16. Photodetection¶
PDF pages 685–694
Chapter 16 Photodetection In Chapter 2, we considered the coherence of classical light, and, for example, what this meant for the classical Hanbury-Brown–Twiss experiment. We will now reconsider this experiment using the quantum description of the fields, and examine the most dramatic departures from the classical-field predictions, which occur for fields with one or two photons. 16.1 Counting Photons Let’s start by developing a bit of formalism to handle detectors interacting with the quantum electromagnetic field.1 Recall that the quantized electric field has the form [Eq. (8.61)]
X k,\zeta r ¯h\omegak 2ϵ0
(16.1) or identifying the two terms with E(+) and E(−), we can isolate the annihilation component of the field:
X k,\zeta r ¯h\omegak 2ϵ0
(16.2) We may regard a photodetection event, as a transition in the state of the field, |i\rangle −\rightarrow |f\rangle , where |i\rangle is the initial state of the field before the detection event, and |f\rangle is the final state of the system afterward, where one photon in mode (k, \zeta) was removed from the field at time t. The transition amplitude for this process is proportional to
(16.3) Including the spatial profile of the field,
(16.4) where the field here is the single, relevant term in the mode sum (16.2), and ˆ\epsilon\zeta is the unit polarization vector of the mode at the location of the detector. Of course, we can include the entire field,
(16.5) since in view of the definition of |f\rangle only one of the field annihilation operators contributes to the matrix element. We have also dropped the subscript on the polarization vector, so that ˆ\epsilon represents the field polarization sensed by the detector. Now the probability for detecting a photon is given by summing the 1Here we are following Roy J. Glauber, ‘‘The Quantum Theory of Optical Coherence,’’ Physical Review 130, 2529 (1963) (doi: 10.1103/PhysRev.130.2529).
Chapter 16. Photodetection squares of the amplitudes (each amplitude corresponds to a final state where a photon is lost from a particular mode; each mode is orthogonal and thus each final state is distinguishable): P(t) ∝ X f
= X f
(16.6) The initial state |i\rangle is arbitrary, and we can think of the expectation value even for a mixed state by performing an ensemble average over initial states. We can also carry out a sum over polarizations, so that P(t) ∝ D E(−) \alpha (r, t)E(+) \alpha (r, t) E , (16.7) (photodetection probability) with an implied sum over \alpha. This expression for the photodetection probability motivates the definition of a field correlation func- tion, the degree of first-order coherence:
D E(−) \alpha (r1, t1)E(+) \alpha (r2, t2) E , (degree of first-order coherence) (16.8) which is the quantum analog of the classical field correlation function, which we saw in normalized form in Eq. (2.23), which gives the fringe visibility in an interference experiment. Note the particular ordering of the field operators in the correlation function and the detection probability, which is of the general form
a\daggera
. (16.9) This ordering is called normal ordering, which refers to having all annihilation operators to the right of all creation operators. This particular ordering is important, as in the vacuum state, the expectation value \langle 0|a\daggera|0\rangle = 0 gives a zero detection probability or zero correlation, both of which are appropriate for the vacuum. The other ordering here would correspond to detection of photons from the vacuum, which is physically nonsensical. Similarly, the joint probability amplitude to detect one photon at (r, t) and (r′, t′) is
(16.10) To compute the detection probability, we square this, sum over all final states, and consider any polarization as before, with the result P(t) ∝ D E(−) \alpha (r, t)E(−) \alpha (r′, t′)E(+) \alpha (r′, t′)E(+) \alpha (r, t) E , (joint photodetection probability) (16.11) again with an implied sum over \alpha. This joint detection probability motivates the definition of a higher-order correlation function, the degree of second-order coherence:
D E(−) \alpha (r1, t1)E(−) \alpha (r2, t2)E(+) \alpha (r2, t2)E(+) \alpha (r1, t1) E . (degree of second-order coherence) (16.12) This is the quantum analog of the classical intensity correlation function, e.g., \langle I(t)I(t + \tau)\rangle , which we saw in normalized form in Eq. (2.68). Note that these joint expectation values are still in normal form with the general form D a\dagger 1a\dagger 2a2a1 E , so that either joint expectation value vanishes unless there are at least two photons around somewhere to be detected. We are also ignoring some subtleties regarding the two field annihilation operators; recall that E(+) \alpha (r1, t1) and E(+) \alpha (r2, t2) commute only if the two respective spacetime points lie on the same light cone. In practice this does not matter, since for example the two detectors will monitor different outputs of a beam splitter, such that they cannot influence each other.
16.2 Beam Splitters 16.2 Beam Splitters For our purposes, a beam splitter is an optical element that transforms two input modes or ‘‘beams’’ into two output modes. We have treated the beam splitter before in Section 12.1.2, but as it is central to our discussion here, we will review the setup with a slightly different notation. Labeling the field at the first and second inputs as E(+) and E(+) , respectively, the transformation properties are characterized by field reflection and transmission coefficients r and t, representing reflection and transmission of E(+) , and coefficients r′ and t′, representing reflection and transmission of E(+) , as shown here. E1 (+) tE1 (+) rE1 (+) E2 (+) roo'E2 (+) too'E2 (+) Assuming the beam splitter is lossless, it must induce a unitary transformation on the two input modes, which we can represent by the matrix U = t r′ r t′ , (16.13) in the sense that the output modes are related to the input modes by this operator: Eout,1 Eout,2 = U E1 E2 . (16.14) However, the fact that U is unitary constrains its form; in fact the general form for a 2 \times 2 unitary matrix is U = t −r∗ r t∗ , (16.15) from which we conclude that
(16.16) which expresses the lossless property of the beam splitter, and r′ = −r∗, t′ = t∗, (16.17) so that the reflection and transmission coefficients from either direction only differ by phases (which we have somewhat arbitrarily fixed here). If we consider only monochromatic fields, with inputs and outputs at the same frequency, then from Eq. (16.2) we see that the fields E(+) differs from a lowering operator a only by a constant factor, which is the same for every mode here. Quantum mechanically, then, we may write Eq. (16.14) in terms of operators as b1 b2 = U a1 a2 = t −r∗ r t∗ a1 a2 , (16.18) (beam splitter transformation) where a1,2 are the annihilation operators for the input modes, and b1,2 are the annihilation operators for the output modes. 16.3 Collision of One Photon and a Beam Splitter Consider a single photon incident on a beam splitter, where we monitor each output of the beam splitter with a detector.
Chapter 16. Photodetection detector 1 detector 2 To the extent that it makes sense to do so, we will consider the input and output fields to be monochromatic as in Eq. (16.18). If we begin with a single photon in input 1, then the initial state is
1|0, 0\rangle , (16.19) where the states |n, m\rangle are joint Fock states for the two inputs. If we solve Eq. (16.18) for the input operators in terms of the output operators, we have a1 a2
b1 b2 = t∗ r∗ −r t b1 b2 , (16.20) so that
(16.21) or a\dagger
- (16.22) To find the output state after the beam splitter, we can use this relation to eliminate the input-field operator in Eq. (16.19)
tb\dagger
(output state for single-photon input) (16.23) The output state is thus an entangled state, with a superposition of having a single photon in each mode. The result is now fairly obvious, but from Eq. (16.11), the joint photodetection probability is P(t) ∝ D b\dagger 1b\dagger 2b2b1 E = 0, (16.24) which of course vanishes for the output state (16.23), since there is only one photon. Experimentally, it is difficult to prepare an input state of exactly a single photon. In practice, a highly attenuated classical field
crystal, where correlated pairs of photons are generated from a laser field via parametric downconversion, then the detection of a photon in the other output can be used to ‘‘herald’’ the presence of a single photon in the beam-splitter setup. Thus measurement is used to convert the coherent state into a one-photon state. The real situation is more complicated due to ‘‘accidental’’ coincidences (since there is a small probability of having two photon pairs present simultaneously), finite detection times, and ‘‘dark counts,’’ or spurious photodetection events due to thermal fluctuations in the detectors. However, this can be done, and is one of the simplest demonstrations of the manifestly quantum-mechanical nature of the electromagnetic field:2 a classical field can be divided arbitrarily, so a classical Hanbury-Brown–Twiss experiment always gives a signal for arbitrarily weak fields. In terms of the normalized degree of second-order coherence, the quantum version of this experiment violates the inequality (2.72), since g(2) can be much smaller than unity. 2P. Grangier , G. Roger and A. Aspect, ‘‘Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences,’’ Europhysics Letters 1, 173 (1986) (doi: 10.1209/0295-5075/1/4/004); J. J. Thorn, M. S. Neel, V. W. Donato, G. S. Bergreen, R. E. Davies, and M. Beck, ‘‘Observing the quantum behavior of light in an undergraduate laboratory,’’ American Journal of Physics 72, 1210 (2004) (doi: 10.1119/1.1737397).
16.4.1 Simple Theory¶
16.4 Two-Photon Interference 16.4 Two-Photon Interference 16.4.1 Simple Theory Suppose we now treat the case of two incident photons on a beam splitter. Again treating the fields as monochromatic (and identical), we can model a photodetection experiment with this particular input. There are two general cases we can consider. The first is when both photons are incident in the same input—here, input 1. detector 1 detector 2 Then the input state is
1)2 \sqrt 2 |0, 0\rangle , (16.25) and again using Eq. (16.22) to eliminate the input operator, we find the output state
tb\dagger
2 \sqrt |0, 0\rangle . (16.26) We can write this out to obtain
\sqrt
(16.27) This is not too surprising. Identifying the probabilities for two photon transmitted, one photon transmitted and zero photons transmitted as |t|4, 2|rt|2, and |r|4, respectively, this is just the classical transmission prob- ability of two independent particles according to the binomial distribution, where the ‘‘success probability’’ for a single particle is |t|2. The other case, where one photon is incident in each input, is quite different, however. detector 1 detector 2 Here, the input state is
1a\dagger 2|0, 0\rangle . (16.28) We can again use Eq. (16.22) to eliminate a\dagger 1, and to eliminate a\dagger 2, we can use Eq. (16.20) to write a\dagger
- (16.29) Thus, the output state is
tb\dagger
−r∗b\dagger
|0, 0\rangle . (16.30)
16.4.2 Coherence Effects¶
Chapter 16. Photodetection Multiplying all this out,
\sqrt
|t|2 −|r|2
\sqrt 2rt∗|0, 2\rangle (output state, for one photon in each input) (16.31) The difference here is that the |1, 1\rangle term exhibits destructive interference. The classical probabilities for two photons in output 1 is |rt|2, which is the same as the probability for two photons in output 2; the remaining probability for one photon in each direction is 1 −2|rt|2 = |r|4 + |t|4. In both the classical and quantum case, the probability for coincidence detection—corresponding to one photon in each output—is minimized for an equal beam splitter with |r| = |t| = 1/ \sqrt 2. In this case, the classical probability is 1/2, while the quantum probability is zero. This is thus a quantum interference effect between the two photons, which rules out the photons leaving the beam splitter in different outputs. This tendency of the photons to ‘‘stick’’ together is a nice demonstration of the bosonic nature of the quantum electromagnetic field. Had the photons been fermions, the requirement of antisymmetry of the state would have actually produced the opposite prediction: coincidences would happen with unit probabilities, since the outcome must always have one photon in each output. The classical prediction is somewhere in between. In this sense, the bosonic case can be regarded as constructive interference for the two non-coincidence outcomes, increasing the probability of finding the photons to be together; this is consistent with our discussion of the exchange interaction in Section 4.4.4.1. This effect is known as the Hong–Ou–Mandel effect, after the first experimental demonstration.3 16.4.2 Coherence Effects Though the Hong–Ou–Mandel effect is due to interference, it turns out that it does not sensitively depend on the relative phase of the two input photons. That is, varying the relative phase by \pi does not necessarily cause a large change in the interference effect, as it would in an interferometer. To see this, we must relax the monochromatic idealization of the input light.4 16.4.2.1 Quantum Beam Recall again from Eq. (16.2) that the annihilation part of the electromagnetic field has the form
X k,\zeta r ¯h\omegak 2ϵ0
(16.32) If we consider the light to be in the form of a ‘‘beam,’’ as in the output of a laser, then we should regard the wave vector k to point along a particular direction, say the x-direction. Thus, ky = kz = 0 and we have only a one-dimensional set of modes. Recall that in calculations with th e three-dimensional field, in the 3C. K. Hong, Z. Y. Ou, and L. Mandel, ‘‘Measurement of subpicosecond time intervals between two photons by interference,’’ Physical Review Letters 59, 2044 (1987) (doi: 10.1103/PhysRevLett.59.2044). 4Here, we are following H. Fearn and R. Loudon, ‘‘Theory of two-photon interference,’’ Journal of the Optical Society of America B 6, 917 (1989) (doi: 10.1364/JOSAB.6.000917).
16.4 Two-Photon Interference continuum limit we made the replacement X k −\rightarrow V (2\pi)3 Z all space d3k, (16.33) since the spacing between modes in any direction in k-space was 2\pi/L, where L3 = V . Then the quantization volume V canceled the corresponding factor from the squared mode functions |fk,\zeta|2 ∝1/V for the free-sapce modes
eik\cdotr \sqrt V . (16.34) In the one-dimensional case, we will similarly have X kx\ge 0 −\rightarrow L 2\pi Z \infty dk, (16.35) taking the beam to point along the positive x-axis. For calculations second order in the field, we can modify the field by changing the sum to an integral, tacking on the square root of the discretization factor L/2\pi, and assume a particular polarization along ˆ\epsilon:
Z \infty d\omega r ¯h\omega
(16.36) (quantized beam) Here, we have changed the integration variable to \omega = \omegak = ck, defined the mode area A = V /L, and written out explicitly the time dependence of the mode annihilation operator. Recall that in the continuum
be tightly localized near some ‘‘laser frequency’’ \omegaL. Since the factor of \sqrt\omega should vary slowly over this spectrum, we can replace it by its value at the laser frequency, so that
r ¯h\omega 4\piϵ0cA Z \infty
(quantized, quasi-monochromatic beam) (16.37) We thus have essentially a Fourier transform of the monochromatic mode operators a(\omega). 16.4.2.2 Pulse-Annihilation Operators Again, the above expression (16.37) shows that the time-dependent electric-field operator for a quasi- monochromatic beam appears as a one-dimensional Fourier transform of the field operators a(\omega). We can take this as a motivation to define the creation operator
Z \infty
(creation operator, pulsed excitation) (16.38) where \alpha(\omega) represents the spectrum of the excitation, which is normalized according to Z \infty
(16.39) Since we are assuming a quasi-monochromatic beam, whose spectral width is much smaller than \omegaL (as in a laser field), we can extend the lower limit of the integral, so that Z \infty −\infty
(16.40)
Chapter 16. Photodetection Thus, this creation operator creates a photon similar to a\dagger(\omega) in the sense that A\dagger(\alpha)|0\rangle represents a normalized, one-photon state, but in a superposition of different frequencies. Thus, emulating the form of the field operator (16.37), we can define a temporal envelope
\sqrt 2\pi Z \infty −\infty
(16.41) (pulse envelope) and it is not hard to show by direct substitution that Z \infty −\infty
(16.42) so that the envelope function created by A\dagger(\alpha) is also normalized. 16.4.2.3 Detection We can now also replace the full field (16.37) with the normalized, time-dependent annihilation operator
\sqrt 2\pi Z \infty −\infty
(16.43) (pulse-annihilation operator) within the same narrowband approximation. This is proportional to the full field except that we have dropped the dependence on the spatial coordinate x, since for propagation in vacuum it can be absorbed into the temporal phase factor. We need not assume any particular frequency dependence for the annihilation operator, and in fact we will need this operator for detection. For a wide-bandwidth detector, this operator corresponds to annihilating a photon at the particular time t. (A finite detector bandwidth corresponds to some uncertainty in the time of annihilation.) Then we can use this operator in place of the full field in Eq. (16.7) for the detection probability, and integrate over the detection time interval T to find the total (average) number of detected photons:
Z T dt
a\dagger(t) a(t)
. (16.44) (mean number of detected photons) We have replaced the proportionality by an equality here; this expression is scaled properly, as we can see by considering the state of n excitations |n\rangle , assuming a sufficiently long detection time T (and assuming
Similarly, based on Eq. (16.11), we can write down the mean cross-correlation for the photocounts for two detectors:
Z T dt Z T dt′ D a\dagger 1(t) a\dagger 2(t′) a2(t′) a1(t) E . (joint detection average) (16.45) This expression is normalized properly as in the average number of detected photons, and for the two- photon input states, that we will consider below, corresponds to the joint detection probability over the (long) detection time. 16.4.2.4 Interference of Coherence Now back to the problem of two-photon interference. The input mode, now with two (possibly different) quasi-monochromatic photons, is
(16.46) where the subscripts A\dagger \beta label the mode on which the creation operator acts. This expression generalizes the monochromatic expression (16.28). The same beam-splitter-transformation relations (16.22) and (16.29)
16.4 Two-Photon Interference hold here (assuming the action of the beam splitter is frequency-independent and nondispersive), so that A\dagger
2(\alpha1) A\dagger
2(\alpha2). (16.47) We have thus connected the input operators A\dagger \beta to the output operators B\dagger \beta, which are defined in exactly the same way. We can obtain the output mode by using these relations in the input state (16.46):
h −r∗tB\dagger
2(\alpha2) i |0, 0\rangle . (16.48) Only the middle two terms correspond to one output photon in each mode, and thus these will give the only contribution to \langle N1N2\rangle . To compute the detector cross-correlation (16.45), we can simply consider the post-detection state,
h |t|2B\dagger
1(\alpha2) i |0, 0\rangle , (16.49) where again we need only the middle two terms of Eq. (16.48). In this state, we have parts that refer to either mode; for example, the part of this state that refers to wave packet 1 in mode 1 is b1(t)B\dagger
\sqrt 2\pi Z d\omega Z
= \sqrt 2\pi Z d\omega Z
= \sqrt 2\pi Z
(16.50) where in the first step we used Eq. (16.43) for b1(t) and Eq. (16.38) for B\dagger 1(\alpha1); in the second step we used
pulse envelope \alpha(t). The other parts of Eq. (16.49) follow from this result simply by relabeling the arbitrary indices, and finally we may use the norm of the resulting post-detection state to write
Z T dt Z T dt′ D b\dagger 1(t) b\dagger 2(t′) b2(t′) b1(t) E = Z T dt Z T
. (16.51) Multiplying out the square, we obtain the squares of each of the terms in the absolute value, which have time dependence of the form |\alpha1(t)|2|\alpha2(t′)|2; due to the normalization of these pulse profiles, the integrals give |t|4 and |4|4 for these two terms. The remaining two cross terms give −|r|2|t|2\alpha1(t)\alpha∗ 2(t)\alpha∗
which when integrated, combine to give −2|r|2|t|2
Z dt \alpha∗ 2(t)\alpha1(t)
. (16.52) Combining terms, we finally find
Z dt \alpha∗ 2(t)\alpha1(t)
, (two-photon cross-correlation) (16.53) where the overall result is automatically positive, since the modulus of the integral at at most unity, since the pulse profiles are normalized.
Chapter 16. Photodetection The last term in the cross correlation is the overlap integral of the two input pulses. For identical, perfectly overlapping pulses, the integral reduces to unity, and thus
|r|2 −|t|22 , (16.54) which recovers the simple result (16.31) from the monochromatic theory. If the pulses are widely separated, then the overlap integral vanishes, and we recover the classical expectation
(16.55) as is appropriate for distinguishable pulses. Finally, if the two input pulses are identical, but one is delayed
Z
, (16.56) and thus the interference term reduces to the degree of first-order coherence (normalized autocorrelation function) for the input pulse. While this modulates the fringe visibility in an interferometer, it represents the entire interference in the Hong–Ou–Mandel experiment. Thus as a function of the time delay of one of the pulses, the coincidence probability exhibits a ‘‘dip,’’ konwn as the Hong–Ou–Mandel dip, whose profile is the pulse-field autocorrelation function (something like the convolution of the pulse with itself). This is illustrated below for the case of a symmetric beam splitter and a Gaussian pulse envelope. t \cdotN1N2‚ 0.5 The width of the dip is of the order of the coherence length, which can be very long for narrow-line lasers, or much shorter for pulsed lasers.