two different measurements by the two observers. The case of inefficient detection is exactly the case of a single observer in the presence of a second observer, where the observer does not have all the possible information and needs to trace over all the undetected information. Then the observer’s SME becomes
p
(inefficient homodyne detection SME) (18.127) and the measurement record is
dt + s \Gamma \eta dW, (inefficient homodyne detection measurement record) (18.128) where \eta is the efficiency of the detector (i.e., the fraction of total intensity that is actually registers on the detector). 18.3 Conditioned Dynamics and Squeezing Now that we have fairly general forms of the master equation for homodyne and heterodyne measurement, we would like to interpret the measurement terms in the master equations to see their physical meaning. In particular, the H[c]\rho terms (i.e., the noise terms) represent the information gain due to the measurement process, while the D[c]\rho terms represent the disturbance to, or the backaction on, the state of the system due to the measurement. Of course, as we see from the dependence on the efficiency \eta, the backaction occurs independently of whether the observer uses or discards the measurement information (corresponding
Interpreting the master equation in this way is an important exercise because the measurement record dr tells us what the observer actually measured, but to find out what the observer actually knows about the system in light of the measurement, we must actually solve the master equation. Of course, this is quite difficult to do in general, but we can consider the evolution of the lowest moments (expectation values of powers of x and p) of the canonical variables. This will give us the observer’s time-dependent estimates of position and momentum, as well as the associated uncertainties. Let us consider the case of homodyne detection of a cavity field—which we recall from Chapter 12 is equivalent to homodyne detection of the atomic field under the replacements \sigma −\rightarrow a, \Gamma −\rightarrow \kappa—with the additional evolution under the action of a Hamiltonian H:
(SME for cavity homodyne detection) (18.129)
18.3.1 Moment Equations¶
Chapter 18. Quantum Trajectories for Photodetection Correspondingly, the measurement record for measurement efficiency \eta is
dt + r\kappa \eta dW. (measurement record for cavity homodyne detection) (18.130) Here, a is the cavity annihilation operator, and we are only considering a measurement of the X1 quadrature (proportional to a + a\dagger, as we will define below) to simplify things. For an arbitrary operator A, we can use the master equation and d\langle A\rangle = Tr[A d\rho] to obtain following equation of motion for the expectation value
+\kappa
a\daggerAa −1
dt
dW. (expectation-value evolution under SME) (18.131) The first line gives the Hamiltonian evolution, the second line the effect of the dissipation/disturbance D[a]\rho, and the last line is the effect of the measurement information H[a]\rho. The evolution of the isolated cavity is then given by the harmonic oscillator Hamiltonian H = p2 2m + 1 2m\omega 2x2, (18.132) (cavity Hamiltonian) which we write in canonical coordinates rather than the raising and lowering operators (m here is an ‘‘effective mass,’’ which while quantizing the field we decided was the permittivity ϵ0). We will thus derive the the lowest few moments of x and p using the above formula for d\langle A\rangle . We will also make the simplifying assumption that the initial state is Gaussian, so that we only need to consider the simplest five moments: the means \langle x\rangle and \langle p\rangle , the variances Vx and Vp, where V\alpha :=
\alpha2 −\langle \alpha\rangle 2, and the symmetrized covariance
mixed states). Recall from Section 5.6.1 that we already decided that the Gaussian state was a ‘‘natural’’ state for the damped harmonic oscillator, where D[a]\rho is precisely the damping term that we used then. 18.3.1 Moment Equations To set up the calculation, we recall for the harmonic oscillator that the annihilation operator is related to the canonical coordinates by a = \sqrt 2x0 x + i x0 \sqrt 2¯hp, (18.133) where the length scale x0 is defined by x0 := r ¯h m\omega . (18.134)
18.3.2 Quadrature Moments¶
18.3 Conditioned Dynamics and Squeezing Evaluating the terms in the above evolution equation for the various moments gives the following moment equations for the conditioned evolution in this case:
r
¯h Vx − ¯h 2m\omega dW
r
¯h Cxp dW
mCxp −\kappa Vx − ¯h 2m\omega
¯h Vx − ¯h 2m\omega 2
Vp −m\omega¯h
¯h C2 xp
¯h Cxp Vx − ¯h 2m\omega . (moment evolution under SME) (18.135) Here, we have used the following moment relations, valid for a Gaussian state:11
x3
[x, p2]+
h
h
. (18.136) This approximation decouples the variances from any higher-order moments and removes any noise terms from the variance equations. 18.3.2 Quadrature Moments We can make these equations look a bit more symmetric by defining the stationary quadrature operators X1 := 1
= rm\omega 2¯h x X2 := 1 2i a −a\dagger = r 2m\omega¯h p, (18.137) (field quadratures) in which case the moment equations transform to
p
VX1 −1 dW
p
VX1 −1
VX1 −1 2
VX2 −1
X1X2
VX1 −1 . (quadrature-moment evolution) (18.138) Here it is more obvious that X1 and X2 are treated symmetrically, and the Hamiltonian evolution simply involves a rotation in the X1-X2 plane at a frequency \omega, corresponding to the free evolution of the cavity field. 11Salman Habib, ‘‘Gaussian Dynamics is Classical Dynamics,’’ arXiv.org preprint (arXiv: quant-ph/0406011); though the relations here simply reflect the fact that for Gaussians, odd-order centered moments always vanish.
18.3.4 Squeezing (or Lack Thereof)¶
Chapter 18. Quantum Trajectories for Photodetection 18.3.3 Interpretation Now on to the interpretation of the moment equations (18.138). First, consider the unconditioned evolution of the means\langle X1\rangle and\langle X2\rangle , where we average over all possible noise realizations. Again, since \langle \langle \rho dW\rangle \rangle = 0, we can simply set dW = 0 in the above equations, and we will drop the double angle brackets for brevity. The Hamiltonian evolution terms are of course the same, but now we see extra damping terms. Decoupling these two equations gives an equation of the usual form for the damped harmonic oscillator for the mean position: \partial 2
(18.139) The same equation of motion follows for the other quadrature X2. Note that we identify the frequency \omega here as the actual oscillation frequency \omega\kappa of the damped oscillator, given by \omega 2
the resonance frequency \omega that appears the usual form of the classical formula. Then both quadratures undergo damped harmonic oscillation, so that the trajectory in the X1-X2 plane is a radially symmetric
The noise terms in these equations correspond to nonstationary diffusion, or diffusion where the trans- port rate depends on the state of the system. Note that under such a diffusive process, the system will tend to come to rest in configurations where the diffusion coefficient vanishes, an effect closely related to the ‘‘blowtorch theorem.’’12 Here, this corresponds to VX1 = 1/4 and CX1X2 = 0, or Vx = ¯h/2m\omega and Cxp = 0 in the original coordinates, which correspond to the values of the ground state (or any coherent state). The variance equations also contain unconditioned damping terms (proportional to \kappa but not \eta). These damping terms cause the system to equilibrate with the same variance values as noted above; they
\eta) in the equation for VX1 merely accelerates the contraction of VX1, and thus represents information gain in the X1 quadrature (i.e., the one we are measuring). It also accelerates the settling to the equilibrium value VX1 = 1/4. The measurement term in the VX2 equation involves only the covariance: this says that if CX1X2̸ = 0, then the two quadratures are correlated, and thus a measurement on X1 also provides some information about X2. 18.3.4 Squeezing (or Lack Thereof) Thus, we see that the essential effect of the antihermitian measurement operator is to damp the energy from the system, whether it is stored in the centroids or in the variances. In fact, what we see is that this measurement process selects coherent states, states that have the same shape as the harmonic-oscillator ground state, but whose centroids oscillate along the classical harmonic-oscillator trajectories. One thing that we can immediately conclude from this analysis is that even though the homodyne measurement obtains information about the ‘‘x quadrature,’’ since the measurement accelerates the decay of Vx, we can see that the measurement does not squeeze the quadradure—that is, the uncertanty does not become smaller than that of the coherent state (ground state) in steady state. Squeezing can be produced by a measurement, but it requires the measurement operator to be of the form a + a\dagger (i.e., we must realize a direct, Hermitian ‘‘position measurement’’) rather than simply a measurement via a. That is, the master equation should be of the form
(18.140) to produce squeezing in X1, but this does not correspond to a photodetection measurement that we have considered thus far. Because the measurement operator is Hermitian in this case, the measurement would not cause damping, and would have to be realized by a dispersive (i.e., nonabsorbing) measurement interaction, say by firing a beam of atoms through a lossless cavity and measuring atomic phase shifts due to nonresonant interaction with the cavity field. This master equation also has the form of a position measurement, which we will consider in depth in the next chapter. 12Term coined by Rolf Landauer, ‘‘Statistical physics of machinery: forgotten middle-ground,’’ Physica A 194, 551 (1993)
18.3.6 Heterodyne Detection¶
18.3 Conditioned Dynamics and Squeezing 18.3.5 Homodyne Detection Now recall that in homodyne detection, the frequency of the local oscillator matches that of the cavity, which has the same effect in the above equations of setting \omega −\rightarrow 0:
p
VX1 −1 dW
p
VX1 −1
VX1 −1 2
VX2 −1
X1X2
VX1 −1 . (quadrature-moment evolution under homodyne detection) (18.141) In this way, the measurement always gets information about a single quadrature (in this case, the measure- ment provides information about the X1 quadrature). 18.3.6 Heterodyne Detection To treat heterodyne detection, note that the local oscillator and cavity frequencies do not match, which amounts to letting \omega −\rightarrow ∆in Eqs. (18.138), where ∆is again the detuning between the local oscillator and the cavity. Then defining the corotating quadratures ˜X1 := X1 cos ∆t −X2 sin ∆t ˜X2 := X1 sin ∆t + X2 cos ∆t, (18.142) (corotating quadratures) we can transform the variances to the new variables using V ˜ X1 = VX1 cos2 ∆t + VX2 sin2 ∆t −2CX1X2 sin ∆t cos ∆t V ˜ X2 = VX1 sin2 ∆t + VX2 cos2 ∆t + 2CX1X2 sin ∆t cos ∆t C ˜ X1 ˜
(18.143)
Chapter 18. Quantum Trajectories for Photodetection to rewrite the moment equations as d D ˜X1 E
D ˜X1 E dt + p
VX1 −1 cos ∆t dW − p 4\eta\kappaCX1X2 sin ∆t dW d D ˜X2 E
D ˜X2 E dt + p
VX1 −1 sin ∆t dW + p 4\eta\kappaCX1X2 cos ∆t dW \partial tV ˜
V ˜ X1 −1
VX1 −1 2
X1X2 sin2 ∆t
VX1 −1 sin ∆t cos ∆t \partial tV ˜
VX2 −1
VX1 −1 2
X1X2 cos2 ∆t
VX1 −1 sin ∆t cos ∆t \partial tC ˜ X1 ˜
X1 ˜
VX1 −1 2
X1X2 sin ∆t cos ∆t
VX1 −1 (cos2 ∆t −sin2 ∆t). (18.144) In doing so, we have eliminated the Hamiltonian free-evolution terms, but we have introduced some explicit time dependence in the equations. We will now consider the variance equations in the limit of large ∆, and we will replace terms oscillating at frequencies of order ∆by their time-averaged values, being careful to implement the inverse relations for the original variances, VX1 = V ˜ X1 cos2 ∆t + V ˜ X2 sin2 ∆t + 2C ˜ X1 ˜ X2 sin ∆t cos ∆t VX2 = V ˜ X1 sin2 ∆t + V ˜ X2 cos2 ∆t −2C ˜ X1 ˜ X2 sin ∆t cos ∆t CX1X2 = −V ˜ X1 sin ∆t cos ∆t + V ˜ X2 sin ∆t cos ∆t + C ˜ X1 ˜ X2(cos2 ∆t −sin2 ∆t), (18.145) so that we obtain \partial tV ˜
V ˜ X1 −1
˜ X1 ˜
V ˜ X1 −1 2 \partial tV ˜
V ˜ X2 −1
˜ X1 ˜
V ˜ X2 −1 2 \partial tC ˜ X1 ˜
X1 ˜
2 C ˜ X1 ˜ X2 V ˜ X1 + V ˜ X2 −1 . (variance evolution under heterodyne detection) (18.146) The heterodyne variance equations are now relatively simple. In particular, notice that in homodyne detec- tion, the measurement was represented in the X1 quadrature by the term
V ˜ X1 −1 2 , (18.147) while in heterodyne detection, both quadratures have measurement terms of this form, but with an overall factor of 2\eta\kappa instead of 4\eta\kappa. So again, while heterodyne detection provides measurment information about both quadratures, it does so at only half the rate at which homodyne detection provides information about a single quadrature. Note that you can get similar results by splitting the field on a 50/50 beam splitter, and use two homodyne detection setups, set to monitor complementary quadratures, to monitor each output field of the beam splitter. In this case, the factor of 1/2 in the information rate is more obvious.
18.3.8 Phase Estimation¶
18.3 Conditioned Dynamics and Squeezing 18.3.7 Explicit Solutions for the Uncertainty Dynamics Incidentally, for homodyne detection, the X1 variance evolves from Eqs. (18.141) as
VX1 −1
VX1 −1 2 . (18.148) If we assume the variance is very broad, the measurement term dominates the dissipation term, so that
VX1 −1 2 , (18.149) and this equation has the solution
VX1(0) −1/4
- (18.150) That is, the variance decreases like 1/t, which makes sense since the uncertainty should decrease as 1/ \sqrt t for averaging a noisy process. Of course, as the variance approaches the steady-state value of 1/4, the damping term becomes dominant, and the decay becomes exponential. This is represented by the more general solution of (18.148),
(VX1(0) −1/4)
4, (18.151) which contains the 1/t behavior at short times as well as the exponential decay at long times. 18.3.8 Phase Estimation As an example of an application that illustrates the difference between homodyne and heterodyne detection, we will consider the problem of estimating the phase of a pulse of light. The phase uncertainty and quantum noise here, for example, put limits on how much information can be encoded in the phase of an optical pulse. To keep this treatment simple, we will consider as as simple model the field pulse to be modeled in the same way as the single-mode cavity field above. First, consider the homodyne detection of the cavity phase in terms of the two complementary quadra- ture variables X1 and X2. We will suppose the phase of the field to be reasonably well defined and the field amplitude to be larger than its uncertainty. We will assume that\langle X1\rangle = 0 initially; we can treat the general case by simply applying a rotation to this basic configuration. We will further assume a Gaussian state for simplicity, and to capture the essence of the problem (recall that the state will be asymptotically Gaussian under the measurement anyway). We can then represent the quantum state by a distribution in phase space (e.g., a Wigner distribution, as in Section 4.3). X¡ X™ dof The question of the phase of the quantum state is then essentially recasting the same problem of measuring the field quadratures into polar coordinates. In the particular case shown here, the uncertainty in the phase ϕ of the quantum state is clearly related to the uncertainty in X1. But it should also be clear that the phase uncertainty ∆ϕ is related to the amplitude of the field (i.e., the distance of the wave-packet centroid from the origin): a larger amplitude implies a smaller phase uncertainty, given that the uncertainty ∆X1 is fixed.
Chapter 18. Quantum Trajectories for Photodetection The measurement, as we discovered above, causes the X1 uncertainty to contract, assuming the local- oscillator phase \phi = 0. For the particular case we are considering now, this means that the phase uncertainty decreases—we gain knowledge about the phase. X¡ X™ dof The other effect of the measurement is that the amplitude of the field decreases, so that the centroid of the wave packet moves towards the origin. After all, photodetection proceeds as the photodetector absorbs the field. (This is assuming that the cavity is not driven to counteract the cavity damping.) The dissipation has the opposite effect on the phase uncertainty; as the wave packet moves towards the origin with ∆X1 fixed, the angle subtended by the wave packet increases, and thus \deltaϕ increases. Of course, we have seen that the dissipation also reduces ∆X1. However, once the variance VX1 reaches its steady-state value 1/4, any decrease in amplitude increases the phase uncertainty. Of course, the point of all this is that when the phase uncertainty is much larger than the minimum quantum limit and the amplitude of the field is large, the dominant effect of the measurement is the rapid decrease of ∆X1, which reduces the phase uncertainty. Now, however, consider the case where the phase ϕ of the field is near \pi rather than near \pi/2. In this case, a reduction in the uncertainty ∆X1 reduces the uncertainty of the field amplitude, but not its phase. X¡ X™ dof Thus, gaining information on the phase in homodyne detection depends on having a particular phase to begin with. That is, the phase sensitivity of homodyne detection has a phase-dependent sensitivity. Heterodyne detection has the advantage that the measurement reduces the phase uncertainty for any phase, since as we saw above, the uncertainties of both X1 and X2 decrease in response to the meaasurement. X¡ X™ dof The price of this ‘‘omnidirectional’’ phase sensitivity is that the phase uncertainty decreases (assuming the decrease to be dominated by the measurement information) at half the rate for the best case of homodyne detection. Of course, it greatly outperforms the homodyne measurement in its worst case, and the heterodyne measurement simultaneously provides information about the field amplitude.
18.3 Conditioned Dynamics and Squeezing To see this mathematically, note that for the case of homodyne detection in the best case of\langle ϕ\rangle = \pi/2, to see the reduction in phase uncertainty, we need the equation for VX1 in Eqs. (18.141):
VX1 −1
VX1 −1 2 . (18.152) If the wave packet is localized and the amplitude of the field is large, then to lowest order in the phase uncertainty we may convert between the Cartesian and polar variances according to
where
(18.153) Then assuming the measurement-induced reduction of Vϕ is much faster than the damping of \langle R\rangle , we may write
Vϕ − 4\langle R\rangle 2 !
Vϕ − 4\langle R\rangle 2 !2 . (18.154) We can see here again the first, damping term (which should also be negligible in this limit), and the second, information term. We can explicitly see how the quantum-limited phase uncertainty is related to the field amplitude: the best phase uncertainty is \delta\phi \approx 1/2\langle R\rangle , at least in the absence of squeezing. Furthermore, the rate of information collapse increases with the field amplitude, which again reflects the fact that the phase uncertainty \deltaϕ is related both to ∆X1 and the field amplitude. Then to treat the general case, we must rotate the phase space while maintaining the measurement of X1. We have already done this in Eq. (18.144), and adapting the variance equation for V ˜ X1 in the same way, we replace ∆t by \phi and for simplicity ignore any covariance between the amplitude and phase to find
Vϕ − 4\langle R\rangle 2 !
Vϕ − 4\langle R\rangle 2 !2 . (18.155) Thus the measurement-induced collapse rate is now modulated by cos2 \phi, where \phi now represents the phase change of the local oscillator from our base case above. That is, this is the sensitivity of the homodyne measurement when the expected phase is \langle ϕ\rangle = \phi + \pi/2 and the measurement is of the X1 quadrature. The heterodyne case follows from adapting the same variance equation in Eq. (18.146):
Vϕ − 4\langle R\rangle 2 !
Vϕ − 4\langle R\rangle 2 !2 . (18.156) This is, of course, the same as the homodyne expression averaged over the local-oscillator phase \phi. Again, we have lost the dependence on the local-oscillator phase, but at the cost of a factor of 2 in the information- collapse rate. 18.3.8.1 Adaptive Measurements Then in making a phase measurement, how is it possible to take advantage of the extra phase sensitivity in homodyne detection, when it seems to require already knowing the phase to begin with? One strategy is to use both heterodyne and homodyne detection in an adaptive phase measurement.13 The idea is to start out a measurement of a light pulse without any knowledge of the phase using heterodyne detection. As the observer begins to get an idea of the phase, the observer switches the local oscillator to homodyne detection, with a local-oscillator phase set to maximize the sensitivity based on the heterodyne estimate. As the homodyne measurement continues, the observer feeds back to the local oscillator phase to track the estimated phase (which is diffusing stochasically due to the measurement) to ensure that sensitivity is always maximized. 13H. M. Wiseman, ‘‘Adaptive Phase Measurements of Optical Modes: Going Beyond the Marginal Q Distribution,’’ Physical Review Letters 75, 4587 (1995) (doi: 10.1103/PhysRevLett.75.4587); this adaptive scheme was implemented experimentally by Michael A. Armen, John K. Au, John K. Stockton, Andrew C. Doherty, and Hideo Mabuchi, ‘‘Adaptive Homodyne Measurement of Optical Phase,’’ Physical Review Letters 89, 133602 (2002) (doi: 10.1103/PhysRevLett.89.133602).
Chapter 18. Quantum Trajectories for Photodetection 18.4 Exercises Problem 18.1 Verify that the stochastic Schrödinger equation (SSE) [Eq. (18.18)] for quantum jumps,
\sigma p
−1 !
(18.157) is equivalent to the stochastic master equation (SME) [Eq. (18.158)]
¯h[H, \rho]dt −\Gamma
dN. (18.158)