Quantum Mechanics by Konstantin K. Likharev - HTML preview
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2
2
1
f
1
H
f
1
H
Fig. 8.4. The simplest (a) classical and (b) quantum ways to classify a single-bit Boolean function f.
On the other hand, as will be shown below, any of the four functions f may be implemented quantum-mechanically, for example (Fig. 5a) as a unitary transform of two input qubits, acting as follows on each basis component j 1 j 2 j 1 j 2 of the general input state (134): ˆ f j j j j f ( j ) ,
(8.144)
1
2
1
2
1
where f is the corresponding classical Boolean function – see the table in Eq. (141).
(a)
(b)
j
j 1
j
j
1
1
1
Fig. 8.5. Two-qubit quantum gates: (a) a
C
two-qubit function f and (b) its particular
f
case C (CNOT), and their actions on a
j
j f ( j )
j
j j
2
2
1
2
2
1
basis state.
In the particular case when f in Eq. (144) is just the YES function: f( j) = f 2( j) = j, this “circuit” is reduced to the so-called CNOT gate, a key ingredient of many other quantum computation schemes, performing the following two-qubit transform:
52 Alternatively, we may perform two sequential experiments on the same black box f, first recording, and then recalling the first experiment’s result. However, the Deutsch problem calls for a single-shot experiment.
53 The XOR sign should be by no means confused with the sign of the direct product of state vectors (which in this section is just implied).
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ˆ
C j j j
j j .
(8.145a) CNOT
1 2
1
2
1
function
Let us use Eq. (142) to spell out this function for all four possible input qubit combinations: ˆ
ˆ
ˆ
ˆ
C 00 00 , C 01 01 , C 10 11 , C 11 10 .
(8.145b)
In plain English, this means that acting on a basis state j 1 j 2, the CNOT gate leaves the state of the first, source qubit (shown by the upper horizontal line in Fig. 5) intact, but flips the state of the second, target qubit if the first one is in the basis state 1. In even simpler words, the state j 1 of the source qubit controls the NOT function acting on the target qubit; hence the gate’s name CNOT – the semi-acronym of
“Controlled NOT”.
For the quantum function (144), with an arbitrary and unknown f, the Deutsch problem may be solved within the general scheme shown in Fig. 3, with the particular structure of the unitary-transform box U spelled out in Fig. 4b, which involves just one implementation of the function f. Here the single-qubit quantum gate H performs the Hadamard (or “Walsh-Hadamard“ or “Walsh”) transform,54 whose operator is defined by the following actions on the qubit’s basis states:
1
ˆ
H 0
0 1
1
ˆ
,
H 1
0 1 ,
(8.146) Hadamard
2
2
transform
– see also the two left state-label columns in Fig. 4b.55 Since this operator has to be linear (to be quantum-mechanically realistic), it needs to perform the action (146) on the basis states even when they are parts of a linear superposition – as they are, for example, for the two right Hadamard gates in Fig.
4b. For example, as immediately follows from Eqs. (146) and the operator’s linearity,
1
1
1 1
1
ˆ
H ˆ
H
ˆ
0 H
0 1
ˆ
H
ˆ
0 H 1
0 1 0 1 0 . (8.147a)
2
2
2 2
2
Absolutely similarly, we may get56
ˆ
H ˆ
H 1 1 .
(8.147b)
Let us carry out a sequential analysis of the whole “circuit” shown in Fig. 4b. Since the input states of gate f in this particular circuit are described by Eqs. (146), its output state’s ket is
1
1
1
ˆ f ˆ
ˆ
H 0 H ˆ
1 f
0 1 0 1 ˆ
ˆ
ˆ
ˆ
f 00 f 01 f 10 f 11 .
(8.148)
2
2
2
Now we may apply Eq. (144) to each component in the parentheses:
54 Named after mathematicians J. Hadamard (1865-1963) and J. Walsh (1895-1973). Note that to avoid any chance of confusion between the Hadamard transform’s operator H ând the general Hamiltonian operator H ˆ , in these notes, they are typeset using different fonts.
55 Note that according to Eq. (146), the Hadamard operator does not belong to the class of transforms described by Eq. (140) – while the whole “circuit” shown in Fig. 4b, does – see below.
56 Since the states 0 and 1 form a full basis of a single qubit, both Eqs. (147) may be summarized as operator equality ˆ 2
H I ˆ . It is also easy to verify that the Hadamard transform of an arbitrary state may be represented on the Bloch sphere (Fig. 5.3) as a -rotation about the axis that bisects the angle between the x- and z- axes.
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ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
f 00 f 01 f 10 f 11 f 0 0 f 0 1 f 1 0 f 1 1
0 0 f (0) 0 1 f (0) 1 0 f )
1
(
1 1 f )
1
(
(8.149)
0 0 f (0) 1 f (0) 1 0 f )
1
(
1 f )
1
( .
Note that the contents of the first parentheses of the last expression, characterizing the state of the target qubit, is equal to (0 – 1) (–1)0 (0 – 1) if f(0) = 0 (and hence 0 f(0) = 0 and 1 f(0) = 1), and to (1 – 0) (–1)1(0 – 1) in the opposite case f(0) = 1, so both cases may be described in one shot by rewriting the parentheses as (–1) f(0)(0 – 1). The second parentheses are absolutely similarly controlled by the value of f(1), so the two outputs of gate f are unentangled: 1
f (0)
f )
1
(
1
F
1
ˆ f ˆ
ˆ
H 0 H 1 ( )
1
0 ( )
1
1 0 1
0 ( )1 1 0 1 , (8.150)
2
2
2
where the last step has used the fact that the classical Boolean function F defined by Eq. (142) is equal to [ f(1) – f(0)] – please compare the last two columns in Eq. (141). The front sign in the last form of Eq. (150) may be prescribed to any of the component ket-vectors – for example to that of the target qubit, as shown by the third column of state labels in Fig. 4b.
This intermediate result is already rather remarkable. Indeed, it shows that, despite the superficial impression one could get from Fig. 5, the gates f and C, being “controlled” by the source qubit, may change that qubit’s state as well! This fact (partly reflected by the vertical direction of the control lines in Figs. 4 and 5, symbolizing the same stage of the system’s time evolution) shows how careful one should be interpreting quantum-computational “circuits”, thriving on qubits’ entanglement: the “signals” on different sections of a horizontal “wire” may differ – see Fig. 4b again.
At the last stage of the circuit shown in Fig. 4b, the qubit components of the state (150) are fed into one more pair of Hadamard gates, whose outputs therefore are
1
F
1
F
1
1
ˆ
H
0 ( )1 1 ˆ
ˆ
H 0 ( )
1 H 1
ˆ
,
and H
0 1 ˆ
ˆ
H 1 H 0 . (8.151)
2
2
2
2
Now using Eqs. (146) again, we see that the output state ket-vectors of the source and target qubits are, respectively,
1 ( )
1 F
1 ( )
1 F
0
1 ,
and 1 .
(8.152)
2
2
Since, according to Eq. (142), the Boolean function F may take only values 0 or 1, the final state of the source qubit is always one of its basis states j, namely the one with j = F. Its measurement tells us whether the function f, participating in Eq. (144), is constant or balanced – see Eq. (141) again.57
Thus, the quantum circuit shown in Fig. 4b indeed solves the Deutsch problem in one shot.
Reviewing our analysis, we may see that this is possible because the unitary transform performed by the quantum gate f is applied to the linear combinations (146) rather than to the basis states 0 and 1. Due to this trick, the quantum state components depending on f(0) and f(1) are processed simultaneously, in 57 Note that the last Hadamard transform of the target qubit (i.e. the Hadamard gate shown in the lower right corner of Fig. 4b) is not necessary for the Deutsch problem’s solution – though it should be included if we want the whole circuit to satisfy the condition (140).
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parallel. This quantum parallelism may be extended to circuits with many ( N >> 1) qubits and, for some tasks, provide a dramatic performance increase – for example, reducing the necessary circuit component number from O(2 N) to O( N p), where p is a finite (and not very big) number.
However, this efficiency comes at a high price. Indeed, let us discuss the possible physical implementation of quantum gates, starting from the single-qubit case, on an example of the Hadamard gate (146). With the linearity requirement, its action on the arbitrary state (133) should be 1
1
1
1
ˆ
ˆ
ˆ
H a H 0 a H 1 a
a
a a
a a
(8.153)
0
1
0
0 1 1 0 1
0 1 0
0 11 ,
2
2
2
2
meaning that the state probability amplitudes in the end ( t = T) and in the beginning ( t = 0) of the qubit evolution in time have to be related as
a ( )
0 a ( )
0
a ( )
0 a ( )
0
a ( T )
0
1
, a ( T )
0
1
.
(8.154)
0
2
1
2
This task may be again performed using the Rabi oscillations, which were discussed in Sec. 6.5, i.e. by applying to the qubit (a two-level system), for a limited time period T, a weak sinusoidal external signal of frequency equal to the intrinsic quantum oscillation frequency nn’ defined by Eq. (6.85).
The analysis of the Rabi oscillations was carried out in Sec. 6.5, even for non-vanishing (though small) detuning = – nn, but only for the particular initial conditions when at t = 0 the system was fully in one on the basis states (there labeled as n’), i.e. the counterpart state (there labeled n) was empty. For our current purposes, we need to find the amplitudes a 0,1( t) for arbitrary initial conditions a 0,1(0), subject only to the time-independent normalization condition a 02 + a 12 = 1. For the case of exact tuning, =
0, the solution of the system (6.94) is elementary,58 and gives the following solution:59
a ( t) a ( )
0 cos t
ia ( )
0 ei sin t
,
0
0
1
(8.155)
a ( t) a ( )
0 cos t
ia ( )
0 e i
sin t
,
1
1
0
where is the Rabi oscillation frequency (6.99), in the exact-tuning case proportional to the amplitude
A of the external ac drive A = Aexp{ i} – see Eq. (6.86). Comparing these expressions with Eqs.
(154), we see that for t = T = /4 and = /2 they “almost” coincide, besides the opposite sign of a 1( T). Conceptually the simplest way to correct this deficiency is to follow the ac “/4-pulse”, just discussed, by a short dc “-pulse” of the duration T’ = /, which temporarily creates a small additional energy difference between the basis states 0 and 1. According to the basic Eq. (1.62), such difference creates an additional phase difference T’/ between the states, equal to for the “-pulse”.
Another way (that may be also useful for two-qubit operations) is to use another, auxiliary state with energy E 2 whose distances from the basic levels E 1 and E 0 are significantly different from the difference ( E 1 – E 0) – see Fig. 6a. In this case, the weak external ac field tuned to any of the three potential quantum transition frequencies nn’ ( En – En’)/ initiates such transitions between the corresponding states only, with a negligible perturbation of the third state. (Such transitions may be 58 An alternative way to analyze the qubit evolution is to use the Bloch equation (5.21), with an appropriate function ( t) describing the control field.
59 To comply with our current notation, the coefficients an’ and an of Sec. 6.5 are replaced with a 0 and a 1.
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again described by Eqs. (155), with the appropriate index changes.) For the Hadamard transform implementation, it is sufficient to apply (after the already discussed /4-pulse of frequency 10, and with the initially empty level E 2), an additional -pulse of frequency 20, with any phase . Indeed, according to the first of Eqs. (155), with the due replacement a 1(0) a 2(0) = 0, such pulse flips the sign of the amplitude a 0( t), while the amplitude a 1( t), not involved in this additional transition, remains unchanged.
(a)
(b)
(c)
E 2
2
11
11
1
2
20
21
10
01 , 10
E
01
1
1
10
1
2
E
00
0
0
00
Fig. 8.6. Energy-level schemes used for unitary transformations of (a) single qubits and (b, c) two-qubit systems.
Now let me describe the conceptually simplest (though, for some qubit types, not the most practically convenient) scheme for the implementation of two-qubit gates, on an example of the CNOT
gate whose operation is described by Eq. (145). For that, evidently, the involved qubits have to interact for some time T. As was repeatedly discussed in the two last chapters, in most cases such interaction of two subsystems is factorable – see Eq. (6.145). For qubits, i.e. two-level systems, each of the component operators may be represented by a 22 matrix in the basis of states 0 and 1. According to Eq. (4.106), such a matrix may be always expressed as a linear combination ( b I + c), where b and three Cartesian components of the vector c are c-numbers. Let us consider the simplest form of such factorable interaction Hamiltonian:
1 2
ˆ
ˆ
,
0
for t
z
z
T ,
ˆ
H
t
(8.156)
int
,
0
otherwise,
where the upper index is the qubit number and is a c-number constant.60 According to Eq. (4.175), by the end of the interaction period, this Hamiltonian produces the following unitary transform:
i
i
1 2
ˆ
ˆ
U
exp H T exp
ˆ
ˆ
(8.157)
z
z T
.
int
int
Since in the basis of unperturbed two-bit basis states j
1
2
1 j 2, the product operator
ˆ
ˆ
is diagonal, so is
z
z
the unitary operator (157), with the following action on these states:
60 The assumption of simultaneous time independence of the basis state vectors and the interaction operator (within the time interval 0 < t < T) is possible only if the basis state energy difference of both qubits is exactly the same. In this case, the simple Eq. (156) follows from Figs. 6b, which shows the spectrum of the total energy E
= E 1 + E 2 of the two-bit system. In the absence of interaction (Fig. 6b), the energies of two basis states, 01 and
10, are equal, enabling even a weak qubit interaction to cause their substantial evolution in time – see Sec. 6.7. If the qubit energies are different (Fig. 6c), the interaction may still be reduced, in the rotating-wave approximation, to Eq. (156), by compensating the energy difference (1 – 2) with an external ac signal of frequency = (1 –
2)/ – see Sec. 6.5.
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ˆ
U
j j
i
j j
(8.158)
int
1 2
exp
)
1
(
(2)
z
z
,
1 2
where – T/, and z are the eigenvalues of the Pauli matrix z for the basis states of the corresponding qubit: z = +1 for j = 0, and z = –1 for j = 1. Let me, for clarity, spell out Eq. (158) for the particular case = –/4 (corresponding to the qubit coupling time T = /4): ˆ
i / 4
ˆ
i / 4
ˆ
i / 4
ˆ
U
00 e
00 , U
01 e
01 , U 10 e
10 , U 11
i / 4
e
11 . (8.159)
int
int
int
int
In order to compensate for the undesirable parts of this joint phase shift of the basis states, let us now apply similar individual “rotations” of each qubit by angle ’ = +/4, using the following product of two independent operators, plus (just for the result’s clarity) a common, and hence inconsequential, phase shift ” = –/4:61
1
2
ˆ
U
exp i ' ˆ
ˆ
i " exp i
ˆ
exp i
ˆ
i
e
.
(8.160)
com
z
z
1
2
/ 4
4 z
4 z
Since this operator is also diagonal in the j 1 j 2 basis, it is easy to calculate the change of the basis states by the total unitary operator ˆ
ˆ
ˆ
U
U U :
tot
com
int
ˆ
ˆ
ˆ
ˆ
U
00 00 ,
U
01 01 ,
U 10 10 ,
U 11 11 .
(8.161)
tot
tot
tot
tot
This result already shows the main “miracle action” of two-qubit gates, such as the one shown in Fig.
4b: the source qubit is left intact (only if it is in one of the basis states!), while the state of the target qubit is altered. True, this change (of the sign) is still different from the CNOT operator’s action (145), but may be readily used for its implementation by sandwiching the transform U tot between two Hadamard transforms of the target qubit alone:
1
ˆ
ˆ 2 ˆ
ˆ 2
C H U
.
(8.162)
tot H
2
So, we have spent quite a bit of time on the discussion of the very simple CNOT gate,62 and now I can reward the reader for their effort with a bit of good news: it has been proved that an arbitrary unitary transform that satisfies Eq. (140), i.e. may be used to implement the general scheme outlined in Fig. 3, may be decomposed into a set of CNOT gates, possibly augmented with simpler single-qubit gates.63 Unfortunately, I have no time for a detailed discussion of more complex circuits.64 The most 61 As Eq. (4.175) shows, each of the component unitary transforms exp{ i ' ˆ
} may be created by applying to
z
each qubit, for time interval T’ = ’/ ’, a constant external field described by Hamiltonian H ˆ '
ˆ . We
z
already know that for a charged, spin-½ particle, such Hamiltonian may be created by applying a z-oriented external dc magnetic field – see Eq. (4.163). For most other physical implementations of qubits, the organization of such a Hamiltonian is also straightforward – see, e.g., Fig. 7.4 and its discussion.
62 As was discussed above, this gate is identical to the two-qubit gate shown in Fig. 5a for f = f 3, i.e. f( j) = j. The implementation of the gate of f for 3 other possible functions f requires straightforward modifications, whose analysis is left for the reader’s exercise.
63 This fundamental importance of the CNOT gate was perhaps a major reason why David Wineland, the leader of the NIST group that had demonstrated its first experimental implementation in 1995 (following the theoretical suggestion by J. Cirac and P. Zoller), was awarded the 2012 Nobel Prize in Physics – shared with Serge Haroche, the leader of another group working towards quantum computation.
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famous of them is the scheme for integer number factoring, suggested in 1994 by Peter Winston Shor.65
Due to its potential practical importance for breaking the broadly used communication encryption schemes such as the RSA code,66 this opportunity has incited much enthusiasm and triggered experimental efforts to implement quantum gates and circuits using a broad variety of two-level quantum systems. By now, the following experimental options have given the most significant results:67
(i) Trapped ions. The first experimental demonstrations of quantum state manipulation (including the already mentioned first CNOT gate) have been carried out using deeply cooled atoms in optical traps, similar to those used in frequency and time standards. Their total spins are natural qubits, whose states may be manipulated using the Rabi transfers excited by suitably tuned lasers. The spin interactions with the environment may be very weak, resulting in large dephasing times T 2 – up to a few seconds. Since the distances between ions in the traps are relatively large (of the order of a micron), their direct spin-spin interaction is even weaker, but the ions may be made effectively interacting either via their mechanical oscillations about the potential minima of the trapping field or via photons in external electromagnetic resonators (“cavities”).68 Perhaps the main challenge of using this approach to quantum computation is poor “scalability”, i.e. the enormous experimental difficulty of creating and managing large ordered systems of individually addressable qubits. So far, only some few-qubit systems have been demonstrated.69
(ii) Nuclear spins are also typically very weakly connected to their environment, with dephasing times T 2 exceeding 10 seconds in some cases. Their eigenenergies E 0 and E 1 may be split by external dc magnetic fields (typically, of the order of 10 T), while the interstate Rabi transfers may be readily achieved by using the nuclear magnetic resonance, i.e. the application of external ac fields with frequencies = ( E 1 – E 0)/ – typically, of a few hundred MHz. The challenges of this option include the weakness of spin-spin interactions (typically mediated through molecular electrons), resulting in a very slow spin evolution, whose time scale / may become comparable with T 2, and also very small level separations E 1 – E 0, corresponding to a few K, i.e. much smaller than the thermal energy at room temperature, creating a challenge of qubit state preparation.70 Despite these challenges, the nuclear spin option was used for the first implementation of the Shor algorithm for factoring of a small number (15 =
53) as early as 2001.71 However, the extension of this success to larger systems, beyond the set of spins inside one molecule, is extremely challenging.
64 For that, the reader may be referred to either the monographs by Nielsen-Chuang and Reiffel-Polak, cited above, or to a shorter (but much more formal) textbook by N. Mermin, Quantum Computer Science, Cambridge U. Press, 2007.
65 A clear description of this algorithm may be found in several accessible sources, including Wikipedia – see the article Shor’s Algorithm.
66 Named after R. Rivest, A. Shamir, and L. Adleman, the authors of the first open publication of the code in 1977, but actually invented earlier (in 1973) by C. Cocks.
67 For a discussion of other possible implementations (such as quantum dots and dopants in crystals) see, e.g., T.
Ladd et al., Nature 464, 45 (2010), and references therein.
68 A brief discussion of such interactions (so-called Cavity QED) will be given in Sec. 9.4 below.
69 See, e.g., S. Debnath et al., Nature 536, 63 (2016). Note also related work on arrays of trapped neutral atoms –
see, e.g., J. Perczel et al., Phys. Rev. Lett. 119, 023603 (2017); D. Bluvstein et al., Nature 604, 451 (2022).
70 This challenge may be partly mitigated using ingenious spin manipulation techniques such as refocusing – see, e.g., either Sec. 7.7 in Nielsen and Chuang, or J. Keeler’s monograph cited at the end of Sec. 6.5.
71 B. Lanyon et al., Phys. Rev. Lett. 99, 250505 (2001).
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(iii) Josephson-junction devices. Much better scalability may be achieved with solid-state devices, especially using superconductor integrated circuits including weak contacts – Josephson junctions – see their brief discussion in Sec. 1.6. The qubits of this type are based on the fact that the energy U of such a junction is a highly nonlinear function of the Josephson phase difference – see Sec.
1.6. Indeed, combining Eqs. (1.73) and (1.74), we can readily calculate U() as the work W of an external circuit increasing the phase from, say, zero to some value :
'
'
'
2 eI
d '
eI
c
2
U U 0
d W
IVdt
sin
c
'
dt
1 cos
.
(8.163)
' 0
' 0
'0
dt
There are several options for using this nonlinearity for creating qubits;72 currently, the leading option, called the phase qubit, is using the two lowest eigenstates localized in one of the potential wells of the periodic potential (163). A major problem with such qubits is that at the very bottom of this well the potential U() is almost quadratic, so the energy levels are nearly equidistant – cf. Eqs. (2.262), (6.16), and (6.23). This is even more true for the so-called “transmons” (and “Xmons”, and “Gatemons”, and several other very similar devices73) – the currently used phase qubits versions, where a Josephson junction is made a part of an external electromagnetic oscillator, making its relative total nonlineartity (anharmonism) even smaller. As a result, the external rf drive of frequency = ( E 1 – E 0)/, used to arrange the state transforms described by Eq. (155), may induce simultaneous undesirable transitions to (and between) higher energy levels. This effect may be mitigated by a reduction of the ac drive amplitude, but at a price of the proportional increase of the operation time and hence of dephasing effects – see below. (I am leaving a quantitative estimate of such an increase for the reader’s exercise.) Since the coupling of Josephson-junction qubits may be most readily controlled (and, very importantly, kept stable if so desired), they have been used to demonstrate the largest prototype quantum computing systems to date, despite quite modest dephasing times T 2 – for purely integrated circuits, in the tens of microseconds at best, even at operating temperatures in tens of mK. Several groups have announced chips with a few dozen of such qubits, but to the best of my knowledge, only their smaller subsets could be used for high-fidelity quantum operations so far.74
(iv) Optical systems, attractive because of their inherently enormous bandwidth, pose a special challenge for quantum computation: due to the virtual linearity of most electromagnetic media, the implementation of qubits requires relatively large components and high optical power.75 In 2001, a very 72 The “most quantum” option in this technology is to use Josephson junctions very weakly coupled to their dissipative environment (so the effective resistance shunting the junction is much higher than the quantum resistance unit R Q (/2) / e 2 ~ 104 ). In this case, the Josephson phase variable behaves as a coordinate of a 1D quantum particle moving in the 2-periodic potential (163), forming the energy band structure En( q) similar to those discussed in Sec. 2.7. Both theory and experiment show that in this case, the quantum states in adjacent Brillouin zones differ by the charge of one Cooper pair 2 e. (This is exactly the effect responsible for the Bloch oscillations of frequency (2.252).) These two states may be used as the basis states of charge qubits.
Unfortunately, such qubits are rather sensitive to charged impurities, randomly located in the junction’s vicinity, which may cause uncontrollable changes of its parameters, so currently, this option is not very actively pursued.
73 For a recent review of these devices see, e.g., G. Wendin, Repts. Progr. Phys. 80, 106001 (2017).
74 See, e.g., C. Song et al., Phys. Rev. Lett. 119, 180511 (2017); S. Krinner et al., Nature 605, 669 (2022); R.
Acharya et al. , Nature 614, 676 (2023).
75 For a state-of-the-art recent work in this direction see, e.g., X. Qiang et al., Nature Photonics 12, 534 (2018).
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smart way around this hurdle was invented.76 In this KLM scheme (also called “linear optical quantum computing”), nonlinear elements are not needed at all, and quantum gates may be composed just of linear devices (such as optical waveguides, mirrors, and beam splitters), plus single-photon sources and detectors. However, estimates show that this approach requires many more physical components than those using nonlinear quantum systems such as usual qubits,77 so right now it is not very popular.
So, despite three decades of large-scale (multi-billion-dollar) experimental and theoretical efforts, the progress of quantum computing development has been rather gradual. The main culprit here is the unintentional coupling of qubits to their environment, leading most importantly to their state dephasing, and eventually to errors. Let me discuss this major issue in detail.
Of course, some error probability exists in classical digital logic gates and memory cells as well.78 However, in this case, there is no conceptual problem with the device state measurement, so the error may be detected and corrected in many ways. Conceptually,79 the simplest of them is the so-called majority voting logic – using several similar logic circuits operating in parallel and fed with identical input data. Evidently, two such devices can detect a single error in one of them, while three devices in parallel may correct such an error, by taking two coinciding output signals for the genuine one.
For quantum computation, the general idea of using several devices (say, qubits) for coding the same information remains valid; however, there are two major complications. First, as we know from Chapter 7, the environment’s dephasing effect may be described as a slow random drift of the probability amplitudes aj, leading to the deviation of the output state fin from the required form (140), and hence to a non-vanishing probability of wrong qubit state readout – see Fig. 3. Hence the quantum error correction has to protect the result not against possible random state flips 0 1, as in classical digital computers, but against these “creeping” analog errors.
Second, the qubit state is impossible to copy exactly ( clone) without disturbing it, as follows from the following simple calculation.80 Cloning some state of one qubit to another qubit that is initially in an independent state (say, the basis state 0), without any change of , means the following transformation of the two-qubit ket: 0 . If we want such transform to be performed by a real quantum system, whose evolution is described by a unitary operator u ˆ , and to be correct for an arbitrary state , it has to work not only for both basis states of the qubit:
ˆ u 00 00 ,
ˆ u 10 11 ,
(8.164)
but also for their arbitrary linear combination (133). Since the operator u ˆ has to be linear, we may use that relation, and then Eq. (164) to write
76 E. Knill et al., Nature 409, 46 (2001).
77 See, e.g., Y. Li et al., Phys. Rev. X 5, 041007 (2015).
78 In modern integrated circuits, such “soft” (runtime) errors are created mostly by the high-energy neutrons of cosmic rays, and also by the -particles emitted by radioactive impurities in silicon chips and their packaging.
79 Practically, the majority voting logic increases circuit complexity and power consumption, so it is used only in the most critical points. Since in modern digital integrated circuits, the bit error rate is very small (< 10-5), in most of them, less radical but also less penalizing schemes are used – if used at all.
80 Amazingly, this simple no-cloning theorem was discovered as late as 1982 (to the best of my knowledge, independently by W. Wooters and W. Zurek, and by D. Dieks), in the context of work toward quantum cryptography – see below.
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ˆ u 0 ˆ u a 0 a 1
a u
a u
a
a
.
(8.165)
0
1
0
ˆ 00
ˆ 10
00
11
0
1
0
1
On the other hand, the desired result of state cloning is
a 0 a 1 a
a
a
a a
a
,
(8.166)
0
1
0
1
0
1
2 00
0
0 1 10
01
2 11
1
i.e. is evidently different, so, for an arbitrary state , and an arbitrary unitary operator u ˆ , ˆ u 0 ,
(8.167) No-cloning
theorem
meaning that the qubit state cloning is indeed impossible.81 This problem may be, however, indirectly circumvented – for example, in the way shown in Fig. 7a.
(a)
(b)
a 0 a 1
0
1
H
H
a 0
a 1
0
1
a 00
0
0
0
a 11
1
0
A
B
C
D
E
F
Fig. 8.7. (a) Quasi-cloning, and (b) detection and correction of dephasing errors in a single qubit.
Here the CNOT gate, whose action is described by Eq. (145), entangles an arbitrary input state (133) of the source qubit with a basis initial state of an ancillary target qubit – frequently called the ancilla. Using Eq. (145), we can readily calculate the output two-qubit state’s vector: ˆ
C a 0 a 1
a C
a C
a
a
.
(8.168) Quasi-
0
1
ˆ
ˆ
0
00
10
00
11
0
1
0
1
N 2
cloning
We see that this circuit does perform the desired operation (165), i.e. gives the initial source qubit’s probability amplitudes a 0 and a 1 equally to two qubits, i.e. duplicates the input information. However, in contrast with “genuine” cloning, it changes the state of the source qubit as well, making it entangled with the target (ancilla) qubit. Such “quasi-cloning” is the key element of most suggested quantum error correction techniques.
Consider, for example, the three-qubit “circuit” shown in Fig. 7b, which uses two ancilla qubits
– see the two lower “wires”. At its first two stages, the double application of the quasi-cloning produces an intermediate state A with the following ket-vector:
A a 000 a 111 ,
(8.169)
0
1
which is an evident generalization of Eq. (168).82 Next, subjecting the source qubit to the Hadamard transform (146), we get the three-qubit state B represented by the state vector 81 Note that this does not mean that two (or several) qubits cannot be put into the same, arbitrary quantum state –
theoretically, with arbitrary precision. Indeed, they may be first set into their lowest-energy stationary states, and then driven into the same arbitrary state (133) by exerting on them similar classical external fields. So, the no-cloning theorem pertains only to qubits in unknown states – but this is exactly what we need for error correction
– see below.
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1
1
B a
a
.
(8.170)
0
0 1 00 1 0 1 11
2
2
Now let us assume that at this stage, the source qubit comes into contact with a dephasing environment – in Fig. 7b, symbolized by the single-qubit “gate” . As we know from Chapter 7 (see Eq. (7.22) and its discussion, and also Sec. 7.3), its effect may be described by a random shift of the relative phase of two states:83
0
i
e 0
i
,
1 e
1 .
(8.171)
As a result, for the intermediate state C (see Fig. 7b) we may write
1
i
i
1
i
i
C a
e
e
a
e
e
.
(8.172)
0
0
1 00
1
0
1 11
2
2
At this stage of this simple theoretical model, the coupling with the environment is completely stopped (ahh, if this could be possible! we might have quantum computers by now :-), and the source qubit is fed into one more Hadamard gate. Using Eqs. (146) again, for state D after this gate we get D a
i
a i
.
(8.173)
0 cos
0
sin 1 00
1 sin
0
cos 1 11
Next, the qubits are passed through the second, similar pair of CNOT gates – see Fig. 7b. Using Eq.
(145), for the resulting state E we readily get the following expression: E a cos 000 a i sin 111 a i sin 011 a cos 100 , (8.174a) 0
0
1
1
whose right-hand side may by evidently re-grouped as
E a 0 a 1
a
a
i
.
(8.174b)
0
1
cos 00 0
1
1
0
sin 11
This is already a rather remarkable result. It shows that if we measured the ancilla qubits at stage E, and both results corresponded to states 0, we might be 100% sure that the source qubit (which is not affected by these measurements!) is in its initial state even after the interaction with the environment.
The only result of an increase of this unintentional interaction (as quantified by the r.m.s. magnitude of the random phase shift ) is the growth of the probability,
2
W sin ,
(8.175)
of getting the opposite result, which would signal a dephasing-induced error in the source qubit. Such implicit measurement, without disturbing the source qubit, is called quantum error detection.
An even more impressive result may be achieved by the last component of the circuit, the so-called Toffoli (or “CCNOT”) gate, denoted by the rightmost symbol in Fig. 7b. This three-qubit gate is conceptually similar to the CNOT gate discussed above, besides that it flips the basis state of its target qubit only if both source qubits are in state 1. (In the circuit shown in Fig. 7b, the former role is played 82 This state is also the three-qubit example of the so-called Greenberger-Horne-Zeilinger (GHZ) states, which are frequently called the “most entangled” states of a system of N > 2 qubits.
83 Let me emphasize again that Eq. (171) is strictly valid only if the interaction with the environment is a pure dephasing, i.e. does not include the energy relaxation of the qubit or its thermal activation to the higher-energy eigenstate; however, it is a reasonable description of errors in the frequent case when T 2 << T 1.
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by our source qubit, while the latter role, by the two ancilla qubits.) According to its definition, the Toffoli gate does not affect the first parentheses in Eq. (174b), but flips the source qubit’s states in the second parentheses, so for the output three-qubit state F we get
F a 0 a 1
a
a
i
.
(8.176a)
0
1
cos 00 0
1
0
1
sin 11
Obviously, this result may be factored as
F a 0 a 1
i
,
(8.176b)
0
1
cos 00
sin
11
Quantum
showing that now the source qubit is again fully unentangled from the ancilla qubits. Moreover, by error calculating the norm squared of the second operand, we get
correction
cos 00 i sin 11 cos 00 i sin 11 cos2 sin 2 1, (8.177)
so the final state of the source qubit exactly coincides with its initial state. This is the famous miracle of quantum state correction, taking place “automatically” – without any qubit measurements, and for any random phase shift .
The circuit shown in Fig. 7b may be further improved by adding Hadamard gate pairs, similar to that used for the source qubit, to the ancilla qubits as well. It is straightforward to show that if the dephasing is small in the sense that the W given by Eq. (175) is much less than 1, this modified circuit may provide a substantial error probability reduction (to ~ W 2) even if the ancilla qubits are also subjected to a similar dephasing and the source qubits, at the same stage – i.e. between the two Hadamard gates. Such perfect automatic correction of any error (not only of an inner dephasing of a qubit and its relaxation/excitation but also of the mutual dephasing between qubits) of any used qubit needs even more parallelism. The first circuit of that kind, based on nine qubits, which is a natural generalization of the three-qubit circuit discussed above, was invented in 1995 by the same P. Shor.
Later, five-qubit circuits enabling similar error correction were suggested. (The further parallelism reduction has been proved impossible.)
However, all these results assume that the error correction circuits as such are perfect, i.e.
completely isolated from the environment. In the real world, this cannot be done. Now the key question is what maximum level W max of the error probability in each gate (including those in the used error correction scheme) can be automatically corrected, and how many qubits with W < W max would be required to implement quantum computers producing important practical results – first of all, factoring of large numbers.84 To the best of my knowledge, estimates of these two related numbers have been made only for some very specific approaches, and they are rather pessimistic. For example, using the so-called surface codes, which employ many physical qubits for coding an informational one, and hence increase its fidelity, W max may be increased to ~10–3, but then we would need ~108 physical qubits for the Shor’s algorithm implementation.85 This is very far from what currently looks doable using the existing approaches.
Because of this hard situation, the current development of quantum computing is focused on finding at least some problems that could be within the reach of either the existing systems, or their immediate extensions, and simultaneously would present some practical interest. Currently, to the best 84 In order to compete with the existing classical factoring algorithms, such numbers should have at least 103 bits.
85 A. Fowler et al., Phys. Rev. A 86, 032324 (2012).
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of my knowledge, all suggested problems of this kind address either specially crafted mathematical problems,86 or properties of some simple physical systems – such as the molecular hydrogen87 or the deuteron (the deuterium’s nucleus, i.e. the proton-neutron system).88 In the latter case, the interaction between the qubits of the computational system is organized so that the system’s Hamiltonian is similar to that of the quantum system of interest. (For this work, quantum simulation is a more adequate name than “quantum computation”.89)
Such simulations are pursued by some teams using schemes different from that shown in Fig. 3.
Of those, the most developed is the so-called adiabatic quantum computation,90 which drops the hardest requirement of negligible interaction with the environment. In this approach, the qubit system is first prepared in a certain initial state and then is let to evolve on its own, with no effort to couple-uncouple qubits by external control signals during the evolution.91 Due to the interaction with the environment, in particular the dephasing and the energy dissipation it imposes, the system eventually relaxes to a final incoherent state, which is then measured. (This reminds the scheme shown in Fig. 3, with the important difference that the transform U may be not fully unitary.) From numerous runs of such an experiment, the outcome statistics may be revealed. Thus, in this approach, the interaction with the environment is allowed to play a certain role in the system evolution, though every effort is made to reduce it, thus slowing down the relaxation process – hence the word “adiabatic” in the name of this approach. This slowness allows the system to exhibit some quantum properties, in particular, quantum tunneling92
through the energy barriers separating close energy minima in the multi-dimensional space of states.
This tunneling creates a substantial difference in the finite state statistics from that in purely classical systems, where such barriers may be overcome only by thermally-activated jumps over them.93
Due to technical difficulties of the organization and precise control of long-range interaction in multi-qubit systems, the adiabatic quantum computing demonstrations so far have been limited to a few simple arrays described by the so-called extended quantum Ising (“spin-glass”) model
j j'
j
H ˆ J σ ˆ σ ˆ h σ ˆ ,
(8.178)
z
z
j z
{ j, j }
'
j
where the curly brackets denote the summation over pairs of close (though not necessarily closest) neighbors. Though the Hamiltonian (178) is the traditional playground of phase transitions theory (see, e.g., SM Chapter 4), to the best of my knowledge there are not many practically important tasks that 86 F. Arute et al., Nature 574, 505 (2019). Note that the claim of the first achievement of “quantum supremacy”, made in this paper, refers only to an artificial, specially crafted mathematical problem, and does not change my assessment of the current status of this technology.
87 P. O’Malley et al., Phys. Rev. X 6, 031007 (2016).
88 E. Dumitrescu et al., Phys. Lett. Lett. 120, 210501 (2018).
89 To the best of my knowledge, this idea was first put forward by Yuri I. Malin in his book Computable and Incomputable published in 1980, i.e. before the famous 1982 paper by Richard Feynman. Unfortunately, since the book was in Russian, this suggestion was acknowledged by the international community only much later.
90 Note that the qualifier “quantum” is important in this term, to distinguish this research direction from the classical adiabatic (or “reversible”) computation – see, e.g., SM Sec. 2.3 and references therein.
91 Some hybrids of this approach with the “usual” scheme of quantum computation have been demonstrated, in particular, using some control of inter-bit coupling during the relaxation process – see, e.g., R. Barends et al., Nature 534, 222 (2016).
92 As a reminder, this process was repeatedly discussed in this course, starting from Sec. 2.3.
93 A quantitative discussion of such jumps may be found, for example, in SM Sec. 5.6.
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could be achieved by studying the statistics of its solutions. Moreover, even for this limited task, the speed of the largest experimental adiabatic quantum “computers”, with several hundreds of Josephson-junction qubits94 is still comparable with that of classical, off-the-shelf semiconductor processors (with the dollar cost lower by many orders of magnitude), and no dramatic change of this comparison is predicted for realistic larger systems.
To summarize the current (circa mid-2024) situation with the quantum computation development, it faces a very hard challenge of mitigating the effects of unintentional coupling with the environment. This problem is exacerbated by the lack of algorithms, beyond Shor’s factoring, that would give quantum computation a substantial advantage over the classical competition in solving real-world problems, and hence a much broader potential customer base that would provide the field with the necessary long-term motivation and resources. So far, even the leading experts in this field abstain from predictions on when quantum computation may become a self-supporting commercial technology.
There may be somewhat better prospects for another application of entangled qubit systems, namely to telecommunication cryptography.95 The initial motivation here was much more modest: to replace the currently dominating classical encryption, based on the public-key RSA code mentioned above, which may be broken by factoring very large numbers, with a quantum encryption system that would be fundamentally unbreakable. The basis of this opportunity is the measurement postulate and the no-cloning theorem: if a message is carried over by a qubit, it is impossible for an eavesdropper (in cryptography, traditionally called Eve) to either measure or copy it faithfully, without also disturbing its state. However, as we have seen from the discussion of Fig. 7a, state quasi-cloning using entangled qubits is possible, so the issue is far from being simple, especially if we want to use a publicly distributed quantum key, in some sense similar to the classical public key used at the RSA encryption.
Unfortunately, I would not have time/space to discuss various options for quantum cryptography and public distribution of quantum keys,96 but cannot help demonstrating how inventive and counterintuitive they may be, on the famous example of the so-called quantum teleportation (Fig. 8).97
Suppose that some party A (in cryptography, traditionally called Alice) wants to send to party B
( Bob) the full information about the pure quantum state of a qubit, unknown to either party. Instead of sending her qubit directly to Bob, Alice asks him to send her one qubit () of a pair of other qubits prepared in a certain entangled state, for example in the singlet state described by Eq. (11); in our current notation
1
'
01 10 .
(8.179)
2
The initial state of the whole three-qubit system may be represented in the form a
a
a
a
' a 0 a 1 '
, (8.180a)
0
1
0
001
0
010
1
010
1 111
2
2
2
2
94 See, e.g., R. Harris et al., Science 361, 162 (2018). Similar demonstrations with trapped-ion systems so far have been on a smaller scale, with a few tens of qubits – see, e.g., J. Zhang et al., Nature 551, 601 (2017).
95 This general field was pioneered in the 1970s by S. Wisener.
96 Two of them are the BB84 suggested in 1984 by C. Bennett and G. Brassard, and the EPRBE suggested in 1991 by A. Ekert. For details, see, e.g., the review by N. Gizin et al., Rev. Mod. Phys. 74, 145 (2002).
97 This procedure had been first suggested in 1993 by the same Charles Bennett and then was repeatedly demonstrated experimentally – see, e.g., L. Steffen et al., Nature 500, 319 (2013) and literature therein.
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which may be equivalently rewritten as the following linear superposition,
1
'
a 0 a
a
a
1
0
1
1
0
1
1
0
s
s
2
2
(8.180b)
1
a 0 a
a
a
0
1
1
1
0
1
0
1
,
2
e
2
e
of the following four states of the qubit pair :
1
1
00 11 ,
01 10 . (8.181)
s
2
e
2
Alice
Bob
'
(a)
Fig. 8.8. Sequential stages of a “quantum
qubit
1
'
teleportation” procedure: (a) the initial state
(b)
with entangled qubits and ’, (b) the back
'
transfer of the qubit , (c) the measurement of
(c)
the pair , (d) the forward transfer of two
bits
2
'
(d)
classical bits with the measurement results, and
(e) the final state, with the state of the qubit ’
' (e)
mirroring the initial state of the qubit .
After having received qubit from Bob, Alice measures which of these four states the pair
has. This may be achieved, for example, by measurement of one observable represented by the operator
ˆ ând another one corresponding to
ˆ ˆ – cf. Eq. (156). (Since all four states (181)
z
z
x
x
are eigenstates of both these operators, these two measurements do not affect each other and may be performed in any order.) The measured eigenvalue of the former operator enables distinguishing the couples of states (181) with different values of the lower index, while the latter measurement distinguishes the states with different upper indices.
Then Alice reports the measurement result (which may be coded with just two classical bits) to Bob over a classical communication channel. Since the measurement places the pair definitely into the corresponding state, the remaining Bob’s bit ’ is now definitely in the unentangled single-qubit state that is represented by the corresponding parentheses in Eq. (180b). Note that each of these parentheses contains both coefficients a 0,1, i.e. the whole information about the initial state that the qubit
had initially. If Bob likes, he may now use appropriate single-qubit operations, similar to those discussed earlier in this section, to move his qubit ’ into the state exactly similar to the initial state of qubit . This fact does not violate the no-cloning theorem (167) because the measurement has already changed the state of .) This is, of course, a “teleportation” only in a very special sense of this term, but a good example of the importance of qubit entanglement’s preservation at their spatial transfer.98
Returning for just a minute to quantum cryptography: since its most common quantum key distribution protocols require just a few simple quantum gates, whose experimental implementation is 98 For this course, this was also a good primer for the forthcoming discussion of the EPR paradox and Bell’s inequalities in Chapter 10.
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already not a large technological challenge, the main focus of the current effort in the field is on decreasing the single-photon dephasing in long electromagnetic-wave transmission channels, with sufficiently high qubit transfer fidelity. The recent progress was rather impressive, with the demonstrated transfer of entangled qubits over landlines longer than 100 km,99 and over at least one satellite-based line longer than 1,000 km;100 and also a whole quantum key distribution over a comparable distance, though at a very low rate yet.101, 102
As an alternative, several classical encryption algorithms that are currently deemed quantum-resistant have already been developed103 for a near-future replacement of the RSA. This is why the first practical applications of the long-range quantum information transfer are perhaps still to be found.104
8.6. Exercise problems
8.1. Prove that Eq. (30) indeed yields E (1)
g
= (5/4) E H.
8.2. For a dilute gas of helium atoms in their ground state, with n atoms per unit volume, calculate its weak-field
(i) electric susceptibility e, and
(ii) magnetic susceptibility m,
and compare the results.
Hint: You may use the results of the variational description of the helium atom’s ground state in Sec. 2, and the model solutions of Problems 6.8 and 6.15.
8.3. Calculate the expectation values of the observables s1s2, S 2 (s1 + s2)2, and Sz s 1 z + s 2 z, for the singlet and triplet states of the system of two spins-½, directly – without using the general Eq. (48).
Compare the results with those for the system of two classical geometric vectors of length /2 each.
8.4. Discuss the factors 1/2 that participate in Eqs. (18) and (20) for the entangled states of the system of two spins-½, in terms of Clebsh-Gordan coefficients similar to those discussed in Sec. 5.7.
8.5.* Use the perturbation theory to calculate the so-called hyperfine splitting of the ground energy of the hydrogen atom,105 due to the interaction between the spins of its nucleus (proton) and electron.
99 See, e.g., T. Herbst et al., Proc. Natl. Acad. Sci. 112, 14202 (2015), and references therein.
100 J. Yin et al., Science 356, 1140 (2017).
101 H.-L. Yin et al., Phys. Rev. Lett. 117, 190501 (2016).
102 A comprehensive review of the quantum cryptography work was recently given by S. Pirandola et al., Adv.
Opt. Photon. 12, 1012 (2020).
103 See, e.g., https://en.wikipedia.org/wiki/NIST_Post-Quantum_Cryptography_Standardization.
104 For the progress of research in this direction, adequate measures of the informational capacity of quantum channels are clearly necessary. Due to lack of time/space, I have to refer the interested reader to literature – for example, to Chapter 12 in Nielsen and Chuang, or to S. Barnett, Quantum Information, Oxford, 2009.
105 This effect was discovered by A. Michelson in 1881 and explained theoretically by W. Pauli in 1924, with the first quantitative calculation made in 1930 by E. Fermi.
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Hint: The proton’s magnetic moment operator is described by the same Eq. (4.115) as the electron, but with a positive gyromagnetic ratio p = g p e/2 m p 2.675108 s-1T-1, whose magnitude is much smaller than that of the electron (e 1.7611011 s-1T-1), due to the much higher mass, m p
1.67310-27 kg 1,835 m e. (The g-factor of the proton is also different, g p 5.586.106) 8.6. In the simple case of just two similar spin-interacting particles, distinguishable by their spatial location, the famous Heisenberg model of ferromagnetism107 is reduced to the following Hamiltonian:
H J ˆs ˆs B ˆs ˆs ,
1
2
1 2
where J is the spin interaction constant, is the gyromagnetic ratio of each particle, and B is the external magnetic field. Find the stationary states and energies of this system for spin-½ particles.
8.7. Two spins-½, different gyromagnetic ratios 1 and 2, are placed in an external magnetic field B. In addition, the spins interact as in the Heisenberg model:
ˆ
H
J ˆs ˆs .
int
1
2
Find the stationary states and energies of the system.
8.8. Two similar spins-½ with a gyromagnetic ratio , localized at two points separated by distance a, interact via the field of their magnetic dipole moments. Calculate the stationary states and energies of the system.
8.9. Consider the permutation of two identical particles, each of spin s. How many different symmetric and antisymmetric spin states can the system have?
8.10. For a system of two identical particles with s = 1:
(i) List all spin states forming the uncoupled-representation basis.
(ii) List all possible pairs { S, MS} of the quantum numbers describing the states of the coupled-representation basis – see Eq. (48).
(iii)Which of the { S, MS} pairs describe the states symmetric, and which the states antisymmetric, with respect to the particle permutation?
8.11. Represent the operators of the total kinetic energy and the total orbital angular momentum of a system of two particles, with masses m 1 and m 2, as combinations of the terms describing the center-of-mass motion and the relative motion. Use the results to calculate the energy spectrum of the so-called positronium – a metastable “atom”108 consisting of one electron and its positively charged antiparticle, the positron.
106 The relatively large value of the proton’s g-factor results from the quark-gluon structure of this particle. (An exact calculation of g p remains a challenge for quantum chromodynamics.) 107 It was suggested in 1926 independently by W. Heisenberg and P. Dirac. A discussion of thermal effects in this and other similar systems (especially the Ising model of ferromagnetism) may be found in SM Chapter 4.
108 Its lifetime (either 0.124 ns or 138 ns, depending on the parallel or antiparallel configuration of the component spins), is limited by the weak interaction of its components, which causes their annihilation with the emission of several gamma-ray photons.
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8.12. Calculate the energy spectrum of the system of two identical spin-½ particles moving along the x-axis, which is described by the following Hamiltonian:
2
2
2
ˆ p
ˆ p
m
1
2
0
0
ˆ
H
2 2
x x x
x ,
1
2
1 2
2 m
2 m
2
0
0
and the degeneracy of each energy level.
8.13. Two particles with similar masses m and charges q are free to move along a planar circle of radius R. In the limit of very strong Coulomb interaction of the particles, find the lowest eigenenergies of the system, and sketch the system of its energy levels. Discuss possible effects of particle indistinguishability.
8.14. Low-energy spectra of many diatomic molecules may be well described by modeling the molecule as a system of two particles connected with a light and elastic but very stiff spring. Calculate the energy spectrum of a molecule within this model. Discuss possible effects of nuclear spins on spectra of the so-called homonuclear diatomic molecules formed by two similar atoms.
8.15. Two indistinguishable spin-½ particles are attracting each other at contact: U x , x W x x
W
1
2
1 2 ,
with
0,
but are otherwise free to move along the x-axis. Find the energy and the orbital wavefunction of the ground state of the system.
8.16. Two indistinguishable spin-½ particles are confined to move around a circle of radius R, and interact only at a very short arc distance l = R(1 – 2) R between them, so the interaction potential U may be well approximated with a delta function of . Find the ground state and its energy, for the cases of:
(i) the orbital (spin-independent) repulsion: U ˆ W ,
(ii) the spin-spin interaction: ˆ
U W ˆs ˆs
,
1
2
both with W > 0. Analyze the trends of your results in the limits W 0 and W .
8.17. Two particles of mass M, separated by two much lighter particles of mass m
M
m << M, are placed on a circle of radius R – see the figure on the right. The particles fully repulse each other at contact, but otherwise, each of them is free to move along M
the circle. Calculate the lower part of the system’s energy spectrum.
m
8.18. Two spin-½ particles are confined in a spherically symmetric potential well U 0(r) =
m 2
0 r 2/2. In addition, they directly interact via a short-range potential that may be described as U int =
W(r1 – r2). In the first approximation in small W, calculate the energies of (i) the ground state, and
(ii) the lowest excited states of the system.
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8.19. N indistinguishable spin-½ particles are placed into the spherically-symmetric potential well U(r) = m 2
0 r 2/2. Neglecting the explicit interaction of the particles, find the ground-state energy of the system.
8.20. Use the Hund rules to evaluate the quantum numbers L, S, and J in the ground states of carbon and nitrogen atoms. Write down the Russell-Saunders symbols for these states.
8.21. N >> 1 indistinguishable, non-interacting quantum particles are placed into a hard-wall rectangular box with sides ax, ay, and az. Calculate the ground-state energy of the system and the average forces it exerts on each face of the box. Can we characterize the forces by certain pressure P?
Hint: Consider separately the cases of bosons and fermions.
8.22. A system of three spins-½ is described by the Heisenberg Hamiltonian
ˆ
H J ˆs ˆs ˆs ˆs ˆs ˆs , 1
2
2
3
3
1
where J is a spin interaction constant (cf. Problems 6 and 7). Find the stationary states and energies of this system, and give an interpretation of your results.
8.23. For a system of three spins-½, find the common eigenstates and eigenvalues of the operators S ˆ
ˆS ˆs ˆs ˆs
z and
2
ˆ S , where
is the vector operator of the total spin of the system. Do the
1
2
3
corresponding quantum numbers S and MS obey Eqs. (48)?
8.24. Explore basic properties of the Heisenberg model (whose few-spin versions were the subjects of Problems 6, 7, and 23), for a 1D chain of N spins-½:
ˆ
H J
ˆ
s ˆs B
ˆs
j
j
, with J ,0
'
j
j, j '
j
where the summation is over all N spins, with the symbol { j, j’} meaning that the first sum is only over the adjacent spin pairs. In particular, find the ground state of the system and its lowest excited states in the absence of external magnetic field B, and also the dependence of their energies on the field.
Hint: For the sake of simplicity, you may assume that the first sum includes the term ˆs ˆs as N
1
well. (Physically, this means that the chain is bent into a closed loop. 109)
8.25. Calculate commutators of the following operators:
†
†
1
ˆ
ˆ a ˆ a ,
ˆ
ˆ a ˆ a ,
ˆ
ˆ†
a ˆ a ˆ†
a ˆ ,
a
1
2
2 1
z
1
1
2
2
2
where †
ˆ a and ˆ a are the operators of the creation and annihilation of bosons in two different states.
,
1 2
,
1 2
109 Note that for dissipative spin systems, differences between low-energy excitations of open-end and closed-end 1D chains may be substantial even in the limit N – see, e.g., SM Sec. 4.5. However, for our Hamiltonian (and hence dissipation-free) system, the differences are relatively small.
Chapter 8
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8.26. Compose the simplest model Hamiltonians, in terms of the second quantization formalism, for systems of indistinguishable particles placed in the following external potentials: (i) two weakly coupled potential wells, with on-site particle interactions giving additional energy J per each pair of particles in the same potential well, and
(ii) a periodic 1D potential, with the same particle interactions, in the tight-binding limit.
8.27. For each of the Hamiltonians composed in the previous problem, derive the Heisenberg equations of motion for particle creation/annihilation operators.
8.28. Express the ket-vectors of all possible Dirac states of three indistinguishable (i) bosons, and
(ii) fermions,
via those of the single-particle states , ’, and ” they occupy.
8.29. Explain why the general perturbative result (8.126), when applied to the 4He atom, gives the correct110 expression (8.29) for the ground singlet state, and correct Eqs. (8.39)-(8.42) (with the minus sign in the first of these relations) for the excited triplet states, but cannot describe these results, with the plus sign in Eq. (8.39), for the excited singlet state.
8.30.* Explore the Thomas-Fermi model 111 of a heavy atom, with the nuclear charge Q = Ze >> e, in which the interaction between electrons is limited to their contribution to the common electrostatic potential (r). In particular, derive the ordinary differential equation obeyed by the radial distribution of the potential, and use it to estimate the effective radius of the atom.
8.31.* Use the Thomas-Fermi model explored in the previous problem to calculate the total binding energy of a heavy atom. Compare the result with that of the simpler model, in that the Coulomb electron-electron interaction is completely ignored.
8.32. Suggest and explore a simple model of dephasing in a system consisting of N similar, distinct, non-interacting components. How does the dephasing time scale with N?
8.33. The notion of the reduced density operator w ˆ defined by Eq. (7.160) is sometimes used for the characterization of entanglement in multi-qubit systems. Calculate w ˆ for one qubit of a two-qubit system that is in an arbitrary pure state, and analyze the result.
8.34. For a system of two distinct qubits (i.e. two-level systems), introduce a reasonable uncoupled-representation z-basis and write, in this basis, the 44 matrix of the operator that swaps their states.
8.35. Find a time-independent Hamiltonian that causes the qubit evolution described by Eqs.
(155). Discuss the relation between your result and the time-dependent Hamiltonian (6.86).
110 Correct in the sense of the first order of the perturbation theory.
111 It was suggested in 1927, independently, by L. Thomas and E. Fermi.
Chapter 8
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Chapter 9. Introduction to Relativistic Quantum Mechanics
This chapter consists of two very different parts. Its first part is a discussion of the basic elements of the quantum theory of the electromagnetic field, usually called quantum electrodynamics (QED). We will see, in particular, that the QED may be viewed as the relativistic quantum theory of particles with zero rest mass – photons. The second part of the chapter is a brief review of the relativistic quantum theory of particles with non-zero rest mass, including the famous Dirac’s theory of spin-½ particles such as electrons. These theories mark the point of entry into a more complete relativistic quantum theory – the quantum field theory – which is beyond the scope of this course.
9.1. Electromagnetic field quantization1
Classical physics gives us2 the following general relativistic relation between the momentum p and energy E of a free particle with rest mass m, which may be simplified in two limits – non-relativistic and ultra-relativistic:
Free
mc
p
m
p
mc
1/
particle’s
2
2 2
2
2
2 / 2 , for
,
relativistic
E pc ( mc )
(9.1)
energy
pc,
for
p mc.
In both limits, the transfer from classical to quantum mechanics is easier than in the arbitrary case. Since all the previous contents of this course were committed to the first, non-relativistic limit, I will now jump to a brief discussion of the ultra-relativistic limit p >> mc, for a particular but very important system – the electromagnetic field. Since the excitations of this field, called photons, are currently believed to have zero rest mass m,3 the ultra-relativistic relation E = pc is exactly valid for any photon energy E, and the quantization scheme is rather straightforward.
As usual, the quantization has to be based on the classical theory of the system – in this case, the Maxwell equations. As the simplest case, let us consider the electromagnetic field inside a finite free-space volume limited by ideal walls, which reflect incident waves perfectly.4 Inside the volume, the Maxwell equations give a simple wave equation5 for the electric field
2
1
2
E
E
,
0
(9.2)
2
2
c t
and an absolutely similar equation for the magnetic field B. We may look for the general solution of Eq.
(2) in the variable-separating form
1 The described approach was pioneered by the same P. A. M. Dirac as early as 1927.
2 See, e.g., EM Chapter 9, in particular Eq. (9.78).
3 By now, this fact has been verified experimentally with an accuracy of at least 10-18 eV/ c 2, i.e. ~10-24 m e – see, e.g. the review by C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1 (2008).
4 In the case of finite energy absorption in the walls, or in the wave propagation media, the system is not energy-conserving (Hamiltonian), i.e. interacts with a dissipative environment as was discussed in Chapter 7. Specific cases of such interaction will be considered in Sections 2 and 3 below.
5 See, e.g., EM Eq. (7.3), for the particular case = 0, = 0, so v 2 1/ = 1/00 c 2.
© K. Likharev
QM: Quantum Mechanics
E r, t E
t
E
t p t
.
(9.3)
j r, ,
with j r,
( )e (r)
j
j
j
Physically, each term of this sum is a standing wave whose spatial distribution and polarization (“mode”) are described by the vector function e j(r), while its temporal dynamics follows the function pj( t). Plugging an arbitrary term of this sum into Eq. (2), and separating the variables exactly as we did, for example, in the Schrödinger equation in Sec. 1.5, we get
2
e
p
j
1
j
const
2
k ,
(9.4)
2
j
e
c p
j
j
so the spatial distribution of the mode satisfies the following 3D Helmholtz equation: Equation
2
2
e k e 0.
(9.5)
for spatial
j
j
j
distribution
The set of solutions of this equation, with appropriate boundary conditions, determines the set of the functions e j, and simultaneously the spectrum of the wave number magnitudes kj. The latter values determine the mode eigenfrequencies, following from Eq. (4):
p 2
p ,
0
with
k c .
(9.6)
j
j
j
j
j
There is a big philosophical difference between the quantum-mechanical approach to Eqs. (5) and (6), despite their single origin (4). The first (Helmholtz) equation may be rather difficult to solve in realistic geometries,6 but it remains intact in the basic quantum electrodynamics, with the scalar components of the vector functions e j(r) still treated (at each point r) as c-numbers. In contrast, the classical Eq. (6) is readily solvable (giving sinusoidal oscillations with frequency j), but this is exactly where we can make the transfer to quantum mechanics because we already know how to quantize a mechanical 1D harmonic oscillator, which in classics obeys the same equation.
As usual, we need to start with the appropriate Hamiltonian – the operator corresponding to the classical Hamiltonian function H of the proper set of generalized coordinates and momenta. The electromagnetic field’s Hamiltonian function (which in this case coincides with the field’s energy) is7
2
2
E
B
H 3
0
d r
.
(9.7)
2
20
Let us represent the magnetic field in a form similar to Eq. (3),8
B (r, t) B
t
B
t
q t
.
(9.8)
j r, ,
with j r,
( )b (r)
j
j
j
j
6 See, e.g., the cases discussed in EM Sec. 7.9.
7 See, e.g., EM Sec. 9.8, in particular, Eq. (9.225). Here I am using SI units, with 00 c–2; in the Gaussian units, the coefficients 0 and 0 disappear, but there is an additional common factor 1/4 in the formula H. However, if we modify the normalization conditions (see below) accordingly, all the subsequent results, starting from Eq.
(10), look similar in any system of units.
8 Here I am using the letter qj, instead of xj, for the generalized coordinate of the field oscillator, in order to emphasize the difference between the former variable and one of the Cartesian coordinates, i.e. one of the arguments of the c-number functions e j(r) and b j(r).
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Since, according to the Maxwell equations, in our case, the magnetic field satisfies the equation similar to Eq. (2), the time-dependent amplitude qj of each of its modes b j(r) obeys an equation similar to Eq.
(6), i.e. in the classical theory also changes in time sinusoidally, with the same frequency j. Plugging Eqs. (3) and (8) into Eq. (7), we may recast it as
2
2
2
p
q
j
2
3
j
j
1
H
e (r)
2
d r
b r d r
(9.9)
j
j
0
3 .
j 2
2
0
Now note that the distribution of constant factors between two operands in each product of Eq. (3) is so far arbitrary, so we may fix it by requiring the first integral in Eq. (9) to equal 1. Since according to the Maxwell equations, there is a specific relation between the field vector amplitudes, B j/E j = (00)1/2
1/ c,9 this normalization makes the second integral in Eq. (9) equal 1 as well, and Eq. (9) becomes 2
2
2
p
q
H H ,
j
j
j
H
.
(9.10)
j
j
j
2
2
Note that pj is the legitimate generalized momentum corresponding to the generalized coordinate q j, because it is equal to L / q , where L is the Lagrangian function of the field – see EM Eq. (9.217): j
2
2
2
2
2
E
p
q
3
0
L d r
B
L ,
j
j
j
L
.
(9.11)
2
2
j
j
j
2
2
0
Hence we can carry out the standard quantization procedure, namely declare Hj, pj, and qj to be quantum-mechanical operators related as in Eq. (10),
Electro-
magnetic
ˆ 2
2
p
ˆ2
q
mode’s
ˆ
j
j
j
H
.
(9.12)
j
Hamiltonian
2
2
We see that this Hamiltonian coincides with that of a 1D harmonic oscillator with the mass mj formally equal to 1,10 and the frequency equal to j. Moreover, since the Lagrangian function L has the same form (11) as in a set of independent harmonic oscillators with coordinates qj and momenta pj, the corresponding operators satisfy the following natural generalization of the commutation relations (2.14):
ˆ q , ˆ p i ,
(9.13)
j
j '
jj '
As the reader already knows, Eqs. (12) and (13) open for us several alternative ways to proceed: (i) Use the Schrödinger-picture wave mechanics based on wavefunctions j( qj, t). As we know from Sec. 2.9, this way is inconvenient for most tasks, because the eigenfunctions of the harmonic oscillator are rather clumsy.
(ii) A substantially more efficient way is to write and use the Heisenberg-picture equations for the time evolution of the operators ˆ q ( t) and ˆ p ( t) .
j
j
9 See, e.g., EM Eqs. (7.6)-(7.7) with = 0, = 0, and H = B/0.
10 Selecting a different normalization of the function e j(r), we could readily arrange any value of mj, so the choice corresponding to mj = 1 is the best one just for the notation simplicity.
Chapter 9
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(iii) An even more convenient approach is to use equations similar to Eqs. (5.65) to decompose the Heisenberg operators ˆ q ( t) and ˆ p ( t) into the creation-annihilation operators a†
ând a ˆ
.
j t
j t
j
j
In this chapter, I will mostly use the last route. After the replacement of m with mj 1, and 0
with j, the last forms of Eqs. (5.65) become
1/2
p ˆ
1/2
p ˆ
j
j
†
j
a ˆ
q ˆ i
,
a ˆ
q ˆ
j
i
.
(9.14)
j
j
j
j
2
2
j
j
Due to Eq. (13), these creation-annihilation operators obey the commutation similar to Eq. (5.68),
a ˆ , a ˆ† Iˆ .
(9.15)
j
j'
jj'
As a result, according to Eqs. (3) and (8), the quantum-mechanical operators of the electric and magnetic fields may be represented as sums over all field oscillators:
1/ 2
j
Ê (r, t) i
e (r) ˆ†
a ˆ a
(9.16a)
j
j
j
,
Electro-
j
2
magnetic
1/ 2
field
j
ˆ
B (r, t)
b (r) a ˆ† a ˆ
,
(9.16b) operators
j
j
j
j
2
and Eq. (12) for the j th mode’s Hamiltonian becomes
†
1
1
H ˆ a ˆ a ˆ I ˆ n ˆ I ˆ , with n ˆ a ˆ† a ˆ , (9.17)
j
j
j
j
j
j
j
j
j
2
2
absolutely similar to Eq. (5.72) for a mechanical oscillator.
Now comes a very important conceptual step. From Sec. 5.4, we know that the stationary (Fock) states nj of the Hamiltonian (17) have energies
Electro-
1
magnetic
E
n
n
(9.18) mode
j
j
j
,
,
1
,
0
,...
2
2
j
eigen-
energies
and, according to Eq. (5.89), the operators †
ˆ a and a âct on the eigenkets of these partial states as
j
j
ˆ a n n
n
a n
n
n
,
(9.19)
j
j
j 1/2 1 ,
ˆ†
j
j
j
j
11/2 1
j
regardless of the quantum states of other modes. These rules coincide with the definitions (8.64) and (8.68) of bosonic creation-annihilation operators, and hence their action may be considered as the creation/annihilation of certain bosons. Such a quasiparticle (actually, an excitation, with energy j, of the j th field oscillator) is exactly what is, strictly speaking, called a photon. Note immediately that according to Eq. (16), such an excitation does not change the spatial distribution of the j th mode of the field. So, such a “global” photon is an excitation created simultaneously at all points of the field confinement region.
If this picture is too contrary to the intuitive image of a particle, please recall that in Chapter 2, we discussed a similar situation with the fundamental solutions exp{ ikx} of the Schrödinger equation of a free non-relativistic particle: they represent sinusoidal de Broglie waves existing simultaneously at Chapter 9
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Essential Graduate Physics
QM: Quantum Mechanics
all points of the particle confinement region. The (partial :-) reconciliation with the classical picture of a moving particle might be obtained by using the linear superposition principle to assemble a quasi-localized wave packet, as a group of sinusoidal waves with close wave numbers. Very similarly, we may form a similar wave packet using a linear superposition of the “global” photons with close values of k j (and hence j), to form a quasi-localized photon. An additional simplification here is that the dispersion relation for electromagnetic waves (at least in free space) is linear:
2
j c const,
i.e.
j ,
0
(9.20)
2
k j
k j
so according to Eq. (2.39a), the electromagnetic wave packets (i.e. space-localized photons) do not spread out during their propagation. Note also that due to the fundamental classical relations p = n E/ c for the linear momentum of the traveling electromagnetic wave packet of energy E, propagating along the direction n k/ k, and S = n E/ j for its angular momentum,11 such a photon may be prescribed the linear momentum p = n j/ c k and the angular momentum (essentially, the spin) S = n, with the sign depending on the direction of its circular polarization (“helicity”).
This electromagnetic field quantization scheme should look very straightforward, but it raises an important conceptual issue of ground state energy. Indeed, Eq. (18) implies that the total ground-state (i.e., the lowest) energy of the field is
Ground-
state
energy
E
.
(9.21)
g
( E )
g
j
j
of EM field
j
j
2
Since for any realistic model of the field-confining volume, either infinite or not, the density of electromagnetic field modes only grows with frequency,12 this sum diverges on its upper limit, leading to infinite ground-state energy per unit volume. This infinite-energy paradox cannot be dismissed by declaring the ground-state energy of field oscillators unobservable, because this would contradict numerous experimental observations – starting perhaps from the famous Casimir effect.13
The simplest implementation of this effect involves two parallel, perfectly conducting plates of area A, separated by a vacuum gap of thickness t << A 1/2 (Fig. 1).
z
t
Fig. 9.1. The simplest geometry of
the Casimir effect manifestation.
11 See, e.g., EM Sections 7.7 and 9.8 (where the angular momentum of the field is denoted L).
12 See, e.g., Eq. (1.1), which is similar to Eq. (1.90) for the de Broglie waves, derived in Sec. 1.7.
13 This effect was predicted in 1948 by Hendrik Casimir and Dirk Polder, and confirmed semi-quantitatively in experiments by M. Sparnaay, Nature 180, 334 (1957). After that and several other experiments, a decisive error bar reduction (to about ~5%), providing a quantitative confirmation of the Casimir formula (23), was achieved by S. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997) and by U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 004549 (1998).
Note also that there are other experimental confirmations of the reality of the ground-state electromagnetic field, including, for example, the experiments by R. Koch et al. already discussed in Sec. 7.5, and the recent spectacular direct observations by C. Riek et al., Science 350, 420 (2015).
Chapter 9
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Rather counterintuitively, the plates attract each other with a force F proportional to the area A and rapidly increasing with the decrease of t, even in the absence of any explicit electromagnetic field sources. The effect’s explanation is that the energy of each electromagnetic field mode, including its ground-state energy, exerts average pressure,
E
j
P
,
(9.22)
j
V
on the walls constraining it to volume V. While the field’s pressure on the external surfaces on the plates is due to the contributions (22) of all free-space modes, with arbitrary values of kz (the z-component of the wave vector k j), in the gap between the plates, the spectrum of kz is limited to the multiples of π/ t, so the pressure on the internal surfaces is lower. This is why the net force exerted on the plates may be calculated as the sum of the contributions (22) from all “missing” low-frequency modes in the gap, with the minus sign. In the simplest model when the plates are made of an ideal conductor, which provides simple boundary conditions E = B n = 0 on their surfaces,14 such calculation is quite straightforward (and is hence left for the reader’s exercise), and its result is
2
A c
F
.
(9.23) Casimir
4
240 t
effect
Note that for such calculation, the high-frequency divergence of Eq. (21) is not important, because it participates in the forces exerted on all surfaces of each plate, and cancels out from the net pressure. In this way, the Casimir effect not only confirms Eq. (21), but also teaches us an important lesson on how to deal with the divergences of such sums at ωj → . The lesson is: just get accustomed to the idea that the divergence exists, and ignore this fact while you can, i.e. if the final result you are interested in is finite. However, for some more complex problems of quantum electrodynamics (and the quantum theory of any other fields), this simplest approach becomes impossible, and then more complex, renormalization techniques become necessary. For their study, I have to refer the reader to a quantum field theory course – see the references at the end of this chapter.
9.2. Photon absorption and counting
As a matter of principle, the Casimir effect may be used to observe not only the electromagnetic field’s ground state but also the field arriving from active sources – lasers, etc. However, usually, such studies may be done by simpler detectors, in which the absorption of a photon by a single atom leads to its ionization. This ionization, i.e. the emission of a free electron, triggers an avalanche-like chain reaction (e.g., an electric discharge in a Geiger-type counter), which may be readily registered using 14 For realistic conductors, the reduction of t below ~1 μm causes significant deviations from this simple model, and hence from Eq. (23). The reason is that for gaps so narrow, the depth of field penetration into the conductors (see, e.g., EM Sec. 6.2), at the important frequencies ω ~ c/ t, becomes comparable with t, and an adequate theory of the Casimir effect has to involve a certain model of the penetration. (It is curious that in-depth analyses of this problem, pioneered in 1956 by E. Lifshitz, have revealed a deep relation between the Casimir effect and the London dispersion force which was the subject of Problems 3.16, 5.15, and 6.18 – for a review see, e.g., either I.
Dzhyaloshinskii et al., Sov. Phys. Uspekhi 4, 153 (1961), or K. Milton, The Casimir Effect, World Scientific, 2001. Recent experiments in the 100 nm – 2 m range of t, with an accuracy better than 1%, have enabled not only a clear observation of field penetration effects on the Casimir force but even a selection between some approximate models of the penetration – see D. Garcia-Sanchez et al., Phys. Rev. Lett. 109, 027202 (2012).
Chapter 9
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appropriate electronic circuitry. In good photon counters, the first step, the “trigger” atom ionization, is the bottleneck of the whole process (the photon count), so to analyze their statistics, it is sufficient to consider the field’s interaction with just this atom.
Its ionization is a quantum transition from a discrete initial state of the atom to its final, ionized state with a continuous energy spectrum, induced by an external electromagnetic field. This is exactly the situation shown in Fig. 6.12, so we may apply to it the Golden Rule of quantum mechanics in the form (6.149), with system a associated with the electromagnetic field, and system b with the trigger atom. The atom’s size is typically much smaller than the radiation wavelength, so the field-atom interaction may be adequately described in the electric-dipole approximation (6.146) ˆ
H
Ê dˆ ,
(9.24)
int
where dîs the dipole moment’s operator. Hence we may associate this operator with the operand B în Eqs. (6.145)-(6.149), while the electric field operatorEîs associated with the operand A în those relations. First, let us assume that our field consists of only one mode of frequency . Then we can keep only one term in the sum (16a), and drop the index j, so Eq. (6.149) may be rewritten as 2
2
ˆ
ˆ
2
Γ
fin
E (r, t) ini
fin
d t n ini
e
a
(9.25)
2
2
2
fin
ˆ†
a
t ˆ a t
ˆ
e(r) ini
fin
d t n ini
,
2
e
a
where n e e(r)/ e(r) is the local direction of the vector e(r), symbols “ini” and “fin” denote the initial and final states of the corresponding subsystem (the electromagnetic field in the first long bracket, and the atom in the second bracket), and the density a of the atomic state continuum should be calculated at the final energy E fin = E ini + .
As a reminder, in the Heisenberg picture of quantum dynamics, the initial and final states are time-independent, while the creation-annihilation operators are functions of time. In the Golden Rule formula (25), as in any perturbative result, this time dependence has to be calculated ignoring the perturbation – in this case, the field-atom interaction. For the field’s creation-annihilation operators, this dependence coincides with that of the usual 1D oscillator – see Eq. (5.141), in which 0 should be, in our current notation, replaced with :
ˆ a( t) ˆ a( )
0 i t
e , ˆ†
a ( t) ˆ†
a ( )
0 i t
e
.
(9.26)
Hence Eq. (25) becomes
2
2
fin
ˆ†
a (0) i
e t ˆ
i t
ˆ
a(0) e
e(r) ini
fin
d( t) n ini
.
(9.27a)
e
a
Now let us multiply the first long bracket by exp{ i t}, and the second one by exp{- i t}: 2
2
fin
ˆ†
a (0) 2 i t
e
ˆ
ˆ
a(0) e(r) ini
fin
d( t)
i t
n e
ini
.
(9.27b)
e
a
This, mathematically equivalent form of the previous relation shows more clearly that at resonant photon absorption, only the annihilation operator gives a significant time-averaged contribution to the Chapter 9
Page 7 of 36
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first bracket matrix element. (As a reminder, the quantum-mechanical Golden Rule for time-dependent perturbations is a result of averaging over a time interval much larger than 1/ – see Sec. 6.6.) Similarly, according to Eq. (4.199), the Heisenberg operator of the dipole moment corresponding to the increase of the atom’s energy by , has the Fourier components that differ in frequency from only by ~ << , so its time dependence virtually compensates the additional factor in the second bracket of Eq. (27b), and this bracket also may have a substantial time average. Hence, in the first bracket, we may neglect the fast-oscillating term, whose average over time interval ~1/ is very close to zero.15
Now let us assume, first, that we use the same detector, characterized by the same matrix element of the quantum transition, i.e. the same second bracket in Eq. (27), and the same final state density a, for measurement of various electromagnetic fields – or just of the same field at different points r. Then we are only interested in the behavior of the first, field-related bracket, and may write
fin
ˆ ae(r) ini 2 fin
ˆ ae(r) ini
fin
ˆ ae(r) ini
* ini
ˆ† *
a e (r) fin
fin
ˆ ae(r) ini
, (9.28)
where the creation-annihilation operators are implied to be taken at t = 0, i.e. in the Schrödinger picture, and the initial and final states are those of the field alone. Second, let us now calculate the total rate of transitions to all available final states of the given mode e(r). If such states formed a full and orthonormal set, we could use the closure relation (4.44) applied to the final states to write Photon
† *
†
2
Γ ini
ˆ a e (r) fin
fin
ˆ ae(r) ini
ini
ˆ a ˆ a ini
*
e (r) e(r) n
e(r) ,
(9.29) counting
ini
rate
fin
where, for a given field mode, n
†
ini is the expectation value of the operator n ˆ a ˆ a ˆ for the initial state of the electromagnetic field. In the more realistic case of fields in relatively large volumes, V >> 3, with their virtually continuous spectrum of the final states, the middle equality in this relation is not strictly valid, but it is correct to a constant multiplier,16 which we are currently not interested in. Note, however, that Eq. (29) may be substantially wrong for high- Q electromagnetic resonators (“cavities”), which may make just one (or a few) modes available for transitions. (Quantum electrodynamics of such cavities will be briefly discussed in Sec. 4 below.)
Let us apply Eq. (29) to several possible quantum states of the mode.
(i) First, as a sanity check, the ground initial state, n = 0, gives no photon absorption at all. The interpretation is easy: the ground state field, cannot emit a photon that would ionize an atom in the counter. Again, this does not mean that the ground-state “motion” is not observable (if you still think so, please review the Casimir effect discussion in Sec. 1), just that it cannot ionize the trigger atom –
because it does not have any spare energy for doing that.
(ii) All other coherent states (Fock, Glauber, squeezed, etc.) of the field oscillator give the same counting rate, provided that their nini is the same. This result may be less evident if we apply Eq. (29) to the interference of two light beams from the same source – say, in the double-slit or the Bragg-scattering configurations. In this case, we may represent the spatial distribution of the field as a sum 15 This is essentially the same rotating wave approximation (RWA) that was already used in Sec. 6.5 and beyond
– see, e.g., the transition from Eq. (6.90) to the first of Eqs. (6.94).
16 As the Golden Rule shows, this multiplier is proportional to the density f of the final states of the field.
Chapter 9
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e(r) e (r) e (r) .
(9.30)
1
2
Here each term describes one possible wave path, so the operator product in Eq. (29) may be a rapidly changing function of the detector position. For this configuration, our result (29) means that the interference pattern (and its contrast) are independent of the particular state of the electromagnetic field’s mode.
(iii) The last statement is also valid for a classical mixture of the different eigenstates of the same field mode, for example for its thermal-equilibrium state. Indeed, in this case, we need to average Eq.
(29) over the corresponding classical ensemble, but it would only result in a different meaning of averaging n in that equation; the field part describing the interference pattern is not affected.
The last result may look a bit counter-intuitive because common sense tells us that the stochasticity associated with thermal equilibrium has to suppress the interference pattern contrast. These expectations are (partly :-) justified because a typical thermal source of radiation produces many field modes j, rather than one mode we have analyzed. These modes may have different wave numbers kj and hence different field distribution functions e j(r), resulting in shifted interference patterns. Their summation would indeed smear the interference, suppressing its contrast.
So the use of one photon detector is not the best way to distinguish different quantum states of an electromagnetic field mode. This task, however, may be achieved using the photon counting correlation technique shown in Fig. 2.17
controllable
semi-transparent
delay
mirror
light
source
detector 2
detector 1
count
statistics
Fig. 9.2. Photon count
calculation
correlation measurement.
In this experiment, the counting statistics may be characterized by the so-called second-order correlation function of the detector count rates,
Second-
order
( t) ( t )
correlation
(2)
g ( )
1
2
,
(9.31)
function
( t) ( t)
1
2
where the averaging may be carried out either over many similar experiments, or over a relatively long time interval t>> , with usual field sources – due to their ergodicity.18 Using the normalized correlation 17 It was pioneered as early as the mid-1950s (i.e. before the advent of lasers), by Robert Hanbury Brown and Richard Twiss. Their second experiment was also remarkable for the rather unusual light source – the star Sirius!
(Their work was an effort to improve astrophysics interferometry techniques.)
18 This is why the argument t in Eq. (31) is just symbolic, and for the rates 1,2, we can use Eq. (29), even though its derivation involved averaging over times much longer than 1/.
Chapter 9
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function (31) is very convenient because the characteristics of both detectors and the beam splitter (e.g., a semi-transparent mirror, see Fig. 2) drop out from this fraction.
Very unexpectedly for the mid-1950s, Hanbury Brown and Twiss discovered that the correlation function depends on the time delay in the way shown (schematically) with the solid line in Fig. 3. It is evident from Eq. (31) that if the counting events are completely independent, g(2)() should be equal to 1
– which is always the case in the limit . (As will be shown in the next section, the characteristic time of this approach is usually between 10-11s and 10-8s, so for its measurement, the delay time control may be provided just by moving one of the detectors by a human-scale distance between a few millimeters to a few meters.) Hence, the observed behavior at 0 corresponds to a positive correlation of detector counts at small time delays, i.e. to a higher probability of the nearly simultaneous arrival of photons to both counters. This rather counterintuitive effect is called photon bunching.
(2)
g
2
1
Fig. 9.3. Photon bunching (solid line) and
antibunching for various n (dashed lines). The
n 1
lines approach the level g(2) = 1 at (on
the time scale depending on the light source).
0
Let us use our simple single-mode model to analyze this experiment. Now the elementary quantum process characterized by the numerator of Eq. (31) is the correlated, simultaneous ionization of two trigger atoms, at two spatial-temporal points {r1, t} and {r2, t – }, by the same field mode, so we need to make the following replacement in the first of Eqs. (25):
Ê (r, t) const Ê (r , t) Ê (r , t ) .
(9.32)
1
2
Repeating all the manipulations done above for the single-counter case, we get
Γ
†
†
*
*
t
t
a t a t a t a t
e r e r e r e r
(9.33)
1 Γ 2
ini
ˆ( ) ˆ(
) ˆ(
) ˆ( ) ini
( ) ( ) ( ) ( ).
1
2
1
2
Plugging this expression, as well as Eq. (29) for single-counter rates, into Eq. (31), we see that the field distribution factors (as well as the detector-specific brackets and the density of states a) cancel, giving a very simple final expression:
ˆ†
a ( t) ˆ†
a ( t ) ˆ a( t ) ˆ a( t)
(2)
g ( )
,
(9.34)
2
ˆ†
a ( t) ˆ a( t)
where the averaging should be carried out, as before, over the initial state of the field.
Still, the calculation of this expression for arbitrary may be quite involved because in most practical cases, the relaxation of the correlation function to the asymptotic value g(2)() is due to the interaction of the light source with its environment, and hence requires the open-system techniques that were discussed in Chapter 7. However, the zero-delay value g(2)(0) may be calculated straightforwardly, because the time arguments of all operators are equal, so we may write
Chapter 9
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ˆ†
a ˆ†
a ˆ ˆ a
a
Zero-delay
(2)
g ( )
0
.
(9.35)
2
correlation
ˆ†
a ˆ a
Let us evaluate this ratio for the simplest states of the field.
(i) The n th Fock state. In this case, it is convenient to act with the annihilation operators upon the ket-vectors, and by the creation operators, upon the bra-vectors, using Eqs. (19): Photon
n ˆ†
a ˆ†
a ˆ ˆ a
a n
n 2 n( n )
1 1/2 n( n )
1 1/ 2 n 2
(2)
n( n
)
1
1
anti-
g ( )
0
1 .
(9.36)
bunching
n ˆ†
a ˆ
2
a n
n 1 1/ 2 1/ 2
n n
n 1 2
2
n
n
We see that the correlation function at small delays is below 1, i.e. below the value for fully independent counts – see the dashed lines in Fig. 3. This photon antibunching effect has a very simple handwaving explanation: a single photon emitted by the wave source may be absorbed by just one of the detectors.
For the initial state n = 1, this is the only option, and it is very natural that Eq. (36) predicts no simultaneous counts at = 0. Despite this theoretical simplicity, reliable observations of the antibunching have not been carried out until 1977,19 due to the experimental difficulty of driving electromagnetic field oscillators into their Fock states – see Sec. 4 below.
(ii) The Glauber state . A similar procedure, but now using Eq. (5.124) and its Hermitian conjugate,
†
*
ˆ a , yields
Glauber
ˆ†
a ˆ†
a ˆ a ˆ
* *
a
field
(2)
g (0)
,
1
(9.37)
statistics
ˆ†
a ˆ
2
a
( *
)2
for any parameter . We see that the result is different from that for the Fock states, unless in the latter case n . (We know that the Fock and Glauber properties should also coincide for the ground state, but at that state, the correlation function’s value is uncertain because there are no photon counts at all.) (iii) Classical mixture. From Chapter 7, we know that such statistical ensembles cannot be described by single-state vectors, and require the density matrix w for their description. Here, we may combine Eqs. (35) and (7.5) to write
Tr ˆ w ˆ†
a ˆ†
a ˆ a ˆ a
(2)
g (0)
.
(9.38)
Trˆ w ˆ†
a ˆ a 2
Spelling out this expression is easy for the field in thermal equilibrium at some temperature T, because its density matrix is diagonal in the basis of the Fock states n – see Eqs. (7.24):
E
w
W , W exp
(9.39)
nn
n nn
n
n
Z n
n ,
where
exp
.
'
'
k T
n
k T
B
0
B
19 By H. J. Kimble et al. , Phys. Rev. Lett. 39, 691 (1977). For a detailed review of photon antibunching, see, e.g., H. Paul, Rev. Mod. Phys. 54, 1061 (1982).
Chapter 9
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So, for the operators in the numerator and denominator of Eq. (38) we also need just the diagonal terms of the operator products, which have already been calculated – see Eq. (36). As a result, we get
W n( n )
1
n n
n
n ( )1 n
(2)
g ( )
0
n0
n0
n0
.
(9.40)
2
2
n
W n
n
n
n0
n0
One of the three series in the last expression is just the usual geometric progression, n
1
,
(9.41)
n
0
1
and the remaining two series may be readily calculated by its differentiation over the parameter :
n
n
d
n
d
1
1
n n
,
n
n
d n
d
0
0
0
1
1
(
)2
(9.42)
2
2
n
n
d
n
d
2
2
2
2
1
2 2
n( n )1 n( n )1
,
2
2
n
n
d n
d
0
0
0
1
1
(
)3
and for the correlation function, we get an extremely simple result independent of the parameter and hence of temperature:
(2
2 2
)
1
/(
)3 1 1
/(
)
g (0)
(9.43) Photon
1
/( )
2.
2
2
bunching
This is exactly the photon bunching effect first observed by Hanbury Brown and Twiss – see Fig.
3. We see that in contrast to antibunching, this is an essentially classical (statistical) effect. Indeed, Eq.
(43) allows for a purely classical derivation. In the classical theory, the counting rate (of a single counter) is proportional to the wave intensity I, so Eq. (31) with = 0 is reduced to 2
I
(2)
g (0)
, with
2
I E ( t)
*
E
.
(9.44)
2
E
I
For a sinusoidal field, the intensity is constant, and g(2)(0) = 1. (This is also evident from Eq. (37), because the classical state may be considered a Glauber state with .) On the other hand, if the intensity fluctuates (either in time or from one experiment to another), the averages in Eq. (44) should be calculated as
I k
(
w I ) I k dI, with
(
w I ) dI ,
1
and k ,
1 2
,
(9.45)
0
0
where w( I) is the probability density. For classical statistics, the probability is an exponential function of the electromagnetic field energy, and hence its intensity:
w( I ) Ce I ,
where 1/ k T ,
(9.46)
B
so Eqs. (45) yield:20
20 See, e.g., MA Eq. (6.7c) with n = 0 and n = 1.
Chapter 9
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C
exp I dI C / ,
1
hence
and
C
,
0
(9.47)
1/ , for k
1,
I k
(
w I ) I k dI C
I I kdI 1
exp
exp k
d
k
0
0
2/ 2
0
for
,
k
2.
Plugging these results into Eq. (44), we get g(2)(0) = 2, in complete agreement with Eq. (43).
For some field states, including the squeezed ground states discussed at the end of Sec. 5.5, the values of g(2)(0) may be even higher than 2 – the so-called super-bunching. Analyses of two cases of such super-bunching are offered for the reader’s exercise – see the problem list at the chapter’s end.
9.3. Photon emission: spontaneous and stimulated
In the last section, we considered how the counter’s trigger atom absorbs a photon. Now let us have a look at the opposite process of spontaneous emission of a photon by an atom at its transition from an excited quasi-stationary (metastable) state to a state with an energy lower by E = .21 By using the same electric-dipole approximation (24) for the atom-to-field interaction, we may still use the Golden Rule for the model depicted in Fig. 6.12, but now the roles of its two component systems change: we have to associate the operator A ˆ with the electric dipole moment of the atom, while the operator B ˆ , with the electric field, so the continuous spectrum of the system b represents the plurality of the electromagnetic field states into which the spontaneous radiation may happen. Since now the transition increases the energy of the electromagnetic field, and decreases that of the atom, after the multiplication of the field bracket in Eq. (27a) by exp{– i t}, and the second, by exp{+ i t}, we may keep only the photon creation operator whose time evolution (26) compensates this additional fast “rotation”. As a result, the Golden Rule takes the following form:
Spontaneous
2
2
photon
fin
ˆ†
ˆ
a 0
fin d e(r) ini
,
(9.48)
s
f
emission
rate
where all operators and states are time-independent, and f is the density of final states of the electromagnetic field – which in this problem plays the role of the atom’s environment.22 Here the electromagnetic field oscillator has been assumed to be initially in the ground state – the assumption that will be changed later in this section.
This relation, together with Eq. (19), shows that for the field’s matrix element to be different from zero, the final field has to be its first excited Fock state, n = 1. (By the way, this is exactly the most practicable way of generating an excited Fock state of a field oscillator.) With that, Eq. (48) yields 21 A straightforward Fourier transform of Eq. (6.114) (which is the inseparable counterpart of the Golden Rule we are going to use) shows that the emitted radiation has the Lorentzian line centered to the frequency , with the half-width equal to the transition rate s that we will calculate.
22 Here the summation over all electromagnetic field modes j may be smuggled back. Since in the quasistatic approximation kja << 1, which is necessary for the validity of Eq. (24), the matrix elements in Eq. (48) are independent of the field vector direction, and their magnitudes are fixed by , this summation is reduced to the calculation of the total f for all modes, and the spatial averaging of e 2(r) – see below.
Chapter 9
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2
2
ˆ
ˆ
Γ fin
d e(r) ini
fin
e
d (r) ini
,
(9.49)
s
f
d
f
where ed is the Cartesian component of the vector e(r) along the electric dipole’s direction, and f is now the density of electromagnetic modes at the frequency . The expression for it follows from our first formula in this course – see Eq. (1.1):
2
dN
V
,
(9.50)
f
2
3
dE
c
where the volume V should be large enough to ensure the spectrum’s virtual continuity:23 V >> 3 =
(2 c/)3. Because of that, in the normalization condition used to simplify Eq. (9), we may consider e 2(r) constant. Let us represent this square as a sum of squares of the three Cartesian components of the vector e(r), with one of those ( ed) aligned with the dipole’s direction; due to the space isotropy we may write
2
2
2
2
e e e e 3 2
e .
(9.51)
d
1
2
d
As a result, the normalization condition yields
2
1
e
.
(9.52)
d
3 V
0
and Eq. (49) gives the famous (and very important) formula24
1
4 3
Free-space
2
1
4 3
*
ˆ
ˆ
ˆ
Γ
fin
d ini
fin d ini
ini
d fin
.
(9.53) spontaneous
s
4 3 3
c
4 3 3
c
emission
0
0
rate
Leaving a comparison of this formula with the classical theory of radiation,25 and the exact evaluation of s for a particular transition in the hydrogen atom, for the reader’s exercises, let me just estimate its order of magnitude. Assuming that d ~ er B e2/ m e( e 2/40) and ~ E H m e( e 2/40)2/2, and taking into account the definition (6.62) of the fine-structure constant 1/137, we get 3
Γ
2
e
s ~
3
~ 3107.
(9.54)
4
c
0
This estimate shows that the emission lines at atomic transitions are typically very sharp. With the present-day availability of high-speed electronics, it also makes sense to evaluate the time scale = 1/
of a typical quantum transition: for a typical optical frequency ~ 31015 s-1, it is close to 1 ns. This is exactly the time constant that determines the time-delay dependence of the photon counting statistics of the spontaneously emitted radiation – see Fig. 3. Very colloquially, this is the temporal scale of the 23 In the opposite case when the same atom is placed into a high- Q resonant cavity with a discrete spectrum (see, e.g., EM 7.9), the rate of its photon emission is strongly suppressed at frequencies between the cavity resonances (where f 0) – see, e.g., the review by S. Haroche and D. Klepner, Phys. Today 42, 24 (Jan. 1989). On the other hand, the emission is strongly (by a factor ~ (3/ V) Q, where V is the cavity’s volume) enhanced at resonance frequencies – the so-called Purcell effect, discovered by E. Purcell in the 1940s. For a brief discussion of this and other quantum electrodynamic effects in cavities, see the next section.
24 This was the breakthrough result obtained by P. Dirac in 1927, which jumpstarted the whole field of quantum electrodynamics. An equivalent expression was obtained from more formal arguments in 1930 by V. Weisskopf and E. Wigner, so sometimes Eq. (53) is (very unfairly) called the “Weisskopf-Wigner formula”.
25 See, e.g., EM Sec. 8.2, in particular Eq. (8.29).
Chapter 9
Page 14 of 36
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photon emitted by an atom. Note that the scale c of the spatial extension of the corresponding wave packet is surprisingly macroscopic – of the order of 30 cm. Such a “human” size of spontaneously emitted photons makes the usual optical table, with its 1-cm-scale components, the key equipment for many optical experiments – see, e.g., Fig. 2.
Note, however, that the above estimate of is only valid for a transition with a non-zero electric-dipole matrix element. If it equals zero, i.e. the transition does not satisfy the selection rules,26 – say, due to the initial and final state symmetry – it is “forbidden”. The “forbidden” transition may still take place due to a different, smaller interaction (say, via a magnetic dipole field of the atom, or its quadrupole electric field27), but takes much longer. In some cases the increase of is rather dramatic –
sometimes to hours! Such long-lasting radiation is called fluorescence. (If the initial excitation is followed first by a series of non-radiative transitions down the energy level ladder, which delay the final radiative transition, the resulting radiation is called either phosphorescence or luminescence.) Now let us consider a more general case when the electromagnetic field mode of frequency is initially in an arbitrary Fock state n, and from it may either get energy from the atomic system ( photon emission) or, vice versa, give such energy back to the atom ( photon absorption). For the photon emission rate, an evident generalization of Eq. (48) gives
2
fin
ˆ†
a n
e
nfin
,
(9.55)
2
s
0 1
1 ˆ†
a 0
where both brackets should be calculated in the Schrödinger picture, and s is the spontaneous emission rate (48) of the same atomic system. According to the second of Eqs. (19), at the photon emission, the final field state has to be the Fock state with n’ = n + 1, and Eq. (55) yields Stimulated
photon
( n )
1 .
(9.56)
e
s
emission rate
Thus the initial field increases the photon emission rate; this effect is called the stimulated emission of radiation. Note that the spontaneous emission may be considered a particular case of the stimulated emission for n = 0, and hence interpreted as the emission stimulated by the ground state of the electromagnetic field – one more manifestation of the non-trivial nature of this “vacuum” state.
On the other hand, following the arguments of Sec. 2,28 for the description of radiation absorption, the photon creation operator has to be replaced with the annihilation operator, giving the rate ratio
26 As was already discussed in Sec. 5.6, for a single spin-less particle moving in a spherically symmetric potential (e.g., a hydrogen-like atom), the orbital selection rules are simple: the only allowed electric-dipole transitions are those with l l fin- l ini = 1 and m m fin- m ini = 0 or 1. The simplest example of the transition that does not satisfy this rule, i.e. is “forbidden”, is that between the s-states ( l = 0) with n = 2 and n = 1; because of that, the lifetime of the lowest excited s-state of a hydrogen atom is as long as ~0.15 s.
27 See, e.g., EM Sec. 8.9.
28 Mind, however, a major difference between the rate discussed in Sec. 2, and a in Eq. (57). In our current case, the atomic transition is still between two discrete energy levels (see Fig. 4 below), so the rate a is proportional to f, the density of final states of the electromagnetic field, i.e. the same density as in Eq. (48) and Chapter 9
Page 15 of 36
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2
fin
ˆ a n
a
.
(9.57)
2
s
1 ˆ†
a 0
According to this relation and the first of Eqs. (19), the final state of the field at the photon absorption has to be the Fock state with n’ = n – 1, and Eq. (57) yields
Photon
n .
(9.58)
a
s
absorption
rate
The results (56) and (58) are sometimes formulated in terms of relations between the Einstein coefficients A and B defined in the way shown in Fig. 4, where the two energy levels are those of the atom, a is the rate of photon absorption from the electromagnetic field, e is that of photon emission into the field, and u is the electromagnetic field density in thermal equilibrium – see Eq. (1.3). Since per Eq. (7.26b), the ratio u/ n is a constant independent of n (and temperature), the statistical averaging of Eqs. (56) and (58) yields29
u
u
Einstein
A B
B
,
(9.59)
21
21
coefficients’
n
12
n
relation
because each of these expressions equals the spontaneous emission rate s.
W
2
n
1
a
s
e
s
n
E
Fig. 9.4. The Einstein coefficients
B u
12
A B u
21
21
on the atomic quantum transition
diagram – cf. Fig. 7.6.
W 1
I cannot resist the temptation to use this point for a small detour – an alternative derivation of the Bose-Einstein statistics for photons. Indeed, in the thermodynamic equilibrium we have just discussed, the average probability flows between levels 1 and 2 (see Fig. 4 again) should be equal:30
W W ,
(9.60)
2 e
1 a
where W 1 and W 2 are the probabilities for the atomic system to occupy the corresponding levels, so Eqs.
(56) and (58) yield
W
n
W n W n
.
(9.61)
2 s
1
,
i.e.
2
1 s
W
n 1
1
But, on the other hand, for an atomic subsystem only weakly coupled to its electromagnetic environment, we ought to have the Gibbs distribution of these probabilities:
W
exp{ E / k T}
E
2
2
B
exp
exp
.
(9.62)
W
exp{ E / k T}
k T
k T
1
1
B
B
B
beyond, while the rate (27) is proportional to a, the density of final (ionized) states of the photon counter’s
“trigger” atom – more exactly, of it’s the electron released at its ionization.
29 These relations were conjectured, from general statistical arguments, by Albert Einstein as early as 1916.
30 This is just a particular embodiment of the detailed balance equation (7.198).
Chapter 9
Page 16 of 36
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By requiring Eqs. (61) and (62) to give the same result for the probability ratio, we readily get the Bose-Einstein distribution for the electromagnetic field in thermal equilibrium:
1
n
,
(9.63)
exp{